New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources

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Series E: Applied Sciences - Vol. 254

New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources

edited by

A. s. Schlachter Advanced Light Source, Lawrence Berkeley Laboratory, Berkeley, California, U.S.A.

and

F. J. Wuilleumier Laboratoire de Spectroscopie Atomique et lonique, Universite Paris Sud, Orsay, France

Springer-Science+Business Media, B.V.

Proceedings of the NATO Advanced Study Institute on New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources Maratea, Italy June 28-July 10, 1992

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-94-010-4375-5 ISBN 978-94-011-0868-3 (eBook) DOI 10.1007/978-94-011-0868-3

Printed on acid-free paper

AII Rights Reserved © 1994 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1994 Softcover reprint of the hardcover 1 st edition No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photo­copying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

CONTENTS

Third-Generation Synchrotron Light Sources ..................................................................... 1 A. S. Schlachter

Investigation of Atomic Structure Using Synchrotron Radiation ..................................... 23 M. YaAmusia

Photoionization of Atoms and Ions Using Synchrotron Radiation ................................... 47 Fran\(ois J. Wuilleumier

Two-Color Experiments on Aligned Atoms ................................................................... 103 B. Sonntag and M. Pahler

Two-Color Experiments in Molecules ............................................................................ 129 I. Nenner, P. Morin, M. Meyer, J. Lacoursiere, and L. Nahon

Electron Correlation in Ionization and Related Coincidence Techniques ...................... 161 G. Stefani, L. Avaldi, and R. Carnilloni

Soft X-Ray Emission Spectroscopy Using Synchrotron Radiation ................................ 189 Joseph Nordgren

Spin Analysis and Circular Polarization ......................................................................... 203 N. V. Smith

X-Ray Magnetic Circular Dichroism: Basic Concepts and Theory for 3d Transition Metal Atoms ............................................................................................. 221

J. Stohr and Y. Wu

High-Resolution Soft X-Ray Absorption Spectroscopy and X-Ray Circular Dichroism ........................................................................................................................ 251

Francesco Sette

Research Opportunities in Fluorescence with Third-Generation Synchrotron Radiation Sources ....................................................................................... 281

D.L. Ederer, K.E. Miyano, W.L. O'Brien, T.A. Callcott, Q.-Y. Dong, J.J. Jia, D.R. Mueller, J.-E. Rubensson, R.C.C. Perera, and R. Shuker

Photoemission Spectromicroscopy ................................................................................. 299 Gelsomina DeStasio and G. Margaritondo

The Properties of Undulator Radiation ........................................................................... 315 M. R. Howells and B. M. Kincaid

Mirrors for Synchrotron-Radiation Beamlines ............................................................... 359 Malcolm R. Howells

Index ................................................................................................................................ 387

PREFACE

The NATO Advanced Study Institute (ASI) "New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources" was held on June 28-July 10, 1992, at Hotel Villa del Mare in Maratea, Italy. The Institute was sponsored by the Scientific Affairs Division of NATO, and additional support was provided by the Region of Basilicata, Italy. A total of 16 lecturers and guest lecturers and 66 participants attended. Selected from more than 120 applicants, the participants represented 18 nations of the world.

Synchrotron radiation has been in use for 30 years to explore, among many other subjects, the interactions of photons with atoms. Since the pioneering experiments that demonstrated the autoionization profiles predicted by Fano in photoabsorption of the rare gases, first- and second­generation synchrotron radiation sources have been used-first in the parasitic mode and then in the dedicated mode-to systematically investigate photoionization and relaxation processes in many atomic and molecular systems. In 1975, almost twenty years ago, an early NATO Advanced Institute, "Photoionization and Other Probes of Many-Electron Interactions," was held in Carry-Ie-Rouet, France, to review the different fields in which the use of synchrotron radiation could yield some new information.1 At that time, the performance of synchrotron radiation was still rather poor: typically, 109 to 1010 photons were available in a 1 % bandwidth. But progress in the field was continuous and spectacular. Many exciting discoveries were made during the following 15 years.

Low-energy, third-generation synchrotron radiation sources are now being built allover the world. Two such facilities were operating by 1993-Super ACO in Orsay and the Advanced Light Source in Berkeley-and a third, ELETTRA in Trieste, is expected to open in 1994. These sources are based on storage rings with low-emittance electron or positron beams and long straight sections containing insertion devices (undulators or wigglers). As electrons pass through a linear array of permanent-magnet dipoles in an undulator, they emit photon beams characterized by extremely high brightness, partial coherence, narrow line width, and collimation in both the horizontal and vertical directions. Up to 1013 photons/s will be available in a 0.01 % bandwidth over a wide energy region, from 10 eV to about 2 keY for a 1.5-GeV storage ring. The considerable increase in brightness over radiation from other sources offers the opportunity for dramatic advances in many scientific disciplines.

Thanks to the wealth of new and exciting experimental data, great progress has also been made over the last 15 years in the theoretical description of photoexcitation and photoionization processes in atoms and solids. In particular, the utility of many-body theories such as the Many­Body Perturbation Theory or Random Phase Approximation with Exchange was demonstrated.

This Institute gave us the opportunity to bring together theoreticians and experimentalists to present in detail the state of the art in experiment as well as in theory, to promote discussions between experimentalists and theorists, and to suggest new directions in research using these low-energy, third-generation storage rings. An important theme of the Institute was the description of basic photoionization processes in atoms and molecules in the ground state, in excited states, and in some ionic states. Radiative and nonradiative relaxation processes following inner-shell ionization were also extensively treated. Special attention was paid to the

1 FJ. WuilIeumier, ed., Photo ionization and Other Probes of Many-electron Interactions (plenum Press, New York and London, 1976).

vii

viii

use of circularly polarized light to study spin effects and magnetic dichroism in surfaces and interfaces. Several lecturers presented the latest developments in x-ray optics, monochromators, and undulators. They described how this advanced technology was incorporated in the design of new facilities such as the Advanced Light Source and the consequent advantages for researchers.

We would like to thank the members of the Scientific Committee for contributing to the success of the Institute by proposing an excellent selection of lecturers. Special thanks are due to Professor J.P. Briand and to Ms. A. de Corte for their efficient help in managing the financial aspects of the Institute. We also thank the lecturers for preparing and delivering their presentations. Especially appreciated is the work of those who made themselves available for the two-week duration of the Institute, thus giving participants the opportunity for frequent informal and lively discussions with them. Likewise we wish to give ample credit to those lecturers who prepared manuscripts for this volume, which is, after all, the final product of the Institute.

The tasks involved in organizing the Institute were extremely complex. Ms. Gloria Lawler must receive special thanks for keeping track of and communicating with applicants, participants, and lecturers prior to the Institute, and for tracking of grants. Ms. Lawler also deserves principal credit for editing, indexing, and coordinating the publication of this volume. We also wish to thank those at Lawrence Berkeley Laboratory who prepared the promotional materials for the Institute and the camera-ready manuscript for this volume: Ms. Connie Silva and Ms. Jean W olslegel for their diligent, skillful word processing of the text and Ms. Linda Geniesse for designing the ASI brochure and for her excellent enhancements of most figures herein. Ms. Geniesse with Ms. Marilee Bailey also designed the ASI poster.

We would like also to acknowledge the valuable help given by Ms. Fran~oise Schont in the selection of the Institute participants. Finally, Ms. Valerie Giardini was responsible for all organizational tasks during the Institute. Everyone there appreciated her efficiency and kindness, and we are sure that all participants would join us in expressing our warmest gratitude.

We wish to thank Dr. L. V. Da Cunha, Director of the NATO ASI Program, for his very helpful assistance. We also gratefully acknowledge the NATO Science Committee and the NATO Scientific Affairs Division. The financial support of NATO, as well as the help from the Region of Basilicata made it possible to invite a panel of outstanding lecturers and to provide a substantial number of grants for participants.

To conclude, we wish to express our appreciation to Ms. Maria Armiento and the staff of the Hotel Villa del Mare, who contributed to making our stay highly agreeable and worthwhile, and to Ms. Barbara Kester of ITST for her assistance in arranging the Institute.

Alfred S. Schlachter Lawrence Berkeley Laboratory

Fran~ois J. Wuilleumier University of Paris

LECTURERS

Dr. M. Ya Amusia, loffe Institute, St. Petersburg, Russia

Dr. G. De Stasio, CNR, Frascati, Italy

Dr. M. Howells, Lawrence Berkeley Laboratory, Berkeley, California, U.S.A.

Dr. B. M. Kincaid, Lawrence Berkeley Laboratory, Berkeley, California, U.S.A.

Dr. G. Margaritondo, Ecole Poly technique Federale, Lausanne, Switzerland

Dr. I. Nenner, CEN Saclay, Gif sur Yvette, France

Dr. J. Nordgren, University of Uppsala, Sweden

Professor Y. Petroff, University of Paris, Orsay, France

Professor G. Sawatzky, University of Groningen, Groningen, The Netherlands

Dr. A. S. Schlachter, Lawrence Berkeley Laboratory, Berkeley, California, U.S.A.

Dr. F. Sette, European Synchrotron Radiation Facility, Grenoble, France

Dr. N. Smith, AIT Bell Laboratories, Murray Hill, New Jersey, U.S.A.

Professor B. Sonntag, University of Hamburg, Hamburg, Germany

Dr. G. Stefani, University of Rome, Rome, Italy

Dr. J. Stohr, IBM Almaden, San Jose, California, U.S.A.

Professor F. Wuilleumier, University of Paris, Orsay, France

SCIENTIFIC COMMITTEE

Dr. A. S. Schlachter (Director), Lawrence Berkeley Laboratory, Berkeley, California, U.S.A.

Professor F. Wuilleumier (Assistant Director), University of Paris, Orsay, France

Professor Y. Petroff, University of Paris, Orsay, France

Dr. Manfred Krause, Oak Ridge National Laboratory, Oak Ridge, Tennessee, U.S.A.

Professor G. Sawatzky, University of Groningen, Groningen, The Netherlands

Professor B. Sonntag, University of Hamburg, Hamburg, Germany

Professor F. Yndurain, University of Madrid, Spain

ix

TIDRD-GENERATION SYNCHROTRON LIGHT SOURCES

A.S. SCHLACHTER Lawrence Berkeley Laboratory University of California Berkeley, CA 94720 USA

ABSTRACT. X rays are a powerful probe of matter because they interact with electrons in atoms, molecules, and solids. They are commonly produced by relativistic electrons or positrons stored in a synchrotron. Recent advances in technology are leading to the development of a new third generation of synchrotron radiation sources that produce vacuum-ultraviolet and x-ray beams of unprecedented brightness. These new sources are characterized by a very low electron-beam emittance and by long straight sections to accommodate permanent-magnet undulators and wigglers. Several new low-energy light sources, including the Advanced Light Source, presently under construction at the Lawrence Berkeley Laboratory, and ELETTRA, presently being constructed in Trieste, will deliver the world's brightest synchrotron radiation in the VUV and soft x-ray regions of the spectrum. Applications include atomic and molecular physics and chemistry, surface and materials science, microscopy, and life sciences.

1. Introduction

Light is one of the most important tools of science. It is the key to viewing the universe-from distant galaxies to cells, molecules, and even atoms. Light has a dual nature, behaving both as a stream of massless particles (photons) and as electromagnetic waves moving through space.

Visible light, which enables us to see the everyday objects around us, is easily generated and easy to detect. The sun, electric lamps, and fire produce it. We can see visible light with our eyes and detect it with photographic film; however, it constitutes only a tiny fraction of the full electromagnetic spectrum (see Fig. 1).

The remainder of the spectrum consists of light with wavelengths longer or shorter than those of visible light. On the longer side are radio waves, microwaves, and infrared radiation. Shorter­wavelength light includes ultraviolet, x rays, and gamma rays. These regions of the spectrum are invisible to the eye and must be detected by special means. Each region has a characteristic range of wavelengths and photon energies that determine the degree to which the light will penetrate and interact with matter. Light sources relevant to this institute produce radiation in the vacuum­ultraviolet and soft x-ray regions of the spectrum. This light is useful for several reasons:

• It can penetrate materials opaque to visible light (see Fig. 2). • It has the right wavelengths-from about 10-7 to 10-10 meter-for exploring the atomic

structure of solids, molecules, and important biological structures. The sizes of atoms, molecules, and proteins as well as the lengths of chemical bonds and the minimum distances between atomic planes in crystals are in this range (see Fig. 3). High-resolution x-ray microscopy is one technique used for such exploration. The combination of wavelengths shorter than visible light and the possibility of obtaining contrast through the

A.S. Schlachter and F.J. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 1-22. © 1994 All Rights Reserved.

2

RADIATION SOURCES TYPE OF DETECTABLE RADIATION OBJECTS

1O-9 t- - 103

~ H+-/

~ / 10-7 ~ - 101 House

RadIo antenna \ Q

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~ ~ • Nucleus ... ~/ 107 r- i - 10-13 0

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~~ .. ~~

nl 8>

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Panicle accelerators \

Figure 1. The electromagnetic spectrum covers a wide range of wavelengths and photon energies. (From "Synchrotron Radiation," by H. Winick. Copyright © 1987 by Scientific American, Inc. All rights reserved.)

Figure 2. This image was made in 1896 by Wilhelm C. Roentgen, who discovered x rays and put them to practical use. (From O. Glasser, Wilhelm Conrad Roentgen, The Early History o/the Roentgen Rays, 1934.)

Atom

0---0

(f---0---O J 6---0---0': ~O--.Q---0: I

I I I ,.6 X I I 1,Iv I

I I I U I

-0---0---6/: I

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Crystal

-./ x ~ x = a few angstroms Chemical

bond

Figure 3. Atoms, chemical bonds, and the distances between atomic planes in crystals all measure a few angstroms­corresponding to the wavelengths of light in the x-ray energy range.

3

interaction of x rays with atoms in the material being examined makes this technique feasible (see Fig. 4). It has photon energies from about 10 to 10,000 electron volts. This energy range corresponds with the binding energy of many electrons in atoms, molecules, solids, and biological systems. Absorption of photons by an atom shows a large increase when the photon energy is sufficient to remove an electron from a given shell of that atom. The photon energies (or wavelengths) at which such increased absorption occurs--called absorption edges-are characteristic of the atom (see Fig. 5). Experiments using this principle can not only determine which atoms are present in the material under study but also can reveal information about the chemical state of the atoms (see Fig. 6).

2. Synchrotron Radiation

Whenever a charged particle such as an electron is accelerated or decelerated, it produces photons. At low velocities, electrons in a curved trajectory emit light of low intensity and low frequency in all directions; however, at relativistic speeds, the intensity, frequency, and collimation of the emitted light increase dramatically. Light generated by bending the path of relativistic electrons is called synchrotron radiation (see Fig. 7).

The natural emission angle for radiation emitted by a relativistic electron is 1Iy, where y is the ratio of the moving mass to the rest mass of the electron. In practical units, this is 1957 E (where E is the electron energy in Ge V). The value of y is approximately 3000 for a 1.5-Ge V electron; thus the natural angle for photons emitted at this energy is of the order of 0.3 milliradian.

The power P emitted by a relativistic electron is proportional to the fourth power of the electron energy E:

where m is the mass of the electron. As shown in Fig. 8, the peak of the emitted energy spectrum increases with E. Note that power emitted is proportional to the inverse fourth power of the electron (or positron) mass, which is the reason that electrons (positrons) rather than heavier particles are used to produce synchrotron radiation.

Early synchrotrons were used primarily for particle physics, and synchrotron radiation was an undesired energy-loss mechanism. Any research done with the x rays was parasitic. These facilities have been called the first generation of synchrotrons. A second generation of synchrotrons was dedicated to the production of synchrotron radiation. In these facilities, electrons are held for many hours in a storage ring, providing a steady source of x rays for research. Figure 9a shows schematically some characteristics of the first- and second-generation synchrotron facilities.

A new, third generation of synchrotrons (see Fig. 9b) is presently under construction and will come on line in the United States, Europe, and Asia, starting in 1993. These facilities are characterized by small electron-beam size, low electron-beam emittance, and long straight sections in which are placed undulators and wigglers, so-called "insertion devices." The result will be x-ray beams of very high spectral brightness; several examples are shown in Fig. 10, along with a conventional x-ray tube. Brightness is defined as flux per unit area of the source, per unit solid angle of the radiation cone, and per unit bandwidth; thus, high brightness is

4

Figure 4. Image of a chromosome from a larva of the midge was obtained through x-ray microscopy. Although the banding structure can be seen through a visible-light microscope, an x-ray microscope was required to capture the filamentary structure between the bands. (Produced by G. Schmahl and M. Robert-Nicoud, University of Gottingen, at the BESSY synchrotron radiation facility, Berlin, Germany.)

Photon energy

Figure 5. At absorption edges, the absorption of photons by an atom increases sharply because the photon energy is sufficient to remove an electron from a given shell of that atom. Thus photons can be used for element-specific detection and imaging.

(a)

(b)

(c)

(d)

Aluminum

/ Silicon / -..,;.,

-

~ilic.on Aluminum dioxide /

/ SijCOn ;"""-"--

-

Figure 6. The chemical composition of a microfabricated sample (a) was determined through a technique called spectromicroscopy. After absorbing x rays, the sample emitted electrons with energies characteristic of its components. Analysis of these energies identified and mapped sections of aluminum (b) and pure silicon (c). Silicon atoms in silicon dioxide molecules (d) were mapped at a different location from pure silicon, indicating two distinct chemical states of silicon. (Based on work done by scientists from the State University at Stony Brook, IBM, and the Lawrence Berkeley Laboratory. Data were taken at the National Synchrotron Light Source, Upton, NY.)

Electron beam

Photon beam

Figure 7. A beam of electrons travelling in a curved path at nearly the speed of light emits photon beams. The beams fan out at angle 9, the natural emission angle.

5

characterized by a high flux of radiation into a small spot and with a small spectral bandwidth. Undulators and wigglers both produce high flux (number of photons delivered per second). In the past, order-of-magnitude increases in brightness have led to qualitatively new developments in spectroscopic and structural studies of both gas-phase and condensed matter. No less is expected at these third-generation synchrotron facilities.

3. Third-Generation, Low-Energy, Synchrotron-Radiation Facilities

Several modern, low-energy, synchrotron­radiation facilities are presently planned or under construction around the world. Figure 11 shows their locations and domains of brightness compared with existing facilities. Table 1 lists important parameters for four low-energy facilities.

One example of a third-generation, low­energy synchrotron-radiation facility is the Advanced Light Source (ALS). At the ALS, an electron gun shoots electrons into a linear accelerator (or linac), which accelerates them to an energy of 50 MeV. The linac then injects the electrons into a booster synchrotron for further acceleration-to 1.5 GeV. At this energy, the electrons are moving at 99.999996% of the speed of light. From the booster, the electrons enter a storage ring with a circumference of 200 meters, where they circulate for hours at constant energy.

6

Photon energy (eV)

Figure 8. Photon emission as a function of photon energy for electrons (positrons) with an energy E of 1.5, 2.5, and 4 GeV. Maximum photon energy increases with increasing electron energy.

(a) Yesterday's synchrotrons

Continuous circular trajectory

Bending magnet radiation

Circular electron

"X ray

p~.r hv

(b) Today's synchrotrons

• Many straight sections containing periodic magnetic structures

• Tightly controlled electron beam

Undulator radiation

1 ~T' : Laser-like

PLLTunable

hv

Figure 9. Evolution of synchrotron-radiation facilities: (a) characteristics of early synchrotrons versus (b) third-generation facilities.

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Photon energy

7

Figure 10. Spectral brightness as a function of photon energy for sample modern synchrotron-radiation sources (ALS and APS) and a conventional x-ray tube. Third-generation synchrotron-radiation sources are characterized by the high brightness of the light they deliver.

In the storage ring, the electron beam travels through a vacuum chamber in a sequence of 12 arc-shaped sections alternating with 12 straight sections. The arc sections are imbedded in a lattice of bending magnets and focusing magnets that force the beam into a curved trajectory and constrain it to a tight ellipse approximately 100 microns in vertical dimension in the straight sections. The three bending magnets in each arc have ports through which beams of synchrotron radiation pass as the electrons curve through the arc. Figure 12 shows the arc sections and magnet lattice.

The straight sections, which have no focusing or bending magnets, are used for other purposes. One is the site of electron injection from the booster. Another is surrounded by radio-frequency (rf) cavities in which electromagnetic fields oscillate at a frequency of 500 MHz. These fields

8

1 eV

United States

1 keV

Photon energy

100 keV 1 eV

Elsewhere

1 keV

Photon energy

100 keV

Figure 11. Many low- and high-energy synchrotron-radiation sources have been built or are under construction around the world. The light delivered by third-generation, low-energy, synchrotron-radiation sources will be up to several order of magnitude higher than that from existing facilities.

Table 1. Salient parameters for four third-generation, low-energy, synchrotron-radiation facilities.

Super ACO ALS ELE1iRA SOLEIL

Energy 0.8GeV 1.5 GeV 1.5-2GeV 2.15 GeV Circumference 12m 197m 259m 200m Emittance 35 nm rad < 10 nm rad 4-7 nmrad 17-36 nmrad Critical energy 0.66keV 1.56keV l.36keV 5keV Critical wavelength 18.6A 7.9A 9.1 A 2.5 A Particle positron electron electron positron Injector linac synchrotron linac synchrotron Straight sections 6 10 11 12 Undulator length 3.2m 4.5m 6m 4-5 m Bunch length lOOps 35-50ps 12-20 ps 70-80 ps Bunch number 24 max 250 max 432 max 240 max Beam lifetime 6 hours 4-6 hours 4-10 hours 14-24 hours Begin operation 1988 1993 1995 2000

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Figure l2. The ALS storage ring has 12 arc-shaped and 12 straight sections. The arc sections are imbedded in a magnetic lattice, which consists of 12 magnet sequences, one for each arc. Each sequence contains three bending magnets (B), six quadrupole magnets (Q), and four sextupole magnets (S). Relativistic electrons, accelerated by the Iinac and booster synchrotron, travel through the storage ring generating synchrotron radiation.

9

Bending magnet

til Quadrupole magnet

Sextupole magnet

10

Electron bunch

\

Synchrotron radiation pulse

\ ... --~.-

Figure 13. Schematic illustration of the bunched structure of an electron beam circling in a storage ring and the corresponding pulsed nature of the synchrotron radiation. At the ALS, for example, 250 electron bunches circle the ring. Each has a duration of about 35 picoseconds. The spacing between bunches, dictated by the rf frequency, is 2 nanoseconds.

replenish the electron-beam energy lost to synchrotron radiation. Synchrotron radiation emitted by the bunched electron beam is not emitted continuously but in a pulsed fashion, as shown in Fig. 13.

The 10 remaining straight sections at the ALS are designed to accommodate devices called undulators and wigglers, which generate synchrotron radiation with enhanced characteristics. Collectively termed "insertion devices," undulators and wigglers consist of a linear array of north-south magnetic dipoles of alternating polarity (see Fig. 14). The normal vertical orientation of the dipoles causes relativistic electrons of energy E to undergo a nearly sinusoidal electron trajectory of period lu in the horizontal plane, causing the emission of synchrotron radiation. The peak magnetic field Bo on the undulator axis depends exponentially on the ratio R of the gap between the dipole north and south pole faces g to the period of the

undulator lu. For hybrid devices made with the permanent-magnet material neodymium-iron­boron, BO is approximately

Bo[Tesla] = 3.44 e-R(5.47-1.8R)

for 0.07 < R<0.7. The deflection parameter K is the ratio of the maximum angular deviation of the electron

trajectory from the insertion-device axis to the natural opening angle of the synchrotron-radiation cone. The cone has a natural opening angle that is inversely proportional to the electron-beam energy. Expressed in terms of undulator parameters, K is approximately

K=0. 934Bo[TeslaJ}"u[cm] .

A K value around 1 defines the breakpoint be­tween an undulator and a wiggler. However, an insertion device retains significant undulator properties at higher K values, and most of the undulators planned for the new synchrotron sources operate in this intermediate range with K > 1. When K » 1, the structure is called a wiggler.

In the undulator regime, an important effect comes into play that gives insertion devices special properties. Because of the rather gentle perturbation of the electron trajectory in an undulator (several micrometers amplitude), the electron trajectory during a pass through the device lies within one radiation-cone opening

Radiation from an insertion device

Figure 14. An insertion device has permanent magnets of alternating polarity that cause electrons moving at nearly the speed of light to follow a sinusoidal path perpendicular to the magnetic field.

11

angle. This condition means that the radiation emitted from successive undulator periods adds coherently. The interference behavior due to the coherence gives rise to a sharply peaked radiation spectrum consisting of a fundamental and several harmonics. The photon energy En of the nth harmonic is

E [keV] = 0.949nE2[GeV] ( 1 ), n Au[cm] I+K2/2+y2e2

where y is the ratio of the electron relativistic and rest masses and e is the angle of emission relative to the undulator axis. By contrast, the lack of interference in a wiggler means that its synchrotron-radiation spectrum is like the broad, continuous spectrum from the dipole or bending magnets in the curved sections of the storage ring.

It is important to the experimenter to be able to select the photon energy of the undulator light-that is, the light should be tunable. In principle, the photon energy is tuned from high to low values primarily by decreasing the undulator gap g from a maximum to a minimum distance, thereby increasing the field Bo and the value of K. Both the minimum and the maximum energies are arbitrarily set by the drop-off of the photon flux at low and high gap values, but are also subject to constraints such as the vertical diameter of the storage ring vacuum chamber. At the ALS, use of the third and fifth harmonics of the undulators is planned to extend their spectral range to higher photon energies than can be reached with the fundamental alone.

While tunable, undulators retain their most desirable properties in a comparatively small photon-energy range. For K « 1, the only significant photon flux is in the fundamental. As K is increased, the number of harmonics with measurable photon flux grows rapidly. It becomes useful to define a critical harmonic nc above which half the total radiated power occurs. The critical harmonic is given by

The cubic dependence of nc on K means that most of the undulator radiation is in the harmonics when tuning the undulator to lower photon energies. Essentially, increasing K too far turns the undulator into a wiggler, as the higher harmonics blend together into a broad spectrum.

A summary of the properties of undulator and bending-magnet radiation is shown in Figs. 15 and 16, in which the data apply to the ALS. The undulator radiation is seen to have both spatial and spectral properties, which enhance its usefulness for research.

Low-emittance storage rings and insertion devices have created new challenges for designers of XUV optics. First, the source size and divergence have become smaller. For ALS undulators at the high-photon-energy end of the spectral range, the rms size is typically 330 J.lm horizontal by 65 J.lm vertical and the rms divergence is typically 40 J.lrad horizontal by 30 J.lrad vertical. The smaller source size requires tighter tolerances for relay optics and monochromator components in both optical figure and finish to avoid loss of light (e.g., rms surface roughness"" 0.5 nm and tangential slope error <1 J.lrad for a condensing mirror). The attainment of higher resolution by the use of smaller slits also becomes practical (the spectral-resolution goal of monochromators in undulator beamlines is AEIE "" 10-4). Monochromator components, therefore, need tighter tolerances to avoid loss of resolution. Finally, the photon-beam power is several kW/cm2. The

12

I l

Bending-magnet radiation (sweeping searchlight)

./ /'

---- -

Beamline optics acceptance angle

Undulator radiation

Nperiods~

~ e-

~'-J!J

1 A..~ 1 Y 1/

acen = yIN r

A,. = ~ (1 + 152 + y2ij2) 2 y2 2

in the central radiation cone:

(0 1 =

(0 N

1 acen = yIN

Figure 15. Characteristics of the synchrotron radiation produced by bending magnets and undulators.

~------ 10 mrad -----\--.j

)(

(@ c: o (5 .s::: 0..

16mW @ 130A, 1% BW

X-ray light bulb

Photon energy

'" U! c: o (5 .s::: 0..

40llrad

- 1 W @ 130 A, 1 % BW

500eV Photon energy

Figure 16. Angular divergence and spectral distribution of bending magnet and undulator radiation. The data apply to the ALS.

13

requirement that thermal distortions and stress be controlled complicates the design [for example, water cooling in a UHV ('" 1 ntorr) environment is required] and limits the choice of materials.

The most serious limitation is that of optical fabrication tolerances. It is difficult to fabricate aspheric optical surfaces (such as paraboloids, ellipsoids, and toroids) sufficiently accurately in mirrors, monochromators, and other optical elements to take full advantage of the undulator source. One way to address this problem is to avoid the use of aspheric surfaces and to build beamlines entirely with plane and spherical surfaces.

4. Special Characteristics of Synchrotron Radiation

Synchrotron radiation has certain characteristics that, individually or combined, allow researchers to perform experiments not otherwise possible:

Very high brightness - high spatial resolution - high spectral resolution

• Tunability • High degree of coherence

A pulsed nature • Linear or circular polarization • High flux.

Of these, the most prized characteristic is high brightness: the light has a high photon flux per unit source area and per unit solid angle into which the source radiates.

A major benefit of brightness is high spatial resolution (or focusability). High spatial resolution is achieved because many photons can be focused on an extremely small spot, in which they can generate characteristic absorption, photoelectron, or other types of spectra. In many cases, the smaller the focal spot, the smaller the object that can be distinguished from its surroundings. With help from special optical devices, a third-generation light source like the ALS is expected to achieve spot sizes as low as 200 A in diameter (see Figs. 17 and 18).

High spectral resolution is a second major benefit of high brightness. High brightness allows a major portion of the photon beam to be focused through the monochromator slits, thereby increasing the resolution at fixed measurement time or decreasing the measurement time at fixed resolution. High spectral resolution is useful for separating chemical shifts and for resolving narrow spectral features.

Chemical shifts can be used to determine the chemical state of atoms in solids. For example, measurement of the electron-energy spectrum emitted by photons hitting Si(100) with a 5-A surface of Si02 allows the oxidation state of Si in the interface region to be determined (see Fig. 19). An example of imaging with chemical-state selectivity has been shown in Fig. 6.

Another example of the use of high spectral resolution is in measuring narrow spectral features. For example, resonant states of the He atom were first measured in 1963 by Madden and Codling, as shown in Fig. 20. These resonant states exhibit the characteristic Fano profile arising from interference between two paths to reaching the same final state. With improved resolution and much higher photon fluxes, many additional states can be seen in the ionization spectrum of He (1992 measurement also shown in Fig. 20).

Higher resolution can yield even more information, as shown in the more detailed spectrum in Fig. 21. The spectral region between two peaks (3+ and 4+) is seen to contain an additional small

14

Processes at surfaces and interfaces

Defects

Growth 0 (

Figure 17. Many physical, chemical, and biological processes occur at solid surfaces and interfaces between materials. With light from third-generation synchrotron sources, scientists can study surface areas only 200 A in diameter. By using techniques such as microprobe analysis or x-ray microscopy, they can obtain information about the growth of layers on the surface, reactions, diffusion, nucleation, and defects (missing or misplaced atoms) in such small areas.

peak (4-) and an even smaller peak (2p3d) next to it. These peaks and the physics underlying them could not be accessed without very high spectral resolution.

Apart from the issue of spectral resolution, the tunability of synchrotron radiation is important in itself. From the range of available wavelengths in a beam, one can select a specific wavelength (or photon energy). A wavelength would be selected, for example, because an atom in a sample exhibits a sharp increase in the absorption of light at this wavelength. X-ray absorption spectroscopy, a family of analytical techniques based on the absorption of light by atomic species, can reveal the identity, electronic structure, and chemical bonding state of the absorbing atom and provide information about the identity, number, and arrangement of the atoms around it.

A significant fraction of the long-wavelength radiation from undulators in a third-generation synchrotron is spatially coherent. The criterion for spatial coherence is that the product of the area of the light source and the solid angle into which it emits must be no larger than the square of the wavelength of the light. Since this is the diffraction condition, spatially coherent light is also said to be diffraction-limited. In accordance with the diffraction condition, the electron-beam emittance e sets the minimum wavelength at which all the radiation can be diffraction-limited, according to the relation

'"

1.0

0.8

E 0.6

~ 0.4 0.2

Solid state Si (Li) detector

y

15

Figure 18. Diagram of an x-ray microprobe built at Lawrence Berkeley Laboratory shows how a pair of grazing-incidence mirrors coated with multilayers can be used to focus a beam of x rays to a spot several microns in diameter. The beam apertures further reduce the spot size to about 21lm. The three­dimensional graph illustrates the capability of the microprobe to detect trace amounts of impurities. In this example, the tall peaks represent iron impurities detected in a silicon carbide ceramic substrate. They came from stainless steel tweezers used to handle the ceramic. (The data were taken at the National Synchrotron Light Source by A. Thompson and K. Chapman of Lawrence Berkeley Laboratory.)

e = Imin/4p .

Even at wavelengths below the minimum, part of the radiation remains diffraction-limited, the fraction decreasing as the square of the wavelength.

Although not as coherent as the visible light from most lasers, undulator radiation has much more coherence than ever before available in the vacuum ultraviolet and soft x-ray regions of the spectrum. One of the most exciting uses for coherent synchrotron radiation is holography (see Fig. 22). While laser light can be used to make holograms of objects that we can see, coherent x rays can image objects that are far smaller.

A more general virtue of coherent radiation is the ability to focus. For example, a Fresnel zone plate can focus a coherent beam of soft x rays to a spot with a radius approximately 1.2 times the width of the outermost zone. With state-of-the-art microfabrication techniques, such as electron­beam holography, it is possible to make zone plates with outer zone widths of about 300 A.. This capability can be exploited in scanning or imaging systems to generate spatially resolved information with a comparable resolution.

Coherent synchrotron radiation can also be used to test the quality of optical lenses and mirrors used in x-ray applications-for example, x-ray astronomy, x-ray lithography for manufacturing microchips, and x-ray imaging of microstructures in biology and materials science. The smaller

16

Si 2P3/2 5 A oxide

hv=130eV

~

!!l ·2 ::J

.e

.!!!. ~ Si(100) ·iii c: .lB .!: c: 0 ·iii en ·e Q)

~ ~ a..

Si(111)

-7 -6 -5 -4 -3 -2 -1 0 2 3

Initial-state energy (eV relative to bulk Si2P3I2)

Figure 19. Photoemission from Si(lOO) with a 5-A oxide surface, shown as a function of initial-state energy. The oxidation states of Si are shown to be resolved. From Himpsel, McFeely, TaIeb-Ibrahimi, Yarmoff, and Hollinger, Physical Review B 38, 6084 (1988). (Measurements made at the National Synchrotron Light Source.)

n =2+

~ 1963 en c: Q)

"C

~ a:

1 - Wavelength

"C 100 6+ Qj 1992 .~- 7+ c: en o:!:: .- c: a;::J

50 N • .- .0 c: .... o res ~-~ 0 a..

64.9 65.0 65.1 65.2 65.3

Photon energy (eV)

Figure 20. Synchrotron radiation absorbed at certain photon energies ionizes helium gas atoms and excites the helium ions to a series of distinct quantum states (as revealed by the peaks). The high resolution and capability to tune the radiation to the discrete photon energies required for excitation allowed the quantum states to be seen. This figure demonstrates the improvement in resolution since the pioneering work of Madden and c:odling in 1963, in which the series of peaks was first observed. (Measurements in 1992 made by Z. Hussein, T. Reich, D. Shirley, et a!., at Stanford Synchrotron Radiation Laboratory.)

- - - - - - - - - - - - -1- - - r T 1111; 1········3·+····r···T·r·~···~

3-.I,

~ 4meV~ ~ I I I I I

60 61 62 63 64 65 64.11 64.13 64.15

Figure 21. Doubly excited states of He+ lying below the n = 2 threshold, including three Rydberg series, have been observed. The 2p3d peak is 4 meV wide, demonstrating the excellent spectral resolution. From Domke et at., Physical Review Letters 69, 1171 (1992). (Measurements made at BESSY.)

Figure 22. The ripple-like pattern in the x-ray hologram (left) contains the information needed to create the reconstructed image (right) of zymogen granules, which appear as dark spherical objects. These are storage vesicles for digestive enzymes found in a pancreatic cell. Such images can reveal information about digestion mechanisms. The shading of the zymogen granules indicates their density. The darker a granule, the more dense it is, indicating a larger enzyme content. (Produced by Stephen S. Rothman, Malcolm Howells, Chris Jacobsen, and Janos Kirz at the National Synchrotron Light Source, Brookhaven National Laboratory.)

17

the object imaged, the greater the demand for high-quality optics polished to eliminate virtually all defects. Figure 23 shows the principal tool to be used at the ALS for testing x-ray optics for lithography.

Time resolution, a feature of all synchrotron sources, follows from the bunched character of the electron beam in the storage ring. Standard pulses delivered by the ALS are 35 ps wide at half maximum and occur at intervals of 2 ns. This time structure can be varied by injecting only one or a few electron bunches into the storage ring. In this few-bunch mode, the ALS delivers pulses at longer intervals.

The time structure of synchrotron radiation can be used for a variety of timing experiments. The structure is different from that of a laser in that the synchrotron radiation has a low energy per pulse but a very high repetition rate (500 MHz for the ALS): therefore, the time-averaged power is high. Because the per-pulse power is low, harmonics cannot be generated with the UV photons produced by a storage ring; however, an undulator can directly produce harmonics.

The pulsed nature of synchrotron radiation can be used to observe short-lived or transient systems by means of time-resolved spectroscopic, scattering, and imaging experiments. For example, if radiation damage due to x-ray exposure can be avoided or minimized, it might be possible to image changes in functioning biological cells and cellular structures in near-natural environments. Also, one might study the kinetics of a chemical reaction or the lifetime of excited states of atoms or molecules (see Fig. 24). The ultimate time resolution, possible only if there are enough photons in a single pulse of bright synchrotron light to generate a useful signal, would be to follow events in real time on the sub-nanosecond time scale.

18

CCO detector

Test mirror

Figure 23. The main tool for determining the quality of optics is an interferometer. Coherent x rays are directed through this device onto the object being tested. Inconsistencies in the curvature or smoothness of its optical surface cause the reflected x rays to change phase with respect to reference x rays from the same source, which do not change phase. The phase difference between the object and reference waves provides a measure of the quality of the optic.

Synchrotron radialion pulse

Photoinduced isomerization

Figure 24. The kinetics of chemical reactions can be investigated with pulsed light from two sources in "pump-probe" experiments. A pump pulse of synchrotron radiation initiates a chemical reaction that is interrogated by a probe pulse from a laser.

19

Time resolution can also be achieved by operating in synchrony with other light sources, such as a high­speed laser in a pump-probe mode. The extremely narrow width and rapid frequency of the standard pulses make it possible to investigate ultrafast processes, whereas the ability to lengthen the interval between pulses would allow the study of processes with relatively long lifetimes.

x \ Left-handed

y/~cHH x

\ Right-handed

-z

Polarization control is another feature that is associated with synchrotron sources. Undulator radiation is completely linearly polarized. Radiation from bending magnets is largely linearly polarized when viewed in the plane of the electron orbit, but it is elliptically polarized when viewed at angles above or below the plane. With special insertion devices, e.g., crossed-field undulators and elliptical wigglers, it is also possible to generate elliptically polarized beams, in particular, circularly polarized beams (see Fig. 25), which can be used to investigate structures

y/Qj z

Figure 25. The unique property of circularly polarized light is the spiral path of its electric-field component. This component maps out a clockwise or counterclockwise circular path to form a right-handed or left-handed spiral.

COOH I

H ..... C""" "/ NH2

CH3

Figure 26. Like left and right hands, chiral molecules are mirror images that cannot be superimposed. The pair of chiral molecules shown here are examples of amino acids, the building blocks of proteins.

with a handedness (chiral structures) and magnetic materials.

Polarized synchrotron radiation can be used to investigate anisotropic molecules-those that exhibit different responses to the light, depending on the direction of the E-vector. For example, one molecule of a chiral pair (see Fig. 26) might absorb more right than left circularly polarized light than its counterpart. An experiment based on this phenomenon could elucidate the molecular structures. Many biological molecules such as DNA or amino acids (the building blocks of proteins) lend themselves to experiments with polarized synchrotron radiation.

Because of their inherent directional spin, magnetic materials can also exhibit dichroism. Scientists have exploited this phenomenon by using circularly polarized synchrotron radiation in conjunction with an imaging microscope to obtain pictures of bits, as small as 10 ~m x 1 ~m, on a computer's magnetic storage disk (Fig. 27). It was possible to obtain these images because each bit corresponds to a magnetic region with its own direction of magnetization. This type of research can lead to a greater understanding of the magnetic characteristics of materials and possibly to the development of new materials with greatly increased magnetic storage capacity.

20

Figure 27. Image of magnetic bits on a storage disk was obtained by subtracting two images recorded with right circularly polarized synchrotron radiation tuned to the cobalt L3 and L2 x-ray absorption edges. The magnetization direction of the bits lies along the rows but points alternately to the right and to the left in the picture. The dimensions of the bits in the three rows are (from the top row, in microns) 10 x 10, 10 x 2, and 10 x I. (Produced by scientists from the IBM Almaden Research Center and the University of Wisconsin, Milwaukee, at Stanford Synchrotron Radiation Laboratory, Stanford, CA.)

High flux is necessary for many experiments. For example, the number of atoms or molecules, such as impurities in a semiconductor or ionized atoms in a vapor, may be too small to generate a measurable signal in a practical time without sufficient photon flux. A high photon flux effectively compensates for the small number of signal generators by exciting them more often, thereby making impractical experiments feasible. An example of an experiment requiring high flux, but not particularly high brightness, is the photoionization of ions without narrow spectral features, as demonstrated by electron-spectroscopic measurements of Ca+ ions (Fig. 28). The ion beam is large, and the width of the spectral feature observed is not narrow; hence, high brightness would not be important in this experiment. High flux, however, is essential because of the low density of the ions in the beam, resulting in a low signal rate.

Many additional types of experiments will utilize the properties of modern third-generation synchrotron light sources. An example is photoelectron holography (see Fig. 29). This figure shows a reconstruction of a copper lattice based on emitted photoelectrons, demonstrating spatial resolution on an atomic level.

30

25 ;;-0

~ 20 f/)

0 '<I' 15 --Ul E :l

10 0 u

5

;;-0 0

~ 12 Ul

0 C\I 8 --~ 4 :l 0 U 0

CMA

Ca+ + hy ~ Ca++ + e-

III hy = 33.20 eV

I

III III I IIIII11111111111111111

hy = 32.67 eV

12 14 16

Lab. kinetic energy (eV)

TGM SuperACO

e+

21

Figure 28. Apparatus used by Bizau et al. to measure the electron spectrum of photoexcited Ca+ ions. Both a photon beam and an ion beam are focused into the entrance volume of a cylindrical mirror analyzer. Also shown is the photoelectron signal at two different photon energies, one on a Ca+ resonance, the other off the resonance. From Bizau et aI., Physical Review Letters 67, 576 (1991). Measurements made at Super ACO.

22

[001]

[110]

Figure 29. Holographic reconstruction of Cu(OOI). From Harp et aI., Physical Review Letters 65, 1012 (1990); Physical Review B 42,9199 (1990).

5. Summary

A new generation of low-energy synchrotron storage rings will produce extremely bright beams of vacuum-ultraviolet and soft x-ray radiation. The high spatial and spectral resolution, along with polarization control, coherence, and a short-pulse time structure present new opportunities for research in a wide variety of fields, including atomic and molecular physics and chemistry, surface and materials sciences, biology and life sciences, and technology. This NATO Advanced Study Institute is devoted to exploring some of these opportunities.

Acknowledgements

My thanks to Gloria Lawler and Arthur Robinson for their kind assistance in writing this manuscript. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, of the U.S. Department of Energy, under contract DE-AC03-76SFOOO98.

INVESTIGATION OF ATOMIC STRUCTURE USING SYNCHROTRON RADIATION

M. Y A. AMUSIA A. F. loffe Physical-Technical Institute St. Petersburg, Russia 194021

Institute for Theoretical Physics, Frankfurt University 6000 Frankfurt am Main 11, Germany

ABSTRACT. Photoionization of atoms as a source of information on their electron structure is discussed. It is demonstrated how the direct interaction between atomic electrons modifies the total and partial photoionization cross sections, which determine the production of multiply charged ions and sateIIite states. The spectrum of vacancies created in photoionization and their decay, affected by electron correlations, is considered. Specific features of near- and subthreshold photoabsorption processes are presented. A number of processes that could be studied in the future using synchrotron radiation are mentioned and briefly discussed.

1. Introduction

Synchrotrons and storage rings are sources of continuous-spectrum radiation which permit the study of the photoionization of atoms. Photo ionization is a process in which one or more electrons are removed from the target due to the action of the electromagnetic field. Because the interaction between a photon and an electron is comparatively weak, the photoionization cross section gives, almost directly, information on atomic structure.

There are a number of reasons that justify the investigation of the electron structure of atoms: They are the "elementary" particles of our surrounding macroscopic world, which is formed from molecules, solids, and liquids, each of them consisting of atoms. So, to understand the properties and behavior of these objects, i.e., to have a deep insight into biology, materials behavior, and chemistry, one needs to know rather thoroughly the structure and properties of isolated atoms. There are many processes in multiatomic formations (e.g., molecules, solids, and liquids), such as high-energy excitations and x-ray production, that are almost completely determined by the electronic structure of the individual atoms. The spectra of isolated atoms and ions, their photoionization cross sections, and the frequencies of discrete and autoionizational transitions are the "fingerprints" of these species and, in a number of cases, present the only possibility for deriving information about the constituents and their states in a multiatomic system, its structure, and the fields acting inside it. The precision with which calculations may be performed for atoms is almost in any case much higher than in other multi particle systems, not only in the domain of the semimacroscopic world but also for microscopic objects. Therefore, highly accurate

23

A.S. Schlachter and F.J. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 23-46. © 1994 Kluwer Academic Publishers.

24

investigation of atomic structure and comparison of the results of calculations and measurement can help to demonstrate whether some extra force or forces exist, apart from pure electromagnetic, that act between the nucleus and the electrons and between the electrons themselves. It is worth noting that, in such a way, the neutral currents of electro­weak interaction were discovered.

Because atoms are systems in which the interconstituent interaction, i.e., that between electrons and the nucleus, is known quite accurately, everything about their properties that differs from the properties of a simple hydro genic picture, including of course the photoionization and photoexcitation cross sections, comes from the multielectron structure of the atom. In the hydrogenic picture, all atomic electrons are considered to be moving independently, without interaction, in the Coulomb field of the nucleus. Thus, it is clear that investigation of atomic structure means, in fact, the study of the role of interelectron interaction. The investigation of atoms as many-electron systems led to the development of approaches and to the discovery of effects that proved to be essential in consideration of other systems:

The Hartree-Fock (HF) method was developed for the first time in investigating atoms. The random phase approximation (RP A) reached, in atomic calculations, a level of sophistication and depth of understanding not yet achieved in other domains of physics. Quite essential is the discovery of the very important role of exchange in the framework of RP A, which led to the invention of a new method, random phase approximation with exchange (RPAE).

• Recently, the post-collision interaction phenomenon was discovered in atomic processes, including photoionization. A similar effect is now under discussion in nuclear physics.

To study the role of multielectron effects in atomic structure and processes, one must improve as much as possible the single-electron picture. This is achieved by performing calculations in HF approximations, which take into account a rather important multielectron effect: the creation of a common field describing the action of all atomic electrons upon the considered one. In this approach, each atomic electron is influenced by the action of all others, but its own motion is not disturbed by direct interaction with other electrons. Therefore, this approach is also called the self-consistent field (SCF) approximation. This field is rather complex because it is nonlocal due to exchange between atomic electrons.

The earlier studies of photoionization cross sections in the VUV frequency region completely ruled out the possibility of limiting ourselves not only to a simple hydrogen-like picture of the process, but also to a more sophisticated one such as the Hermann-Skillmann approximation to the HF field, which includes exchange and nonlocality in the self-consistent potential.

As the initial step in a theoretical investigation, the HF method is now used. Only that which is definitely impossible to describe in the HF frame is attributed to multielectron correlations. We will demonstrate here that by going beyond SCF approximation, one encounters a multitude of interesting and exciting effects, thus confirming the important role of correlations.

2. Main Scientific Results Manifesting the Role of Multielectron Correlations

Photoionization studies have presented a great deal of data that disclose the important and, in many cases, decisive role played by direct interelectron interactions between atomic electrons, which are completely neglected in all one-electron approaches, including the HF method:

It was demonstrated that, in a number of atoms (for instance, those with a filled 4d1O shell), the cross section of photoabsorption is dominated by a large maximum called "giant

25

resonance." It came to be understood that this could be described qualitatively only if multielectron effects were accounted for. It was demonstrated theoretically and then proven experimentally using synchrotron radiation that multielectron shells qualitatively modify the photoionization cross section of the neighboring few-electron shells, leading to new continuous spectrum resonances. It was shown that the mutual influence of two or even three multielectron shells far from their thresholds is very strong, completely modifying the partial (from a given shell) contribution to the total cross section. For example, there are many cases in which the single-charge ion yield, well above the intermediate shells' ionization threshold, is completely determined by this intershell interaction. It was shown that, together with the main lines in photoelectron spectra, there are satellite (or shadow) lines of a multielectron nature. In contrast to the main ones, these lines do not correspond to the elimination of a single electron from a level that it occupied. The observation of these lines (satellites and shadows) and the demonstration of their comparatively large contribution to the total photoabsorption cross section present direct evidence of the importance of the interelectron interaction in atoms. The prominent role played by interelectron interaction in atoms and their rather complex electron structure is clearly demonstrated by the observation of different kinds of autoionizational resonances, i.e., structures in the photoionization cross section arising from the interaction of a discrete excitation, mainly from the inner and intermediate shells, with the outer-shell continuous spectrum. It was demonstrated that two-electron discrete excitations interact rather strongly with the one-electron photoionization continuum, leading to prominent structure even in the total photoionization cross sections. It was observed that in the photoionization process, multiply charged ions are formed with a rather high probability, not only as a direct consequence of step-by-step Auger decay but also via other mechanisms such as shake off of electrons due to vacancy creation and direct knock-out of outer electrons by the photoelectron on its way out of the atom. It was experimentally demonstrated that the photoionization cross section and the probability for the decay of vacancies formed depends considerably on the "doorway" states of the process, namely on whether it proceeds directly or via discrete spectrum excitation. It appeared that an electron excited to a discrete level, even far removed, can play an impressively important role in both the ionization and vacancy decay process. It was shown that the Auger line is shifted to the higher-energy side when an inner vacancy is created near its threshold due to instant alteration of the atomic field acting upon the outgoing photoelectron at the moment of Auger decay. It became clear that there is a high probability of a photoelectron from one of the intermediate or inner shells colliding inelastically with the outer electrons and leading to their excitation or ionization.

Certainly, electron correlations are demonstrated in a number of atomic processes and have a considerable effect on the characteristics of ground and excited states. Most transparent and impressive are their effects in photoionization, different features of which is the subject of this paper.

The following aspects of this process will be considered at length: Photoionization with formation of differently charged ions, A +, A ++, A +++, etc. Creation of vacancies and their position and decay, including different correlative decays.

26

• Near-threshold effects. Secondary processes, such as photon and electron emission, will be discussed.

In what follows, we apply the many-body theory to atoms by using a diagrammatical technique that is most convenient in describing the electron correlational effects.

3. Correlational Effects in Photoionization: RP AE Frame

Random phase approximation with exchange (RP AE) is an approach that takes into account both the direct elimination or excitation of an atomic electron after it absorbs a photon and an indirect effect, namely the elimination of the electron under consideration by an alternative part of the atomic self-consistent field resulting from virtual or real excitation of any other electrons after absorbing a photon. RPAE accounts for the time- (or frequency-) dependent polarization of atomic electron distribution by the incoming photon and for the action of this polarized or deformed distribution upon all atomic electrons, which leads to their excitation and ionization.

In this paper, many-body theory diagrams will be introduced intuitively. They will be used to a large extent as pictures describing most transparently the considered physical mechanisms. Corresponding analytical expressions will also be used. More details about the diagrams in connection with photoionization can be found in [1].

The essential element for constructing a diagram that represents a physical process is a dashed line

(1)

standing for a photon. A solid line with an arrow to the left

2--.. --·1 (2)

represents the propagation of a vacancy (HF occupied state) from point 2 in space and time to point 1. A solid line with an arrow directed to the right

2 ..... -...,.-_. 1 (3)

denotes an electron (HF continuum state) propagating from point 2 to point 1. In some cases it is essential to emphasize that the electron is on a discrete level that was vacant

in the atomic initial state. This is done by a solid line with a doubled arrow:

(4)

The interelectron interaction in its most simple form is a Coulomb potential acting between electrons. It is denoted by a vertically oriented wavy line.

(5)

Its vertical orientation demonstrates that this interaction is time-independent and transfers the connection between particles without retardation.

27

A diagram presents a picture of a physical process as it is developing in time, considering the time increasing from left to right.

In a one-electron HF approximation, the amplitude of photon absorption is presented by a diagram describing a transition of an electron from its initially occupied level to the final state E,

or the creation of an electron-vacancy pair

---< (6)

to which a matrix element of length r *

(7)

and velocity V

(8)

corresponds. Here Ii) and (EI are the HF wave functions describing the electron in its initial Ii) and final (E I states.

RPAE corrections include the following infinite sequence of diagrams:

e 00--< . I

(a)

While (9a) presents the direct photoionization amplitude, the other diagrams in (a) describe indirect mechanisms, for example (9b), in which initially any electron-vacancy pair ET is created. Then this pair, due to electron-electron interaction, annihilates, giving life to the real final state with vacancy i and electron E. Diagram (9bl) presents inelastic vacancy-electron scattering. Diagram (9q) shows the simplest example of the influence of ground-state correlation upon the photoionization process. After the interaction of two electrons in their initial state i, i', they occupy two vacant levels E', E. Then interacting with the photon, one electron-vacancy pair annihilates while the other becomes a real final state. This process is referred to as a time-reverse of that given in (9bl), which is called time-forward. Using the definition of the RPAE amplitude, its much more compact expression via a diagrarnmatical equation can be presented as:

* The atomic system of units is used in this paper: e = h = me = 1.

28

E'

ro E ~ E

~£ +

-~£ (e,)

--~.=-- + I

(a) (~:) i E'

(10)

--~ ~ : E I

+ + -[Zt I E i' E' i (c2) (b2)

Analytically, the total RPAE photoionization amplitude D(ro) is represented by the integral equation

(EID( ro )Ii) = (Eldli) + t E'>F,i'~F [(E'ID( ro)li')( ro - E' -Ii' ) -I (E'iluli'E)

+ (i'ID( ro )IE')( ro + E' + Ii' ) -I (i'iluIE'E)] . (11)

Here u stands for the combination of the direct and exchange Coulomb potential 11\ - f2rl matrix elements:

The symbol f denotes integration over the continuous spectrum and summation over the discrete levels: i':5: F over occupied and E > F over vacant in the initial state of the atom.

If the sum over i' in (10) neglects all terms but that corresponding to the same ionization potential/r = Ij. only intrashell RPAE correlations are included. Other terms, with Ii';;!; I, present the intershell correlations. When i belongs to the outer shell, the production of singly-charged ions A + is described by the (iID( ro)IE) amplitude. However, for deeper vacancies that can decay via an Auger process, RPAE describes A++ formation. If a doubly charged ion is created in a real two-step process, the second being inelastic photoelectron scattering leading to double ionization of the atom, it is also described by the RPAE amplitude.

It is essential to have in mind that a considerable fraction of diagrams (10) may be included by a proper choice of the self-consistent field acting upon the outgoing electron. If single-electron wave functions calculated in the term-dependent version of the HF approximation are used, the following RP AE matrix elements are included in this one-electron approach:

E

---c::~} E'

ro_--e::::=.= + --~ I E } i' L=1

E' i L = 1 (12) 5=0 ---.c: J : E

} 5=0 Ii' = Ii +

L=1 i' 5=0

29

where L = 1 is the total angular momentum of the electron-vacancy pair and S = 0 is its spin.* Thus, all intrashell time-forward RPAE diagrams may be taken into account by a proper choice of visibly one-electron wave functions describing an electron moving in the vacancy i field, which far off the atom has a simple (-lIr) form. But it is essential to have in mind that, in the creation of this field, not only one electron (or bette~ to say, vacancy i) participates. On the contrary, all electrons of the ionized shell participate in the formation of this field together and usually coherently. Indeed, (12) may be iterated, leading to the sequence:

---<~ -I

__ ~E. = __ <~ ~l ,,1

+ ---0 E + __ E

7~ i

+

(13)

+ the time-forward exchange terms Ir = Ii" = ... =/j.

It is seen that this sequence, conserving the total angular momentum and spin of the pair, allows the inclusion of all terms with different i' (or in, ... ) states that have the same ionization potential I;. It means that the field that acts upon the outgoing electron I:: is created by all electrons of the ionized subshell, for example, by the 10 electrons if the dlO subshell is ionized. Therefore, the results of calculations performed in term-dependent HF approximations include, in fact, an essential fraction of correlational effects and cannot be considered as purely one-particle.

4. Beyond the RPAE Frame: Effects of Rearrangement

While in RPAE the photoelectron leaves the atom moving in a frozen core potential with a single vacancy i (i being the state of the ionized electron), in reality the situation is different because the residual ion is able to rearrange during the ionization process. If the photoelectron is slow enough, the ion has time to change its state because other atomic electrons, not only the photoelectron itself, are affected by the vacancy. As a result, they feel some extra attraction of the core, come closer to the nucleus, and screen the field that acts upon the outgoing electron. Consequently, the photoionization cross section near threshold and the ionization potential decreases. These effects present the statical (or adiabatical) rearrangement (SR), which may be exemplified by the following diagrams:

(J) -- (14)

* L = 1 and S = 0 are quantum numbers of a dipole photon.

30

ro --

(15)

E'" all i", E" included

In (14) the interaction between the photoelectron and the vacancy is "intercepted" by virtual excitations of other atomic electrons, thus leading to the screening of the direct Coulomb photoelectron-vacancy interaction. In the SR approach, this screening is taken into account by calculating the photoelectron wave function in the field of a completely rearranged (i.e., self­consistent HF) ion with a vacancy i instead of the frozen core of an atom with this vacancy.

Diagram (15) exemplifies the sequence of corrections that leads to the shift of the ionization potential Ii from the modulus of the HF one-particle energy eigenvalue leil. The difference between these originates from the alteration of all one-electron states due to the formation of vacancy i described above. Note that rearrangement is represented only by parts of entire diagrams: The vacancy i remains the same in intermediate states of (14) and (15) instead of summation over all i' and (".

If SR corrections are included in all intermediate states of (10), the corresponding method is called generalized RPAE or GRPAE I.

Another kind of rearrangement effect is connected with the Auger decay of a vacancy formed in an inner shell in the photoionization process. After inner-vacancy decay, the photoelectron instantly finds itself in a double-vacancy field instead of a single-vacancy field. This is called dynamical rearrangement (DR). Diagrammatically, it is represented by the following picture:

ro --i' i"

EA

(16)

Here the doubled line, which starts from the moment of decay, denotes that the photoelectron moves in the field of a doubly charged ion with vacancies iT'. If e is small, the effect of field alteration can be rather large. Crudely, the influence of this Auger decay can be taken into account by considering that the outgoing electron moves in the double-vacancy field from the moment of its creation. If this is done not only for the final state but also for all intermediate states in (10), and if these vacancies are able to decay, another generalized RP AE-GRP AE 11-is created.

Instead of demonstrating numerous results of calculations, qualitative effects for the considered types of correlations are schematically presented below. Most transparent is the RPAE effect in the vicinity of the 4d vacancy in Xe, Cs, Ba, etc. In Fig. I, the effect of RP AE intrashell correlations is presented.

61 (00) f' I , I , I \ , \ HF , \ I \ , \ ,

~ ______ -L ________ ~_--__ --~--~~~~~~ ro, Ry I

Figure 1. RPAE intrashell correlation effect (schematic).

31

A good example of intershell RP AE correlations is the action of the outer 5p6 and inner 4d 10

shells upon two 5s electrons [1]. Illustrated in Fig. 2, this effect is typical and, like the previous effect, is universally important for many atoms. Qualitatively, the effect of intershell correlations also looks similar in those cases when the inner-shell excitation is dominated by a discrete excitati~n instead of by continuous spectrum resonance, as in the case of atoms with semifield shells such as Cr, Mn, and Eu [2].

For intermediate shells, RPAE leads frequently to narrow "spikes" in the photoionization cross section at threshold. These are not observed experimentally. A typical example is the 2p6 shell of Ar [1]. Figure 3 illustrates the role of statical rearrangement in the case when RPAE correlations are small. Another interesting example is the 4d-shell photoionization of the La atom, in which the influence of RP AE correlations is large but not sufficient to push the cross-section maximum into the continuum, which is done by inclusion of SR [3]. This is represented by Fig. 4.

61 (00)

o Experiment

-----~~~ __ ~ ________ ~ ______ ~ ______ ~oo,Ry

Figure 2. RPAE intershell correlation effects (schematic).

32

61 (0) 61 (0)

Experiment

RPAE

~. / Experiment

~ GRPAEI

L-______ ~ ______ ~~ ____ ~ 0)

1

Figure 3. Effect of statical rearrangement (SR) upon HF cross section: Ar 2p6 subshell.

Figure 4. Effect of statical rearrangement upon RPAE cross section: La 4d10 subshell.

The dynamical rearrangement is important mostly for the inner Is shells. This is illustrated by Fig. 5, which shows that the increase of the field acting upon the outgoing electron leads to an increase of the cross section at threshold.

The inclusion of these effects-RPAE, SR, and DR using RPAE or GRPAE I and II-achieves reasonable agreement with experimental data or even permits one to make predictions. It is seen that by investigating the photoionization of different atomic shells, information is obtained on atomic structure, namely on the ability of electron shells to be polarized and to rearrange themselves as a result of the action of vacancies created in the process of photoionization.

5. Beyond RPAE Frame: Double-Vacancy and Satenite Formation

Here we will discuss the possibility of creating double vacancy states in the photoionization process. Until now, in this domain, most of the results have been obtained using the lowest-order perturbation theory to describe the formation of these states, or within the shake-off approximation.

61 (0) Jt' Experiment

,/HF "- ............ --DR ------------L _______ _

'_'-'-'-'-SR ._._._._._._.-

'-Hydrogen~like .--;~~ 11s ~ __ ~~ ____ ~ ____ ~ __ ~~ __ ~ro,Ry

1

Figure 5. Effect of dynamical rearrangement (schematic).

33

Double ionization can proceed by four basic mechanisms

(a) IDt2e, (b) C2 i'

£'

i1 £1

i2 £2

£2 i2 (17)

(c) i2 (d) £2 £1

£1 i1

i1 i2 £2

which account for ground (a,b) state and final (c,d) state correlations, respectively. Here the dashed circle describes the RPAE (GRPAE) correlations.

Diagrams (17) take into account the direct Coulomb interaction between two electrons participating in the considered process. Diagram (17d) represents real Auger and quasi-Auger effects, depending on whether or not vacancy r is really able to decay into the ili2£2 state. If r = iI, this diagram presents the lowest-order term of the shake-off. Diagram (17d) describes a process that can really proceed in two steps: first, the formation of vacancy i' and, second, the Auger decay. The same is true for (17c), which also includes a real two-step process: creation of an electron £', which then inelastically collides with other atomic electrons leading to the final £1£2ili2 state. Process (17c) is called direct knock-out.

As was mentioned above, the formation of doubly charged ions by two-step processes is included automatically in the RPAElGRPAE cross section. The (17a,b) terms as well as the part of (17c,d) that cannot be presented as a two-step process describes that which proceeds via virtual but not real states.

For high 0) (as compared to the ionization potential 1++), the main contribution to the double­electron photoionization cross section cr++ is given by (17a,b), i.e., by ground-state correlations, and partly by (17d) [4], whereas the contribution of (17c) is smaller by a factor (1++/00)2. For 00 ~ 1++, the predominant contribution comes from (17c), mainly its two-step part, so that a rather simple formula is valid [5]:

(18)

where W~ + ~A ++ (00 -Ii) is the probability of ionization (and excitation) of a singly charged ion A + in its collision with an electron whose energy is equal to £' = 00 - h

This mechanism contributes considerably to A ++ formation. It is therefore possible that at least one of the electrons £1 and £2 can have subsequent inelastic collisions with an outer atomic shell, thus producing A+++ and ions with higher charges. The mechanism shown in (17c) can also be called internal atomic friction because it describes the dissipation of £' electron energy.

34

The direct knock-out process also modifies considerably the A + formation cross section. For inner-shell ionization, (17c) becomes a real two-step process, so that the 0'+ cross section is given by [6]:

(19)

where aE is the elastic scattering phase shift of the ionized electron. Note that [maE "# 0 only if an intermediate inelastic collision takes place:

0) --(20)

where the vertical line denotes an intermediate state £'£"j that must have the same energy as £.

For the concrete case of Xe just above its 4d threshold [7-9] and for the same region in Ba [9], the corrections due to direct knock-out proved to be as large as 20-60%, and the account of this effect, presented in Fig. 6, gives very good agreement between the results of calculations and measurements.

Direct knock-out essentially alters not only the photoionization cross section but also the angular distribution because it leads to variation of the photoelectron angular momentum.

Of course, the four diagrams in (17) are unable to describe A ++ formation at any 0), £1, £2. In general, the following correlational corrections may be important:

The interaction between the two particles must be modified by taking into account virtual excitation of other electrons, i.e., the polarization of other electron shells due to energy transfer. The interaction between the electrons in the ground state .

61 (co)

I I

o I 000 /

........... I '<,RPAE

I \ I \ I \ l \

\ \ \

(very close to absolute photoionization cross section)

I L-~ ______ ~ ______ ~ ______ L-____ ~co,Ry

Figure 6. Yield of singly charged ions in Xe (schematic).

35

• The same in the final state. The role of other intermediate states, mainly the resonant ones. The respective diagrams are exemplified by:

(b) 101

(a)

i1

i1 i2

/f ., i2 102 Screening I (21) or polarization 102

(c)

~.' (d)

........... 11 i2 i2

102 i1

The role of corrections (2Ia,b) is most important when the transferred energy [00 - (£1 + II)] is close to the ionization threshold of an easily polarizable multielectron shell or to its discrete excitation. The correction (2Id) is most important when the intermediate state i'iln becomes real instead of being virtual and then decays by autoionization into £2ili2. The intermediate state i'iln is a satellite of the vacancy i. The important role of (2Id), specifically for Ar, with i = r = 3s, n = 4p, il = h = 3p, giving about 40-60% of the doubly charged ions, was experimentally demonstrated recently [10].

The formation of A+n, with n > 2, if it can not be achieved via step-by-step Auger decays, is quite far from being explained theoretically. A rather high degree of ionization is observed, so it is almost evident that the role of other mechanisms, such as direct knock-out and shake-off must be important [11].

The role of virtual excitations of intermediate multielectron shells (or their powerful discrete excitations) is universally important in A+n formation. It indicates the importance of RPAE (GRP AE) corrections presented in (17) by dashed circles. For the 4d 10 sequence of elements (Xe, Cs, Ba, ... Eu), it leads, as was recently observed experimentally [12], to a prominent increase of A+n production in the region of 4d ~ £f (4f) excitation on the way from Xe (4d~£f) to Eu (4d~4f). Except for achieving a general qualitative understanding, not enough was done in this interesting domain of research.

6. Creation of Vacancies and Their Spectra

Here we will consider single-electron ionization accompanied by discrete excitation, i.e., the creation of satellites, the simplest structure of which is a vacancy together with this discrete excitation. The formation of satellites is described basically by the same diagrams (17), but instead of the continuum states £1 or £2, a discrete excitation n is considered. The same higher­order corrections as (21) are important for this process also. If a discrete state is excited by the (17d) mechanism, its photoionization cross section is similar to that of the r vacancy creation.

36

The term "satellite" was initially applied only to the case in which i' = i 1 so that i2n was a monopole excitation. If i' #; i2, but (17d) is still most important, a "shadow" state is formed. A good example of such a state is 3p-23deS) in Ar, being a "shadow" of the 3s-1 vacancy. Its cross section is closely connected with that of 3s [13]. The discrete excitation in shadows is not a monopole. Their formation resembles an Auger decay, although energy forbidden. It is important to have in mind that for high co, the ratio of satellite (or shadow) line intensities created by the (17a,b,d) mechanisms to those of the main line becomes frequency independent. On the contrary, the same ratio for satellites excited via the (l7c) mechanism decreases rapidly with an increase of co.

Studied at high co, the spectrum of atomic states reveals that the combination of the main and satellite lines, their position, intensity, and widths are determined by electron correlations, including those that are out of the frame of diagrams (17) and (21).

If in HF one has an isolated vacancy state with energy EHF, after inclusion of electron correlations, a number of adjusted satellite and continuous spectrum excitations appear (Fig. 7), which, however, obey certain sum rules.

It was proven [14, 15] that the following sum rules are valid:

JHF = '" Jj I +J""J(E)dE , ~ rea I (22)

j

which means conservation of total intensity, j being the number of a discrete satellite,

EHF = LERjZj + J EZ(E) dE , (23) j I

where ZAZ(E)] is the relative intensity (the total intensity being normalized to 1) of main and satellite lines. So Eq. (23) presents the conservation of the spectrum center-of-gravity position.

The normalization condition is given by

1= LZj+ JZ(E)dE j I

Intensity, J

~--~~~~--~----------~E

Figure 7. Correspondence between inner HF level and the satellite spectrum.

(24)

37

Note, however, that the direct derivation of Zj from photoelectron spectroscopy experimental data is impossible due to the important role played by a group of diagrams exemplified by (25) and representing a mixture of ground- and final-state correlations:

E

(25)

E', n where ili2n(E') are satellites to i. With an increase in 0), the ratio of the excitation intensity of such states to that of i is frequency-independent and cannot be eliminated, leading to pure values of Zj [16].

It was demonstrated that double-vacancy creation and excitation with ionization processes give information on the role played by the direct electron-electron interaction in atoms.

7. Decay of Vacancies

7.1. ONE-VACANCY DECAY

7.1.1. Important information on atomic structure can be obtained by investigating even one­vacancy decay processes. Diagram (26a) represents a simple Auger amplitude:

(a) (b) i -4--.---_-- i'

(26)

j

E

There are quite a few examples of investigations of corrections similar to (26b). One of them is the Xe case, with i = 4s, r = 4p,j = 4d, where inclusion of (26b), iN = 4d, decreases the Auger decay probability by a factor of 3.

7.1.2. The interference between direct and indirect amplitudes, including polarization of other shells, may also affect the radiative-transition amplitude:

(a) -_"I-~,~_"I-- i'

....... .........

(b) .. e; i' f" --.

(27)

Up to now, the most impressive example is the Xe case, with i = 4p, i' = iN = 4d, in which the inclusion of (27b) decreases the radiative decay probability by a factor of 10. Constructive interference, which increases the decay probability, is also possible [I].

38

7.1.3. A strong correlative may be semi-Auger decay, in which the emission of an electron is accompanied by the radiation of a photon. This process is presented in (28) together with its correlative correction (28b):

(a) i' (b) i i'

£ £ (28)

\ j j

\

7.1.4. An essential correlative process is step-by-step Auger decay, which leads to multiple-atom ionization, reaching up to A + 10. An example of this process is presented in (29):

i' £1

i1 (29) £2

i2 £3

i3

Of course, the screening of interelectron interactions and other similar corrections may considerably modify the decay probability, affecting electron-vacancy "tree" formation.

7.2. DECAY OF SATELLITE AND TWO-VACANCY STATES

7.2.1. The presence of an extra vacancy can essentially modify the decay probability, leading to prominent alteration of the field acting on both the emitted electron and on the virtual excitation that modifies the interaction leading to the decay:

q q q q

i(q) i'(q) i(q) i'(q) i"(q) (30)

Eq i"(q)

Eq

The index (q) emphasizes the alteration of the respective vacancy (electron) state due to the presence of vacancy q.

7.2.2. Two vacancies created in an atom after photoionization can decay almost independently, via an Auger process or radiatively, leading to the emission of the either two electrons or an electron and a photon. However, even this process can be essentially modified by the direct

39

interaction between these initially formed vacancies or by the interaction of the outgoing electron and the final state created after the second vacancy decay.

But there are decay processes that are entirely determined by interelectron interaction leading to the emission of a single electron (31a) or a single photon (31b):

(a) i1 i'1 (b) i1 : ~ : . i'1

i2 i'2 i2 .,

(31) 4 12 j .......

...........

Old

E<!

The most transparent manifestation of such a process is the emission of an electron or a photon whose energy is considerably higher than the maximum electron or photon energy emitted in a one-vacancy decay. This takes place when iI, i2 are inner vacancies. If i1 and i2 belong to intermediate shells, process (31) can make the vacancies closer, shifting them away from each other or permit both of them to go up [17]. The probability of such a process for intermediate shell vacancies can be rather high, leading to a number of new lines in the Auger electron and the photon emission spectra. Most essential is the contribution of the decay mechansism (31) in cases when the energy of the intermediate state iii is close to that of the initial state i1 h.

7.2.3. Rather interesting are pure cooperative (or correlational) decays of multivacancy states, for which the independent decay of each of the vacancies is forbidden due to energy or angular­momentum conservation, while the entire process is allowed. The following radiative decay in N can serve as an example [18]:

(32)

N (nitrogen): 1sJ.., 1si ; 2sJ.. ,2si ; 2p3i .

Here the vertical arrow denotes the spin projection, the ground state electron configuration being presented in (32). The IsJ.. radiative decay is forbidden by angular momentum and spin conservation: There are no 2p electrons in the ground state. The 2pi state decay is prohibited by the energy-conservation law. However the entire correlational process is allowed.

7.2.4. A number of rather complicated decays take place when a satellite state is excited. Consisting of two vacancies and one excited electron, the state has many more channels for decay than even the two-vacancy state. Indeed, the excited electron can easily change its energy and angular momentum, thus either being emitted out of the atom or opening other channels for vacancy decay. An example of such a process is given by (33):

40

Satellite state: (33)

where, because of a spin-flip interaction of n and iI, the term of a two-vacancy state is altered, leading to the emission of a low-energy electron [19].

8. Near and Subthreshold Formation and Decay of Vacancies

8.1. POST-COLLISION INTERACTION

One of the interesting phenomena connected with the formation of inner vacancies near their threshold is the modification of the Auger-electron line position and shape due to alteration of the field acting upon the slow photoelectron after the vacancy Auger decay. The corresponding essential diagram is (16), but contrary to the dynamical rearrangement (DR) case, attention is concentrated on the energy distribution of both emitted electrons. Due to the attractive nature of the extra field (two vacancies instead of one), eA increases while the Auger line becomes broader and asymmetric.

The exchange of angular momentum between a slow photoelectron and the i'i N state is also allowed. Therefore, IE can be different from IE' [see (16)]. Thus, even for i = Is, the angular anisotropy parameter is altered and becomes photon-frequency dependent.

If the velocities of slow photoelectron Vs and fast Auger electron VA appeared to be comparable, the latter energy shift &A starts to be dependent upon the angle between v s and v A

[20]. The simple classical expression for this energy shift is

(34)

where ri is the i-vacancy total width. In principle, radiative decay also can affect the slow photoelectron, changing the field acting

upon it. However, the corresponding frequency shift of the emitted photon is much smaller than after Auger decay.

8.2. THE INFLUENCE OF AN EXCITED ELECTRON

Below the photoionization threshold, the excited electron can be either a passive spectator or an active participant in the inner-vacancy decay process. The decay alters the field, acting upon the photoelectron, which may considerably influence the decay probability itself. The corresponding diagrams are:

41

(a) (b) n

---i1 (35)

i2 i'

E

where the doubled line stands for the excited electron in the modified field ili2 and r instead of i, respectively. Here, the 2p -+ 4d transition in Ar can serve as an example in which the 4d­electron wave function is so strongly modified by the possible change of the field that (4dl4d) "" 0 and the Auger-decay of the 2p-l state proves to be forbidden in the presence of the 4d excited electron. Instead, the decay of the 2p-l state accompanied by 4d electron excitation to the 5d,6d state is much more probable [21].

The presence of an excited electron can affect considerably the correlational correction contribution of a decay process, which is presented by (36):

(a) ~ nQ (b) nQ

+ i' (36) ~ "

.. i' ....... ......

--For the Xe case, when i = 4p and r = i" = 4d, the presence of the excited nQ electron can change the (36b) contribution not only in magnitude but also in sign, thus considerably affecting the decay process.

9. Emission of Secondary Particles in Near-Threshold Processes

Slow photoelectrons are always under the strong action of not only the vacancy potential, but also of the ion polarization potential created as a result of virtual excitations of the core:

E'

---. ~---.. - E o (37)

The effect is also strong when the photoelectron energy E is close to the threshold of a new channel E = Ij. For E ~ Ij, and in the region E "" Ij, the motion of photoelectron E is strongly modified, acquiring a cusp, i.e., a Baz-Wigner singularity. For E > Ij, emission of secondary electrons becomes possible.

Two basic mechanisms are responsible for the emission of secondary photons and electrons­either a direct one, i.e., simple bremsstrahlung and inelastic scattering, or more complicated processes, which account for correlation of atomic electrons.

42

The case of photon emission is presented by diagrams (38):

(a) £' - " - £ ,

"-'00

(b) £' - .... ----:~~£ ... " __ , •

(38)

.... 00

Here (38a) describes simple intra-atomic bremsstrahlung, while (38b) represents the effect of the residual ion polarization by the photoelectron. The prominent contribution of (38b) is known for electron-atom scattering [22] but has not been investigated for photoelectrons leaving the atom. If £ is small, the contribution of (38b) is affected by the interaction between the slow photoelectron and the virtual core excitation. If there are elastic scattering resonances for the outgoing photoelectron, they affect both terms of the bremsstrahlung amplitude, (38a) and (38b).

Rather interesting are cases in which the photoelectron's energy is close to the autoionizational resonance energy of the core. Then a phenomenon that one can call internal post-collision interaction takes place. This is illustrated by (39):

(39)

This effect must strongly modify the energy distribution of both outgoing electrons £1 and £2. It will be most impressive when the autoionizational width is large.

Of interest are also cases in which the large width is due to the interaction of the discrete excitation ni with some other intermediate state instead of the final £2i2 one. The role of the autoionizing state can be played also by any comparatively narrow, continuous-spectrum, electron-vacancy excitations.

A good example of the three-step process is the case of Mn or Eu. The corresponding diagram for Mn is given by (40):

(40)

£2} any outer channel

The connection between 3p-l 3d and 3d-1 ef excitations give a width of about 2-3 eV, which must affect the energy £1 and £2 distributions considerably.

Of course, there are many other correlational effects that can manifest themselves in the photoionization process.

43

10. Other Objects of Photoionization Studies

In the preceding sections, only atoms in their ground states were considered as objects of photoionization studies. Photoionization cross sections and other characteristics of this process (angular anisotropy parameters, photoelectron polarization) were calculated in the RPAE and out of its frame, using GRP AE I, GRP AE II, or perturbation theory. The calculations were performed for a number of atoms-their outer, intermediate, and, in some cases, inner shells. Much attention was given to noble gases, but also to other atoms with filled and semifilled electron shells [1]. The accuracy achieved looked satisfactory at the beginning but became insufficient with the improvement of experimental techniques: Detailed and reliable data are much more demanding on the theoretical description, forcing it to take into account more sophisticated correlational effects.

Meanwhile, new objects become available for photoionization studies: positive and negative ions and atoms in their excited states in positive ions. At first glance, electrons seemed to be subject to a stronger action of the nucleus field in positive ions than in atoms, so correlations should less important. It appears, however, that this is not the case, at least for degrees of ionization that are not too high. For these objects, the electron correlations in photoionization proved to be very important. So it was not a big surprise to learn that, for photoionization of negative ions, the role of different kinds of electron correlations is more important than for atoms, leading to rather sophisticated cross sections [23] and even to the very existence of the negative ions themselves (e.g., Ca-) [24].

Promising are such objects of photoionization as excited, metastable, or short-lived states of atoms. There correlations mainly manifest themselves at frequencies that are comparable with the core excitation and ionization potentials.

Rather exciting as objects of photoionization studies are atom-like multiatomic formations, such as clusters and fullerenes, whose ionization potentials are close to those of the constituent atoms, while the number of electrons on a given level is much larger, thus leading to much stronger correlational effects-more pronounced collective resonances, etc. These new objects, mainly those that are unstable, require for their investigation more and more intensive sources of radiation such as the new synchrotrons.

11. Further Investigation of Atomic Structure Using Synchrotron Radiation

We would like to finish this paper by mentioning some interesting and important problems for future investigation of atomic structure using synchrotron light:

11.1. Systematic study of the partial cross sections of different atoms, comparing the results obtained by electron spectroscopy and photo-induced fluorescence spectroscopy [25] with those obtained by measuring the multiply charged ion yield. The aim is to study the role of intra- and intershell as well as other types of correlations and of such processes as shake-up and shake-off.

11.2. Systematic investigation of outer-shell ionization near the inner-shell thresholds using either electron spectroscopy techniques or by measuring the singly charged ion yield. Close to the inner threshold, the post-collision interaction effect is important; whereas far away, only intershell correlations are essential.

44

11.3. Investigation of satellite levels produced by ground-state electron correlations, by real or virtual decay of vacancies, or by direct excitation of the residual ion by the photoelectron. Photoionization with formation of these states is a direct manifestation of the role played by electron correlations.

11.4. Measurement of the angular distribution and polarization of photoelectrons emitted from inner s, mainly Is, shells. The aim is to study the role of nondipole components of the photon and the angular-momentum exchange between the photoelectron and the core after Auger decay of the inner vacancy. The nondipole part of the angular distribution comes from the nondipole component of the photon and momentum exchange between the core and the photoelectron.

11.5. Photoionization near the inner or intermediate shell threshold. The aim is to investigate post-collision interaction, mainly for those shells and frequency regions where the velocities of two outgoing electrons, so-called "fast" and "slow," are about the same. Until now, mainly the fast electron energy distribution, but not the angular correlations, was measured. It is essential to find the cross section and angular distribution variation due to radiative vacancy decay.

11.6. Photoionization producing electrons with energy close to the outer-shell ionization or excitation potentials. The aim is to study the cross section variation due to the opening of some other channels; this is known for systems with short-range interaction (Wigner-Baz singularity) but is almost not considered for the case of Coulombic interaction.

11.7. The investigation of double-electron photoionization for almost all atoms, starting from the lightest to medium and heavy. The aim is to study different mechanisms of interelectron interaction leading to double-electron photoionization (initial- or final-state correlation). The energy and angular distribution of electrons as a function of incoming photon energy give information on interaction in both the initial and final states, starting from the threshold region where the Wannier regime dominates to short-range correlations for ro > 137Z, Z being the nucleus charge. Double-electron photoionization gives information on the role of different kinds of virtual excitations: discrete and continuous spectra of other than the ionized atomic electrons.

11.8. Study of the photoionization of excited (including highly excited) atoms with one electron out of the core. The aim is to find not only the deviations from simple hydrogen-like behavior of the photoionization cross section, but also to disclose the influence of the core upon this cross section. It is of interest to learn how the probability of inner-shell ionization is modified by the excited electron and how the latter affects the double-ionization cross section of the excited atom.

11.9. Investigation of negative-ion photodetachment. The aim is to fmd specific features of this process in the threshold region of photodetachment and near inner-shell thresholds. Of special interest are such objects as Si-, Ca-, and Ba-, for which quite interesting cross-section variations near threshold are predicted. Also of interest is the threshold low of the photodetachment, which is strongly affected by the polarization of the atom by the detached electron. Inner-electron elimination deserves to be investigated because it may be affected by the presence of the extra electron in negative ions.

11.10. Investigation of positive-ion photoionization. The aim is to study the role of electron correlations in ions, which seems to be rather important. With the growth of the degree of

45

ionization, the contribution of correlations in general decreases, but not linearly. In A+ or A++, it may be even stronger than in A.

11.11. Creation and then photoionization of exotic atomic states such as those with all electrons having the same spin projection, planetary-type, and hollow structures. All these states are so unusual that their photo ionization characteristics will have little in common with what is already known.

11.12. The study of ion formation in coincidence with secondary photons that originate from internal bremsstrahlung. The most transparent example is He++ formation in He ionization. It permits the study of the double-electron interaction after He ionization in cases when the pair's total momentum is zero, while in double-electron photoionization itself, the total momentum is equal to 1.

11.13. Observation of "drag" currents of photoelectrons (ordered motion of photoelectrons) in the direction of (or opposite to) the photon flux. The aim is to study the nondipole part of the angular photoelectron's distributions. It permits one to obtain information on the rather complicated linear (not angular) momentum exchange between the photoelectron and the remaining ion.

11.14. Investigation of elastic and inelastic photon scattering by atoms. The aim is to measure the frequency dependence of the atomic polarizability, which is strongly affected by electron correlations. Inelastic scattering permits the study of all mechanisms of photoemission and the role of electron correlations in these processes.

11.15. The study of high-energy (relativistic) photoionization. The aim is to disclose the role of nondipole components of the photon and possible effects of virtual excitations of the nucleus. The fast (relativistic) photoelectron on its way out of the atom can essentially interact with other atomic electrons, becoming a probe of a number of processes connected with rapid elimination of an inner electron, including shake-off and direct knock-out.

11.16. Investigate, using the electron-spectroscopy method, the high-energy (ro » l) part of the photoelectron spectrum of single and double ionization in order to derive from the experiment the Fourier image of the one-electron and correlational two-electron initial-state wave functions.

11.17. Consideration of photoionization of multiatomic formations well above the inner- or intermediate-shell thresholds of the constituent atoms. The aim is to study the extra multiatomic potential by a procedure similar to Fourier transformation, but on the basis of accurate numerical wave functions for isolated constituent atoms.

Of course, there are other interesting problems, those for which the emphasis would be on the role of electron correlations. Similar processes are of interest not only for isolated atoms in their ground and excited states and positive and negative ions, but also for multiatomic formations, such as molecules and clusters.

From the purely theoretical point of view, development along the directions presented above would help to improve considerably the accuracy of ab initio calculations for isolated atoms, enabling it to reach the level of 1%. Such development will also contribute considerably to deepening the understanding of multiatomic-formation structure, in order to reach the level already achieved for isolated atoms.

46

Acknowledgement

This work was performed during the author's stay in Germany and finished while he was at University of Nevada, Reno. The author is grateful to Alexander-von-Humboldt Foundation, which made his stay and research in Germany possible. He is thankful to the Department of Physics of the University of Nevada, Reno, for hospitality and to Mary Kliwer for preparation of the manuscript.

References

1. Amusia, M.Ya., Atomic Photoeffect (Plenum Press, New York, 1990). 2. Amusia, M.Ya., Dolmatov, V.K., and Mansurov, M.M., J. Phys. B: At. Mol. Opt. Phys. 23,

LA91 (1990). 3. Amusia, M.Ya., Ivanov, V.K., and Kupchenko, V.A., Zeitschrift ftir Physik D: Atoms,

Molecules and Clusters 14, 215 (1989). 4. McGuire, J.H., Adv. At. Mol. Opt. Phys. 29, 217 (1992). 5. Samson, J.A.R., Lee, E.-M., and Chung, Y., Physica Scripta 41,850 (1990). 6. Amusia, M.Ya., Chemysheva, L.V., Gribakin, G.F., and Tsemekhman, K.L., J. Phys. B: At.

Mol. Opt. Phys. 23, 393 (1990). 7. Shannon, S.P., Codling, K., and West, J.W., 1. Phys. B: At. Mol. Opt. Phys. 10,825 (1977). 8. Kammerling, B., Kossmann, H., and Schmidt, V., J. Phys. B: At. Mol. Opt. Phys. 22, 841

(1989). 9. Becker, U., Szostak, D., Kerkhoff, H.G., Kupsch, M., Langer, B., Wehlitz, R., Yagishita, A.,

and Hayaishi, T., Phys. Rev. A 39, 3902 (1989). 10. Becker, U., and Wehlitz, R., Physica Scripta T41, 127 (1992). 11. Zimmerman, P., Comm. At. Mol. Phys. XXIII, 45 (1989). 12. Nagata, T., Yoshino, M., Hayaishi, T., Itikawa, Y., Itoh, Y., Koizumi, T., Matsuo, T., Sato,

Y., Shigemasa, E., Takizawa, Y., and Yagishita, A., Physica Scripta 41, 47 (1990). 13. Wijesundera, W., and Kelly, H.P., Phys. Rev. A 39,634 (1989). 14. Manne, R., and Aberg, T., Chern. Phys. Lett. 7, 282 (1970). 15. Kheifets, A.S., and Amusia, M.Ya., Phys. Lett. 82A, 407 (1981). 16. Kheifets, A.S., and Amusia, M.Ya., Phys. Rev. A 46 (3), 1261 (1992). 17. Amusia, M.Ya., and Lee, I.S., Phys. Rev. A 45 (7), 4576 (1992). 18. Amusia, M.Ya., Kolesnikova, A.N., and Lee, I.S., Zh. Tech. Fiz. 57 (6), 1228 (1987) (in

Russian). 19. Annen, G.B., and Larkins, F.P., J. Phys. B: At. Mol. Opt. Phys. 24, 741 (1991). 20. Kuchiev, M.Yu., and Sheinerman, S.A., Sov. Phys. Usp. 32, 569 (1989). 21. Meyer, M., Raven, E.V., Sonntag, B., and Hansen, J.E., Phys. Rev. A 43,177 (1991). 22. Polarizational Radiation of Atoms and Particles, edited by V.N. Tzitovih and J.M. Oiringel

(Plenum Press, New York, 1992). 23. Amusia, M.Ya., Gribakin, G.F., Ivanov, V.K., and Chemysheva, L.V., J. Phys. B: At. Mol.

Opt. Phys. 23, 385 (1990). 24. Gribakin, G.F. Gul'tsev, B.V., Ivanov, V.K., and Kuchiev, M.Yu., 1. Phys. B: At. Mol. Opt.

Phys. 23, 4505 (1990). 25. Schartner, K.-H., Lenz, P., Mabus, B., Schmoranzer, H., and Wildberger, M., Phys. Rev.

Lett. 61, 2744 (1988).

PHOTOIONIZATION OF ATOMS AND IONS USING SYNCHROTRON RADIATION

Fran~ois J. WUILLEUMIER Laboratoire de Spectroscopie Atomique et /onique, Unite Associee au CNRS, URA n° 775, Universite Paris-Sud, BtU. 350, 91405 ORSAY Cedex, FRANCE

ABSTRACT. Progress made over the last thirty years in the use of synchrotron radiation for atomic photoionization studies is described, and is illustrated with selected examples.

1. Introduction

The first observation of synchrotron radiation (SR) occurred more than 40 years ago, with the 70-MeV General Electric electron synchrotron [1]. Later on, several groups of experimentalists determined the unique characteristics of this radiation, using the 250-MeV synchrotron at the Lebedev Institute in Moscow [2, 3] and the 300-MeV synchrotron at Cornell University [4, 5]. During the same period, a number of theoretical works were successful in calculating the properties of this radiation [6, 7,8].

We will celebrate in 1993 the thirtieth anniversary of the first use of SR in research, namely the investigation of atomic photoabsorption in the ultraviolet region [9]. Over the last three decades, continuous improvements have been achieved in the production and in the use of synchrotron radiation. Nowadays, the number of scientists making use of this powerful tool and the number of

Electron orbit

Figure 1. Experimental arrangement for utilizing synchrotron light for absorption spectroscopy. The gas to be studied fills the entire spectrograph. A 3-meter, 600-lImm concave grating is used at 84.5 degrees angle of incidence (from Ref. 10).

47

A.S. Schlachter and F.J. Wuilleumier (eds),

synchrotron radiation centers throughout the world has grown exponentially. Pioneering experi­ments that were at the ultimate edge of technical possibility have become, twenty to thirty years later, routine experiments. Now, they are systematically perform­ed in many places to go deeper and deeper into the detailed knowledge of atomic properties and of the dynamics of excitation and relaxation in photoionization processes. Other atomic systems, more difficult to

New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 47-102. © 1994 Kluwer Academic Publishers.

48

70

--.J--

110

Electron voila 15 10

;r - ... . 200 220

Wavelength (A)

55 50

J1 ,--

240

Figure 2. The absorption coefficient of helium in the 50-70-eV region. The solid curve connects the points at which the data were reduced in the experiment. The dashed curve is obtained from the interpolation of the data taken off resonance [10].

produce and to study, have become amenable to exploration.

To illustrate the qualitative and quantitative changes that occurred in the use of synchrotron radiation, in atomic physics as well as in many other fields, we would like to compare two experiments that are quite representative of the state of the art in 1963 and in 1993. In Fig. 1 [9, to], we show the experimental arrangement for utilizing synchrotron radiation in these first photoabsorption spectroscopy experiments. The light emitted by the NBS lSD-MeV electron synchrotron in Washington fell upon the entrance slit of a grazing incidence monochro­mator. The rare gas under study, helium here, was introduced into the whole spectrograph. A photographic plate was used to detect the intensity of the radiation without and with the gas present in the spectrograph. An example of the absorption spectrum, which was obtained in the 165-200-A region, is visible in Fig. 2 [to].

Several series of two-electron excited resonant states were revealed, showing characteristic Beutler-Fano profiles due to the interference between the discrete state of the two-electron excitation channel and the continuum of the single ionization channel, as predicted by Fano [11]. In these two-electron, one-photon transitions to He** autoionizing states, the two-electron excited states are produced by correlation effects.

In Fig. 3 [12}, one can see the experimental setup used in 1991 for the PISA experiment to study resonant photoionization in singly-charged Ca+ ions with photoelectron spectrometry [12, 13]. Here, the synchrotron radiation emitted by the SU6 undulator of the Super ACO storage ring is monochromatized with a toroidal-grating monochromator and is focused into the source volume of an electron spectrometer, the cylindrical mirror analyzer (CMA). A beam of singly­charged Ca+ ions produced in a Nier-Bemas-type ion source [14] is transferred through an ion beam line using magnetic dipoles and an electrostatic deflector, and is finally focused with electrostatic quadrupoles into the same source volume of the CMA. The interaction between the

Deflector Magnet 2 CMA \ e+

SuparACO

~ Ion Source Magnet 1 Quadrupole. Faraday cup.

Figure 3. Layout of the PISA (nphotoionization of Ions with Super Acon) experimental setup. Ions propagate from left to right; photons travel the other way. Both beams are focused into the source volume of the CMA [12].

25

hV.33.20eV

l! ~ 10 U

5

0L-~14~~--~18~~--~22~~~~~

lab. Kinetic Energy ( eV )

i; - 12 .. i 8 o ..

o

hVa32.67eV

14 18 22 28 lab. Kinetic Energy ( .v )

Figure 4. Top: photoelectron spectrum measured in the photoionization of a 20-keV, 35-JlA. Ca+ ion beam with 33.20-eV (33.17-eV in the laboratory frame) photons. The Ca+ photoelectron line appears at 19.2 eV. The average current in Super ACO was 200 mAo Bottom: photoelectron spectrum observed under the same experimental conditions, except for the photon energy which is detuned from the maximum of the resonance by 0.5 eV. The Ca+ line is not visible because the direct photoionization cross section of Ca+ is very weak at this photon energy [Ref. 12].

49

monochromatic photon beam and the ion beam causes the excitation of an inner-shell 3p electron onto the 3d orbital to create a core-excited ionic state, according to 3s23p64s 2S~3s23p53d4s Zp. The electrons emitted in the autoionization of these Ca+* ions to the ground state of Ca++ ions are energy-analyzed within the CMA and are detected with a battery of channeltrons. The spectra in Fig. 4 [12] summarize the various

effects taking place in the interaction. They are explained in the caption. This experiment is heavy and expensive; it provides, however, the unique possibility to probe, with synchrotron radiation, a new class of atomic species: atoms in ionic states.

2. Observable Parameters in Atomic Photoionization

2.1. PHOTOABSORPTION AND PHOTOIONIZA nON

During the first ten years after the historical experiment at the National Bureau of Standards, experiments in atomic physics were mostly confined to total photo absorption or reflectivity studies. Although such experiments are quite useful, they are somewhat limited in several aspects. In a simple photoabsorption experiment, the excited states of the neutral atoms show up as resonance transition series and the total photoabsorption cross section can be measured. Thus, information can be gained only on the sum of all partial cross sections for the various processes contributing to the attenuation of the photon beam. These processes are single ionization of an electron into the continuum, simultaneous excitation and ionization of the atom leaving the residual positive ion into various excited ionic states, double photoionization, and eventually triple photoionization. If hv is the energy of the monochromatic photon beam, all atomic electrons whose binding energy is lower than hv are involved in these processes. Thus, more sophisticated spectrometries must be used to explore these various channels. We illustrate this need in Fig. 5 [15]. It shows the photoabsorption spectrum of lithium, which has been measured using SR. Several photoionization processes can occur, depending on the photon energy. They are listed in Table 1. At 92-eV photon energy, e.g., the single photoionization of the Is electrons

50

TABLE 1. Photoionization processes in lithium.

Type of process Transition from Ionization threshold Peak number in IS22s 2S to (eV) Fig. 6

2s photoionization IS2+ £ 5.39 1

Is single photoion- Is2s 3S + £ 64.41 2 ization (main lines) Is2s IS + £ 66.31 3

Is photoionization + Is2p 3p + £ 66.67 4 2s--+2p excitation Is2p lp+£ 67.61 5 (n = 2 satellites)

Is photoionization + Is3s 3S + £ 74.17 6 2s--+3s excitation Is3s IS + £ 74.67 6 or 2s--+3p excitation Is3p 3p + £ 74.76 6 (n = 3 satellites) Is3p lp + £ 75.04 6

(ls2s 1,3S Li+ final states) contributes the most to the total cross section. However, weaker ionization processes occur also in the Is subshell, producing correlation satellites with the outer

electron being excited to the 2p orbital (ls2p 1,3p final states) or to the (n = 3) orbitals (ls3s 1,3S and

c o

4

13 I 2

I o

115 125 135 145 155

Wavelength (A)

Figure 5. Photoabsorption cross section of atomic lithium above the Is ionization thresholds. The black squares are the results

Is3p 1,3p final states), as well as in the 2s subshell (ls2 IS final state). The 2s photoionization cross section is very weak so far above the 2s ionization threshold (5.39 eV). These lower cross-section processes can be studied only with electron spectrometry, as exemplified in Fig. 6 [16]. Here, electrons ejected from each subshell in a specific photoionization process have different kinetic energies, and thus these photoionization processes can be analyzed subshell by subshell. Partial cross sections can be determined on a relative scale from

166 the corrected integrated area measured under each photoelectron line. In the general case, when a hole is produced in an inner shell, the decay of the inner vacancy may occur via the nonradiative Auger or autoionization processes. Both types of processes

of a theoretical calculation using Hartree- yield electrons that are also seen in the electron Fock wave functions [15]. spectrum, bringing information on the decay of the

core-ionized ionic or excited atomic states.

2.2. OBSERVABLE PARAMETERS

Different techniques of increasing sophistication have been developed to explore the various ionization and relaxation channels: energy-resolved, angle-resolved, and spin-resolved electron

51

spectroscopies; ion spectroscopy; and fluorescence spectroscopy. More recently, coincidence techniques have been introduced to allow for more selective experiments to be done, such as electron-ion, photon-ion, photoelectron-Auger electron, photon-photon coincidences. Nowadays, it is possible to measure parameters related to the atomic structure, such as the binding energies of the atomic electrons, the energies of resonant transitions, the natural width of the levels, and parameters describing the dynamics of photon-atom interactions. The latter include partial photoionization cross sections, oscillator strengths in the case of resonant transitions, angular

Binding energy (eV) distribution of photoelectrons and Auger

2000 U

I ~ 1500

• C ~ 1000 o

500

0-15

1a 3P.3L 6 ,..

, ' 20

, , 1\ t •

5 .. :. :

25 30

92.4.V

as

1 '"

90

Kinetic energy (eV) Figure 6. Photoelectron spectrum following photoionization of Li atoms in the Is22s 2S ground state by 92-eV photons. See Table 1 for assignment of the photolines [16].

electrons, fluorescence yields, Auger rates, and spin polarization parameters [17]. These quantities are related to the wave functions of the initial and final states of the atomic system and to an operator that describes the interaction. In this context, the growing use of synchrotron radiation has provided a large amount of data showing evidence of phenomena that could not be explained in the one-electron model, but required the introduction of electron correlations and also relativistic formalism into the theoretical models. Thus, theory has also greatly benefitted from the development of synChrotron radiation, since ex­perimental results have inspired theore­ticians to develop more and more accurate approximations to describe

better the atomic structure and the dynamics of photon-atom interaction. The aim of this chapter is to show how atomic photoionization studies developed over the last

three decades and to address the present status of the field, but not to make an exhaustive presentation of all experimental and theoretical aspects of photoionization. Thus, we will give a few indications only about the two main parameters that are measured worldwide in most of the experiments, i.e., partial photoionization cross sections and angular-distribution parameters. We will also comment on a recent breakthrough in the study of alignment effects. For a detailed description and formulation, we refer the reader to various experimental [17-24] and theoretical [25-27] reviews.

2.2.1. Photoionization Cross Sections. The probability describing an atomic transition between an initial state i and a final state f with photoionization of an electron can be expressed as:

(1)

52

with: 'l'i = initial-state wave function 'l'f = final-state wave function of the system (ion + photoelectron)

ct = Dirac matrix "t = photon polarization It = momentum vector of the incident photon n = number of atomic electrons ~ = vector of the nth electron

Since the photon-energy range explored in most of the SR experiments is the soft x-ray region (a few hundred electron volts at most), the dipo~~approximation is considered to be valid, i.e., the retardation effects are neglected, and exp (-ik.rn ) = 1. Thus, in the length formulation, the transition probability can be written as:

p oe 1/'1' ; r 'Pi drl2 =R 2 , (2)

and the cross section:

afi = 4n2a 2R2/hv ,

where a is the fine structure constant.

(3)

How is it possible to determine experimentally absolute values of this cross section in order to test the various theoretical approximations? In an electrostatic electron spectrometer without accelerating or decelerating lenses, the number of photoelectrons being ionized by an incident photon flux N(hv)dE with energy hv in the dE interval energy and detected into the solid angle dO at the angle 9 is given by:

with:

N(E)dE = Gn1ll(E)T(E)(daJdO)a dON(hv)dE (4)

G = geometrical factor n = atomic density in the source volume I = length of the source volume T)(E) = detection efficiency of the electron detector T(E) = transmission of the electron spectrometer (daru'dO) 9 = differential photoionization cross section for nl atomic electrons at

the angle 9 9 = angle between propagation directions of the photon and photoelectron

In the nonrelativistic dipole approximation, which is valid at low photon energy, the differential photoionization cross section of (n,1) atomic electrons in the dO solid angle is given, for an elliptically polarized photon beam, by [28]:

with:

daJdO = aJ4n {I - ~J2 [P2(cose) - 3/2 cos 2m cos 2cp sin2e] } (5)

P2= Legendre polynomial of 2nd order = (3 cosZO - 1)/2 ro = ellipticity of the polarization ellipse = tan b/a, b and a being the length of

the small and large axis of the ellipse 9 = angle between the directions of photons and photoelectrons cp = azimuthal angle of the photoelectron referred to the main axis of the ellipse

53

If the target atoms are randomly oriented, collecting all photoelectrons emitted in the (O,21t) q> angular range and choosing 9 = 54°44' (the magic angle) makes the differential photoionization cross section equal to :

(6)

In this geometry, one measures N(E)dE and one extracts dan/dil. In fact, the determination on an absolute scale of all terms in Eq. (5) is extremely difficult and practically impossible when one aims for a 10%-20% accuracy. But it is easier to determine on a relative scale 11 and T. In that case, for all photoelectron lines measured in the same electron spectrum, one has:

(7)

K is a proportionality factor, the same for all lines within a given spectrum. T(E) can be determined using photoelectrons emitted from rare-gas subshells with known cross sections or Auger electrons. Thus, all a nl can be compared on a relative scale. For other geometrical configurations, pseudo-magic angles can be defined.

Th obtain absolute values, a normalization procedure must be used. When total photoabsorption cross sections have been measured and if the branChing ratios for all processes contributing to the total photoabsorption cross section, including multiple ionization, are known, absolute values of a nl can be determined. However, there are only few cases for which absolute values of total photo absorption cross sections have been measured: the rare gases [29], lithium [15], and sodium [30]. In all other cases, partial nl cross sections are made absolute by normalization to a theoretical calculation or by use of some sum rules, which are not always accurate enough because all relative oscillator strengths for resonant and continuum transitions have to be known. Thus, most of the time, one is able to test only the energy dependence of the theoretical calculations and not the absolute values of their results. It must be well understood, in particular by the theoreticians, that the vast majority of the partial cross sections that have been measured and published are not the results of abolute determinations and, thus, should not be used to test the validity of the absolute values of calculated partial cross sections. Let's mention, however, that in the case of a gaseous sample, a determination of the pressure of a rare gas introduced in the source volume of the electron analyzer can help to get absolute cross sections, using theoretical formulas for electron elastic-scattering cross sections. This has been achieved once, in the case of xenon atoms in the ground state [31].

In the case of resonant transitions, one can make use of the fact that the oscillator density df/dE is connected to the cross section through the following relation, in atomic units:

df/dE = c121t2 x a . (8)

Integration of df/dE along the line profile gives the oscillator strength after comparison with a partial photoionization cross section.

2.2.2. Angular Distribution Parameter. The angular distribution parameter ~ can be determined by using relations similar to the one shown in Eq. (5). Whatever the experimental configuration is, it is always necessary to determine the polarization rate of SR and, often, the relative transmission of several spectrometers, including electron-detector efficiencies. Photoelectron lines with known angular distribution parameters and cross sections must be used, such the He $ = 2

54

for the He+ls 2S final state in the nomelativistic approximation) line and/or photoelectron lines from other rare gases.

1301 depends on the radial matrix elements R and on the phase shift S of the continuum waves. The value of 13 is always between -1 and +2. When only one wave is necessary to describe the continuum, 13 = 2. Thus, for s states, 13 = 2 in the nonrelativistic case, such as for He left in the Is 2S He+ state. For higher Z values, relativistic effects contribute to the existence of two waves in the continuum, Ej _l12 and Ej + 1/2' Thus, even for an s state, the 13 value may not be 2, because of possible interferences between two continuum waves .

Angular distribution of the photoelectrons characterizes the symmetry of the final state, and its shape helps to identify the final ionic states. For higher photon energies, let's say typically 1 keV, multipole contributions may become important. Higher-order Legendre polynomials have to be included in the expression of the angular-distribution parameter [32]. Transfer of angular momentum and parity conservation have been studied in detail by Fano and Dill [33] to determine the favored and unfavored transitions. Their results are very useful for predicting or interpreting the experimental data.

2.2.3. Alignment of Photoions and Auger Electrons and Polarization of Fluorescence Photons. Photoionization of nonoriented atoms with linearly polarized photons can produce aligned photoions, because the magnetic sublevels are nonstatistically populated as soon as the inner vacancy is created in a subshell with a j quantum number higher than 112. This alignment can be observed either in the measurement of a nonisotropic angular distribution of the Auger electrons or fluorescence x-ray photons emitted in the relaxation of the vacancy or in the measurement of the polarization of the emitted photons. Predicted by Mehlhorn [34] and theoretically described by Cleff and Mehlhorn [35, 36] and by Fliigge et ai. [37], these effects were first observed in the fluorescence radiation emitted in the decay of a 4d vacancy in Cd [38], then in the Auger lines following creation of a 3d vacancy in Kr [39], a 2p vacancy in Ar and a 4d vacancy in Xe [40]. Very recently, alignment effects have also been measured in the autoionization of laser-excited­and-aligned lithium atoms [41, 42]. Alignment of the photoions may even lead to spin­polarization of the Auger electrons through spin-orbit interaction [43]. If the inCOming photon beam is circularly polarized, the orientation and the alignment of the photons can be transferred to the spin of the Auger electrons [44, 45]. Studies of these effects will greatly benefit by the availability of helical undulators providing an intense flux of circularly polarized radiation.

2.3. THEORETICAL CONSIDERATIONS

In the beginning of the 60's, the only theoretical model routinely used to describe the photoionization process was the central-field Hartree-Slater approximation. A better description of the photon-atom interactions was obtained by taking into account the many-body effects, i.e., the correlations between the atomic electrons, which are strictly forbidden in the frozen-core central independent-particle model. Relativistic effects are important for mainly high-Z elements. Correlation effects have been extensively studied over the past ten years: multiple photoexcitation and photoionization, post-collision interaction, autoionization, and Raman processes are a manifestation of their importance. Some of these effects are strongly energy­dependent and their study was greatly favored by the continuous spectral distribution of SR. The large amount of experimental data, that has become available, was of great help for the theoreticians in order to better describe the interactions to be taken into account in the models. It

55

has been slowly understood that single photoionization processes could also be strongly influenced by correlation effects in such a way that some features, which were originally explained in a one-electron approximation, have been reproduced theoretically only when electron correlations were introduced in the theoretical model. A typical example is the behavior of the 4d cross sections of xenon and barium atoms, which were correctly described only three years ago (See Sections 3,4, and 6 of this chapter).

Electron correlations may be described alternatively in the configuration interaction scheme or in the many-body theories (MB1). While the configuration interaction procedure tries to optimize separately the initial and final state wave functions, the MBT focuses on the dipole matrix elements involving simultaneously initial- and final-state correlation effects. Correlations in the ground state or initial-state configuration interaction (ISCI) are, to a large extent, independent of the photoelectron kinetic energy as well as correlations in the final ionic state (FISCI). Both ISCI and FISCI may be considered to be due to "intrinsic" [46] correlations; Le., they are present even in the absence of photoionization. However, this is true only in the sudden approximation limit, Le., when the relaxation, during and after ionization, of the passive electrons has not had enough time to occur because the electron leaves the atom very quickly. In this approximation, the probability that the atom will be found in a particular shake-up state after the ionization can be determined by projecting the initial wave function on the final states in the relaxed ion. Thus, the shake-up (shake-oft) intensity is proportional to the square of the overlap of the matrix elements between the relaxed and the unrelaxed orbitals of the remaining electrons. However, when the passive electrons have enough time to adjust adiabatically to the change in the potential, Le., when the photoelectron leaves the atom slowly, the satellite (double-ionization) intensity should decrease with decreasing photoelectron kinetic energy. In this case, the shake-up (shake-off) process is a dynamic [46] process. Another example of dynamic correlations can be found in the inelastic scattering process [47], also called internal electron scattering. While leaving the atom, the photoelectron can excite or ionize another atomic electron. In the early days, the satellites due to this effect have been named "conjugate shake-up satellites". This is not considered to be a good designation by the theoreticians, but, qualitatively, it describes well the difference between this type of satellite (the excited electron changes its orbital quantum number by one unit) and the shake up satellites (no change of the orbital quantum number in these monopole transitions). The relative intensity of the conjugate shake-up satellites is strongly dependent on the energy of the photoelectron. A close similarity has been found recently between the energy-dependence behavior of this process and of the ionization of singly charged ions by electron impact [48]. In the configuration-interaction approach, this process is described as arising from continuum­state configuration interaction (CSCI).

From a theoretical point of view, the CI description is not satisfactory because it is not possible to somewhat arbitrarily distinguish between interactions in the initial and final states. The many-body theories prefer to consider various types of diagrams to take into account ground- state correlations, core relaxation, polarization effects, intrachannel and interchannel coupling, spin-orbit coupling, relativistic corrections, and inelastic scattering of the photoelectron. Indeed, the many-body perturbation theory [49] and the random phase approximation with exchange in its non-relativistic (RPAE, [50]) and relativistic (RRPA, [51]) formulations have been the most successful in reproducing the relative and absolute intensities of one-electron and two-electron processes, as they have been experimentally measured.

56

2.4. EXPERIMENTAL CONSIDERATIONS

From the experimental point of view, a much more intense and well-collimated photon beam is required for selective spectroscopies than for photoabsorption experiments because the pressure in the interaction zone of a differential analyzer generally has to be lower by at least four orders of magnitude and the length of the effective source volume is typically a hundred times lower. In addition to these requirements, no window can be used to contain the sample when electrons or ions have to be detected. This explains why the types of experiments that can be conducted at a given time depend critically on the quality of the synchrotron radiation source and on the techniques used to provide the experimentalists with a monochromatic photon beam of well­defined characteristics. In view of the future use of new electron (positron) storage rings, it is the purpose of this chapter to briefly review how each new advance made in the design of synchrotron radiation sources and/or in monochromatization techniques has been immediately followed by a qualitative jump in the capabilities of the experiments to go deeper and deeper into a more selective analysis of the photoionization processes.

One can distinguish four phases in the history of synchrotron radiation. We will illustrate in the following how each of them can be related to new progress being made in atomic photoionization. We refer the reader to experimental [17-24] and theoretical [25-27] reviews for a more exhaustive description of the state of the art in the many different fields.

3. Phase 1: Electron Synchrotrons and Photoabsorption Experiments (1963-+)

For synchrotron-radiation users, this phase can be characterized by the parasitic use of electron/positron synchrotrons built for high-energy collision physics, namely for studying (e+e-) interactions. In these machines, the energy of the charged particles was continuously changing, the stability of the beam was not great and the photon flux available was low, typically lW to 108 photons/sec, depending upon the desired resolution. The use of these machines for synchrotron-radiation experiments started in the early 60's. Fig. 2 describes the very first results. The number of synchrotron-radiation facilities was small at that time. In addition to the NBS synchrotron in Washington, other locations where synchrotron radiation was used were Frascati, Hamburg (DESY), and, later on, Glasgow (for a short while), Bonn, and Daresbury (NINA). The characteristics of the synchrotron radiation beams available allowed mostly photo absorption experiments to be performed for the reasons previously mentioned. Data were obtained first on rare-gas atoms because they are easier to produce and to handle, and then on thin metallic films. Later on, experiments were performed on metallic vapors, and a very large number of well­resolved spectra were obtained, covering most of the Mendeleev Table (see Ref. 18 for a detailed review of photo absorption data). Presently, experiments of this sort are still carried out in a few places and are extended into the higher photon energy range. It should be noted again, however, that absolute photoionization cross sections in the gas phase have been measured, up to now, in a limited number of cases: the rare gases (see Ref. 29 and references therein) and some alkali atoms [15, 30]. Autoionizing atomic states produced by excitation of an inner electron or two outer electrons have also been extensively studied [18].

As an example of the results obtained in the rare gases, we have already seen in Fig. 2 [10] the variation of the absorption coefficient of helium in the 6O-eV photon-energy range as obtained with the experimental setup shown in Fig. 1. The asymmetric shape of the resonance profiles due to interference effects between the direct photoionization into the continuum (dashed line) and

- 400 c CD ·u

==

, ~ T I II i 4d9 582 5p6 (205l2)np

, ~ ',"i (203/2)

CD 0 U c 200 0

-+-Slit

a ... 0 (/) .a cC

64 66 68 70

Photon energy (eV)

Figure 7. Total photoabsorption cross section of Xe between 64 and 70 eV. The members of the two series are indicated by vertical marks [53].

57

the resonant channel via the (sp, 2n +), (sp, 2n -), (sp, 3n +), and (sp, 4n +)lPO two-electron excited states was analyzed in terms of the Fanoparameters [52]: reduced energy e, width r, and index q of the resonances. Good agreement was obtained with the theoretical predictions [11]. Another example, this time involving the excitation of a single electron from an inner-shell to produce a highly excited autoionizing state is presented in Fig. 7 in the case of the 4d subshell of Xe [53]. The Lorentzian shape of the resonances is completely different from the Fano profile observed in He. Several members of the 4d95s25p6 CZD3/2, 2DS12) np series are identified, and the series limits are determined. For the first and most intense resonance, the values of q and p2 are close to 200 and 0.003, respectively, indicating that very little interference occurs between the direct process and the resonant decay path. This is due to the fact that the off-resonance 5p photoionization cross section (4d105s25p5 2PI12,3/2 final ionic states) is weak at this photon energy so far from threshold.

The discovery of the non-hydrogenic behavior of some inner-shell photoionization cross sections in the soft x-ray range was made at that time, namely in the case of the 4d photoionization cross section in xenon [54], which is shown in Fig. 8. Instead of the maximum to be expected at threshold in the hydro genic approximation, the experimentally observed photo absorption spectrum (curve b) showed that the photoionization cross section of the 4d electrons in Xe rises to a large maximum about 30 eV above threshold. This unusual behavior was soon interpreted, at least qualitatively, by theory (curve a in the figure) [55]. It results from the fact that the potential barrier existing at threshold, because of the high value of the centrifugal force for high-l electrons, prevents the ef wave from penetrating the region of maximum charge density of the 4d electrons. Only as the photon energy increases, can the f­wave penetrate the centrifugal barrier so that the partial cross section for 4d ~ eftransitions rises rapidly and becomes the dominant contribution, giving rise to what was called a delayed onset of the cross section. Quantitative agreement between experiment and theory was not good, neither for the energy position nor for the cross section value at the maximum because the theoretical model used at that time was a rather crude, frozen-core, one-electron model. Only recently, Le.,

58

-.a 120 :::1!iE -c • 0

~ 80 Q) U)

U) U) 0 40 ... 0

80 100 120 140 160

Photon energy (eV)

Figure 8. Photoabsorption cross section of Xe above the 4d ionization thresholds (2Ds12 at 67.55 eV, 2D312 at 69.52 eV). (a) is the curve calculated in the one-electron Hartree-Slater approximation [55]; (b) is the experimental curve [54].

25 years later, a close agreement has been finally obtained between the newest photoelectron data [31] and the best many-electron relativistic calculations including relaxation and overlap effects [56], and inelastic scattering processes [57]. Many photoabsorption studies were carried out for the rare gases during this phase, showing interesting features in the discrete part as well as in the continuum part of the spectra [18-20].

In the ultrasoft x-ray range, another interesting behavior was observed in the absorption spectra of some heavy elements [58]. The spectrum of each of these elements showed large variations in the absorption coefficient well away from the absorption threshold, i.e., pronounced minima followed by some maxima. These observations were theoretically explained in summing the calculated contributions due to mainly 5d and 4f subshells. Although these results were obtained for samples in the solid phase, they are a manifestation of the atomic properties of the elements.

Among the first photoabsorption data obtained on metals in the vapor phase are the results of a photoabsorption experiment on Na vapor, which are shown in Fig. 9 [59]. The total photoabsorption cross section exhibits deviation of the atomic potential from the Coulomb potential and also demonstrates the influence of correlation effects. In particular, this spectrum clearly reveals the importance of inner-shell excitations involving one electron (below the 2p threshold at 38 eV and below the 2s threshold at 71 eV) and two electrons (below and above the 2p threshold), leading to numerous autoionizing states of excited sodium atoms.

Many absorption spectra have been measured with a high resolution (up to ')JM = 2xlOS) on a large number of atomic vapors, using the synChrotron radiation from the Bonn electron synchrotron [18]. Data obtained with such an improved resolution show the need for more and more input into the theoretical interpretation of rather complicated systems such as in the barium case [60].

Again, absolute determinations of total photoabsorption cross sections are greatly needed, if one wants to obtain partial cross sections on an absolute scale and to compare them to theoretical

:s 12

!. 10 c

. 2 U 8 III I)

I) In 6 2 u c 4 :8 a. ... 0 2 III

.Q cC

Na

.............. '" ' .... .... , .... .... .......

-- vapour

.............. atomic calc •

.... ....

..... .... : ...............

i • I I lfiI Na+ atatee "'1

2p53p 48 511 2Fi 3d

40 50 60 70 80 90 100 120 140 160

Photon energy (eV)

Figure 9. Spectral dependence of the absorption cross section of atomic sodium in the 30--150 eV energy range measured in arbitrary units and normalized to the atomic calculation. Note the large numbers of two-electron excitation lines above the Ip threshold at 38.0 eV [59].

59

calculations independently of any normalization to the calculated values. It is a field where the experimental investigations should be augmented to ensure most rigorously the validity of the comparison of the experimental data with the theory.

Details about the photo absorption data measured over this period can be found in several reviews [18-20].

4. Phase 2: First Electron Storage Rings and Photoionization Experiments (1974--+)

The following generation of atomic photoionization experiments, extending beyond total photo absorption measurements, was initiated with the Glasgow and Daresbury electron synchrotrons. But detailed studies were made possible only when the first storage rings, built for the study of (e+e-) collisions, became available to synchrotron-radiation users, initially in the parasitic mode. With the advent of ultra-high vacuum technology, the level of the photon flux available at the exit slit of a monochromator in the soft x-ray range (10-150 eV) reached 1010 to 1011 photons/sec in a 1 % bandwidth. These numbers were still quite low for the high demand of selective and differential spectroscopies, and most of the experiments performed during this period were basically exploratory experiments carried out to demonstrate that new effects were amenable to experimental studies. Counting times were very long, but new experiments using ion and electron spectrometries were successfully conducted in the photoionization area. Also, the unique time structure of synchrotron radiation (typically nanosecond pulses separated by hundreds of nanoseconds) was fully exploited to develop new time-of-flight electron and ion spectrometers. For electron spectrometry studies, the CMA proved itself to be a good compromise (intensity versus resolution) to take the best advantage of the available synchrotron­radiation flux.

The main places where such storage rings were used throughout the 70's for atomic

60

photoionization studies are Hamburg (DORIS), Stanford (SSRL), Stoughton (TANTALUS I, fully dedicated to synchrotron radiation use), Washington D.C. (SURF n, also fully dedicated to synchrotron radiation), and Orsay (ACO and DCI). Of course, photoabsorption experiments were continuously carried out also over this period, taking advantage of the improved photon flux to study atomic systems more difficult to vaporize such as transition metals and rare earths.

In almost every subject involving photoionization processes, each experiment provided new and interesting data, and demonstrated that synchrotron radiation was an extraordinary tool to investigate the dynamics of photon-atom interactions. Every topic of possible interest was investigated at least once: single electron subshell properties, i.e., partial cross sections and angular distributions in the case where only one electron is removed from the orbital occupied in the initial state; two-electron transitions, including correlation satellites (e, n processes) and double-ionization phenomena (e, e processes); resonant processes (autoionization); and threshold effects (post-collision interaction). These pioneering experiments were, of course, performed mostly on the atomic species that are the easiest to produce in sizable densities, i.e., the rare-gas atoms and some other closed-shell systems, such as cadmium and mercury [61] and lead [62].

4.1 SUBSHELL PHOTOIONIZATION CROSS SECTIONS

In the study of subshell photoionization properties, partial cross sections and branching ratios were measured for the ns and np outer shells of Ne [63, 64], Ar [~8], Kr [69], and Xe [66, 67, 70-73] and for the 4d inner-subshell of Xe [67, 70, 74]. It is fair to emphasize that, in fact, the

initial results were obtained using the synchrotron radiation emitted by old electron

30 -..a :!E -c: 20 o ~ ~

= 10 e o

Partial cross section for

Xe 4d10 Sal' Sp6 ~ 4cP SS2 Sp6 ...

" I \ I \ I ,

I \ I ,

\ \ \ \ \ \ \

\ \

\ \

synchrotrons, namely the Glasgow [75-77] and Daresbury [61, 63, 65, 70] synchrotrons. Some of the most interesting data were obtained in the case of Xe. In Fig. 10, we show the Xe 4d partial cross section determined by West et al. in photoelectron spectrometry [70]. Comparing these results to the photoabsorption data of Ederer [54] (see Fig. 8), one sees immediately that, even though the qualitative variation of the photo absorption and photoemission results is the same, the 4d single photoionization cross section is only approximately 2/3 of the total cross section. This is also obvious when comparing the 4d results with the RPAE

0 ........ --+----..--,--..-....-......,.-....-....... calculations [78] which are very close to the 60 80 100 120 140 total photo absorption curve and in marked

Photon energy (eV) disagreement with the partial 4d cross section.

Figure 10. Photoionization cross section of Xe for the 4d1O subshell. The data points are from West et aI. [70]; the dashed curve represents the results of an RPAE calculation by Amusia [78].

Also of great interest were the results for the 5s cross section of Xe, measured in two different experiments [66, 70]. They are shown in Fig. 11. First, in the low-energy part of the photon range, one observes a deep

-U).Q

~~ ... c 00

1.5

~:g ~ 0.5

Xe ~:r-""". ,.' 'i.

f ,'. f X'. ,'.

';'. , ....

40 80 120 Photon energy (eV)

Figure 11. Single 5s-photoionization cross section of Xe. The experimental results are from Adam et al. [66] (white square, small error bars) and from West et al. [70] (black points, large error bars). Data at low energy are from Samson and Gardner [110]. RPAE theoretical results are from Amusia [78], without (dashed line) and with (dotted line) account for relaxation of the 4d-subshell.

61

minimum around 33 eV, which is caused by intrashell correlations within the n = 5 shell: the 5s subshell is totally screened by the 5p subshell, and the 5s electrons cannot be ionized in this photon-energy region. This behavior is similar to that observed for the 3s cross section in argon [65,66] as can be seen in Fig. 12. The experimental results are not reproduced by Hartree-Fock calculations [79], while all many-body theories [80-83], taking into account intra-shell correlations between the ns and np electrons, reproduce the variation of the ns cross section in both cases, in particular the existence and, to a certain extent, the position of the Cooper minima. We see how sensitive the variation of a low-subshell cross section is to a correct description of correlation effects. It should be noted that the calculations have a tendency to give results higher

U) M

0.4

0.2

40

/

o

Photon energy (eV) 60 80 100

Ar

20 40 60 80 Photoelectron energy (eV)

Figure 12. Single 3s-photoionization cross section of Ar from threshold to 70 eV. Experimental data are from Adam et at. (white squares with small error bars) [66, 67], Houlgate et al. (black circles) [88], and Tan and Brion (crosses), measured in quasi-photon (e, 2e) experiments [111]. Theoretical results are from Kennedy and Manson (HF) [79], Amusia et al (RPAE) [80], Burke and Taylor (R-Matrix) [81], Lin (SRPAE) [82], and Johnson and Chen (RRPA) [83] (from Ref. 67).

62

Rydberg 2 3 4 5 6 7 8 9 10 -.a

X.- 51!, + excited ionic state. :::i -&: 1.5 0 :g CII U)

U) U) 5. 0

~ ... u 0.5 ii 1:: ca ~ 0 --

60 80 100 120

Photon energy (eV)

Figure 13. 5s-photoionization cross section of Xe, including main line plus satellite channels [84]. The RP AE theoretical results [78] are still too high because the coupling of the 5s channel with the two­electron channels is neglected in the calculation.

than the experimental values. This might come from neglecting the coupling with the weak two­electron satellite channels, as will be emphasized later.

For the 5s cross section in Xe, an additional effect is seen in Fig. 11 at the onset of the 4d ionization. Instead of a monotonic decrease, as expected in the one-electron model, the 5s cross section increases again, being enhanced via intershell correlations with the 4d subshell. Such an enhancement was qualitatively predicted by the RPAE calculations [78], but the value of the cross section was overestimated by a factor of two. The disagreement is partly reduced when the experimental contribution to the satellite channels, measured in one of the experiment [66, 84], was included in the 5s cross section, as can be seen in Fig. 13. The remaining discrepancy, due again to the neglect of the two-electron processes in the calculations, was solved later by Amusia [85], who introduced a so-called spectroscopic factor to account for these processes.

Some attempts were also made to extend the measurements of partial photoionization cross sections, in Ne and Ar, to higher photon energies [64, 86]. Fig. 14 shows an example of a photoelectron spectrum following photoionization of 3s and 3p electrons in Ar obtained with 208-eV photons, and Fig. 15 shows the variation of the 3s photoionization cross section [64]. The results of RPAE calculations show the same energy dependence as the experimental results but are still a little higher, as already observed at lower photon energies (see Fig. 12) and probably for the same reasons.

4.2. ANGULAR DISTRIBUTIONS

Angular distribution of the photoelectrons was also investigated for the rare gases [62, 65, 68, 75-77, 87, 88] and was shown to be generally in good agreement with RPAE calculations. Here again, we would like to comment on a case where the neglect of two-electron processes in the calculations leads to a strong disagreement with the experimental result, namely the 5s angular distribution parameter j3 in Xe. This case kept the attention of experimentalists and theoreticians over many years and is now a showcase of synchrotron-radiation history. In the

63

Photon energy (eV) Electron energy (eV) 170 180 190 200 100 140 180 220

U II ., 400 II) ,... ..... ., .. C ::l 0200 U

Ar hv = 208.0 .v 3p B.P ... 2.0 .v

38

40 30 20 10 Binding energy (eV)

Figure 14. Photoelectron spectrum following photoionization of Ar in 3s and 3p subshells by 208 eV photons [64, 86].

:a ."'. 2!0.2 RRP~ C o ;: g U)

U) U) o 0 0.1

U) C")

Ar

70 110 150 Photoelectron energy (eV)

Figure 15. 3s-single photoionization of Ar from 70 eV to

200 eV. RPAE theoretical results are still higher than the experimental data [64].

left part of Fig. 16, we show the status of experiment and theory before synchrotron radiation could be used to clarify the situation. Deviation of a ~ value of 2 for s electrons is possible only when spin-orbit interactions between the continuum wave and the ionic core are taken

2.0

1.0

o

-1.0

30 40 50 60 30 40 50 60 70

Photon energy (eV) Photon energy (eV)

Figure 16. Asymmetry parameter P for angular distribution of 5s photoelectrons in Xe. In the left part, the only result available prior to SR experiments was measured by Dehmer and Dill [90], using the 4O.8-eV He line. In the right part are shown the first SR measurements of White et al. [91]. Theoretical results are from various approximations: DF (ICDF-V and L) [89], RRPA (5s + 5p) [89] and (5s + 5p + 4d) inter-shell correlatious [89], RPAE [92], DS [93], DF [94], and K-Matrix [95].

64

into account. This leads to two dipole-allowed continuum channels (nsl12 ~ £P1l2 and eP3/2)

which can interfere, resulting in ~5s values lower than 2. Several different calculations [89]: Dirac-Fock (ICDF-V and L), and relativistic random phase approximation (RRPA), including two-shell (5s,5p) and three-shell (5s,5p,4d) correlations, gave very different results, particularly in the placement of the Cooper minimum. However, most calculations were in fairly good agreement with the only experimental value [90] available, which has been measured with Hell radiation, at 40.8-eV photon energy. Part of the answer to this intriguing question came when some measurements [91] could be made at SSRL. The new results, shown in the right part of Fig. 16, seemed to favor the RPAE [92] and RRPA [89] calculations that included the full 5s-5p and 4d interchannel couplings. The additional theoretical results shown in this figure are the Dirac Slater (DS) [93], Dirac-Fock (DF) [94], and K-Matrix [95] calculations. However, measurements at the Cooper minimum were still needed. To get the final answer, one had to wait several more years until synchrotron radiation was capable of providing more intense photon flux and theory was capable of introducing coupling to the two­electron satellite states in the calculations.

4.3. TWO-ELECTRON TRANSITIONS

Exploratory experiments were also carried out on two-electron transitions. Correlation satellites were observed and studied, following photoionization in the outer shell of helium [67,96-99], argon [100] and xenon [84] for the first time in this energy range. As an example, we present in Fig. 17 the now well known He case. On the left, the earliest photoelectron spectrum [96] taken in He with synchrotron radiation is shown, and on the right, is another spectrum measured under better intensity conditions, i.e., with a toroidal lnirror (instead of a cylindrical mirror) used in the Rowland circle monochromator to focus the incoming light onto

Binding energy (eV)

M a 22 , Binding energy (eY) 68 6t 28 24

He hv=72.6eV He He'1. 5000

hv=93.OeV ~,2p 18 H.'

-.----------- i

4000 Ahv=O.8OeV

800 4000 CJ II • He' 2~,2p :g600 3000

j .... .a c .-....

• C 2000 :::1400

8 2000

8 200 1000

o . Y-.,. ....{ \.,. 0

26 30 68 72 Kinetic energy (eY)

4 8 48 50 Kinetic energy (eV)

Figure 17. First SR photoelectron spectra of He showing the main line (He+ Is) and the flfst correlation satellites (He+ 2s or 2p). Left: the first measurement, with very low SR intensity, at 72 eV [96]; right: improved spectrum obtained with a toroidal mirror in the Rowland circle monochromator at 93 eV photon energy [67,97].

i!-c .--"ii 12 C -t J: -N"8 II C -t J:

60 80 100 120 140 Photon energy (eV)

Figure 18. Variation of the He+ (n = 2)/He+ (n = 1) branching ratio measured as a function of photon energy [98]. Theoretical results are from ISCI (SZ) [103] and close-coupling (JB) [104] calculations (from Ref. 98).

65

the entrance slit of the monochromator [97,98]. The He+ ion can be left in the n = 1 (He+ Is) or in the n =2 (He+, 2s or 2p) states. Previous experiments with discrete soft x-ray lines [101, 102] left a large gap between 1l0-eV and 70-eV photon energy. The branching ratio between the n = 2 (2s or 2p) and the n = 1 final states of He+ is visible in Fig. 18. The experimental results [98] were between the theoretical predictions based on initial-state configuration interaction [103] (SZ) and the close-coupling calculations of Jacobs and Burke [104]. Double ionization of an atom in the same shell is another straightforward example of correlations effects. Double ionization in the outer shell of the rare gases [105, 106] as well as in several metallic vapors [107-109] was also studied during this period by ion spectrometry and the newly obtained results contributed to removing the uncertainty existing on the probability of these two-electron transitions. Good agreement was obtained for Ne with the MBPT calcula-tions of Chang and Poe [47], who took

into account several correlation effects, in particular the inelastic scattering of the photoelectron.

4.4. RESONANT AND THRESHOLD EFFECTS

Resonant effects were also observed at that time, taking advantage of the continuous nature of synchrotron radiation. Resonant Auger spectra were measured following excitation of 3d and 4d electrons in krypton and xenon, respectively [112]. Autoionization was demonstrated to change the photoelectron spectrum drastically, depending upon the character of the resonantly excited state. Other new data obtained in the case of Ba vapor [113], are presented in Fig. 19. The excitation of a 5p electron to two different autoionizing states results in photoelectron spectra showing greatly enhanced intensities of the electron lines for different final states of the Ba+ ion. This experiment was the starting point of the resonance photoelectron spectroscopy method, which was widely used later in many cases.

Finally, as a last example of these pioneering experiments, we would like to mention the first use of synchrotron radiation to study threshold effects in photoionization, as they were observed in post-collision interaction (PCI) studies [114]. When the photon energy is close to the ionization energy of an inner electron, the nonradiative decay of the inner hole via Auger effect may occur while the slowly receding photoelectron is still in the vicinity of the ion. Then, the Auger spectrum may be influenced by the presence of the slow ejected electron which partially screens the ionic

66

Sa (sp6 6s2) + h v ~ Sa* [ ]

[{(sp5scJ30)&s2Daf2}5d ] hv= 19.94eV

[{(sp5scJ30)6s203f2}6d ] ~ :: hv = 21.48 eV

12 10 8 6 4

Binding energy (eV) Figure 19. Photoelectron spectra of atomic barium taken at the energy of two autoionizing resonances: 19.94 eV for the upper spectrum and 21.48 eV for the lower spectrum [113].

field; therefore, the ejected faster Auger electron sees a smaller attractive potential.

This PCI leads to an energy loss for the slow photoelectron balanced by an equal gain in energy for the fast Auger­electron. Experimentally, this is seen as a shift and broadening of the corres­ponding lines in the electron spectrum. The two main effects of PCI were actually observed in the experimental spectra for the N5-02,3 °2,3 lSo Auger line resulting from the nonradiative decay of a vacancy formed in the 4d subshell of Xe by photons of 68.3 eV, i.e., 0.8 eV above the 4d 2D512

ionization threshold. Several series of experiments [114, 115] led to values of the energy shift of this Auger line which were found to be in good agreement with the semi-classical model developed by Niehaus [116].

To conclude with this period, it was an exciting time when almost every experiment was bringing something new and sometimes unexpected, to the basic knowledge of photoionization.

S. Major Advances in the Production and in the Use of Synchrotron Radiation (1977 --+)

Four series of events occurred over a period of a few years, namely from the end of the 70's until the beginning of the 80's. First of all, major progress was made in monochromatization techniques developed to make the best possible use of synchrotron radiation: the toroidal grazing-incidence monochromator, the TGM, was invented, bringing to experimentalists the most suitable monochromator to study photoetnission from low-denSity targets. This apparatus has a very high throughput because of its optical design, which is actually well-adapted to experiments requiring a high photon flux in a small source volume. Typically, photon flux in the 1012 photons/sec range over a bandwidth of about 0.5 % was made available at the exit slit of this monochromator with the existing facilities. Thus, most of the pioneering experiments that have been described in Section 4 of this chapter could be repeated with much improved experimental conditions. In the meantime, new experiments were thought of to explore new ways such as photoetnission from metallic vapors, photoionization from excited atoms and ions, or studies of double photoionization, and, more generally, of low-cross-section processes in the rare gases.

Second, some of the frrst-generation storage rings, to which only litnited and parasitic access was previously available, became partly or fully dedicated to the exclusive use of synchrotron

67

radiation. Having full control of the accelerator and being able to choose the characteristics of the electron beam best adapted to synchrotron-radiation experiments (and not to the study of e+e­collisions) extended considerably the yet-limited community of potential users in atomic physics (as well as in all other domains). The possibility of not working only during the nights or over holiday weekends made the use of these large instruments more attractive and brought a new generation of old and young scientists into the field.

Third, synchrotron radiation was no longer considered to be a by-product of accelerators primarily built for particle physicists. The funding agencies, especially, considered that it could be more useful to build accelerators providing intense radiation beams covering continuously the spectral range from the UV to the x-ray region. A much larger community of physicists, chemists, and biologists could thus do fundamental and applied research. Several tens of experimental stations can receive synchrotron radiation beams simultaneously compared to only one or two experimental sections in the case of experiments involving particle physics. Doing "light" physics with "heavy" tools appeared to be a cheaper investment that could also be more efficient for general activities of research and development within a nation. Thus, the decision to contruct second-generation storage rings, i.e., accelerators designed and optimized for the exclusive production of synchrotron radiation, was made in many countries all over the world.

Fourth, insertion devices were designed, tested, and shown to be able to further enhance the performance of synchrotron radiation. Wigglers and especially, undulators are able to bring considerable improvements in the photon flux and in the spectral brightness available to experimentalists, opening again new areas of investigation and leading to the design of third­generation storage rings.

In the following, we will briefly describe the major characteristics of these new devices and facilities, and we will illustrate the deep impact they had in atomic physics with some selected examples.

5.1. THE TOROIDAL-GRATING MONOCHROMATOR

Following a previous idea of Madden and Ederer [117], the grazing-incidence monochromator developed by Lepere and Petroff in 1977 is based on a toroidal holographic grating [118]. The scheme of the LHT Jobin-Yvon prototype [119] is shown in Fig. 20. Radiation from the storage ring (ACO for these first experiments) is focused by a toroidal mirror onto the entrance slit of the monochromator. Then, the radiation is diffracted and focused by the holographic toroidal grating onto the exit slit of the monochromator. Finally, the monochromatic radiation beam is refocused

Toroidal

s

Toroidal grating

Photoemission experiment

Figure 20. Experimental scheme of the toroidal-grating monochromator (LHT 30) prototype [119].

by a second toroidal mirror into the experiment. Both entrance and exit arms have a length of 0.30 m. The system has the same simplicity as the Seya-Narnioka type. The entrance and exit slits are fixed, and the photon-energy range is scanned by a simple rotation of the grating. The widths of the slits can be continuously varied from the outside, and the whole apparatus is operated under ultrahigh

68

vacuum conditions. Since the synchrotron radiation is strongly polarized in the horizontal plane, the monochromator is placed vertically, in order to maintain and even to improve the rate of polarization. The enormous advantage of this system is the high photon flux it delivers into the experiment with a good resolution. The resolution is governed by the diffraction pattern of the grating and by the dispersion into the planes of the entrance and exit slits. Since the optical object is the point source formed by the electron beam in the storage ring, every focus point (at the slits or in the experiment) is the image of this point source through the optical system. These images can be made very small with good optics, which explains the tremendous success of this monochromator for experiments in which a small focusing point and a high intensity are required. Electron spectrometry on dilute atomic targets is a typical example of such experiments.

After the successful tests of the prototype, a larger version of this instrument was designed and built [119]. It is shown in Fig. 21. The optical design is basically the same as previously chosen. The length of the entrance arm is 2.9 m, and the length of the exit arm could be 3.4 m (with a total deviation angle of 150°) or 4.3 m (deviation angle of 166°). It was designed to cover

I

~ A.c.o pulse

5x5 mracJ2 a entJ'!-nce ,L:.?s slit

s To.roidal.J ~ O:t. 1.5 mm

mirror: 750m s=4m s'=1.33 m a= 52

entry optics

51 =2960 mm 5 _3530 mm (0 = 15°)

2 - 4323 mm (0 = 7°)

monochromator

experimental setup

Figure 21. First full size TGM instrument. The entrance ann has a length of 3 meters. Two exit anns allow optimization of the monochromator for two different photon energy ranges (from Ref. 119).

an extended photon-energy range from about 30 eV to the CK edge around 280 eV. It has been successfully operated over many years at the ACO storage ring for photoemission experiments.

Some atomic physicists had the privilege of early access to this instrument. The results they obtained were an illumating demonstration of the capabilities of the new monochromator design. An example of them is given in Fig. 22, which shows a photoelectron spectrum of He produced by 69-eV photons, i.e., less than 4 eV above the (Is, He+ n = 2) second- ionization threshold [98, 99]. The satellite line corresponding to the case where the He+ ion is left in the (n = 2) excited state has a high experimental intensity, even at 4-eV kinetic energy. This spectrum must be compared to the spectra shown in Fig. 17 at a slightly larger photon energy using a Rowland circle monochromator and the same electron spectrometer. The counting rate for the n = 1 photoline is now 5000 counts/sec instead of 60 counts/sec previously and 160 counts/sec for the n = 2 photoline as compared to 1-5 counts/sec. The comparison of these figures illustrates perfectly the advantages of the new system. Simpler to operate than the Rowland circle system, it covers a large

-.. o 30

1Il~ 20 CD -~ :I 10 o (J

He

68

Kinetic energy (eV) 2 4

30

20

10

66 64 26 24 Binding energy (eV)

Figure 22. First photoelectron spectrum of He measured with a TOM, at 69 eV-photon energy. Compared to the spectra previously obtained (see Figure 17), the counting rate has been increased by a factor of about 100 [98,99, 120].

69

photon-energy range while keeping a good alignment. The improvement by a factor of about 100 in the photon flux routinely available for experiments made possible a host of new investigations. For instance, using spectra such as the ones shown in Fig. 22, it was possible to solve the pending question of the threshold behavior of the (n = 2) He+ partial photoionization cross section. The question had been raised by earlier investigations using discrete soft x-ray lines but was not solved, because of the lack of lines between 70 eV and 100 eV. The branching ratio between the (n = 2) and the (n = 1) final state was already shown in Fig. 18. Then, it was possible to determine, with good accuracy, the photon-energy dependence of this cross section

[98, 99, 120] as shown in Fig. 23. Among the various theoretical predictions, the new results demonstrated that a fairly good agreement is obtained with the results (C) of a many-body perturbation theory calculation by Chang [121]. The curve marked B on the figure is the result of

,g 0.12 ~

.5 i 0.08

S +

:! 0.04 tJ

JB

70 80 eo 100 110

hv, Photon energy (eV) Figure 23. Absolute partial photoionization cross section of He leaving the residual He+ ion in the (n = 2) final state [98, 99]. SZ = ISCI results [103]; JB = early close-coupling calculations [104]. The best agreement between theory and experiment is given by the MBPT results of Chang (C) [121] and the more recent close coupling calculations by Berrington et al. (B) [122] (from Ref. 120).

a more recent Close-coupling calculation by Berrington et al. [122], which shows the best overall agreement with the experimental data not only for this (2s+2p)He+ cross section but also for the 0(2s) and 0(2p) partial cross sections, as demonstrated later by angular distribution measurements of the n = 2 photoelectron line [123-125] and by the measurement of the fluorescence radiation emitted in the decay of the (He+ n = 2) excited state [126, 127].

The TOM design was soon adopted at several experimental stations, in Orsay [128, 129], in Stoughton (Wisconsin) [130], and finally in a large number of synchrotron-radiation centers.

70

5. 2. DEDICA JED FIRST-AND SECOND- GENERATION STORAGE RINGS

With the new interest brought into the use of synchrotron radiation, several accelerators, that were previously built as collision rings to study (e+ e-) reactions became partly or fully dedicated

Table 2.- First generation storage ring sources (from Ref. 131).

Location Name of the ring Electron energy Comments (GeV)

DENMARK Aarhus ASTRID 0.6 Partly dedicated

FRANCE Orsay ACO 0.54 Fully dedicated from 1978 until

1988 DCI 1.85 Fully dedicated since 1985

GERMANY Bonn ELSA 2.5 Partly dedicated

0.5 Fully dedicated Hamburg DORIS 3.5 Partly dedicated

ITALY Frascati ADONE 1.5 Partly dedicated

USA Gaithersburg SURF 0.28 Dedicated Ithaca CHESS 5.5 Partly dedicated Stanford SPEAR 3.5 Now dedicated Stoughton TANTALUS I 0.24 Dedicated since 1970

RUSSIA Novosibirsk VEPP-2P 0.7 Partly dedicated

VEPP-3 2.2 Partly dedicated VEPP-4 5 Partly dedicated

over the 80's. A list of them is given in Table 2 [131]. The emittances of these rings are generally poor, in the hundred-nanometer-radian range, except for the large colliders (PEP, PETRA, TRISTAN), which can achieve very low emittance when operated in a dedicated mode. Only those which have been or are still actively involved in synchrotron radiation-experiments are listed in the table. Moreover, starting at the end of the 70's, new storage rings were designed and built for the use of synchrotron radiation only. They were optimized to deliver intense beams of synchrotron radiation in the bending magnets. They have emittances in the 40--150 nanometer­radian range. They usually have large capacity, supporting over 20 beam lines and 50 experimental stations with hundreds of users every year. One or two straight sections were let free for further insertion of undulator or wiggler devices. Several hundreds of milliamperes are routinely injected into these machines. Only those that have been heavily involved in

71

Table 3. Second generation synchrotron radiation sources. All machines are dedicated storage rings (from Ref. 131).

Location Ring Electron energy Year of first experiments (GeV)

CHINA Hefei HESYRL 0.8 1990

ENGLAND Daresbury SRS 2 1980

GERMANY Berlin BESSY 0.8 1985

INDIA Indore INDUS-l 0.45 in construction

JAPAN Okasaki UVSOR 0.75 1988 Tokyo SOR 0.38 1982 Tsukuba TERAS 0.6 1987

PHOTON FACTORY 2.5 1983

SWEDEN Lund MAX 0.55 1991

USA Baton Rouge CAMD 2 1993 Stoughton ALADDIN 1 1988 Upton NSLS-I 0.75 1982

NSLS-I1 2.5 1988

synchrotron radiation experiments are indicated in Table 3 [131]. Most of the new data obtained since the beginning of the 80's have been the results of experiments carried out in these synchrotron-radiation centers.

5.3. UNDULATORS

An undulator is a periodic electromagnetic structure of N permanent dipole magnets which is installed along a straight section of a storage ring. The vertical magnetic field forces the relativistic electrons to oscillate transversely in the horizontal plane. Constructive interference can occur between the waves emitted by the electrons at each undulation. In principle. a perfect undulator emits a series of narrow spectral lines.

The emitted radiation is concentrated in a very small opening angle. The wavelength of the nth harmonic is given by [132]:

(9)

72

where y = FlllloC2 (with E being the energy of the electrons), ~ is the magnetic period, 9 is the observation angle referred to the axis of the straight section, n is the harmonic number, and K is a constant characterizing the strength of the interaction between the magnetic field and the electron beam. K = eBA../2moc, where B is the peak magnetic field along the vertical axis. When B and ~ are expressed in units of tesla and centimeters, respectively, then K = 0.934 BA.u.

The characteristics of the radiation emitted by an insertion device depend strongly on the value of K. The undulator regime corresponds to low-K values (1 to 2). Under these conditions, the perturbation in the electron orbit introduced by the magnetic field is weak. The emitted radiation appears at discrete photon wavelengths given by relation (9). The s~tral width of the emission at q = 0 results from interference effects and is of the order of IINVn. In the low-field case (K < I), mainly the fundamental frequency is emitted through a small pinhole placed on the axis of the undulator. For higher values of the magnetic field (K > I), harmonics of the fundamental frequency appear as K increases while the peak energies shift to lower values. In the undulator mode, the increase in the photon flux is supposed to be proportional to N2 for a perfect undulator, as compared to the emission from a bending magnet. In fact, when one takes into account the different collection angles for the radiation emitted from a bending magnet and from an undulator, the actual gain in the photon flux is close to N. The wiggler regime is characterized by high values of K (10 to 100). With increasing values of K. the number of harmonics increases

-=! 50 .!!.

50 100

Photon energy (eV)

Figure 24. Spectral distribution of the photon flux emitted by the SU6 undulator (N = 16) of Super ACO (E = 0.8 GeV, average positron current of 200 mA). The measurements have been made with a CMA electron analyzer using the Ne 2p photoelectron line to calibrate the monochromatized photon flux [13].

and the peaks broaden in such a way that they start to overlap until they merge completely, producing a continuous spec­tral distribution of the emitted radiation. This distribution is similar in shape to the radia­tion emitted from bending magnets, but is shifted to higher (eventually lower) photon energies, depending on the radius of curvature of the electron trajectories and with an N-increase in intensity. A wiggler is thus similar to a series of bending magnets adding incoherently their emitted radiation.

In reality, there are many causes of broadening and the actual spectral distribution

looks quite different. As an example, Fig. 24 presents the energy dependence of the intensity emitted by the SU6 undulator of the Super ACO storage ring (N = 16, ~ = 7 cm, E = 0.8 GeV). The ideal spectrum from an undulator may be observed for the first time at the third-generation storage ring ALS (Advanced Light Source in Berkeley), which is just coming into operation with a very low-emittance electron beam and carefully designed undulators (See chapter by Kincaid in this volume).

The absolute photon flux emitted by the SU6 undulator confIrms that the gain in photon flux is on the order of N. Values of a few 1013 photons/sec in a 0.3-eV bandwidth were measured

hv=33.2eV

Ne 300

40 45 50

GAP(mm)

Figure 25. Variation, as a function of the undulator gap, of the photon flux emitted by the SU6 undulator of Super ACO, at a fixed photon energy, 33.2 eV here [13].

73

[13], in agreement with the predicted estimations. The gain at higher resolution, i.e., the spectral brightness, is higher when it is compared to the radiation emitted from a bending magnet.

One key parameter is the capability of varying the energy of the radiation emitted by the undulator. Changing the energy of the electrons in a storage ring can cause severe disturbances in the electron orbit; thus, it is practically impossible to play with this parameter in order to tune the photon energy. With permanent-magnet undulators, the only way to modify the K value is to change the gap of the undulator, i.e., to vary the distance between the upper and the lower benches of magnets. For low-K values, this may be done without noticeable perturbations in the other parts of the electron orbit. As an example, we show in Fig. 25 how the measured intensity emitted at a given photon energy, here 33.2 eV, varies dramatically with the values of the gap [13]. For each photon energy, the optimum gap has a well defined value and experimental results demonstrate how accurate the tuning of the undulator gap must be. The only way to

take full advantage of these devices is thus to design undulators whose gaps are continuously variable during-storage ring operations without perturbation of the electron orbit.

The systematic use of undulator radiation will be possible only with third-generation storage rings. At most of the presently existing facilities, only one or two straight sections are available to insert wigglers or undulators. Thus, the access to these devices is quite limited since they have to be shared with many other scientists, in particular with solid-state and surface physicists. To illustrate the qualitative step introduced by the use of undulator radiation in atomic physics, we would like to show one example, namely the photoelectron spectrum of the Ar 3s, 3p correlation satellites. In the "early days" (1978 ... ), the satellite structure was observed with electron spectrometry for the first time using SR emitted by the ACO storage ring in Orsay [100], at several photon energies between 50 and 100 eV. One of these spectra, recorded with a l-eV bandwidth, is shown in the upper part of Fig. 26. The gross features of the satellite distribution, mainly the final ionic state configuration interaction between the 3s3p6 2S and 3s23p44s (or 3d or 4d) 2S final states, were correctly determined, but an individual analysis of each satellite line was not possible. The lack of SR intensity prevented the monochromator slits from being closed in order to work with a better resolution. This was made possible twelve years later with the undulator photon source of the ALADDIN storage ring [133], as can be seen in the lower part of Fig. 26. The spectral detail given by 100 meV-resolution combined with good signal strength and nearly 100% polarization of the photons afforded a highly detailed understanding of electron correlation processes, while confirming the results of the earlier experiments.

74

1200

800

1500

500

Ar

3P ______ ..... "L.' -

1 0 ...... _ .... ....111'---

18 ---"'--

4cf2s 3cPs 4a2s 4p2P

\ ~ J I

3p

3s

P SATELLITES ~ ~ '\ 3d

4d

5d

44 40 36 30 28 16 14 ELECTRON BINDING ENERGY (eV)

Figure 26. IlIustration of the progress made in photoionization studies with the use of an undulator. Upper part: 3p-3s photoelectron spectrum of Argon measured in the "early days" (1978, Ref.1(0); lower part: the same photoelectron spectrum recorded twelve years later with the ALADDIN undulator [133].

6. Phase 3: New Storage Rings and New Experiments (1981 ~)

During the previous decade, experiments on the rare gases have revealed the main features of the photoeffect and emphasized the importance of correlation and relativistic effects. The agreement between experiment and many-body theories was generally better than 10-15 % for single­electron properties in the soft x-ray range. The discrepancies observed in several cases were attributed to the neglect of two-electron processes in the theoretical approximatons. Cooper minima and shape resonances were among the most prominent issues studied both experimentally and theoretically. Development of theoretical methods revealed the impact of electron correlations upon partial single photoionization cross sections and angular distributions of the emitted photoelectrons.

With the dramatic improvements in experimental conditions offered by SR, an impressive extension of the experimental activity in atomic physics occurred within a few-year period. The

75

number of laboratories involved in such studies expanded from less than 10 during the previous decade to more than 30 at the beginning of the 90's. Over a period of 10 years, several hundreds of new results were published, most of them related to the determination of partial and differential cross sections. Several opportunities were open to the new (and early) investigators: to improve the old experiments and to carry out new studies in the rare gases; to develop new experiments in the widely unexplored fields of the open-shell atoms, especially in atomic vapors; to pave the way to more challenging goals such as photoionization of atomic species, not in the ground state, but brought into excited or ionic states. All these themes were studied, and it is difficult and somewhat arbitrary to select a few examples from the mass of new experimental results of high quality level.

To briefly summarize, the study of two-electron processes has become a major topic. Two­electron excitations, photoionization accompanied by the simultaneous excitation of another electron leaving the residual positive ion in an excited state, double and multiple photoionization were studied at length, especially in the rare gases. Extension of the experiments into the x-ray region was achieved. Autoionization resonances were widely studied. Several experiments on atomic excited states and on singly-charged ions were successful. Fluorescence studies, in the VUV as well as in the x-ray range, were initiated and complemented the results obtained with ion and electron spectrometries to study relaxation effects. Alignment effects were observed, either in the polarization of the fluorescence light or in the nonisotropic angular distribution of Auger electrons.

To meet these new challenges and to take the best advantage from the progress made in the production and monochromatization of SR, it was also necessary to design and build new experimental devices. Before illustrating some of the new results, we will describe briefly a few experimental techniques which have been successfully used over the past few years.

6.1. NEW EXPERIMENTAL DEVICES

6.1.1. Coincidence Experiments in the Rare Gases. In order to study the threshold behavior of single-and double-photoionization processes, several apparatuses involving two charged-particle detectors of different types have been designed to allow coincidence measurements to be made between electrons and ions. One of these [134] is shown in Fig. 27. It consists of an electrostatic electron-energy analyzer mounted opposite a drift tube that can serve either as a time-of-flight mass spectrometer for ions or as a low-energy electron selector. Ionization occurs at the intersection of a focused monochromatic photon beam with an effusive beam of argon. Zero­

Gas Inlet Fllgth tuba for Ions

InteracHon zona, ,t or low energy electrons

~/rj;rrr~ * f': Electron Electron anergy .' ~ ~: ,! : ~ or

analyser (, i dlL L\j;ai L1 ,I- Ion .'gna'

~ Electron .'gna'

Figure 27. Scheme of one of the first coincidence apparatuses. A cylindrical 1270 electron-energy analyzer and a time-of-flight tube for ions and low-energy electrons are mounted on opposite sides of the interaction volume [134].

energy electrons can be selected either by use of the electrostatic energy analyzer or by their collimation after acceleration in a weak uniform field together with their measured time of flight. Most of the experiments involved coincidence counting, which was done by time-to-amplitude conversion followed by multichannel analysis. In the first experiment [134], the

76

Side view gas

"---' ....... l====iI Inlet

Top view distribution of low-energy electrons in coincidence with doubly-charged Ar++ ions brought information about the population of the various excited Ar++ ionic states and about the near­threshold behavior of the double­photoionization cross section. In an improved version of this apparatus, a similar instrument [135] was mounted on a turntable, which

photon beam Scm I rotates in a plane perpendicular to

Figure 28. Schematic illustration of the spectrometer used for double-ZEKE coincidence spectroscopy. In the side view, the photon beam intersects the drawing plane perpendicularly at the point indicated by the crossed circle [139].

the photon direction. The 12r cylindrical-deflector analyzer consisted of a double-electrostatic aperture-lens system that collected and focused the particles out the entrance slit. Both of these analyzers could be operated in the threshold

mode using the penetrating-field technique for the selective collection of zero-energy electrons. Partial and differential cross sections for double photoionization were measured for the rare gases [135-138] using so-called TPEPECO (threshold photoelectron-photoelectron coincidence) and TPEPICO (threshold photoelectron-photoion coincidence) modes. A different technique was also used, involving so-called double ZEKE (zero kinetic energy)

1200 100

! G

700

I !

coincidence spectroscopy to study state selectivity in the double photoionization of argon [139]. A schematic representation of the spectrometer is shown in Fig. 28.

~ I I I The photon beam enters and exits

07.3 07.1 07.7 the source region through two IS. I

1500

'. 43 44 45 46 47

Photon energy (eV)

Figure 29. Double-ZEKE coincidence spectrum of Ar. The lowest Ar++ thresholds are indicated. For the spectra shown in the lower part of the figure, the monochromator slits were set at 200 /lm. The spectra shown in the upper part are coincidence results measured at the Ar++ ePJ and 1 D2)

thresholds with 100 !lID slits (left and middle insets), and at the Ar++ ISo threshold with 200 !lID slits (left inset) [139].

opposite apertures. There are four different spatial regions with specific electrical fields: source region, acceleration region, drift region, and post-acceleration region separated from each other by high-transmission gold meshes connected to the respecti ve V 1 to V 3 potentials.

The two zero-energy electrons emitted at the threshold of double­photoionization processes are detected in coincidence by the two channeltrons whose pulses are analyzed in appropriate time windows. As an example of the results, Fig. 29 shows the double-

synchrotron radiation

Figure 30. Schematic diagram of the electton-electron coincidence experiment involving the use of two CMAs. The coincidence detection and data logging electtonics are also shown [141].

77

ZEKE coincidence spectrum for double photoionization in the outer shell of argon. The measured intensity ratios at the 3pJ (J = O. 1.2) levels indicate a disturbance of the direct double­photoionization process by the presence of indirect autoioniza­tion processes. In addition. clear experimental evidence was found for the existence of the ISo ionic state. which could not be detected in previous experi­ments. The intensity ratio between 1 D2 and ISo states was even measured [139].

Another type of electron­electron coincidence spectro­scopy was developed to unravel first- and second- step Auger

lines in complex Auger spectra. Either two time-of-flight electron analyzers [140] or two cylindrical mirror analyzers [141] were used to detect in coincidence the two electrons emitted upon the two-step decay of the photoexcited Xe 4d96p resonances. Here we like to show in Fig.

-:; cO -~ 'iii c: S c:

o

I II I I I II I ~I--~I~ilrrl----------~

1 p 1 3p 1r--i-.:;,-+------~-

0.5 10.0 15.0 20.0

Kinetic energy (eV) 20.5

Figure 31. Low energy part of the resonant Auger spectrum of Xe at the 4d512 ~ 6p resonance (65.1 eV). The marks indicate corresponding first- and second- step Auger lines [141].

30 the schematic diagram of the electron-electron coincidence experiment involving the use of the two CMAs. By using the pulsed time structure of SR emitted by the DORIS storage ring. low-energy electron spectra of xenon. such as the one shown in Fig. 31. were taken at the 4dS/2 --+ 6p resonance (studied long time ago in photoabsorp­tion; see Fig. 7). The three strongest lines arise from the resonant Auger decay of the 4d95s25p66p core-excited state into 4d105s5p66p (at 24.38-eV kinetic energy). 5s5p4 5d 6p (at 18.98 eV-19.20 eV) and 5s05p66p (at 12.96 eV) singly charged ionic states. Then. these states Auger decay into the 4d10

5s25p4 Xe++ final states. Coincidences between the electrons emitted upon the first-and second­step decays were registered for the strongest second-step lines. In this way. it was possible to measure the energy and the fractional intensities of two-step Auger transitions.

78

6.1.2. Photoionization of Atoms in Excited States. Photoionization studies of atoms in excited states were initiated in 1981 [142]. They involve the combined use of a dye laser and SR. Fig. 32 shows the experimental setup used in the beginning [142, 143, 145, 146] for angularly integrated electron spectroscopy and now also for angular-distribution studies [144]. An effusive beam of

M M " Ar I

55 4D

:--~-~~ Monochromatised light

35 I I I I

M I

I~A~r 1I1I-1=f"[i""L~~~~ Ikl-L Iv.s.1 Electron

spectrum

Na beam

Figure 32. Experimental setup for photoelectron spectrometry studies of excited atoms. SR and laser beams are focused into the source volume of a CMA. An oven is mounted on the axis of the CMA and emits an atomic vapor-beam (sodium, here). A second laser can be used to produce highly-excited optical atomic states [146].

atoms is sequentially excited by two beams, a cw laser beam (DL I) to pump part of the atoms in the first excited state and a monochromatized SR photon beam to probe the excited atom by exciting or ionizing an inner electron. This excitation takes place in the source volume of a CMA. The laser beam is perpendicular to the CMA axis in order to minimize the Doppler effect, whereas the monochromatized SR is colinear to the CMA axis. The dye laser is pumped with an argon or krypton ion laser (Ar I in the figure). The polarization of the laser beam is linear. Because of the poor tunability of the cw dye lasers and of the relatively low values of SR flux available, only alkali-and alkaline-earth atoms have been investigated so far. In the case of sodium atoms, higher optically excited states were populated in some of the experiments [145, 146], using a second laser system (DL II and Ar II in the figure). The synchrotron radiation producing the second step of excitation is monochromatized with a TGM. Another experimental setup was specially designed for angular distribution measurements on laser-aligned atoms. It is described in another chapter (see Sonntag and Pabler in this volume), together with the results that have been obtained on excited-and aligned-lithium atoms.

6.1.3. Photoionization of Singly Charged Ions. The first measurements of partial photoionization cross sections for atomic ions involved the use of SR in merged photon beam-ion beam experiments [147]. The experimental setup is shown in Fig. 33. Singly charged ions were produced in a surface ionization source, accelerated, and deviated to be colinear with a beam of monochromatized photons from the Daresbury storage ring. The ion beam and the

PO

-+--~--E-------!-------3---Figure 33. Scheme of the first experimental set up used to study photoionization of singly charged atomic ions with ion spectrometry. The singly charged ion beams are produced in a surface-ionization ion source and are deflected to merge with a monochromatized photon beam. After the interaction zone, the unaffected singly charged and photoionized doubly charged ions are magnetically separated and counted in two different ion detectors [147].

79

monochromatic photon beam were merged over a length of about 10 cm. 1ben, the unaffected singly charged ions and the doubly charged ions formed by photoionization were deflected into two different detectors. The measured signals served to determine the absolute cross sections of some singly charged ions [147,148].

6. 2. RARE GASES REVISITED

In face of the wealth of new experimental results, it is extremely difficult to select only a few examples to be presented in the limited number of pages available to the author. The final selection is strongly influenced by the personal feelings of the writer. Even though two-electron processes were the major topics of this past decade, it was also possible to elucidate some of the yet-unSOlved "single" photoionization properties, i.e., interaction processes in which the electronic configurations of the initial and final atomic or ionic state differ only by the orbital occupied by one electron.

6.2.1. Single Photoionization Events. We would like to comment in some detail on the solving of two problems we have already described in Section 4 of this chapter: the behavior of the 4d photoionization cross section of Xe over the region of the delayed maximum and the behavior of the Xe 5s ~ parameter through the energy region of the Cooper minimum. The extension of photoionization studies above the keY photon energy-range has also been of great interest, mainly for argon, krypton, and xenon.

The Xe single-electron 4d photoionization cross section was remeasured several times [31, 149-151], up to high photon energies [150, 151]. Using a novel normalization procedure that makes the result independent of the partition of the measured total photo absorption cross section, the most accurate study of this case [31] has obtained the results (black squares labeled

80

30 Xe

-..a :E 20 -

10

70 90 110 130 150

Photon energy (eV)

Figure 34. Single-electron 4d-photoionization cross section in Xe: 1, black circles, experiment from West et al. [70); 2a, open diamonds, experiment from Becker et al. [149); 2b, black squares, best experimental results from Kiimmerling et al. [31). Theoretical results are from RPAE [57) taking into account SEP only (3, thin line) and SEP plus relaxation (4, dashed line) and from MBPT with relaxation (5, dashed-dotted line, ref. [56]). Total photoabsorption cross section (labeled 6, thick line) is from Haensel et al. [152) (from Ref. 57).

2b) shown in Fig. 34 . The new results are in good agreement with earlier data on the-low energy side of the 4d shape resonance, with one exception [149], and are, on the high­energy side, significantly higher than the results of the first new measurements (open diamonds labeled 2a in the figure) of this cross section [149] (which were, in fact, corrected later [151]). The agreement with the latest theoretical calculations [56, 57] is now quite impressive, if one remembers the first calculation of this cross section 25 years ago (see Fig. 8). These new theoretical results fully de­monstrate the importance of correlation and relaxation effects. They were obtained first in the framework of the many­body perturbation theory [56] by including relaxation and overlap factors (curve labeled 5 in the figure). But the best

overall agreement is observed with the most recent RPAE calculations [57]: the new interaction taken into account is the proper self-energy part (SEP) of the photoelectron's Green function. The SEP describes the polarization potential acting upon the photoelectron. Taking SEP into account means a change in the photoelectron wave function and a loss of flux of photoelectrons in the single 4d ionization channel because of the inelastic scattering by the outer-shell electrons. On its way out of the atom, the 4d-photoionized photoelectron may excite or ionize an outer electron, changing the final ionic state. The excitation produces ions excited in the outer shell (the previously so-called "conjugate shake-up satellite", that it would be better to name "collision correlation satellites" ). The ionization produces doubly charged ions. The calculations taking into account SEP and relaxation effects (dashed curve labeled 4 in the figure) are evidently in beautiful agreement with the experimental values of the single 4d photoionization cross section. When relaxation is not taken into account, agreement is not as good (thin solid curve labeled 3 in the figure). It should be noted, however, that coupling with the other two-electron channels is not yet taken into account in these calculations, while an upper limit of about 20% has been established for the relative intensities of these two-electron processes involving the 4d shell [31]. This result was recently confirmed by direct measurements [151].

The second pending question, i.e., the angular distribution of 4s and 5s electrons in Kr and Xe, respectively, seems now to be definitively solved. In Fig. 35, we show two ,series of new experimental data [153, 154] obtained in 1983, in the region of the Cooper minimum. Both sets

~ 2.0TIrffHO+----+-H+r------;~-----+~ ... CD -~ 1.8 ca ... ca C. 1.6

>-... -CD E E

1.4ITh ... hOld

>- 1.2 U)

« 1.0~~----------+-----+-----+---~

30 40 so 60 70

Photon energy (eV) Figure 35. Comparison of the latest experimental data with the theoretical predictions for the Xe 5s ~ parameter in the region of the Cooper minimum. Experimental results are from Refs. 153 and 154. RTDLDA calculations are from Ref. 156 (from Ref. 17).

81

of ~ values agree well with each other and do not confirm the pronounced minimum predicted by RPAE or RRPA calculations (See Fig.16). Suggestions were made at that time that the neglect of two-electron processes in the theoretical calcula­tions was the cause of the persisting discrepancy [155]. A first answer was given one year later with the relativistic, time-dependent, local density approximation (RTDLDA) calculations of all photoionization parameters for the outer shells of the rare gases [156]. These results were the closest to the experimental data in the region of the Cooper minimum. They include implicitly, in an average way, the previously neglected interaction effects between the photoionization main line and the satellites, which turned out to be effectively important because the

single 5s photoionization transition amplitude is small. A final confirmation was given very recently by the results of a 23-channel multiconfiguration Dirac-Fock calculation including explicit coupling to doubly excited states having a 5p45d configuration [157]. The latest calculated curve is in excellent agreement with the experimental data.

As for the first photo absorption measurements in the 10-100 eV photon-energy region, the extension of the studies to the x-ray domain started with the rare gases. L-shell photoabsorption of Xe was studied by Breinig et al. [158] and by Koizumi et ai. [159], K-shell photoabsorption of Ne and Ar were measured by Esteva et ai. [160] and by Deslattes et al. [161], respectively. Partial cross sections for inner shells in xenon were measured up to 1 keY, using electron spectrometries [150,151].

6.2.2. Two-Electron Processes. Photoabsorption was successful in reopening the field of two­electron excitations, 30 years after the initial experimental [9] and theoretical work [162]. The most spectacular case is the study of these processes in He [163, 164]. Considerable attention has recently been paid to autoionizing states of He also from the theoretical [165-169] point of view. Upon photoexcitation of both electrons to states below the (n = 2, He+) ionization threshold, the number of observable Rydberg series is reduced to three [170]. Only two of them, the (sp, 2n +)lpo and the (sp, 2n _)lpo series have been observed until recently [165, 169], while the third (2p, nd)lpo series was missing [171]. The latest photo absorption experiments [164], with a resolving power of about 16000 combined with high photon flux, allow resolution of the (sp, 2n +)lpo and (sp, 2n _)lpo series up to the n = 20 and the n = 11 state, respectively. In addition, the observation of the four lowest states of the (2p, nd) 1 Po series was reported for the first time. The lowest six of the (sp, 2n -) resonances are shown in detail in Fig. 36. Except for n = 3, all of the (sp, 2n -) signals exhibit an additional peak in the leading-edge region, marked by solid vertical

82

bars. For n = (4 -), this additional peak is well separated, by about 16 meV, from the main resonance, while it gradually approaches the (sp, 2n -) state with increasing n. For n = 7, their separation is only 4 meV. The spectra in Fig. 36 were least-squares fitted by assuming Fano-type profiles for the (sp, 2n -) lines, convoluted with the monochromator function. A list of the results for the resonance energies of the (sp, 2n -) and (2p, nd) states is given in [164]. The best agreement with theory was found for the old calculation with the truncated diagonalization

" CD ">­c o ~ "~ C o "0 15 .c Q.

62.73

, I I I I 4-

,'--' , . , '-----.finev ~ ~ '--------

(I I I I I I

62.75 62.n ' 64.11 64.13 64.15' 64.63

r l

I I.

' ............ ..

64.65 64.67

64.89 64.91 64.93 65.05 65.06 65.07 65.13 65.14 65.15

Photon energy (eV) Figure 36. (sp, 2n -)Ipo and (2p, nd)IPoresonances of doubly excited He [164].

method, with 83 configurations included [172], and recent calculations with the complex rotation method [173]. Finally, an upper limit of 0.05 meV was estimated for the width of the (2p, 3d) state.

The study of two-electron excitation processes was also extended into the high x-ray energy region. Here, we would like to illustrate these quite new investigations, with one of the first experiments to be performed with the help of an insertion device (a multipole wiggler), in showing the study of photo absorption processes in the K and M shells of Ar [161]. An expanded scale of the multi-vacancy excitation region is shown in Fig. 31. All energy scales are adjusted so that the onset of the one-electron continuum (K-shell ionization threshold) is at 3206.0 eV. A local scale is shown relative to this photon energy. Previous photoabsorption studies, using the bremsstrahlung emitted by an x-ray tube, have already measured the single-resonant and continuum photoabsorption spectra [174-178], and some of them were able to see weak features

-.c ::::& 0.105 14- ~1s3p4sns -c 0 ;:; u CD 0.100 rn rn rn 0 ~

u c 0.095 0 ;:; ftS

\ "'\

':

\, ::~CD E ... A·· V~ F .... • -.........;-.. G ,~. ~

~\ ! 1 4- ~ 3p 4p np •••. ~. 14 15 1~3pnsES •..

14 15 ~1s3pnsES

14151~1S 3p 4p np 8:, 14 15 1l§1s3pnpEP

~1s3s4snp

1 4 I~ 1s 3s ns £P

H

::J C CD 0.090 := c:(

f"4l5=I~ 3p 4s ns 14,4 10 2 30 40 o 1~3s ns mp

50 60

3210 3220 3230 3240 3250 3260

Energy (eV)

Figure 37. Argon photoabsorption around the K edge. Structures due to two-electron excitation/ionization are labeled A to H [161].

83

due to simultaneous excitation of Is and 3p electrons [176-178] and Is and 3s electrons [177, 178] in 1963. Indeed, the existence, energies, and identification of these two new double­excitation features are confirmed here with quite improved resolution and contrast. In Fig. 37, they are labeled B and E. Even weaker two-excitation lines are observed here for the first time, as other members of the Is-13s-1 and Is-13p-1 excitation and/or ionization series (structures labeled A, C, D, F-H in the figure). Then, in the same experiment, the radiative decay of these one inner-and one outer-shell vacancies was measured as a function of photon energy in analyzing the relative intensity of the I</3v and 1</3" satellite lines (two-hole state decay) versus the diagram 1</31,3 emission line (one-hole state decay) observed in fluorescence spectrometry. Later on, weaker photoabsorption processes involving simultaneous excitation of two electrons belonging to two inner shells were detected in Ar (ls2p and Is2s electrons, [179]), in Xe (2s3s electrons [179]), and in Kr (ls2s,ls2p, 1s3p, ls3d, ls4p and 2s4p electrons, [180, 181]).

Correlation satellites were further studied in the rare gases over a much broader energy range than during the previous period [182-196]. New satellites were observed, particularly in Ne and Ar, whose intensity was found to be important only near threshold [187-190]. In He, the threshold intensity of the (He+, n) satellites was measured [191, 194] up to n = 10. A cross section as low as 0.4 Kb could be measured, for the n = 10 He+ state. Autoionization effects in the excited (ns,np) resonance region were continuously measured [188, 191, 194] up to the He++ threshold at 79.0 eV. In particular, the Beutler-Fano parameters for the weak (3s, 3p) resonance were determined with good accuracy in two different experiments [197, 198] and were found to disagree with some theoretical predictions. The relative intensity of shake-up and shake-off processes following Is photoionization in Ar was measured as a function of photon energy from 3220 eV to 5000 eV from the observation of diagram and satellite Auger lines [199]. The results present different energy dependences, the intensity of the shake up being finite at threshold while the intensity of the shake off goes to zero. The theoretical calculations reproduce well these different energy behaviors of the experimental data.

84

4 -,g ..lII: -c:: 0 t;

2 CD fI)

:l 0 ... (.)

81 83

Photon energy (eV)

Figure 38. Double-photoionization cross section in He just above the threshold. The solid line represents the least-square fit to the Wannier expression with the coefficient a = 1.05. The Wannier law seems to be valid up to only 2 eV above threshold [200].

Direct double-ionization phenomena were also studied with more scrutiny. In particular, the threshold behavior of the double photoionization cross section in He was measured for the first time [200]. The results, shown in Fig. 38, check the validity of the threshold law established by Wannier accord­ing to: (T(He++) = (To AE1.056. The measured exponent was found to be equal to 1.05 (0.02), but the experiment demonstrated that the range of validity was smaller than expected, extending only over a 2 eV interval above the double-ionization threshold. Other experiments measuring electrons in coincidence gave the same value for the expon­

ent[138, 201, 202]. Triple ionization in Ne (and in 0) was also measured to be in good agreement with the Wannier predictions [203]. The relative probability of the double photoionization process in He has been found [120] to be in good agreement with MBPT calculations up to 280 eV photon energy [204]. It has also been recently measured up to 12 keY photon energy [205, 206], questioning some of the previously predicted branching ratios as well as the analogy established by Samson [48] between the behavior of the double-photoionization process and the electron-impact ionization of singly-charged ions.

PCI was also reinvestigated with a greater accuracy and a new effect was observed: the vanishing of the post collision interaction [207-209]. For many years, the theoretical treatments of PCI in inner-shell photoionization were based on a high velocity of the Auger electron; this approximation is now called sudden PCI. However, it has been shown recently [210] that it was necessary to take into account the finite velocity of the Auger electrons for a better description of PCI. In this so-called retarded PCI model, the energy distributions have different shapes and energy shifts. Furthermore, it requires that PCI must vanish when the photoelectron becomes faster than the Auger electron, since, in this situation, the Auger electron can never pass the photoelectron. This vanishing PCI is also called the no-passing effect and must be taken into account for accurate determination of binding energies. It has been observed in two new measurements of Auger spectra following photoionization in the 4d [207, 209] and in the 2p [208] subshells of Xe, respectively. In Fig. 39, the measured energy for the N5-02, 3 02, 3 ISo Auger line in Xe is shown as a function of photon energy. Earlier determinations are in good agreement with the new set of data. For photon energies above 100 eV, i.e., when the photoelectron energy is higher than the Auger energy, the measured values of the Auger electron energies are observed to scatter around an asymptotic value that is equal to the Auger energy without influence of PCI. The predictions of the retarded PCI model are in agreement with these observations, while the sudden PCI does not correctly describe the situation. Confirmation of the vanishing PCI effect came also from the investigation of the L2-L3N4 Coster-Kronig line of

~ 30.2

Xe Ns - ~3 023150

80 100 120

Photon energy (eV)

Figure 39. Variation of the absolute kinetic energy of the NS-02• 3

O2• 3 Auger line of xenon ionized by photon impact. Experimental data are in agreement with the theoretical prediction of the retarded PCI model (solid line) and not with the sudden model (dashed line) [209].

85

xenon (at 226-eV kinetic energy) in the photon-energy region extending from 5150 to 6007 eV, i.e., from 46 to 900 eV in excess of the XeL2 binding energy [208,210].

Another manifestation of PCI was also experimentally observ­ed, namely the recapture of the photoelectron through post­collision interaction. When the Auger electron overtakes the photoelectron, a sudden change in screening occurs, causing energy transfer from the photo­electron to the Auger electron. The photoelectron can thus be recaptured by the atom from which it was emitted, an effect that changes the measured rates of singly and doubly charged

ions. The experimental confirmation of this effect was obtained in measuring the partial Ar+ and Ar++ ion yields in the region of the Ar2p photoionization threshold [211]. The Ar+ yield was measured to increase when the photon energy was decreased, while still being above the L2 ionization threshold. The theoretical analysis made it possible to extract the recapture probability from the experimental data.

Before leaving the rare gases, we should also mention a number of experimental results investigating the effect of resonant excitations of inner-shell electrons on the autoionization and Auger decay spectra of excited atoms [213-221]. Evidence was found that shake-up and shake­off processes occur during the decay. X-ray Raman scattering has been studied in great detail [222]. Photoion spectra mostly observed in coincidence with threshold electrons have been extensively measured [223-232] to determine the yield of multiply charged ions following inner­shell excitation or ionization in the rare gases. Also, two-electron excited states have been studied through the observation of the light emitted in the radiative decay of the innershell vacancy [233, 234].

6.3. ATOMIC VAPORS AND OPEN-SHELL ATOMS

A significant part of the periodic table has been the subject of experimental investigations over the past few years. For the alkali metals (Li, Na), partial photoionization cross sections and two-electron transitions have been observed and studied in the continuum photoionization as well as in the region of autoionizing resonances, from 15-eV up to 150-eV photon energy [235-242]. In Fig. 40, we show the partial cross sections for photoionization of atomic lithium in the Is shell, the residual Li+ ion being left in one of the Is2plp and 3p "conjugate shake up states," as measured in Refs. 236 and 237. The fact that some oscillator strength was given to the production of these states, especially the 3p state, was surprising at that time. Further

86

\ \ LI+ 1a2p 1P ,

" ' ...... ---___ RHF(l) -.... --

RHFM

O~---..~------L-----~100=-----J

hv, Photon energy (eY)

..........

---- LI+1a2p 3P --I III II -------~~ it! f f

RHFM

O~--~.~---'~~----1~OO~---'1~10r-~

hv, Photon energy (eY)

Figure 40. Photoionization cross sections of atomic lithium in Is shell leaving the residual positive ion in the excited Is2p Ip state (left curve) and in the Is2p 3p state (right curve) [236, 237].

experiments [238, 239] confirmed these first observations. Partial subshell cross sections, multiple excitation and ionization processes, autoionization, and Auger decay have been extensively measured in the alkaline earths (Be [243], Mg [244], Ca [245, 246], Ba [247-249]). The case of the 4d photoionization cross section in Ba is of special interest for testing a great number of theoretical calculations involving different approximations. The variation of this cross section is shown in Fig 41. The results of several recent photoemission experiments [248-249] were considerably smaller than predicted by most of the existing theories. The earliest nonrelativistic RPAE calculations [250] included only intrashell correlations within the 4d shell; the calculated cross section peaked at about 80 Mb. The relativistic version of this approximation (RRPA (4d» [251] lowered the peak to about 60 Mb. Further inclusions of inters hell

80 correlations in the RRPA :" (4d+5s+5p+6s) calculations I \( RPAE [251] lower the peak even more,

:c 60 : \ Ba 4d to about 46 Mb. Several RPAE !. .i':~F ~l'\ \/"RRPA(4d) and MBPT calculations (not c 4<IY> 4<IY>. I VI shown in the figure), including .2 111 l:'-"\~\/ RRPA (4d + 55 + 5p + 6s) the use of relaxed orbitals but in U 40 i! \" 1\ a nonrelativistic approximation, :1:cn 14<M 4<M!l "~~"'.. RRPA-R

y '" further decrease the peak value, ~ ~ ', .. , RPAE-5EP to about 30 Mb, while the ... 20 • experimental cross section peaks (,) RPAE-SEP-R

100 120 140 Photon energy (eV)

Figure 41. Single photoionization cross section of the 4d subshell of atomic Ba. Experiments are from Refs. 248 and 249. The best agreement is obtained with the RRPA-R results [251] and with relaxed RPAE calculations including self-energy correction (RPAE-SEP-R in the figure) [57].

in the vicinity of 20 Mb. Only with the introduction of relaxed orbitals in the relativistic RRPA­R [251] and MBPT [252] calculations, the theoretical cross section is approaching the experimental values. However, it should be noted that double­electron excitations have not yet been incorporated into the calculations. In this context, it is

87

worthwhile to note the strong relative intensity of the 4d satellite channels: their sum is about 70% of the 4d main lines [249]. The double-ionization channels, involving simultaneous ionization in the 4d subshell and in one of the more outer subshells, are also likely to contribute significantly to the total photo absorption cross section. Introducing the coupling to these channels into the theoretical calculations will be necessary before any final conclusion can be drawn. Finally, as in the case of the 4d subshell in Xe, the introduction of the self-energy correction, taking into account both inelastic scattering and relaxation processes [57], gives theoretical results that are the closest to the experimental data.

Several open-shell atoms such as the III-A (AI [253], Ga [254], In [255]) and IV-A (Sn [256, 257], Pb [258,259]) metals were investigated. In these atomic systems, transitions of the type (n­l)d ~ P were likely to be strong due to the presence of unoccupied levels in the np subshell. TIle effect of autoionization resonances was expected and was actually measured to dominate the photoionization spectra in many cases. Ag [260, 261], Cd [262] and Hg [263] have also been investigated and relative cross sections measured for single and double [261] photoionization.

Major progress has been made in the study of the 3d and 4f metal atoms [264]. Some of these metals are difficult to vaporize, which explains why photoemission studies on this species in the atomic phase were carried out only during the past few years. In the 3d transition metals, the partial cross sections are dominated by strong asymmetric resonances that are not described by one­electron models. The importance of many-electron phenomena is due to the existence of unoccupied 3d levels that spatially overlap with the occupied 3p-core levels. The strong interference between the discrete 3p63dn ~ 3p53dn+ 1 excitations and the direct ionization channels gives rise to the asymmetry of the resonance profiles. Most of the transitions metals have been

(Xe)6. Z

La ~(Xe)(5d,6.)3 ~4f(5d'6.)3 ~)4f36aZ

~(Xe)4f46.Z

Ce

Pr

Nd

Sm

Eu

Gd

Tb

(Xe)4f 7 6aZ

(Xe)4f75d 6.zfi

(Xe)4f8~

100 140 180

Photon energy (eV) Figure 42. Photoabsorption spectra of Ba and eight rare-earth elements in the region of the giant resonances [272].

studied so far: Ti [264], Cr [265], Mn [266-269], Fe [270], Co [270], Ni [270], and Cu [271].

Of great interest are recent photoabsorption and photoemission results obtained for some of the rare-earth metals [272-275]. Atomic rare earths exhibit giant resonances above and below the 4d threshold, which may be described mainly by the excitation of inner-shell 4d electrons to unoccupied (4f, et) levels. With increasing values of Z along the series, the maximum gets narrower and the main peak shifts towards threshold. TIle main features can be explained by the double-valley potential caused by the attractive Coulombic field and the repulsive centrifugal term. In neutral atoms, the collapse of the 4f wave functions is supposed to take place between Z = 56 and Z = 57 [276]. However, the 4d94f IPI state is shifted to the outer well because of a large exchange interaction. Therefore, the oscillator strength of the discrete transition to this state is very small for the first elements of the series. Photo absorption spectra of eight of these elements, shown in Fig. 42, clearly illustrate

88

this behavior [275]. One observes distinct differences between light and heavier elements. In light elements, the so-called giant resonances occur above the ionization limits of the 4d electrons (indicated by arrows in the figure), whereas, for increasing nuclear charge, the resonances are shifted into the direction of these limits in such a way that, for heavier elements, the main part occurs below the ionization limits. The inner well gets deeper for heavier elements in such a way

-::;; .!t tl

40

0(40) 0

30 o (5P) -..c :E - 20 tl

10

0 100 120 140 160

Photon energy (eV)

Figure 43. Photoion yield spectra and partial photoionization cross sections of La [275].

1+

-::;; ~ tl

30

o(4F) * 0(40) 0

_ 20 a(5p) • ..c o(SS) 0

:E -tl 10

0 100 140 180

Photon energy (eV)

Figure 44. Photoion yield spectra and partial photoionization cross sections ofDy [275].

that the 4f wave functions can collapse into the inner well. The major part of the oscillator strength is thus transferred from the continuum into the discrete 4d -+ 4f part of the spectrum. This behavior is fully confirmed by the results of photoionization [272] and photoemission [273, 277] measurements and calculations [277, 278]. In the case of La, the photoion yield spectrum [272] is dominated by the La3+ signal and the dominating cross section is the 4d cross section [273, 274], as can be seen in Fig. 43, in the upper and lower parts, respectively [275]. The final products of the Auger decay following photoemission of a 4d electron are La3+ and La4+ ions. Therefore, these ion signals are correlated with the 4d cross section. In comparison with La, the case of Dy, shown in Fig. 44, is completely different The dominant ion yield is the singly charged ion signal [272] and the most intense cross section is the 4f cross section [276, 277]. The partial cross sections show characteristic Fano profiles: the localization of the 4f wavefunctions in the inner well gives rise to strong discrete transitions 4d104flo -+ 4<194fll , with subseqUent decay by autoionization into the continuum of the singly-charged ion.

Photoion spectra have been also measured for alkali- and alkaline-earth atoms to determine the yield of singly and doubly charged ions following inner-shell excitation or ionization [279-283].

-.a :E 101 -c .2 1 (jJ -(,) CD fI) 1 cr1 fI) fI) o .. 1cr2 (J

5d

I I

: :

Ba 6s5d

~ f ,:: I.'- '-' /." IJ'··t .....

.,' ft 6S + 5

5d

. ,

1cr3~-L~--~-L~--~-L~~~-L~~~-L--U

o 20 40 60 80 100 120 140

Photon energy (eV) Figure 45. Photoionization cross sections in 5d and 6s subshells of 5p66s5d 1.3D excited barium atoms. Experimental data and many-body calculations are from ref. 289, theoretical Hartree-Slater calculations are from Ref. 296 [296].

6. 4. ATOMS IN EXCITED AND IN IONIC STATES

89

Photoionization of atoms with one of the outer electrons in an excited orbital are of interest for a number of reasons. Excited atoms allow us to examine the dynamic properties of electrons and, hence, the wave functions at large radial distances. They play an important role in chemical reactions and in media of astrophysical interest. They have been predicted to display some features not present in the ground state, offering a reliable test of atomic theory. The first successful experiment combining the use of laser and synchrotron radiations to study photoionization processes in an excited atom was conducted in 1981. The direct photoionization of a 2p electron in a laser-prepared sodium atom in the 3p-state and the nonradiative decay of 2p53s 3p autoionizing states formed by photoexcitation of a 2p electron in the excited sodium atom were observed at that time [142, 284]. Later on, oscillator strengths for these resonant transitions could be measured [143, 285, 286]. More recently, it was possible to prepare high enough populations ofNa atoms in more highly excited states, by using two lasers in cascade, to study their resonant photoionization with synchrotron radiation [145, 146, 287]. Other photoionization processes in some laser-excited atoms, namely Ba [288-290], Li [291], Ca [292] and Na [293] have also been investigated. Alignment effects have been observed in the autoionization spectra of core-excited Na [294], Li [42, 291], and Ba [295] atoms simultaneously laser-excited in the outer shell. Several partial cross sections have been measured [289, 290] in Ba over an extended photon-energy range. As an example, we show, in Fig. 45, the 5d and 6s cross sections of the 5p6 6s 5d 1,3D excited states of Ba measured [289] between 20 e V and 150 eY. The theoretical results obtained with the random phase approximation based on a local density approximation potential (TDLDA) are in satisfactory agreement with experiment at high

90

-U) .:t::: C ::s .a ... as -~ U) C

50

Na (3s)

nt =~s Ii

4d 4p

200

Na (3p)

2pS nt from 2p6 38 2~12 4s • IIiI i

3d 3p

2pS nt from 2p6 3p 2p312

G) 40 - nt = i~S,~~ "iff3d 4f 4p

C -20

hv =75.6 eV

~s

Figure 46. Photoelectron spectra following photoionization of Na atoms in the ground state (upper part) and in the the fIrst 2p63p 2P312 excited state (lower part) [241].

photon energies, both theory and experiment reproducing the enhancement of the 5d cross section above the 4d ionization thresholds via interchannel coupling. At low photon energies, the one-electron calculations [296] appear to be closer to the experimental data.

Another category of two-color experiments, involving also the combined use of SR and laser radiations, is the production in situ of atoms difficult to study otherwise. In these experiments, an argon ion laser is used to photodissociate a diatomic molecule such as, e. g., 12, and SR is used to photoionize the I atoms (297). These two-color experiments are described in another chapter of this volume (see Nenner el al.).

Photoionization of positively charged ions is an almost new field to be explored, since mostly photoabsorption spectra have been measured after the early absorption experiments (298-300). We have already shown, in Figs. 3 and 33, the first experimental setups used for photoelectron spectrometry (12) and photoion spectrometry (146) on singly charged ions. As mentioned in Section 6.1.3, ion spectrometry has been recently used with great success at the SRS in Daresbury to measure the photoionization cross sections of a few singly charged ions, taking advantage of the enhancement of these cross sections by autoionization of ionic states resonantly excited in inner-shell transitions [147, 148, 301-303]. Some of these resonant cross sections reach very large values, up to 3000 Mh. More information about the theoretical and experimental aspects of the photoionization of singly and multiply charged ions can be found in recent reviews [304-307].

91

7. Conclusion: Towards The Golden Age With Undulator Radiation and Third-Generation Storage Rings (1989~)

The next generation of electron/positron storage rings is being built with a large number oflong straight sections where insertion devices, such as undulators and multipole wigglers, could be inserted. Even if the bending magnets of trese rings will still be used to do more routine experiments, they will serve primarily to bero the trajectory of the electrons and to force trem to pass through the straight sections. Photon flux enhanced by about one order of magnitude and brightness enhanced by at least two orders of magnitude are expected from these devices. For instance, to develop experiments on ions, photon flux of typically 1013 photon/sec in a O.3-eV bandwidth are already available with the newly built positron storage ring Super ACO in Orsay. This ring is the first one of this generation already in operation, with space available for up to six undulators, four of them in operation. At the end of this review, we would like to show, in Fig. 46, the full photoelectron spectrum (main line and satellites) measured with Super ACO in the photoionization of excited Na atoms and, in Fig. 47, the first new results that trese experiments were able to provide: a strong enhancement of the shake-up correlation satellites in inner-shell photoionization of excited Na atoms [241,242]. This result has been confirmed by similar observations made on excited U and K atoms [308, 309]. With Super ACO also, the first photoelectron spectra on singly charged ions have been measured (See Fig. 4, Ref. 12).

Other new SR projects are at different stages of preparation, mainly the European Synchrotron Radiation Facility (ESRF) in Greooble, tre Advanced Ught Source (ALS) in Berkeley, the Advanced Photon Source (APS) in Argonne, ELE1TRA in Trieste, BESSY IT in Berlin, MAX IT in Lund, and some additional rings in Japan. A few wigglers and undulators are also in operation at some existing facilities and have already been used with efficiency for a few of the photoionization experiments presented in this review. But their wider availability in atomic physics will certainly cause a new expansion of research activities in this field and will give access to new areas in the continuously {rogressing studies of photon-atom interactions.

The first NATO Advanced Study Institute devoted to atomic physics studies with SR was reId in 1975, in Carry-Ie-Rouet (France), at the time wren first-generation storage rings became available. The

60

~ I::

II I ~ fl ::::;'40 II i i ~ aI I:: GI ]l ~20 if ! t t I

i I I i I I I it -

i GI a:

0 50 70 90 110

Photon energy (eV)

Figure 47. Relative intensity of the shake-up correlation satellites following 2p inner-shell photoionization of Na atoms in the ground state (lower series of points) and in the first excited state (upper series). These results demonstrate [241] the strong enhancement of this category of satellites in an excited atom.

second of these NATO-ASI devoted to atomic physics with third-generation storage ring was held in 1992, in Maratea (Italy). Comparison between the information contained in the {roceedings of the Carry-Ie­Rouet meeting [310] and the results described in the {resent chapter will give to the reader an additional overview on the {rogress made in the field over a period of 17 years. With the use of new third-generation storage rings, another Advanced Study Institute is likely to be held before a new period of time of 17 years has elapsed.

92

8. Acknowledgements

The author would like to thank M. Ya Amusia for helpful discussions. He is very grateful to M. Berland and to D. Huissier for their great help in preparing this manuscript. He also warmly thanks E Schlachter for his patience in waiting for this chapter, G. Lawler for a critical reading and T. Morgan for a final check of the manuscript.

9. References

1. Elder, E R., Langmuir, R. v., and Pollock, H.C. (1948) Phys. Rev. 74, 52. 2. Ado, I. M. and Cherenkov, P. M. (1956) Sov. Pbys. Dokl. 1,5. 3. Korolev, EA, Markov, V. S., Akimov, E. M., and Kulikov, O. E (1956) Dokl. Akad.

Nauk 110, 542. 4. Hartman, P. L. and Tomboulian, D. H. (1952) Phys. Rev. 87, 233. 5. Thmboulian, D. H. and Hartman, P. L. (1956) Phys. Rev. 102, 1423. 6. Ivanenko, D. and Sokolov, AA (1948) Dokl. Akad. Nauk. 59, 1551. 7. Schwinger, 1. (1949) Phys. Rev. 75, 1912. 8. Sokolov, A A and Ternov, I. M. (1957) Sov. Phys. JE1P 4, 396. 9. Madden, R. P. and Codling, K. (1963) Phys. Rev. Lett 10,516. 10. Madden, R. P. and Codling, K. (1965) Astrophys. 1. 141, 364. 11. Fano, U. (1961) Pbys. Rev. 124, 1866. 12. Bizau, 1.-M., Cub aynes, D., Richter, M., Wuilleumier, E 1., Obert, 1., Putaux, I.-C.,

Morgan, T. I., Kiillne, E., Sorensen, S., and Damany, A (1991) Pbys. Rev. Lett 67,576.

13. Bizau, 1.-M., Cubaynes, D., Richter, M., Wuilleumier, E I., Obert, 1. and Putaux, I.-C. (1992) Rev. Sci. Instrum. 63,1389.

14 Putaux, I.-C., Obert, I., Boissier, G., and Paris, P. (1987) Nucl. Instrum. Methods B 26, 213.

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Similar spectra were already measured by P. Gerard, Thesis (1985), University Paris-Sud, and by several other authors.

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TWO-COLOR EXPERIMENTS ON ALIGNED ATOMS

B.SONNTAG II. Institut flir Experimentalphysik Universitat Hamburg GeT1lUJny

M.PAHLER Laboratoire de Spectroscopie Atomique et Ionique Universite Paris Sud, Orsay France

ABSTRACT. Vacuum-ultraviolet (VUV) photoelectron spectroscopy is well established as an extremely powerful method for investigating many-electron effects in atomic photoionization. Energy- and angle­resolved measurements allow detailed probing of the dynamics of the ionization but still do not provide all the information needed for a full quantum-mechanical characterization of the photoionization process. Additional information can be obtained by preparing the atoms so that they are in a well defined initial state. This can be achieved by pumping a resonance transition with a laser. Combining a dye laser with a tunable undulator source opens the VUV range to extremely powerful. angle-resolved, photoelectron and fluorescence studies of laser-aligned or laser-oriented atoms. Such experiments yield information on the symmetry of core resonances, on transition matrix elements, and on the relative phases of the outgoing electron wave functions. Theoretical and experimental approaches will be discussed. The fascinating possibilities offered by the intense, well collimated, linearly or circularly polarized photon beam emitted by tunable undulators mounted in "third-generation" storage rings will form the focus of this discussion. Experiments using CW and pulsed lasers synchronized with the storage ring are described, and first results are presented.

1. Introduction

Atoms are the basic building blocks of matter; therefore, a thorough understanding of the properties of atoms provides a sound basis for the description of the properties of molecules, clusters, and solids. The correlated motion of the atomic electrons often causes dramatic deviations from the predictions of single-particle approaches. Atoms are ideal model systems for the study of many-electron effects. In particular, the excitation spectra of subvalence and outer core electrons are strongly influenced by electron correlation. For many years, vacuum­ultraviolet (VUV) photoelectron spectroscopy has been established as an extremely powerful method to investigate many-electron effects in atomic photoionization (see, for example, Refs. 1-9). Energy and angle-resolved measurements allow for a detailed probing of the dynamics of the ionization process, but still they do not provide all the information needed for a full quantum­mechanical characterization. The missing information can be obtained by detecting the polarization of the outgoing electrons [10-12], by measuring the angular distribution of the Auger and photoelectrons [13-14], or by determining the polarization or the angular distribution of the fluorescence radiation [15-16].

Recent progress in laser systems provided the alternative possibility of preparing the atoms so that they are in a well defined initial state. The combination of laser excitation and synchrotron­radiation ionization has opened the field of VUV photoelectron spectroscopy of excited atoms

103

A.S. Schlachter and F.l. Wuilleumier (eds). New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 103-127. © 1994 Kluwer Academic Publishers.

104

(see, for example, Refs. 17-19 and references therein). Using a line source, Kerling et al. [20] made the first successful attempt to determine the angular distribution of VUV photoelectrons emitted by laser-aligned atoms. The synchrotron-radiation studies on laser-aligned atoms were pioneered at BESSY [21-27].

2. Atomic Alignment or Orientation

Alignment or orientation is characterizeq by a nonstatistical occupation of the magnetic sublevels. Let us use a 2P312 state as an example, because later we will often discuss aligned Li or Na 2P3/2

states. Figure 1 displays the population of the MJ states for a J = 3/2 system. To describe the states of

the ensembles of atoms, we will use the density matrix formalism [28]. The density operator p is defined by

The diagonal elements Pii represent the probability that an atom is in the state 1$;>:

Ipii=Trp=N ,

c: .Q

.i :::l a. o a..

I I

I I 3 "2

Figure 1. Population versus MJ for a J = 3/2 system: (a) unpolarized ensemble, (b) aligned ensemble, and (c) oriented ensemble.

105

where N is the total number of atoms of the ensemble. The off-diagonal elements Pij give the phase relation between the states I$i> and I$}>. If Pi} :#; 0, the phases of the two states are coupled. If only one Pii :#; 0 and all the other Pi} = 0, then the system is in a pure state. All other states are called mixed states.

The averaged expectation value of an operator A is given by

<A> = Tr (p A) .

This relation shows that the density matrix contains all the relevant information about the system. To make the concepts of alignment and orientation more concrete, let us again consider an

ensemble of atoms with total angular momentum 1 = 3/2. If we want to account only for the occupation probabilities, a four-dimensional vector suffices:

where NM is the number of atoms in the state 11M>. In writing down this vector, we assumed cylindrical symmetry.

NM = PMM <JMlp1M> represents the populations in the M levels, and PM'M = <JM'lp1M> represents the coherence terms between the M' and M levels. In order to make use of their transformation properties, it is advantageous to introduce the spherical tensor operators:

[1' 1 K]

T (J'l)KQ = L(-I)J'-M' ~2K + 1 11' M' >< 1 MI M'M M' -M -Q

K is the rank and Q the component. The tensor operators behave under rotation like the spherical harmonics Y LM. The density operator can be decomposed as

P= L p(J'l)KQT(1'l)KQ ' J'JKQ

where the expansion coefficients p(J' J)KQ are given by

[1' 1

p(J'l)KQ = L(-l)J'-M' ~2K + 1 M'M M' -M

K] <l'M'lp1M> . -Q

For our ensemble of atoms, there is only one degenerate level with total angular momentum 1 = 3/2, K ~ 21. Therefore, the relation reduces to

106

P(J)KQ=L(-l)'-Mvf2K+I(J J K]<JMIPJM> " M M -M -Q

For axial symmetry Q = 0,

p(~) = L(-1)312-M vf2K +1 "2 "2 [3 3

2 KO M-M

K] 3 3 <-Mlp-M> " o 2 2

To bring out the symmetry of the system, it is convenient to express this relation by using "spherical-basis" vectors [29]:

3 "2

_ 1-1 ; T2="2

-1

is proportional to the total number of atoms; p(3/2)oo is equal to the monopole moment

( 3) 1 {3 1 1 3 } - -p - =- -N3/2+-Nl/2--Nl/2--N3/2 =Tl"N ; 21O..J52 2 2 2

p(3l2ho is proportional to the magnetic dipole moment of the ensemble and is known as the orientation:

p(~) '" <1 > ; 2 10 Z

is proportional to the quadrupole moment of the ensemble and is known as the alignment

107

3. Production of Alignment or Orientation by Laser Excitation

The basic principle for the production of atomic alignment or orientation in the experiments with laser and synchrotron radiation is the absorption of polarized light. Because of the selection rules /lM] = 0 for 1t transitions and /lM] = ±1 for 0 ± transitions, the sub states 11M]> of the excited atoms are unequally populated. Figure 2 shows as an example the excitation of the substates 13I2M]> from 1112 M]'>, which can be realized by the resonance transitions of the alkaline atoms from the ground state 2S1l2 to the excited state 2P3/2.

In the case of 1t transitions, only the M] = ±1I2 sub states are populated, and, in the case of 0+

transitions, only the M]= 1/2 and 312 sub states are populated. Therefore, 1t excitation yields an atomic alignment, and crt or 0- excitation an atomic orientation. For the 0+ transitions in Fig. 2, another aspect of laser excitation can be studied: As the excited M] = 112 substate can also decay by spontaneous emission via a 1t transition to the M] = 112 substate of the ground state, the final effect of continuous laser excitation is the optical pumping of the atoms into the M] = 312 substate. Therefore, one has to distinguish between an initial orientation after one absorption process and a "stationary" after many absorption processes. The number of absorption processes depends on the intensity of the laser field, on the interaction time of the atom with the laser field, and on the natural lifetime of the excited state that is due to spontaneous transitions. In our experiments in which the laser beam is focused (diameter about 1 mm) on an atomic beam with thermal velocities, up to about 100 absorption processes per atom can take place. In the following, we will concentrate on excitations by linearly polarized light.

f1MJ= 0 1t transitions

Alignment

I \ 'I I", ~ I I \ / I

t 1\ yl I 1 \

I" \1

2P3/2

f1MJ= ±1 o± transitions

Orientation

Figure 2. Alignment or orientation produced by the absorption of polarized light

108

4. Determination of the Alignment by Measuring the Angular Distribution or the Polarization of the Fluorescence Radiation

The angular distribution and the polarization of the fluorescence radiation is determined by the alignment of the excited state, which in tum is determined by the polarization of the exciting laser light. The laser light can be characterized by the Stokes parameters [28] as defined in the coordinate system in Fig. 3.

I (13) denotes the intensity transmitted through a linear polarizer oriented at an angle 13 with respect to the x axis. The statistical tensor of the photons expressed in terms of the Stokes parameters is given by [30]:

ph I (p _.p ) P2±2 = - '2 1 + 1 2 .

For the description of the two-photon experiment, we will use a different coordinate system, which is displayed in Fig. 4. In this coordinate system, the photon statistical tensor obtained by a coordinate transformation is

ph 1 ph 1 ( . 2 ) Poo = ..J3 ; P20 =- {6 2-3sm 11

LS coupling is assumed to hold. For the 2S 112 ~ 2P3/2 excitation displayed in Fig. 2, the statistical tensor of the excited state is

proportional to the statistical tensor of the exciting photon:

Photon x 1(0°) - 1(90°) P,=

1

Py 1(45°) - 1(135°) P2 =

1 Z

P3 = 1+-L

1

Figure 3. Photon coordinate system.

109

Synchrotron PSR

Figure 4. Schematic representation of the laboratory coordinate system used for the description of the two­photon experiment. The angle <p is referred to the x axis, and the angles e and TJ are referred to the z axis.

The angular distribution of the fluorescence radiation is given by

where e is the angle between the direction of the outgoing photon and the polarization axes of the laser light. The asymmetry parameter ~ is related to the statistical tensor of the excited state by

I pph I __ -.1!l..- __ 2·.fi pljg - 2

The polarization of the fluorescence radiation observed perpendicular to the laser beam and the polarization axis of the laser (see Fig. 5) is given by

In Fig. 6, the experimentally determined polarization P of the fluorescence radiation is displayed as a function of the atomic density in the interaction region. The highest polarization of 40% achieved in the experiment lies considerably below the 60% predicted by model calculations. The model neglected the following effects, which can result in a change of the

llO

Laser

Figure 5. Geometrical arrangement for the determination of the polarization of the fluorescence radiation.

alignment and consequently in a change of the polarization: radiation trapping, hyperfine splitting, and optical pumping.

A limiting factor for the production of aligned or oriented atoms by the absorption of polarized radiation is the effect of radiation trapping. If one increases the intensity of the atomic beam, the probability of reabsorption of the fluorescence increases, and thus the degree of alignment or orientation of the excited atoms decreases. In Fig. 6, one can see that for particle densities up to 10 (which corresponds approximately to several times 1010 atoms/cm), there is a constant degree of polarization; but for higher densities, the degree of polarization decreases. For strong resonance transitions, therefore, particle densities of about 10 11 atoms/cm is a compromise between high densities of excited atoms and their alignment or orientation. As an example, the level scheme of 7Li (/ = 3/2) is shown in Fig. 7.

c 60 o ~ N 'i: al

281/2 F = 1 ,2 ~ 2P3/2 F = 0,1 ,2,3 ~ 281/2 F = 1,2

• • • o 40~~~----~L-----------------------___ a. -c

Q) o '-Q)

a. 20

• •••

A AA O~-'------------------~~-----'--------~

F= 1 ~ 2P1/2 F= 1,2 ~ 281/2 F= 1,2

0.5 5.0

Density of atoms (arb. units)

10.0

Figure 6. Degree of polarization of the Li fluorescence radiation as a function of the density of atoms in the target region. Density of atoms in arbitrary units.

7Li (l = 3/2) F

+ .r--r-------E- --1r---- 18 MHz

+ 2p_-r--<

A. = 6708 A E= 1.85 eV

2sLl---( 2

Figure 7. Energy-level diagram of 7U [31].

t 92 MHz

~ B

1 820 MHz

1

111

Under the assumption that all hyperfine transitions can selectively be excited (Le., no power and/or Doppler broadening), it would be advantageous to use either the transition from the hyperfine level F' = 2 in the ground state 2S ll2 to the hyperfine level F = 3 in the excited state 2P3/2 or the transition F' = 1 to F = O. In both cases, the excited atoms decay to the same initial levels, whereas, for all other hyperfine transitions, there is always the possibility of branching decay into the other hyperfine level in the ground state, with the consequence that those atoms are lost for further excitation. Under actual experimental conditions, however, there is an urgent need for a high density of excited atoms. Therefore, large power and Doppler broadening of the transitions must be taken into account. This can be seen in Fig. 8, in which the intensity of the reemitted fluorescence is shown as a function of the laser frequency. The line width does not allow a selective excitation of the different hyperfine levels in the excited states, and even the hyperfine splitting of the ground state is not completely resolved.

The exciting laser photon aligns or orients the electronic cloud. Due to the hyperfine interactions, J is no longer a constant of motion, but precesses around the total angular momentum F. The alignment is reduced [29] according to the relation

112

1000 2000

l!.v (MHz)

Figure 8. Reemitted fluorescence from the excited Li 2P1I2 (upper curve) and 2P3/2 (lower curve) states as a function of the laser frequency.

(J)_~(2F+l)(2F'+I)(FF'k) 1 . (J) ~o 1 - £... (21 + 1) '1 '1 I 1 + W2 't2 ~o 1 '

FF' FE"

where W FE" represents the hyperfine splitting, and 't the lifetime of the excited state. Taking the Rabi oscillations into account by reducing the lifetime 't [32], one obtains for Li a reduction of the alignment by a factor of = 0.7, which corresponds to a polarization of approximately 45%. In order to obtain a better description of the alignment of the excited state achieved by pumping by a CW laser, one has to solve the coupled Bloch equations [33-37].

5. Two-Photon Ionization of Li: Laser Plus VUV Radiation

The three-electron atom Li is the simplest open-shell, many-electron system and therefore an excellent model system for a thorough investigation. The excitation and decay channels are displayed in Fig. 9.

The experimental arrangement is schematically depicted in Fig. 10. The lithium atoms emanated from a resistively heated oven. After passing an interaction region about 1 mm in diameter, the atoms were condensed on a liquid-nitrogen-cooled copper plate (not drawn in Fig. 10). At an oven temperature of =500°C, the atomic density in the interaction region amounted to approximately 1010 cm-3. The radiation from a CW dye laser pumped the Li Is22s 2S1l2 ~ Is22p 2P3/2 transition at 670.8 nm with a density of=l00 mW/mm2. Approximately 10% of the atoms could be prepared in the excited Li Is22p 2P312 state. Transitions from both hyperfine levels of the 2S 1I2 F::: 1,2 ground state (Llli::: 820 MHz) were pumped simultaneously by using an electro­optical modulator, which splits the frequency of the exciting photon beam. Because of the Doppler and saturation broadening of the 2P3/2 resonance (= 960 MHz), the hyperfine structure of the excited state could not be resolved.

E(eV)

66.67 ~1S2P3p

64.42

61.06

r::: o ~ ~ CD

5.39 a:

1.85

C/)

iii CIl tU

O.OO...J

1s nl n'l' L = 0,1,2

113

Figure 9. Schematic energy-level diagram for the Li even-parity states reached by two-photon excitation.

Synchrotron p SR

radiation

Atoms 11

~ Oven

Laser radiation

Figure 10. Scheme of the experimental arrangement. The synchrotron and the laser radiation propagate in opposite directions along the y axis. Both photon beams intersect the beam of Li atoms in the source volume of the electron-energy analyzer, which can be rotated around the y axis.

114

From the aligned 2P312 state, the Li IsnlnT 2S1I2; 2D312,512 core resonances were excited by the high-flux undulator radiation available at the TGM6 station of the electron storage ring BESSY with a bandpass of 0.3 eV at hro = 60 eV. The degree of linear polarization at this photon energy determined from the angular distribution of the xenon 5s photoline was 98%. The electrons were analyzed by a simulated hemispherical spectrometer [38] (angular acceptance ±3°) with a four-element electrostatic entrance lens. By operating the analyzer in a nonretarding mode, energy resolution of 2% was achieved. This spectrometer could be rotated around the photon beams in a plane perpendicular to the beams. In this plane, e measured the angle between the outgoing electrons and the polarization axis of the undulator radiation. The angle between the two polarization axes was given by T]. The system was operated in two modes: 1) by setting the spectrometer at a fixed angle 9 and recording the electron intensity as a function ofT] by rotating the polarization axis of the laser, and 2) by keeping T] fixed and registering the electron intensity as a function of e by rotating the electron analyzer. Additional experiments have been performed using a cylindrical mirror energy analyzer in the same way as described by Meyer et at. [21]. With this setup, the spectrum given in Fig. 11 was obtained.

Since the Is22p 2P1I2 state cannot be aligned, the spectrum in Fig. 11 represents the relative partial autoionization cross section. In contrast to this, excitation via Is22s 2S 112 -t Is22p 2P3/2 -t Is nl nT -t Is2 ISOE I results in alignment of the laser-excited state and consequently in a dramatic variation of the intensity with the angle T]. The spectra depicted in Fig. 12 demonstrate that the intensity of the outgOing electrons critically depends on the symmetry character of the

3 1s2s (18) ns -$ ·c

1 ::J

nd 3 4 5

.ci .... ttl -~

3 4 5 ~ 1s2s e8) Ins I

nd I I I 3 4 5 ·00

c Q) -c

58.5 64.5

Photon energy (eV)

Figure 11. Intensity of the electrons emitted upon the Is22s 2S 112 ~ Is22p 2P1l2 ~ Is nl n'l' ~ Is2 lSOE I excitation and autoionization sequence. The stick diagram gives the relative strengths of the Is22p2p ~ Isnl n'l' absorption lines detected by Mcilrath and Lucatorto [39].

- 11 = 0° .-----. 30° --_. 60° ---- 90°

60.5

115

5

61.0 61.5 62.0

Photon energy (eV)

Figure 12. Intensity of the electrons emitted upon the Li Is22s 2S 1l2 --+ls22p 2P3/2 --+--+ lsnl n'r --+ls2 IS OE I excitation and autoionization sequence for different angles 1\ between the polarization axis of the two radiation fields.

core-excited state, the alignment of the intermediate state, and the relative orientation of the polarization axes of the two radiation fields. In a simple model, the intensity variation of the photoelectron lines with 11 can be explained by the different population of the M sublevels in the coordinate system of the VUV radiation [21]. 'This population is obtained from the population of the laser frame by coordinate transformation. As an example, the populations in both coordinate systems are depicted in Fig. 13 for an Li 1s2 2s 2SI12 ~ 1s2 2p 2p 312 ~ Is nl n'L 2S1l2 ~ ls2 ISO E 1 process for 11 = 0° and 90°. As 11 increases from 0° to 90°, the model predicts an increase in the intensity of 2D core resonances and a drastic decrease for 2S resonances.

6. Angular Distribution of the Electrons

The information on the angular distribution of the photoelectrons is contained in the statistical tensor of the core excited state p.~Q. 'This tensor is determined by the tensor of the ground state, the tensor of the laser photon Pk~' the tensor of the intermediate state PK'Q', and the tensor of the VUV photon Pk~2. 'This relation is schematically given in Fig. 14.

We have already shown the relation between P K' Q' and PkQI. The equations relating P KQ with the other statistical tensors are very lengthy, and therefore we refrain from giving them here. For details, the reader is referred to Refs. 26 and 27. For the elements of the statistical tensor P KL, one obtains:

116

11 2S1/2 JiJi

_11 2P3l2A 1111 !

II 2S1/2 11 3 1 1 3 3 1 1 3 -2-2 2 2 -2 -2 2 2 PSR PL PSR t ~ - ~ -=-- PL

T\ = 0° T\ = 90°

Figure 13. Population of the M sublevels in both photon coordinate systems for a process Li Is22s 2S 1I2 -? Is22p 2P312 -? Is nl n'l-? 2S 112 -? Is2 ISOEI for T] = 0° and 90°.

PKO J2L2 --~------~-------- -----~ --------------------

vuv

PK'O'

Laser

PK"O"

Figure 14. Scheme of the excitation and decay.

a) 2S1I2 resonance

b) 2D312,5/2 resonance

Poo =~[4XO +.!.(2-3sin21J)X2] 54-v5 5

P20 =~[-4Xo -(2-3sin211)X2] 270-v2

P2+2 = CM1(-sin21J)X2 - 20v21

__ C_(+ . 2 ) P 4±1 - r:iA _sm 11 X2 15v14

P4±2 = C r-:;(sin2 1J)X2 . 30-v7

117

In the above expressions, C is a normalization constant that is not important for the angular distribution measurement. The parameters XKL (KL = 0, 2) take the coherent excitation of the unresolved hyperfine structure of the first excited state into account [40].

In the LS coupling approximation, the probability for the emission of an electron in the direction iie is given by

W(iie ) = I.J2it·CJ.KL . IPKLQL .YKLQL(iie ) , KL QL

where CJ.KL is a kinematic parameter including the square of the Auger matrix elements IMLI2, PK&L are the statistical tensor components of the decaying states, and YK&L the spherical harmonics. The tensor rank KL is restricted to 0 ~ KL ~ 2L and is even. L is the total orbital angular momentum of the decaying state. In the case of the Li 1snln'l' core resonances, the complexity is greatly reduced. There is only the decay of the 2S1I2 and 2D312,512 states into the ISO state of the Li+ 1s2 ion. The autoionization of the 2P1l2,3/2 resonances is forbidden because of parity and angular -momentum selection rules. The angular distribution of the outgoing electrons in the x-z plane (see Fig. 10) is given by

118

b) 2D312,5/2 ~ ISO + E I

with PKLQL the normalized associated Legendre functions. The angle e is defined in Fig. 10. The spectrum of the electrons emitted upon the Li Is22s 2S1/2 ~ Is2 2p 2p1/2 ~ IsnlnT

2S1/2; 2D312,512 ~ Is2 1 So + E I excitation and autoionization sequence in the energy range E = 60.35-66.35 eV (with respect to the ground state) is presented in Fig. 11. This spectrum comprises a series of discrete autoionization resonances. The Li Is2p2 2D312,512 resonance at E = 61.06 eV is by far the strongest and energetically well separated from the other resonances. There is general agreement on the assignment of this resonance [39, 41, 42, 43]. In order to test their approach, Pahler et al. [27] started with the investigation of the angular distribution of the electrons emitted upon the Li Is2p2 2D3/2,512 ~ Li Is2 1 So + Ed autoionization. The intensity of the outgoing electrons as a function of the spectrometer angle e (see Fig. 10) for three different orientations of the two polarization axes (11 = 0°,45°, and 90°) is shown in Fig. 15. Between the angles e = 80° and e = 115°, the spectrometer was blocked by the cold trap; thus, in this range, no data could be taken. The bottom spectrum in Fig. 15 was recorded at an angle 11 = 90°. The spectra shown at the center and at the top of this figure were obtained by changing the angle to 45° and 0°. All the other experimental parameters were kept fixed. For the center spectrum, the symmetry with respect to e = 90° is lost. The solid lines in Fig. 15 represent least-squares fits of the experimental data by the theoretical angular distribution W(e).

These results already prove the D character of the resonance, because the angular distribution of the outgoing electrons for an S resonance should not depend on the angle e. Another possibility for checking the assignment of the resonance is to measure the electron intensity as a function of the angle 11 (see Fig. 10) between the polarization axes of the two radiation fields. Figure 16 shows the intensity of the emitted electrons upon the decay of the same resonance at E = 61.06 eV as a function of the angle 11. For the lower spectrum in Fig. 16, the spectrometer was positioned in the horizontal plane e = 0°. The error bars are of the magnitude of the spot size. The upper spectrum in Fig. 16 shows a measurement made under the same conditions, but with the spectrometer set at an angle e = 135° with respect to the z axis. Again the solid lines represent a least-squares fit of the data to W(e). Within the experimental uncertainties, the five sets of data obtained in different measurements by varying 11 or e can be well approximated by the theoretical distribution W(e) for a D resonance. The consistency of the approximation is corroborated by the small variation of the ratio X2/XO of the fitting parameters X2 and Xo, which correct for the influence of the hyperfine interaction.

119

1.2

0.8

0.4

0.0 -CJ) ~ C :::l 0.8 .c .... CIS ->-~ 0.4 CJ) c Q) -£:

0.0

0.8

0.4

0.0

Spectrometer angle e (deg.)

Figure 15. Intensity of the electrons emitted upon the process Is22p 2P312 -+-7Is(2p2) 2D312,512 ~ Is2 1 SOE I with respect to the spectrometer angle E> for three different angles between the polarization axes (1] = 0°, 45°, 90°). The solid lines represent a fit to the data by W(E».

Confident in this approach, we turn to the resonance at E = 63.54 eV (line 9 in Figs. 11 and 12), which is well suited for demonstrating the potential of angle-resolved photoelectron spectroscopy of laser-aligned atoms. Measurements of the e-dependence show an intensity variation of the outgoing electrons that is definitely larger than expected for an S resonance but at the same time significantly smaller than for aD autoionization resonance. Also the 11-dependence of the electron intensity cannot be reconciled with the decay of a pure S or D resonance. The upper spectrum in Fig. 17 shows the electron intensity as a function of 11 measured for a spectrometer angle e = 45°. A comparison with the corresponding spectrum in Fig. 16

120

1.2 8 = 1350

1.0

0.8

0.6

Cil 0.4 -'c ::J 0.2 .ci .... ~ 0.0 ~ 8=00

'iii 1.0 c: Q) -.E 0.8

0.6

0.4

0.2

0.0 -90 0 90 180 270

Polarization angle 1'\ (deg.)

Figure 16. Intensity of the electrons emitted upon the process 1s22p 2P312 --t Is(2p2) 2D312,512 --t 1s2 1 SOE I with respect to the angle 11 between the two polarization axes. Upper spectrum: spectrometer angle E> = 135°. Lower spectrum: E> = 0°. The solid lines represent a fit to the data by W(E».

demonstrates the unexpected small variation of the intensity with 1'\. The experimental results can be explained by assuming that the decay of two closely spaced Sand D resonances gives rise to the electron emission. The existence of an S and a D resonance in this energy range is supported by several experimental [39, 44-47] and theoretical [39, 41, 42, 47] results, although the assignments of the resonances differ. Based on calculations of Weiss [see Ref. 39], Wakid et at. [41], and Chung [42], we expect the transitions Li Is22p 2P312 ~ Is2p2 2S1I2; Is2s( 3S)4d 2D312,5/2 to contribute. Because of the O.3-e V bandwidth of the VUV radiation, both resonances are excited in our experiment.

By superimposing the angular distributions of the electrons emitted upon the decay of an Sand a D resonance, reasonable agreement with experimental results can be achieved. The two distributions and their sum are given in Fig. 17. The relative weight of the two distributions is determined by the Li Is2 2p 2P3/2 ~ Is2p2 2Sl/2; Is2s(3S)4d 2D312,5/2 excitation probabilities. Based on the assumption that the resonances coincide, the ratio of the dipole matrix elements 1< rPjlr(l)Irs >12 and 1< rPlir(l)I r'D >12 is close to 1. This result is also supported by Hartree-Fock calculations. A comparison of the experimental resonance profile with the superposition of two Gaussians of the deduced intensity ratio and a half width of 0.16 eV shows no marked deviation if the two Gaussians are separated by less than 0.1 eV. Chung [42] has

1 .2 ,...,.,rrn"TTTTTT"rTT"1"TTTTTT"rTT"1"TTTT"M"T"TT"IrTTTTTT".,

1.0

0.8

0.6

i' 0.4 '2 ::::I 0.2 .e

,-' ,-, , " , , \ .......... / \ ....... , \\ / .. / ,l.... .\ .. \ ... \ ... ~:, ... ~ ... ~ )< ............. .

.. / .. /~' ...... / ............. --' ......... . ....

S 0.0 ....... ~u....uu..u ............... .L..U-u....uu....u ....................... .r....L..I-L...L...L.j

1.0

0.8

0.6

0.4

0.2

0.0 -90 o 90 180 270

Polarization angle Tl (deg.)

121

Figure 17. Intensity of the electrons emitted upon the decay of the resonance at E = 63.54 e Vasa function of the angle 11 between the polarization axes of the two radiation fields. The solid line represents the superposition of the angular distributions calculated for the decay of an 5 and a D resonance. The two distributions are given separately by the dotted (5 resonance) and the dashed (D resonance) curves. Upper spectrum: spectrometer angle e = 45°. Lower spectrum: e = 0°.

calculated the oscillator strength of the Li Is2 2P312 ~ Is2s(3S)4d 2D3/2,5/2 transition, but there is no /value given for the transition to the Li Is2p2 2S1I2 state.

Table 1 summarizes the results for the most prominent resonances. For comparison, the assignments and the energies presented by several calculations are included. The results confirm most of the assignments. Even for the resonance at 63.54 eV. the theoretical predictions are consistent with the interpretation of the experimental results discussed above.

122

TABLE 1. Li 1 s nl n'( resonance energies and assignments determined in this work. For comparison, the results of several theoretical studies are included: [39], [41], [42], and [43]. The energies are relative to the IsQs 2S1l2 ground state. The numbers in parentheses indicate the uncertainty of the last digit.

Resonance energy from ground state (eV)

This work Assignment [39] [42] [43] [41] Assignment Energy (eV)

Is2p22D 61.11 61.071 61.177 61.064 2D 61.06(1) Is2s (3S) 3d 2D 62.98 62.909 63.008 62.914 2D 62.90(2) Is2s (IS) 3s 2S 63.23 63.425 63.159 2S 63.11(2) Is2p22S 63.50 63.485 } 2S+ 2D 63.54(2) Is2s (3S) 4d 2D 63.62 63.567 63.676 Is2p (lp) 3p 2D 65.654 65.625 2D 65.60(2)

7. Photoelectron Angular Distributions from Laser-Aligned Atoms Directly Ionized by VUV Radiation

In the preceding section, we have demonstrated the power of the method for resonant excitations. Now let us turn to experiments in which the laser-aligned state is directly ionized by VUV photons. Up to now, most of these experiments have been limited to excitation and ionization by visible or ultraviolet laser radiation [40, 48-51]. Kerling et al. [20] performed angle-resolved studies of nonresonant photoemission from laser-aligned Yb atoms using linearly polarized VUV radiation from an intense Ar discharge source. The experimental setup is shown in Fig. 18. Yb atoms were aligned by pumping the Yb 6s2 ISO ~ Yb * 6s6p 3Pl transition by a CW laser. The excited states were subsequently ionized: Yb* 6s6p 3Pl ~ Yb+ 6s 2S1I2+ ES,Ed. Because the polarization axes of both radiation fields could be varied, the position of the electron

~Electro" _ J"Z?y_ spectrometer

! @ Channeltron I

Fresnel I, / Vb atoms

rhomb E t; L. it. LaSl!r ____ tG __ / __ L. ~ • ¢= _____ !~~~-r f)..-r-Il--n VUV beam f S " ~J~lamp

• Rotatable I reflecting

Aperture XL polarizer I ens

Photodiode 0

Figure 18. Experimental setup for measurements of photoelectron angular distributions and excited-state fluorescence.

123

spectrometer could be kept fixed. The angular distribution of emitted electrons was determined for different angles 11 between the polarization axes as a function of the angle e between the direction of the outgoing electron and the polarization axis of the VUV radiation. A general theoretical expression for the photoelectron angular distribution was given by KIar and Kleinpoppen [52]. For the geometry used by Kerling et al. [20], the angular distribution can be written in the following form:

The coefficients (1.ij depend on the alignment A of the intermediate state on the angle 11, the transition amplitudes Ds (outgoing E s electron) and Dd (outgoing Ed electron) of the reduced dipole matrix elements, on their phase difference A, and on the degree of polarization P of the VUV radiation.

(1.00 = ~C{[8+2(1 + 3Pcos211)A ]D; +(5 - P)[2+...!...(5 - 3COS211)AJD~ 24 10

-..J2 (1- P)[4+(I- 3cos211)A] DsDd cos A} .

(1.20 = ~C{[ 2P+ l~ (7P+21COS211)-18PCOS211AJD~ -..J2[4P+(P+ 3cos211)A] DsDd cos A} .

0.21 = ~C[7 - 3P)sin21]AD~ -..J2 sin21]ADsDd COSL1]

18 0.40 =-CPcos21]AD~

35

0.41 =-.2...CPsin21]AD~ 70

For P = I, these equations correspond to those of Hansen et al. [40]. The experimental results reported by Kerling et al. are shown in Fig. 19.

From the fits of the data points from a series of angular distributions, the ratio of the transition amplitudes Ds/Dd and the absolute value of the cosine of the phase difference A could be determined. The average results are

IcosAI=0.83±0.14 .

The results show that the amplitude of the p ~ Ed transition dominates.

124

A = -0.61 ± 0.02 A = -0.67 ± 0.01 1X:20 = 1.22 ± 0.05

1140 = -0.69 ± 0.06 1X:20 = 1.51 ± 0.08

4 1X:21 = -0.05 ± 0.03 3 1140 = -0.17 ± 0.08 1141 = 0.02 ± 0.02 1X:21 = -0.33 ± 0.04

3 + + 1141 = -0.12 ± 0.03 2

2

f 'TI = 0° 'TI = 30° ::J 0

0 0 S-

6 A = -0.60 ± 0.01 A =-0.71 ±0.02 1X:20 = 2.05 ± 0.09 8 1X:20 = 2.18 ± 0.06

5 1140 = 0.66 ± 0.10 1140 = 0.84 ± 0.08

4 1X:21 = -0.34 ± 0.04 1X:21 = 0.04 ± 0.02 1141 = -0.13 ± 0.03 1141 = 0.03 ± 0.02

3 4 2

2 1 0

0° 90° 180° 0

0° 90° 180° AngleS

Figure 19. Angular distributions obtained at phase angles 'TI of 0°, 30°,60° and 90° [20].

Using the BESSY undulator radiation, Lorenz et al. [53] recently succeeded in investigating the angular distribution of the photoelectrons created upon the ionization of laser-aligned Na atoms in the range of the 2s excitations, where, in addition to the direct Na 2s2 2p6 3p ~ Na+ 2s2 2p5 3p + E I photoionization, the ionization via Na 2s2 2p6 3p ~ Na 2s2p6 3p2 ~ Na+ 2s2 2p5 3p + E I contributes. These experiments proved that the limitations imposed by the discrete wavelengths of the discharge sources can be overcome by modem undulator sources, which allow for continuous tuning over a large photon-energy range. These experiments promise a wealth of very detailed information on the dynamics of photoionization and therefore will allow for stringent tests of theoretical approaches.

8. Perspectives, Dreams

For the future we foresee the following developments: • Experiments will be extended toward higher photon energies. Progress in lasers will allow

for the excitation and alignment/orientation of atoms up to hv = 6 eV. The restriction on alkali and alkaline earths will be lifted.

• New undulator sources will provide sufficient photon flux for experiments up to several keY; thus, the excitation of more tightly bound shells will be possible.

• The availability of circularly polarized XUV radiation will be increased.

125

• Ideally, the new insertion devices will make it possible to change quickly from linear to left or right circular polarization.

• The size of the focal spot of the XUV radiation will go down to ::;;100 x 100 ~2. This development will allow for better collimation of the atomic beam (e.g., reduction of Doppler broadening) and an increase in angular and energy resolution.

• Progress in laser technology will make it feasible to synchronize a high-power, high­frequency laser with an electron storage ring. This development will create a new dimension for time-resolved experiments.

• Detection of the polarization and angular distribution of XUV fluorescence emitted by core-excited, aligned/oriented atoms will become possible.

• The detection of two outgoing particles (electrons, photons, ions) in coincidence should become possible for aligned species.

These experiments will contribute considerably to our understanding of the electron dynamics of isolated, many-electron atoms and of atoms aligned/oriented in solids and on surfaces.

Acknowledgments

The authors thank their collaborators C. Lorenz, E. von Raven, J. RUder, S. Baier, B.R. MUller, M. Schulze, H. Staiger, P. Zimmermann, and N.M. Kabachnik. The financial support of the Bundesministerium fiir Forschung und Technologie and the European Economic Community is gratefully acknowledged.

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Lett. 12, 35 (1990). 20. C. Kerling, N. B(jwering, and U. Heinzmann, J. Phys. B 23, L629 (1990). 21. M. Meyer, B. Muller, A Nunnemann, T. Prescher, E. von Raven, M. Richter, M. Schmidt,

B. Sonntag, and P. Zimmermann, Phys. Rev. Lett. 59,2963 (1987). 22. M. Meyer, M. Pahler, T. Prescher, E. VOn Raven, M. Richter, B. Sonntag, S. Baier, W.

Friedler, B.R. Muller, M. Schulze, and P. Zimmermann, Physica Scripta, T31, 28 (1990). 23. T. Prescher, Thesis, Universitiit Hamburg (1988). 24. B. Muller, Thesis, TU Berlin (1990). 25. M. Pahler, Thesis, Universitiit Hamburg (1991). 26. S. Baier, W. Fiedler, B.R. Muller, M. Schulze, P. Zimmermann, M. Meyer, M. Pahler, T.

Prescher, E. von Raven, M. Richter, J. Ruder, and B. Sonntag, 1 Phys. B 25,923 (1992). 27. M. Pahler, C. Lorenz, E. von Raven, 1 Ruder, B. Sonntag, S. Baier, B.R. Muller, M.

Schulze, H. Staiger, P. Zimmermann, and N.M. Kabachnik, Phys. Rev. Lett. 68, 2285 (1992).

28. K Blum, Density Matrix Theory and Applications (Plenum Press, New York, 1981). 29. C.H. Green and R.N. Jake, Annu. Rev. Phys. Chern. 33, 119 (1982). 30. AT. Ferguson, Angular Correlation Methods in Gamma Ray Spectroscopy (North Holland,

Amsterdam, 1965). 31. E. Arimondo, M. Inguscio, and P. Violino, Rev. Mod. Phys. 49, 31 (1977). 32. C.R. Stroud, Phys. Rev. A 3, 1044 (1971). 33. A Fischer and I.V. Hertel, J. Phys. A 304, 103 (1982). 34. P.M. Farrell, W.R. MacGillirray, and M.e. Standage, Phys. Rev. A 37, 4240 (1988). 35. J.J. McClelland and M.H. Kelley, Phys. Rev. A 31,3704 (1985). 36. M. Wedowski, Diplomarbeit, TU Berlin (1991). 37. W.R. MacGillirray, M.C. Standage, P.M. Farrell, and D.T. Pegg, J. Mod. Opt. 37, 1741

(1990). 38. K Jost, J. Phys. E 12, 1006 (1979). 39. TJ. McIlrath and T.B. Lucatorto, Phys. Rev. Lett. 38, l390 (1977). 40. J.e. Hansen, J.A DUncanson, R. Chien, and R.S. Berry, Phys. Rev. A 21, 222 (1980). 41. S. Wakid, A.K. Bhatia, and A Temkin, Pbys. Rev. A 21, 496 (1980). 42. KT. Chung, Phys. Rev. A 24, l350 (1981). 43. AK Bhatia, Phys. Rev. A 18, 2523 (1978). 44. M. Rodbro, R. Bruch, and P. Bisgaard, J. Phys. B 12,2413 (1979). 45. S. Mannervik and H. Cederquist, Physica Scripta 31, 79 (1985). 46. D. Rassi, V. Pejcev, and Kl Ross, 1 Phys. B 17, 3535 (1977). 47. M. Simsek, S. Simsek, and S. Erhoc, Chern. Phys. Lett. 91,456 (1982). 48. R.L. Chien, O.e. Mullins, and R.S. Berry, Phys. Rev. A 28, 2078 (1983). 49. A Siegel, 1. Ganz, W. Bussert, and H. Hotop, 1. Phys. B 16,2945 (1983). 50. O.C. Mullins, R.L. Chien, J.E. Hunter, J.S. Keller, and R.S. Berry, Phys. Rev. A 31, 321

(1985). 51. lS. Keller, lE. Hunter, and R.S. Berry, Pbys. Rev. A 43, 2270 (1991).

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52. H. Klar and H. Kleinpoppen, 1. Phys. B 15,933 (1982). 53. C. Lorenz, M. Pahler, J. Riider, B. Sonntag, S. Baier, M. Schulze, H. Staiger, P.

Zimmermann, J.T. Costello, and L. Kiernan in BESSY Annual Report 1991 (Berliner Elektronenspeicherring-Gesellschaft fiir Synchrotronstrahlung, Berlin, 1991), p. 103.

TWO-COLOR EXPERIMENTS IN MOLECULES

I. NENNER, P. MORIN, M. MEYER, J. LACOURSIERE ANDL.NAHON LURE, Laboratoire mixte CNRS, CEA, MENC Biitiment 209D Centre Universitaire 91405 Orsay cedex, France

and

Service des Plwtons, Atomes et Molecules Biitiment 522 Centre d' Etudes de Saclay 91191 Gifsur Yvette cedex, France

ABSTRACf. This paper describes two-color experiments in which a laser in the visible or near ultIaviolet is combined with synchrotron radiation (SR) in the vacuum ultraviolet (VUV) to study: (1) the photoionization of radicals produced by the photodissociation of a molecule, and (2) the dynamics of the photodissociation itself. Various lasers and existing SR sources are compared. Pump-probe experiments with two photons, either in the cw mode or in a pulsed synchronized mode, are briefly described. A general description of the physics of photodissociation, covering diatomic to large molecules, is briefly presented. Selected applications of laser-induced photodissociation of molecules with subsequent photoionization of the fragments are presented. These studies investigate electron relaxation in open-shell, core-excited halogen atoms and the molecular photodissociation dynamics of a simple aromatic molecule. Also examined are possible future experiments employing the pump (SR) - probe (laser) combination, synchronization of both sources, and the use of the infrared from a free-electron laser associated with SR.

1. Introduction

"Two-color experiments in molecules" refers primarily to photon-induced dissociation through a pump-probe arrangement (Figs. la and Ib), in which the energy of one photon lies in the UV, VUV, or soft x-ray range, and that of the second photon is in the visible or near-UV range. The ftrst photon (laser or SR) is the pump and is used to photodissociate (Fig. la) or to photoexcite (Fig. Ib) a molecule. The second photon (laser or SR) is used to photoionize (or photoexcite) the fragments (Fig. la) or photoionize (or photoexcite) the excited molecule (Fig. Ib). This is called pump-probe, resonance-enhanced, multi-photon ionization (1+1 REMPI) or (1+1) double­resonance photodissociation.

Molecular photodissociation dynamics have been studied by a variety of techniques for many years and have been the subject of many outstanding reviews in the late 1970's (Simons, 1977;

129

A.S. Schlachter and F.1. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 129-160. © 1994 Kluwer Academic Publishers.

130

(a) Molecule Fragments

~ ~ ~ 0

Electrons

-{ 0

Photon 1 Photon 2

(b)

o

o

Photon 1 Photon 2

Figure 1. Schematics of a two color (1+1) experiment in a molecule. Figure la refers to pump-probe photodisSOCiation. Figure 1 b refers to (1 + 1) double-resonance pholo<iissociation.

Gelbart, 1977; Freed and Band, 1977; Ashfold et aI., 1979) and textbooks (Okabe, 1978). Photofragment dynamics, which represents an important subfield of photodissociation with special emphasis on fragmentation details such as final state distributions, dissociation lifetimes, translational-energy distributions, angular distribution of fragments, and fluorescence polarization of fragments, appeared later (Simons, 1984; Bersohn, 1984; Leone, 1982; Hodgson et al., 1985; Vasudev et al., 1984; Welge, 1984; Andresen et al., 1984, 1985), mainly because of the advent of commercially available UV laser sources, i.e., tunable dye lasers, frequency-doubled dye lasers, or rare-gas halide excimer lasers. Two-photon photodissociation teChniques were introduced next (Andresen et al., 1984, 1985; Schinke et aI., 1985; Bigio and Grant, 1985) and are widely developed even in small-size molecules (see, for example, Kung et aI., 1986; Trick! et aI., 1989) and medium-size molecules (see, for example, Song et al., 1990; Page et al., 1988; Stolow et al., 1992). Picosecond and femtosecond pump-probe experiments (for a review, see Zewail, 1988, 1992) also developed successfully, allowing for the first time the investigation of photodissociation processes in real time. One-photon VUV photodissociation experiments developed in the 1980's (see Baer, 1986, and Nenner and Beswick, 1987, for a review) because of the development of coherent laser sources (Marinero et aI., 1983; Hilbig et al., 1984) and also SR sources with beam lines suited for gas-phase experiments. However, double-resonance photodissociation experiments in which one of the photons is SR in the UV or soft x-ray region have barely developed, since the first work was reported only in the last few years by Tremblay et ai., 1988; Nahon et al" 1990a, 1990b, 1991, 1992a, 1992b; and Nahon and Morin, 1992. This specific aspect of photo fragment dynamics, based on experiments associating SR and lasers, is

131

the main subject of this article. The problems of combining lasers with existing and future SR sources are reviewed, with special attention to their time structure and intensity per pulse.

The purpose of (1+1) double-resonance experiments is twofold. First, molecular photodissociation is used as an efficient way to produce spectroscopically clean radicals (atoms or molecules with an open valence shell). The investigation of their electronic properties is the central interest and is achieved by photoionization. Second, the study of the molecular photodissociation itself is the main goal, and photoionization is the method to probe the fragments. The investigation can be carried out by starting with a neutral molecule in the ground state but also by starting with a photoexcited molecule.

1.1. PHOTOIONIZATION OF RADICALS

Radicals are unstable species having an incomplete valence shell. The presence of this vacancy gives them very specific properties compared to species having a closed valence shell. They are either open-shell atoms (any element of the periodic table except the rare gases) or molecular radicals (for example, OH, CH3, or NH4+). Their electronic properties are difficult to predict because of strong electron correlations. However, the investigation of these species allows for a severe test of theoretical descriptions and extends our understanding of electron-correlation effects. For polyatomic molecular radicals, their equilibrium geometry may be quite different from their closed-shell neutral or ionic counterparts. Both atomic and molecular radicals are very reactive in collisions and playa very important role in chemistry and in nature (for example, in earth and planetary atmospheres, interstellar media, combustion, and radiation damage). In the laboratory, one produces them by sublimation of a solid material, by discharges in a gas, or by laser ablation of a solid. However, their intrinsic reactivity sometimes makes it difficult to produce a "radical gas" in a chemically and spectroscopically pure phase. Photodissociation of a molecule is one of the gas-phase chemical reactions that can provide an alternative method, as long as the molecular precursor exists and the photon source reaches the required photon energy. The photoionization of atomic iodine or bromine, produced by photodissociation of molecular iodine or bromine (12 or Br2), is an example of such a study.

1.2. MOLECULAR PHOTODISSOCIATION

The fragmentation of a molecule induced by photon impact is the most basic process in photochemistry. It is considered to be one of the fundamental elementary chemical acts, together with bond formation, electron transfer, and environment effects (for example, solvent effects). The understanding of the dynamics, energetics, nature, and internal energy of the products and the kinetic-energy release is an important step toward a comprehensive view of natural photochemical processes and laser control of photochemical phenomena. Among the latter, let us mention the importance of selective photolysis in biochemistry, isotope separation, laser design, and laser-assisted synthesis, as well as the role of photo fragmentation in photo desorption from surfaces.

The challenge is to carry out a "state-to-state" photodissociation experiment in which the initial state of the molecule and the final state of the products are well defined. One should keep in mind that a photodissociation process is complex, because the relevant time scale ranges from femtosecond to millisecond, and the process depends strongly upon the energy stored in the molecule. Furthermore, there is a difficulty related to the size of the system and the various

132

situations encountered with bonding. Restricting the problem to a diatomic species AB, we show in Fig. 2 the various photodissociation processes and the competing phenomena (radiative decay, single and double valence ionization, and core ionization), as the photon energy increases from the UV to the VUV and (soft) x-ray region.

In the following, we will concentrate on the dynamics of the dissociation of a polyatomic system, the s-tetrazine molecule, in its electronic ground state by probing the internal energy of the fragments. We also discuss the motivation for developing such multiphoton excitation for investigating the dynamics of fragmentation of molecular ions and for the photoexcitation and

Dissociation

A+ 8* ~

A+ + 8- hv

Single ionization

Double ionization

Core ionization

Figure 2. Schematics of photodissociation processes in a diatomic molecule AB, as the incident photon energy increases from the UV (no ionization), to the VUV (single and double ionization), and then to the soft x-ray range (core ionization). The typical dissociation rates show that ionization generally precedes dissociation, but there are exceptions (see text). From Nenner and Beswick, 1987.

133

photoionization of excited molecules. Finally, future trends are briefly outlined, with an emphasiS on the capabilities and limits of third-generation SR sources and the rationale for combining an infrared source with UV or VUV sources.

2. Combination of Laser and Synchrotron Radiation

These experiments are based on the combination of two sources with very different characteristics to excite a molecule in a jet (effusive, supersonic expansion). They employ various types of detection techniques currently used in molecular physics: ion, electron, and fluorescence spectroscopy, and numerous coincidence techniques.

2.1. SOURCE CHARACTERISTICS

Table 1 summarizes the tunability, spectral bandwidth, and power properties of tunable lasers and synchrotron radiation sources. These data are partly extracted from Koch, 1982; Balcou et at., 1992; Meyer et aI., 1990; and Sonntag, 1990. We have also reported in Table 2 and Fig. 3 a comparison of the temporal characteristics of selected pulsed lasers and SR. We have considered commercially available lasers (cw and pulsed dye lasers, mode-locked dye lasers) that can be brought to a synchrotron radiation facility and operated within the limited space available near a beam line. We have also considered harmonic generation from high-power lasers, which already provide extremely interesting capabilities out of reach by SR (Cromwell et at., 1989; Balcou et at., 1992). It is clear that the great current progress in the laser field makes this comparison hazardous for more than ten years in the future. For example, it is difficult to predict the performance of cw lasers extended to the UV or the repetition rate of excimer lasers beyond the kilohertz range. On the SR-source side, progress is expected, and many new sources now being built or proposed promise great improvement in brightness. Progress in the quality of optics and in the resolving power of monochromators will certainly be made, but it is difficult to visualize the limits. Finally, we have considered free electron lasers (FEL), which are available for users today. One is a FEL injected with relativistic electrons produced in a linear accelerator; this provides near- or far-infrared light (Ortega et at., 1989; Prazeres et at., 1992). Another is a FEL injected with relativistic electrons produced in a storage ring; this provides near-UV radiation (Couprie et at., 1992; Billardon et at., 1992). Note that these rather heavy devices are located at the SR-source site or nearby and can be combined with synchrotron-radiation sources with no major difficulty.

The pulsed structure of synchrotron radiation is directly related to the number n of circulating bunches of electrons (or positrons) in the machine, the length AT of each bunch, and the interpulse period T, which is correlated directly to the length L of the machine by the simple relation

T=Unc , (1)

where c is the speed of light. Present facilities at which the electron beam has a pulsed structure are operated with a small number of bunches to keep the interpulse period T long enough compared to the pulse width. On the other hand, for a given machine, the duty factor ATIT

134

TABLE 1. Comparison of the tunability, bandwidth, and power of selected lasers and synchrotron radiation.

Bandwidth Power or Source Tunability (aE/E) energy/pulse

Continuous-wave dye laser IR to visible 10-9 lW (nearUV)

Pulsed dye laser IR to near UV 10-6 1 mJ/pulse to lQ4 J/pulse

Mode-locked dye l~.ser IR to visible 10-4 1 nJ/pulse (nearUV)

Excimer laser NearUV 3 x 10-3 200 mJ/pulse

Harmonic generation IRtoVUV 10-3_10-7 1.5 nJ/pulse to from high-power laser 0.3 mJ/pulse

(4 x 108 photons! pulse at 22 eV) to 0.3 mJ/pulse (1011 photons/pulse at 16 eV)

Synchrotron radiation IR to x-ray 10-3_10-5 105-106 photons! with monochromator pulse

Storage ring FEL Visible to near 10-4 0.5 ~/pulse UV

Linear accelerator FEL IR 4x 10-3 20 J.1J/pulse (65 mW average}

TABLE 2. Comparison of temporal characteristics of selected pulsed lasers and synchrotron radiation (few-bunch operation).

Pulse width Interpulse FWHM period

Source (seconds) (seconds) Repetition rate

Dye laser 10-14_10-5 10-4-10 0.1-10 kHz

Mode-locked dye laser 10-12 10-8 80 MHz

Excimer laser 10-8 10-2 100Hz

Harmonic generation from 10-14_10-12 0.5-10 0.1-20 Hz high-power laser

Synchrotron radiation few 10-10 few 10-7 few MHz

Storage-ring FEL 25 x 10-12 1.2 X 10-10 8 MHz

Linear-accelerator FEL

micropulses 8 x 10-12 3.2 X 10-8 25-30 MHz

macropulses 10-5 2 x 10-5 50Hz

10-15

SR mode

SR time structure

Laser time structure

• Pulse width

/I Intetpulse period

Time resolution

of detectors

10-12

I

Time (seconds)

10-9 10~ I I

135

10-3 1 I I

< PULSED I CO~TINUOU9

Pulse width

~ Intetpulse period

~ .... -)' _ .... -,

Harmonic generation • • ~--_. Pulsed dye

~. ~ Mode locked dye

~--- Excimer dye ---+-1il!llfI

• liJJ- UV· FEL

Micropulses Macropulses • • '--IR.FEL- •

Synchronization required

----- Streak camera Channel plate

Photomultiplier

Direct dissociation ---Physical

phenomena

Predissociation ----.....;..;.~

Electronic relaxation ---...: Vibrational relaxatio!? ....... .

____ ---'Autoionization ____________ ~R~a~d~~~liv~1~~y

Isomerization, atomic rear~.'!fl.~ment

Figure 3. Relevant time domains for the pulse width and repetition rate of selected pulsed lasers and synchrotron radiation, and for the time resolution of detectors and physical phenomena in molecular photophysics.

136

typically ranges from 10-4 to 10-3 with T in the 0.1- to I-microsecond range, as shown in Table 3. Notice that third-generation facilities will operate the ring with many bunches because of high-brightness requirements. Even though the pulse width will be reduced below 0.1 ns, the interpulse period will also decrease to a few nanoseconds because of the large number of bunches (Nenner, 1992). Therefore, in third-generation machines, even if the length of the machine tends to increase, the duty factor ATIT will be basically the same as that of second-generation facilities. For this reason, in third-generation facilities, experiments requiring a long interpulse period (typically around 100 ns or more) should have dedicated running time with a smaller number of bunches, although the photon intensity and the mean lifetime of the beam will be reduced. In practice, this few-bunch operation should be announced very early in the conception of the machine to obtain the best compromise between photon intensity and beam lifetime.

TABLE 3. Time structure of selected facilities using the pulsed structure of SR.

Interpulse Length Number of Pulse width period Duty factor

Facility (m) bunches * AT (ns) T(ns) ATIT

Super ACO 74 2 (24) 0.25 120 2.1 x 10-3

BESSY 62 1 (70) 0.07 208 3 x 10-4

DESY 288 1 (4) 0.13 960 1 x 10-4

ALst 197 1 (250) 0.03-0.05 656 4.6 x 10-5

SOLEIU 216 6 ~240l 0.07 120 6 x 10-4 * The value in parentheses is the number of bunches nonnally filled for a high-intensity

t operation. Under construction.

* Projected.

2.2. CW OPERATION AND DETECTION TECHNIQUES

2.2.1. CW Operation. Continuous-wave operation refers to the use of a cw laser with SR and, of course, no synchronization between the two sources. As shown in Fig. 3, this mode of operation is well suited for pump-probe experiments in which the "potential" delay between the pump and the probe is long (Le., beyond milliseconds). The best operation is obtained with SR at the highest repetition rate (Le., with the maximum number of bunches filled in the machine). Then the duty factor is the highest. For example, in Super ACO with 24 filled bunches, the duty factor amounts to 0.002 and offers the best match with that of the cw laser, which equals unity. For future third­generation machines, the usual choice of a high number of bunches in the ring increases the duty factor and favors this type of operation. One can distinguish two classes of experiments. One is the pump Oaser) - probe (SR); the other is the reverse: pump (SR) - probe (laser). Although the first has been extensively used since the early eighties (Wuilleumier, 1982; Meyer et at., 1990; Sonntag, 1990), the second has yet to be developed.

The application of the pump Oaser) - probe (SR) is guided primarily by the wavelength range covered by the laser. For photodissociation experiments, it can be applied if the bands of the photo absorption spectrum are reached by the laser. But in many small molecules, the system

138

fragments. Furthermore, it provides the release of translational energy, as one can see from the symmetrical broadening of the time-of-flight peak. For slowly dissociating ions, the parent ion travels some distance in the acceleration region before dissociation occurs. The fragment and parent ion signals are asymmetrically broadened. One extracts the dissociation rate for a given channel. Time-of-flight mass spectrometry can be used in the photoion-photoion coincidence mode or in the photoelectron-photoion-photoion coincidence mode. These methods are especially well suited for studying the fragmentation pattern of doubly (or multiply) charged ions. As for monocations, the coincidence peak shape is broadened because of translational energy. For slowly dissociating ions, the coincidence peak shape is asymmetrically broadened and information on the dissociation rate can be obtained. Note that, in all these coincidence techniques, no information about the internal energy of the fragment can be obtained.

When the fragments are photoexcited, the detection technique can be photon-induced fluorescence. Fluorescence-photoion coincidence techniques can be also used. The advantage of fluorescence detection is the access to the internal energy of the fragments. Although it is restricted to small species, the method offers high sensitivity and selectivity.

2.3. SYNCHRONIZATION

The synchronization of a pulsed laser train with synchrotron-radiation pulses is required when the lifetime of the excited species is shorter than the interpulse period of the photon source (see Fig. 3). With SR sources operating in a single-bunch mode, this limit amounts to about 100-1000 ns (Table 3). The synchronization offers several advantages over cw operation, especially for short-lived excited states. The peak power obtainable with pulsed lasers is much higher (about 200 times higher for a picosecond mode-locked laser) than the "useful" peak power resulting from the sampling of a cw laser by the pulse train of synchrotron radiation. Consequently, the pulsed-laser power is more efficiently used than in the cw case because the cw laser flux is useless between two consecutive synchrotron pulses.

Another advantage of such synchronization is to dephase one photon pulse with respect to the other and to take advantage of the maximum interpulse period of SR to investigate real-time (nanosecond or subnanosecond) phenomena (see Fig. 3). Finally, the large difference in repetition rate can be compensated by the extremely large energy per pulse attainable by the laser. A very high density of excited states can be produced.

However, the synchronization of SR with a laser is not an obvious task when one compares the repetition rates and the pulse widths (see Table 2). This synchronization has been achieved to date in very few cases. Mitani, 1988, has synchronized a pulsed dye laser with a UV SR beam from the UVSOR machine. Mills, 1991, has synchronized a frequency-doubled Nd:YAG laser at 530 nm, 20 Hz, with an x-ray SR beam at CHESS (7.08-7.2 keY).

Generally, synchronization requires that the following difficulties be overcome:

• The laser should oscillate at the same frequency as the SR or at a multiple of that frequency. Thus, the cavity length must be adjusted and/or the mode-locker crystal selected for the right frequency.

• The phase between both sources should be adjusted. One could use a single reference clock provided by the radio frequency of the storage ring. Alternatively, the phase matching could be done in real time through a feedback procedure.

137

absorbs only in the near-UV. Only large molecules and very specific medium ones absorb in the visible. The high mean power of the laser provides a high density of photoexcited molecules, and the density of fragments is controlled only by the quantum yield for dissociation.

The advantage of SR is the possibility of ionization at different, well-selected wavelengths so that the fragments can be identified with maximum efficiency. The arrangement chosen at Super ACO by Nahon et al., 1992b, is shown in Fig. 4. One of the difficulties is aligning the laser with synchrotron radiation in a vacuum vessel at the area of maximum density in an effusive jet of gas. When the alignment is made in a quasilinear arrangement, a small angle is desirable to avoid damaging the refocussing mirror for the SR.

The interest in the pump (SR) - probe (laser) is much greater because the infinite range of wavelengths reached by SR does not limit the systems to be studied. The limitations again come from the laser (Le., the types of fragmel}ts that can be investigated).

2.2.2. Detection Methods. The choice of the detection technique is guided by its sensitivity and its selectivity. When the fragments are photoionized, the detection technique can be photoelectron spectroscopy or mass spectrometry.

Photoelectron spectroscopy with electrostatic analyzers allows the determination of the signature of the products, as well known in chemielectron spectroscopy (Cockett et al., 1990; Bock and Dammel, 1987) and the estimation, in some favorable cases, of the internal (vibrational) energy of the fragments (e.g., when the bands of the parent molecule are negligible and the fragment bands are well separated). On the other hand, photoelectron spectroscopy is not a sensitive method because transmission does not usually exceed 10-3• Additionally, it is not selective (Le., in the common case where the dissociation yield is not unity, there is sometimes an overlap between the parent and fragment spectra that makes data analysis difficult). In contrast, threshold electron spectroscopy is a very sensitive technique, especially well suited for a tunable photon source in the VUV. It provides the spectroscopy of monocations. When associated with time-of-flight mass spectrometry, coincidences between threshold electrons and fragment ions provide the fragmentation pattern of state-selected ions.

Time-of-flight mass spectrometry is a sensitive detection technique because of its high transmission (0.1 % or better). It is also selective since it provides the mass and the charge of the

VUV synchrotron light

Interaction zone

Electron analyzer

Oven

Figure 4. Schematic of the experimental set-up used by Nahon et al., 1992b.

139

• The SR pulse width varies with the current. In other words, when the current decreases, the bunch length decreases. Thus, it is necessary to have a reliable system to control the synchronization.

• The stability of the interpulse period of the SR pulse train is critical. Typically, the jitter should not exceed a few tenths of the pulse width.

• If the frequency of the laser is a multiple of the SR frequency (because of the limited length of the laser cavity), one may select some pulses of the laser train and operate the laser in a Q­switch mode.

3. Dynamics of Molecular Photodissociation

3.1 SMALL MOLECULES (OR MOLECULAR IONS)

According to the scheme of Fig. 2, a molecule undergoes dissociation in a very large range of photon energy. Notice that when ionization is allowed, dissociation generally occurs after ejection of the electron(s). There are exceptions, especially when autoionization of the molecule becomes slower than direct photodissociation (see, for example, Morin and Nenner, 1986; Cafolla et al., 1989), but this aspect remains beyond the scope of this paper.

The simplest case of photodissociation is a direct process (see also Fig. 5a) with a photon energy hv below the first ionization threshold:

AB + hv -+ A + B .

The aim is the identification of A and B and the partitioning of energy between the translational energy and the internal energy of the fragments, keeping in mind that by conservation of energy, we have

Eavail == Eint (A) + Eint (B) + Euansl .

A further question concerns the nature of the internal energy: Is it electronic, vibrational, or rotational? A slightly more complex case is predissociation (see Fig. 5b). The process can be separated into the initial stage of photoexcitation of the molecule into a "prepared" state and its subsequent evolution into dissociation:

AB + hv -+ AB* -+ A + B

Here the predissociation process is delayed compared to the direct one. The questions to ask concern the nature and the lifetime of the AB* state. Depending on the coupling scheme, one classifies predissociation into electronic (Herzberg's type I), vibrational (type II), or rotational (type III). In type I predissociation, the intermediate AB* and the final A + B belong to different Born-Oppenheimer electronic configurations. In type II predissociation (see Fig. 6a), the intermediate and final states have the same electronic configuration, but they differ in the amount of energy in vibrational degrees of freedom other than the dissociative one. Therefore, this process can occur only in polyatomic systems. In type III predissociation (see Fig. 6b), the

140

(a) v

- - - - -1I.->.'dtNIIWIftNI\MII\MM-

I I

- - - - - - +'\-'''<:A.-:A:I~-:A:I~

I - - - - -!. +/\-->,;~~~r-

: I I I I I

E~ I g)

R

(b)

R

Figure 5. (a) Schematic of direct photodissociation; (b) schematic of electronic predissociation (type J). Reproduced by courtesy of Beswick, 1992.

(a) E

A B\C RBC

A +BC (v= 2)

A +BC (v= 1)

A +BC (v= 0)

(b)

E

R

Figure 6. (a) Schematics of vibrational predissociation (type II); (b) schematics of rotational predissociation (type III). Reproduced by courtesy of Beswick, 1992.

141

potential presents a centrifugal barrier resulting from the competition between the attraction of the potential and the repulsion of the centrifugal term J(J+l)I2JlR2. Quasi-stationary states AB* may exist above the dissociation threshold and decay by tunneling through the barrier. AB* states appear as sharp resonances in contrast to the rather smooth line shape observed for direct dissociation. Therefore, scalar properties, such as time dependence and conservation of energy, are the first to be established to obtain detailed "state-to-state" measurements. Examples of such properties are:

• Total photodissociation and photoabsorption cross sections • Appearance energies for each dissociation channel • Branching ratio into different channels • Lifetimes of photoexcited states • Translational energy distributions of fragments • Vibrational and rotational distributions of fragments.

Because the incident light, by its polarization and direction of propagation, introduces a preferential axis in space, one expects a strong correlation between the electric vector of the light, the electric dipole moment, the internuclear axis, and the relative velocity of the fragments. The angular distribution of photofragments is generally anisotropic. Another consequence of the anisotropy of the photon absorption is that an alignment or orientation is generated in the electronic and rotational angular momentum of the photofragments. Therefore, the following vector properties are interesting to measure:

• Angular distributions of fragments • Alignment and orientation • Angular momenta correlations • Polarization of fluorescence.

More details, including the theory of photo fragmentation in small molecules, can be found in Freed and Band, 1977; Beswick and Durup, 1979; and Beswick, 1992.

3.2. LARGE MOLECULES

For large polyatomic systems, the density of states is very high, typically 5-10 states per cm-l ,

and it is impossible to distinguish between type I or type II predissociation (see Figs. 5 and 6). The problem is generally solved within the statistical theory. The statistical theory known as the RRKM (after Rice, Rampsberger, Kassel, and Marcus) or the quasi-equilibrium theory (QET) supposes that the dissociation proceeds on a single potential-energy surface with the activation of all vibrational modes. In other words, the electronic energy is readily converted into vibrational energy of the ground state. The diSSOciating molecule is taken as a microcanonical ensemble in terms of the density of internal energy states. The fragmentation rate and the product energy distribution depend first on the ground-state molecular frequencies and the total energy E. They depend also on the activation energy Eo, which corresponds to a saddle point in the potential surface and a point of no return along the reaction coordinate (see Fig. 7): the transition state. Depending on the shape of the potential surface, the transition state may be "loose" or "tight" (see, for example, Stolow et ai., 1992). The loose case refers to a simple bond breaking, and the tight case refers to one in which two (or more) bonds break and new bonds are formed. Notice that the available energy for the fragments near threshold is small for a loose complex and large

142

(a) (b)

A+B

Reaction coordinate

Figure 7. Dissociative potential surface of a large molecule for a loose (a) and tight (b) transition state; A and B may be complex molecular assemblies.

for a tight one. Probing the transition state (geometries, frequencies, density of states) is a challenging question (Lovejoy et aI., 1992).

Let us consider a population of photoexcited molecules A * with internal energy E that undergoes a fragmentation process. One defines the fragmentation rate k(E) as

-d[A *(E)]/dt = k(E)A *(E) .

In the statistical theory, the rate k(E) for a system with s oscillators, a total energy E, and an activation energy Eo, is given by

kTIE-EoI S-

1 k(E)=---

h E (2)

This is a classical expression, but it is easily derived assuming s quantized harmonic oscillators. In this theory, one introduces different vibrational frequencies between the structure of the transition state and the molecule. The useful expression of k(E) is then

k(E)=crG*(E-EO) , hN(E)

(3)

where G* represents the sum of states from 0 to E - Eo of the transition state, N(E) is the density of states of the molecule, and cr represents the number of equivalent ways the molecule can dissociate. The statistical theory can be extended to a polyatomic system for which the ground­state potential curve shows two or more wells, each of them associated with a different isomer of the molecule (see Fig. 8). Isomerization may precede dissociation or be in competition with it. Consequently, in polyatomics, the goal is to measure the photon-energy dependence of the

143

A+B

Reaction coordinate

Figure 8. Different cases of dissociative potential surfaces in a large molecule, where A and B represent polyatomic radicals.

dissociation rates and the main features of the ground-state potential of the molecule: potential wells and barriers along isomerization and fragmentation coordinates, and the partition of the available energy into the translational and internal energy of the fragments. Furthermore, the number of dissociation pathways increases with the size of the molecule. Finally, the vector properties listed above, in the case of small molecules, are also important, especially when the molecule's behavior departs from statistics. .

As a result, photo fragmentation is a complex process. The elementary fragmentation event always occurs on the time scale of a vibrational period. However, the fragmentation rate is often longer because it is not a direct process. The complexity is also due to the difficulty of separating molecules into small and large species.

4. Photodissociation of Halogen Molecules

Photodissociation of the 12 and Br2 molecules are well documented cases. The iodine molecule is known to absorb light in the visible, as shown by the photo absorption spectrum of Calvert and Pitts, 1966, reported in Fig. 9. The shape of the photo absorption spectrum is easily explained by the potential curves of the molecule, shown in Fig. to. When the molecule is excited just below the second dissociation limit, it is photoexcited into the bound B3n state. This state is known to fluoresce (Broyer, 1973) and is predissociated by lnu. Dissociation produces ground­state iodine atoms. The quantum yield for dissociation depends on the vibronic state. For the S14-nm laser exciting wavelength, it amounts to 0.93 (Brewer and Tellinghuisen, 1972). When the molecule is photoexcited at a shorter wavelength, one reaches the continuum of the bound B3n state correlated to two iodine atoms, one in the ground state and another in the metastable excited state. This process competes with the direct photodissociation through the repulsive 1 nu state correlated with two ground-state iodine atoms. The competition is in favor of the direct

144

600

400 E

200

o~--~----~~~----~--~~--~~~ 1000 4000 5000 6000

Wavelength (A)

Figure 9. Photoabsorption spectrum of 12 (Calvert and Pitts, 1966). The arrow corresponds to the dissociation limit of the B3rr state.

process (Oldman, 1971). Therefore, below 499.5 nm, fragment iodine atoms are essentially found in their ground state. One can readily see the value of photodissociation for producing spectroscopically pure open-shell atoms.

Photodissociation of the bromine molecule is similar to that of iodine. The photoabsorption spectrum of Br2 is shifted towards shorter wavelengths (Fig. 11) because the molecule is made of lower-Z elements. The smaller value of the cross section at maximum compared to 12 is also due to the Z effect. Here the dissociation limit of the B3n state lies at 510.8 nm. According to the potential curves of Fig. 12, the two dissociation limits are close to each other and photoexcitation with 514 and 488 nm (the principal wavelengths of the laser) leads to transitions into the A and B states. According to previous studies (Smedley et al., 1987), the dissociation quantum yield is close to unity and the fluorescence yield is negligible. The role of the repulsive state in the predissociation of the B state is not as strong as for the iodine molecule. Consequently, when exciting the continuum of the B state, one expects that the photoexcitation will lead more efficiently to the population of the continuum associated with the second dissociation limit than in the iodine case.

4.1. PHOTOIONIZATION OF ATOMIC IODINE, I 5s2 5p5 2P312

We have seen that the use of a visible laser to photoexcite the iodine molecule is an efficient way to produce dissociation into two iodine atoms in their ground state. Combining such a laser in the cw mode and the 21.21-eV radiation from a helium lamp, using the arrangement of Fig. 4, and detecting the photoelectron spectrum led to the results shown in Fig. 13. With the laser off, the valence photoelectron spectrum of molecular iodine is obtained. When the laser is turned on, providing a power of 6 W, new lines appear because of the photoionization of atomic iodine. With an 8-W laser, the molecular lines disappear. These results demonstrate that such a two-color experiment allows the investigation of the electronic properties of a very reactive species under "clean" conditions, Le., with more than 95% of the iodine atoms in the interaction zone.

In the following, we show measurements performed upon excitation of the 4d core level of iodine in order to point out open-shell effects, absent in the photoionization of closed-shell systems such as xenon, the rare gas following iodine in the periodic table. In Fig. 14, we present

3

2

eV

1

o

2 4

A

2p 1/2 + 2p 3/2

Figure 10. Potential curves of the iodine molecule (Brewer and Tellinghuisen, 1972).

100 E

O~~~~~~ __ ~-L __ ~~ 2000 3000 4000 5000

Wavelength (A)

Figure 11. Photoabsorption speclnUn of Br2 (Calvert and Pitts, 1966).

145

146

u 4

eV 3

2

1

o

2

\ \ 1n, (Iu) \ \ \ \ \ \ \ \ ,

"

3 4

A

2p 1/2 + 2p 3/2 (2 Br)

2p 312 + 2p 3/2 (2 Br)

5

Figure 12. Potential curves of the bromine molecule (Brewer and Tellinghuisen, 1972).

the total ion-yield spectrum of atomic iodine, recorded in the energy region 40-140 eV (Nahon et al., 1990b). 1bis spectrum presents three domains ofinterest. The first one, located around 46 eV, shows the two strong resonances corresponding to the transitions into the valence 5p orbital, I 4d1O 5s2 5pS (2P3/2) ~ 1* 4d9 5s2 5p6 (2DSI2, 2D312) which are, of course, absent in the photo­absorption spectrum of xenon. The second one corresponds to transitions into the Rydberg np (n > 5) orbitals I 4d1O 5s2 5p5 (2P312) ~. 1* 4d9 5s2 5p5 np. The third one is asSigned to a giant shape resonance in the 4d ~ d channel.

We wish to focus here on the relaxation of the 4d ~ 5p (2DSI2) resonance. Two photoelectron spectra have been recorded on and off this resonance at 45.5 and 46.2 eV, respectively (Nahon et ai., 1990b), and are reported in Fig. 15. In the first part of the spectrum (l0-15-eV binding energy), one can clearly assign the lines to the different components of the 5p4 multiplet. The most striking feature is that these lines are strongly enhanced on the resonance, especially in the singlet channels, indicating that autoionization into outer-shell channels is a main decay process of the excited state. Note that in xenon photoexcited in the 4d ~ 6p resonance, the dominant relaxation channel is the resonant Auger process, in which the promoted electron remains as a spectator or is involved in a shake process (Southworth et al., 1983; Becker et al., 1986). We thus conclude that the decay mechanisms of core excitations are strongly dependent on the nature (Rydberg or valence) of the orbital into which the core electron is promoted.

147

(a) 2ng1/2 I Laser off I 2n '2n 2 2+

g3l2 u3l2 nu1/2 l:g

12+ lines

. '.'

I Laser 6 wi

(c) I Laser 8 wi 3PO.1 1D2 180

1+ lines

8 10

Binding energy (eV)

Figure 13. Photoelectron spectra of iodine, recorded with 21.21-eV radiation under laser excitation: (a) laser off; (b) laser on, 6 W; (c) laser on, 8 W.

148

~ "iii c: Q) -c:

!Valence I 4d -? 5p

!Rydberg!

40 60

! Continuum I 4d -? Ef

80 100

Photon energy (eV)

Figure 14. Total ion-yield spectrum of atomic iodine (Nenner et at., 1991b).

On resonance

o ®

>. -'iii c: Q) -c:

5 10 15 25 Binding energy (eV)

120 140

40

Figure 15. Photoelectron spectra of atomic iodine recorded on (46.2 eV) and off (45.5 eV) resonance. Shaded area indicates the double-ionization limit (Nahon et al., 1991).

149

In the second part of the spectrum (20-30 eV, peaks labeled 1-10), the situation is more confusing, since in a one-electron picture, one would expect to observe only the few lines corresponding to the 5s-1 5ps (3p,1p) state. We thus deduce that the numerous lines spread over more than 8 eV are due to strong electronic correlations. Because these numerous satellite lines also appear in the "off-resonance spectrum," we conclude that they are also produced by autoionization, with a redistribution of the intensity of the parent lines into satellite lines principally by a final-state configuration interaction (FISCI) process.

Finally, the peak labeled 11 is assigned to the 5s-2 5p6 state, which is the only one produced by a resonant Auger process with a 5p spectator electron. This peak, located above the double­ionization threshold, can relax by a two-step autoionization process into the double-ionization continuum.

4.2. PHOTOIONIZATION OF ATOMIC BROMINE, Br 4s24p5 2P312

Nahon and Morin, 1992, using a visible laser to photodissociate the bromine molecule, obtained bromine atoms with a 75% efficiency. The percentage of metastable bromine was found low enough (5%) to consider the atomic fragments to be in their ground state. The total-yield spectrum of atomic bromine is shown in Fig. 16. One observes the strong resonance 3d ~ 4p, which does not exist in krypton. The Rydberg transitions 3d ~ np are present as in iodine, but there is no resonance in the continuum.

Let us analyze the electronic relaxation of the main resonance Br 3d9 4s2 4p6 (2DS/2, 2D3/2).

Two photoelectron spectra recorded on and off resonance, at 64.54 and 63.45 eV, are reported in Fig. 17. One observes qualitatively the same behavior as for the iodine case, i.e., a strong autoionization in the outer-valence ionization continua and numerous enhanced lines associated

!Valence!

3d~4p

! Rydberg!

3d~np

70 80

! Continuum I 3d ~Ef

Photon energy (eV)

Figure 16. Total ion yield of atomic bromine in the region of the 3d edge.

120

150

30 2 2ITg1/2

en 25 ITg3/2 . ;t::: C ::J

..c ... 20

~ 15 Z' ·iii 10 c Q)

C 5

o 16

en 14 :g 12 ::J

.e ~ >--·iii c Q) -

10 8 6

4

3p 2

3PO,1

I Laser off I

I Laser on I

c 2

O~:"'---'--::=--~--r----=~~~:==:=; 10 11 12 13 14 15 16

Binding energy (eV)

Figure 17. Photoelectron spectra of atomic bromine recorded on (64.54 eV) and off (63.45 eV) resonance.

with the ionization of the 4s electron. In contrast, the decay of core-excited krypton is quite different and resembles the xenon case. Therefore, the difference between halogen and its rare­gas neighbor stays constant and does not depend on Z.

Laser-induced dissociation appears to be a very powerful method for producing radicals. The behavior of these radicals is different from that observed in closed-shell species with regard to photoionization dynamics.

5. Photodissociation of Polyatomic Molecules

We wish to focus here on a specific aspect ofphotodissociation, three-body dissociation:

ABC+hv ~A+B+C,

in which A. B. and C are molecules or radicals. Generally. a three-body dissociation process is a sequential event, i.e., a stepwise two-body reaction, as schematically shown in Fig. 18. The

151

(a) (b)

ABC

Reaction coordinate

Figure 18. Schematics of a three-body fragmentation process: (a) sequential and (b) concerted.

criterion for a stepwise reaction (Strauss and Houston, 1990) is that the intermediate AB has a lifetime much longer than a rotational period (typically in the nanosecond range and longer). Therefore, one can define for each step a transition state (loose or tight). One can readily see from Fig. 18 that the activation energy for a three-body process is higher than that for a two-body process. In contrast, a concerted three-body mechanism occurs when the intermediate lives on the time scale of its rotational period (or shorter), Le., in the picosecond or sub-picosecond range. The original classification by Strauss and Houston of stepwise and concerted reactions can be reconsidered with a better time resolution. A stepwise reaction is recognized if the intermediate survives even for a period much less than one rotation, whereas concerted reactions require a lack of both rotation and large-amplitude bending vibrations so as to retain fixed angular relations among the fragment trajectories (Nenner and Eland, 1992). For concerted three-body reactions, one defines a Single transition state along the reaction coordinate (Fig. 18) with an activation barrier that can be sometimes lower than that for a two-body mechanism. Very few cases are established to date (see Leone, 1982; Baer, 1986; Strauss and Houston, 1990).

'S.1. DYNAMICS OF TIlE PROCESS

We take the photodissociation of s-tetrazine C2N4H2 into three body as the first example (Nahon et aI., 1992b):

N2 + 2 HCN.

This aromatic model system, isoelectronic with the benzene molecule, is known to fragment into a single channel. The question was to find out whether this process is a concerted three-body reaction in accordance with the recent photo fragment translational spectroscopy measurements of

152

Zhao et al., 1989, or a stepwise reaction as suspected by Glownia and Riley, 1980, who found the HCN fragments with a dual translational-energy distribution. Meanwhile, the ab initio calculations of Scheiner et al., 1986, showed that, indeed, the potential barrier for the three-body concerted reaction is the lowest,and they calculated the geometry of the tight transition state. A schematic of the potential curves is shown in Fig. 19. Nahon et al., 1992b, performed a two-color experiment of the 1 + 1 REMPI type to revise this problem by measuring the vibrational-energy content of the fragments.

Experimentally, a cw Ar+ laser was used to photodissociate the molecule in the gas phase (effusive jet), and VUV SR was used to photoionize the fragments. An electron analyzer provided the signature of the fragments with energy resolution good enough to resolve stretching modes of the electronic bands. We show in Fig. 20 the photoelectron spectrum of C2N4H2 with the laser on and off. The laser power (3 W) is set to obtain the maximum efficiency for dissociation. The VUV wavelength was chosen to ensure the maximum of sensitivity for photoionization of the fragments. The results of Fig. 20 show that, under laser irradiation, the molecule dissociates almost completely and the fragments are indeed nitrogen and hydrogen cyanide molecules. When running the spectrum at higher resolution, one detects hot vibrational bands. As seen in Fig. 21 for nascent nitrogen, the additional peak expanded in the insert corresponds to a transition from N2(X) (v = 1) to N2+ A (v = 0), which appears at a lower energy than the normal 0--0 transition lying at 16.7 eV. Given the Franck-Condon factors, one can extract the vibrational-energy distributions. Five percent of the nascent nitrogen is found in the v = 1 level. Similar results (not shown here) have been obtained for HCN, and hot vibrational bands are observed. Although the bending vibrational peaks are not resolved, we have been able to establish that nascent HCN is found essentially hot in the bending mode (26%) with almost no C-N stretching excitation. In order to correlate these vibrational energy distributions with the dynamical quantities of the process, namely the geometry of the initial molecule and the transition state, the potentials of N2 and HCN were used. The maxima of the distribution reflect the N-N, C-N distances and the HCN angle of the tight transition state, as calculated by Scheiner et al., 1986. In other words, the

53.8 (2.34)

Figure 19. Energy diagram of s-tetrazine. Energy values are given in keal/mol and in eV (in parentheses) (Nahon et al., 1992b).

153

s-Tetrazine:C2N4H2

I Laser off I (a) 40

30

20

Ul ~ 10 :::J

.e 0 ~ !!3 12 I Laseronl N; (X)

(b) c:: :::J 10 0 HCN+ (X,A) ()

8

6

4

2

0 8 10 12 14 16 18 20 22

Binding energy (eV)

Figure 20. Photoelectron spectrum of s-tetrazine recorded with 23-eV photon energy: (a) laser off and (b) laser on (3 W).

350

300 Nascent N2

50 -CIl 250 -·c :J

.e 200 ~ CIl 150 -c:: :J

40

rr 30 20

10

0+-.--.--.-..,......, . 16.2 16.4 16.6

0 100 ()

50

o .. -: •. : .. -. 15 16 17 18

Binding energy (eV)

Figure 21. Photoelectron spectrum of nascent nitrogen (X and A bands) recorded with 23-eV synchrotron radiation, 0.13-eV total energy resolution, and laser on (3 W) (Nahon et aI., 1992b).

154

perpendicular coordinates in the transition state, which do not participate in the dissociation itself, govern the vibrational excitation of the fragments. These results are in good agreement with the translational energy distribution of Zhao et al., 1989, which represents some 73% of the excess energy. In conclusion, these results support a concerted theee-body mechanism.

Notice that because the dissociation time is on the picosecond or sub-picosecond time scale, the time resolution accessible by SR is of no value for real-time measurements.

6. Future Trends

In this section, we sketch only a few possibilities based on the very rich situation afforded by the extremely broad tunability of SR (see Fig. 3). As indicated in Section 3.1, we do not describe experiments aimed to investigate the vectorial properties in a "state-to-state" fragmentation process. Moreover, we do not consider the specific problems of free-cluster fragmentation, which is in itself a new field, nor specific experiments aimed to understand the fragmentation of molecules adsorbed on surfaces. We do consider separately the problem of photodissociation of core-excited molecules, of molecular ions, and of photoexcited molecules.

6,1. PHOTODISSOCIATION OF CORE-EXCITED MOLECULES

6.1.1. Pump (SR) - Probe (visible or UV laser). Investigation of the fragmentation of core­excited molecules has been a subject of great interest in recent years (for example see Nenner et al., 1990), because tuning the photon energy to a resonance near a core edge enables one to excite a molecule near a given atomic site or along a specific bond, as long as there are no other chemically equivalent atomic species in the system. The idea is to analyze the specificity of the dissociation channels according to the original photoexcitation.

A great deal of effort has been made to analyze the abundance of ionic fragments, their momentum distribution, and the kinetic energy release (Morin et al., 1992). The concept of Coulomb explosion has often been rejected because other forces of a chemical nature dominate. This problem is far from being understood because little is known about the scalar or vectorial properties of the photodissociation processes, as listed in Section 3.1.

The photoexcited molecule relaxes first by the electronic relaxation of an inner-shell vacancy, and many electrons are ejected. Each of the doubly, triply, etc. charged ions is formed with a large internal energy distribution and dissociates efficiently into two or many fragments. The first difficulty is to identify the dissociation channels for a given charge state of the molecular ion. This is being solved by detecting an ion time-of-flight spectrum in coincidence with an Auger electron (Shigemasa et al., 1992). Such measurements also provide the kinetic energy released in each dissociation channel. However, in polyatomic systems, the fragments may be molecular radical species, and their internal-energy content should also be known. If the fragments are electronically excited, they may fluoresce spontaneously, and this fluorescence can be detected (Poliakoff et al., 1987). This case does not require a two-color experiment. In contrast, if the fragments are produced without electronic excitation, but with vibrational and rotational energy, one should use a laser-induced fluorescence technique. One can imagine different types of experiments, all based on pump (SR) - probe (laser) double excitation.

Such experiments can be conducted in the cw mode because the core-hole relaxation is a fast process (on the femtosecond time scale), and the dissociation processes are also fast (on the

155

picosecond time scale). The lifetime of the excited products is limited by their radiative lifetimes (see Fig. 3), and the cw mode is adequate for rovibrational excitation. For electronic excitation of the fragments, the synchronization mode may be preferable. Notice that a UV FEL, with its high repetition rate, is totally adequate.

A plain laser-induced fluorescence experiment associated with core excitation of the molecule will provide global information on the internal energy of selected fragments. However, this information will not be associated with a given charge state nor with the internal energy of the parent molecular ion. To reach this level of information, one must detect the fluorescence and the Auger electron energy. The feasibility of such experiments depends strongly on the number of photoexcited species because of the poor efficiency of detecting fluorescence and especially of electron analyzers. Thus, the initial pump (SR) intensity should be maximized by using an undulator.

6.1.2. Pump (SR) - Probe (IR laser or FIR SR). The molecular structure of fragments that are formed after the core-hole relaxation can be investigated by probing with an IR or far-infrared (FIR) laser. Such experiments have been performed by Andresen et al., 1984, 1985. Recently, Cohen et al., 1989, used a pump (excimer) - probe (FIR laser) arrangement that could be used with SR and an IR FEL because of the good match between the repetition rate of the two sources.

Synchrotron light is also an intense FIR source that surpasses conventional ones. Since a FIR laser is easily transported, one can imagine performing time-resolved Fourier transform studies to probe the nascent fragments. Such experiments performed by using a pulsed excimer laser as the pump (Hall et al., 1992) could be extended to similar ones based on the combination of normal VUV SR and FIR SR, keeping in mind that both sources have the same time structure.

6.2. PHOTODISSOCIATION OF MOLECULAR IONS

We consider the problem of fragmentation of ions separately from those of neutral species because the experimental techniques should include the detection of the ejected electron(s) in order to know the internal energy of the species under consideration.

6.2.1. Pump (SR) - Probe (visible or UV laser). First, we will consider singly charged ions that can be produced by one-photon ionization of the neutral molecule. The threshold ionization for presently considered molecules lies generally above 9 eV, and the region for ionizing the outer valence electrons (for example, 2p-like for molecules containing first-row elements) extends to about 20 eV. Above this, is the region of inner-valence ionization (for example, 2s-like). A normal photoelectron spectrum measured to 50-eV binding energy shows numerous lines (satellite lines) indicating a complete breakdown of the independent-particle model (Cederbaum et ai., 1986). Very little is known about the dissociation dynamics of these highly excited ions because they cannot be produced by laser sources. Although one suspects that most of them are dissociative states, observations in medium-size molecules show that some of these states can be bound or weakly predissociated because they are found to radiate (Aarts et al., 1987). It would be very interesting to investigate such states by photoionization with SR, using threshold electron­ion coincidence detection. A study of this type would provide information on the nature of ionic fragments and kinetic-energy release as a function of the internal energy of the monocation. However, it would be important to probe the neutral fragment(s) at the same time. A visible or near-UV laser (FEL) associated with SR in a pulsed mode with a proper delay between pulses, in

156

conjunction with threshold electron and laser-induced fluorescence detection, could be the appropriate method.

6.2.2. Pump (SR) - Probe (visible or IR laser). Doubly charged ions, which can be formed efficiently by one-VUV -photon excitation of the neutral molecule, are quite intriguing objects. Many of them have a bound ground state showing that chemical forces overcome the Coulomb forces that tend to produce two singly charged fragments. Moreover, the equilibrium geometry of such dications is significantly different from that of the original molecule. Bond distances and bond angles are different, and severe atomic rearrangement may occur (Lammertsma et at., 1989). It would be of great interest to investigate those wells by performing new kinds of "hole burning" experiments. Synchrotron radiation could be used to produce the dication; a hole could be burned by a visible laser; and an IR fEL could be used to analyze the resulting change in nuclear configuration.

6.3. PHOTOEXCITATION AND PHOTOIONIZATION OF EXCITED MOLECULES

1bis a very rich subject in itself because of the different types of possible excitation: vibrational or electronic. In the following, we consider only two-color experiments based on synchronization of a laser with SR, for which we intend to discover new phenomena that cannot be attacked with one-photon excitation. More ambitious developments can be imagined with higher-order multi­photon ionization like 1 + 1 + 1 REMPI. The third photon would be used to probe the fragmentation dynamics.

6.3.1. IR Laser with VUV SR. Infrared radiation makes it possible to excite vibrations and rotation if the frequency is tuned on the appropriate value. An IR fEL is the ideal instrument to achieve such excitation. One can excite a given rovibrational mode. VUV SR would make it possible to promote the molecule to a super-excited state. Because the ion would be produced in specific rotational and vibrational states that cannot be reached by one-photon ionization, one would expect new dynamical processes to occur. For example, one may observe new dissociation routes that could compete efficiently with ionization, keeping in mind that such competition only occurs in very specific cases when the molecule is vibrationally cold (Morin and Nenner, 1986; Cafolla et at., 1989).

Infrared radiation can be used to photoexcite a molecule in a very high vibrational state, even near the dissociation limit. Then VUV SR allows transitions from a species having large­amplitude motion. Therefore, new phenomena (Kligler et at., 1978; Zittel and Little, 1979) can be investigated because of this wide Franck-Condon zone. One can reach states for which the minimum corresponds to an equilibrium geometry quite different from the ground state. Ion-pair states (associated with the formation of A+ + B-) or doubly excited states are very interesting candidates because their dissociation dynamics are widely unknown and may reveal surprises. One also can increase the dissociation cross section by large factors.

6.3.2. Visible Laser with VUV SR. Visible light is absorbed only by polyatomic molecules or heavy diatomic ones. Small ones require UV radiation. The photoabsorption band corresponds to an electronic excitation into the various optically allowed states. By combining a visible laser with SR, one can reach states that are difficult to reach by one-photon excitation. Those are

157

doubly excit!!d states: satellites in core-level photo absorption spectra, satellites in valence and core-level photoelectron spectra, and ion-pair states seen in negative-fragment ion spectra.

6.3.3. UV Laser with VUVor Soft X-Ray SR. UV light is absorbed efficiently by many systems, especially hydrocarbons. The combination of a UV laser (the UV fEL) with VUV SR will allow all studies briefly mentioned in the preceding paragraph (6.3.2). Moreover, because this region corresponds to the fragmentation of the molecule, similar experiments can be used to photoionize the fragments.

6.4. CAP ABILmES AND LIMITS OF TIlIRD-GENERA TION SYNCHROTRON RADIATION SOURCES

Third-generation synchrotron radiation sources, which are built for higher brightness than second­generation sources, are certainly optimized for providing a higher average photon flux after monochromatization under high-resolution conditions. They have this advantage because of the small transverse dimension of their electron (or positron) beam and the use of undulators. However, the requirements of high brightness and a mean beam lifetime of several hours make it necessary to split the electron beam into many bunches. As a result, the photon flux per bunch decreases in comparison with second-generation machines. For this reason, it is desirable to operate third-generation machines in two different modes. One is the multi-bunch mode, which is adequate for cw two-color experiments. The other is the few-bunch mode, in which the machine is operated with a small number of bunches. This mode is used when synchronization is necessary and provides the capability of introducing a delay. Even so, the finite size of the electron bunch limits the temporal width of the SR source. As shown in Fig. 3, any phenomenon below 20 ps is totally out of reach when one of the two photons is SR Finally, two-color experiments in which an IR FEL is used require the length of the two machines to be matched properly to allow synchronization. In contrast, the combined use of a UV FEL with VUV SR does not require any special care because both sources are based on the same machine.

References

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ELECTRON CORRELATION IN IONIZATION AND RELATED COINCIDENCE TECHNIQUES

G. STEFANI Dipartimento di Fisica Universita' di Roma La Sapienza P. le A. Moro 2 1-00185 Roma, Italia

L. A V ALDI AND R. CAMILLONI Instituto di Metodologie Avanzate Inorganiche CNR Area della Ricerca di Roma CPlO 1-00016 MonterotoOOo, Italia

ABSTRACT. Correlation effects in bound states and in the continuum have been the subject of several recent investigations using both electrons and photons as probes. Single- and double-ionization coincidence experiments in atoms, molecules, and solids are reviewed with an emphasis on correlation effects. The relevance of angle-resolved, time-correlated spectroscopies is discussed, and experimental techniques relevant to coincidence experiments are presented. Emphasis is placed on new opportunities and problems created by the use of these techniques in conjunction with third-generation synchrotron radiation sources.

1. Electron-Collision Coincidence Experiments

1.1. ELECrRON-CORRELATION AND TIME-CORRELAlED EXPERIMENTS

Many of the spectroscopies currently used to investigate the electronic structure of atoms, molecules, and solids rely upon single-particle models for interpretation of the results. Beyond the hydrogen atom, the nature of the Coulomb potential limits the validity of this approximation. Indeed, even in the simplest case of two charged particles in a potential well, the infinite range of the Coulomb potential prevents them from being considered as truly independent, no matter how far apart they are. Furthermore, a satisfactory solution to the problem of describing n interacting charged particles doesn't yet exist, not even for n as small as 3. Hence comes the motivation for investigating correlations, defined as whatever is not accounted for by independent-particle approximations. Usually, radial electron-electron (e-e) correlations are described by self-consistent field (SCF) treatments, whereas angular e-e correlations require the use of wave functions fully dependent on the relative electron positions. In particular, two classes of e-e correlations are recognized to be relevant to electron spectroscopies: 1. Internal state correlations (ISC), which are present in the initial and/or in the final ionic

state of the target. They modify the structure of the wave functions and the properties predicted by independent-particle models.

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A.S. Schlachter and F.J. Wuillewnier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 161-188. © 1994 Kluwer Academic Publishers.

162

2. Dynamical correlations in the continuum final state (CFSC), which are switched on during the ionization process. They involve short- and long-range correlations of the free electrons with each other and with the bound target electrons.

It is not yet completely understood to what extent ISC can modify the single-particle description and provide a better understanding of the electronic properties of complex systems, for example, transition metals and their compounds. To improve on this situation, it is essential to devise spectroscopic methods that allow one to obtain information about the correlated nature of the ground state.

CFSC mostly affect the dynamicS of the break-up processes induced either by photons or by charged particles, in which three or more charged fragments are present in the continuum of the final state. Ionization is the most relevant process among break-up reactions. It was first studied in the early 1930s and since then has continuously attracted the interest of both experimentalists and theorists [1]. In particular, molecular photoionization

(1)

and electron-impact core ionization

(2)

are typical processes with several unbound charged particles in the final state. Current ionization experiments detect and analyze only one of the final particles of the reaction. In such experiments, correlation effects in the ionization cross sections are averaged out over the dynamical variables of the undetected particles. In recent years, coincidence experiments have been conducted in which the averaging effect has been completely or partially eliminated by measuring multiply differential cross sections. A schematic of such an experiment is given in Fig. 1. Briefly, the experiment consists of collecting time-coincident outgoing pairs of particles at selected scattering angles and kinetic energies. The probability distribution of the particle pairs is then measured versus the energy and/or the momentum balance of the reaction.

163

Figure 1. Schematic diagram showing the principle of a coincidence experiment in which two of the final reaction particles (D 1) and (D2) are analyzed in angle and energy and their detection is correlated in time.

1.2. ELECTRON SCATIERING AND PHOIDIONlZAlION

Among coincidence experiments, those involving electron impact were developed first, whereas those related to photoionization processes are limited by the low intensity of conventional photon sources until the new intense synchrotron radiation (SR) sources become available.

At first glance, it might appear surprising to treat electron-impact and photoionization processes on an equal footing. In fact, sharp similarities and interesting complementarity do exist between the two classes of experiments [2]. This is apparent from the quantum­mechanical description of the photoabsorption and of the angle-resolved electron-energy loss (AREEL). The photo absorption cross section is proportional to

(3)

where hv is the photon energy, e is the polarization, and If} and Ii) are the final and initial N-electron states, respectively. The AREEL cross section is written as [3]

(4)

where q is the momentum transfer of the collision, E is the energy loss, Eo is the incoming electron energy, and the summation is over the atomic electron coordinates. The q value is determined, for fixed E and Eo, by the scattering angle 81. Note that, in the follOwing, the index 1 will always be used for kinematic quantities related to the scattered electron and 2 for those related to the ejected one, although they are indistinguishable.

In the limit of vanishing momentum transfer, it can be shown that the two expressions, (3) and (4), become very similar. The following approximation holds [4]:

164

(tlexp(iq • r)li) '" (tliq. rli) . (5)

The same matrix element then characterizes both photon- and electron-induced ionizations, with the direction of q corresponding to the photon electric vector. This is the dipolar regime of the collision.

If the value of the momentum transfer increases, higher terms in the expansion of the exponential exp(iq. r) become non-negligible (for instance electric quadrupole and magnetic dipole terms). Thus, dipole-forbidden transitions can be excited, and electron-impact ionization can complement photoionization in the investigation of the electronic structure of matter.

For large enough momentum transfer (q2/2 '" E2), the similarity is restored because the electron-atom collision can be approximated with a two-body, electron-electron collision, as happens in Compton scattering (photon collisions with single electrons) [5]. This is the impulsive regime of the collision.

1.3. 1lIE (e, 2e) EXPERIMENTS

Among electron-impact ionization experiments, the coincidence type, usually abbreviated (e,2e), are able to completely determine the kinematics and to give evidence of correlation effects [6].

The (e, 2e) technique consists of measuring simultaneously the energy Eo of the incident electron, the energies Eland E2 of the two final electrons, and the probability of their being emitted into solid angles 01(810«1>1) and 02(82,«1>2). By chooSing the energy loss E and the momentum transfer q, a large variety of kinematics can be selected. This allows a choice of conditions for which correlation effects are enhanced or diminished [7].

Figure 2a shows (e,2e) kinematics. The coincidence spectrometer used for these experiments is usually a crossed-beam apparatus like the one sketched in Fig. 2b. It features a well-collimated beam of monochromatic electrons that crosses the target (an effusive gas jet in the figure). Two electron analyzers rotate independently around the scattering center and detect pairs of electrons coincident in time and carefully selected in energy and scattering angle. During the collision, the primary electron can exchange a continuum of energy and of momentum. In general, the measured coincident ionization cross section (IDCS) will contain a coherent superposition of different features. If the kinematics are appropriately chosen, the effects of either one of the aforementioned correlations can be studied. Namely, the dipolar regime is more appropriate for studying CFSC and the impulsive regime is preferred for ISC.

In first-order interaction models"the (e, 2e) IDCS reduces to

d3a a k k k -11,;12 An An de 1 2 0 '·1 • ~"1·~"2·

(6)

where kl. k2' and ko are the momenta of the unbound electrons. The factor IFI2 can be greatly simplified in the two limiting cases [8]:

165

Figure 2. (a) Kinematics of an (e, 2e) experiment. (b) Schematic view of an (e, 2e) spectrometer. The components illustrated include: (1) Faraday cup, (2) electron analyzers, (3) electron gun, (4) gaseous beam, and (5) independently rotatable turntables.

166

1. In the dipolar limit:

(7)

where rj indicates the ith atomic electron coordinate;

2. In the binary limit (impulse approximation):

(8)

where M is simply the free-electron scattering matrix element. In both cases the angular distribution of the TDCS is characterized by cylindrical

symmetry around q. Factorizing the cross section amounts to separating the dynamical terms from the ones due to the electronic structure of the target. As a consequence, when the dynamics of the process are known, (e, 2e) experiments are a useful spectroscopic tool for collecting information on the electronic structure of the target. Vice versa, when the collisional approximations of (7) and (8) are not valid, (e, 2e) on simple targets can be used to seek information on the ionization mechanism. In particular, upon validity of the impulse approximation and within the single-particle framework for the bound-electron wave furictions, the IFI2 factor is further simplified as follows:

(9)

where k = ko- k1 - k2. and the form factor l<Pn(k)12 is the squared Fourier transform of the single-particle electron wave function. The validity of the impulse approximation has been established for a variety of atomic orbitals. For example, Fig. 3 shows the binary form factor as measured on helium in coplanar symmetric geometry (El = E2, 9 1 = 92, <PI = <P2 = 0) and for various kinematics [9]. It is evident that the result does not depend on collision dynamics and is in full agreement with the Hartree-Fock SCF calculation (solid line in the figure). The impulsive (e, 2e) experiments are therefore a unique spectroscopy for measuring orbital­resolved spectral-momentum densities in atoms, molecules, and solids [10].

1.4. IN1ERNAL-STATECORRELATION

In the study of ISC, the structural aspects take precedence over the dynamical ones; therefore, high-energy impulsive (e, 2e) experiments are to be performed. It can be shown [8] that when ISC are present, the energy spectrum of the TDCS shows manifolds of transitions associated with each occupied orbital of the neutral target. Usually, the most intense transition (main peak) corresponds with the ground state of the ion, and the other transitions (satellite peaks) correspond with the excited states of the ion.

167

1.0

0.5

Figure 3. Form factor measured on helium in coplanar symmetric (e, 2e) kinematics [9]. The solid line is the He Is Hartree-Fock momentum distribution.

Whenever initial-state correlations are dominant, the angular distribution of the satellites is a combination of the various single-particle hole-state configurations contributing to the excited ion state. Namely, the angular distribution of the main peak differs from those of the satellite peaks. When final-state correlations are dominant, all peaks belonging to the same manifold are characterized by identical angular distribution. Therefore, (e, 2e) spectroscopy is a unique way to disentangle initial- and final-state correlation effects. In the case of He, only initial-state correlations are present, and in the energy-separation spectrum, the n = 2 and n = 3 satellites have been observed [10] in addition to the main peak (ion left in the n = 1 state). The aforementioned difference between the angular distributions of the main and satellite peaks was indeed observed. While the Hartree-Fock description of the single-particle, one-hole Is-1 is adequate to describe the n = 1 transition, correlated wave functions must be used to describe the TDCS for n = 2 and n = 3. In particular, the n = 2 IDCS is capable of discriminating between different He correlated wave functions [10]. This is a clear example of how a coincidence experiment emphasizes characteristics that are hidden in an integrated cross section.

1.5. DYNAMICAL CORRELATION

CFSC have been extensively studied in atomic hydrogen with dipolar (e, 2e) experiments. In this case, ISC are absent. A comprehensive review on this subject was written by Joachain [11 ].

On the other hand, helium is the simplest target for displaying both types of correlations, and it will be used as a demonstration case. Stefani et al. [12] have recently investigated the n = 2 satellite, when the electron is left in the He ion in the degenerate 2s12p state. The chosen collisional regime approaches the dipolar limit, and the region of energy transfer investigated is free from resonances, i.e., only direct ionization processes are energy-allowed. The angular dependence of the n = 2 satellite (64.5-eV transition energy) and the n = 1 main peak

168

(24.6 eV) have been measured in asymmetric kinematics at two incident energies (570 and 1500 eV) and at small momentum-transfer values. As an example, the absolute TDCS mea­sured at 570-eV scattered-electron energy and 40-eV ejected-electron energy for the main and satellite transitions are reported in Fig. 4. All the measured IDCS are described by two lobes, the binary and the recoil lobe, which are roughly parallel and antiparallel to the mo­mentum-transfer direction. Comparison of the results obtained for the transition to the n = I and to the n = 2, shows that a more intense recoil lobe is present in the n = 2 IDCS. Furthermore, while the angular distribution for the n = I state is quite symmetric around the q direction, the binary and recoil lobes for the n = 2 state deviate from the q symmetry. These differences are very noticeable, because the two cross sections were measured for similar kinematics, i.e., similar collisional regimes. In order to interpret these results, one must bear in mind that the prevalence of the binary lobe over the recoil lobe usually observed in the TDCS is due to the coherent superposition of many partial waves in the expansion of the Coulomb operator and is peculiar to a binary collision. In this case, initial-state correlations alone are not sufficient to explain the observed differences between the main and satellite peaks, as is the case for fully symmetric experiments [10]. The absence of cylindrical sym­metry and an intense recoil lobe for the satellite peak are evidence for many-body collisions, with a large fraction of q transferred to the residual ion. Therefore, to account for correla­tions in the continuum, it is crucial to describe the ionization process correctly.

The many-body character of He ionization is also clearly demonstrated in photoionization experiments. For instance, a recent investigation at very high energy resolution [13] has shown that the He II ionization spectrum converges with a series of Rydberg states (two­electron processes) to the double-ionization threshold. In this region of the He excitation function, autionizing processes are possible in addition to direct ionization. Some (e, 2e) experiments have also been conducted in this region to study events in which direct and autoionizing channels are open at the same time. The angular distributions of the IDCS measured by these experiments [14] have shown very detailed features of the interference between continuous and resonant channels that demand to be explained by the current theories.

Evidence for interference between direct and resonant ionization has been observed in the core and valence IDCS of molecules whenever the energy transferred in the collision coincides with a shape-resonance excitation energy [15]. Xenon is certainly a good candidate for the study of this phenomenon in atoms. Photoionization experiments [16] have shown that coupling between the valence and 4d electrons exists at the excitation energies where the 4d shape resonance occurs, and it persists well beyond the shape resonance region. The TDCS for ionization of the Xe 4d and 5p orbitals has been measured in coplanar asymmetric kinematics. The scattered electron energy was held at 1000 eV, while the ejected electron energy was changed so as to select an energy loss value in or out of the region characterized by the giant resonance in the 4d~£f channel. The interference between direct and resonant processes was studied by varying q at each ejected-electron energy. In Fig. 5, the Xe 5p IDCS as measured for E2 = 80 eV [17] are reported for different values of momentum transfer q. At the smallest momentum transfer, i.e., in the dipolar regime, a strong interchannel coupling between the 5p and the 4d shells was observed for the first time in an (e, 2e) experiment. The coupling amounts to virtual excitation of the 4d orbital that leads to 5p ionization. It disappears when the momentum transfer is large, i.e., in the binary regime, where the measured IDCS is mostly determined by the single-particle properties of the 5p orbital [see Eq. (9)].

0.7

0.6

0.5

0.4

0.3

0.2

0.1

," t \

l ~ { \

# ~ {

# {

# {

#

(a)

0.0 4-,.."T"T" ......... r-r-lr-T""T~~~M-,r-r-r""T""T"T"'I"-r-r-rT!~~:,...,

C\I

b 0.0045 ,.... -en 0.0040 o ~ 0.0035

0.0030

0.0025

0.0020

0.0015

0.0010

0.0005

o 50 100

•• • • •

150 200 250 300 350

(b)

•

•• • • • ••• •

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o 50 100 150 200

92 (deg) 250 300 350

169

Figure 4. The (e, 2e) TDCS plotted against the ejected-electron angle for the ionization of He, without (a) and with (b) excitation of the residual ion to the state n = 2. The energy of the scattered electron was 570 eV; the energy of the ejected electron was 40 eV; and the scattering angle was 4°. The dashed line is a plane-wave Born approximation, and the solid line is a first Born calculation using ground-state and continumn-correlated wave functions [12].

170

0.7 r---------:------, 0.325 r--~,;-----------~

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o 90 180 270 360 o 90 180 270 360

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o 90 180 270 360 0 90 180 270 360

92 (deg)

Figure 5. Relative IDeS of Xe 5p at El = 1000 eV, E2 = 80 eV and (a) q = 0.45 a.u., (b) q = 0.9 au., (c) q = 1.9 au., (d) q = 2.4 a.u. The solid and dashed lines are distorted-wave Born calculations, with the scattered-electron wave function calculated in the target or in the ion potential, respectively [17].

171

1.6. INNER-SHEIL IONIZATION

When ionization involves an inner or intermediate orbital, the primary ion eventually decays by photon or Auger-electron emission. In the latter case, three free electrons are present in the field of the residual ion in the final state of the reaction, and correlations ~ong the secondary electrons should largely influence the behavior of the TDCS. Experiments in which an Auger electron is detected in coincidence with a primary electron that has suffered a selected energy loss and momentum transfer were proposed as a tool to investigate CFSC among secondary electrons whenever the energy transfer is close to the ionization threshold [18]. In passing, it should be mentioned that similar experiments, hereafter termed (e, e' Auger), were also proposed by Berezhko et al. [19] for studying the alignment of the primary ion.

The first successful attempt to carry out (e, e' Auger) experiments was reported by Sewell and Crowe [20], and in the following years, two other groups succeeded in similar experiments [21, 22]. Even though different energies and kinematics were used, all the measurements were related to the L3M23M23(lSO) Auger transition of argon. The energy loss and momentum transfer were comparable, thus constituting a homogeneous body of measurements. The simplest first-order interaction model that can be used in interpreting these experiments without taking into account final-state correlations is the first Born two-step model (IB2S). The basic assumptions are: (1) complete independence of the Auger relaxation process from the ionizing collision (two-step model), and (2) validity of the first Born approximation in describing the primary ionization collision. Investigations of the (e, e' Auger) type have clearly shown that the IB2S model is inadequate in several kinematics. The continuum correlations between the Auger and the slow emitted electron are responsible for the energy shift and the profile distortion of the Auger line observed by the (e, e' Auger) experiments at a few eV above threshold and an incident energy of 1 KeV [20, 23].

A similar energy shift of the whole LMM Ar Auger spectrum was found by (e, e' Auger) experiments performed at 8-KeV incident energy, with an energy loss that is 7 eV (excess energy) larger than the L ionization threshold [22] and a scattering angle of 1.5°. These kinematics were chosen in order to approach as much as possible the conditions for validity of the first Born approximation, i.e., the dipolar limit. An energy shift of roughly 0.15 eV toward higher energies was detected, which agrees with several predictions [18, 2~].

Nevertheless, the energy resolution was not sufficient for determining the Auger line shape, which has been shown to bear further traces of final-state correlations, as well as information about the interference of the Auger decay channel with direct double ionization [21].

Better energy resolution was achieved in a recent (e, e' Auger) experiment on the Xe N4,SOO spectra at l-KeV incident energy [22]. In Fig. 6, two coincident Auger spectra are reported, together with the noncoincident ones, for an excess energy of 20 eV (a) and right at the Ns threshold (b). The threshold spectrum displays a large energy shift of the NsOO component, whereas the N400 component is absent. Furthermore, at threshold, the line shape of the well-resolved NsOO (ISo) transition (= 30 eV) is well described by semiclassical models that take into account CFSC. The observed correlation effects among secondary electrons, which have been shown to be relevant for kinematics with low excess energy [23], constitute a violation of assumption (1), thus preventing the interpretation of the coincident Auger angular distributions in terms of the simple IB2S approximation.

172

730

600

470

340

210

30.4 32.3 34.2 36.1 38

~ Energy (eV)

I/) c ~ 420~--------------,l-N-50-0-----(b-')

I II ~ '-( ----i-:---:;II---;' N400

280

140

OL-~~L-~~~~~~~~

27 30 33

Energy (eV)

36 39

Figure 6. Xe N4,5 (e, e' Auger) spectra measured at an energy loss of 20 eV above threshold (a) and at the N5 threshold (b). The abscissa is the kinetic energy of the slower electron. The solid line is the noncoincident Auger spectrum as recorded during the coincidence measurement. The incident energy of the electron beam was 1 KeV [22].

173

Coincident Auger angular distributions were measured for several coplanar and non­coplanar kinematics. Some results agree with IB2S predictions [21], while others exhibit a substantial violation of the expected symmetry around the direction of the momentum transfer [20, 26]. This work, however, does not allow for assessing whether a more refined description of the unbound electron wave functions (e.g., the distorted-wave Born approximation [27]) would improve the agreement between theory and experiment or whether the Born approximation itself is to be questioned. To gain a better understanding of this problem, angular correlations have been separately measured [23] for the two coincident pairs, (e, 2e) and (e, e' Auger), in the same process (Le., Ar 2p ionization), leading to L3M23M23(lSo) Auger relaxation. The kinematics chosen for measuring these coincident angular distributions are the same as those used in measuring the coincident Auger spectrum, Le., high incident energy, small momentum transfer, and two different excess energies (7 eV and 60 eV). Two different energies were chosen in order to show final-state correlation effects. Contrary to the IB2S theory, the Auger angular distribution is not symmetric around the direction of the momentum transfer. Moreover, not even a common symmetry axis exists for the angular distributions in the two opposite half-planes. These results have been interpreted in terms of final-state correlation effects as has the shift of the coincident energy spectrum observed under identical kinematics. Also the TDCS is not symmetric around the q direction, and, moreover, the recoil lobe is larger than the binary one. These features, already observed in several other (e, 2e) experiments, provide evidence for second-order interactions in the ionizing collision and correlations among the final continuum electrons and the residual ion.

The analogous experiments performed at higher ejected energy (60 eV) show an unquestionable symmetry around the q direction and, in the (e, e' Auger) experiment, a fairly good agreement with the 1 B2S predictions. Thus, a correct interpretation of the experiments at lower excess energy requires the introduction of higher-order terms in the description of the ionizing collision as well as correlation of the secondary electrons.

2. Photoionization Coincidence Experiments

Pairs of unbound electrons in the final state are also generated in double-photoionization events (DPI). The angular and energy distributions of the time-correlated pairs of final charged particles will provide information on ISC and CFSC. Over the past years, quite a few coincidence experiments have been performed at photon energies ranging fr~m threshold to a few eV above threshold. In the following paragraphs, a few selected examples are given. They are meant to provide a summary of the obtainable physical information but do not extensively cover the subject.

The general scheme for a DPI process is

(10)

Most of the experiments performed to date detect only selected pairs of final particles, i.e., electron-electron, electron-ion, or ion-ion coincidences. The examples presented in the following sections are grouped on the basis of photon energy and of the types of detected­particle pairs.

174

2.1. 1lIRESHOlD OOUBLE PHOTOIONIZATION

Performing DPI experiments close to threshold gives the advantage of a very large collection efficiency. This advantage has been recently exploited by the apparatus shown in Fig. 7. This was designed as a versatile electron-ion coincidence spectrometer for photoionization studies [28] and was recently modified to perform electron-electron coincidences. The apparatus is based on the idea of selectively collecting low-energy charged particles by means of an electrostatic field penetrating into the interaction region [29].

Photoelectron-photoion coincidence experiments (PEPICO) have been performed on a variety of noble gases and molecules by a number of researchers using time-of-flight (TOF) andlor electrostatic energy dispersive spectrometers [30]. In Fig. 8a, a recent PEPICO result on argon is shown [31]. The experiment was performed with a photon resolution of 60-100 meV and involved the collection of photoelectrons in the energy window 0-20 meV. The curve (a) is the threshold electron spectrum, while (b) and (c) show, respectively, the yield of doubly and singly charged ions coincident with threshold electrons as a function of the photon energy. From the threshold-PEPICO spectrum, it is evident that as soon as the double-ionization 3p channel is open (hv ~ 43.5 eV), virtually all of the Ar+ excited states are bound to decay to Ar++. From the curve (b) a cusp-like feature at the ID threshold is observed as well. The interplay between direct and resonant DPI channels is also noteworthy. The direct channel is responsible for the smooth continuum, on top of which the contribution of the resonant channels appears with sharp peaks, particularly in the energy region between the 3p and ID DPI thresholds.

Threshold photoelectron-photoelectron coincidences (TPEsCO) in molecules are a unique tool for measuring the onset energies and vibrational structures of doubly charged ions. An example of such spectroscopy on CO++ is shown in Fig. 8b, in which the yield of zero­kinetic-energy electron pairs is plotted versus the photon energy. The two manifolds of peaks (s and p in the figure) are interpreted as vibrationally excited states of the CO++ lI.+ and X31t states, respectively [32]. This experiment represents the first application of TPEsCO.

Photon beam TOF axis D fl t 1270 cylindrical

Gas e ec ors deflection

CEM ~I r aoa".' II ~ ~ ~r I nI rotn aoalyze'

particle II 1/~I-l~LJ~lrnIU detector 7 )screenin9T 'Lens' 1\

Electrostatic lens electrodes \j Deflector CO extractor ~

TOF ~ 'R~ extractor Rotation about CEM ~

photon beam axis particle detector

Figure 7. Schematic diagram of a coincidence spectrometer for double photoionization experiments [31].

175

2

1

0

0

1

0 In

41.5 -c: 43.5 45.5 47.5 49.5 ::::l 0 U

1200

800

400

! p S,l p

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co++

(b)

tf+fH""'+'H~~ O~~~~J-~~~~~~~~~~~~~~~~~

40.5 41.5 42.5 43.5

Photon energy (eV)

Figure 8. (a) Threshold PEPICO spectra of doubly (ii) and singly (iii) charged Ar; (i) is a noncoincident photoelectron spectrum. Arrows indicate double-ionization potentials. Total dwell time was .. 10 min/channel for all three spectra [31]. (b) TPEsCO spectrum for doubly charged CO. Total dwell time was 7 min/channel. Letters ''p'' and "s" indicate possible series of 1t and L states [32].

176

It suffered from poor discrimination against background energetic electrons, which is characteristic of the time-of-flight (TOF) spectrometer (see Fig. 7). The apparatus was later improved by replacing the TOF with an electrostatic-energy-dispersing analyzer.

A more recent investigation on molecular oxygen [33] constitutes a good example of the potential of TPEsCO spectroscopy. Oxygen DPI has been studied repeatedly over the past 60 years, and the question of whether or not the local minimum of the potential curve of 02++ XIIg+ substains vibrational states had always remained unanswered. The coincidence experiment has clearly shown vibrational levels of the 0++ ground state up to v = 18 with an energy resolution of 20 meV that was limited by the light monochromator, not by the coincidence spectrometer. The vibrational constants of the double ion were also derived from the experiment. Some evidence for competition between single-step and double-step ionization was found at photon energies higher than 38 eV. The latter process appears in the energy spectrum as a structureless continuum contribution to the coincidence rate.

The molecular hydrogen DPI is the prototype for a complete break-up reaction:

(11)

Energy and angular distributions of the final charged particles will bear deep traces of both ISC of the bound electrons and CFSC of the four charged particles in the continuum. Kossmann et al. [34] have studied the H2 dissociation through a photoion-photoion (PIPICO) experiment initiated by photons in the energy range 52-110 eV. The coincident ions were collected by TOF analyzers operating with pulsed electric fields synchronized with the time structure of the SR source. The measured angular distribution of photo-fragments was used to derive the asymmetry parameters of the fragmentation. In dipole approximation, two transition amplitudes (for molecular axes oriented perpendicular and parallel to the photon electric vector) contribute to the process. The experiment has shown that, close to threshold, the perpendicular component becomes dominant and thus gives insight into the correlations existing between the bound electrons in the molecule.

Angular distributions of coincident pairs of electrons in DPI experiments close to threshold were measured for the first time by Mazeau et al. [35]. The experiment was done on Kr at a photon energy exceeding threshold by 2.26 eV and with an overall energy resolution of 300 meV. A similar experiment, but for electron impact, had already been performed with an (e,2e) spectrometer [36]. In that case, because of the mixing of several final states, it was not possible to discriminate between direct and shake-off ionization processes nor to assess whether the Wannier theory [37] predicts the correct angular distributions. On the contrary, the photoionization experiment, with a He lamp used as the photon source, identified direct DPI as the dominant process and indicated that the angular correlation resulting from the interaction of three charged bodies is compatible with the Wannier theory.

2.2. DIRECf OOUBLE PHOlOIONlZATION

If DPI experiments were performed at energies of 500 eV or higher, the final particles would be properly described in an independent particle scheme. CFSC effects would be negligible, and the cross section of the process would be determined by ISC alone. This idea underlies the experiments first proposed by Neudatchin et al. [38]. These experiments could be performed either with electrons or photons and are usually indicated by the abbreviations

177

(e, 3e) and (hv, 2e), respectively. In the high-energy limit, the DPI cross section factorizes like that of the impulsive (e, 2e) case, allowing for direct determination of the following quantities [39]: (1) the spectral density of Green's two-particle function (from the energy separation spectrum) and (2) the Fourier amplitude of the pair correlator (from the angular distribution of the final unbound electrons detected in coincidence). Levin et al. have also shown the sensitivity of the method to different types of correlated wave functions [40].

High-energy, double-ionization experiments have not yet been carried out, even though their feasibility at much lower energies has already been shown for both (e, 3e) [41] and (hv, 2e) [35] experiments.

The first attempt to perform direct DPI on solids was made on van der Waals solids of rare gases using a photon energy of 20-56 eV [42]. The spectrometer that was used permitted the measurement of electron-electron coincidence without energy or angular discrimination. Pairs of electrons were produced with a surprisingly high efficiency. Single photoionization followed by cascade electron-impact ionization was supposed to be responsible for such a copious production of electron pairs, rather than direct-valence DPI.

The smallness of the high-energy DPI cross section, which is expected to be 103 times smaller than the (e, 2e) cross section, puts the high-energy DPI experiments below the present sensitivity limit for coincidence spectrometers. Third-generation SR sources could make these experiments feasible. This will be of great relevance, .because ISC are believed to playa key role in determining the electronic properties of transition metals and their compounds, oxides and superconducting oxides, and surface- and adsorbate-induced states.

2.3. INNER-SHEIL OOUBLE PHOlDIONIZATION

At high photon energies, direct DPI from a core level is an improbable phenomenon. Conversely, a two-step process via core photoionization followed by Auger decay is a much more probable event.

Angular distributions of pairs of electrons resulting from a core photoionization process have been recently reported for the Xe DPI [43]. Namely, the coincidence angular distribution of Xe 4dS12 photoelectrons and NS02.302.3 ISO Auger electrons was measured with linearly polarized light of 94.5 eV. This work has established the first complete fragmentation pattern for two-electron emission and has provided information on the momenta and the spin projection of both electrons.

Auger and photoelectron spectroscopy are important sources of information on the electronic structure of the target. Auger spectroscopy on solids provides a useful tool for investigating the two-particle density of states, hole-hole Coulomb interactions, and hence correlation effects. Unfortunately, the complexity of the spectra, due to overlap of adjacent lines and to the fact that the initial state is not a simple one-hole state, often obscures the information. Furthermore, it is not always obvious how to distinguish Auger transitions from shake-up, shake-off, and Coster-Kronig processes.

Detecting Auger and photoelectrons that are correlated in time and originate from the same core ionization event (APECS) removes most of these uncertainties. The first experiment of this kind to be performed on a solid target was reported by Haak et al. [44]. The Cu L23M4SM4S Auger spectrum was measured in coincidence with the 2Pl/2 and with the 2P312 photoelectrons. With APECS, the Coster-Kronig-preceded Auger lines were unambiguously separated from the main L23M4SM4S transitions. It was also noticed that the

178

background due to multiple scattering of photoelectrons is strongly reduced in the APECS spectrum with respect to that in the conventional Auger and photoelectron spectra.

In the same paper [44], it was suggested that APECS should have enabled core photoelectron spectroscopy with an energy resolution better than the natural line width of the core-hole state. This effect was indeed found by Jensen et al. [45] in an APECS experiment on copper performed with an SR source. The coincident 31>312 photoelectron line was found to be narrower than the corresponding noncoincident transition. This is clear experimental evidence of the inadequacy of the two-step model in which the Auger decay is considered to be independent from the photoionization event. As a matter of fact, the energy conservation of the APECS event reads

(12)

where Ep and EA are the kinetic energies of the photoelectron and Auger electron, respectively, and £A~A++ is the separation energy between the initial neutral and final doubly ionized state. Because of the short lifetime of the core hole, the energy profile of the photo emitted electron undergoes a Lorentian broadening. On the contrary, when detection of both electrons is correlated in time, the uncertainty about their kinetic energies cannot exceed the line-broadening of the final double-ionized state, which is much narrower than the core-hole state. The same experiment has also revealed a hole cascade process from the 3P1l2 to the 3p312. followed by an M3 VV transition.

APECS has also shown its utility in discriminating surface from bulk processes. Coincidence measurements between 4f photoelectrons and NVV Auger electrons on Ta [46] have indicated a possible hole hopping from the surface to the second layer that takes place before the conventional Auger decay occurs. Furthermore, APECS experiments have been used to measure the L VV Auger spectra of an Al(111) sample with a lOO-L oxygen overlayer, discriminating among transitions generated in pure metal, chemisorbed sites, and ionic ally oxidized sites [46].

In atoms and molecules as well as in solids, core DPI is an interesting process. In particular, the tunability of SR allows atomic and molecular core electrons to be excited into bound or continuum orbitals that will decay through multiple ionization and/or fragmentation. To elucidate the various dissociation pathways of the core excited states, coincidence techniques are a great help. Among them, PIPICO has been largely applied, for instance, to the study of dissociative ionization in poly atomic molecules. Photoelectron­photoion-photoion (PEPIPICO) triple-coincidence spectroscopy has been used to study these processes [47). A discussion of this subject is beyond the purpose of these lectures, and we refer the interested reader to the review papers in the literature [48).

3. Coincidence Techniques

3.1. A COINCIDENCE SPECTROMETER

We have been discussing several types of coincidence experiments without describing any experimental detail. It is now time to discuss in some detail the key parameters for a

179

coincidence measurement. The (e, 2e) spectrometer shown in Fig. 2 can be considered a prototype for such apparatus, the main components being:

1. The primary beam (electrons, photons, ions) 2. The target (gas or solid) 3. Two (or more) independently tunable analyzers to detect, correlated in time, the final

reaction products.

To describe the technique of preparing, transporting, and detecting charged particles and photons is beyond the purpose of these lectures. The reader is referred to the many excellent review papers existing in the literature on this subject; see for instance Ref. 49. From now on, it will be assumed that electrons and ions are analyzed and detected by current single-pulse counting techniques. The simplest electronic circuitry for correlating in time the events detected by the two analyzers is shown in Fig. 9. The automatic control system for the experiment is also shown. The fast electronic pulses coming from the particle detectors (electron multipliers in this case) are fed through fast amplifiers to constant-fraction discriminators in order to separate the true pulses from the background noise. (Threshold level for these discriminators is usually set in the range of a few millivolts.) The shaped pulses generated by the discriminators are then sent directly to the scalers to measure the "single" count rate of each channel and then to a time-to-amplitude converter (T AC) to establish the time correlation between pulses. The T AC generates an output pulse for each pair of pulses acting as start and stop inputs. The

Figure 9. The electronic scheme of the coincidence spectrometer of Fig. 2.

180

amplitude of the TAC output pulses is proportional to the time interval (delay) between start and stop. Then a multichannel analyzer (MCA) records the frequency of appearance for the different amplitudes (Le., delays), thus yielding the probability distribution of delay time between pulses ("time spectrum" of the experiment). If all of the pulses present in the two channels were due to truly uncorrelated, randomly distributed events, the resulting distribution of delay time should be constant. The rate of probability for two pulses to be coincident within a time window (rate of "chance" coincidence) is given by

(13)

where Rl,2 are the single count rate in two channels, and tw is the minimum delay time resolved by the experiment. Particles correlated in time, i.e., generated in the same physical event, wiIl result in pulses always delayed by the same amount of time (td) and generated at a rate of "true" coincidence Ry that is proportional to the differential cross section under measure. A typical time spectrum for such an experiment is reported in Fig. 10.

In Fig. 10, the shaded area Nc and the area under the peak Nt are, respectively, the total "chance" and "true" coincidences accumulated within a set time window in a given integration time ti (N i = R iti). From this figure, it is evident that several subtraction procedures can be used to amend the time spectrum from the "chance" coincidence. The simplest one is the subtraction of the linear interpolation of the "chance" coincidence

800

600

Q) 0 c: Q) "0 400 '0 c: '0 u

200

TAC channel (1 ns per channel)

Figure 10. Time spectrwn of a coincidence experiment (continuous source).

181

frequency. The position of the true coincidence peak on the time axis is due to several experimental parameters discussed in the following paragraph and can be controlled by inserting a variable electronic delay in the "stop" branch of the coincidence electronic chain. A typical coincidence experiment will consist of recording the "true" coincidences Nt versus the energy balance and/or the angular distributions of the selected pair of particles. The "true" rate Rt usually ranges between few tenths and few tens of Hertz; it implies that the duration of a coincidence experiment ranges from a few hours to a few weeks. It is therefore vital to have the coincidence setup controlled by a computer and the main experimental parameters (e.g., beam current, detector efficiency, and target density/thickness) continuously monitored. In this way, it is possible to renormalize the full set of experimental results, since it is virtually impossible to rely on constant (within a few percent) experimental conditions over such long periods of time. A scheme for the data-acquisition and control system for an (e, 2e) apparatus is given in Fig. 9. The beam energy and intensity, the energy and angular setting of the two analyzers, and the gas target density are controlled and monitored by a personal computer via a CAMAC interface system. The computer also provides data collection. As the "true" coincidence count rate can be very low, it is mandatory to implement an electronic chain identical to the ones used to process start and stop signals, which inhibits the time-to-amplitude conversion Whenever an electromagnetic noise is picked up by the aerial to which it is connected. In that way, pairs of signals originated by noise and falling within the "true" coincidence time window are not processed. There is no other method to distinguish them from truly coincident signals.

3.2. OPTIMIZATION OF A COINCIDENCE SPECIROMETER

The key characteristic of a coincidence spectrometer is its capability for discriminating "true" from "false" coincidences, because ultimately this will set the limit for the feasibility of a coincidence experiment. Therefore, it seems obvious that, in designing the experiment, we should aim for a "true"-to-"chance" coincidence ratio (r = RtlRc) as large as possible. In reality r is not a useful figure of merit for such experiments. Let us assume, for simplicity, an experiment in which a Single pair of spectrometers is used to detect events in coincidence. Each analyzer is characterized by the energy resolution BE1,2, the accepted solid angle 01,2, and the detection efficiency <Xl,2. The current of incoming particles/photons is 10 , while the density of target-scattering centers per unit area is (dl).

The cross section that yields the "single" events detected by each channel is indicated with 0 2 = d201,2/dOl,2dE, and the coincidence cross section with 0 3 = d30JdOld02dE. One obtains the following results:

(14)

(15)

where:

182

d 2ol2 A12 = , (dl)·M12 ·Bn12 '(X12 , dn dE ' , ,

1,2 (16)

(17)

BEe is the "effective" energy resolution for the coincidences and comes from the overlap integral of the two analyzer response functions, taking into account energy conservation. When the response function is Gaussian in shape (this is a good assumption for most electron spectrometers), BEe becomes:

(18)

It is to be noted that BEe is not the energy resolution to be expected for the coincidence energy spectrum, i.e., the coincidence rate versus the energy balance E. In this latter case, the energy resolution is simply the convolution of the two response functions:

(19)

The r parameter can then be written as

(20)

and it is at maximum for vanishing incoming current or vanishing density of scattering centers. Obviously, to aim for maximum r value does not help in planning the experiment. To this end, it is meaningful to consider the time (t%) required to achieve a given statistical error expressed as a percentage of "true" coincidences (B%) for a given experimental condition. It has been shown that:

From this relation, it is evident that, no matter how large 10 or (dl) is, t% will asymptotically reach a limit that is inversely proportional to the accepted solid angle and directly proportional to the coincidence-resolving time two Therefore, in order to reduce t%, the n's should be as large as possible, while tw should be kept to a minimum.

Figure 11 shows the variation of t% with 10 for a hypothetical experiment on copper involving photoelectron-Auger coincidence. The experimental parameters are: B% = 10%, BEl = BE2=100 meV, tw from 2 to 50 ns, and 2° accepted angles. Clearly there would be no point in increasing the incoming current beyond the value at which t% starts to level off, and it has been shown that usually this happens for r values close to unity. Hence, r "" 1 is the best compromise between large "true"-to-"chance" ratio and short integration time. It is

183

-*-*-~ fw = 50 ns -x-x-x- fw = 10 ns

--o---o--<r- fw = 2 ns

105

-~ CI)

E :0:;

c 104 0

~ C) CI) -£:

103 '0- - ~ - -0- - -0- - ~

1010 1011 1012 1013 1014 1015

Photon flux (phis)

Figure 11. Example of the variation with the photon flux of the measurement time required to achieve a given statistical uncertainty in a coincidence experiment (see text).

finally to be noted that, as far as energy resolution is concerned, relation (18) shows that the setting El = E2 is the most sensible choice.

3.3. MINIMIZATION OF TIlE ACCUMULATION TIME

Maximizing the accepted solid angle and, at the same time, minimizing the resolving time are conflicting requests, mainly in the case of electron spectrometers. Among the various sources of time uncertainty in a coincidence spectrometer, the spread in time walk for the various particle trajectories inside each analyzer is certainly the most relevant. This contribution to the overall tw value has never been made smaller than 200 ps for electrons and 100 ns for ions, while the contribution from the electronic circuitry and particle detectors can be kept below 100 ps. To further reduce the time-walk spread, either the accepted solid angle must be reduced or suitable time-compensation techniques must be applied. If time compensations are not applied, the resolving time degrades quickly as the solid angle increases. For instance, in the photoelectron-Auger coincidence experiment by Haak et al. [50], with Q "" 5xlO-2 sr and r"" 1, tw was not better than 16 ns. When time-compensation techniques were used,

184

either by placing the detector along the isochronal surfaces [51] or by applying a software correction [52], a better overall time resolution of 500 ps was achieved with Q "" 2xlO-2 sr.

To reduce the measurement accumulation time without enlarging tw too much, multichannel devices were developed in which individual detectors are combined to yield a coincidence matrix that allows up to a few hundred coincidence experiments to be performed in parallel [53]. One possible and successful approach consists of using position-sensitive detectors behind a broadband, energy-dispersive, electron analyzer [54]. Coincidence spectrometers of this kind provide parallel acquisition of the coincidence energy spectra and have been used for both gas and solid targets and for both (e, 2e) and (e, e' Auger) experiments. An alternative approach is represented by the "angular multi-parameter coincidence" devices that have been constructed to perform parallel acquisition of the coincidence angular distributions. The first apparatus of this kind was built by Skillman et al. [55]. In this device, an array of ten individual electron detectors were placed at the exit of an electrostatic-energy-dispersing element and allowed to collect 25 angle-resolved coincidence spectra at once. A similar apparatus employing a position-sensitive detector instead of an array of individual electron multipliers was used by Schnetz and Sandner [51] for (e, e' Auger) experiments and achieved a multiplicity of 144 channels.

Multiple-coincidence experiments have been reported as well. Lahmam-Bennani et al. have shown that, at least for triple-coincidence experiments [(e, 3e) processes], a minimum value of the integration time does exist for a specific value of the beam intensity [56]. In the case of triple-coincidence experiments, distinguishing the "true" counts from the background of chance coincidences is a cumbersome task. The chance-coincidence background is made of a uniform distribution due to the arrival of fully uncorrelated triplets of particles to the detectors. Superimposed on this background are "ridges" generated by the triplets, where two of the electrons are correlated and the third one is not [56]. The aforementioned (e, 3e) experiment made use of two TACs for the triple-coincidence circuitry.

3.4. CONTINUOUS AND PULSED BEAMS IN COINCIDENCE EXPERIMENTS

Up to this point, coincidence experiments have been discussed with the assumption that the primary beam is a continuous one. This assumption is appropriate for the charged-particle sources currently used and for laboratory photon sources. As the intensity and brightness of SR sources increases, it will be possible to use them for coincidence and multi coincidence experiments. Indeed, as shown in the previous paragraphs, SR has been already used for electron-electron, electron-ion, ion-ion, and electron-ion-ion coincidence experiments.

We should then ask ourselves whether the optimization criteria established for the continuous beams are also valid for pulsed sources. Let us assume T as the period of the pulsed beam and tp the duration of the beam pulse. Two extreme cases are to be discussed:

1. The resolving time tw is greater than the period T. In this case, the beam time structure is a quasi-continuum, and the relations derived for r and t% for a continuous beam can be extended to this case, assuming 10 as the average beam current.

2. The resolving time tw is shorter than the period T. In this case, the time distribution of the "chance" coincidences is not uniform, but is modulated with a time structure identical to that of the pulsed beam [57, 58]. Given these conditions, the average rates of "chance" and "true" coincidences are, respectively,

185

(22)

and

(23)

where io is the peak current and 10 is the average current. The A's are the same as defined in (16) and (17). As a consequence,

(24)

To optimize r, the period T should be as short as possible. It can be shown that for identical values of the average current 10 the minimum t% is achieved with a continuous beam.

The advantage of the pulsed beam comes from the fact that unless the period is larger than the resolving time of the spectrometer, the coincidence time window is determined by the pulse duration tp. Hence, if SR sources are used, coincidence time windows as short as 30 ps are possible, whereas with continuous beams and state-of-the-art analyzers and electronics, 800 ps is the shortest time achievable.

A further advantage of using high-brightness SR is due to the small section of the beam (in the range of few tens of ~m). This allows for good energy resolution and good collection efficiency. Indeed, an intense and localized source of events makes analyzers easier to use with high energy resolution and still allows for an excellent overlap of their field of view, which is an important condition to be fulfilled by any coincidence spectrometer.

Acknowledgments

Work partially supported by the EEC Contract: Science M.SCI.*0175-C(EDB) and by the NATO grant C.O.G.920101.

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(1992).

SOFT X-RAY EMISSION SPECTROSCOPY USING SYNCHROTRON RADIATION

JOSEPH NORDGREN Uppsala University, Physics Department Box 530, S-751 21 Uppsala Sweden

ABSTRACT. This presentation describes soft x-ray emission spectroscopy and its recent developments based on the use of synchrotron radiation for the excitation. A brief account of instrumentation employed in such experiments is given, and a few areas of application in physics research are discussed. In particular. experiments in threshold and resonant soft x-ray emission spectroscopy are presented. and application to surface physics is briefly discussed. Also. the direction of future developments is indicated.

1. Introduction

Soft x-ray spectroscopy has a long history that dates back to the third decade of this century when it became feasible to extend the wavelength range of x-ray spectroscopy into the softer part of the spectrum by using grazing incidence gratings [1] or by employing organic Bragg crystals [2]. In x-ray spectra, chemical effects had already been observed in 1920 by Bergengren, who studied phosphorous compounds in K absorption [3]. A few years later, Lind confirmed these findings in observations of chemical shifts in chlorine and sulphur compounds [4]. Absorption studies of molecules in the ultra-soft range using the grazing­incidence grating technique were made by Magnusson in 1942. For example, he recorded the N K absorption spectrum of N20, clearly separating the terminal and center nitrogen wl).ite lines [5]. Important contributions to the understanding of solids in terms of band theory were made by x-ray spectrometry, starting in 1933 with the work of Sommerfeld and Bethe [6] and continuing with the work of Seitz [7], and O'Bryan and Skinner [8] in 1940. The particular ability of soft x-ray emission to provide a projection of the valence band density of states with respect to atomic site and angular momentum was further exploited in studies of metals and alloys in many laboratories [9]. Studies of soft x-ray emission of free molecules were started in a few different laboratories in the mid-sixties [10, II, 12] and brought to a level of high resolution in the beginning of the seventies [13]. This experimental work was successfully based on molecular orbital schemes for interpretation of x-ray spectra put forward by several authors [14, 15, 16]. Steady progress has since been made in these fields, including the study of solid molecular systems [17, 18, 19].

The introduction of synchrotron radiation in spectroscopic studies of electronic structure during the seventies using the so-called first-generation sources did not have great impact on soft x-ray emission spectroscopy immediately, although one group was successful in setting up an experiment for white-light excitation of solid samples [20]. For some years, the main obstacles preventing this method from further developing its potential in this new

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A.S. Schlachter and F.J. Wuilleumier (edr J, New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 189-202. © 1994 Kluwer Academic Publishers.

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environment of synchrotron radiation were shortcomings in source intensity and detection efficiency. In the x-ray range, work in the field of tunable synchrotron-radiation-excited emission of free molecules was started in the mid-eighties [21], whereas for the soft range, this step was taken in 1987, when it was demonstrated for the first time that monochromatized synchrotron radiation could be used to excite threshold and resonant soft x-ray emission [22]. This achievement was largely based on the use of insertion devices rather than bending magnets and recent advances in instrument technology [23]. Simultaneously, another group started work in the field using white light from a bending magnet [24], and later also employing monochromatization [25]. Since then, further development in source technology has taken place, and the present and foreseen situation for synchrotron-radiation-excited soft x-ray emission spectroscopy is quite advantageous. This fact is reflected in a substantial number of new experiments of this kind under construction or planned at storage rings.

The use of synchrotron radiation for the excitation of soft x-ray emission spectra adds several important qualities to this spectroscopic method. Firstly, it provides a yery intense photon-excitation source, comparable in excitation efficiency with electron-impact excitation, and with the useful features of deep sample penetration and reasonable softness with respect to sample degradation. Secondly, monochromatized soft x-ray photons, which can presently be generated at sufficient intensities only from synchrotron radiation sources, offer a higher degree of energy selectivity than do electrons. Electrons are subject to inelastic scattering, giving rise to a range of excitation energies below the primary electron energy. Thirdly, synchrotron radiation offers the possibility of exciting soft x-ray emission by polarized light, a feature marginally exploited so far, but likely to play an important role in future work in this field. In addition, one can mention the time structure of synchrotron radiation as a useful property, for example, because of the feasibility of background suppression and its potential in coincidence experiments.

There are several research areas in which soft x-ray emission has not been used for reasons of experimental nature or has played only a minor role, but for which this spectroscopy now may offer new possibilities due to the introduction of synchrotron radiation. One such field is the study of surface adsorbates, which has recently been demonstrated feasible for soft x-ray emission [26]. Another such area of research is the study of resonant processes and threshold effects in core excitation, in which tunability of the excitation energy is required [22, 27]. A third area is polarization-resolved soft x-ray emission, which is likely to benefit very much from the use of monochromatized synchrotron radiation [28]. An additional example is the study of buried systems, in which the depth-probing capability in combination with the tunability to a certain species and state are desirable assets.

2. General

Soft x-ray emission spectra are generated upon the decay of localized inner vacancy states by the transition of electrons from the valence band. More specifically, we are dealing with transitions between the two outermost principal electron shells of atoms, molecules, or solids. The fact that the chemically active electron shell is engaged is important and is a basis for the usefulness of this method for studies of chemical bonding. For light atoms, there are obviously no harder x rays than the soft ones since these atoms do not have more than two main shells of electrons, whereas, for heavier atoms, harder valence-core x-ray transitions are also possible. The deexcitation of inner vacancies in the soft x-ray energy range by photon

191

emission is facing strong competition from nonradiative deexcitation channels such as Auger emission and autoionization, i.e., the fluorescence yield is normally very small. Typically, this yield is in the sub-percent range, increasing with energy, and for the same photon energy, K emission shows higher yields than L emission, L emission higher than M emission, etc. [29]. The low yield of soft x rays together with the low luminosities associated with high-resolution detection at these wavelengths has been an obstacle to widespread application of soft x-ray emission spectroscopy up to the present.

Apart from the atomic-species selectivity of x-ray spectra, originating in the energy separation of the inner vacancy states of different atoms, there is a state selectivity due to the localization of the initial state and the electric dipole nature of the x-ray transitions. For soft x-ray emission, this can be used to separate the contributions of the different atoms to the valence electronic structure of a bonded system. For a molecule in which the initial state of the x-ray transition can be represented by an atomic orbital centered on one particular atomic site, and the final-state molecular orbital by a linear combination of atomic orbitals centered on the various atomic sites, the x-ray intensity can be approximated by the weights of the dipole-allowed atomic orbitals centered on the same atomic site as the initial inner vacancy. This approximation is often referred to as the one-center model for x-ray intensities [30], because it is based on the neglect of the two-center terms in the transition moment integral. Thus, for K emission in a molecule, this leads to the following intensity expression for a particular molecular orbital

where cp represents the weights of atomic p orbitals in the LCAO expansion of the molecular orbital in question, centered on the proper atom. This expression assumes that the cubic energy factor can be neglected, which is often the case. For L emission, the expression takes the form

where the weighting factors come from the integration of the angular part of the wave functions, and assuming the 3s and 3d radial parts to be equal. The applicability of the one­center intensity model for molecular x-ray intensities has been investigated for small molecules. This study shows that the neglect of two-center terms is a fairly good approximation for K emission and is often still reasonable for L emission spectra [31]. It turns out that inclusion of relaxation and correlation effects is often of greater importance in achieving agreement between experiment and theory [32, 33]. However, current advanced computational methods used for calculations of the x-ray spectra of molecules take both relaxation and correlation into account in quite an adequate manner, although for larger, more complicated systems less accessible to accurate ab initio calculations, the simple model can be of considerable value.

For so-called broadband systems, like metals and other periodic solids, in which the valence electrons form broader energy distributions and are represented by density-of-state functions rather than molecular orbitals, the application of the local and state-selective probing capability of x-ray emission provides a decomposition of the total density of states in terms of projected partial densities of states. Thus, the intensity distribution expression reads

192

/(E) - V3 P(E) N(E),

where P(E) is the transition matrix element squared and N(E) is the occupied valence band density of states. In the free-electron model, the intensity distribution reduces to the well known forms

and

for K and L emission, respectively [34]. The shortcomings of the one-electron model often limit its use for obtaining information about the valence band, though, and more refined methods taking transition probabilities and many-body effects into account have to be considered [35].

One point worth mentioning with respect to core excitation and deexcitation spectroscopy studies is the effect of the inner hole on the valence electronic structure. In x-ray absorption experiments, the final state as represented by the x-ray absorption spectrum is modified with respect to the "unperturbed" unoccupied state by the presence of the core hole if the screening is not efficient, as may be the case, for instance, in transition-metal L spectra. Therefore, one has to exercise care in the comparison of x-ray absorption spectra with ground-state calculations of the partial density of states. For emission spectra, on the other hand, the x-ray transition closes the core hole and the spectrum is more likely to be well represented by ground-state calculations.

3. Instrumentation

Soft x-rays are very strongly absorbed in matter, and, unlike shorter and longer wavelengths of the electromagnetic spectrum, they can only penetrate windows of submicron thickness, a condition that has considerable experimental implications. Although the soft x-ray range historically has its short wavelength limit around 2 A, where the radiation starts to penetrate macroscopic distances at atmospheric pressures, one often refers to soft x rays as the range above 10 A or below I keY (a range that is sometimes called the ultra-soft x-ray range, USX). There are reasons for making this distinction, partly of an experimental nature. Firstly, x-ray transitions between the two outermost main electron shells fall in this region. Secondly, the greater part of the x-ray range is accessible to spectroscopy by instruments based on Bragg diffraction; however, below the I-keV region, a lack of suitable crystals and the increasing absorption start to present problems that make this part of the spectrum less accessible. Especially below the oxygen K edge, the power of the Bragg technique is limited because of the lack of natural crystals. (This condition may change in the future when sufficiently thick, high-quality, synthetic, multilayer structures become available as dispersive elements.) Grating diffraction, on the other hand, which is the commonly used technique in spectroscopy for visible and vacuum ultraviolet wavelengths, looses power going to very short wavelengths because of decreasing efficiency and resolution. It seems that, for many experiments, the crossing point, at which one or the other method becomes more advantageous, occurs around I keY.

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Focusing grating instruments are the principal choice for high-resolution emission spectroscopy in the ultra-soft range, especially in connection with sources of limited extension, such as are often encountered with synchrotron radiation. Then the advantage of simultaneous dispersion of all wavelengths offered by grating diffraction, as opposed to Bragg reflection, can be fully utilized, provided that multichannel detection is employed. Going one step further by arranging detection in two dimensions offers increased sensitivity without significant losses in resolution because the part of the astigmatic imaging errors associated with the curvature of the spectral lines can be accounted for and larger acceptance in the sagittal plane is allowed. In other words, long, slightly curved spectral lines can be recorded without loosing appreciable resolution. Below is a brief description of a soft x-ray spectrometer for high-resolution spectroscopy in the 50-1000 e V range. This instrument has played a key role in recent progress in soft x-ray spectroscopic studies. A more detailed description can be found in Refs. 23 and 36.

The basic idea behind the design of this instrument is the use of several fixed spherical gratings and a large, position-sensitive detector (see Fig. 1). The gratings have different radii and groove densities chosen to optimize the performance over a wide wavelength range, and they work at different incidence angles chosen to match the respective wavelength ranges covered.

The detector can be positioned so that it is tangential to the Rowland circle of the respective gratings by precision movement in a three-axis coordinate system (two translations, one rotation) . This arrangement allows some important qualities to be attained. The resolution and luminosity remain high, while the instrument size is brought down to a minimum. Simple

Figure 1. Photograph of a multi-grating, grazing-incidence spectrometer used for synchrotron-radiation-excited fluorescence spectroscopy.

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flange-mounting is feasible, allowing convenient use of the instrument at different sources. Another advantage is the freedom of configuring the instrument with virtually any gratings in order to optimize performance for a particular wavelength region.

The luminosity of the instrument is high owing to the use of a large, efficient detector and a multi-grating arrangement that keeps the distance between source and detector small. The detector is based on 50-mm multichannel plates (MCP) and two-dimensional readout is accomplished by a resistive anode technique. The efficiency of the detector is enhanced by coating the front MCP with a layer of CsI, and an entrance electrode is introduced to capture electrons ejected from the interstitial surfaces of the MCP, which are the active surfaces at grazing incidence. Holographic, ion-etched gratings have been used throughout, offering typical diffraction efficiencies of 10%.

Most of the work presented in this review was done by using monochromatized synchrotron radiation for the excitation. For this purpose, two insertion-device sources were used, at HASYLAB in Hamburg and at NSLS in Brookhaven. At HASYLAB, a 32-pole wiggler in one of the straight sections of the DORIS storage ring was used. A plane-grating monochromator (FLIPPER I) was used for wavelength selection in the range 20-1500 eV [37]. This monochromator has a number of different pre-mirrors, which, one at a time, can be put into position in order to illuminate the grating at the most suitable angle for the desired wavelength range. The source point is about 35 m away from the monochromator, making the beam nearly parallel, and a parabolic mirror is used as the post-mirror to focus on the exit slit of the monochromator. The output flux is of the order of 1013 photons per second at a bandpass of about 0.5%. These high flux levels are necessary because of the low fluorescence yield for ultra-soft x-rays and the limited solid angle of acceptance for grazing incidence spectrometers. For some of the work, the Xl undulator at the x-ray ring at NSLS was used. This beamline is equipped with a very high-resolution, spherical-grating monochromator employing a movable exit slit [38].

An important point concerning the detection system of the present instrument in connection with the use of storage-ring sources is related to the time structure of the ring. The duty cycle of such sources is sometimes quite low and consequently may allow the electronic noise of the detection system to be virtually eliminated by synchronizing the detector with the time structure of the ring. This is particularly true for a few-bunch storage ring like DORIS. By the use of standard electronics (ECL) to obtain a time window of 7 ns, the dark noise was reduced to less than 3% in the four-bunch mode used for dedicated synchrotron-radiation operation of DORIS.

4. Selectively Excited Soft X-Ray Emission

X-ray satellites, which notoriously accompany the diagram lines in x-ray emission spectra excited at excess energies, were observed and to a certain extent understood a long time ago [39]. In many cases, especially for metallic systems in which screening is efficient, x-ray satellites tend to overlay the diagram lines in a manner that obscures the band-structure features and obstructs proper analysis. An obvious remedy for this problem is, of course, to use more energy-selective excitation because a large portion of the x-ray satellites originate in multiple excitation/ionization processes. Such selectivity is one of the key features offered by synchrotron radiation, which is intense and collimated enough to allow monochromatized, high-flux beams to be obtained.

Separation of satellite structure by selective excitation is demonstrated in Fig. 2, in which the Cu L emission spectra of metallic copper and the YBa2Cu307_x superconducting oxide (so called 1-2-3 superconductor) are shown [40]. One notices that reducing the excitation energy from 970 eV to 935 eV, i.e., to just above the L3 threshold, makes the L2 line disappear, as expected, and it also leads to a different form of the L3 band.

This difference comes about by the removal of the satellites caused either by shake-up!shake-off processes in the L3 excitation or by Coster-Kronig decay of L 2 vacancies, both processes being energetically forbidden at the lower excitation energy. In the local one­electron approximation, the threshold­excited L3 spectrum then reflects the "clean" 3d partial density of states. Subtracting the "935-eV spectrum" from the "970-eV spectrum" yields the pure satellite spectrum, as seen in the figure. In comparing the satellite spectra of the metal and the oxide, one notices that they have different intensity and shape, reflecting the modification of the valence band in forming the oxide and, in

920

Photon-excited Cu L emission

hVexc = 970 eV

930

hVexc = 935 eV

Ditt. spectrum

940 I

950 Emission energy (eV)

195

Figure 2. Copper L2.3 emission spectra of Cu metal (above) and YBa2Cu307_x (below), excited at energies near the L3 threshold and well above the L2 threshold, respectively. The difference spectrum is also plotted.

particular, a quenching of the Coster-Kronig decay of L2 holes. In Fig. 3, the Cu L3 satellite intensity (somewhat arbitrarily expressed in terms of half width

20

15

Cu L3 satellite intensity vs. excitation energy

10+----,----.----.,----.---. 920

at quarter maximum) has been plotted against excitation energy. One observes a gradual increase of the intensity between the energy thresholds of the subshells, whereas at the thresholds, in particular the L2 threshold, a rapid increase occurs. This step can be explained in terms of the opening of the Coster-Kronig decay channel for the L2 hole state, which leads to L3M double-hole states, i.e., initial states for satellite transitions [27]. Figure 4 shows the L2.3 emission spectrum of Zn

960 1000 1040 1080 1120 metal, excited at a variety of energies from Excitation energy (eV)

Figure 3. Variations in Cu L3 satellite intensity vs. excitation energy.

near the L3 threshold up to and above the Ll threshold. One can clearly observe how the satellite structure, in this case more

196

discrete lines on the high-energy side of the L3 peak, grows with increasing excitation energy.

The ability to tune the excitation energy to certain resonance energies offers the possibility for studies of the dynamics of excitation and relaxation processes on the time scale of the core-hole lifetime, and it can help to focus on finer details in the electronic structure. Resonance lines in x-ray emission have been studied by using electron excitation in solids [41] and also in free molecules [42, 43], corresponding to a reversed absorption process. Generally, of course, the decay of x-ray absorption states can be studied in emission, provided that these states can be populated at sufficiently high rates. This type of study is possible through the use of monochromatized synchrotron radiation. Figure 5 shows such a case, in which two YBa2Cu307_x super­conductors differing slightly in critical tem­perature were studied by using varying energies of the exciting photons.

For a wide range of energies, and in particular for white-light excitation, the two

Zn L fluorescence from metallic zinc

1010 1020 1030 1040 Emission energy (eV)

Figure 4. Zn L2,3 emission spectra of Zn metal excited at a variety of energies between the L3 and Ll thresholds.

o K emission '. '.,

hVexc = 541 eV /\ . L

i\ \ Oxygen rich i I. \. ~ .. ,,-!;..v.~.~.;wt::/: i \ .I ...... #~r.--:~.

I ' \" .:.":'.~.

Oxygen poor ,i -----'<'

samples show almost identical oxygen K emission spectra. At certain distinct energies, though, differences appear. For example, at 535 eV, a structure above the Fermi level appears, and at 541 eV, shown in the figure, a dramatic change of the valence-band spectrum is observed. This sensitivity of the x-ray spectrum to the functionally crucial oxygen content may provide useful information for the understanding of these systems. It should be mentioned, though, that there is a possibility that such effects are also due to inhomogeneities with respect to sample surface versus bulk, because the penetration depth is energy-dependent in the near-edge energy region.

I 535

I 525

Emission energy (eV)

I 515

Figure 5. 0 K emission spectra of two YB a2C U3 0 7 -x samples with slightly different oxygen content. The emission was excited at 541 eV.

In one of the early experiments with tunable photon excitation of soft x-ray emission spectra, the N K and Ti L emission of TiN were studied [22]. By tuning the excitation energy to below the Ti 2p threshold, the N K spectrum could be recorded for the first time without the accidentally overlapping inner-core transition in Ti (3s-2p) (see Fig. 6, bottom). The middle

spectrum in the figure shows the Ti L emission excited just above the L3 threshold, consequently showing the bare L3 contribution to the spectrum. When the 6-eV spin-orbit splitting between the L2 and the y states is taken into account, an apparent agreement with previously reported, nonselectively excited spectra was thus observed. However, one realizes the falseness of the interpretation of this agreement by comparing the L2 and the L3 emission, as can be done by looking at the uppermost spectrum in Fig. 6, the derived L2 emission spectrum (derived by taking the difference between spectra excited above and below the L2 threshold). One notices that the two spectra are in fact quite different, and therefore the observed emission peak at 460 eV [22] had to be given a different explanation from the previously accepted one.

Recently, unexpected excitation-energy dependence was observed in the K emission of diamond up to 30 eV above the threshold of C Is excitation (see Fig. 7). These observations were unexpected because diamond is regarded to be rather well described in a one-electron band model, and both x-ray absorption and emission have been found in concordance with this view. An

197

-15 -10 -5 EF 5 Energy (eV)

Figure 6. Soft x-ray fluorescence spectra of TiN. Bottom, N K emission excited by 41O-e V photons. Middle, Ti L emission excited at 458 eV. Top, difference spectrum between Ti L excited at 480 eV and 458 eV, respectively. The spectra have been aligned by means of XPS data.

explanation proposed to account for these findings is based on the presence of inelastic scattering, which is enhanced in the vicinity of the core threshold [44]. Earlier observations of

t:: o ·iii (/)

·E w

Diamond C K emission

Excitation 350.0 304.5 302.5 299.5 294.0 291.0 289.5

256 262 268 274 280 286 292

Energy (eV)

Figure 7. C K emission of diamond excited at various photon energies.

energy dependence in the Si L emission of crystalline Si had been explained in terms of local excitations similar to the behaviour observed in so-called narrow-band materials [25]. Later observations suggest, though, that this interpretation is incorrect, because this dependence was not found in the L emission of amorphous Si [45].

5. Surface and Bulk Probing in Soft X-Ray Emission

The comparatively deep penetration of x rays, as compared to electrons of comparable energy, offers excellent opportunities for studies of bulk properties. For example, See Fig. 8, which shows the L3 emission of Ni in two phases. The solid curve

198

represents pure Ni, and the dotted curve shows Ni atoms diffused into a Cu crystal. X-ray photoemission (XPS) showed no trace of nickel in this case; thus, the diffusion depth was more than several tens of angstroms. The figure shows that the diffused nickel L emission spectrum is narrower than the pure Ni spectrum, reflecting the modification of the valence band in the diluted alloy phase and suggesting that the method could be used to study segregation processes [46].

Tuning the photon energy to certain resonances and thereby increasing the absorption cross section allows the probe depth of soft x-ray emission to be decreased. Even with white-light excitation,

846 848

Emission energy (eV)

Figure 8. L3 emission of Ni. Solid curve represents pure Ni metal; dotted curve represents diffused Ni in Cu.

soft x-ray emission can be used in the study of some extreme surface problems, by virtue of the fact that the surface sensitivity can be enhanced by grazing illumination. In addition, the energy separation of soft x-ray emission pertaining to different core-hole states provides a means to record separately the emission from an adsorbed species and from the substrate. As a matter of fact, it has recently been shown [26] that highly resolved soft x-ray fluorescence spectra can be obtained from fractions of monolayers of adsorbed atoms and molecules on surfaces by using grazing-incidence excitation. Figure 9 shows oxygen K emission spectra of a Ni(loo) surface exposed to 50 Land 1000 L of oxygen. In the former case, a c(2 x 2)

1 .2 .----,--.-----.-,-----y---,--.-----.---,

~ 1.0 'c ::J 0.8 .e ~ 0.6

0.2

01000 L • 50 L

23.2 23.4 23.6 23.8

Wavelength (angstroms)

Figure 9. 0 K emission spectra, excited with white­light synchrotron radiation, from a Ni(100) surface exposed to 50 L and 1000 L of oxygen.

monolayer adsorbate structure is formed, giving rise to a spectrum with a distinct Fermi edge, directly reflecting the hybridization between the 0 2p and the Ni 3d electrons. In the latter case, the high exposure forces NiO to be formed rather than only an adsorbate monolayer, and indeed a different spectrum, characteristic of NiO, is observed. To date, a number of different adsorbate systems have been studied, including the COINi (100) system (see Fig. 10). The results of these studies indicate that there may be very interesting prospects for soft x-ray emission spectroscopy in surface physics research, especially with the even higher brightnesses expected from the third-

generation synchrotron-radiation sources. Obviously the ability of this method to probe the local partial density of states is of great value in this kind of study, since one escapes the difficulty of sepa­rating the adsorbate-derived states encountered in photoemission experiments. The method then allows a direct means to in­vestigate the bonding of the adsorbate in terms of orbital hybridization. In addition, the method allows core-hole relax­ation processes to be studied on the timescale of the lifetime of the core vacancy.

A direct effect of alignment is observed in the 0 K emission spectrum of CO adsorbed on Ni(100) (see Fig. 10). It turns out that the a-orbitals do not con­tribute to the emission spectrum in the direction of the surface normal, as expected considering

199

OK Emission

CO/Ni(100)

~ 00

•••• h h •• I I •• o:::x:x:D

c(2 x 2)

,-. '. ", ~ '-- CO (gas)

\ .. /

520 522 524 526 528 530 532 534 536 Emission energy (eV)

Figure 10. 0 K fluorescence spectrum from CO adsorbed on Ni and the corresponding electron-excited spectrum from gaseous CO.

the perpendicular bonding geometry of this system and the dipole intensity distribution. Furthermore, given the ability to record soft x-ray emission spectra from sub-monolayer systems, it is clear that the proper use of the polarization of the synchrotron radiation used for excitation adds an important element of selectivity. Such studies have not yet been made, but are being planned for the near future.

6. Future Prospects

It is obvious that the third-generation synchrotron-radiation sources presently being constructed at various locations in the world will offer many new possibilities for using soft x­ray emission spectroscopy. Experiments that have been demonstrated barely feasible at the present facilities will be conducted on a routine basis, and new experiments that are presently beyond reach will become feasible. Ultimately, though, there will be limitations; for example, certain fragile systems may not be able to tolerate the high brightness of the new sources.

One exciting prospect is the ability to excite SXES spectra of organic molecules at excitation bandpasses of 100 meV and below. This would allow the separation of emission spectra pertaining to identical but chemically shifted species. With nonselective excitation, the different spectra would appear superimposed because of poor resolution, and detailed analyses would be obstructed. Having access to sufficiently narrow excitation may also offer possibilities for the study of coherent processes in x-ray emission, such as the lifetime­vibrational interference observed in small molecules with vibrational splitting and core-state

200

lifetime widths of comparable magnitude [47]. Such narrow excitation would also provide ways to perform detailed studies of surface effects and even buried layers and interfaces, such as studies of the bonding of polymer layers to metal surfaces.

Polarization-resolved x-ray spectroscopy has been pursued in several laboratories. One has studied the angular dependence and the polarization of the K emission in noncubic crystals [48, 49] and those properties of the L emission in free molecules [50]. In the latter case, polarized excitation has been employed, and the nonisotropic emission distribution, which is retained because of the short lifetime of the core states, has been recorded. The use of synchrotron radiation may. offer interesting possibilities with regard to ordered materials in this respect, for example, transition-metal systems. By exciting with linearly polarized photons, one can envision separating the m-resolved d-band contributions to the partial density of states by detecting the angular distribution of the L emission. It has also been shown that grazing-incidence detection allows a certain degree of polarization-state detection because of the difference in photoyield between perpendicular and parallel polarized photons illuminating a photocathode at grazing angles [51].

Recently, L emission data for Fe suggesting that magnetic dichroism could be studied in emission spectra were presented [52]. Because the experiments were conducted with a white beam of circularly polarized synchrotron radiation, they suffered from a considerable background of nondichroic contributions. Applying monochromatized synchrotron radiation is likely to improve the conditions for such experiments considerably and possibly will provide a means to obtain information about magnetic properties and, in particular, about spin densities of occupied states.

References

1. T.H. Osgood, Phys. Rev. 30, 567 (1927). 2. M.A. Dauvillier, J. Phys. Radium, Ser. VI T VIII, No.1, 1 (1927). 3. J. Bergengren, Compt. Rend. 171,624 (1920); Z. Phys. 3, 247 (1920). 4. A.E. Lind, Compt. Rend. 172, 1175 (1921); ibid. 175,25 (1922). 5. T. Magnusson, doctoral thesis, Uppsala University, 1942. 6. A. Sommerfeld and H. Bethe, Handbuch der Physik 24, 333 (1933). 7. F. Seitz, The Modern Theory o/Solids (McGraw Hill, New York), 1940. 8. H.M. O'Bryan and H.w.B. Skinner, Proc. R. Soc. London A 176, 229 (1940). 9. See, for example, Soft X-ray Band Spectra, edited by DJ. Fabian (Academic Press, New

York, 1968). 10. R.A. Matson, Adv. X-Ray Analysis 8, 333 (1965); R.A. Matson and R.c. Ehlert, Adv. X-

Ray Analysis 9, 471 (1966). 11. R.E. LaVilla and R.D. Deslattes, J. Chern. Phys. 45, 3446 (1966). 12. B.L. Henke, Adv. X-Ray Analysis 7, 430 (1964). 13. L.-O. Werme, B. Grennberg, J. Nordgren, C. Nordling, and K. Siegbahn, Nature 242,

453 (1973) 14. P.E. Best, J. Chern. Phys. 49, 2797 (1968). 15. V.1. Nefedov and V.A. Fomichev, 1. Struct. Chern. 9, 107 (1968). 16. R. Manne, J. Chern. Phys. 52, 5733 (1970). 17. A. Meisel, G. Leonhardt, and R. Szargan, Rjjntgenspektren und Chemische Bindung

(Akad. Verlagsgesellschaft, Leipzig, 1976).

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18. A.V. Kondratenko, L.N. Mazalov, and I.A. Topol in Vysokovozbyzjdennye Sostojanija Molekul, edited by E.V. Cobolev (Nauk, Novosibirsk, 1982).

19. J. Nordgren and C. Nordling, Commun. At. Mol. Phys. 13, 229 (1983); J. Nordgren, and H. Agren, ibid. 14, 203 (1984).

20. N. Kosuch, E. Tegeler, G. Wiech, and A. Faessler, Nucl. Instrum. Methods 152, 113 (1978).

21. P.L. Cowan, S. Brennan, R.D. Deslattes, A. Henin~, T. Jach, and E.G. Kessler, Nucl. Instrum. Methods A 246, 154 (1986).

22. J.-E. Rubensson, N. Wassdahl, G. Bray, J. Rindstedt, R. Nyholm, S. Cramm, N. Martensson, and J. Nordgren, Phys. Rev. Lett. 60, 1759 (1988).

23. J. Nordgren, G. Bray, S. Crainm, R. Nyholm, J.-E. Rubensson, and N. Wassdahl, Rev. Sci. Instrum. 60, 1690 (1989).

24. T.A. Callcott, K.-L. Tsang, C.H. Zhang, D.L. Ederer, and E.T. Arakawa, Rev. Sci. Instrum. 57, 2680 (1986).

25. J.-E. Rubensson, D. Mueller, R. Shuker, D.L. Ederer, C.H. Zhang, J. Jia, and T.A. Callcott, Phys. Rev. Lett. 64, 1047 (1990).

26. N. Wassdahl, A. Nilsson, T. Wiell, H. Antonsson, L.C. Duda, J.H. Guo, N. Mmensson, J, Nordgren, J.N. Andersen, and R Nyholm, Phys. Rev. Lett. 69, 812 (1992).

27. N. Wassdahl, G. Bray, S. Cramm, P. Glans, P. Johansson, R. Nyholm, N. Mmensson, and 1. Nordgren, Phys. Rev. Lett. 64, 2807 (1990).

28. S.H. Southworth, D.W. Lindle, R Mayer, and P.L. Cowan, Phys. Rev. Lett. 67, 1098 (1991).

29. M.a. Krause, 1. Chern. Phys. Ref. Data 8,307 (1979). 30. R Manne, J. Chern. Phys. 52, 5733 (1970). 31. H. Agren and J. Nordgren, Theor. Chim. Acta 58, 111 (1981). 32. F.P. Larkins and A.J. Seen, Physica Scripta 41, 827 (1990). 33. H. Agren and A. Flores-Riveros, J. Electron. Spectrosc. 56, 259 (1991). 34. H.W.B Skinner, Philos. Trans. R. Soc. London A 239,95 (1940). 35. P. Longe, in Advances in X-ray Spectroscopy, edited by C. Bonnelle and C. Mande

(Pergamon Press, 1982). . 36. 1. Nordgren and R Nyholm, Nucl. Instrum. Methods A 246, 242 (1986). 37. F. Senf, K. Berens v. Rautenfeldt, S. Cramm, C. Kunz, J. Lamp, V. Saile, J. Schmidt-May,

and J. Voss, Nucl. Ins.trum. Methods A 246, 314 (1986). 38. K.J. Randall, J. Feldhaus, W. Erlebach, A.M. Bradshaw, W. Eberhardt, Z. Xu, Y. Ma, and

P.D. Johnson, Rev. Sci. Instrum. 63, 1367 (1992). 39. G. Wentzel, Z. Phys. 31,445 (1925); M.J. Druyvesteyn, ibid. 43, 707 (1927). 40. N. Wassdahl, Thesis, Uppsala University, Acta Universitatis Upsaliensis 114 (1987). 41. C. Bonelle in Advances in X-ray Spectroscopy, edited by C. Bonnelle and C. Mande

(Pergamon Press, Oxford and New York, 1982). 42. RE. LaVilla, J. Chern. Phys. 56, 2345 (1972); ibid 57, 899 (1972). 43. L.a. Werme, J. Nordgren, H. Agren, C. Nordling, and K. Siegbahn, Z. Phys. 272, 131

(1975). 44. Y. Ma, N. Wassdahl, P. Skytt, 1. Guo, 1. Nordgren, P.D. Johnson, J.-E. Rubensson, T.

Boske, W. Eberhardt, and S.D. Kevan, submitted to Phys. Rev. Lett. 45. D. Ederer (private communication). 46. T. Wiell (private communication).

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47 A. Flores-Riveros, N. Correia, H. Agren, L. Pettersson, M. Backstrom, and J. Nordgren, J. Chern. Phys. 83, 2053 (1985).

48. J.A. Leiro, F. Werfel, and G. Drager, Phys. Rev. B 44, 7718 (1991). 49. E. Tegeler, N. Kosuch, G. Wiech, and A. Faessler, J. Electron. Spectrosc. 18, 23 (1980). 50. D.W. Lindle, P.L. Cowan, R.E. LaVilla, T. Jach, R.D. Deslattes, B. Karlin, J.A. Sheehy,

TJ. Gil, and P.W. Langhoff, Phys. Rev. Lett. 60, 1010 (1988). 51. G.W. Frazer, M.D. Pain, J.F. Pearson, J.E. Lees, C.R. Binns, P.S. Shaw, and J.R.

Fleischmann, Production and Analysis of Polarized X-Rays, SPIE, 1548 (1991). 52. C. Hague, presented at the Int. Conf. on the Physics of Vacuum Ultraviolet Radiation

(VUVI0), Paris, July 13-17, 1992.

SPIN ANALYSIS AND CIRCULAR POLARIZATION

N.V. SMITH AT&T Bell Laboratories Murray Hill, NJ 07974, USA

ABSTRACT. The use of spin analysis and circular polarization in spectroscopies of condensed matter using synchrotron radiation is surveyed. Recent examples of spin-polarized photoemission work are taken from work on the U5U beamline at Brookhaven National Laboratory and cover studies of chemisorption and magnetic multilayers. Examples of soft x-ray magnetic circular dichroism studies are taken from work on the "Dragon" beamline and cover elemental ferromagnets and "fingerprinting" of magnetic materials. Recommendations are offered on the operation of the third-generation sources and on the production of circularly polarized synchrotron radiation.

1. Introduction

The aim of these lectures is to offer a survey and some recommendations on the use of spin analysis and circular polarization in spectroscopies of condensed matter using synchrotron radiation. We shall begin with some basics and with some history of the field starting circa 1970. Illustrative examples of more recent work will be taken from results obtained on the U5U (undulator) and on the U4B (Dragon) beamlines at the National Synchrotron Light Source at Brookhaven. It should be emphasized that this selection is for the convenience of the lecturer and represents only a small part of a large worldwide effort. For more comprehensive surveys, the reader is referred to the recent books by Feder [1] and by Kirschner [2].

2. Some Basics and Some History

The phenomenon of spin polarization in photoemission from solids has two physical origins: (1) relativistic effects, i.e., spin-orbit interaction; and (2) magnetic effects, i.e., exchange interaction. The primary emphasis in this paper will be on magnetic effects, but let us first consider some relativistic effects.

2.1. RELATIVISTIC EFFECTS

An excellent illustrative example of relativistic effects in photoemission is the negative-electron­affinity (NEA) GaAs source of spin-polarized electrons [3,4]. Figure 1 shows the band structure of GaAs in the region of the fundamental gap and also the selection rules for optical transitions across the gap. The bottom of the conduction band is s-like. The top of the valence band is p-like

203

A.S. Schlachter and F.J. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 203-219. © 1994 Kluwer Academic Publishers.

204

V 8112 I7lj = -1/2 mj = +1/2

r'r ~ ~ ~

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \

Eg = 1.52 eV \ \ \ \

J \ \ \

\(f) @I 0+ \ \ 0- \

0\ \ \ \

\ \ \

P3I2 \ \ \

~ mj = -3/2 mj = -1/2 mj = +1/2 I7lj = +3/2 L \\\ A Pl/2 I7lj = -1/2 mj = +1/2

Figure 1. Optical selection rules underlying the operation of the negative-electron-affinity GaAs spin­polarized electron source. On the left is the band structure of GaAs showing the bottom of the conduction band, which is s-like, and the top of the valence band, which is p-like and spin-orbit split. The dipole­permitted optical transitions are shown on the right for both helicities, 0+ and (L, of the circularly polarized infrared light. Relative intensities are given by the circled numbers.

and is spin-orbit split into P312 and P1l2 states. If photoemission from the P1l2 level is suppressed, it is evident from Fig. 1 that one helicity of incident radiation will deliver a 3:1 ratio of up/down spin photoelectrons, whereas the other helicity will deliver a 1:3 ratio. This condition is achieved by using a sufficiently low infrared photon energy. It is also necessary to lower the vacuum level so that electrons excited into the s level can escape from the crystal. This is done by depositing overlayers of Cs and 0 on GaAs surfaces. The invention of the NEA GaAs spin-polarized source came out of the group of Siegmann in Zurich. It is now a standard tool in condensed matter physics and high-energy physics [5].

There is much activity, notably at BESSY, in the further study of relativistic effects in the vacuum-ultraviolet (VUV) range, particularly on transition metals. The interested reader is referred to the articles of Heinzmann [6] and Kirschner [2] for further details.

2.2. MAGNETIC EFFECTS

The first observations of spin-polarized photoelectrons from ferromagnets used Mott scattering and came also from Siegmann's group in Zurich [7]. The group's results on Ni caused a sensation because they indicated a positive asymmetry in the photoyield at threshold rather than the expected negative asymmetry. It has turned out that these early results were insufficiently resolved. If one approaches close to threshold on very well characterized samples, the spin

205

asymmetry is indeed negative [8]. The importance of the early work lies not in its correctness but in the demonstration that spin analysis is possible in photoemission.

Spin-polarized photoemission studies of magnetism then languished for more than a decade (1970-1983). The main obstacle to progress was an incompatibility between the needs for spin analysis (magnetized samples) and the needs for photoelectron energy analysis (very low ambient magnetic fields).

This incompatibility was eventually resolved by Raue et al. [9]. Using a "picture-frame" sample (a single crystal with a hole cut in it), this group confined the magnetic flux largely within the sample, thereby minimizing the deleterious influence on energy selection. The basics of the experimental configuration are shown in Fig. 2. The "transverse" configuration, i.e., magnetization in the plane of the surface, also confers advantages because it eliminates the need for a 90° deflection in order to convert longitudinal to transverse polarization.

Mott detector

[110]

Transfer manipulator to Leed! Auger and sputter gun

LJ------

Rotatable polarizer

Energy analyzer

<\\. Resonance V lamp

Figure 2. The experimental configuration of Ref. 9, which permitted both energy analysis and spin analysis of the photoelectrons. An important feature is the use of a "picture-frame" sample.

206

2.3. FERROMAGNETIC IRON

The culmination of this era is best exemplified by the results of Kisker and co-workers [10] on Fe at BESSY. By exploiting the continuum nature of synchrotron radiation, these workers were able to tune to the r-point of the Brillouin zone; this occurs at lim = 60 eV, and the results are shown in Fig. 3. Peak A can be identified with the r25' level in the minority-spin (.1) band structure. Peaks Band C correspond to the r12 and r25' levels in the majority-spin (i) band structure. One can read off immediately the exchange splitting from the energy separation between peaks A and C, and it is in reasonable agreement with first-principles band calculations. Perhaps more interesting is what happens on approaching the Curie temperature Tc. First, the exchange splitting does not go away, as might be expected from a Stoner model. Second, and even more interesting, is the observation that peak C starts to appear in the .1-spin spectrum, and a hint of peak A can be discerned in the i-spin spectrum. This intermixing is what one expects from the local band-structure picture of ferromagnetism, and the experimental results compare favorably with tight-binding simulations by Haines et al. [11], which are also shown in Fig. 3.

3. Spin-Polarized Photoemission at the U5U Beamline at Brookhaven

A facility has been established at Brookhaven that is dedicated to spin-polarized photoemission studies of bulk and surface magnetism, magnetic thin films, and magnetic multilayers [12]. Offered here is a description of this bearnline, some results, and some recommendations. The U5U beamline is fed by synchrotron radiation from an undulator. It is therefore a prototype of the kind of beamline that will become standard at the third-generation synchrotron radiation sources presently under construction.

3.1. SPIN ANALYSIS

A principal instrumental innovation in this facility is the use of the miniature spin detectors invented and further perfected by Pierce and Celotta and coworkers [13, 14]. The original experimental configuration is illustrated in Fig. 4. Electrons emerging from the exit slit of the hemispherical energy analyzer impinge on a gold scattering target at an energy of -150 e V. This is to be contrasted with the 100 keVin conventional Mott scattering. There is still a significant asymmetry in the backscattering, and from this the polarization of the incoming beam can be inferred. The obvious advantage of this detector is its small size. It is fitted onto the back end of a commercial analyzer, which is itself mounted on a goniometer. Thus the full power of angle­resolved photoemission can be retained, but with the added capability of spin analysis. A slight disadvantage is the need to calibrate the detector against a detector of known spin asymmetry. This is readily done, however, by using the dramatic results on clean Fe(OOI), shown in Fig. 3, as a calibration standard.

3.2. CHEMISORPTION STUDIES

A demonstration of the performance of this facility is illustrated in Fig. 5, which shows spin­resolved photoemission spectra taken on Fe(OOI) with one monolayer of adsorbed oxygen in the

207

Theory Experiment

A-" " T=O •• tl T= 0.3 Tc • • I I • I t

• 1 • I I I

• I • I

C I I

I I I I • •

en • I • . t:: • C :::l I

.ci , • . ." ,. ,." . .,..,. . . ,."

,.

.... , ~ .. ~

T= 0.85 Tc 'en c " (J) , ~ - , • c , • •

\

-3 -2 -1 o -3 -2 -1 o Initial state energy (eV)

Figure 3. Comparison between the angle-resolved photoemission experiments or Fe(OOl) in Ref. 10 with the theoretical simulation in Ref. II.

Fe(OOl) p(l X 1)0 configuration [15]. The features within 3 eV of the Fermi level arise from the Fe 3d valence band and are essentially the same as those seen on clean Fe(OOl).

The features in the region 4-8 e V below Ep are due to the adsorbed oxygen. By using photon polarization selection rules, the two peaks can be identified with the pz or Px orbitals. For angles

208

(a)

Gold-scattering target

(b)

Energy­filtering grids

UV light

Channel plates

Anodes

Figure 4. Experimental arrangement on the U5U beamline at Brookhaven: (a) schematic of the picture­frame sample and the hemispherical analyzer with the miniature spin detector mounted after the exit slit; (b) an expanded schematic of the miniature spin detector.

of p-polarized photon incidence close to nonnal (see 8j = 35° spectra in Fig. 5), there is a smaller component of the electric vector perpendicular to the surface, and this will tend to suppress emission from the perpendicularly oriented pz orbital. Larger angles of incidence will favor the Px orbital. (A coordinate system has been chosen in which the photons are incident in the xz­plane; so the Py orbital is never excited.)

The principal observation is that the 0 p orbitals are themselves exchange-split, and that the splitting is different for pz and Px symmetry. By using the angle-resolving capability of the instrument, it is possible to map the O-derived bands across the surface Brillouin zone; the results are also shown in Fig. 5. It is seen that there is a reversal in the ordering of the pz and Px orbitals

Binding energy (eV)

1

2

~ 3.:/ >. ~ Q) s:: Q)

Cl s:: '6 s:: ill

9

10

o

209

0.25 0.75 1.00

Figure 5. Spin-polarized photoemission studies of the chemisorption system Fe(OOI)p(1 x 1)0. On the left are photoelectron spectra separated into i-spin and .I.-spin for two angles of photon incidence: (a) 9j = 70° near grazing incidence favors emission out of the 0 Pz orbital, (b) 9j = 35° near normal incidence tends to suppress the Pz orbital and favor emission out of the in-plane Px orbital. On the right, are the derived spin­separated E(kll) dispersion relations for adsorbed 0 p orbitals; the shaded area represents the projection of the bulk Fe valence bands.

and that the large exchange splitting of the pz orbital at the zone center diminishes on approaching the zone boundary. This effect is not reproduced in first-principles theoretical calculations [16, 17], and remains unexplained.

3.3. MAGNETIC MULTILAYERS

There is much current excitement about magnetic thin films [18] and periodic multilayer structures having layers of magnetic material separated by spacer layers of nonmagnetic material [19]. It has been observed that the coupling between successive magnetic layers alternates between ferromagnetic and antiferromagnetic, depending on the thickness of the nonmagnetic spacer layers [20-23]. The additional observation of "giant magnetoresistance" in the Fe/Cr system [21] is of interest to the developers of pick-up heads in the magnetic recording industry.

As an example of a relevant study, we examine here some results obtained on the U5U beamline on Ag overlayers on Fe(OOI). This system has the advantage that the Ag d band lies 4-7 eV below EF. Thus the magnetically active Fe 3d states closer to EF can be studied with minimal interference from the substrate d bands.

210

Figure 6 summarizes results obtained by Brookes et al. [24] for progressive additions of monolayers of Ag. On the clean substrate, there is a minority-spin surface state at -2 eV below Ep. With successive addition of Ag monolayers, this feature survives as an interface state and moves upwards in energy, crossing Ep at -3 monolayers. As inferred from the spin-polarization asymmetry, also shown in Fig. 6, this state retains its minority spin.

Tight-binding slab calculations reproduce this behavior and indicate that the interface states permeate the entire Ag film. The states are perhaps more appropriately regarded as standing waves or "quantum-well states" within the film [25]. The Ep crossover may have some relevance to the alternation between ferromagnetic and antiferromagnetic coupling in multilayers.

~ c: 0

~ N .;:: ell (5 a.

40

20

0

-20

-40

40

20

0

-20

-40

40

-20

-40

1¥W&a\r.:,,;J - - ----+3 MI Ag !

111- .!.A .... -.. ...... ~~ +2 MI Ag

•

+1 MI Ag ~--~--~---+--~~--+-~

o -------20 t -40 Clean Fe(001)

-4 -3 -2 -1 EF

Binding energy (eV)

t Clean Fe(001)

-4 -3 -2 -1 EF

Binding energy (eV)

Figure 6. Development of the minority-spin interface state with increasing thickness of Ag on Fe(100). Spin-integrated spectra are shown on the right for the clean Fe(100) surface and for Ag coverages of 1, 2 and 3 monolayers. The corresponding spin-asymmetry spectra are shown on the left and demonstrate the J,-spin character of the interface state and its evolution from a J,-spin surface state of clean Fe(100).

211

3.4. UNDULATORS!SUGGESTIONS

The U5U beamline at Brookhaven has an interesting history, since none of its original components were intended for the purpose for which they were eventually used. It took its radiation originally from an undulator intended as part of a free-electron laser. The monochromator was built by Malcolm Howells for an entirely different project. The input and output focusing optical elements were acquired for other bearnline projects. This is not a recommended procedure, especially for the third-generation synchrotron radiation sources!

The U5U beamline is being progressively upgraded. The first upgrade is the installation of a new undulator under the direction of S.D. Bader. The characteristics of this undulator are shown in Fig. 7, and it has already been demonstrated that it meets its specifications. At the lower gap spacing (K = 3), the first harmonic can be forced down to about firo = 10 eV. At the higher gap spacing (K = 1), the third harmonic can be stretched out to firo - 150 eV. The range of the U5U undulator fills a niche not well serviced by the third-generation synchrotron radiation sources as presently conceived. This is evident from the comparison shown in Fig. 8 of the calculated output fluxes of the Advanced Light Source (ALS), the Advanced Photon Source (APS), and other sources. The output of the U5U undulator is in the upper left comer. It is seen that the U5U undulator beamline should remain competitive even after the third-generation sources have come into operation.

This leads to the suggestion that serious consideration be given to operating the ALS at Lawrence Berkeley Laboratory at two energies: the nominal energy, 1.5 GeV, to service the soft x-ray community and the minimum design energy, 1.0 GeV, to better service the vacuum ultraviolet community. Such a scenario has been proposed for the Daresbury Advanced Photon Source (DAPS) [26]. If such a mode of operation is indeed possible, there would be an obvious disposition of branch lines. On each straight section, the principal beamline would deliver photons using the higher operating energy. This line could then branch downstream from the monochromator. When operating at the lower machine energy, one would branch upstream from the principal monochromator, delivering beam to a secondary low-energy monochromator whose output could itself be branched.

4. Circularly Polarized Synchrotron Radiation

Closely related to the detection of photoelectron spin is the issue of photon circular polarization. It is well known that the out-of-plane radiation from bend-magnet synchrotron radiation sources has a high degree of circular polarization and that it is possible to alternate left/right helicity. In the low-photon-energy regime, this property has been exploited for some time, and the reader is referred to the review by Heinzmann and Schtinhenze [6] for further details. More recently, this method has been extended to the soft x-ray and hard·x-ray ranges. We discuss here some recent soft x-ray results obtained on the Dragon monochromator at Brookhaven.

4.1. MCD ON ELEMENTAL FERROMAGNETS

The total x-ray absorption (0+ + 0_) and the magnetic circular dichroism (MCD) (0+ - 0_) at the L2 (2p1I2 ~ 3d) and L3 (2p3/2 ~ 3d) edges in iron are shown in Fig. 9. Qualitatively, the results

212

(a)

§' co ~ 0 ,.... ci "0 Q)

~ .J::: a. ->< ::J

u:::

(b) -C\I

E E ~ "'0 e E ~ co ~ 0

~ Q)

~ .J::: a. -(/J (/J Q) c: -.J::: Cl ·c co

1015

1014

1013

1012

1011

1010

109

0

5e + 16

4e + 16

3e + 16

2e + 16

1e + 16

0

Pinhole = 0.14 mrad x 0.14 mrad

_ FluxK=1.0 - FluxK=3.0

50 100 Energy (eV)

First harmonic

Third harmonic

50 100 Energy (eV)

150

150

Figure 7. Characteristics of the new undulator on the U5U beamline at Brookhaven: (a) theoretical output flux for two values of the gap; (b) anticipated brightness out of the first and third harmonics as the gap is swept.

u~,_o_ _ ~3.9 W13.6 ~~W - - -, '- - 1 - -, , -, ,-... - - -'- - "'" - - - - .... -.-

NSLS sources ALS sources APS sources

\

\ X17

\

\

\

\

213

1012~~ __ ~~~ __ ~ __ ~ __ ~ __ ~~~ __ ~~~~~~~~~

101 103

Photon energy (eV)

Figure 8. Comparison of the flux output of various second- and third-generation synchrotron radiation sources. The output of the Brookhaven U5U undulator, shown earlier in Fig. 7, is seen on the upper left.

are consistent with expectations; the absorption displays "white lines," and the MCD displays a negative spike at the L3 edge and a positive spike at the L2 edge. Quantitatively, however, the results are at variance with the simplest model, that of Erskine and Stem [29]. In that model (more appropriate to Ni than Fe), it is assumed that the unoccupied d holes are entirely of J, spin. It then follows that the intensity I(L3)II(L2) ofthe white lines should be 2:1 in the total absorption and -1: 1 in the MCD. One can easily see that the results of Fig. 9 are in disagreement with this simple model.

One is released from the inflexibility of the Erskine-Stem model by introducing spin-orbit interaction into the 3d valence band. The author and his colleagues have recently attempted to quantify this effect using a relativistic tight-binding representation [30]. The calculational results, shown in Fig. 10, are seen to be in disagreement with the experimental data in two ways. First, the calculation predicts structure in the spectra; this is not observed even though it is within the resolution capability of the Dragon. Second, there is no plausible value of ~ that can reproduce the L31L2 intensity ratio in the MCD spectrum. This can be traced to the positive excursion of (0'+ - 0'_) just above EF, which thwarts attempts to reproduce a ratio departing significantly from -1:1.

214

100

80

(a) I ron L2,3 edges

60

40

20

0L-~ ______ L-____ ~ ______ J-____ ~

5 (b) L3

o

-e -5 Cll - -10

-15

-20

700 710 720 730 Photon energy (eV)

Figure 9. Experimental data on Fe for (a) the total x-ray absorption cross section (0+ + 0_) and (b) the MCD cross section (0+ - 0_).

4.2. MAGNETIC FINGERPRINTING

Even without a quantitative understanding of soft x-ray MCD, we can assemble a repertoire of qualitative experience. This will be treated in more detail by other speakers at this institute. As a particularly toothsome example, we consider here the results of Rudolf et al. [31] on the magnetic dynamics of the gadolinium iron garnet Gd3Fe6012. This material has two antiferromagnetically coupled sublattices of Fe3+ ions, three per formula unit on tetrahedral sites and two on octahedral sites. The Gd3+ ions are weakly coupled antiferromagnetically to the net moment of the Fe3+ system. At room temperature, the Gd3+ system is essentially disordered, and the net magnetization is carried by the Fe3+ system. This is evident in Fig. 11 in which the MCD spectrum shows a strong negative spike at the Fe L3 edge at T = 300 K. The antiferromagnetic

(a)

60

~ .§ 40

.ri ... <U - 20 tl + -',: +

0 0 ;!. ·00

(b) c: Q) -60 .5 c: 0

:;::::

e- 40 0 II) .c <{

20

0

~ = 0.00 Ry 16

~ = 0.01 Ry

0.6 0.8 Energy (Ry)

1.0

8

-~ 0 ·2 ::J

.e -8 ~ tl -16 I

b ;!. ·00 16 c: ~ .5 8 Cl o 0 ~

-8

-16

215

Energy (Ry)

Figure 10. Calculated L3 (solid curves) and L2 (dashed curves) spectra for Fe using a tight-binding representation and without regard for the exclusion principle. [Only that part of the spectrum above the Fermi level (indicated as the thin vertical line) is observable in x-ray absorption measurements.] Panels (a) and (b) show the total absorption cross section for two values of the 3d valence spin-orbit parameter ,;. Panels (c) and (d) are the corresponding MCD cross sections. The exchange splitting was taken as Aex = 0.125 Ry.

coupling between the Gd3+ and Fe3+ systems is also evident as the weak positive spike at the Gd Ms edge at T = 300 K.

At lower temperature, the Gd system becomes ordered. Since the Gd magnetic moment (-21 J.lB per formula unit) is much larger than the net Fe moment (-5 J.lB per formula unit), there is a temperature-the "compensation temperature" (actually 288 K)-below which the Gd system dominates and the macroscopic magnetization signatures flip direction. This is clearly evident in the Fe L3 and Gd Ms MCD spectra at T = 77 K shown in Fig. 11.

Thus, just the sign and magnitude of the MCD spike offers a "fingerprinting" measure having atomic specificity for the study of the magnetic dynamics of compound materials. This topic is taken up in greater detail by other lecturers at this institute.

216

60

~ 60 :Fe L2,3 T= 300 K (a)

~ 50 I/) 50

l I/) c: c:

CI)~ -ii CI)~ 40 - I/) 40 : i - I/) c:~ c::t:::: -- c: ---- i J, -- c: 30 c: ::J c: ::J o - 30 o -:;::::;€ :;::::;€ 20 e-as 20 e-as o~

~ o~

I/) I/) 10 .c 10 .c « « 0 0

Gd M4,5 T = 300 K (a)

-it ---- i J,

-.

~ ~ ~ (b) ~ -2 -2 15 ::J ::J

(b)

.e 0 .ci 10 .... ~ ~

~ -10 ~ 5 I/) -iii c: c: 0 ~ .s

-20 -~ c c -5 () ()

~ -30 ~ -10

~

60

~ 60 Fe L2,3 T= 77 K (c)

~ 50 I/) 50 I/) c: c: .s I/) -it CI)~ 40

- I/) c.:t:::: 40 c:.:t:::: -- c: ---- i J, -- c: c: ::J c: ::J 30 0 30

i~ 0 :;::::;.e +=i.e e- as 20 e-ii; 20 o ~ o~

I/) I/) .c 10 .c 10 « «

0 0

I T=77K (c) a Gd M45 fl ' :z: -it Jl \ ----f!

-~ Iii" ~ ~ (d) c:

5 ::J ::J

.e 20 .e

..!!!. ~ 0 ~

~ (d)

~ ~yI

~ 10 ~ -5 -iii I/)

c: c: .s ~ -10 .5 0 C c -15 () ()

~ -10 ~ -20 705 710 715 720 725 730 1180 1200 1220 1240

Photon energy (eV) Photon energy (eV)

Figure 11_ Fe L2,3 and Gd M4,S absorption and Men spectra measured with circularly polarized soft x rays in the magnetically oriented single-crystal garnet Gd3Fes012- Results are shown for two temperatures (300 K and 77 K) above and below the "compensation temperature" (288 K)_

217

4.3. INSlRUMENTAL CONSIDERATIONS/RECOMMENDATIONS

Alternation between above-plane and below-plane acceptance to acquire circular polarization works only for synchrotron radiation from bend magnets. Out-of-plane emission from conventional undulators is complicated and offers no circular-polarization advantage. Therefore, there is a strong case for the construction of advanced insertion devices dedicated to the generation of circularly polarized synchrotron radiation.

But, we must separate off a low-photon-energy region (tiro below, say, 50 eV) in which an advanced circularly polarized source would be inappropriate. We make an excursion here into elementary metal optics. The triple-reflection polarizer, routinely used to convert unpolarized light into linearly polarized light [32], can convert linearly polarized light into circularly polarized light. The principle is illustrated in Fig. 12. The Fresnel formulae for the reflectances of s- and p-polarized light are:

where 9 is the angle of incidence and E(= EI - iE2) is the dielectric function of the reflecting material. Figure 12 shows the dependence on 9 of r s, rp, and the differential phase change

\ I \ I

\ M I

e'~ -#. ~ ~. M1 I Ma /

I I

180r-~~---------.-.-. 1.0

0.8

0.6 ~c.

~ih 0.4

Ci Q) :g. 90 <l

0.2

OL-____ ~ ______ ~~-L~. o 30 60

9 (de g)

Figure 12. Operating principle of the triple-reflection quarter-wave retarder .. The geometry of the device is shown on the left and indicates the two positions about the beam axis for left and right circular polarization. On the right is shown the reflectances for s- and p-polarized light as a function of angle of incidence on a gold mirror at If<o = 15 eV.

218

A( = Op - Os) for a gold mirror at lim = 15 e V. By appropriate choice of incidence angles, one can arrange for a cumulative phase difference of 900 • By appropriate choice of orientation (a) of the device about the axis of the beam, one can arrange for equal s and p amplitudes in the emerging beam. One therefore has a simple quarter-wave retarder. Such a device was proposed a decade ago by Johnson and Smith [32] and has recently been demonstrated in Japan by Koide et al. [33].

The closing recommendation is that special insertion devices for the production of circularly polarized undulator radiation would be desirable in the higher photon-energy range (lim ~ 50 eV). In the lower photon-energy range, a low-technology alternative is available.

References

1. R. Feder (editor), Polarized Electrons in Surface Physics (World Scientific, Singapore, 1985).

2. J. Kirschner, Polarized Electrons at Surfaces (Springer-Verlag, Berlin, 1985). 3. D.T. Pierce, F. Meier, and P. Zurcher, Appl. Phys. Lett. 26, 670 (1975). 4. D.T. Pierce and F. Meier, Phys. Rev. B 13,5484 (1976). 5. D.T. Pierce and R.J. Celotta, in Optical Orientation, edited by F. Meier and B.P.

Zaharchenya (Elsevier, Amsterdam, 1984). 6. See for example U. Heinzmann and G. Schonhenze, Chapter 11 of Ref. 1. 7. U. Banninger, G. Busch, M. Campagna, and H.C. Siegmann, Phys. Rev. Lett. 25, 585

(1970). 8. W. Eib and S. F. Alvarado, Phys. Rev. Lett. 37,444 (1976). 9. R. Raue, H. Hopster, and R. Clanberg, Phys. Rev. Lett. 50,1623 (1983). 10. E. Kisker, K. SchrOder, W. Gudat, and M. Campagna, Phys. Rev. B 31, 329 (1985). 11. E. Haines, R. Clanberg, and R. Feder, Phys. Rev. Lett. 54, 932 (1985). 12. P.D. Johnson, N.B. Brookes, S.L. Hulbert, R. Klaffky, A. Clarke, B. Sinkovic, N.V. Smith,

R. Celotta, M.H. Kelly, D.T. Pierce, M.R. Scheinfein, B.J. WacIawski, and M.R. Howells, Rev. Sci. Instrum. 63,1902 (1992).

13. J. Unguris, D.T. Pierce, and R.I. Celotta, Rev. Sci. Instrum. 57, 1314 (1986). 14. M.R. Scheinfein, D.T. Pierce, J. Unguris, J.J. McClelland, and R.I. Celotta, Rev. Sci.

Instrum. 60, 1 (1989). 15. A Clarke, N.B. Brookes, P.D. Johnson, M. Weinert, B. Sinkovic, and N.V. Smith, Phys.

Rev. B 41, 9659 (1990). 16. H. Huang and J. Hermanson, Phys. Rev. B 32, 6312 (1985). 17. S.R. Chubb and W.E. Pickett, Phys. Rev. Lett. 58,1248 (1987). 18. D. Pescia (editor), Magnetism in Ultrathin Films, special issue of Appl. Phys. A 49, (1989). 19. L.M. Falicov, D.T. Pierce, S.D. Bader, R. Gronsky, K.B. Hathaway, H.I. Hopster, D.N.

Lambeth, S.S.P. Parkin, G. Prinz, M. Salamon, I.K. Schuller, and R.M. Victora, J. Mater. Res. 5, 1299 (1990).

20. P. Grunberg, R. Schreiber, Y. Pang, M.B. Brodsky, and H. Sowers, Phys. Rev. Lett. 57, 2442 (1986).

21. M.N. Baibich, J.M. Broto, A. Fert, F. Nguyen Van Dan, F. Petroff, P. Eitenne, G. Creuzet, A. Friederich, and J. Chazelas, Phys. Rev. Lett. 61, 2472 (1988).

22. B. Heinrich, Z. Celinski, J.F. Cochran, W.B. Muir, J. Rudd, Q.M. Zhong, A.S. Arrott, K. Myrtle, and J. Kirschner, Phys. Rev. Lett. 64, 673 (1990).

23. W.R. Bennett, W. Schwarzacher, and W.F. Egelhoff, Jr., Phys. Rev. Lett. 65, 3169 (1990).

219

24. N.B. Brookes, Y. Chang, and P.D. Johnson, Phys. Rev. Lett. 67, 354 (1991). 25. J. Ortega and F.J. Rimpsel, Phys. Rev. Lett. 69, 844 (1992). 26. The Daresbury Advanced Photon Source (DAPS) Feasibility Study, Report to The Science

Board of the (UK) Science and Engineering Research Council (1991). 27. See for example: C.T. Chen, F. Sette, Y. Ma, and S. Modesti, Phys. Rev. B 42, 7262 (1990). 28. See for example: G. Schiitz, W. Wagner, W. Wilhelm, P. Kienle, R. Zellar, R. Frahm, and

G. Materlik, Phys. Rev. Lett. 58, 737 (1987). 29. J.L. Erskine and E.A. Stern, Phys. Rev. B 12,5016 (1975). 30. N.V. Smith, C.-T. Chen, F. Sette, and L.F. Mattheiss, Phys. Rev. B 46,1023 (1992). 31. P. Rudolf, F. Sette, L.R. Tjeng, G. Meigs, and C.T. Chen, J. Magn. Magn. Mat. 109, 109

(1992). 32. P.D. Johnson and N.V. Smith, Nucl. Instrum. Methods 214, 505 (1983). 33. T. Koide, T. Shidara, M. Yuri, N. Kandaka, and M. Fukutani, Appl. Phys. Lett. 58, 2592

(1991).

X-RAY MAGNETIC CIRCULAR DICHROISM: BASIC CONCEPTS AND THEORY FOR 3D TRANSITION METAL ATOMS

J. STOHR AND Y. WU IBM Research Division Almaden Research Center 650 Harry Road San Jose, CA 95120, USA

ABSTRACT. We discuss the basic theory underlying the x-ray magnetic circular dichroism technique with emphasis on 3d transition-metal atoms. The basic concepts are developed from an elementary level and illustrated by model calculations using an atomic picture and a simple Hamiltonian that accounts for two key interactions associated with the dichroism effect: the spin-orbit and exchange interactions. We present calculations for a one-electron model that is shown to be equivalent to a configuration-based model for the tP configuration and discuss the relationship between the measured dichroism signal and the spin, orbital, and total magnetic moments. Our results can be cast into a simple two-step model that considers the core shell as the "source" of photoelectrons with polarized spin and/or angular momentum and the valence shell as a spin- or angular-momentum resolving "detector." We also point out that, according to the initial-state rule, the dichroism intensity is linked to the ground-state d-shell occupancy and moments.

1. Introduction

Michael Faraday's discovery in 1846 that the polarization of visible light may be changed upon transmission through a magnetic material opened the door for future investigations of the magnetic properties of matter by means of electromagnetic radiation [I]. Today's laser methods for investigating magnetic properties are based on the observation of changes in the light's polarization upon transmission (the Faraday effect) or upon reflection (the magneto-optical Kerr effect). Because typical laser light has energies in the 1-4-e V range, the Faraday and Kerr effects involve electronic transitions from filled to unfilled electronic valence states, as schematically shown in Fig. 1 (a).

One can qualitatively understand the change in polarization of the light by considering an electronic transition between two magnetic band states characterized by quantum numbers L, S, J, and MJ, as illustrated in the figure. We assume that the incident laser light is linearly polarized, i.e., that it is a coherent sum of left and right circularly polarized light. Because the dipole selection rule is different for right (!!.MJ=MJ- Mj =+1) and left (dMJ=-l) circularly polarized light, the components may be absorbed differently, depending on the nature of the two band states. The emitted radiation will then reflect this imbalance and will therefore be elliptically polarized [2], with the major polarization axis rotated relative to that of the incident light.

In 1975, Erskine and Stem [3] considered excitations between a core state and a valence state and performed a simple calculation on the expected effect for the M3,2 edge in Ni metal. This paper constitutes the birth of what is now known as x-ray magnetic circular dichroism (XMCD)

221

A.S. Schlachter and F.J. Wuilleumier(eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 221-250. © 1994 Kluwer Academic Publishers.

222

(a) Magneto-optical Kerr effect and Faraday effect

Energy

Conduction band

n Valence

band

-> H

t

(b) X-ray magnetic circular dichroism

-> M

t Spin "up" Spin "down"

Conduction

* t

EF band

~ 2P3l2 2P1/2 Core

15

Figure 1. Schematic of energy bands and electronic transitions involved in (a) the magneto-optical Kerr effect and Faraday effect and (b) in x-ray magnetic circular dichroism. In (a), the selection rules for left (llMJ = -1) and right (llMJ = 1) circularly polarized light are indicated, and transitions occur between filled and unfilled electronic band states. In (b), transitions occur from a core shell (typically L shell) to empty conduction-band states above the Fermi level labeled EF. For magnetic materials, spin-up and spin-down bands are unequally populated. In an external magnetic field H in the down direction, the magnetization M is also in the down direction as is the electron spin of the band with the smaller electron population (minority band). The electron spin in the majority band is up, as shown.

spectroscopy. The first experimental demonstration of the effect, however, did not occur until 1987 when Gisela Schlitz and co-workers [4] measured the XMCD signal at the K edge of Fe metal. In the meantime, Thole, van der Laan, Sawatzky, and co-workers had proposed a different (and, at the time, experimentally more convenient) way to obtain magnetic information through x­ray absorption. In 1985, they published calculations for rare-earth materials [5] that revealed a strong dependence of the near-edge x-ray absorption fine structure (NEXAFS) on the orientation of the magnetization direction relative to the E vector for linearly polarized x rays. The experimental proof was provided a year later [6]. This method is now called x-ray magnetic linear dichroism (XMLD) spectroscopy, and it provides information complementary to that from XMCD spectroscopy.

Besides the XMCD and XMLD techniques, which are based on x-ray absorption, other x-ray techniques have been used to obtain magnetic information. Siddons et al. [7] have measured the rotation of the plane of polarization following the transmission of linearly polarized x rays through a thin magnetic sample, which is the x-ray analogue of the Faraday effect. As reviewed by Gibbs [8], many x-ray scattering experiments, both nonresonant and resonant, i.e., near an absorption edge, have also been performed. The Kerr and Faraday as well as the XMCD and XMLD effects originate from electric dipole transitions driven by the electric field of the electromagnetic radiation. The electric field E does not act on the electron spin directly but only indirectly through the orbital momentum and the spin-orbit coupling. This is in contrast to nonresonant magnetic x-ray scattering, which arises from the direct interaction of the magnetic field vector B with both the spin and the orbital moments [8]. Magnetic x-ray scattering is much weaker than conventional charge scattering, and therefore such experiments are much more demanding in terms of photon flux. On the other hand, magnetic x-ray dichroism is a relatively large effect, of similar magnitude to conventional x-ray absorption structure, and therefore has the same requirements in photon flux as conventional x-ray absorption measurements.

223

The use of x rays for studies of the magnetic properties of matter is more than a simple extension of laser-based investigations because it offers unique capabilities. Most conventional techniques used for the study of magnetic materials measure the total magnetic response, which, for complex materials, contains contributions from the various magnetic elements. The conventional Faraday and Kerr techniques are no exception since the band structure of the material is a complicated function of the elemental composition. On the other hand, by use of tunable x rays, one can select specific elements in the sample through one of their characteristic absorption edges. Since the XMCD and XMLD effects are associated with the fine structure near such edges, one can, through x rays, study the magnetic properties of a complex material element by element. Better yet, different edges of the same element provide information on the magnetic contributions of different kinds of valence electrons through the dipole selection rules. For example, as shown in Fig. 1 (b), excitation near the L3 and L2 edges will yield information on the empty d-like density of states because of the p ~ d dipole selection rule. Similarly, K-edge studies are sensitive to the p-like component of the empty density of states. Another significant advantage of the magnetic x-ray dichroism technique is that it can be used for element-specific magnetic imaging [9]. These unique capabilities promise to make the XMCD and XMLD techniques powerful tools for the study of complex magnetic materials and to provide important scientific and technological information not obtainable by other methods.

The purpose of the present article is the review of the basic principles of the XMCD effect. The emphasis will be on elementary physical insight by model calculations. We emphasize atomic theory, first, because rigorous inclusion of the various solid-state effects is difficult and, second, because x-ray absorption is a local atomic probe and therefore atomic effects still play a significant role when the atom is embedded in a solid. We emphasize the use of circularly polarized x rays since, in our view, they provide the most powerful and versatile tool for the investigation of magnetic materials. XMCD is the technique of choice for the investigation of ferro magnets, ferrimagnets, and paramagnetic systems. Because it depends on the existence of a net magnetic moment along a preferred magnetic direction (magnetic handedness), it fails for certain antiferromagnets (i.e., systems in which, for a given atom in a specific sublattice, the magnetic coupling is antiferromagnetic). In this case, the complementary XMLD technique may still be used because it only requires the existence of a magnetic anisotropy axis and the effect depends on the angle between the x-ray E vector and the anisotropy axis.

Here we shall further restrict our discussion to d transition metals and compounds. Conventional L-edge absorption spectra of such systems have previously been treated theoretically by Zaanen et al. [10] and by de Groot et al. [11]. Dichroism spectra have been treated by Ebert et al. [12, 13], van der Laan and Thole [14], Smith et al. [15, 16], and Jo [17]. Experimental XMCD spectra for 3d metals have been reported by Chen et at. [18, 19, 15,20], Tobin et al. [21], and Wu et at. [22]; for 4d metals, by Wu et al. [23]; and for 5d metals, by SchUtz et al. [24, 25]. Extension of the presented concepts to rare earths and actinides is straightforward, and in fact, the rare earths have been extensively treated in the literature in theory [5,26-28] and experiment [6, 29-31].

2. Interaction Energies for 3d Transition Metals

It is well known that the magnetic properties of the transition metals are mostly due to their d electrons [32]. The most important magnetic transition-metal atoms are Fe, Co, and Ni, which, in the elemental metals, have valence-shell configurations close to 3d', 3fi8, and 3cfJ, respectively.

224

In describing the magnetic properties, it is important to first consider the relative size of the various interactions that affect the d shell in a solid, e.g., for the metals. There are five interactions whose relative importance may be judged by their characteristic energies:

• The interaction of the d electrons on one site with those on neighboring sites, resulting in a bandwidth of3-5 eV [33]

• The Coulomb and exchange interactions between the d electrons on the same site, resulting in a multiplet splitting of about 2-5 eV [10, 11]

• The magnetic exchange interaction giving rise to an exchange splitting in the d band of 1-2 eV [32]

• The spin-orbit interaction within the d shell, which for the 3d metals is 0.05-0.1 e V [34] • The interaction of the d electrons with the local point charge potential of the surrounding

atoms, characterized by a crystal field splitting of about 0.05-0.1 eV [33]. Since we are interested in probing the magnetic characteristics of the d valence shell, we need

to excite p core electrons, i.e., utilize p ~ d dipole transitions. It is therefore important to consider the interactions affecting the p core shell. For the 3d elements Fe, Co, and Ni, it is advantageous to excite the 2p shell for XMCD experiments because L3,2 edges exhibit much larger edge jumps or signal-to-background ratios than the shallower 3p or M3,2 edges. Also, the M3 and M2 edges overlap energetically. The largest splitting of the p shell is caused by the spin-orbit interaction, which leads to the well known separation of the L3 (2p3/2) and L2 (2P1l2) edges. The measured binding energies for the edges are: Fe: 707 eV (L3), 720 eV (L2); Co: 778 eV (L3), 793 eV (L2); Ni: 853 e V (L3), 870 e V (Lz). Hence the spin-orbit splitting ranges from 13 to 17 e V. Also important are the p - d Coulomb and exchange interactions, which are about 3-6 e V [11] and for transition-metal compounds give rise to detailed multiplet splitting in the spectra [35, 11, 14].

At present, it is difficult if not impossible to treat all the interactions listed above from first principles in a single calculation. In this paper, we will concentrate only on those interactions that are of fundamental importance for the origin of the XMCD effect, i.e., the exchange and spin­orbit interactions. Band-structure effects will be dealt with mostly in a split atomic level model, similar to the rigid-band model used by Stoner [36] to explain ferromagnetism. Only in the last section shall we consider a wave-vector-dependent band model. Exchange and correlation effects within the d shell and between the p core and d valence shell will only be touched upon for the example of the simple J8 initial-state configuration. Such effects are absent in the one-electron model used here or in a configuration model for the special case of a cP ground-state configuration. We will ignore crystal field effects altogether, since their consideration is not needed for the basic understanding of the XMCD effect.

3. L-Edge X-Ray Absorption: One-Electron and Configuration Picture

In the literature, two pictures are commonly used to describe the x-ray absorption process, and it is important to understand their difference since terms like "initial" and "final" state have opposite meanings.

The first picture, the so-called one-electron picture, focuses on the transition of an electron from one orbital to another. This picture ignores what happens to all other electrons during the excitation process and is therefore also referred to as the "active electron" approximation [37]. The beauty of this picture is its intuitive nature and simplicity. When it is applied to L-edge absorption spectra, the electron is excited from the spin-orbit split 2P312 and 2p 112 levels to empty

225

d valence states, as shown in Fig. 1 (b). The splitting arises from the spin-orbit interaction with the Hamiltonian

(1)

As discussed above, ~p is large for the 2p shell, and therefore the splitting is always observed in L-edge spectra for d transition metals. Because of the localized nature of the d valence states, such transitions are very intense and are often referred to as "white lines" owing to their first observation on photographic plates used in early spectrographs. Because of the dominance of the p ~ d over the p ~ s transition intensity, we shall ignore p ~ s transitions in the following. L­edge spectra of the heavy 3d transition metals Fe, Co, Ni, and Cu are shown in Fig. 2. Except for Cu, which has the electronic configuration dlO and therefore a filled d shell, the spectra exhibit strong white lines at the L3 and L2 absorption edges, as expected from their unfilled d shell configurations. It is the polarization-dependence of the white-line resonance intensities that is the topic of the present paper.

The one-electron diagram shown in Fig. 1 (b), however, is misleading, especially to the photoemission community, because it depicts the spin-orbit splitting of the p core shell as an "initial-state" effect. It is clear that in reality the p shell is filled in the ground state, and therefore there is no observable effect of the spin-orbit interaction. In the proper description of the x-ray absorption process, based on a configuration picture, an atom is excited from a ground or initial­state configuration to an excited or final-state configuration. For L-edge absorption, the initial

10

'0 05 ';:;'" :J « -- Fe '0

Co Ni Cu 05 ';:;'" 5 r:::: 0 .... -0 (])

05 tU -0 I-

0 700 750 800 850 900 950

Photon energy (eV)

Figure 2. X-ray absorption spectra recorded by total electron-yield detection near the L3 and ~ edges for Fe, Co, Ni, and Cu atoms in a NiFe/Co/Cu multilayer, showing the existence of white lines for Fe, Co, and Ni and their absence for Cu, due to its filled d shell.

226

state is characterized by the atomic electron configuration 2p6dn and the final state by the configuration 2p5dn+ I. For transition metals with configurations 1 ::;; n ::;; 9, it is easiest to deal with the configuration cfJ in x-ray absorption. This is approximately the configuration for Ni metal. In this case, the initial state contains a filled p shell (p6) and one hole in the d shell (cfJ electron or d l hole configuration). Because of the cancellation of angular momenta in a filled shell, we ~an simply describe the initial state as a dl hole configuration. The fmal state p5d lO has a closed d shell and therefore can be described by a pI hole configuration. Hence in the configuration picture, for a eft ground state, L-edge spectra are described by a transition from the configuration d l to p I. The spin-orbit splitting of the p shell is therefore a final-state effect, as shown in Fig. 3 (a); and in the Russell-Saunders coupling scheme [38], the L3 and L2 edges correspond to 2D ~ 2P3/2 and 2D ~ 2P1I2 transitions, respectively. Note that, in this scheme, the J = 3/2 state is the lower energy state as required by Hund's rule for a more than half-filled shell.

While, in general, the one-electron and configuration pictures are not equivalent, equivalence does exist for the case of a eft ground state, as illustrated in Fig. 3. This, in fact, justifies use of the one-electron model. This figure depicts three cases that differ in the way the d valence states are treated. The reason for the choice of the particular examples will become clear later. At this stage, it suffices to treat them as hypothetical cases. In the first case, all 10 d states are assumed to be degenerate and one of the states is empty. In the one-electron model, an electron is excited from either the P3/2 or the P1l2 core state into the d hole state. The transitions in the one-electron picture can be envisioned in an inverted level scheme as shown in Fig. 3 (b).

In the second case illustrated in Fig. 3, we have assumed that the d states are split into spin-up and spin-down states. As discussed below, this happens in magnetic materials as a result of the exchange interaction, and we have therefore labeled this case "exchange." If the magnetic moments of the ferromagnetic domains of the sample are aligned by an external magnetic field in the "down" direction, the spin-up states are lower in energy [39]; and in the one-electron picture, there is one empty spin-down hole, which can accommodate the excited photoelectron. In the

2P1/2

2P3/2

L3

20

(a) Configuration picture p6(j9 -7 pSd10 or d1 -7 P 1

! !

forbidden

203/2

205/2 exchange spin-orbit

d

P3/2

P1/2

I I I

L3 : I I I I

(b) One-electron picture p -7 d transition

I , ---iI ... !eO-- d5/2 -If , -_ .. '+-- d3/2

L2 forbidden

exchange spin-orbit

Figure 3. ElIergy-level diagrams and electronic transitions for L-edge x-ray absorption processes in (a) a many-electron picture for a;P configuration and (b) a one-electron picture. The figure establishes a one-to­one correlation between the two pictures for three special cases. In the first case, only the spin-orbit interaction in the p core shell is considered. The second case considers an additional exchange interaction in the d shell, leading to a separation of spin-up and spin-down states. In the third case, the exchange interaction in the d shell is replaced by a spin-orbit interaction in this shell. Now the transition between j = 112 andj = 5/2 states is dipole-forbidden because of the /:ij = 0, ±1 selection rule.

227

corresponding configuration picture, the transitions occur from the lowest-energy spin-up state. At first sight, this seems incorrect since, in the one-electron picture, the transitions go to the unfilled spin-down states. Both models are equivalent, however, because in the one-electron picture, the spin refers to that of a single electron, whereas in the many-electron-configuration picture, the spin refers to the total spin of the atom. For a cfJ configuration, the total spin is that of a single hole and therefore opposite to that in the one-electron model. We note, however, that the magnetic moment is the same in both cases because spin and magnetic moment are antiparallel for an electron but parallel for a hole (see Section 3.1).

The third case assumes a spin-orbit splitting in both the d and the p states, and again the equivalence of the two pictures is apparent. Because of the /)J = 0, ± I dipole selection rule, the 112 ~ 512 (or inverse) transition is forbidden. This can have a dramatic effect on the L31Lz white­line and dichroism intensity ratios, as discussed in Section 7.

Since, in the tJ9 ground-state configuration, the relevant transitions correspond to the simple d l ~ pi "one-hole" transition picture shown in Fig. 3 (a), all exchange and correlation effects are absent. These latter effects complicate the x-ray absorption spectra in that they introduce multiplet structure. Although a detailed treatment of these effects is beyond the scope of this paper, let us, nevertheless, briefly illustrate how the multiplet splitting arises for the simple case of a d8

configuration, i.e., Co metal. In the configuration picture, exchange and correlation effects introduced through the

Hamiltonian

H=Ie2

i<j rij (2)

need to be considered in both the ground state (i.e., between the d electrons) and the excited state (between the core p and valence d electrons). The simplest case is that of a p6d8 ground-state configuration leading to a p5tJ9 excited state. The ground-state configuration is equivalent to a d2 hole configuration, and the excited state to a pd hole configuration. Hence in both cases, we have to consider the exchange and correlation between two holes, which is the simplest multielectron (multihole) case.

The interaction of the two d holes results in Russell-Saunders multiplets: 3 F, I D, 3 P, I G, and IS [38] with 3 F being lowest in energy according to Hund's rule. Since the d spin-orbit interaction is small compared to the multiplet splitting, one can, to first order, ignore mixing of the multiplets and simply consider the spin-orbit splitting of the lowest multiplet only. Of the resulting 3F2,3,4 terms, 3 F 4 is lowest and hence the ground state. For the excited pd state, the p spin-orbit splitting is larger (about 15 eV) than the multiplet splitting (about 3 eV), and thus the latter interaction leads to a further splitting of the spin-orbit components. The transition from the pure spin-orbit to the pure multiplet case has been treated by van der Laan and Thole [40]. For example, in the absence of the spin-orbit interaction, the excited pd configuration splits into the multiplets: I P, ID, IF, 3p, 3D, and 3 F; and in the LS coupling scheme, only the transitions 3F ~ 3D and 3 F ~ 3 F are dipole-allowed (IlL = 0, ±1).

For the general treatment of multielectron effects, it is most convenient to use tensor algebra [41], in which wave functions and matrix elements can be written in terms of 3j, 6j, and 9j symbols, which are readily evaluated by computers. We shall present the formalism and the result of such multielectron calculations elsewhere [42].

228

3.1. ONE-ELECTRON CALCULATIONS FOR SPECIAL CASES

Below we shall present model calculations for two special cases that are of fundamental importance for understanding the basic concepts of the dichroism effect. We shall calculate the dichroism effect in a one-electron model for p ~ d transitions, under two assumptions: fIrst, that the d band is split by the exchange interaction into unequally populated spin-up and spin-down states; and, second, that the d band is split by the spin-orbit interaction and a magnetic fIeld into unequally populated components.

The fIrst case is of fundamental importance because it underlies the most basic band picture of itinerant ferromagnetic metals, the so-called Stoner model. In this model, the d band in transition metals is split by the exchange interaction

(3)

into spin-up and spin-down bands. In an external magnetic fIeld in the down direction, the center of gravity of the spin-up band is lower in energy than that of the spin-down band by the exchange splitting. When the Fermi level intersects the d band, there will be more fIlled electronic states with spin up (hence the name majority spin band) than with spin down (minority spin band). As depicted in Fig. I (b), the magnetic moment M in this picture is parallel to the external magnetic fIeld and is in the spin-down direction because, for electrons, spin S and magnetic moment are antiparallel according to the relationship [38, 39]

(4)

where JLB is the Bohr magneton. In relating spin and magnetic moment directions, it is convenient to think of the magnetization as arising from the holes in the minority band since M and S are parallel for holes. In the simple Stoner model, the magnetic moment is solely due to the electron spin without any orbital contribution.

The second case considers the effect of the spin-orbit interaction

(5)

in the d band. This interaction, although relatively small, may have a profound effect on the relative transition intensities at the L3 and L2 edges, and in the presence of a magnetic fIeld, a particularly illustrative case is obtained for the origin of the dichroism effect. This latter case is also of interest because it allows one to explore how the existence of an orbital magnetic moment

(6)

influences the dichroism signal. The orbital moment contributes to the total moment according to [38,39]

(7)

and it is interesting to explore how the dichroism effect is related to the spin, orbital, and total moments. In practice, the orbital moment is typically reduced in transition metals by crystal fIeld

229

effects such that the spin contribution to the total moment dominates. Nevertheless, the orbital moment is of considerable importance because its direction is "locked in" by anisotropies in the lattice, which through the spin-orbit interaction leads to an anisotropy of the total moment. Hence, even though the orbital moment is small in transition metals, its quantitative determination is of fundamental importance for the understanding of the magnetocrystalline anisotropy. In more general terms, the spin-orbit interaction in both the initial and final states of the electronic transition is of fundamental importance for the existence of the magnetic x-ray dichroism effect and therefore needs to be discussed.

4. Dipole Matrix Element for Polarized X Rays and Quantization Axes for Photon Spin and Magnetization

The x-ray absorption intensity per atom I for a transition between an initial state Ii) and a final state If) is a dimensionless quantity, often called optical oscillator strength, which, with the dipole matrix element written in the "length" form, is given by [37]

(8)

Here Ef and Ej are energy eigenvalues of the states If) and Ii), and their difference may differ somewhat from the absorbed photon energy 11m. In practice, we assume Ef- Ej = 11m and neglect the remaining dependence of the intensity on 11m, in particular, on the difference in the L3 and L2 energies, which for 3d elements is of the order of 2%. We note that the use of Eq. (8) is advantageous when considering transitions between two bound electronic states, as will be done in our simple model calculation below. In other cases, e.g., when considering the intensity of resonances that lie above the ionization potential, it is appropriate to calculate the x-ray absorption cross section and determine the resonance intensity by energy integration over the resonance width [37]. In Eq. (8), the direction of the electric-field vector of the x rays is characterized by the unit vector e, and r is the electron-position vector which in Cartesian coordinates is given by

(9)

If we define the z axis of our coordinate system to point into the x-ray propagation direction, as shown in Fig. 4, the unit electric field for right circularly polarized radiation can be expressed as

(10)

The electric-field vector rotates clockwise as the wave travels along the z axis. For right circular polarization, the photon spin points into the propagation direction, and the reverse is true for left circularly polarized light (see Fig. 4). The unit electric field vector for left circularly polarized radiation can be expressed as

230

(a) Polarization and photon spin

z

x " ,,--)

E

"

x rays

Right Photon F spin

ill rp (). t

" y ,,-

Left Photon z+ spin

ill qJ ~ !

(b) Magnetization and electron spin

-)

H

t Minority

Magnetic electron moment spin

t (). t

-)

H

! Minority

Magnetic electron moment spin

! ~ !

Figure 4. (a) Coordinate system and conventions for right and left circular polarization and photon-spin directions. (b) Correspondence between directions of an external magnetic field H and magnetic moment and minority electron spin vectors in a magnetic material that has been fully aligned by the external field.

(11)

Therefore, the respective x-ray absorption oscillator strengths for the two polarization cases are

(12)

where the plus sign refers to right and the minus sign to left circularly polarized light. The corresponding equation for linearly polarized light with the E vector of the x rays along

the z axis of our coordinate system (i.e., x-ray propagation along an axis in the x-y plane) is given by

(13)

The polarization-dependent dipole operators above can be expressed in terms of Racah's spherical tensors C~) [38], with 1= 1 and ml = 0, ±1. The ~~ are related to the well known spherical harmonics Yil (9,1/», and the dipole operators for right (P}1», left (P~\», and linearly (P61» polarized radiation can be written:

Right: (14)

231

Left: p(1) = _1_(x_ iy) = re(1) = r [41r y-l -\ -fi -\ ~3 1 ' (15)

and

(16)

In the above discussion, we have defined the x-ray polarization and therefore the dipole operator in the coordinate system shown in Fig. 4 (a). In calculating the dipole matrix element, we need to consider the coordinate system of the wave functions I i) and I f). For the calculation of the magnetic x-ray dichroism effect, the wave functions are best chosen in a coordinate system where the z axis is parallel to the magnetization direction M, i.e., parallel to the direction of the magnetic moment on the atom of interest. In the following, we shall assume that both the photon spin and electron spin lie either parallel or antiparallel to the z axis of our coordinate system. In particular, we shall assume that for right (left) circularly polarized x rays, the photon spin points up in the +z (down in the -z) direction. If we place our sample in an external field H in the +z (up) direction, the sample magnetization M and the minority electron spin direction will also be up, and correspondingly for H along -z, as shown in Fig. 4.

We will show below that it is equivalent for the consideration of the dichroism amplitude whether the photon spin direction is reversed for a fixed magnetization direction or the magnetization direction is reversed for a fixed photon spin. Furthermore, we shall fix the sign of the dichroism effect M by adopting the definition of previous authors [3, 18, 19, 16] who used the relative orientation of photon spin and majority electron spin direction according to:

photon spin relative to majority electron spin

(17)

This is normally equivalent to the following definition using the relative alignment of photon spin and magnetization, photon spin and external field, or photon spin and minority electron spin directions:

photon spin relative to magnetization direction

(18)

We note that, for the above definitions to be equivalent, the spin moment needs to be larger than and parallel to the orbital moment such that the magnetization direction is determined by the direction of the spin moment.

232

5. Dichroism Effect in the Stoner Model: Spin-Only Magnetic Moment

As mentioned in Section 3, we shall present simple model calculations for a tfl configuration. In particular, we shall use the notation of the one-electron picture and perform calculations for an atomic equivalent of the Stoner model.

Here we shall assume that the exchange interaction splits the d band into a spin-up and spin­down component, that the spin-up states are filled, and that there is one hole in the spin-down states as shown in Fig. 3. The angular part of the spin-dependent d functions can be represented by basis functions I/,m[,s,ms) or the product of spherical harmonics and a spin-dependent function. If we use the notation a for spin up (s = 112, ms = 112) and f3 for spin down (s = 112, ms = -112), the spin-down states are represented by the five functions y~l f3 with m[ = 2, 1,0, -1, -2. Because the five states are degenerate in the present case and anyone of the states may contain the hole, we need to calculate the transition probabilities to all five states and average over them.

In a one-electron picture, we calculate the dichroism effect by considering transitions from the P312 and PII2 states to the empty spin-down states of the d band. Thus we need to write the P3/2 and PII2 wave functions in the basis I/,m[,s,ms) and then consider transitions to the empty spin­down d states. The relevant wave functions are listed in Table 1. For convenience, the table lists the correspondence of wave functions written in the basis II, s, m[, ms) with those in the basis I/,s,j,mj). We shall use the special form of the d functions listed in the table later when considering the spin-orbit interaction.

Since the dipole operator does not act on spin, the initial and final states of allowed transitions have the same s and ms values. The relevant expressions for the dipole matrix elements for transitions from a state characterized by angular momentum I to the state 1+ 1 (e.g., for the case of the P ~ d transitions of interest here) are given in Bethe and Salpeter, Chapter IV [43]:

and

(n',l + l,ml + Ilpp)ln,l,ml) = _ (I + ml +2)(1 + ml + 1) R 2(21 + 3)(21 + 1)

( n',1 + l,ml -IIP:~)ln,l,ml) = _ (/- ml + 2)(1- ml + 1) R . 2(21 + 3)(21 + 1)

All other matrix elements are zero. Here

(19)

(20)

(21)

is the radial matrix element. We shall assume that the radial wave functions and therefore the radial matrix elements are identical for the spin-orbit split states.

Equations (19) and (20) allow us to derive the important fact that, for the measurement of the dichroism effect, a change in the photon spin relative to the fixed magnetization direction is equivalent to a change in the magnetization direction relative to the fixed photon spin. This fact is of great practical importance since it is usually simpler to reverse the magnetization direction in a

233

TABLE 1. One-electron s, p, and d wave functions.

One-Electron Label Configuration Label Ilsjmj) Basis Ilsmpns) Basis

lj 2S+1Lj mj Y;nl cl>ms * S1l2 2S1l2 1 yga "T

1 ygp -2

2P1l2 1 .h (Y?a - ..fiYIP) P1I2 2 1 ...1..( ..fiy-Ia - Y0fJ) -2 ..J3 I I

2P312 3 Yla P312 2 1 ...l...(..fi yOa+ yiP) 2 ..J3 I I

1 ...l...(y-Ia+..fiyoP) -2 ..J3 I I

3 YI-1p -2

d312 2D312 3 i(Yla - 2YiP) 2

1 ...l...(..fiyoa-.fjylp) 2 ..[5 2 2

1 ...l...(.fjy-Ia-..fiyoP) -2 ..[5 2 2

3 i(2Y-2a - y-IfJ) -2 5 2 2

dSI2 2DS/2 5 Yia 2 3 ...l...(2y1a+ y2fJ) 2 ..[5 2 2

1 ...l...(.fjyOa+ ..fiylfJ) 2 ..[5 2 2

1 ...l...(..fiy-Ia+.fjyoP) -2 ..[5 2 2

3 ...l...(y-2a + 2y-IP) -2 ..[5 2 2

5 y22P -"2

* cl>ms=1I2 = a (spin up), cl>ms=-1I2 = f3 (spin down).

dichroism experiment than to reverse the photon spin. The reason is that change from left to right circularly polarized synchrotron radiation from a bending-magnet source requires a change in the optical path of the x rays through the monochromator. This typically results in energy shifts and/or resolution changes and leads to problems when the spectra for different polarizations are normalized relative to each other. The proof that it is equivalent to switch either the magnetization or the photon spin can be directly read from Eqs. (19) and (20), namely (n',l + 1,-ml + 11 pfl)ln,I,-ml) = (n',l + 1,ml -IIP~~)ln,l,ml)' i.e., changing the magnetization or the quantum number ml to -ml for right circularly polarized light (operator pfll) is equivalent to changing to left circularly polarized light (operator p~~» and keeping the magnetization or quantum number ml fixed.

234

We have written the matrix elements above for the one-electron model, i.e., p ~ d transitions. Figure 3 shows that, in the configuration picture, transitions occur from D to P states and originate from spin-up (a) states. It is easy to see that these two cases are equivalent for a tfJ configuration. In this case, the level schemes for the one-electron and configuration pictures are inverted. The one-electron scheme corresponds to p ~ d spin-down electron transitions, and the configuration scheme to d ~ p spin-up hole transitions. Since the magnetic moment is in the same direction for both schemes, the dichroism effect is given by the difference of squared matrix elements of the same operators p~l) and p~~), respectively. A hole has opposite spin and orbital momentum from the missing electron, and the corresponding matrix elements in the two cases are (p,-mzIP?)ld,-(mz +1) = -(d,mz +IIP?)lp,mz) and (p,-mzIP~~)ld,-(mz-l) = -(d,mz-lIP~~)lp,mz); hence, the transition intensities are the same no matter whether the hole or the electron wave functions are used. Therefore, the two pictures lead to the same dichroism result.

The angular-momentum matrix elements reveal the following selection rules: N = 1, I1mz = +1 for right circularly polarized light, and I1mz = -1 for left circularly polarized light. The above selection rules apply to our special case of p ~ d transitions and circularly polarized light. The general dipole selection rules are 111 = ±l and I1mz = 0, ±l. The case 111 = -1, corresponding to p ~ s transitions, is not considered here, and the case I1mz = ° applies for linearly polarized light.

In considering p ~ d transitions in a one-electron picture, we need to evaluate matrix elements for the six p functions listed in Table 1. Since we have assumed that all spin-up d states are full, no transitions of spin-up electrons can occur and we only need to consider the spin-down parts of the p wavefunctions, i.e., six functions of the form

(22)

where the coefficients ami are given in Table 1. For the d states, we use five spin-down functions of the form

(23)

with mz = 2, 1, 0, -1, -2. We could as well have used the 10 spin-orbit-split basis functions in Table 1, but the above choice simplifies the calculation and gives the same result.

We obtain for the L3 edge (P312 ~ d transition)

and

It3 = LIUIP?)li}12 = ~R2 i,f

Ii3 = LIUIP~~)li}12 = % R2 . i,f

(24)

(25)

235

The summation in i is over the four p functions in the P312 manifold, and that infis over the five d functions. For the L2 edge (P1l2 ~ d transition), we obtain

and

/t2 = .rIUIP?)li}12 = ~R2 i,f

/"4 = LIUIP~~)li)12 = i R2 . i,f

(26)

(27)

The dichroism effect is then given by M = /+ - /-, since in our case /+ corresponds to photon spin up and majority electron spin up, and /- to photon spin down and majority electron spin up (see Fig. 4). We obtain for the L3 and L2 dichroism effects ML3 = (-2/9)R2 and I1h2 = (2/9)R2. The dichroism signals at the L3 and L2 edges are identical in magnitude but of opposite sign. At the L3 edge, left circularly polarized light excites more spin-down electrons than right circularly polarized light, and at the L2 edge right circularly polarized light excites more spin-down electrons than left circularly polarized light.

Within the simple atomic model used above, one can derive other interesting results for the dependence of the dichroism effect on the spin «sz», orbital «/Z», and total «2sz + Iz») magnetic momenta; the relevant quantities are listed in Table 2 for the five d states. In this table, we have also listed the calculated dichroism effects M for excitation from the p shell, in the presence and absence of its spin-orbit splitting, to the various substates of the d shell. Comparison of the respective dichroism effects to the individual orbital, spin, and total momenta reveals that the sum of the L3 and L2 dichroism effects is directly proportional to the corresponding orbital momenta of the d subshell. This is a special case of a more general sum rule [44] to be discussed in more detail later. In particular, an average over the five d states yields (lz) = 0, and in this case the L3 and L2 dichroism effects cancel. On the other hand, there is no obvious relation between the spin and total momenta and the L3, L2, or total (L3 plus L2) dichroism effects. This is another important result to be discussed in more detail below.

TABLE 2. The d-orbital angular-momenta and dichroism effects for P312~ d, P1I2 ~ d, and P ~ d transitions.

d orbital (lz)(lf) (sz)(lf) (2sz + lz)(lf) ML/R2 ML/R2 (ML3 + M4)/R2

yiP 2 1 1 2 4 .Q.. -"2 15 15 15

yiP 1 1 0 2 1 3 -"2 15 15 15 ygp 0 1 -1 2 2 0 -"2 45 - 45

YZ1p -1 1 -2 2 1 3 -"2 -15 -15 -15 Yz2p -2 1 -3 6 0 6 -"2 -15 -15

Sum 0 5 -5 2 2 0 -"2 -9 9

236

6. Simple Two-Step Dichroism Model

The above results lead to a particularly simple two-step model for the origin of the XMCD effect. In the first step, the interaction of circularly polarized x rays with the p shell leads to the excitation of spin-polarized electrons. In XMCD spectroscopy, the core shell can therefore be viewed as an atom-specific, localized "source" of spin-polarized electrons. Let us first assume that both spin-up and spin-down p ~ d transitions are allowed, i.e., we assume that the d band is not exchange-split and therefore has an equal number of spin-up and spin-down vacancies. As depicted in Fig. 5 for the P312 initial state (L3 edge), right circularly polarized light then excites

Band picture

2p -15 eV

Dipole ~l = ±1 ~s=O

3d

~ ~

H M Atomic picture

1_1 ===m~~/2 Dichroism ~ml=+1 ~ml =-1 ~ms=O

+1/2

62.5%; ; 37.5%

L R

25%, ,75%

LJ?J kR Figure 5. Correspondence between the rigid-band picture of a magnetic material (Stoner model) and its atomic analogue and relevant energy levels for L-edge x-ray absorption. In the atomic picture. the origin of the dichroism effect can be visualized in a two-step model. In the first step, the absorption of circularly polarized photons by the spin-orbit split p core shell leads to the excitation of spin-polarized electrons. The spin polarization of the photoelectrons is analyzed by the valence shell, which for magnetic materials has a spin imbalance and is therefore a more effective detector for one kind of spin. Relevant selection rules and spin polarizations are also given.

237

62.5% spin-up and 37.5% spin-down electrons, and the reverse holds for left circularly polarized light. For the P1I2 initial state (L2 edge), right circularly polarized light excites 25% spin-up and 75% spin-down electrons, and left circularly polarized light does the opposite. Here "spin-up" and "spin-down" are defmed relative to the photon spin direction, as illustrated in Fig. 4. For a d band with equal spin-up and spin-down occupancy, there is thus no dichroism effect since the total (spin-up plus spin-down) transition intensities for right and left circular polarization are identical. We also note that the sum of the P3/2 and P 112 contributions, which takes into account the fourfold and twofold degeneracy of the respective core states, shows no net spin polarization for either right or left circular polarization.

If the metal is ferromagnetic, an imbalance in empty spin-up and spin-down states will exist, and hence transitions involving one spin orientation will be favored. We can then visualize a second step in which the spin-polarized electrons originating from the P shell are analyzed by a "spin-resolving detector" consisting of the exchange-split d final state. The quantization axis of the detector is given by the magnetization direction, which for maximum dichroism effect needs to be aligned with the photon spin direction, as discussed in Section 4. If there are only unfilled spin-down states, for example, the detector is only sensitive to spin-down electrons and the dichroism effect is maximized. In the presence of unfilled states of spin-down as well as spin-up character, the relative size of the observed dichroism effect is simply proportional to the difference in spin-down minus spin-up holes, i.e., to the magnetic spin moment [39]. We have noted above that for a given polarization, the sum of the P312 and P1I2 contributions shows no spin polarization. Therefore, it is also true that the dichroism effect is zero when the L3 and L2 intensities are summed. This illustrates the important fact that sensitivity to the d electron spin in the Stoner model arises entirely through the spin-orbit coupling (splitting) of the P shell.

Hence in the simplest two-step model of XMCD, circularly polarized x rays generate spin­polarized electrons from a localized atomic inner-shell "source," and by proper alignment of the photon spin with the magnetization direction of the outer valence shell, this shell serves as a spin­resolving "detector."

At this point, it is interesting to consider another important case, that of K -shell excitation. Here the I s core shell has zero angular momentum and hence no spin-orbit splitting. In this case, the existence of an exchange (spin) splitting of the P valence shell is insufficient for the existence of a magnetic dichroism effect, in contrast to the case of the d valence shell and L3 and L2 edge excitation. Hence the simple two-step model developed above no longer applies. It can be easily generalized, however, as we shall discuss below.

Again, the first step is the interaction of the photon with the core electron. Because right and left circularly polarized photons possess a well-defined angular momentum, ti and -ti respectively, its conservation requires that the excited photoelectron carry the respective angular momentum. There are two possibilities for the electron to carry the angular momentum, i.e., either by the spin or by the orbital degree of freedom. Since the spin does not interact directly with the electric field, one can only transfer the photon angular momentum to the orbital part in the absence of spin-orbit coupling. This is the case for the excitation from an atomic s core state; thus, the excited electron will carry orbital momenta ti or -ti for the two polarizations of the light, respectively, and no spin polarization exists.

In the second step, we need to consider the valence shell, which acts as the "detector" for the spin and/or orbital momentum of the excited photoelectron. If the valence shell does not possess a net orbital moment, photoelectrons with orbital momenta ti and -ti cannot be distinguished and no dichroism effect will be observable, even if the valence shell has a net spin polarization as in the Stoner model. If, however, the valence-band density of empty states has an imbalance of

238

angular momenta, a differential "detector" exists for the angular momentum of the photoelectron. Therefore, for K-shell excitation, a dichroism effect exists only if the p valence shell exhibits an orbital moment. Sensitivity to the magnetic properties, i.e., to the spin polarization of the p shell, then arises from the spin-orbit interaction in the p shell.

We can now also explain the dichroism effect in L3 and lJ2 spectra by the same generalized two-step model. If the core state is split by a spin-orbit interaction, the substates in each of the manifolds are no longer pure spin states. As a result, the photon angular momentum is transferred to both the orbital and spin degrees of freedom of the excited photoelectron. In fact, a relatively large portion can be transferred to the spin degree of freedom which, in the case of p ~ d transitions, results in a large net spin polarization of the excited electrons as discussed above. In this case, just a spin imbalance in the valence shell is sufficient for the existence of a dichroism effect, i.e., the "detector" is a pure spin detector. If we assume that the spin-orbit interaction in the p shell is zero (or sum the L3 and L2 intensities), we obtain a case that is similar to the K shell. Then a dichroism effect exists only if the d valence shell has an orbital moment.

7. Spin-Orbit Interaction in the d Shell: Spin and Orbital Moment

Another interesting and instructive case is that of spin-orbit split initial and final states. Let us use the one-electron picture to explore the white line and dichroism intensities in this case, shown schematically in Fig. 3. Furthermore, let us assume that the degeneracy in mj states is lifted through an external magnetic field H in the +z up direction so that the energies are given by

(28)

where g, is the Lande g-factor [38]. The energy-level scheme is pictured in Fig. 6, and the eigenfunctions are listed in Table 1. Here we have assumed that the spin-orbit interaction in the p and the d shells is much larger than the Zeeman energy, i.e., we have ignored the mixing of different j states.

Figure 6 also lists the transition intensities between the various states for right (+) and left (-) circularly polarized x rays. These intensities are readily calculated using the wave functions in Table 1 and the expressions for the matrix elements in Eqs. (19) and (20). One can see in Fig. 6 that the sum of all transition intensities between a given pair of initial and final j states is the same for right and left circular light and, therefore, no dichroism effect exists. Furthermore, the total p 112 ~ d intensity is half of the P312 ~ d intensity, as expected from the degeneracy of the two p states.

It is clear that the existence of a dichroism effect requires an imbalance in the intensities for right and left circular light. In general, this arises if different mj states have different occupation, which in practice can be accomplished by making the magnetic splitting larger than the temperature (J.lB = 6.72 x 10-5 KlOe). In our diagram, transitions would then occur only to the higher-energy Zeeman levels of the d manifold. If we assume that the d3/2 substates have hole occupancies a (mj = -312), b (mj = -112), c (mj = 112), and d (mj = 312), the PII2 or L2 dichroism intensity is given by

1 llh2 =-[3(a-d)+b-c] .

9 (29)

+ -1-1 1'--1

~3 1-"9

One-electron picture: spin-orbit interaction and extemal magnetic field

-+ -+ H M

r r

+

239

Figure 6. Energy-level diagram for the case of p -+ d transitions in a one-electron model for the case of spin-orbit and Zeeman interactions in both shells. We have assumed that the spin-orbit splitting is large compared with the Zeeman splitting. Intensities for all allowed transitions for right (+) and left (-) circularly polarized x rays are given, as are average orbital (lz) and spin (sz) momenta (units of 1i) or moments of the various Zeeman levels in the d shell (units of g] IlB and inverted sign).

(Note that PII2 -+ dS/2 transitions are dipole-forbidden.) In Eq. (29), we have used the fact that, for the energy-level diagram in Fig. 6, H is assumed to be in the up direction such that the dichroism intensity is given by the difference M = r - J+. We note that ML7. is zero or negative since d ~ a and c ~ b. The above expression for III can be compared to the orbital and spin moments of the d312 final state. The individual momenta (units If) or moments (units gJJlB and inverted sign) fo! the mj substates, calculated with the wave functions in Table 1 according to (lz)=(I,s,j,mjI1zll,s,j,mj) and (sz)=(l,s,j,mjlszll,s,j,mj), are listed in Fig. 6, and the resulting momenta are given by

and

3 (lz}d = --[3(a-d)+b-c]

3/2 5

1 (sZ}d =-[3(a-d}+b-c].

312 10

(30)

(31)

240

Hence,

(32)

We can perform a similar calculation for the L3 edge, and the results for the various m j substates are listed in Fig. 6. If we associate hole populations e (mj = -512), f (mj = -312), g (mj = -112), h (mj = 112), i (mj = 3/2), and j (mj = 512) with the various sublevels of the d5/2 state, we obtain for the P3/2 or L3 dichroism intensity

ML3 =~[5(e- j)+3(f -i)+g-h]+~[3(a-d)+b-c] . 25 225

(33)

Because j ~ e, i ~ f, h ~ g, d ~ a, and c ~ b, ML3 is negative. For the total orbital and spin momenta, we obtain

(lz}d = -~[5(e- j)+3(f -i)+ g-h] 5/2 5

(34)

and

(SZ}d = -~[5(e- j)+3(f -i)+ g-h] . 5/2 10

(35)

Hence,

(36)

and 4 4

ML =--(s } +-(S) 3 5 Z d512 45 Z d3/2 (37)

Two extreme cases can be distinguished, depending on the variation of the population of the Zeeman levels with temperature, which in our atomic model is given by Boltzmann statistics. We consider the time average of the occupation of the various states. When the temperature is comparable to the Zeeman energy and much smaller than the spin-orbit splitting, there will be a population difference in the Zeeman levels of the d3/2 subshell but not in those of the d5/2 subshell (all empty). In this case, the dichroism effect is entirely determined by the d3/2 subshell. M~ and ML3 will both be negative, and the L2 dichroism will dominate: M2/M3 = 12.5. In the high-temperature limit, when the temperature is comparable to the spin-orbit splitting but much larger than the Zeeman energy, there will be a population imbalance only in the d5/2 subshell since all Zeeman levels in the d3/2 subshell will be filled. In this case, !!t.L2 = 0 and the L3 dichroism signal (negative) is determined only by the d5/2 subshell. Note that in both cases, the relative sign and magnitude of the L3 and L2 dichroism signals differ substantially from the case of the Stoner-like model in Section 5. This difference, of course, is a direct consequence of the spin-orbit coupling, which gives rise to a sizable orbital moment. In fact, according to Eqs. (30), (31), (34), and (35), the orbital momentum is larger than the spin momentum for the present examples.

We are interested in establishing a correlation between the L3 and/or L2 dichroism signal and the spin moment (Sz)d = (Sz)d512 + (Sz)d312, orbital moment (lz)d = (lz)d512 + (lz)d312, and/or total

241

moment 2(sz)d + (lz)d of the d shell. It is clear from the results derived above that neither Ah3 nor Ah2 is proportional to either (Sz)d or (lz)d but rather to a specific linear combination of the subshell contributions. The question arises whether a simple correlation can be established by using a linear combination of the L3 or ~ dichroism signals. From the above results we fmd

(38)

(39)

and

(40)

As found earlier at the end of Section 5, there is direct proportionality between the orbital moment of the d shell and the sum of the L3 and L2 dichroism intensities. Although the spin moment and total moment can be expressed as a linear combination of the respective dichroism intensities, it can be shown that the coefficients are model-dependent, barring a general correlation between the measured dichroism signals and the spin and total moments. On the other hand, the correlation found here for the orbital moment is a general one, as shown by a powerful sum rule derived by Thole et al. [44].

8. Band-Structure Model with Exchange and Spin-Orbit Interactions

Finally, let us consider the transitions involved in an XMCD experiment within a simple but realistic band-structure model. Specifically, we consider transitions from an np core level to the d band. For simplicity, we assume that the spin-up majority band is completely filled. For transitions involving core states, we can consider the transition originating from a particular atom. To facilitate the calculation of the matrix elements, we expand the valence band state tPik(r) (where i is a band index) in terms of local spherical harmonics centered on the atom, wave vector (k), position (r), and spin (ex, {J) dependent terms, e.g., [45]

tPik(r) = Lalm,ikRnl,ik(r)YF f3 , (41) I,m

and similarly for spin-up bands. Note that the radial part Rnl,ik(r) is independent of m. The above wave function is of a form that results from a muffin-tin approximation of the crystal potential [46]; thus, the discussion below is valid within this approximation. The state characterized by the wave function may have a spin as well as orbital moment.

The coefficients alm,ik can be determined by a suitable band-structure calculation. For the excitation of the np core state, there are six possible initial states whose angular parts are given in Table 1. In the following, we shall assume that the radial parts Rn1 of the spin-orbit split p functions are identical, i.e., we neglect relativistic corrections.

242

In order to calculate the x-ray absorption intensity, we evaluate the matrix elements corresponding to transitions from each of the spin-orbit split p states to the valence states tf>ik(r). For simplicity, we shall assume that all spin-up states in the d band are filled. For transitions from the np3/2 states to a band state i excited with right circular polarization, we obtain, for example,

(42)

where we have omitted the label k and i in the coefficients a for brevity, and CJ (ik) = J R,';2,ik(r)Rn'l (r)r3dr is the radial part of the matrix elements. The various polarization­dependent dipole matrix elements for transitions from the spin-orbit components of the P core states to a band state tPik are listed in Table 3, and the polarization-dependent transition intensities and dichroism intensities for the L3 and L2 edges are given in Table 4. The sum of the dichroism intensities for a given band state ik at the L3 and L2 edges is calculated as

This expression can be compared to that for the orbital moment of a d-band state ik, which can be calculated with its I = 2 projected wavefunction tPfk2 (r) = Rn2,ik(r) Lm a2m,ik YT f3 according to

(Lz(ik») = (<I>~-k2(r)lizl<l>~-k2(r»)

= IC2(ik)12 {21a2d + la2d2 -la2_d2 - 21a2-2n ' (44)

where C2(ik)=# J R~2,ik(r)f?,z2,ik(r)r2dr.

TABLE 3. Matrix elements between the various core states tPc and valence states tPik. All listed matrix elements should be multiplied by the radial part Cl(ik) = J R;'2,ik(r)Rnl (r)r3dr, which has been omitted for brevity.

tPc mj ( tP/k Ip?) ItPc ) ( tP/k Ip~~) ItPc )

2p1I2: 1 ..jl;a21 ..jl;a2-1 -2 1

#Sa22 .fj;a2O 2

2p3/2: 3 ..jl;a2O #a2-2 -2 1 ..[1;a21 ..[1;a2-1 -2

1 {1;a22 .Jtsa20 2 3 0 0 2

243

TABLE 4. Dipole transition intensities N and dichroism intensities M = [+1 _11 between two spin-orbit split np core states and d conduction band states for polarized photons with polarization q. All listed in­tensities should be multiplied by the square of the radial matrix element CI(/k) = f R;'2 ik (r)Rnl (r)r3dr, which has been omitted for brevity. '

q

+1

o

-1

1~ (la2012 + 21a21f + 21a22n

9~ (9Ia2_tl2 + 81a2012 + 31a21n

15 (18Ia2-d + 61a2_tl2 + la20n do [-12Ia22 12 + 4Ia2012-12Ia2_212+

12(2Ia2212 + la2t12 - la2_112 - 2Ia2_212)]

1~ (la21 12 + 41a22n

15 (21a2012 + 31a21n

15 (3Ia2_112 + 21a20 12 )

do [12la2212 - 41a2012 + 121a2_212 +

6(2Ia2212 + la2112 -la2_112 - 2Ia2_212)]

Comparison of Eqs. (43) and (44) reveals a remarkable similarity. If we make the simplifying assumption that the radial part is the same for all bands, Rnl,ik(r) = Rnl(r), a particularly simple and powerful result is obtained. This assumption is a reasonable approximation that underlies tight-binding band calculations [46]. In this case, the ratio ICI/C21 is a constant for all empty band states ik above the Fermi level; and for each empty band state, the sum of the L3 and L2 dichroism intensities is proportional to its orbital moment. Furthermore, if we sum over all empty bands i and integrate over the Brillouin zone, we obtain for the total dichroism intensity M =Li fBZM(ik)dk

(45)

where C is a constant and (Lz) is the orbital moment due to all d-like states around the absorbing atom.

The important result expressed by Eq. (45) is the one-electron band structure analogue of the sum rule derived by Thole et at. [44] for the general multi-electron atomic case. This sum rule is also revealed by previous results obtained with tight-binding band structure schemes. Erskine and Stem's calculation [3] for the 3p ~ 3d and the calculations of Chen et at. [18] and Smith et at. [16] for 2p ~ 3d transitions are special cases of our general case. In Erskine and Stem's model, the expansion coefficients were derived from a band-structure calculation that did not take into account the spin-orbit interaction, and therefore the orbital moment vanishes: (Lz) = O. Chen et al. [18] realized that more generally, in Erskine and Stem's model, the sum of the dichroism signals at two edges is zero no matter what expansion coefficients are used, as long as the coefficient for Yfl is equal to that for Yim1 • This is simply due to the fact that, in the absence of spin-orbit coupling, the orbital moment in the d shell is always zero and, according to Eq. (45), so is the sum of the L3 and L2 dichroism signals. The importance of Eq. (45) lies in the fact that it establishes a direct link between the dichroism signal and the orbital moment of the d band. This

244

is significant, since it is difficult if not impossible to determine orbital moments experimentally by other techniques.

In our calculation above, we have neglected transitions to I = 0 bands, i.e., s bands. It can be shown, however, by means of a similar calculation as done above for transitions to I = 2 bands, that the sum of the L3 and L2 dichroism signals for such transitions is zero, in accordance with the zero angular momentum of an I = 0 state. Our simplifying assumption made above that the empty bands contain only spin-down states is also unnecessary. Equation (45) holds also in the presence of both spin-up and spin-down empty bands.

Finally, we need to make another important point: In general, there is no direct correlation between the measured dichroism signal and the spin or total magnetic moments. The spin moment of band state ik is given by

(Sz(ik» = (tf>/k2 (r)i Szic"/k2 (r»)

= -~ICI (ik)12 {la22 12 + la2112 + la2012 + la2_112 + la2- 2n ' (46)

and this expression needs to be compared to the L3 and L2 dichroism signals, which according to Table 4 are given by

and

(48)

The above expressions are valid for the general case of spin-orbit coupling, and it is clear that, for an arbitrary band state ik, neither the L3 nor the L2 dichroism signal is proportional to the spin moment (and therefore the total moment) because the quantities are determined by different linear combinations of the band structure coefficients alm,ik.

Even in the absence of spin-orbit coupling and, therefore, zero orbital moment in the d shell, a correlation between the dichroism signal and the spin moment does not generally exist. This is­seen from the expressions for l!.h3 and M~ given in Table 4 in which we have factored out the part 21a2212 + la2112 -la2_112 - 21a2_212, which is proportional to the orbital moment [compare Eq. (44)]. If we assume that this part vanishes, i.e., as in the spin-only Stoner picture, we obtain

(49)

and

(50)

245

Again, no direct correlation exists for an arbitrary band state ik between the dichroism signals for the two edges and the spin moment given b1' Eq. (46~. It can be shown that a correlation does exists in special cases, e.g., if la22 I = la2-21 = -la2112/4= -la2_112/4. The above considerations are for a specific band state ik only. The measured dichroism spectrum, of course, corresponds to a sum over all bands and an integration over the Brillouin zone. This, however, does not change our basic conclusion that the measured dichroism signal is, in general, not proportional to either the spin or the total moment.

A similar result has also been derived by Carra et al. [47] in terms of a sum rule which relates the measured dichroism intensity to the spin moment. The sum rule states that, in general, the L3 and L2 dichroism intensities are not simply related to the spin moment alone but rather to a combination of the spin moment and the expectation value of the magnetic dipole operator. Carra et al. argued, however, that in systems with a small spin-orbit coupling in the valence shell and a crystal lattice with high symmetry, e.g., for the 3d transition metals, the dipolar term is small such that an approximate relationship of the form ML3 - 2ML2 = C (Sz) can be established.

The above discussion is equally applicable to x-ray absorption involving other core states. For example, it is rather straightforward to show that the integrated dichroism intensity for s ~ p core-to-valence transitions is proportional to the 1 = I projected orbital moment of the valence band. The fact that, in many cases, the orbital moment is aligned either parallel or antiparallel and proportional to the spin moment might explain why the K-edge dichroism spectra resemble the difference in the p density of states between majority and minority spins.

9. Effect of the Core Hole: Initial- and Final-State Rules

Our model calculations above utilized a simple one-electron model. In this model the L3 and L2 white-line intensities and dichroism intensities are related to the density of empty d states. This model nicely explains the decrease in the white-line intensity as the d shell is filled, as shown in Fig. 2. The problem with the one-electron model, however, is that it neglects all multielectron effects, e.g., the effect of the core hole, which results in considerable electronic rearrangement on the excited atom and its environment in a solid. The question arises whether the white-line and the dichroism intensities are related to the density of empty d states on the core-excited atom in its ground state or its excited state. This fascinating question has been discussed recently by Zdansky et al. [48] for the case of Ni metal. Since, in the equivalent core approximation [37], an atom with atomic number Z, after core excitation, corresponds to a Z + 1 atom, one can rephrase the above question as follows using the specific example of Ni metal. Do the x-ray absorption and dichroism spectra for Ni metal correspond to those of a Ni atom (d9 , or ground state configuration) or a Cu atom (d lO, or excited state configuration) in the Ni host? The general theory was worked out in 1977 by Grebennikov, Babanov, and Sokolov [49] in two important papers. For brevity, we shall only state the answer here. In x-ray absorption, the integrated white-line and dichroism intensities are indeed proportional to the ground-state occupancy of the d shell as assumed in the one-electron model. This result is a consequence of a sum rule that relates the integrated intensity of the dynamical multielectron system to the intensity calculated in the static ground-state picture. The effect of the core hole manifests itself in the shape of the absorption and dichroism spectra, giving rise to a pile-up of oscillator strength at threshold. The important fact that the dichroism intensity in x-ray absorption is related to the ground-state magnetic properties was also pointed out in connection with the recently derived sum rules for the orbital [44] and spin [47] moments.

246

The above results for x-ray absorption may be generalized to include x-ray emission. Grebennikov, Babanov, and Sokolov [49] showed that x-ray absorption and x-ray emission spectra can be described by two general rules. The initial-state rule states that, in a configuration picture, the p ~ d transition intensities are proportional to the initial-state d shell occupancy. In x-ray absorption, the initial state is the ground state, while in x-ray emission, it is the core-excited state. Thejinal-state rule states that the spectral shape ofthe spectrum is determined by the final­state configuration, e.g., the core-excited configuration in x-ray absorption and the ground-state configuration in x-ray emission.

10. Application of XMCD: Enhanced Orbital Moment in Co/Pd

In this last section, we present experimental results for Co metal and a ColPd multilayer, which are interpreted by using the important correlation expressed by Eq. (45). We believe that the quantitative determination of orbital moments and (in the future) of their anisotropy may be one of the most important applications of the XMCD technique. The results presented below have been published by Wu et al. [22], and the interested reader is referred to this paper for more details.

The XMCD experiments were performed at the Stanford Synchrotron Radiation Laboratory (SSRL) on Beamline 8-2, which is equipped with a spherical grating monochromator. Circularly polarized x rays were obtained by moving the prefocusing mirror below the electron-orbit plane yielding a degree of circular polarization of l~-l~ = 90±5%, where [R and]L are the x-ray intensities with right- and left-handed circular ~oUrization. The magnetization direction of the sample relative to the photon spin was then changed by alternately measuring two pieces of the same sample that were remanently magnetized in opposite directions. X-ray absorption was measured by using total electron-yield detection. The nominal structure of the multilayer sample used for the present study was Si(111)/(200 A)Pd/[(lO A)Pd/(4 A)CO]NI(10 A)Rh, where N = 11 is the number of periods. The sample was prepared by electron-beam evaporation in a 1O-8-mbar base-pressure system at a growth temperature below 50°C. X-ray diffraction measurements indicated that the crystal structure of this multilayer is consistent with an fcc lattice with a strong [111] texture. The sample exhibited 100% remanence in the perpendicular direction and a large coercive field of - 2000 Oe. The hcp Co thin film sample of 250-A thickness was grown by DC sputtering as part of a Si(100)/(11O A) NiFe/(lOO A) FeMn/(250 A) Co/(lO A)NiFe sandwich. Through exchange-biasing by the antiferromagnet FeMn, the so­grown Co film exhibited 100% in-plane magnetic remanence with a coercive field of -20 Oe.

Polarization-dependent x-ray absorption spectra for the Co thin film and the Co (4 A)/Pd(lO A) multilayer were recorded at the Co L edges with photon spin and majority electron spin parallel and antiparallel to each other. The dichroism spectra and their differences are shown in Fig. 7. The spectra are normalized to the incident photon flux and rescaled to a constant step height far above the Co L edges. Our dichroism difference spectra for Co metal look similar to those recorded by Sette et al. [19], but the effect is significantly larger in our spectra. For the multilayer sample, the L3 resonance intensity is enhanced relative to that for the Co metal, and so is the L31L2 dichroism ratio, which varies from -1.8:1 for the Co thin film to -2.8: 1 for the multilayer sample.

From the areas of the L3 and ~ peaks in the dichroism difference spectra, we fmd that the sum of the intensities for Co in the ColPd multilayer is larger by a factor of 1.90 than in Co metal.

247

8 Co Co/Pd multilayer

"0 I+1, ii Q) ':;' I-1, i J, c 6 0 .... -0 CD Q) 4 "0 CD

.!::! (ij

2 E .... 0 ... Z

0 L3 L2

1

0

CD 0 c CD -1 .... CD

== i:5

-2

770 790 810 770 790 810

Photon energy (eV)

Figure 7. Top panels: Co L2,3 absorption spectra of Co thin film and ColPd multilayer samples recorded with parallel and antiparallel alignment of majority electron-spin and photon-spin vectors. The ColPd sample was measured at normal, and the Co metal sample at 20° grazing x-ray incidence. The data were scaled so that the jump far above the edge is 1. Bottom panels: Dichroism spectra for the same samples obtained by taking the difference between the absorption spectra according to Eq. (17).

248

Here we have corrected the intensity for Co metal for the fact that the x-ray incidence angle and magnetization directions were at an angle of 20°, i.e., we multiplied the measured intensity by lIcos 20°. If we assume that the constant in Eq. (45) is the same for both systems, we are led to the result that the orbital moment (Lz) for Co in ColPd is 90% larger than that in Co metal. Since the constant in Eq. (45) is given by the ratio of two radial integrals, the above assumption should be valid to a good approximation.

These results give the first experimental confirmation of the suggestion that orbital moments can be greatly enhanced in multilayers relative to those in the pure metal. The results support the first-principle calculations by Daalderop et al. [50], in which orbital moments of 0.13 )IB for bulk hcp Co metal and 0.28 J.LB for a COIPd2 multilayer were predicted. The agreement of the theoretical ratio of 2.15 with the XMCD experimental value (1.90) is very good, especially since perfect agreement is not expected owing to the fact that the crystal structure used for the calculation may differ from that of our polycrystalline sample.

Acknowledgements

We would like to thank M. Samant, B. Hermsmeier, and D. Weller for their involvement with experimental studies that led to the present calculations and for valuable discussions. We especially would like to thank P. Bagus for valuable theoretical advice and comments on the manuscript. One of us (JS) is grateful to A. Nilsson for teaching him the initial- and final-state rules. Experimental spectra were recorded at SSRL, which is operated by the Department of Energy, Division of Chemical Sciences.

References

1. For a review of the magneto-optical Faraday and Kerr effects see: D.E. Fowler, in Encyclopedia of Materials Characterization, edited by C.R. Brundle, C.A. Evans, Jr., and S. Wilson (Butterworth-Heinemann, Boston, 1992), p. 723.

2. E. Hecht, Optics (Addison-Wesley, Reading, PA, 1987). 3. J.L. Erskine and E.A. Stem, Phys. Rev. B 12,5016 (1975). 4. G. SchUtz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Frahm, and G. Materlik, Phys.

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B.P. Tonner, Science 259,658 (1993). 10. J. Zaanen, G.A. Sawatzky, J. Fink, W. Speier, and J.C. Fuggle, Phys. Rev. B 32, 4905

(1985). 11. F.M.F. de Groot, J.C. Fuggle, B.T. Thole, and G.A. Sawatzky, Phys. Rev. B 42,5459

(1990). 12. H. Ebert, B. Drittler, R. Zeller, and G. SchUtz, Solid State Commun. 69,485 (1989). 13. H. Ebert and R. Zeller, Phys. Rev. B 42, 2744 (1990).

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14. G. van der Laan and B.T. Thole, Phys. Rev. B 43, 13401 (1991). 15. C.T. Chen, N.V. Smith, and F. Sette, Phys. Rev. B 43, 6785 (1991). 16. N.V. Smith, C.T. Chen, F. Sette, and L.F. Mattheis, Phys. Rev. B 46, 1023 (1992). 17. T. Jo, Synchrotron Radiation News 5 (1),21 (1992). 18. C. T. Chen, F. Sette, Y. Ma, and S. Modesti, Phys. Rev. B 42,7262 (1990). 19. F. Sette, C. T. Chen, Y. Ma, S. Modesti, and N.V. Smith, in X-Ray Absorption Fine

Structure, edited by S.S. Hasnain (Ellis Horwood Limited, Chichester, England, 1991), p.96.

20. L.H. Tjeng, Y.U. Idzerda, P. Rudolf, F. Sette, and C.T. Chen, J. Magn. Magn. Mater. 109, 288 (1992).

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(1992). 23. D. Weller, Y. Wu, J. Stohr, B.D. Hermsmeier, and M.G. Samant (unpublished). 24. G. Schutz, R. Wienke, W. Wilhelm, W. Wagner, P. Kienle, R. Zeller, and R. Frahm, Z.

Phys. B 75,495 (1989). 25. G. Schiltz, H. Ebert, P. Fischer, S. Ruegg, and W.B. Zeper, Mat. Res. Soc. Symp. Proc. 231,

77 (1992). 26. J.B. Goedkoop, B.T. Thole, G. van der Laan, G.A. Sawatzky, F.M.F. de Groot, and J.C.

Fuggle, Phys. Rev. B 37, 2086 (1988). 27. S. Imada and T. Jo, J. Phys. Soc. Jpn. 59, 3358 (1990). 28. P. Carra, B.N. Harmon, B.T. Thole, M. Altarelli, and GA Sawatzky, Phys. Rev. Lett. 66,

2495 (1991). 29. G. Schiltz, M. Knulle, R. Wienke, W. Wilhelm, W. Wagner, P. Kienle, and R. Frahm, Z.

Phys. B 73, 67 (1988). 30. P. Fischer, G. Schiltz, and G. Wiesinger, Solid State Commun. 76, 777 (1990). 31. H. Maruyama, T. Iwazumi, H. Kawata, A. Koizumi, M. Fujita, H. Sakurai, F. Itoh, K.

Namikawa, H. Yamazaki, and M. Ando, 1. Phys. Soc. Jpn. 60,1456 (1991). 32. B.D. Cullity, Introduction to Magnetic Materials (Addison-Wesley, Reading, PA, 1972). 33. S. Hufner, in Photoemission of Solids II, edited by L. Ley and M. Cardona, Topics in

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(1992); F.M.F. de Groot, J.C. Fuggle, B.T. Thole, and GA Sawatzky, Phys. Rev. B 41, 928 (1990).

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Berkeley, 1981). 42. Y. Wu and J. Stohr (to be published).

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G.H.O. Daalderop (private communication).

IllGH-RESOLUTION SOFT X-RAY ABSORPTION SPECTROSCOPY AND X-RAY CIRCULAR DICHROISM

FRANCESCO SETIE European Synchrotron Radiation Facility BP 220, 38043 Grenoble Cedex, France

ABSTRACT. The recent availability of high-energy-resolution, high-flux, and highly polarized soft x rays has contributed to the development of many novel photoabsorption and photoemission experiments in a wide range of research areas. This paper will give an overview of the present status and potential future developments in high-resolution soft x-ray absorption spectroscopy. A selection of experimental works in the fields of gas-phase spectroscopy, solid-state physics, and surface science are discussed, and novel studies on the detection of x-ray dichrosim with circularly polarized x rays are presented and analyzed. The paper describes the application of circularly polarized synchrotron radiation to the study of transition-metal 2p~3d and rare-earth 3d~4f optical transitions in magnetically oriented ferro- and ferrimagnetic materials. The measured x-ray dichroism is shown to give information on the presence and orientation of local magnetic moments on specific atomic species. Theoretical arguments are presented on a novel magneto-optical sum rule to show that the integrated x-ray dichroic signal of absorption edges is determined by the ground-state orbital angular momentum of the photoexcited atom. This sum rule, relating an experimentally determined quantity to the orbital magnetic moment, opens a new means to investigate directly the orbital magnetism of specific atoms and orbitals and thereby determine independently the spin and orbit contributions to the total magnetization of materials.

1. Introduction

Photoabsorption spectroscopy in the soft x-ray region is primarily concerned with electronic excitations from an atomic core state. The a priori knowledge of such a state allows one to obtain information on electronic states localized around the photoexcited atom. The attractive feature of performing atom-specific spectroscopies has been an important motivation for the construction of existing synchrotron-radiation sources, explains the enormous development of core-level spectroscopies in both the soft and the hard x-ray regions, and strongly contributes to the scientific case for the new third-generation sources.

Among the objectives of core-level spectroscopies, perhaps the most ambitious is to derive ground state electronic properties around the photoexcited atom, i.e., the photoexcited core state is used as a local probe of the electronic structure. Although one must deal with the conceptual problem that the creation of a core state is, in general, a strong local perturbation of the ground state potential, with a careful use of the excitation selection rules and with control of the experimental conditions, it is possible to obtain relevant information on important local properties of the system. For example, the development of the extended x-ray absorption fine structure (EXAFS) technique had a major impact in science. This technique is used to derive from a photoabsorption spectrum above an atomic absorption edge the geometric structure around the

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A.S. Schlachter and F.J. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 251-279. © 1994 Kluwer Academic Publishers.

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photoexcited atom (interatomic distances, the number and kind of atoms in its first few coordination shells) [1].

In this paper, we will be concerned with the information contained in the near-edge region of a core photoabsorption spectrum, which is determined by the lowest-energy excitations of the system in the presence of a core hole. These excitations involve transitions from the ground state to final states in which a core hole exists and the valence occupation increases by one electron. From the symmetry of the atomic core level, the orientation of the sample to the photon polarization, and the dipole selection rules, one can derive specific properties of the excited state and connect them to the electronic properties of the system in the ground state. In these studies, it is of primary importance to separate excitations to different final-state electronic configurations. An important parameter, therefore, is the experimental resolution, but one also must consider limitations intrinsic in the system such as final-state broadening and, most importantly, the natural lifetime of the core hole. As we will discuss in the next paragraph, these lifetime effects are minimized in the soft x-ray region (100-2000 eV) where one can perform studies with increased resolution provided a monochromator with adequate characteristics is used. A further feature of soft x-ray spectroscopy is that, in this photon-energy region, one can excite one or more important core lines for almost every element in the periodic table.

The remainder of this paper consists of three major sections (Sections 2, 3, and 4) and our conclusions. In Section 2, we review the main aspects of high-resolution soft x-ray absorption spectroscopy, with some emphasis on the experimental requirements. In Section 3, a few representative results are discussed. In Section 4, we discuss the recent development of soft x-ray absorption with circularly polarized radiation in magnetic systems. This discussion shows that one can derive important information on the magnetic structure of the system and the ground­state expectation value of its orbital magnetization.

2. Why High Resolution Soft X-Ray Photoabsorption Spectroscopy?

2.1. THE INTERACTION HAMILTONIAN IN X-RAY PHOTOABSORPTION AND ITS MAIN PROPERTIES

The photoabsorption probability is derived considering the interaction Hamiltonian Hi, which couples the electron and photon fields. In the weak relativistic limit, this is given by [2]

Hi = e I (me)l:iPiAi ' (1)

where Pi is the momentum of the jtb electron in the system, Ai is the electromagnetic field vector potential, and e, m, and e are, respectively, the electron charge, the electron rest mass, and the speed of light. In the Coulomb gauge (V A = 0), the vector potential satisfies the wave equation and it can be decomposed in a superposition of transverse plane waves Aoeqeikr, where k is the photon wave-vector k = role, and eq is the transverse polarization vector.

Quite generally, in the photoabsorption process, the photon wavelength A = 21t1k is large compared to the spatial extent of the initial-state, one-electron wave function. The electromagnetic field, therefore, changes little in the absorption region where the electron is localized and can be approximated by A(r) = (l+Vr)A(r)lr=o. The interaction Hamiltonian is then

253

(2)

The first term in the expansion of the interaction Hamiltonian HI is referred to as the dipole approximation. because the transition matrix element between a state la) and a state Ib) is proportional to the transition expectation value of the scalar product between the electric dipole operator er and the photon polarization eq. In fact. using the commutation rule p = i21tm1h[Ho.r].

(3)

HI is the leading term in determining the photoabsorption process. The second term H2 can be decomposed into two parts that are respectively proportional to the electric quadrupole and magnetic dipole operators. An order-of-magnitude estimate of the relative strength between the interactions HI and H2 in core excitations can be given by simply notiCing that they differ by the term (rk) in H2. Roughly speaking (Le .• without taking into account differences in polarization dependence and selection rules. and the non-commutation of p and r). if Ia) is a core state. rkla) -aBw/(ZeffC)la). where aB is the Bohr radius and aB/Zeff is the spatial extent of the core wave function. Zeff is the atomic number Z minus the core electrons in deeper shells. Le .• Zeff= Z at the Kedge. Zeff = Z - 2 at the L edges. and so on. Using aBw/(Zeff c) = hv/(mc2f1Zeff). where a. is the fine-structure constant. we derive

(4)

Le .• second-order transition matrix elements are a factor [hv/(mc2)][1I(a.Zeff)] smaller than first­order transitions.

In Table 1. using expression (4) squared. we give an estimate of the relative transition probability at different edges of various atoms. Here we see that. for x-ray energies below 10 KeV. the dipolar excitations strongly dominate the second-order transitions; therefore, especially in the soft x-ray region. the dipole approximation is valid to a great extent. For this reason, we will consider only dipolar transitions in the following. It is useful, however, to mention that one can find special conditions to observe second-order transitions or even interference effects between second-order and first-order excitations. This kind of spectroscopy is in its infancy. With improved control of the photon polarization and the experimental conditions, one can hope to isolate these weak effects and learn about the peculiar symmetry properties of the system exhibiting them.

In the dipole approximation. the probability w(hv) of absorbing a photon of energy hv in the unit time is given by the Fermi golden rule:

The absorption probability is related to the absorption cross section cr, defined as the energy absorbed per unit time divided by the average energy flux:

(5)

(6)

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TABLE 1. Relative transition probabilities of photoexcitations due to the H2 and HI interaction Hamiltonians for different atoms and edges. The absorption energies in e V are also indicated.

Atom K Edge (Is) Ll Edge (2s) L3 Edge (2P3I2)

C6 2 x 10-4 285

Si14 1 x 10-3 1 x 10-5 5 x 10--6 1839 150 100

Ge32 9x 10-3 2 X 10-4 1 x 10-4 11105 1415 1215

Sn50 2.5 x 10-2 6 x 10-4 5 x 10-4 29200 4465 3930

Pb82 8 x 10-2 2.5 X 10-3 1.5 X 10-3

85530 14840 12285

In photabsorption experiments from a core level, the system is excited from its ground state Ivn) with n electrons in the valence shell into an excited state 1c-1vnf) with a core hole and an electron in the continuum. The transition is governed by the dipole selection rules which in the j-j coupling scheme are AI = 0, ±1; I!lM = 0, ±1; ll.L = ±1; and /!is = 0, where J and M are the total angular momentum quantum numbers, L is the angular momentum, and S is the total spin. We see that by exciting a core level of given symmetry, we select only excited states with symmetry consistent with these selection rules. This state selection becomes even more stringent when one can control the photon polarization and can establish the direction (magnetically oriented atoms) or the orientation (oriented anisotropic crystals or molecules) of the quantization axis in the systems. For this purpose, synchrotron radiation has the unique feature of producing tunable, highly linear, circularly polarized radiation, as will be further discussed later. Furthermore, from Eq. (6), it is obvious that in core excitation, one will select those excited states that have an appreciable overlap with the core-state radial wave function, i.e., states localized around the photoexcited atom.

In summary, core excitations are a means to investigate excited states that are localized at specific atomic sites and have defined symmetry consistent with the selection rules and the photon polarization.

In one-electron language, excitations near a core absorption edge involve the first empty states of the system, which are complementary to the those at the top of the valence band or those below the Fermi level in a metal (Le., those states that contribute the most to the electronic and thermal macroscopic properties of the specific material). Optimally resolved measurements of these near­edge core excitations can be utilized to select and study these states specifically. In the next subsection, we will discuss the issue of optimal resolution, which will lead naturally to the importance of experiments in the soft x-ray region.

2.2. BROADENING MECHANISMS IN CORE EXCITATIONS AND OPTIMAL INSTRUMENTAL

RESOLUTION

The broadening of an x-ray absorption spectrum arises from various sources:

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1. An instrumental broadening due to the response functions of the x-ray monochromator and beamline optics.

2. The so-called initial- and final-state broadening that can arise from various causes, all associated with specific properties of the system. Among them are static disorder due to composition fluctuations, elastic deformations, broken crystalline structure, defects, and impurities, which induce variations in the ligand field of the photoexcited atomic species and change both the ground-state energy (chemical shifts) and that of a specific excited state. These effects are strongly system-dependent and may, in themselves, constitute the aim of a core-photoabsorption study. Similarly, energy dispersion in the final states due to orbital hybridization (band structure) or pile-up around specific excitation energies of the multiplet structure will also broaden the spectral features. These effects can be of interest and may justify a specific study. Finally, the creation of a core hole in the excited state introduces a time-dependent potential that strongly modifies the ground-state potential, allowing in general a rearrangement of the electronic structure with important electron relaxations and core-hole screening effects. They may give rise to multiple spectral features as well as to an energy continuum of excited states that will widen the spectral features. Again, these effects are of great interest in understanding electron-relaxation mechanisms in the presence of a perturbing localized potential.

3. Dynamic disorder due to atom vibrations. These effects are slow on the time scale of the photon absorption event. One may distinguish three different situations:

The vibrational motion of the atoms, which is different from zero even at OK, induces a temperature-dependent distribution of the atoms surrounding the core-excited one, but at the photoabsorption event, the excitation energy will be determined by the specific configuration of the nuclei at that time. This broadening is not necessarily proportional to kT, but it is determined by the dependence of the crystal field on the interatomic distances. Its study can reveal important physics ofthe electron-phonon coupling. Another scheme resulting in the broadening of spectral features and associated with atomic vibrations is the Franck-Condon effect. Considering that the vibrational states of the systems are different in the ground state and in the excited state in which the core­hole potential is turned on, it is possible that the electronic excitation gives rise to vibrational excitations. They result from the projection of the ground-state distribution of vibrational states on the vibrational eigenfunctions basis set of the interatomic potential in the excited state. The study of the vibrational structure associated with a specific electronic transition constitutes, especially in molecules, a very sensitive method for determining the properties of the excited-state interatomic potential, and one can derive new interatomic distances, vibrational frequencies, potential anharmonicities, and changes in molecular symmetry. The Franck-Condon effect is based on the Born-Oppenheimer approximation, according to which the total wave function is assumed to be factorized into an electronic part, which parametrically depends on the nuclei positions, and a purely nuclear part. There are cases in which this approximation breaks down, and a strong coupling between the electronic states and the atomic positions takes place (vibronic coupling). Under these conditions, the Franck-Condon factors are no longer valid and spectral broadening as well as new

256

spectral features may arise from excitation-energy dependence on specific positions of both electrons and nuclei at the photo absorption event.

4. The finite core-hole lifetime. This is generally the most important cause of spectral broadening. In contrast with points 2 and 3, the core-hole lifetime is an almost purely atomic property [2, 4]. The decay of the core hole, which takes place either by emission of a fluorescent photon or of an Auger electron, has a lifetime determined by the number of electrons that can lower their energy, filling up the core hole, with a probability directly related to their overlap with the core-hole wave function. The radiative decay is governed by the same interaction and selection rules responsible for optical excitations, whereas, in the Auger case, the decay channels are dictated by the coulomb and exchange interaction integrals between the decaying electrons and the holes in the core level and in the continuum. The electrons most favored are those close to the core hole, Le., those on the excited atom. For deep core states, Le., when the atom has other core levels at lower binding energies, electrons from these shallower core states constitute the main decay channel, and the core­hole lifetime is almost a purely atomic property. For shallower core levels, where decay involves only more delocalized valence electrons, the core-hole lifetime can depend on the specific electronic structure of the system. This, however, is still not at large variance from the lifetime of the isolated atom.

In summary, the core-hole lifetime is an almost purely atomic property, especially for deep core levels, and is shorter (Le., the spectral broadening is wider) for deep core levels. It follows that to minimize spectral broadening due to the core-hole lifetime and thus optimize the conditions for resolving features in the absorption spectra, one has an advantage in studying shallow core levels. This argument naturally leads to the soft x-ray spectral region to obtain the greatest amount of information in photo absorption experiments.

It must be mentioned that recently it was demonstrated that one can perform photoabsorption experiments in the hard x-ray region without core-hole lifetime broadening [5]. This is possible via the so-called resonant Raman scattering mechanism, which involves only the virtual creation of a deep core hole by the use of photon energies at resonance with the core-hole binding energy. In these experiments, the photon energy is tuned across the deep-core absorption line, while a photon analyzer monitors the radiative-decay photon with a resolution comparable with the lifetime of the shallower core level participating in the decay. Under these conditions, the deep core hole is only virtually excited and a shallower core hole, typically with binding energy in the soft x-ray region, is left in the system. Due to the resonance mechanism, however, the intensity of the fluorescent photon as a function of the primary photon energy reproduces the deep core­level absorption spectrum. Spectral broadening, however, is determined only by the mono­chromator and analyzer resolution and by the core-hole lifetime of the shallower core state [2]. This photon-scattering mechanism, formally described by second-order perturbation theory of the interaction Hamiltonian HI. allows one to perform absorption spectroscopy in the hard x-ray region with the lifetime broadening typical of core levels in the soft x-ray region.

The typical lifetime of core states in the soft x-ray region (100 to 2000 eV) varies with increasing binding energy from minimum values of 50 to 500 meV [4]; that is, they produce a spectral broadening of about 5 x 10-3. This value sets the experimental requirement for the desired resolution for a soft x-ray monochromator. One must aim for a resolution of -10-5 x 10-5

to perform soft x-ray absorption studies with instrumental resolution better than the broadening mechanisms intrinsic in the system.

257

2.3. HIGH-RESOLUTION SOFf X-RAY MONOCHROMATORS

In this subsection, we will not describe the details of monochromator design, but we will make some generalizations about successful philosophies for characterizing an instrument suited for performing high-resolution soft x-ray spectroscopy, i.e., an instrument able to deliver a resolving power ElM in the range of 1()4 with good flux, i.e., with a good match between the monochromator acceptance and the phase space of the synchrotron-radiation source,

In the hard x-ray region, the typical method for producing a monochromatic photon beam is to use crystals, while in the optical and vacuum ultraviolet (VUV) ranges, one uses diffraction gratings in reflective or refractive geometries. The soft x-ray region lies between the VUV and the hard x-ray regions, and one might think it possible to benefit from either technology or from a mixture of the two [6].

With regard to crystal monochromators, the resolving power of a crystal reflection is determined by the crystal's form factor, which determines the reflectivity of each individual plane. This reflectivity, however, must be reconciled with the photoabsorption in the crystal, which must be kept small in the diffraction region. In the soft x-ray region (100-2000 eV), the photon wavelength varies from 120 to 6 A. For a resolving power of 104, it is necessary to have 1()4 crystalline planes contributing to the reflection, with a 2d-spacing comparable to the photon wavelength. This implies crystal volumes with a depth of 60 to 3 ~m. At the longer wavelength, this requirement is impossible to achieve because nature does not give us any crystalline material with absorption small enough to allow such penetration. At energies above 700 eV, i.e., wavelengths smaller than 17 A, there are some crystals-generally quite exotic materials-that meet the requirement for a large d-spacing and low absorption, for example, beryl, l3-alumina, and recently YB66. These materials are at the borderline of fulfilling the resolution requirements. They are being actively investigated, and there are many results promising reliable coverage of the 1000-2000 eV region with a resolving power of 104•

In the lOO-1000-eV region, however, there is not much hope of using crystals, and one is obliged to use diffraction gratings in a reflective geometry and at very glancing angles for optimal reflectivity. For use in the soft x-ray region, diffraction gratings must be coupled with refocusing optics, i.e., with mirrors that are also operated at very shallow incidence angles for good reflectivity. When designing such an optical system, one must face manufacturing constraints that place restrictions on how perfect mirrors and gratings can be constructed in the desired dimensions. In specifying such elements, one must consider the figure error, i.e., the maximum deviation from the theoretical shape of the element. This is the most critical parameter for accomplishing the desired optical performance. One also must specify the microroughness, i.e., the finishing of the element surface on a scale necessarily smaller than the photon wavelength. Typical requirements for a performing instrument are less than 0.5 arc sec of figure error on the whole optical element and a microroughness in the I-A scale. In the case of gratings, one must also specify the line spacing and the groove depth and shape. The shape is critical for the efficiency of the diffraction orders, while the homogeneity of the groove depth and spacing is critical for the quality of the diffracted beam and the minimization of nonmonochromatic scattered light. For beamlines utilizing a bending-magnet synchrotron-radiation source, the typical dimensions of illuminated areas on mirrors and gratings are of the order of 20-50 cm in the beam direction and 2-5 cm in the transverse direction. If an undulator source is used, these dimensions can be reduced, particularly in the transverse direction where the illuminated area can be well below 1 cm. Given the critical requirements for the optical elements, an undulator beam is very attractive because of its collimation, and this, together with the increase in flux, constitutes

258

one of the major motivations for third-generation synchrotron radiation from high-brightness undulator sources.

Presently there are two monochromator designs that have been able to achieve resolving power in the IQ4 range. One is the Dragon monochromator, developed first on the VUV ring at the National Synchrotron Light Source (NSLS) at Brookhaven. This monochromator achieved the desired resolution in 1987 in measurements of the K-edge photoabsorption spectrum of the N2 molecule in the gas phase [7-9}.

A new Dragon monochromator has recently been put into operation on a soft x-ray undulator in the x-ray ring at the NSLS and has obtained the current record for resolving power [10}. Figure 1 shows the N2 Is~ 17tg * absorption spectrum measured on this Xl undulator Dragon beamline.

A resolving power of approximately 2 x 104 has been estimated from the relative peak-to­valley intensity of the vibrational structure. It is obvious, however, that the main source of broadening is the width of the peaks at half maximum, which is in the range of 100 meV at 400 eV. This broadening, due to the Nls core-hole lifetime, would correspond to a resolving power of 4000, a value well below the instrument performance. It is therefore clear that, for this kind of study, we are presently able to obtain the desired resolution with soft x rays.

The second instrument that has been able to achieve a resolving power of 104 is the SX700 at the BESSY ring [II}.

400

~ 350 ·2 :J

.e 300 ~ ~ 250 ·iii I::

~ 200 I::

o 150 e-o (/) 100 ~

50

400.0 400.5 401.0 401.5 402.0

Photon energy (eV)

Figure 1. K-shell photoabsorption of N2 measured on the XlB Dragon undulator beamline at the NSLS.

259

3. Representative Results of High-Resolution Soft X-Ray Absorption Studies

3.1. ABSORPTION MEASUREMENTS IN GAS MOLECULES

An important area for high-resolution soft x-ray spectroscopy is the study of inner-shell excitations in atoms and molecules. Typical interesting applications are the study of the lifetime of an atom in different chemical environments, its Rydberg series, the validity of the Born­Oppenheimer approximation and the use of the Franck-Condon principle to characterize the interatomic potential in the excited state, the core-hole localization-delocalization pictures, and double excitations. In addition, it would be interesting to distinguish specific electronic transitions from their vibrational replica, for example, by measuring marked molecules and determining the isotopic shifts in the vibrational structure of a specific electronic excitation. It would also be interesting to test theoretical models like the equivalent core-hole model (ECM). In the following subsections, we will review some experimental results on gas-phase molecules measured on the U4 Dragon beamline at the NSLS.

3.1.1. K-Shell Photoabsorption of N2. The K-edge photoabsorption spectrum of N2 is a prototype system for s1;!Jdying inner-shell processes and testing theoretical models. Figure 2 shows the K-shell photoabsorption spectrum [9]. The peaks between 400 and 402 eV are Nls~l1tg* excitations and their vibrational sideband. The features observed between 406 and 410 eV are Nls~Rydberg-series transitions. The Nls ionization threshold occurs at 409.94 eV.

K-shell photoabsorption of gas-phase N2

~ .§ N1s ~ 11tg .e N1 s ~ Rydberg series, ~

Shape resonance

~ Double excitations

~ ~ c

~ r o x 10 Ul .0 «

400 405 410 415 Photon energy (eV)

Figure 2. K-shell absorption spectrum of the N2 molecule.

420

260

The area between 413 and 416 eV is due to double excitations that can be viewed as shake-ups of the Nls~ 17tg* transition from valence orbitals to the I1tg* state itself. The broad peak at 420 eV is the cr shape resonance. Blow-ups of the Nls~l1tg* transition, Rydberg series, and double excitations are shown in Fig. 3.

With such improved resolution, new absorption features were resolved and spectroscopic constants determined with high precision, including term values, vibrational frequencies, and internuclear separations for different excitation states. From the comparison with the NO optical transitions, the final-state configurations of the Nls~Rydberg-state excitations (peaks shown in Fig. 3b) were determined, thereby resolving uncertainties existing in the literature. Moreover, the core-hole localization picture and the ECM were found to be valid to a very high degree of precision. One can observe subtle differences in the vibrational frequencies and in the derived internuclear separations for the Nls core excited states of N2 and between the equivalent core states of N2 and NO. They were explained in terms of the properties of the bonding character of the valence orbitals and of the charge differences in the vicinity of the nuclei.

3.1.2. Carbon K-Shell of co and 13C180. Like N2, CO is a prototypical model molecule in gas­phase spectroscopy. In Fig. 4, we show the Cls~21t*, the Cls~Rydberg states, and the double­excitation spectra for CO and 13C180 [12].

-~ c: ::J

(a)

414 415 £ 400 401 402 r-----------~--------------------~

~ (b) ® ~ C/) c: Q)

£ c: o 2-~

N1 s ~ Rydberg series

Photon energy (eV)

Figure 3. Blow-ups of the regions marked in Fig. 2.

(a) C1s ~ 27t*

287

(b) C1 s double excitation

Photon energy (eV)

_12C 160

···-13C 180

261

Figure 4. (a) Cls~21t*, (b) double-excitation spectra, and Cls-7Rydberg states for CO (solid line) and 13C180 (dashed line) [12].

Isotope shlfts of only 20 meV are observed with -300-eV excitation energy and a lifetime broadening of -100 meV. This shift agrees with a -O.09-hv value calculated for the vibrational frequency of a C-O atom pair. This isotope shift allows the identification of the vibrational sidebands of different electronic excitations. The intensity of these sidebands is also very different from the intensity of those in N2 because of the different internuclear separation in the two molecules in both ground and excited states.

Identification of the vibrational sidebands allows a straightforward assignment of the different Rydberg states. Made by the same procedure used for N2 (Le., comparison with the optical spectrum of NO), the peak assignment for the core-excited Rydberg states of CO is given in Fig. 4. Five different electronic excitations are also identified in the double-excitation region. Their aSSignment is more difficult and must be assisted by theoretical calculations. As in the case of N2, they are shake-ups of the CIs~27t* transition from the valence-band orbitals, presumably from the I1t and 5a to the 21t or to Rydberg states.

262

The comparison between the Rydberg states in the C K-sheU of CO and in the N K-shell of N2 is of particular interest. Since the core-excited Rydberg states of CO and N2 have the same final states in the equivalent core model as the 27t valence excited states of NO, one expects very similar Rydberg states in the two molecules.

In Fig. 5, we see the direct comparison of the Rydberg states of CO and N2 plotted as a function of their term value, Le., after subtracting the ionization potential from the excitation energy (296.08 eV in CO and 409.94 eV for N2). This figure clearly shows that the term values of the 3scr, 3p7t, and 4p7t states are essentially the same, confirming once more the high degree of validity of the ECM.

The Rydberg states around 1.5 eV, however, have term values differing by -0.1 eV. This apparent breakdown of the ECM can be resolved by assigning these two peaks to different Rydberg states, namely to the 4scr for N2 and to the 3d7t for CO. This implies that the ECM is valid for the transition energies but cannot be used to interpret the oscillator strength, which can be very different in different molecules. In fact, we can further observe that, while all the np7t Rydberg states are visible in the two spectra, the nscr are not seen in CO for n > 3. Possible reasons for such intensity differences may be that N2 has a mirror-image plane whereas CO does not and that the overlap between the Is and the Rydberg-state orbitals is different in the two molecules. In fact in CO, the core electron is excited from the N site of its equivalent core valence-excited NO molecule; whereas for N2, it is excited from the 0 site.

3.1.3. Symmetry Breaking of Core-Excited Ethylene and Benzene. The study of vibrational structure has been so far restricted to core-excited diatomic molecules that have only one

Core-excited Rydberg states of C*O and NN*

3p7t

4 3 2 1 o Term values (eV)

Figure 5. Comparison of the Rydberg states of CO and N2 plotted after subtracting the ionization potential from the excitation energy (296.08 eV in CO and 409.94 eV for N2).

263

vibrational mode. The extension to more complex molecules has been attempted on core-excited ethylene and benzene, in which a large number of modes can be excited [13]. The spectra are in fact much broader, although a few well-defined spectral features can nevertheless be recognized. Using the isotope shift, one can assign these features to modes that predominantly involve a defined pair of atoms. In Fig. 6, we see the Cls~1t* excitation in hydrogenated and deuterated ethylene and benzene. Comparing the relative positions of the absorption features, one identifies three different vibrational structures: features A and C, which shift consistently with a C-H(D) mode; and feature B, which does not shift as expected for a C-C mode. Features B and C have energies consistent with the C-C and C-H stretching frequencies and are assigned to these two modes. Feature A, however, is at too Iowan energy to be a stretching mode; however, the ground-state values indicate that it is consistent with the first vibrational state of the C-H out-of­plane non-totally symmetric bending mode. Using this tentative assignment, one must observe that, according to the Franck-Condon principle, the observation of such a mode provides definitive evidence that, in both molecules, the ground-state planar symmetry is broken. From this study, one also finds that the C-H stretching frequency is larger than in the ground state, while the opposite happens for the C-C mode. This difference indicates that the internuclear potentials between the core-excited carbon and its C and H neighbors have been modified differently. This symmetry breaking, the stronger C-H potential, and the weaker C-C potential are consistent with the the C-C antibonding character of the 1t* orbital. This analysis also supports the view that the core hole is localized on the photoexcited carbon atom. This analysis is

0 ~B " ,

:§' ·c

C2H4 :::l

.ci ... ~ z. ·w c:: Q) -.E

C2D4

284 285 286 Photon energy (eV)

:§' ·c :::l

.ci ... ~ z. ·w c:: Q)

E

285

0 ,

CaHa

CaDa

286 Photon energy (eV)

Figure 6. Cls~1t* excitation in hydrogenated and deuterated ethylene and benzene.

264

obviously qualitative and based on the Born-Oppenheimer approximation. Sophisticated quantum chemistry calculations have, in fact, shown that a quantitative explanation of the absorption spectrum requires vibronic coupling to be taken into full account [14]. The conclusions of this theoretical work, however, confirm the qualitative analysis presented here and are able to derive in detail both the planar-symmetry breaking indicated by peak A and the changes in the C-C* and H-C* bond lengths indicated by peaks Band C. This work clearly established for the first time vibronic coupling in core excitation and demonstrates the richness of the detailed information that can be obtained from high-resolution soft x-ray spectra on molecules when coupled with theoretical calculations.

3.2. ABSORPTION MEASUREMENTS IN SOUDS

High-resolution soft x-ray spectroscopy has many applications in condensed matter. They range from surface-physics problems to materials science, but most importantly to the study of many fundamental aspects of the electronic structure of solids. In the following subsections, we will review some recent studies of core excitations in potassium halides and in high-T c

superconductors performed on the Dragon monochromator.

3.2.1. Crystal Field Splitting in Core Excitation of Ionic Crystals. Empty or partially empty d states in cubic ionic crystals are split by the crystal field into two states with Eg and T2g symmetry [15]. The L2,3 absorption edges of potassium in potassium halides offer a unique opportunity to study these effects. In fact, the K atoms are in the dO configuration in the ground state, and at the L absorption edge, one expects to observe the 3dO~2p53dl excitations with a multiplet structure dominated by the crystal-field interaction. Figure 7 shows the K L2,3 photo absorption spectra in KMnF3, KF, KCI, KBr, and KI taken at room temperature [16].

Two well-resolved peaks, A, B and A', B' are observed at the two edges of the potassium halide spectra, while only one peak is observed in KMnF3. The two peaks correspond to the 2p53d1 excited state split by the crystal field into the Eg and T2g configurations. The separation between peaks A and B (A' and B') decreases from KF to KI, indicating the reduction of the crystal field with increasing K-halide distance. The absence of split peaks in KMnF3 follows from the unique arrangement of the ions in this crystal: Here the crystal field generated by the twelve F- ions in the first shell is nearly cancelled by the eight Mn+2 and six K+ ions in the second and third shells [16]. These observations not only confirm the origin of the splitting but also allow for a direct measurement of the crystal field strength.

Another important aspect of this study is the investigation of the crystal-field dependence on the interatomic distance and the atomic motion, which directly relates to the electron-phonon coupling [16]. For this purpose, the temperature dependence of the splitting in KCI was investigated [16]. Figure 8 shows the spectra taken at 80,300, and 600 K.

These spectra clearly show that, with increasing sample temperature, the energies of peaks A and B (A' and B') shift toward each other and the line widths of all the peaks are greatly increased. The energy shift is due to the increase of lattice size upon heating, and the broadening is caused by the fact that the photoabsorption process is faster than the vibrational motion of the atoms, so that the crystal-field potential on the 2p53d1 excited states is different for each instantaneous arrangement of the atoms. Further analysis based on the lODq theory allows one to derive from these spectra the average size of the 3d orbitals in the 2p core excited state and Shows

265

Potassium L2,3 edge

Photon energy (eV)

Figure 7. Potassium L2,3 photoabsorption spectra of KMnF3, KF, KCI, KBr, and Kl.

a reduction with respect to their ground-state values. The dynamical character of the process is further demonstrated by the correlation between the spatial extent of the excited-state wave function and the dielectric function at optical frequencies.

3.2.2. Electronic States in La2_xSrxCu04 Probed by Soft X-Ray Absorption at the 0 K-Edge. The origin of carriers in high-Tc superconductors is, at present, one of the most challenging problems in solid-state physics. Spectroscopic probes with chemical specificity such as resonant photoe­mission, Auger spectroscopy, electron-energy loss, and x-ray absorption have provided important insights on the symmetry and nature of the electronic states responsible for conduction and, in particular, have shown the dominant 0 2p character of the carriers in the hole-doped cuprates

266

KCI Potassium L2,3 edge

296 Photon energy (eV)

........... 80° K ------- 300° K -- 6500 K

Figure 8. Potassium L2,3 photoabsorption spectra in KCl taken at 80, 300 and 600 K.

[17]. In this example of how high-resolution soft x-ray spectroscopy can contribute to solid-state physics problems, we show the results of a careful investigation of the 0 K-edge photoabsorption spectra in the near-edge region of La2_.tSrxCu04 [18] measured as a function of Sr concentration in the region of the insulator-metal transition. It was already established that the undoped mate­rial is a correlated electronic system well described by the Charge-transfer Mott insulator model. The important question motivating the 0 K-edge photoabsorption study is whether, after the insu­lator-metal transitions and at concentrations where superconductivity is observed, the material is still a highly correlated system described by a doped charge-transfer insulator model or whether a less-correlated band-like model is recovered for the metallic state.

In Fig. 9, we show the 0 K-edge spectra of La2_.J)rxCu04 for x between 0 and 0.15 and for an oxygen-enriched sample La2Cu04.005' The data were obtained by monitoring the 0 Ka fluorescence yield for increased bulk sensitivity [18]. Two distinct peaks, labelled A and B, are observed at photon energies around 528.8 and 530.3 eV. They show a separation comparable to the optical gap in the insulating phase (1.8 eV). Peak A, absent for x = 0, grows in intensity with the Sr concentration, while peak B loses intensity. The total intensity of the two peaks increases with x. This behavior can be quantitatively explained in the frame of the Hubbard model, and peak B is identified with transitions to the bottom of the upper Hubbard band, which has a predominant Cu 3d character but through hybridization also acquires some 0 2p character [19]. The lower peak, present only in the hole-doped samples, correspOnds to transitions into the carrier states at the top of the charge-transfer band, which has a dominant 0 2p character.

"'C Qi .:;. Q) CJ c:

~ ~ o :2 "'C

.~ a; E o z

267

530 535

Photon energy (eV)

Figure 9. (a) 0 K-edge spectra of La2-xSrxCu04 for x between 0 and 0.15 and for an oxygen-enriched sample La2Cu04.005. The data were measured by monitoring the 0 Ka fluorescence yield. (b) The pre­edge peaks of Fig. 9a are shown here after subtraction of the background [solid lines indicated in Fig. 9(a)].

The absence of any discontinuity at the insulator-metal transition (x = 0.07) and the oscillator­strength transfer from peak B to peak A as a function of doping are characteristic of a highly correlated electronic system, even in the metallic state [19]. Therefore, this study demonstFates that the soft x-ray absorption data are consistent with a picture of the low-energy electronic states derived from a doped charge-transfer insulator over the entire range of carrier concentration from the very dilute insulating regime to the metallic superconducting regime.

268

4. Soft X-Ray Photoabsorption Spectroscopy with Circularly Polarized Synchrotron Radiation

Synchrotron radiation from a bending magnet is linearly polarized in the orbit plane. Similarly, radiation from an insertion device with a vertical magnetic field is linearly polarized. Synchrotron radiation emitted off the orbit plane of a bending magnet or from specific insertion devices with asymmetric vertical fields or with a horizontal component in the magnetic field can acquire a large degree of circular polarization. Circularly polarized light can be very useful for studying systems that do not have inversion symmetry on a specified quantization axis. A typical example is an aligned magnetic system. In this section, we will examine in some detail the use.of soft x-ray absorption spectroscopy with Circularly polarized radiation to study the dichroic response of an aligned magnetic material. As we will see, one can obtain atom-specific information on the magnetic properties of the material. For example, one can establish whether the atom carries a magnetic moment and whether it is aligned parallel or antiparallel to the external magnetic field. Another important piece of information contained in the dichroism of an x-ray absorption spectrum regards the orbital angular momentum of the photoexcited atom. In this respect, circular magnetic x-ray dichroism (CMXD) is a complementary technique to those that are more sensitive to the total magnetic moment of a specific atom, such as magnetic x-ray and neutron scattering.

A specific interest in CMXD studies in the soft x-ray range comes from the fact that one can excite strong dipole-allowed transitions in the most interesting magnetic materials, such as 2p~3d transitions in 3d transition metals, 3d--Hf transitions in rare earths, and 4d~5f and 4f~6d transitions in actinides.

In a CMXD experiment, one measures the photoabsorption spectrum with the polarization vector parallel or anti parallel to the external magnetic field used to align the magnetic material. The CMXD is obtained by taking the difference of the two spectra.. For a magnetically aligned atom, Le., with the direction and sign of the z-quantization axis fixed by the external magnetic field and with specified quantum numbers J, L, S, and M, the dipole selection rule dictates t::.M = 1 or -1 when the circular-polarization vector (photon helicity) is respectively parallel or anti parallel to the z-quantization axis. Electrons photoexcited with circularly polarized light from a spin-orbit split core excitation in an aligned atom therefore have a net magnetization that depends on the specific core level and has opposite values for opposite circular polarizations. In order to observe a dichroic effect for excitations at a given energy, it is necessary that the allowed final states have a net magnetization, so that the transition strengths are different for the two opposite circular polarizations.

Typically, the strongest dichroic effects come from excitations involving final states of a partially filled shell with a magnetic moment in the ground state. This is easily understood considering that the empty states of such a shell will have the opposite polarization from the filled states. Dichroism can also be observed in excitations involving empty shells that are polarized (Le., energy split between spin-up and spin-down transitions) by exchange interactions with the magnetic states existing in the atom. These are the magnetic moment in the valence band as well as the magnetic moment of the core hole and of the photoexcited photoelectron, which have equal and opposite signs but very different spatial distributions that can polarize the system differently.

From the previous introduction on the origin of CMXD, it is understood that dichroic effects in core-excitation spectra are expected from a net magnetic polarization of the photoexcited core level and of the final states. It is then clear that it is necessary to align (by direction and sign) a macroscopic number of core-excited atoms; therefore, CMXD is expected in ferro- and ferri-

269

magnetic materials as well as in para- and diamagnetic systems if the external magnetic field is sufficiently strong to align them (Zeeman effect). No circular x-ray dichroism can be observed, however, in anti-ferromagnetic materials, in which the same atom is both parallel and antiparallel to the external magnetic field in equal amounts.

In the following subsections, we will consider, in a one-electron picture, the information that can be extracted from a dichroism spectrum. A more complete n-electron treatment will be briefly mentioned. A few experimental results will be reviewed to stress the great potential of this spectroscopic technique.

4.1. MAGNETIC X-RAY DICHROISM AT TIlE LZ.3 EDGES OF NICKEL

In the previous section, we mentioned two fundamental requirements for observing CMXD: the natural spin polarization of a spin-orbit split core excitation when using circularly polarized radiation and a spin polarization of the final states. These requirements are met by the 2p~3d excitations of elemental ferromagnetic systems such as iron, cobalt, and nickel.

In Fig. 10, we show the L2.3 soft x-ray photo absorption and circular CMXD of nickel [20]. The two photoabsorption spectra were measured with a magnetic field of 0.3 T aligned along the photon-beam propagation axis, either parallel or anti parallel to the polarization vector. A degree of circular polarization close to 90% was obtained by collecting a small cone of radiation (0.2 mrad) centered 0.75 mrad above the electron-orbit plane in the U4 Dragon bending-magnet beam line at the NSLS VUV ring. We observed a strong dichroic effect of about 10% at the L3 edge and almost twice as much at the L2 edge. At the two edges, the dichroism has opposite sign.

The dichroism of core p to 3d empty states in nickel was predicted with a very simple model by Erskine and Stern [21]. In this model, the empty part of the nickel 3d band was represented as a combination of 3d atomic states with spin down (minority band). A combination of 3d orbitals, symmetric in the spherical harmonics Y 2.m and Y 2.-111' was used to impose the quenching of the orbital angular momentum from the crystal field. They considered a spin-orbit split 2p core level and calculated the transition strengths from the 2p levels to the 3d final state with circularly polarized radiation. The transition operator eqr, in the spherical harmonic formalism, can be conveniently expressed as rY I.q' provided that the quantization axis in the sample coincides with the photon-beam propagation vector. The values q = 1 or -1 correspond to circularly polarized radiation, and q = 0 corresponds to linear polarization. The calculation of the dichroism in this model is reduced to the evaluation of integrals such as (Y l,m\IY 1,q1Y 2,m2)' The results are reported in Table 2 for the two circular polarizations.

From Table 2 one can calculate the dichroism (~I - ~_I) and the photoabsorption intensity (~l + iLl) at the two edges. It is found that (~I- ~-I)r'2= 2/3(A - C) and ~l- ~-lk3= -2/3(A­C) for the dichroism, whereas (~I + ~-lk2= 2/3(A + B + C) and (~l + iLlk3 =4/3(A +B + C) for the photo absorption intensity.

This simple model predicts correctly the relative sign of the dichroism at the two edges and gives the statistical value of the branching ratio. From the quantitative point of view, however, it is not satisfactory. We see that independently from the values of the parameters A, B, and C, the dichroism ratio is always -1 and its integral is 0; similarly the branching ratio is always 2. lbis is in disagreement with the experimental results, which show well above any possible error a dichroism ratio of -1.6 and a branching ratio of 2.5.

270

.~ I/) c:

~ c: o e o I/)

(a) L2•3 photoabsorption of nickel

120 -it ---- t-l.

80

~ 40

o '---_....L-____ .L...-___ ....L-_-'

+4 (b) Magnetic circular dichroism

o

-4

-8

850 870 890 Photon energy (eV)

Figure 10. Photoabsorption and magnetic circular dichroism (MCD) spectra of the Ni L2.3 edges in magnetically aligned ferromagnetic nickel measured with circularly polarized radiation parallel or antiparallel to the external magnetic field.

On the basis of these results. we can appreciate the role of the spin-orbit interaction. which. by splitting the core excitations into two branches. allows the observation of dichroism. Moreover. we can speculate that to modify the relative dichroism and photoabsorption intensities at the two edges. it is necessary to abandon the assumption of a completely quenched orbital angular momentum. In fact. with a finite value of Lz for the nickel 3d empty band. the optical transitions involving final states Y 2.m with strengths Am. Bm will be different from those involving states Y 2.-m with strengths A-m.B-m. Consequently. one expects that the relative dichroism intensity between the two edges will depend on the value of Lz for the partially filled band. Lz for the 3d states in nickel is in fact different from 0 as a consequence of the spin-orbit interaction in the

271

TABLE 2. Transition strengths with circularly polarized light for the spin-orbit split 2P312, 2P112 core levels into a spin-down 3d..!. final state. There are three possible integrals, which have been parametrized by the constants A, B, and C.

III (Yll) Y22..1. Y21..1. Y20..1. Y2-1..1. Y2-2..1.

Y 10 i - "h Y 11..1. (2P1l2,1I2) 2/3A Y 10..1. - "2 Y 1-1 i (2pI/2,-1I2) 1I3B

Y 11 i (2p312,312) "2 Y 10 i + Y 1l..1. (2p312.1I2) 1/3A "2 Y 10..1. + Y 1-1 i (2p312.-112) 2/3B Y 1 -1..1. (2p312.-312) C

Jl-l(Y 1-1)

Y 10 i - "2 Y 11..1. (2P1l2.1I2) 2/3C Y 10..1. - "2 Y 1-1 i (2P1l2,-1I2) 1/3B

Y 11 i (2p312,1I2) "2 Y 10 i + Y 1l..1. (2p312,112) 1/3C "2 Y 10..1. + Y 1 -1 i (2p312.-1I2) 213B Y 1 -1..1. (2p312.-312) A

valence band. It is evaluated to be about 0.05 IlB in nickel, and it would be very interesting from the experimental point of view to find that such a small value of Lz is able to modify by more than 50% the relative dichroism intensity. This topiC will be discussed in detail in the next section.

To conclude, we want to stress that the simple Erskine-Stern model also shows that the absolute intensity of the dichroism depends on the number and symmetry of the allowed final states that are magnetically aligned (magnetic holes). Once the role of angular momentum is established, one may be able to predict the existence and orientation of a magnetic moment at a specific atomic site from the sign and intensity of the dichroism at a specific edge.

4.2. MAGNETIC X-RAY DICHROISM AND ITS RELATION TO ORBITAL MAGNETIZATION

In this section, we will investigate the relation between the CMXD spectra from core transitions into a partially empty and magnetically polarized valence shell and the orbital angular momentum of these empty states. It is clear that the total magnetic moment, the spin, and the orbital angular momentum of these states are equal in magnitude and opposite in sign to the corresponding quantities of the electrons occupying the considered valence shell.

We will use a one-electron model based on band structure, in which a relation between Lz and the CMXD is very simple to derive and therefore pedagogically valuable. The results obtained are identical to those obtained in a many-electron atomic model. The limitations of our conclusions will be discussed at the end of this subsection.

We consider the valence-conduction band structure of an electronic system, which for simplicity we assume to have only one atom in the unit cell. We construct a crystal state ",(k,r) using the cellular method of Wigner and Seitz, by which one defines as a unit cell a polyhedron

272

obtained by bisecting with perpendicular planes the lines joining the central atom at the origin with its nearest neighbors. One must solve the SchrOdinger equation inside this Wigner-Seitz cell and apply the appropriate boundary conditions at the cell faces. This method is very convenient if the crystal potential inside the cell can be approximated by a spherical potential; then the Schroedinger equation is separable, and the crystal state can be expressed as

(7)

where cl,m(k) are appropriate coefficients and e, <P, and r are the polar coordinates of r with respect to the center of the cell (the atomic site). The function RI(Ek, r) is a solution of the radial wave equation with a spin-independent, spherically symmetric crystal potential and energy eigenvalue Ek. Each solution is associated with a specific k value by imposing the Bloch theorem at the cell boundaries. Xs is the spin wave function. The approximation of a spin-independent, spherically symmetric crystal potential is often appropriate, but this fundamental assumption must be kept clearly in mind because of its effect on our conclusions.

If Eq. (7) is used to represent a crystal state, the electron density in k space for a subshell of angular momentum I, nl(k) is given by

(8)

and the expectation value of the orbital angular momentum Lzl(k) in the same subshell is

(9)

We consider one-electron transitions from a core state 'l'c(r) = Cll (r)Yllml(e,$)XJ)nto the crystal state'l'(k,r). The optical absorption with polarized radiation is proportional to

~q(c,k) -11('I'c(r)lrYl,q\'I'(k,r)}112 =

11~I~mClm(k)( Y1lml (e,<p)IYl,ql Y1m. (e,$»)( Cll (r)HRz. (Ek,r)BslSI12. (10)

Defining P cl(k) = (Cl l (r)lrIRl(Ev» and using the Clebsh-Gordon coefficients for the angular­matrix element, one can write Eq. (IO) as

~q(c,k) - ~('I'c(r)lrYI,q\'I'(k,r»)~2 =

11~I~mClm(k)[3(21 + 1)1 41t(2/\ + 1) f2 (nool/\ o)(nmql/\m\)pcl(k)BsISr .

(11)

The Clebsh-Gordon coefficient (l1mqll\m\) is different from 0 when 11- 11 $lt $1 + 1 and m + q = ml, and so only two terms of the sum may survive, i.e., 1 = 1\ + 1 and I = 1\-1 (dipole selection rule). We will consider them independently and neglect the possible interference term. We also sum on all possible core states, i.e., over ml, and Eq. (11) reduces to

Jlq(c,k) -1('1J"c<r)lrY1•ql'l'(k,r)t =

Lm lIlLmClm(k){ 3(21 + 1) /41t[2(1 + 1)+ IJ} 112

(1l001(1 ± I)O)(llmql(l ±1)(m+Q»)pcl(k)Bml.m+q8s1SI12 =

Ilpcl(k)~2Lmllchll(k)ln3(21+1)/41t(211 +1)]

11(1100111 O)(llmql(l ± 1)(m + Q»)112 •

273

(12)

The sum over ml has been carried out. The sum over m can be explicitly carried out for the two possible cases I = II ± 1.

(a) Case I = II + 1:

Jlq(C,k) -1('I'e(r)lrYl.ql\jl(k,r»)~2 = "Pel (k)112 Lm IIClm (k)112 [3(21 + 1)141t(2/-1)]

11(1l001(1-1)O)(llmql(l-1)(m + Q»)~2 =

IIpcl(k)1I2 {3/ [41t(21 + 1)(21-1) J}Lmlhm(k)1I2 Dq ,

where Dq = [12 + m2 -1- qm(21- 1)]12 for q = ±1, and Do = (12 - m2) for q = O.

(b) Casel=II-I:

Jlq(c,k) - ~('I'e(r)lrYI.ql\jl(k,r»)r =

"Pcl(kf Lm llc/m (k),,2[3(21 + 1)141t(21 + 3)]

II(nOOI(1 + I)O)(llmql(l + l)(m +q»)112 =

"pcl(k)1I2{3/[41t(21+3)(2/+1)J}Lmllclm(k)1I2 Dq ,

(13)

(14)

where Dq = [(I + 1)2 + m2 + I + 1 + qm(21 + 3)]12 for q = ±1, and Do = [(l + 1)2 - m2] for q = O.

We consider (for I = It + 1) the total absorption spectrum:

JlI(c,k)+ Jlo(c,k)+Jl_I(C,k)-

II pcl(k)f {31/[41t(21+1)J}Lm lhll(k)1I2 =

IIPd(k)f{31/[41t(21+1)]}n/(k) ,

and the dichroism spectrum:

(15)

274

JlI(c,k)- Jl_I(c,k)-

-llpel (k)112 {3 / [47t(21 + 1)l}Lmllclm(k)~2 m =

-llpel (k)f{3/[47t(2/+1)l}Lzl (k) . (16)

We see that the unpolarized absorption is proportional to the density of states nl(k), while the dichroism spectrum is proportional to the ground state expectation value of the orbital angular momentum Lzl(k) of the crystal state 'I'(k,r). Most importantly, if we take the ratio of the dichroism and the total absorption, this quantity is completely independent from the core hole and is strictly proportional to the expectation value of Lz divided by the density of states in k-space, with a proportionality coefficient that depends only on the total angular momentum l. From the experimental point of view, such a procedure is not easily applicable; whereas, in general, it is much easier to isolate a feature in the absorption spectrum that can be associated with a specific group of transitions, for example, the white line in the nickel spectra of Fig. 11, which is due to 2p~3d transitions and is responsible for the most intense dichroic signal.

In order to obtain an analytical result in performing such summation on all transitions associated with the empty states of a partially empty valence band, we must make another fundamental assumption: that the radial matrix element P cL<k) is constant for all of the considered empty states. The empty-state radial wave function contributes to this matrix element only in a very localized region around the absorbing atom where the core radial wave function is different from zero. Given that we are considering only a subshell of the partly empty band with angular momentum I, we expect that the corresponding radial wave functions for different k­space values are similar in the immediate surroundings of the central atom. Within this important assumption, we can then carry out the integration in k-space over these empty band states. For the total absorption, we obtain

T= f dk[JlI(C,k)+Jlo(c,k)+Jl_I(C,k)]

II Pel 112 {31 / [47t(21 + I) l} f dknl(k) = II Pel (k)112 {31 / [47t(21 + 1) l}hl '

(17)

where hi is the number of holes with angular momentum I in the band. Similarly, for the dichroism one obtains

d= fdk[JlI(c,k)-Jl_I(c,k)]-

-II Pel 112 {3 / [47t(21 + 1) l} f dkLz(/, k) =

-llpel f{31/[47t(21+1)l}Lzl '

(18)

where LZI is by definition the ground state expectation value of the orbital angular momentum of the empty valence-band subshell with angular momentum I, which, as previously stated, is equal and opposite in sign to the same quantity in the filled part of this valence-band subshell. Taking the ratio of d and T and repeating the same calculation for the case I = II - 1, one obtains a final

275

expression that allows the determination of LZI per hole from the integral of the dichroism spectrum divided by the total absorption intensity:

p=dIT=

{f dk[1l1 (c,k) -11-1 (c,k) 1} I {f dk[1l1 (c, k)+ Ilo(c,k)+ 11-1 (c,k)]} =

[tl(lI-1)-I(l-I)-2]/[21(l-I)]Lz/ I hi . (19)

The quantity p is independent from the core hole and gives the relation between dichroism and orbital angular momentum.

This sum rule has been obtained within two basic assumptions: (a) that the Wigner-Seitz method is applicable to a spin-independent spherically symmetric crystal potential, which allows decoupling of the Schroedinger equation into angular and radial differential equations; and (b) that the radial matrix element in the optical absorption between the core and the valence states of angular momentum I can be considered constant over the k-space range spanned by the empty band states.

Both assumptions should hold in many interesting magnetic materials, particularly in cubic symmetry. However, one can envision cases in which the sum rule can break down. For example, for an exchange-split band with both majority and minority empty states at the same excitation energy spanning different k-space regions, one might expect that the radial-transition matrix element for core excitations is different for the two bands. In such a case, the dichroism spectrum and its integral will be a weighted sum of the Lz expectation values for the two bands, with weighting factors proportional to the respective IIPcl(k)1I2. Such a situation is encountered, for example, in the 2p~3d excitations in elemental Fe; however, in this case, the sum rule seems to hold. Another case is represented by the 2p~5d excitations in elemental Gd, in which the almost completely empty 5d band is polarized by the 4f shell. The validity of the sum rule here is not completely clear and is currently under debate.

The two basic assumptions made in the sum-rule derivation are related. In the presence of a strong eXChange potential in the valence band, for example, the crystal potential is no longer spherically symmetric in the Wigner-Seitz cell; however, one can still represent a crystal state using the spherical harmonics, but must allow more than one radial function for states with the same value of angular momentum /; that is, Eq. (7) must be modified into:

(20)

One can repeat the previous derivation using this generalized crystal state. For example, in the case It = I + I, one obtains the following, instead of Eqs. (IS, 16), for the absorption intensity:

III (c, k) + Ilo(c, k) + 11-1 (c, k) -

1:n Ilpcln (k)f {31 I [41t(21 + 1) 1}1:m Ilclmn (k)112 = 1:n Ilpcln (k)f {31 I [41t(21 + 1) ]}ntn (k) ,

and for the dichroism spectrum, one obtains

(21)

276

~I (c,k) - ~-I (C, k)-

-~nllpc/n(k)112 {3 / [41t(21 + 1)l}~mllclmn(k)112 m =

-~n~pc/n(k)112 {3 / [41t(21 + 1)l}Lzn(l, k) . (22)

The sum rule expressed by Eq. (19), still within the assumption that the core-to-valence radial matrix element is constant in each empty subshell of angular momentum I of a band n containing hln holes, gets modified into:

p=dIT=

{J dk[~I(C,k)-~_I(C,k)1}/ {J dk[~I(C,k)+~o(C,k)+~_I(C,k)]} = (23)

[II(lI-I)-I(I-I)-2]/[21(l-I)](~nllpc/nI12 Lzln)/(~n~Pc/nI2hln) .

Equation (23) is a more general formulation of Eq. (19) but obviously is less valuable from the practical point of view because it explicitly depends on the core hole via the radial matrix elements P c/n. This expression must be considered whenever important band overlaps take place in a certain excitation-energy range. It should be noticed, however, that Eq. (23) depends only on the relative ratios of the P cln matrix elements, and therefore they can be evaluated quite precisely with fairly simple methods when the band wave functions are known. With reasonable assumptions about the relative value of Lz per hole in the different bands, one can then apply Eq. (23) to derive the average ground-state expectation value LZl. It is clear, however, that in such situations, the practical value of the sum rule is diminished.

The derivation presented here is based on a one-electron approximation inherent in the band­structure model. A similar calculation has been carried out in an n-electron atomic model, taking into account all interactions among the electrons in the system [22]. A result identical to Eq. (19) was obtained based on analogous assumptions, Le., to consider constant the radial matrix element in the optical absorption between the ground state with n electrons in the valence band of angular momentum I, Yln, and the excited states with a core hole C-IYln + 1. This assumption also implies an atomic potential with spherical symmetry. In the derivation of Ref. 22, similar to what we derived here, the sum rule was shown to hold also in the case of hybridization (Le., in the presence of more than one partly filled valence subshell) if changes in the radial matrix element and weak direct transitions from the core state to the ligand field can be neglected.

For a specific core state of angular momentum II, valence states with angular momentum I = II + I or I = II - I are automatically selected. In the possible but unlikely situation of valence states degenerating in energy with both angular momenta II + I and 11 - I, one must reconsider both derivations to take into account the effect of the interference term in the sum rule.

The sum rule shown in Eq. (19) can be directly applied to experimental results. Using the nickel data of Fig. 10, one can measure the normalized intensity of the total absorption and of the dichroism for the 2p~3d transitions. Using a tight-binding-derived density of states, the 2p~3d transitions were isolated from the background excitations [23]. An experimental value p = 0.025 ± 0.003 was found, which gave an orbital angular momentum per nickel atom Lz2 = 0.05 ± 0.006 ~B, in very good agreement with band-structure calculations and derivations from experimental neutron form factors. The sum rule has also been successfully applied to other materials such as the 2p~3d photoabsorption and dichroism spectra of iron and cobalt and the 3d~4f of gadOlinium, confirming its general validity and very high sensitivity in determining Lz.

277

We conclude this chapter by noting that the sign of the dichroic signal is determined by the sign of Lz, provided that the angular momenta I and II of the valence shell and of the core level are known. (Obviously the sign of the circular polarization with respect to the sign of the quantization axis in the sample imposed by the external magnetic field must also be known.) From the dichroism spectrum and from Eq. (19), one can derive the sign of the orbital magnetic moment and therefore its coupling with the external magnetic field. In most cases, one also knows the coupling of the orbital magnetic moment with the spin moment (Hund's rules); therefore, in principle, it is possible to derive the direction of the magnetic moment on a specific atom from the sign of the dichroism at a given edge. Applications can be found in the literature [24] and in the contribution of 1. StOhr to these proceedings.

5. Conclusions

High-resolution soft x-ray spectroscopy has advanced substantially in the past few years, thanks to the development of new monochromators. Its application has greatly contributed to many fields of research including solid-state physics, chemistry, and traditional spectroscopy; in many cases, it can be considered to be a powerful analytical tool. We expect that the new third­generation synchrotron radiation sources coming on line in the next few years will contribute to a further substantial development of soft x-ray spectroscopy. In particular, increased flux, source brightness, and control of the polarization of the radiation from undulators will permit the implementation of the ultrahigh-resolution conditions in extreme applications involving very dilute samples, such as impurities, atoms on surfaces, and gaseous systems. The experiments described here as well as more sophisticated studies, including clusters of atoms, electron­photoemission techniques, and the emission of soft x rays from core-excited atoms in specific final states, will become possible. The new third-generation sources, are therefore expected to initiate a completely new and exciting chapter of research based on the use of soft x-ray radiation.

Acknowledgements

In this paper, a selection of experiments on high-resolution soft x-ray spectroscopy was taken from the author's research activity. Many other groups in the world have made and are continuing to make very important contributions to this field, and a large number of these excellent studies are not reviewed in the present paper because of obvious space and time limitations. However, much of this work is represented in these proceedings and in the references.

I am particularly indebted to C. T. Chen, who is co-author of the work done on the Dragon beamline and presented here. I also acknowledge useful discussions with M. Altarelli, P. Carra, J. Goulon, and T. Thole.

278

References

1. For a review on x-ray absorption spectroscopy, see for example: X-Ray Absorption: Principles, Applications, Techniques 0/ EXAFS, SEXAFS and XANES, edited by D.C. Koningsberger and R. Prins (Wiley, New-York, 1988).

2. W. Heithler, The Quantum Theory 0/ Radiation (Oxford University Press, Oxford, 1960). 3. F. Sette, G.K. Wertheim, Y. Ma, G. Meigs, S. Modesti, and C.T. Chen, Phys. Rev. B 41,

9766 (1990), and references therein. 4. D.L. Walters and C.P. Bhalla, Phys. Rev. A 3, 1919 (1971); M.O. Krause, J. Phys Chern.

Ref. Data 8, 307 (1979). 5. K. Hlimliliiinen, D.P. Siddons, J.B. Hastings, and L.E. Borman, Phys. Rev. Lett. 67, 2850

(1991). 6. For a review on crystal optics, see for example: T. Matsushita and H. Hashizume, "X-Ray

Monochromators," in Handbook on Synchrotron Radiation, Vol. lA, edited by E.E. Koch (North Holland, Amsterdam, 1983). For a review on VUVand soft x-ray monochromators, see R.L. Johnson, "Grating Monochromators and Optics for the VUV and Soft X-Ray Region," ibid.

7. C.T. Chen, Nucl. Instrum. and Methods 256,595 (1987). 8. C.T. Chen and F. Sette, Rev. Sci. Instrum. 60, 1616 (1989). 9. C.T. Chen, Y. Ma, and F. Sette, Phys. Rev. A 40,6737 (1989). to. KJ. Randall, 1. Feldhaus, W. Erlebach, A M. Bradshaw, W. Eberhardt, Z. Xu, Y. Ma, and

P.D. Johnson, "Soft X-Ray Spectroscopy at the Xl Undulator," in National Synchrotron Light Source Annual Report (Brookhaven National Laboratory, Upton, NY, 1991).

11. E. Weschke, C. Laubschat, T. Simmons, M. Domke, and G. Kaindl, Synchrotron Radiation News 4,18 (1991).

12. Y. Ma, C.T. Chen, G. Meigs, K. Randall, and F. Sette, Phys. Rev. A 44, 1848 (1991). 13. Y. Ma, F. Sette, G. Meigs, S. Modesti, and C.T. Chen, Phys. Rev. Lett. 63, 2044 (1989). 14. F.x. Gadea, H. Koppel, J. Schirmer, L.S. Cederbaum, KJ. Randall, AM. Bradshaw, Y. Ma,

F. Sette, and C.T. Chen, Phys. Rev. Lett. 66, 883 (1991). 15. F. Sette, B. Sinkovic, Y. Ma, and C.T. Chen, Phys. Rev. B 39,11125 (1989). 16. See for example: S. Sugano, Y. Tanabe, and H. Kamimura, Multiplets o/Transition-Metal

Ions in Crystals (Academic Press, New York, 1970). 17. Important articles on the argument are: A Fujimori et al., Phys. Rev. B, 35,8814 (1987);

J.A. Yarmoff et al., Phys. Rev. B 36, 3967 (1987); Z.-X. Shen et al., Phys. Rev. B, 36,8414 (1987); D. van der Marel et al., Phys. Rev. B 37,5136 (1988); N. Nucker et al., Phys. Rev. B 37,5158 (1988); AJ. Arko et al., Phys. Rev. B 40,2268 (1988); 1. Allen et al., Phys. Rev. Lett. 64, 595 (1990). For a bibliography see F. Al Shamma and J.C. Fuggle, Physica C (Amsterdam) 169, 325 (1990).

18. C.T. Chen et al., Phys. Rev. Lett. 66, 104 (1991). 19. H. Eskes, M.BJ. Meinders, and G. Sawatzky, Phys. Rev. Lett 67, t035 (1991). 20. C.T. Chen, F. Sette, Y. Ma, and S. Modesti, Phys. Rev. B 42, 7262 (1990). 21. 1.L. Erskine and E.A. Stern, Phys. Rev. B 12,5016 (1975).

279

22. B.T. Thole, P. Carra, F. Sette, and G. van der Laan, Phys. Rev. Lett. 68, 1943 (1992). 23. C.T. Chen, N.V. Smith, and F. Sette, Phys. Rev. B 43,6785 (1991). 24. See for example: F. Sette, C.T. Chen, Y. Ma, S. Modesti, and N.V. Smith, in X-Ray

Absorption Fine Structure, edited by S.S. Hasnain (Ellis Horwood Pub!., London, 1991); Y. Wu, J. StOhr, B.D. Hermsmeier, M.G. Samant, and D.B. Weller, Phys. Rev. Lett. 69, 2307 (1992).

RESEARCH OPPORTUNITIES IN FLUORESCENCE WITH THIRD-GENERATION SYNCHROTRON RADIATION SOURCES

D.L. EDERER AND K.E. MlY ANa Tulane University New Orleans, LA 70118, USA

W.L. O'BRIEN: T.A. CALLCOTT, Q.-Y. DONG, AND J.J. JIA University of Tennessee Knoxville, TN 37996, USA

D.R. MUELLER AND J.-E. RUBENSSONt National Institute of Standards and Technology Gaithersburg, MD 20899, USA

R.C.C. PERERA Advanced Light Source Lawrence Berkeley Laboratory Berkeley, CA 94720, USA

and

R.SHUKER Ben Gurion University Beer Shiva, Israel

ABSTRACT. Synchrotron radiation sources have opened a new window on the century-old use of x rays as a scientific tool. X-ray fluorescence, excited by the photoabsorption process, has been a part of this research picture almost since the day that x rays were first discovered. However, the investigation of multi­photon processes in gases and solids had to wait until the second half of the 20th century. The advent of intense synchrotron radiation sources based on the use of specialized insertion devices will provide many new scientific opportunities for the 21 st century. This presentation will outline some of the recent excjting discoveries in soft x-ray fluorescence spectroscopy and discuss a new type of laser-synchrotron hybrid experimental technique based on the time structure of the synchrotron radiation.

1. Introduction

X rays have been used as a scientific tool for almost 100 years [1]. Fluorescence x rays have been detected in the hard x-ray region almost since their discovery, and in the middle of the 20th century, Skinner [2] showed how the soft x-ray portion of the spectrum can be utilized

* Present Address for W.L. O'Brien is: Synchrotron Radiation Center, Stoughton, WI. t Present Address for J.-E. Rubensson is: KFA Jillich, IEEIIFF, Postfach 1913, D 5170 Jillich, Germany.

281

A.S. Schlachter and F.J. Wuilleumier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 281-297. © 1994 Kluwer Academic Publishers.

282

for experiments that map out the localized density of states through transitions between the valence and core levels. Such maps are modified by the dipole selection rule, and one obtains a selected localized density of states of one of the elements in the compound under investigation. Thus, one obtains information complementary to that obtained by photoelectron spectroscopy.

Soft x-ray fluorescence emission is especially powerful because the shallow core levels have a natural width that is about a factor of 10 less than deeply bound levels. Thus, the valence-band spectrum is not unduly broadened by the width of the core level. Furthermore, because the fluorescence photons are not affected by electric and magnetic fields, insulating samples can be studied without experimental complications. Photons produce less damage than other probes, scatter less than electrons and ions, and are more bulk-sensitive than electrons. Fluorescence is sensitive to the electronic structure around selected elemental species and thus facilitates the analysis of multicompound samples. X-ray emission from a specimen also conveys information about the angular-momentum symmetry of the electronic states.

Advancing technology provides new opportunities to make use of the power of photon-in, photon-out methods to study valence electrons in atoms, molecules, and solids. Now that high­brightness synchrotron sources are available, methods for studying the properties of materials by photon-in, photon-out techniques become even more exciting. Furthermore, multicolor photon experiments requiring heroic efforts under present circumstances will become routine at these new sources. Extensive progress has been made in developing multicolor photon experiments to study atomic and molecular processes. These developments have been summarized very well by other speakers at this school [3-5], but a new pump-probe technique that exploits the time characteristics of the radiation will be described in this report.

2. Instrumentation for a Soft X-Ray Excitation Scheme

At present we employ a spectrometer mounted on a beam line at the National Synchrotron Light Source (NSLS) [6]. The spectrometer incorporates toroidal gratings, which have a gain of 10 in efficiency over spherical gratings. A position-sensitive area detector further enhances the sensitivity of the instrument by a factor of 100. At NSLS, white light or monochromatic radiation from the storage ring, as well as electrons, can be used to excite the sample as shown in Fig. 1.

When white light is used for excitation, the reflected beam from the sample can be detected by the fluorescence spectrometer and provides a measure of the absorption coefficient of the sample [7]. At near normal incidence, the reflectivity is given by

For values of the refractive index n near 1, the reflectivity is proportional to k2, the square of the extinction coefficient. Thus, one can measure a quantity proportional to the unoccupied density of states by the square root of the reflectivity measurement, and the occupied density of states can be determined from the intensity of the valence-band emission.

The monochromatized beam from the recently installed variable-line-spaced grating [8] provides a flux to the sample of about 1013 photons per second at 100 A in a O.4-eV bandwidth with an electron current of 500 rnA in the storage ring. This monochromator has several unusual features that are especially important for fluorescence spectroscopy. It has fixed input and output

Soft X-ray Fluoresence Excitation Scheme

Sample

.... .... .... .... .... ....

Fluorescence or scattered SA

SA white or mono

ebeam/ .... ·

283

Figure 1. Soft x-ray fluorescence can be excited by an electron beam or by a broad band of radiation from a synchrotron source. A monochromator can also be used to provide a narrow band of excitation.

axes, good throughput, and simple optics, and it scans wavelength by the translation and rotation of a plane mirror. The monochromator achieves a resolving power of about 500 at 100 A with a 250-llm exit slit. By ray-tracing the optical system, we were able to show that the resolving power is not a strong function of the entrance-slit width. A change in the entrance-slit width by a factor of 4 results in a 20% change in the resolving power. This translates into high throughput at bandwidths that are especially suitable for solid-state experiments. A complete report on this instrument can be found in Ref. 8.

3. Photon-In, Photon-Out Research Opportunities

When a photon excites an electron from a core state to a state in the conduction band, the resulting emission is expected to be independent of the excitation energy. The phenomena that can introduce an excitation dependence are the screening produced by the excitation of an exciton, inelastic photon-scattering processes, many-electron interactions, and phonon relaxation. The soft x-ray spectral-wavelength band is especially suited for studies of these processes. We will show examples of such studies in this and following sections.

284

3.1. THE EXCITA nON OF SILICON ~.3 EMISSION VS. PHOTON ENERGY

Silicon is a good candidate for such an investigation because the band structure is well known [9] and it is easy to obtain high-quality samples. A typical silicon L2.3 emission spectrum is shown at the bottom of Fig. 2. In crystalline silicon, the main features in the emission spectrum are the three peaks shown as a, b, and c in the bottom panel of Fig. 2. Several years ago, the excitation of silicon emission as a function of photon energy was first investigated by using a monochromator

Top Intensity Ratios

-e- alb -B-c/b

- e-gun (alb) -x- e-gun (c/b)

0 .9 --

O.81--....:r--+-i---I

O.71---i--

0 .6 .- .-

0 .5 ---.,.-.- ",-£]

0.4

140 Excitation energy (eV)

-~-Figure 2. Peak heights as a function of excitation energy. In silicon ~.3 emission, the peaks are denoted according to the scheme shown below the graph. The cross and the diamond symbols placed at 160-eV excitation refer to the spectrum excited by 2-keV electrons.

285

with a bandwidth of 4 eV [to]. The clearly visible amplitude variation of the emission peaks labeled a, b, and c is shown as the upper panel in Fig. 2. The largest changes take place near the threshold. At excitation energies 40 e V above the L2,3 absorption threshold, the spectrum takes on the same form as one excited by electron-beam bombardment or by high-energy photons. In that paper [to], the variation with excess energy was attributed to a many-body scattering effect that takes place near threshold excitation energy.

More recently, K emission spectra from a diamond single crystal were obtained by using undulator-produced, narrow-band photon excitation near threshold [11]. The resulting variation of the x-ray emission as a function of excitation energy was interpreted as a resonant inelastic scattering phenomenon, by which the photon momentum and the crystal momentum were conserved. The authors note that the electrons are excited by the photons to unoccupied states with specific values of the electron wave vector k in the Brillouin zone, as allowed by the excitation energy according to the band structure. These authors then proposed that, for the negligible photon momentum in the soft x-ray spectral region, there is an enhanced x-ray emission for valence-band features having the same value of k. If this k-conservation model is one of the factors that produce a variation of the intensity of emission as a function of photon excitation energy, the emission from an amorphous sample, because of the lack of crystalline order, would show little or no change as the photon excitation is changed.

Our group conducted measurements again on amorphous and crystalline silicon [12], using the new monochromator with a narrow bandpass to produce excitation photons in a OA-e V bandpass. The results are shown in Fig. 3. The amorphous sample shows no emission dependence as a function of the excitation energy. On the other hand, emission from the crystalline sample shows a pronounced variation as a function of the excitation energy. Photon excitation energy at 99.7 and to1.2 eV produces transitions to high-density points X and L in the Brillouin zone. Emission from the valence band should be enhanced at these symmetry points, which are indicated by an arrow labeled 1 for the point of X symmetry and 2 for the point of L symmetry. Enhancement in the emission is clearly visible for these symmetry directions. Similarly, other variations in the emission spectrum, such as the changing ratio of peaks A and B, may be explained in the context of this model [12]. The emission spectrum from the valence band gradually resembles the one produced by electron-beam excitation, as the photon excitation energy increases.

It has been suggested that these enhancements be used to map out the density of states in certain crystalline substances [11]. If this is a viable suggestion, there will be yet another reason to use soft x-ray emission (SXE) as a research tool.

3.2. ELECTRON SCREENING AND SPECTATOR DECAY: EMISSION FROM 8203

Our group has observed soft x-ray valence-band emission from a number of insulating compounds, such as MgO and Si02. We have studied intermediate coupling [7], temperature effects [13], and phonon relaxation [14] in these compounds. Insulators such as these also have a core exciton as a bound state. In B203, we have studied the change in the valence emission band as affected by the screening produced by the exciton [15]. Figure 4 is the result of these measurements. Emission spectra from B203 are shown as 'a function of excitation energy, indicated by the numbers ranging from 193.7 to 217 eV. The lowest energy of excitation, 193.7 eV, corresponds to that required for the excitation of a core exciton. At this photon excitation energy, one observes resonant fluorescence from the exciton (omitted for clarity) and emission from the valence band. At resonance with the exciton, emission from the valence band

286

>. ... ·00 c: Q) ... c: "0 Q)

.~ ct:l E ..... o Z

r-""'''''''~r-''"''"''T'"''''''''''''I'''''''''''""''"r""''''''''~'''''''''''''''''~.,.......j Excitation energy

a-Si 99.7 eV

{101.2ev

-- 120 eV 1 S6 eV

r-"""""T'"...........,r-""""'T""...........,r""K'"""T""'.......,.,.,..,...r"TT"',...,..,..,.,..,...~ Excitation ene rgy

c-Si

86

8

1 , c

88 90 92 94 96

Photon energy (eV)

99.7 eV 100.2 eV 101.2 eV

98

Figure 3. Si L2,3 soft x-ray emission spectra are plotted for amorphous and crystalline silicon samples with the photon excitation energy as a parameter. The a-Si data are normalized to equal area, while the c-Si data are normalized to the intensity of peak B. The vertical line near peak A of the crystalline data marks the energy position of this peak for higher hVe.

occurs while the exciton acts as a "spectator." At a slightly higher photon energy, off resonance with the exciton, one observes emission from the valence band. The band shifts significantly in energy (almost 2 eV) and changes shape. The shift is due partially to the change in initial- and final-state screening produced by the absence of the electron excited to the bound excitonic state. The change in shape is possibly due to the inelastic photon-scattering phenomena described in the preceding section, and the large difference in the phonon coupling also contributes to the

287

0.4

0.35

0.3

-f/) 0.25 +-' ·c

::J

.0 0.2 ... ~ w 0.15 X CJ)

0.1

0.05 193.7 eV

0 160 165 170 175 180 185 190 195 200

Photon energy (eV)

Figure 4. Boron K soft x-ray emission spectra from B203 for different photon excitation energies.

difference in shape. Off resonance, the valence emission spectrum remains relatively unchanged as a function of the photon energy until hv = 202 e V, when an inelastic collision of the outgoing electron can excite one of the valence electrons into the excitonic state. Then one observes emission from both the exciton and the valence band.

The branching ratio for radiative decay of the boron K exciton can be determined. This is the ratio of direct excitonic recombination to spectator-valence transitions for the K exciton excited state. Linear combinations of the normal and spectator SXE spectra were compared to the SXE spectra obtained at energies greater than 210 e V, where it is possible to populate the K exciton by an inelastic scattering process. From these comparisons, it was determined that 12% of the valence SXE features was due to spectator emission. The intensity of the exciton peak relative to the intensity of the two main emission peaks is 0.033. Thus the branching ratio (ratio of radiative decay of the core exciton to that of the valence band) is 0.033/0.12 = 0.28. The large value of the branching ratio suggests that the oscillator strength of the exciton is comparable to the integrated number of electrons in the valence band.

SXE can be used to study phonon relaxation, or Stokes shift, in insulating compounds. The main ideas of phonon relaxation were put forth in two papers [16,17]. Measurements of this type were carried out on metals [18, 19] using data from SXE and soft x-ray absorption (SXA) from the same spectrometer. In the case of metals, the shift is less than 0.1 eV, whereas for MgO, the

288

Stokes shift is of the order of 0.5 eV [14]. Mahan [20] has shown that core-electron excitations couple most strongly with longitudinal optical phonons at X. For MgO these phonons have an energy of 0.06 eV. In an independent measurement [13], we have determined the phonon relaxation energy for MgO to be 1.5 eV. This large shift in MgO is due to the longer core-hole lifetime, faster phonon relaxation, and higher phonon relaxation energy.

3.3. ELECTRONIC BONDING AT BURIED INTERFACES

Many techniques are available for the study of adsorbates bonded to surfaces [21], but most of these techniques are not readily applicable to the study of the structure of buried solid-solid interfaces [22]. It is especially important that there is some means to study buried solid-solid interfaces nondestructively, because interfaces are common in semiconductor devices and in multilayer coatings. X-ray fluorescence spectroscopy is not surface-sensitive and provides information about the interior of the sample. Furthermore, it is a site-specific technique. We have shown [23] that x-ray fluorescence has potential as a method for the investigation of buried interfaces.

In our study, alternating layers of carbon and silicon were used as the medium to study the interface. Because each layer pair had two interfaces, the signal was amplified by 2N, where N is the number of layer pairs. If one of the layers was thin enough, the signal was further enhanced because the number of bulk atoms did not overwhelm those from the interface. Furthermore, monochromatic photons were used to excite the sample, and the photon energy was chosen to produce holes in the L2,3 shell of silicon but not in the K shell of carbon. Thus, fluorescence radiation from transitions between the valence band and the carbon K hole did not interfere with the intensity distribution of the valence band to L2,3 hole transitions in silicon. The interference was caused by the overlap of the carbon K radiation diffracted in third order in the spectrometer with that of the silicon L2,3 diffracted in first order.

Figure 5 illustrates the use of this method. The top two spectra are the L2,3 emission of silicon from 9-A Si layers alternating with 30-A carbon layers and from crystalline SiC. The asymmetry in the 90-e V peak and the similarity of the shoulders at 86 e V are obvious in these two spectra. This type of sharp structural feature in the spectra of tetragonally bonded compounds is known to be associated with structures that are ordered both spatially and chemically to at least the second­neighbor positions [9]. This ordering often produces a splitting in the most intense feature in crystalline Si, which disappears in amorphous Si. In this paper, we suggested that for the 9-A layers, the Si was mobile enough to form SiC over the full depth of the layer.

The lower two spectra in Fig. 5 are the L2,3 emission from the 25-A Si layers alternating with carbon layers and from amorphous silicon. The broadening of the 90-e V peak from the samples with silicon layers that increased from 9 to 25 A of silicon is clearly associated with an admixture of the amorphous silicon spectrum from the center of the silicon layer and the SiC-like spectrum at the interfaces.

289

1.0

0.8

0.6

0.4

-en :!: 0.2 c ::l

..0 ..... 0 CCI -->. .... 'w 0.2 c Q) .... c

0

0.2

0

0.2

0 70 80 90 100 110 120

Photon energy (eV)

Figure 5. The top two spectra are the L2.3 emission from silicon layers 9 A thick alternating with carbon layers 30 A thick and from crystalline SiC. The lower two spectra are the ~.3 emission from silicon 25 A thick alternating with carbon and from amorphous silicon. The spectra were excited by a 130-eV monochromatized beam of synchrotron radiation.

290

3.4. MAPPING OF OCCUPIED AND UNOCCUPIED STATES IN BARIUM FLUORIDE

Figure 6 is a spectrum recorded by using the spectrometer to obtain a measurement of the reflection from a barium fluoride sample that was illuminated with white light. As noted, the reflection spectrum is proportional to k2, the square of the extinction coefficient. The structure observed is in good qualitative agreement with measurements of the absorption coefficient of BaF [24] and the absorption coefficient of doubly ionized barium [25].

The sharp features in the spectrum at photon energies near 75 eV are due to fluorescence emission by transitions of electrons in the 5p shell to holes in the 4d shell, produced by excitation photons with energy greater than that required to produce a 4d hole. The fluorescence emission from states of p and f symmetry in the valence band appears between 82 e V and 85 e V. Two bands appear because of the 2.4-e V spin-orbit split 4d core level.

Figure 7 illustrates emission spectra excited by monochromatic phonons of the energies indicated. The features shown in Fig. 7 are due to core-core transitions between the 5p shell and the 4d shell. The three transitions allowed according to the selection rules for j-j coupling are observed. If there were an admixture of f electron character in the 5p shell, a multiplet structure would be produced that would split the 4d-5p transitions into a group of lines with an energy spread of about 5 eV. This is not observed. The spectra have been corrected for spectrometer

en -C ::I o ()

2.5x104

2.0x104

1.5 X 104

1.0 x 104

5.0x103

50

White light excitation

Sa 5p ~4d

'" Valence band ~ Sa 4d

70 90 110 130 150 170 190

Photon energy (eV)

Figure 6. Soft x-ray emission from barium fluoride excited by white light from the NSLS storage ring. The structure at photon energies greater than 95 eV is due to the increased reflection from the barium fluoride.

C') ->-C) .... Q) c: Q)

c: o (5 ..c: Cl.

~ -c: :::J o ()

0.011-

118.4 eV

...... .. , :." ' .

: .. 4d3/2 ~ 5p 1/2

... /

-.. ',"'--..,-~

-

0:_:.-.,."" '. ----------1 , '" .,

100.0 eV ../,- .... ~ OL.~~-L~~~~~~~~~~~~~~~-L~~-~·~~~~~~·--~~~~~~~

291

72 73 74 75 76 77 78 79 80 Photon energy (eV)

Figure 7. Intensity of core-core transitions between the 5p and 4d shells in barium fluoride at the photon energies indicated on the left-hand side of the graph.

response and background, and have been divided by a factor of photon energy cubed to correct for the energy-dependence of the photon density of states within the dipole approximation. The curves have been offset along the vertical axis for clarity. The incident photon beam was characterized by an energy spread corresponding to a full width at half maximum of -0.8 e V.

Since the excitation of the initial core vacancy occurs on a time scale much faster than its subsequent decay, the excitation and emission processes are independent, and changes in the relative intensity of the 4d3!2 and 4dS12 terms reflect the energy-dependence of the relative photoionization cross sections of the 4d3/2 and 4ds12 levels. These cross-section measurements add a great deal more information about the correlated behavior of the electrons than simply a measurement of the absorption cross section, such as the one shown in Fig. 6. The ratio of the partial cross section due to the photoionization of a 4dS12 subshell electron to that for a 4d3!2 subshell electron, for the same binding energy, is called the branching ratio. If electron-electron correlations are small, this branching ratio would be a constant as a function of the energy, equal to the statistical weight of the states. In the case of barium 4d photoionization, the statistical value of the ratio is 1.5. An inspection of Fig. 7 qualitatively suggests that the ratio is not constant, indicating that electron correlations are a factor in the photoionization of barium fluoride. Additionally, if the holes are produced by circularly polarized light, it would be possible to observe the dichroic behavior of the unoccupied states through· a recording of the emission intensity from the spin-orbit split core-core transitions. Furthermore, it would also be possible to

292

observe dichroic x-ray fluorescence from the occupied states through transitions of the valence electrons to the polarized core hole [26, 27].

In the series of examples put forth in the preceding paragraphs, we have illustrated a number of new applications of photon-excited soft x-ray spectroscopy, namely inelastic photon scattering, absorption branching ratios, electronic screening, and phonon relaxation, and we have suggested that soft x-ray dichroic. fluorescence will play an important role in the development of new materials. Strange et al. [26] have shown recently that the density of majority and minority valence-band spin states can be measured by the x-ray emission anisotropy for transitions between the valence and core holes prepared by the absorption of a circularly polarized photon. The tunability of the photon source is crucial to the success of the method. The core electron must be promoted into the empty part of the exchange-split d band within 1.3-2 eV of Ef. According to Strange et al., this will ensure that the core hole will be highly polarized, and the resulting emission spectrum will show a strong dichroic anisotropy. Experiments of this type will enable one to study how the spin polarization changes across the Fermi energy.

With the same apparatus, it will also be possible to study the unoccupied spin states in the conduction band. This study can be done by measuring the magnetic circular dichroism (MCD) of the core absorption by observing the core-core transition as a function of the excitation energy. In fact, the x-ray dichroic fluorescence and the MCD in absorption can be observed simultaneously. If the feasibility of such experiments can be demonstrated, the outcome of such research could have enormous impact on the understanding of new designer magnetic materials.

4. Pump-Probe Experiments Using the Time Structure of Synchrotron Radiation

While pulsed VUV sources were used in conjunction with lasers in the mid-seventies to study the photoionization of excited atoms [28], multicolor photon experiments utilizing lasers and synchrotrons had their beginnings in the late seventies [29]. Other speakers at this conference [3-5] have described multicolor pump-ionize experiments, in which one photon is used to produce an atom or molecule in an excited state and another XUV photon is used to ionize the excited atom or molecule. We wish to describe briefly one type of experiment that will provide new information and that requires circularly polarized synchrotron radiation as well as circularly polarized laser radiation.

4.1. PHOTOIONIZATION FROM STATE-SELECTED ATOMS USING CIRCULARLY POLARIZED LIGHT

Perhaps one of the most elegant experiments to be done with a laser-undulator hybrid source is the study of the photoionization of a state-prepared atom. In an experiment of this type, circularly polarized laser radiation is used to optically pump the atom to a magnetic sublevel with the highest possible magnetic quantum number. Then circularly polarized photons from an undulator ionize the excited atom and, depending on the sense of the polarization, add or subtract one unit of angular momentum to or from the excited atom. This experiment was done for sodium using lasers alone [30]. A circularly polarized dye laser was used to optically pump the 3 2p312 state to ML = + 1 and MF = +3. The two senses of circularly polarized ionizing radiation from another dye laser produced transitions to continuum states with ML = 0 or ML = 2. Only Ed continuum states have ML = 2, so the signal with one degree of polarization is proportional to cr(3p'Ed), and the signal with the other degree of polarization is proportional to a linear combination of cr(3p'Ed) and

293

cr(3p'ES). The quantity cr(3p'El) is the photoionization cross section of 3p electrons to continua with angular-momentum states of El. The narrow wavelength tuning range of the laser limited these measurements to hv < 3.5 eV. Present-day lasers can provide a somewhat broader tuning range, but newly developed undulators producing circularly polarized tunable radiation over a broad photon-energy range (5 eV < hv < 1.5 keY) will enable experiments of this type to be carried out over a very broad energy range. This technique provides a unique way to probe the photoionization cross section of a particular angular-momentum channel. It also adds new information about the cross-section channels, which is obtained by branching-ratio measurements described in the previous paragraphs.

4.2. DYNAMICAL PROCESSES IN ATOMS AND SOLIDS FROM THE TIME STRUCTURE OF

SYNCHROTRON RADIATION

In another group of experiments, the time structure of the laser and the synchrotron source are used directly to study dynamical processes. Table 1 identifies some of the dynamical phenomena that can be studied with pulsed light sources. Third-generation synchrotron sources will have light pulse widths on the order of a few picoseconds, and present-day lasers can produce sub­picosecond pulses. At the present time, most storage rings are limited to pulse widths of a few hundred picoseconds. Third-generation sources will produce a whole new time regime for investigation. They will be especially valuable for dynamical processes such as molecular vibration and rotation, and processes involving the transfer of an electron from one molecular site to another, which have a time duration of the order of a picosecond. Present-day synchrotrons can be used to study the decay of the excited states in atoms or molecular tumbling

TABLE 1. Dynamical phenomena that can be studied with pulsed light sources.

Times( s )nength

10-15/3000 A

10-12/0.3 mm

1O-9/30cm

Phenomenon

Photon absorption­emission

Electron emission

Molecular rotation

Molecular vibration

Electron transfer

Exciton migration

Collisions in gases and liquids

Fluorescence

Molecular tumbling in solution

Study techniques

Laser

Laser

Streak camera

Phase shift

Synchrotron

Laser

Flash lamps

Synchrotron

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in solutions, phenomena that have a time duration of about 10-9 second [31]. Phenomena occurring in -10-6 second take place on a time scale near the upper practical limit amenable to study by pump-probe techniques employing synchrotrons, because the time for an electron bunch to travel around a storage ring is of a similar order of magnitude. This is the time regime for the decay of metastable excited states and phenomena that involve phosphorescence.

In experiments utilizing the time structure of synchrotron radiation, a pulsed laser is used in conjunction with the pulsed synchrotron for studies of energy-transfer mechanisms in gases and condensed matter. These experiments were first done at the Hasylab [29], where UV from the synchrotron was used to produce an exciton through a valence-band excitation, and a pulsed laser was synchronized with the pulses in the synchrotron at 10 Hz to measure the lifetime of the exciton. Similarly, a pulsed laser was synchronized with the x rays from the Cornell synchrotron light source to study the pulsed annealing of silicon [32]. A pulsed copper-vapor laser running at a few kHz has been synchronized with the NSLS x-ray ring running in the single-pulse mode to study band bending in semiconductors [33].

All these experiments utilized either a cw laser or a laser pulsed at a rate slow compared to that of the synchrotron. The first use of a mode-locked laser synchronized with synchrotron pulses was carried out at the UVSOR [34]. In this paper, we shall describe our technique for synchronizing a mode-locked laser to storage-ring light pulses and describe the limitations and advantages of the method by outlining two experiments. In the first, VUV photons were used to populate a core exciton level in an insulator, and then the exciton was photoionized by the laser [35]. In the second, the laser was used to inject carriers in Hgl-xCdxTe grown by molecular beam epitaxy (MBE), and then infrared photons the from NSLS VUV ring were used to observe a change in absorption [36].

Pump-probe techniques can be used to match the high peak flux from a synchrotron source with the high peak power available from mode-locked lasers and to use the time difference between the synchrotron and laser pulses to study the dynamics of physical processes, within the constraints provided by the current generation of synchrotron-radiation sources. The duty cycle is increased, and the maximum intensity is used effectively for the excitation of processes that have a lifetime of the order of nanoseconds. Lasers are powerful but not as broadly tunable as synchrotron radiation. By developing this technique, we will have the technology in place to exploit third-generation synchrotron radiation sources and the faster pulses of greater intensity that will be available from them. With this method, fast events can be studied with a cw detector. All the timing information is carried by the high speed inherent in the laser and synchrotron.

We have synchronized a mode-locked laser to the string of pulses in the ring by using the rf driving the synchrotron source to drive the acoustic modulator in the laser cavity. The laser pulses occur at twice the rf frequency. The two trains of pulses, one from the laser and the other from the electrons in the storage ring, are locked together with some arbitrary time interval between them. One way to change the time interval between the synchrotron-radiation pulse and the laser pulse is to electrically change the phase of the rf driving the synchrotron and the phase of the rf driving the laser acoustic modulator. Voltage control of the phase is especially important for dithering the time interval at a low frequency CO and using phase-sensitive detection methods to detect low-level signals with greatly enhanced sensitivity. The generic setup is shown schematically in Fig. 8, in which a pulse train from the laser and one from the synchrotron radiation source overlap at the sample. The sample produces excitation products, usually electrons or photons, which are detected via a suitable spectrometer.

295

Detector Spectrometer

Computer

Mode-locker

SR-RF Phase shift

Figure 8. Schematic representation of the apparatus used to synchronize aNd: YAG mode-locked laser to the light pulses from a storage ring. SHG is the second-harmonie-generation module. The rf phase is shifted electrically to adjust the time interval between the synchrotron light pulses and those from the laser.

The laser power used at NSLS is about 0.5 W at 530 nm, which corresponds to 1.4 x 1 ()4 watts/cm2 peak: power, yielding a fluence Q of about 3 x 1022 photons/cm2-s at the sample. The fractional change in signal intensity M / I produced by the presence of the laser is to a good approximation given by

M/l=Q(J't ,

where the quantity 't is the lifetime of the excited state expressed in seconds and 0 is the cross section expressed in cm2 to deplete the excited state by interaction with the laser field. The quantities M / I, Q, and 0 can be measured or calculated, thus yielding the lifetime 'to We can estimate the range of 't that is available to this measurement technique by substituting the known value of Q and making an educated guess for the cross section 0. We assume a cross section of about 10-17 cm2, for example, and multiply it by the laser fluence Q. The product of the two numbers is 3 x 105. Therefore, to measure times of the order of picoseconds, the quantity M / I must be measured to one part in 106. Measuring a change in intensity as small as one part in a

296

million is a difficult task and usually requires some sort of phase-sensitive detection scheme. Of course, this constraint is relaxed if the lifetime is longer or the cross section is larger.

We used this hybrid technique on a couple different systems. In the first case, pulses of VUV photons were used to populate the exciton in aluminum oxide [34], and the laser was used to quench the resonance fluorescence by promoting the excited electron into the conduction band. A signal change of one part in a thousand could be observed, which puts an upper limit of 10-8 s for the lifetime of the exciton. However, it is believed that the core excitons have lifetimes several orders of magnitude shorter than this. To push the measurements to shorter times, it is necessary to use phase-sensitive detection schemes that would produce a gain in sensitivity of several orders of magnitude. To obtain more photons, it is possible to Q-switch the laser and gain somewhat in the signal-to-noise ratio at the expense of a considerably longer integration time because of the reduced duty cycle.

The other experiment that was used to test this pump-probe technique involved transporting doubled, mode-locked, synchronized pulses via an optical fiber to the infrared beam line, U4-IR, at the NSLS [37]. Carrier relaxation with nanosecond resolution was measured in MBE-grown Hgl_xCdxTe. The photocarrier decay was shown to be consistent with a simple exponential [35].

The events of the past year have provided additional stimulation to a field that has a long and distinguished history. All of us involved in its development and maturation have been rewarded beyond our wildest dreams with many new and interesting phenomena to study.

Acknowledgments

This research was supported in part by National Science Foundation Grant No. DMR-8715430, by a Science Alliance Center for Excellence Grant from the University of Tennessee, and by the U.S. Department of Energy (DOE) Contract No. DE-AC05-840R21400 with Oak Ridge National Laboratory. One of the authors, R.C.C.P., acknowledges the support of the Director, Office of Energy Research, Office of Basic Energy Sciences, Materials Sciences Division, of the u.S. Department of Energy under Contract No. DE-AC03-76SFOOO98, The research was carried out at the National Synchrotron Radiation Laboratory at Brookhaven National Laboratory, with support from DOE Contract No. DE-AC02-76CHOOOI6. One of the authors, D.L.E., was a visiting scientist at the Laboratoire de Spectroscopie Atomique et Ionique, Orsay, France, during a portion of this work.

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PHOTOEMISSION SPECTROMICROSCOPY

GELSOMINA DE STASIO Instituto di Struttura della Materia Consiglio Nazionale delle Ricerche, Via E. Fermi 38,00044 Frascati, Italy

G. MARGARITONDO Institut de Physique Appliquee Ecole Poly technique Federale, CH-1015 Lausanne, Switzerland

ABSTRACT. We briefly review the recent progress in photoemission spectromicroscopy, the experimental technique that combines synchrotron-radiation photoemission and high lateral resolution. We discuss, in particular, the scanning photoemission spectromicroscope MAXIMUM, its applications in neurobiology, and the future opportunities opened up by the new ultrabright synchrotron sources currently under development at Trieste and Berkeley.

1. An Ideal Way to Utilize the New Synchrotron Sources

The most important improvement of the new synchrotron sources of soft x rays, ELETTRA at Trieste and the Advanced Light Source (ALS) at Berkeley, concerns their brightness. The importance [I] is linked to one of the many versions of Liouville's theorem: phase-space conservation along an ideal beamline. This means that in a loss-free line, brightness is conserved: One cannot focus the beam without increasing the angular divergence and vice versa. A large angular divergence means large-size optical components with increased technical difficulties and costs. Conversely, with a high-brightness source one can concentrate a large photon flux into a small area without insurmountable difficulties.

We know that the increase in brightness brought by ELETTRA and the ALS is one of the most amazing instrumentation accomplishments of all time: an increase by several orders of magnitude with respect to the existing sources. This truly impressive achievement carries a high price, both in terms of human resources and in terms of funds. Synchrotron scientists face, therefore, a great challenge: creating novel experiments to fully exploit these amazing new instruments that their colleagues are constructing for them with much personal sacrifice.

We would like to argue that microscopy, almost any kind of synchrotron-based microscopy, is perhaps the most effective way to meet this challenge. And also that photoemission spectromicroscopy, the class of experiments covered by the present review, further enhances this effectiveness [2]. The development of photoemission spectromicroscopy is, therefore, one of the milestones of today' s experimental science.

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A.S. Schlachlerand F.J. Wuilleumier(edsJ, New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 299-313. © 1994 Kluwer Academic Publishers.

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What, then, is spectromicroscopy? It is a crucial improvement with respect to an already well­established class of experiment collectively known as photoemission spectroscopy [I]. What spectromicroscopy adds to ordinary photoemission [2] is the capability to operate with high lateral resolution: as good as 900 A at present and 100 A or better with ELETIRA and the ALS. This removes one of the major limitations of photoemission experiments in materials science and makes it possible for the first time to extend photoemission to the life sciences [3-6].

In order to understand the magnitude of this breakthrough, we propose a historical perspective [7]. More than sixty years were necessary to move from the discovery of the photoelectric effect to the first real photoemission experiments. In this period, the photoelectric effect had played a major role in the development of modem science, including the birth of quantum physics. But without the major instrumentation advances of the 1950s, it could not lead to real widespread applications in materials science.

Afterwards, it took twenty years to reach the next milestone, the advent of synchrotron radiation [I, 7]. This novel source of photons had two major consequences: first of all, with the brute force of its superior brightness, it unlocked many new research opportunities in photoemission. Second, it made possible the control of all of the photon parameters, which was previously impossible with conventional photon sources.

These factors led to a stampede [1] of new achievements: angle-resolved photoemission and band mapping, cross section techniques and resonant photoemission, photon-polarization techniques, partial-yield spectroscopy, constant-initial-state and constant-final-state spectroscopy, ultrahigh-energy-resolution spectroscopy, spin-polarized photoemission, depth-resolved photoemission, and many others. These accomplishments notwithstanding, photoemission was still affected by the aforementioned major limitation: the lack of lateral resolution [2].

How important is this limitation? In materials science, it is a major problem; but in the life sciences, it is a disaster. We know that many of the important properties of materials are determined by phenomena that occur on a submicron scale; for example, the formation of metal­semiconductor interfaces is thought to be dominated in many cases by localized defects. Ordinary photoemission experiments, however, are blind to phenomena that occur on a scale smaller than 0.1-1 mm.

Suppose, for example [8], that the Fermi-level pinning of a cleaved semiconductor surface is not a global phenomenon, but that it occurs locally on, say, 10% of the surface where there is a high density of defects. The pinning is typically detected by measuring the binding energy of a given core level and seeing whether it reveals any band bending near the surface. But if the pinning is confined to a small portion of the surface, then an ordinary photoemission experime!}t reveals only a small core-level peak superimposed on the main one from the unpinned portion. The sum of the two components simulates a small and de facto undetectable shift of the main one. One would then conclude that the surface is everywhere unpinned. Unfortunately, the formation of interfaces on the same surface may be dominated by its small pinned portion.

Examples of this kind are plentiful in materials science and demonstrate the need for lateral resolution in photoemission. But the need is even more clear-cut in the life sciences [3-6]. The typical spatial reference is the size of a cell or of its components: from microns to the submicron domain. Without being able to detect features on this scale, photoemission is blind to the phenomena of interest in biological specimens-and therefore nearly useless.

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2. Recent Encouraging Progress

The acute need for lateral resolution in photoemission clashes with the reality of the limitations in instrumentation technology [2]. Until three years ago, experiments were mostly limited to total spatial integration in the lateral direction, over a scale of 0.1-1 mm. On the contrary, photoemission has always had spatial resolution along the vertical direction; this is produced by the short mean-free path of excited electrons in materials [I]. Photoelectrons originate from a near-surface region whose depth ranges from a few angstroms to tens or hundreds of angstroms. This vertical spatial resolution can also be tuned by changing the photon energy and therefore the electron kinetic energy on which the mean-free path depends. In a sense, therefore, ordinary photoemission is a "vertical" spectromicroscopy.

As to the lateral directions, there are two main obstacles along the path towards high lateral resolution [2]: first, the limited brightness of the photon source, which limits the signal level in ordinary photoemission and makes it difficult or impossible to operate within small areas; and second, the limitations of the remaining parts of the instrumentation. Soft x-ray optics, in particular, are technologically very complex because these photons are absorbed by all materials and their reflection is very inefficient.

The main breakthroughs concerning the first obstacle were: first, the commissioning of the second-generation sources of synchrotron radiation such as Aladdin at Wisconsin, which were optimized for delivering high flux and brightness; and second, the commissioning of the first undulators, which were capable of producing unprecedented brightness over a specific spectral band. The Berkeley-Stanford undulator [9] on Aladdin, for example, delivers more than two orders of magnitude more brightness than a bending magnet on the same storage ring. The impact of this exceptionally high brightness has been dramatic both in materials science and in atomic and molecular physics [10].

As to the second obstacle, the availability of high-brightness sources has greatly stimulated research to solve instrumentation problems, and this has produced a series of breakthroughs. For example, substantial progress has been made in the construction of soft x-ray optical components and, in particular, in the enhancement of reflection by multilayer coatings [11]. Novel electron­optics devices, capable of producing microimages with spectromicroscopic information using the emitted photoelectrons [12], have been designed and implemented.

At present, there are several active photoemission spectromicroscopy programs in the world, and several more are under preliminary development [2]. A complete description of all these programs would of course be impossible within the boundaries of the present overview and also beyond its scope. In the following sections, we will therefore limit our discussion first to a general overview and then to specialized examples provided by our own research programs.

3. The Two Modes Of Photoemission Spectromicroscopy

Photoemission spectromicroscopy can be implemented with two different approaches that mirror those found in most other microscopies: focusing/scanning and electron-optics imaging (see Fig. 1). The first consists of focusing the x-ray beam onto a small sample area and taking photoelectron spectra of that area [11,13,14]. One can also scan the sample position relative to the focused photon beam while collecting photoelectrons of a fixed energy corresponding to the core-electron photoemission of a given element in a given chemical status. This produces two­dimensional chemical maps with high lateral resolution.

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(a)

X-Y scanning

stage

(b) electron optics

Figure 1. The two modes in which each x-ray spectromicroscopy can be implemented, illustrated in the case of photoemission spectromicroscopy. In the focusing-scanning mode (a), a special device F focuses the x-ray beam onto a small area of the specimen (S). The emitted electrons e are analyzed, providing chemical and electronic information. The sample is mounted on an X-Y scanning stage, and photoelectron microimages are taken by operating the stage while collecting photoelectrons of fixed energy. In the optical imaging mode (b), the x-ray beam covers a larger sample area, and microimages are created with an electron-optics magnifying system.

In the second approach, the photon beam covers a relatively large area of the specimen, and an electron-optics system magnifies the photoemitted electron beam [15-19]. This resembles an ordinary electron microscope, except that the electron source is the specimen itself.

These two approaches are not in competition but are largely complementary [2]. The focusing/scanning approach is preferable for experiments that analyze the electron energy. Some experiments, however, require scanning the photon energy [1], which is difficult with x-ray focusing devices. For them, the electron-optics approach is more suitable. For example, one can scan the photon energy through an x-ray absorption threshold of a given element and reveal the element's spatial distribution from the changes in the magnified image.

Table I shows a list of active photoelectron microscopy and spectromicroscopy programs. Optical-imaging photoemission microscopy in the life sciences was pioneered by O.H. Griffith and coworkers at the University of Oregon [20]. The technical progress has been so rapid that commercial devices not requiring synchrotron radiation are now available. For example, the

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TABLE 1. X-ray spectromicroscopy programs.

ProgramlLocation Type Technical Features

Electron diffraction and photoemission microscopy, Clausthal [18]

Optical imaging

Non-synchrotron photon source

X-ray microscopy, Gottingen and Berlin [16]

Stanford [15] and Minnesota-Wisconsin [17]

HASYLAB-Hamburg, Maxlab-Lund [13]

Brookhaven-SUNY-IBM [14]

Hitachi, Tsukuba [16]

MAXIMUM, Wisconsin [11]

XSEM, Wisconsin [ 12]

Focusing

Optical imaging

Fresnel zone-plate focusing

Magnetic-field electron optics

Focusing- Elliptical-mirror focusing scanning

Focusing- Fresnel zone-plate focusing scanning

Focusing- Walter mirror focusing scanning

Focusing- Schwarzschild lens focusing scanning

Optical imaging

Scienta 300 (Uppsala, Lausanne, Optical Conventional photon sources Lehigh, etc.) [19] and imaging Vacuum Generators

rotating-anode Scienta-Seiko system of the Centre de Spectromicroscopie at the Ecole Poly technique F6d6rale in Lausanne (CS-EPFL) couples a wide spectral range and high energy resolution (290 meV at 1 keV) with an intermediate lateral resolution of 20-30 Ilm. As to synchrotron-radiation instruments, one of the most advanced is the x-ray secondary emission microscope (XSEM) developed by Brian Tonner and his coworkers at the University of Wisconsin [12].

The focusing/scanning programs of Table 1 are based on different solutions of the already mentioned problems encountered in focusing x rays: materials. An effective solution is provided by the Fresnel zone plate [14], consisting of a transparent substrate with a series of opaque circular lines whose width progressively decreases with the diameter. Fresnel zone plates for x rays are exceedingly sophisticated devices: the line width scales down with the photon wavelength, reaching values beyond 0.05 Ilm. An x-ray absorption spectromicroscope and a photoemission spectromicroscope, both based on Fresnel zone plates, operate at Brookhaven's NSLS [14].

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4. An Example of Scanning Spectromicroscope: MAXIMUM

The present (1992) record for lateral and energy resolution in scanning photoemission spectromicroscopy is held by the multiple-application x-ray imaging undulator microscope (MAXIMUM) at the Wisconsin Synchrotron Radiation Center [11]. We will use this as an example to illustrate the state of the art in photoemission spectromicroscopy. MAXIMUM is a collaboration program that involves several groups and institutions [3-6, 11]. The core program was developed at the Center for X-Ray Lithography of the University of Wisconsin-Madison, using the storage ring Aladdin of the university's Synchrotron Radiation Center. Other partners are the Center for X-Ray Optics of the Lawrence Berkeley Laboratory, the Xerox Corporation, the University of Minnesota, and the Ecole Poly technique Federale de Lausanne. The photon source, a 30-period undulator, was originally developed by Stanford and Berkeley [9].

The main objective of the MAXIMUM program is to reach high lateral resolution in established synchrotron-radiation spectroscopies. MAXIMUM brings together two elements of progress: the high brightness of the undulator source and the use of multilayer coatings to enhance the near-normal incidence reflection of spherical surfaces. This last element is used to produce highly efficient Schwarzschild objectives for soft x rays.

The MAXIMUM system, an artist's view of which is shown in Fig. 2, takes the radiation emitted by the Stanford-Berkeley undulator [9] on Aladdin, filters it with a monochromator, and then focuses it onto the sample. This enables one [11] to perform different kinds of synchrotron radiation spectroscopies on a microscopic sample area; such spectroscopies include, for example, absorption, reflection, and desorption techniques. In the photoemission mode, photoelectrons emitted by the small sample area are collected and analyzed by a double-pass, cylindrical-mirror, electron-energy analyzer.

Besides taking spectra from a small sample area, one can also scan the sample position with respect to the focused beam and create two-dimensional microimages. For example, we can scan while measuring the photoemission signal at a fixed photon energy corresponding to the emission from a given core level of a given element in a given chemical status [3-6, 11]. This produces microimages of the lateral distribution of that element in that chemical status. Whereas other techniques exist that can perform microchemical analysis on the scale of MAXIMUM [21-24], no other technique can reach its energy resolution and deliver fine information on the chemical status of elements.

The first stage of the MAXIMUM program adopted several technical compromises to fit a limited budget. For example, the first monochromator was borrowed from the Synchrotron Radiation Center and not optimized for the undulator output. Severe problems were identified, for example, those related to the roughness of the Schwarzschild mirrors' surfaces. These problems notwithstanding, we were able to demonstrate good lateral resolution for total-yield photoelectron microimages [3,11]. In the years 1989-1990, the lateral resolution of these images improved from a few microns to 0.5 /lm. Breaking the micron barrier opened the possibility of using the instrument for life-science experiments on neuron systems [3].

In 1991-1992, the system was rebuilt and optimized. Virtually every portion of the instrumentation was improved, but the most important changes concerned the beamline. The borrowed monochromator was replaced with a spherical-grating instrument from the Lawrence Berkeley Laboratory. After extensive computer simulation of the beamline response, the optical system was optimized to the undulator's output. By the end of 1991, most of the rebuilding work had been completed, and tests of the new instrument's performance were initiated.

305

Schwarzschild objective

Pinhole

Electron Analyzer

Figure 2. Artist's view of the MAXIMUM system at the Wisconsin Synchrotron Radiation Center.

Since then, we completed a long and extensive series of such tests, which demonstrated, on one hand, a marked improvement with respect to the previous performances and, on the other hand, performance levels that are unmatched at present for this kind of instrument [11].

Perhaps the most important element in the new performance level is the lateral resolution. Figure 3 shows the transmission microimage of a portion of a Fresnel zone plate used as a standard for one of the lateral resolution tests. By analyzing the features from progressively smaller zones, we observe that MAXIMUM is capable of imaging features whose size is consistent with a resolution of the order of 900 A.

Excellent resolution was also obtained while taking partial-yield photoemission microimages. Figure 4 shows, for example [11], the three-dimensional reconstruction of the photoelectron micrograph of a series of metal lines on a semiconductor substrate, taken with submicron resolution. The three-dimensional reconstruction emphasizes the fact that photoelectron

Figure 3. Transmission microimage of a portion of a Fresnel zone plate, obtained (Ref. 11) with MAXIMUM. From the smallest distinguishable zones, one estimates a lateral resolution of approximately 900A.

306

microimages carry some topographic information, probably due to the limited angular collection of the electron analyzer and to the fact that the angular distribution of the photoelectrons, as well as the absorption of photons, depends on the local orientation of the photoemitting surface.

Results like those of Figs. 3 and 4 demonstrate the microscopic performance of MAXIMUM but not its spectromicroscopic capabilities. These were tested in a series of preliminary experiments by analyzing core-level and valence-electron spectra of different systems, in particular f and d core levels of semiconductor surfaces and interfaces. These tests demonstrated [11] a record spectromicroscopy energy resolution better than 350 meV. And they already produced some very interesting results on the local pinning of the Fermi level at cleaved semiconductor surfaces.

5. Spectromicroscopy in Neurobiology

Similarly successful spectromicroscopy tests were performed on neuron network specimens [3] . These systems were produced by the technique described in Ref. 25. Cerebellar granule cells from 7-day-old rats were seeded on a gold substrate (approximately 1.5 x 105 cells/cm2) previously treated with a 5 mg/ml of poly-L-Iysine solution. The cells were obtained [25] by enzymatical and mechanical dissociation of the cerebellar tissue and plated in Basal Medium (Eagle's salt) containing 10% fetal calf serum. They were then allowed to grow in an incubator at 37°C in a 5% C02 humidified atmosphere. After 7 days, the neuron cultures were fixed with para-formaldehyde and dehydrated.

The neuron specimens so produced tend to assume a monolayer configuration on the substrate and to form a neuron network. Their suitability for photoemission experiments was demonstrated by the previous tests described in Refs. 3-6.

Figure 4. Three-dimensional reconstruction (obtained with the ©Spyglass software) of a photoelectron­yield microimage of a series of metal lines on a semiconductor substrate (data from Ref. II).

307

An example of photoelectron-yield microimaging is shown in Fig. 5. The micrograph was taken by collecting photoelectrons of 1.3-eV kinetic energy. It shows cell bodies and smaller structures: axons and dendrites interconnecting the cell bodies in the network. Pictures of this type illustrate the neuron culture's capability to grow as a nearly-monolayer architecture on a flat substrate [3-6]. The micrograph was taken using I-Jim scanning steps. Once again, the micrographs obtained in this way contain some topographic information that is emphasized by three-dimensional reconstructed images like that of Fig. 6.

Results like those of Figs. 5 and 6 show that photoelectron microscopy can achieve performance comparable to optical microscopy, but are not new [3]. The novelty of the most recent experiments is the move from mere photoemission microscopy to real spectromicroscopy [26]. A first example of this move is shown in Fig. 7.

The curves in this figure are photoemission spectra (photoemission intensity vs. photoelectron kinetic energy [I]) taken in two different cell-body areas of the same neuron specimen. In each case, the probed area had microscopic dimensions of the order of 1 x 1 Ilm2. 1be spectra, therefore, reflect the chemical composition and properties of extremely localized portions of the neuron networks.

Both spectra in Fig. 7 exhibit characteristic features related to the Ca 3p, K 3p, and Na 2p core levels, plus other features mostly related to oxides. The elements Ca, K, and Na, which playa fundamental role in the homeostasis of each cell and in nerve pulse transmission, are present in the cell membrane ion channels. Figure 7 shows, therefore, that photoemission spectromicroscopy has become capable of detecting and analyzing localized elements that are crucial to the physiology of biological systems.

We note that the spectral features in Fig. 7, although always present, vary substantially in relative intensity from spectrum to spectrum. The causes of such changes are not identified at the

Figure 5. Secondary-photoelectron partial-yield microimage [3] of an 80 x 80 Ilm2 portion of a neuron network.

308

Figure 6. Three-dimensional ©Spyglass reconstruction of the photoelectron-yield microimage of an 80 x 80 J.IlIl2 portion of a neuron network. The kinetic energy window for the photoelectrons was centered at 1.3 eV.

:

. . .. .. , ... t'

20 30 40 50 60 70 80 90 100

Photoelectron Energy (eV)

... ~~ ..

40 50 60 70 80 90 100

Photoelectron Energy (e V)

Figure 7. Photoemission spectra taken on small (1 x Illm2) portions of a neuron cell body, revealing contributions from the ion-channel elements in the cell membrane: Na, K and Ca [26].

309

present time, and the results of Fig. 7 must be interpreted only as a test of our instrument's capability to detect them.

Another type of spectromicroscopy test with MAXIMUM [26] is illustrated in Fig. 8. This test is implemented by setting the electron analyzer to the photoelectron energy of a given core level and then measuring the photoemission intensity point by point while operating the sample scanning stage. This produces microimages of the spatial distribution of the specific element corresponding to the core level. The example shown in Fig. 8 compares a global partial-yield microimage and the specific microimage of the potassium and calcium/sodium distributions in the same area.

Figure 8 demonstrates, therefore, a successful test for the use of photoemission techniques in detecting chemical distributions on a micron and submicron scale. This approach can also be used to image not only a given element, but also the element in a specific chemical status [I]. The core-level energy, indeed, changes slightly with the chemical status [I]; with sufficient energy resolution, therefore, one can distinguish each chemical status from the others of the same element.

Spectromicroscopy experiments on neurobiological specimens were also performed with other instruments besides MAXIMUM [4-6]. Particularly spectacular were the realtime video images obtained by Tonner's XSEM [12]. Figure 9 shows a nice example [4] of chemical contrast obtained with the XSEM.

The chemical contrast is obtained in this case by tuning the photon energy rather than the electron energy. By moving across the gold optical absorption edge of the substrate, the contrast between the gold-rich substrate and the neuron-related structure is reversed. The XSEM was also used to obtain total-yield optical absorption spectra of microscopic areas. These spectra have been able to detect, in particular, the spatial distribution of neuro-poisoning metals such as cobalt and manganese [6].

Figure 8. Three photoelectron micrographs [26] taken in the same small (50 x 80 J..lm2) area of a neuron network. From left to right, the images were formed by 1.3-,70-, and 58.5-eV photoelectrons. The left­hand-side image is a global picture obtained by detecting secondary-electron signal at 1.3 eV. The other two reveal the contributions from the K and Na + Ca features in Fig. 7. The photon energy was 95 eV, and the lateral resolution 0.5 J..lm.

310

Figure 9. X-ray Secondary Emission Microscopy (XSEM) [12] microimages of a neuron aggregate [4], taken at two different photon energies above and below the Au 4f threshold. Note the reversal of the substrate-aggregate contrast.

6. Consulting the Crystal BaIl

Under normal circumstances, predicting the future [27] is a rather difficult task even for the authors of this paper. The specific boundary conditions, however, are so unusual in facilitating future-reading that these authors feel authorized to forget their usual modesty and play the role of wizards.

What are the special circumstances? They are, of course, already known to the reader: the huge investments made in ELETTRA and the ALS that absolutely require exceptional results for their justification. And we feel that spectromicroscopy has an excellent probability of contributing to the justification.

The present limitations in photoemission spectromicroscopy are almost all still related to the signal level, which in turn is due to the limited source brightness. With orders of magnitude more brightness and signal, we will be able, for example, to reach the lOO-A-levellateral-resolution limits of MAXIMUM set by diffraction. And we will have signal to spare even after this accomplishment.

It should be noted, in fact, that even the most advanced spectromicroscopies solve only in part the general problem of the full exploitation of the photoelectric effect. We have seen that until recently no lateral resolution at all could be obtained, that some lateral resolution is available at the present time, and that better lateral resolution is expected. But it would be desirable to

311

combine these achievements with other improvements, such as ultrahigh energy resolution, spin resolution, or angular resolution.

The superior brightness of ELETIRA and the ALS will make it possible in future months to improve lateral resolution to the limit and also one of these other characteristics. In order to simultaneously achieve lateral resolution and excellent results for two or more of the above characteristics, one will encounter problems even with ELETIRA and the ALS. It is therefore necessary to look further in the shadowy parts of the crystal ball, where we can barely distinguish a fourth-generation synchrotron source, perhaps an ultra-ultralow emittance source in Switzerland or a free-electron laser operating in the ultraviolet or soft x-ray region.

But even forgetting these perhaps distant dreams, the near-future reality is extremely exciting: the spectromicroscopy experiments that we will be able to perform a few months from now will truly revolutionize the century-old field of photoemission science and open up a torrent of new opportunities in materials science and the life sciences alike-opportunities of which our overview, we trust, has provided a hint, however partial and pale with respect to reality.

Acknowledgments

Our spectromicroscopy activities are made possible by the collaboration of many excellent colleagues under different programs. We thank, in particular, the two leaders of the MAXIMUM and XSEM programs, Franco Cerrina and Brian Tonner. We are also very grateful to our excellent colleagues Paolo Perfetti, Delio Mercanti, Maria Teresa Ciotti, Cristiano Capasso, Weiman Ng, A. Ray-Chaudhuri, T. Liang, S. Singh, J. Welnack, R. K. Cole, J. Wallace, Carlo Coluzza, Fabia Gozzo, Philippe Almeras, Henri Jotterand, J.-P. Baudat, Marino Marsi, Mario Capozi, Tiziana dell'Orto, James Underwood, Rupert Perera, Jeff Kortright, Scott Koranda, and many others. Our spectromicroscopy research is supported by the Fonds National Suisse de la Recherche Scientifique, by the USA National Science Foundation, by the Ecole Poly technique Federale de Lausanne, and by the Italian National Research Council.

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27. For a general reference on this subject, we refer the reader to anyone of the many editions of Nostradamus' predictions.

THE PROPERTIES OF UNDULATOR RADIATION

M.R. HOWELLS AND B.M. KINCAID Advanced Light Source Lawrence Berkeley Laboratory Berkeley CA 94720 USA

ABSTRACT. A new generation of synchrotron radiation light sources covering the VUV, soft x-ray, and hard x-ray spectral regions is under construction in several countries. These sources are designed specifically to use periodic magnetic undulators and low-emittance electron or positron beams to produce high-brightness near-diffraction-Iimited synchrotron radiation beams. Some of the novel features of the new sources are discussed, along with the characteristics of the radiation produced, with emphasis on the Advanced Light Source, a third-generation 1.5 GeV storage ring optimized for undulator use. A review of the properties of undulator radiation is presented, followed by a discussion of some of the unique challenges being faced by the builders and users of the new undulator sources. These include difficult mechanical and magnetic tolerance limits, a complex interaction with the storage ring, high x-ray beam power, partial coherence, harmonics, optics contamination, and the unusual spectral and angular properties of undulator radiation.

1. Introduction

Undulators are now established as operational sources of ultraviolet and x-ray radiation at many synchrotron radiation facilities around the world. They are providing qualitatively new and better types of radiation beams and have been involved in many of the most creative new experiments. The success of undulators can be credited to the combined efforts of the originators of the undulator concepts (Motz, 1951; Motz et al., 1953; Madey, 1971; Alferov et ai., 1974; Kincaid, 1977), and to more recent activities such as the work of magnet specialists in the realization of practical undulators (Halbach 1981, 1983; Halbach et ai., 1981), accelerator designers (Chasman et ai., 1975; Green, 1977; Vignola, 1985), builders who incorporated wigglers and undulators into real storage rings (Bazin et al., 1980; Artamonov et ai., 1980A, 1980B; Brown et ai., 1983; Krinsky et ai., 1983), and users applying the undulator radiation to scientific problems (Rarback et ai., 1986; Johnson et ai., 1992). A primary motivation for investment in undulators is that undulator beams concentrate the x-ray output into fairly narrow spectral peaks that can be arranged to cover the desired photon energy range. This greatly reduces the amount of unwanted x-ray power and the associated engineering challenges. The experimental benefits of the higher­brightness beams provided by undulators fall into two main classes: (1) the possibility for improved performance of monochromators, and (2) the ability to focus the x-ray beam to a small probe. These are essentially applications of the small optical-phase-space area of undulator beams and, in general, they use a multiplicity of wave modes. A third related benefit, which we consider to be separate, is that a useful amount of power is now available in a single mode. This is one of the qualitatively new features of undulator radiation and opens the way for a class of experiments that use coherent beams.

315

A.S. Schlachter and F.J. Wuilleumier (eds). New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources. 315-358. © 1994 Kluwer Academic Publishers.

316

In this report, we consider the physical basis and characteristics of undulator radiation and the calculation of its spectral and angular distribution. We describe the coherence properties of undulator beams and show how to calculate the coherence functions needed for applications. We examine the effect of real-world variables on the production of undulator radiation, including the beam optics of the storage ring, radiation from the upstream and downstream bending magnets, and failure of the far-field assumption that is conventionally used in calculating undulator output. We give a brief analysis of the effect of undulator magnetic field errors on the electron beam and on the radiated spectrum and discuss several examples from the Advanced Light Source (ALS) undulator program. Finally, we make some comments on the capability of present-day undulator technology and the performance trade-offs now available.

2. Fundamentals of Radiation Emission by Fast Electrons: Time Compression

Following Kim, 1989, we consider an electron with an instantaneous velocity v = pc (c being the velocity of light) on an arbitrary trajectory r(t') relative to an origin 0 as shown in Fig. 1. An observer is located at x, whose position relative to the electron is specified by the unit vector n making an angle e with v. An electromagnetic signal emitted by the electron at time t' and traveling in a straight line will arrive at the observer at a later time t, where

t = t' +,-Ix_-_r_(t'-,-,)I (1) c

The stationary observer sees the electron's motion as a function of time t, which is different from r(t') due to the change in time scale represented by Eq. (1). The scale-change factor is given by

~=1+ dlx-r(t')ll =1-n,p=I-J3cose . dt' dt' c

(2)

If we now define r for the electron as the ratio of its mass to its rest mass, then we have

Figure 1. Electron trajectory, observer and notation for time c,ompression.

1 1 Y - - j==:===:~

- ~1- v2 - ~(1-f3)(1+f3) c2

1 I- f3 =-2y-2 .

If we now expand the cosine in Eq. (2) and use Eq. (3), we arrive at

317

(3)

(4)

which allows us to estimate the size of the "time-compression" effect represented by dt/dt'. For typical storage rings, the electrons are extremely relativistic and y is of the order of a few thousand. This means that if 8 = 0, then the time is compressed by a factor of a few million. On the other hand, if 8 is greater than a few times lIy, then the 8 2 term dominates in Eq. (4) and the time compression is much less. The time compression is the factor by which the wavelength of signals radiated by the electron is shortened. We see from this argument that, in practical cases, the time compression is a very large effect, but it is mainly confined to emission angles within a cone of half angle lIyaround the line from the observer to a "tangent point" on the electron trajectory .

Physically, the time compression is due to the fact that a highly relativistic electron follows very closely behind the signals it emitted at earlier times. Moreover, the strength of the electric field at the observer is proportional to the apparent transverse acceleration of the electron as seen by the observer, which will be large when the time compression is large. Thus, the amount of radiation will be large within the lIy emission cone. To see this more quantitatively, consider a tangent point P on an electron trajectory with local radius of curvature p, and define a curve segment AB centered on P and subtending an angle 2/yat the center of curvature (Fig. 2a). In terms of the emission time, the electron moves from A to P in a time .1t' = ply and, during that time, suffers a transverse displacement L1x of ply 2 (Fig. 2b). In terms of observation time, the displacement L1x happens in the much shorter time.1t = .1t'y2= pl2y3c. Thus, the motion seen by the observer has the form shown in Fig. 2c. The sharp kink at P corresponds to a very large transverse acceleration as seen by the observer,

d 2x L1x '" 4c2 y4

dt2 '" (.1tA~p)2 P (5)

which is of order y4 times larger than the acceleration in the emission time frame. On this basis, the typical frequency of the radiation should be about lI.1t or 2y3c1p . This is in reasonable agreement with the so-called "critical frequency" we = 3y3c12p, which is conventionally used to characterize a bending magnet spectrum.

3. Undulators

3.1. BASIC DESCRIPTION

An undulator is a device intended to drive the electron in a sinusoidal trajectory. Most commonly, this is accomplished by applying an alternating magnetic field in the vertical direction

318

x (b)

x (c)

z

p t ---- ---------~ P

A ---1---- El---f- -y2 1 1 1 1 1 1 1 1 1 1 1 1 1 I ... ..e -I 1 1 1 1 1 yc 1 1 1 1

--- p _____ t ~-P... 1 ----r y2 1 1 ~2p 1 _

1 y3c 1

t,

t

Figure 2. The effect of time compression: (a) the electron trajectory in space, (b) radial coordinate as a function of emission time t', (c) the apparent variation of the radial coordinate as a function of observation time t. See text for further explanation.

319

so that the oscillations lie in the horizontal plane. We begin with the case of an exactly sinusoidal field and trajectory as shown schematically in Fig. 3. For this case,

x = -acos(kuz}

dx = kuasin(kuz} dz

( d2x) = k2a =.!. 2 u ' dz max P

(6)

where z is along the undulator axis, x is horizontal, y is vertical, and ku = 2ntAu. The centripetal force at maximum curvature (radius = p) is that corresponding to the peak field B and is given by

m v2 evxB=_e-

p or (7)

where e, me, and mo are the electronic charge, mass, and rest mass, respectively. Eliminating a and p between Eqs. (7), (8), and (9), we can determine a value for (dxldz)max that we define to be equal to Kly. When defined in this way, the deflection parameter K is given by

eB K = --= O. 934Au (cm)B(T)

ku moe (8)

and is equal to the maximum angular excursion of the beam in units of I/y. From Eqs. (6) and (7), we can also obtain the following expressions for a and f3x = vxic:

I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I

I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I ~ I t I AUeB

K = 2nmc = 0.934 Au [cm]B[7]

III( -'1

Figure 3. Basic undulator layout an«;l notation.

320

K a=--

kur (9)

dx dxdt K. -=--=/3x =-sm(kuZ) , dz dtdz r

The value of a is normally rather small (about 10 !lm or so), which makes it much less than the horizontal width of most storage ring electron beams.

A device that deflects the beam by about lIr or less is known as an undulator (K:;;: 1). One that deflects the beam by much more than lIr is known as a wiggler (K»I). According to our earlier discussion of time-compression, the synchrotron radiation beam can be regarded as a kind of "searchlight," of angular half width about lIr, pointing along a tangent to the electron trajectory. The above definitions, therefore, suggest that the time variation of the electric field as seen by the observer will be roughly sinusoidal for an undulator and will consist of a series of pulses for a wiggler. It is, consequently, quite understandable that the spectrum (the Fourier transform of the field) of an undulator has a sharp peak with a few harmonics while the spectrum of a wiggler has a broad distribution of harmonics. Two representative spectra are shown in Fig. 4. It is noteworthy that the wiggler spectrum extends as far as a harmonic number approximately equal to K3.

3.2. THE FUNDAMENTAL EQUATION

The fundamental equation of undulator action sets a relationship between the wavelength of the undulator and the wavelength of the emitted radiation. The undulator output wavelength is determined essentially by the Doppler shift due to the motion of the radiating electron. The amount of the shift, or "time-compression factor," which is also the compression factor between lengths, is given by Eq. (2) as

(10)

1.0 r-

r-

0.8 r-Undulator Wiggler

K=1 K=3 ~ 0.6 r- r-. .~ -.. .-11N Q)

£ 0.4 r-

5 0 10 20 30 40 50 60

n n

Figure 4. Wiggler and undulator comparative spectra.

321

Here, pz is the average forward velocity of the electron and is given by

(11)

where {3z = {3 coslfl and lfI is the angle of the electron trajectory to the axis. By expanding the cosine in Eq. (10), using Eq. (3) for 1-{3, and allowing for harmonics, we get

Am = __ u_ 1+_+y2(J2 . A (K2 ) 2my2 2

(12)

This is the fundamental equation describing undulator action derived from the principle of time compression. Another point of view, applicable to the on-axis radiation, is that the factor J.u/2y2 represents two separate effects: (1) the Lorentz contraction of the undulator period as seen by the moving electron (a factor lIy), and (2) a relativistic Doppler shift of the emitted wavelength due to the relative velocity of the electron and observer (a factor 1I2y) .

We have introduced the deflection parameter K as a measure of the angular excursion. It is also a dimensionless measure of vector potential and scales as Au' B for a pure sine-wave undulator. In actual undulators, the magnetic field is non-sinusoidal and can be represented as a Fourier series with only odd spatial harmonics, as in Eq. (54). The even spatial harmonics are normally forbidden by the symmetry of the magnetic structure. We can regard our results in Section 3.1 as applying to the first harmonic of such a series and then repeat the development of Eqs. (6) through (8) for the mth harmonic. This leads to

K _ eBm m-

mkumoe (13)

The undulator output wavelength is determined essentially by the time compression due to the motion of the radiating electron. The amount of time compression is given by Eq. (10), which leads to the fundamental equation as shown above. For an undulator field described by a Fourier series like Eq. (54), the average forward velocity is modified. The output wavelength is still determined by K via the same fundamental equation, but K is now defined as

~ eBiff ~2 K = ~ K2 = __ e - ,where Beff = ~-l!!. £... m k moe £... m2 mUm

(14)

The relations in Eq. (14) are proved in Appendix 1. It is noteworthy that Beffis neither the rms field nor the peak field.

3.3. DIFFRACTION LIMITS AND THE CENTRAL CONE

From Eq. (12), we see that the on-axis wavelength is lengthened (red-shifted) if the receiving point moves off the axis or, equivalently, if the electron trajectory has an angle to the axis. We also know that the fundamental wave train radiated by the undulator must have N periods, where NJ.u = L and L is the length of the undulator. Therefore, even a single electron emission pattern on axis must have a spectral spread of about L1.YA = liN for the fundamental or WArn = limN for the mth harmonic. From Eq. (12), we find that the amount of red shift is

322

(15)

Equation (15) defines a useful quantity, r*. Further, let us defme an angle (1; corresponding to a spectral spread mAm = 1I2mN as

(16)

The angle (1; turns out to be important in the analysis of undulator beams. It is the rms width of the one-electron undulator beam due to diffraction. One can see this in a rough way by calculating the angle of the first minimum of the diffraction pattern of an ideal longitudinal line source. Consider a parallel beam of rays emitted coherently at angle () from every point on the source. The diffraction minimum will occur when the path difference between the rays from the upstream and downstream ends of the source (L - Lcos(}) is equal to AI2. This leads directly to e=~A,IL.

Equations (15) and (16) show that, provided the collection half angle (}cc is less than (1;, then the intrinsic spectral width limN is not much spoiled by red shifting. The radiation within (}cc is called the central cone and is the most useful part of the undulator emission. The central cone of an undulator beam is even more highly collimated than normal synchrotron radiation. Equation (16) shows that it has a characteristic angular width 11 (r-fN), which is substantially smaller than the lIr width of a bending magnet beam. Because every harmonic is red shifted according to Eq. (12), the wavelength of each harmonic will equal that of the fundamental at a sufficiently large off-axis angle. The radiation pattern at the fundamental frequency thus consists of a bright central peak on the axis and a series of partially illuminated rings of angular radius .j m -11 r*. A similar argument holds for higher harmonics which have rings due to the harmonics of higher number than themselves.

For the case of a real electron beam, it may happen that the electron beam angular spread (1; is greater than (}cc. In this case, the central cone width has to be defined equal to (1 ; , and this will represent a degradation of the spectral brightness of the undulator. Storage rings such as the ALS, which are intended to operate with undulators, are designed to have electron beam angular spreads that are small compared to (}cc.

3.4. PRACTICAL REALIZATION OFUNDULATORS

The practical realization of undulators is now nearly always by means of permanent magnets following the methods developed by Halbach, 1981, 1983, and Halbach et aI., 1981. We do not have space for a review here, but the most common design for building high-field devices (the so­called hybrid scheme) consists of blocks of permanent magnet material combined with soft iron pole pieces as shown in Fig. 5. The materials used for the recently completed ALS undulators were neodymium-iron-boron blocks and vanadium permendur pole pieces. The ALS devices are the largest and most demanding yet attempted, and their achieved field quality and projected performances are treated in more detail in a later section.

The technology of the undulator magnetic structure and the physics of the resulting magnetic field distributions set limits on the range of devices that can, in principle, be built. Usually, one starts with a knowledge of a photon energy operating range and a magnetic gap defined by the requirements of the storage ring injection system. As a start, we may safely assume that the output of an undulator falls to zero as K approaches zero. In fact, as we shall see in Section 4, it

rJl------ Tuning Stud (Steel) /o-?....."Y?\

Backing Plate (Steel)

Keeper (Alum. Alloy)

-f:ffl--lt----- CSEM Blocks (Magnetization Ori entation Shown)

v,<>1-+----- Pole

~!!!I!"I!:!:!:~~~~:r&~~m"., --L (Vanadium Permendur)

'I+I+I+'~I ~' ~;~:~~.;::.:," Period (I..)

323

Air Region

Pole (Vanadium Permendur)

:J ~--- CSEM (Magnetization

Orientation Shown)

Half Gap

a.) Elevation Cross Section b.) Quarter Period Magnetic Flux Plot

CSEM-Steel Hybrid Insertion Device

Figure 5. Construction of a hybrid undulator from current-sheet-equivalent material (CSEM) and steel.

falls to about half maximum at K = 0.5, and fairly rapidly below that, so that we may reasonably regard K = 0.5 as a limit. Using Eq. (12) with K = 0.5 and ()= 0, we obtain the value for Au that delivers the required minimum wavelength. Accepting this value, we then find that the maximum wavelength obtainable will be determined, via Eq. (12), by the highest achievable value of K, which depends on the field.

A certain amount of information about the field can be calculated from formulas that apply to magnet structures of optimum design (Halbach, 1983);

Neodymium - iron:

Eo = 3. 44exp[ - fu (5.08-1.54 fJ] Samarium - cobalt:

Eo = 3. 33exp[ - fu (5.47 -1.8 fu)]

0.085 <L< 0.8 A.u

(17)

(18)

324

In these formulas, Bo is the peak field and, hence, is an overestimate of Beff. Using Bo to compute K or A would, therefore, be incorrect. An accurate evaluation of the value of K via Eq. (14) (and thence A) requires a knowledge of the Fourier expansion of the field. This can be obtained using a program such as POISSON (see for example, Warren et ai., 1987). In pursuing this type of design exercise, one possible variation is to reduce Au somewhat to gain some brightness (larger N for a given L) and reduce the total power. The trade-off would be a reduction in the spectral range. We consider this question further in Section 7.1.

Although Eqs. (17) and (18) are correctly given in the reference quoted, they have, nevertheless, often been misunderstood. Accordingly, we wish to point out several things that these equations do not imply.

1. They can only be used in the stated ranges of values of the gap-to-period ratio. Outside the stated ranges they can give meaningless results.

2. They give the maximum total field obtainable by good design for a single gap-to-period ratio. They do not predict what field this same well-designed undulator will produce at other gap-to-period ratios.

3. The field given is the peak field, not the rms field or the Be!! used in Eq. (14). Consequently, one cannot obtain a correct value for K using Bo from Eqs. (17) or (18).

4. The total field may contain a strong harmonic content. For the smallest gap-to-period ratios, the field is highly non-sinusoidal. The greater the harmonic content, the greater the difference between Bo and Beff.

4. Characteristics of Undulator Radiation

4.1. CALCULATION OF THE SPECfRAL AND ANGULAR DISTRIBUTIONS

The general problem of calculating radiation from accelerated electrons has received extensive attention in the literature as reviewed as reviewed, for example, by Blewett, 1988. The first derivation of a synchrotron radiation spectrum was in the 1912 publication by G.A. Schott (Schott, 1912), although not much could be done with it at the time. After the experimental discovery of synchrotron radiation, Schwinger derived expressions in terms of known functions describing bending magnet radiation and clarified the physics of the process (Schwinger, 1949). A lucid treatment of the problem, and one which has been widely used by other authors, is provided by Jackson, 1975. If the coordinate system is the one shown in Fig. 6, then according to Jackson, the flux per unit solid angle is given (in SI units) exactly by

dIe (J)

dmdQ

_1_e2 {J)2 +f~ nx[n-Jl]xJl + (n-Jl)c eiW(HR(T)/c)d ' { . } ,2 4n-Eo 4n-2c _~ (l-n.Jl)2 R y2(I-n.Jl)2 R2

which reduces in the far field to

2 dIe (J) 1 e 2 {J)2 +~ --=---- fnx(nxJl)eiW(T-noXeIC)dt dmdQ 4n-Eo 4n-2c _~ ~

(19)

(20)

325

Figure 6. Notation and coordinate systems for radiation calculations.

Not all applications of these equations are in the far field, and, to avoid the complication of Eq. (19), Wang, 1993B, has shown that it can be simplified to

~ 2

4 +00 dI( 0)) _ a 1'10) J (') iolt d --- n t e t dQ 4n2c2

-00

(21)

provided that the distance R to the observation point satisfies R » rA. This requirement is much less restrictive than the far-field condition and is satisfied in all cases of interest in synchrotron radiation applications. Equation (21) has been used for numerical calculations (discussed in a later section) of the radiation pattern to be expected from the ALS undulators based on the actual measured fields of the devices. It also provides a much greater degree of physical insight into the relationship between the electron's trajectory and its radiation pattern.

The full calculation of the angular and spectral distribution of undulator radiation in terms of known functions was first given by Alferov et al., 1974. Helical undulators have been treated by Kincaid, 1977. More recent treatments of the calculations, starting from Eq. (20), have been provided by Hofmann, 1986, and Krinsky et al., 1983, and an ab initio calculation has been given by Kim, 1989. Hofmann's calculation of the spectral power per unit solid angle due to a single electron in an undulator leads to the following expression:

;PPm =

dQdO)

P 3m2r*2 [{2r*ecoSI/>Sml-K*Sm3)2 +{2r*esinI/>Sml)2] N [Sin( ~nN)l2 u n{1+K2/2)2K*2 {1+r*2(2)3 0)1 ,1O)nN '

0)1

(22)

326

where

+00

8ml = IJI(mau )Jm+21(mbu )

1=-00

+00

8m3 = L/I(mau )(Jm+21+1(mbu)+Jm+2l-1(mbu ») 1=-00

K*2 b _ 2K*r(}* costP u- 2 (1 + r*2(}2)

(23)

and (} and tP are the radial and azimuthal angles, respectively. In comparing Eq. (22) with the corresponding equation given by Krinsky et ai., 1983, and after translation of Krinsky's notation to Hofmann's, one finds that there is still an apparent disagreement in the 8m3 term of Eq. (22), which Krinsky expresses in terms of different Bessel function series. However, upon application of the recurrence relation J J1+1(X)+JJ1-1(X) = 2J1.1 J1(x)/x, the two expressions can be seen to be fully identical.

The first term in the square bracket of Eq. (22) describes the (1 polarization and the second term describes the n. Obviously, the 10 contribution is zero in the horizontal plane (tP = 0). In fact, as shown by Kitamura, 1980, the undulator radiation is plane-polarized in the (1 direction out to several central-cone widths so that this is the prevailing form for essentially all applications. The only frequency dependence in Eq. (22) is that of the sinc function, which represents the intrinsic fractional bandwidth limN due to the presence of N undulator periods as we have noted above and can see in Fig. 4.

The shape of the light intensity distribution for the first four harmonics is depicted in Fig. 7. One can see that the strength of the even harmonics is zero on the undulator axis and that the mth pattern has m lobes along the horizontal axis of the receiving plane and none along the vertical axis. The amount of each harmonic present depends, in a complicated way, on all the variables and is not represented in the diagram. An important quantity, in practice, is the on-axis flux per unit solid angle a '!E"IaQ. We can obtain this under the approximation of zero electron beam emittance by setting (} = I/J = 0 in Eq. (22). Considering the exact harmonic frequencies and using the Bessel function properties JJ1(O) = 0J10 and LJ1(x) = (-l)J1JJ1(x), we find that the 81 terms of Eq. (22) vanish and

8m3 = J m+l (mau )-Jm-l (mau ) m odd - -

2 2 bu =0. (24) =0 m even

Using these values in Eq. (22), multiplying by the number of electrons in the undulator at any instant (lL/ec), and changing from power to photon flux, we finally get the flux per unit solid angle

(25)

where

09

0.8

0.7

0.6

';; 0.5 0.4 ~ - 0.3 0.2 0.1

0

0.3

0.25

0 .2

';; 0 .15 ~

0.1

0.05

o

0.9

0.8

7

0.6

0.5 ~

0.4 "">:, ~

0.3 -0.2

0.1 0

I

2nd harmonic I

Figure 7. Intensity distribution (in arbitrary units) for the first four harmonics.

327

328

o.

-,."

!!. -

-,."

!!. -

Figure 7. (continued).

~ ~ ---

3rd harmonic I

0.09

0.08

-,." ~ -

4th harmonic I

329

m odd (26)

This function is easier to calculate and is shown in Fig. 8 for several values of m. We can recast Eq. (25) in a useful way in terms of a; as

(27)

This shows that the denominator of the right hand side of Eq. (27), which we call '2Fm , is approximately equal to the flux in the central cone of the mth harmonic of the one-electron pattern.

4.2. THE EFFECf OF A FINITE ELECTRON BEAM EMITI ANCE

Let us now consider the case when the electron beam has a non-zero emittance. Suppose that the center of the undulator is a waist of the electron beam (implying a vertical phase-space ellipse) with horizontal (x) and vertical (y) rms beam widths (ax, ay) and angular widths (a;, a y) given by

(28)

where ex, loy are the storage ring emittances and /3x, /3y are the electron beam amplitude functions at the waist. Suppose that an electron in the mid-plane of the undulator has phase-space coordinates (x, x', y, y~, and that we regard the coordinates as representing a ray. Let the arrival point of the ray in a receiving plane distance D downstream be (~, 1]). Then ~ = x + x'D and 1] = y + y'D. If each phase-space coordinate is Gaussian-distributed, then the normalized probability that the arrival point will be (~, 1]) is

0.5

0.4

~ 0.3 -..;;..

I.t.;;::' 0.2

0.1

1

Figure 8. The function Fm(K) [Eq. (26)].

2

K

----m=1

----m=3

------. m=5

-.-.-•• m=7

- •• - •• - m=9 3 4

330

where

and (12 = (12 + D2(1'2 1) y y .

(29)

In calculating the total intensity at (~,1]), we need to know the weighted average of lIPua2PlaQam over 0 and tjJ. The weight corresponding to (0, tjJ) is the probability, from Eq. (29), of the ray arrival point (uD, vD), which is the one needed to send light at angles (0 costjJ, o sintjJ) to the point (~, 1]). Thus, the intensity at (~, 1]) per unit area per unit frequency interval is

2 oo2n a I(~,1])=~f f G(K,o,tjJ,m)p(u)p(v)8dOdtjJ ,

amas D 0 0 (30)

where G is essentially the right-hand side of Eq. (22) and p(u) and p(v) are normalized Gaussians like Eq. (29) with

u=I-OcostjJ D

v =.!l_ OcostjJ D

Although this treatment is based on the superposition of the intensities of the one-electron patterns according to the principles of geometrical optics, it remains valid in the far field even when the system is diffraction limited or partly so. However, there are regimes when both diffraction is important and the calculation is in the Fresnel region. In these cases, it is necessary to carry out a superposition of the fields rather than the intensities. This is covered by the so­called "brightness convolution theorem" (Kim, 1989), which requires use of the brightness function that we discuss in Section 5.

Since it is important to be able to model the behavior of undulators in real storage rings, it is necessary to evaluate large numbers of integrals like Eq. (30). In fact, such evaluations pose one of the principal difficulties in designing efficient codes for the frequent "production runs" involved in developing and using undulator x-ray sources. One approach to minimizing the processing time is to use Gaussian quadrature as proposed by Kincaid, 1993. There exist several fairly widely used computer codes capable of implementing the calculations discussed so far (Kim, 1989; Jacobsen and Rarback, 1985; Walker, 1992).

4.3. FLUX AND BRIGHTNESS ESTIMATES FOR REALELECfRON BEAMS

We now tum to assessing the effect of finite emittance on the flux per unit solid angle and on the brightness. The basic approach was worked out on geometrical optics principles by Green, 1977, for bending magnet radiation. To adapt Green's ideas to undulator radiation, we use Eq. (16) for the diffractive angular spread in place of the vertical opening angle of bending magnet radiation. This leads us to a new way to write Eq. (27) for the flux per unit solid angle:

331

(31)

where

(32)

To obtain a similar estimate of the brightness, we need to know the diffraction-limited source size aT corresponding to the diffraction-limited emission angle a;, both of which can be calculated by approximating the one-electron undulator source as a Gaussian laser mode (Kim, 1986). At the wavelength of peak emission, which is slightly longer than Am (see Section 6.2), this results in the following description in terms of rms width and angular width of the radiation beam:

,_ r;:;;; aT -VI:'

, Am e=GTGT =-

4n

The on-axis spectral brightness is then given by

B (00)= ~m m' 4 2L L'L r 1! x x y y

where

and

(33)

(34)

To summarize the present section, we show in Fig. 9 the spectral flux per unit solid angle and the spectral brightness of a variety of synchrotron radiation sources as calculated by Hulbert and Weber, 1992.

5. Coherence of Undulator Radiation

5.1. SPATIALANDTEMPORALCOHERENCE

We are interested in the possibility of interference experiments for which we must create two or more interfering beams with a definite phase relationship so as to allow interference fringes to be formed. There are two ways to do this, and each one challenges the degree of coherence of the x­ray beam in a different way. In the first method, we combine the beam with a delayed copy of itself formed by amplitude division as in the Michelson interferometer. If the delay is greater than the length of the wave train (the "coherence length" of the beam), then we will not see any interference fringes. Thus, for this method, we must have a sufficiently monochromatic beam, which is the same as having high temporal coherence. In the second method, we combine beams of x-rays taken from two different points on the wave front (wave front division) as in a Young's slits experiment. If the distance between the two points is greater than the "coherence width" over which a sufficiently good phase relationship exists, then, again, we will not get the desired fringes. The requirement for this method is to have good collimation (a source subtending a sufficiently small angle at the experiment), which is the same as having high spatial coherence.

332

-~ co .... E LO

~ CO ;:,g 0 ,.... ci --~ !I) c:: 0 (5 L: S X ::l

u:::

N "0 co .... E --(\J

E E --~

CO ;:,g 0 ,.... ci --~ !I) c:: 0 -0 L: S !I) !I) Q) c:: 1: Ol '': CO

1016 - NSLS Sources

U5.0 ,--- ALS Sources

U5U U8.0----- _.- APS Sources

1015

X21,X25

1014

1013

1012L-~ ____ J-__ ~ __ L-__ -L __ -L __ -L~ __ L-~ __ ~-L-LJ-_

101 102 103 104 105

1020

1019

1018

1017

Photon energy (eV)

_ .. -.'---.....

- NSLS Sources ,- - - ALS Sources _.- APS Sources

". \

\. UA \ i

X21,X25 _._._.-..WIGA . .,.,\

W16.0

103 104

Photon energy (eV)

Figure 9. Flux (a) and brightness (b) for various synchrotron radiation sources from Hulbert and Weber, 1992.

333

5.2. DEFINITION OF A MODE OF THE UNDULATOR BEAM

The angle over which a source provides spatially coherent illumination is roughly the wavelength divided by the source size. If only this angle is filled with light in each of the horizontal and vertical directions, then the beam is said to comprise a single mode. Under such conditions, its size-angle product (emittance) is approximately equal to the wavelength. To make the concept of an undulator mode more precise, we represent the undulator radiation pattern in a phase space (x, x', y, y,), which is essentially the same as the phase space used to represent the electron beam. Calculations of the paraxial ray optics of the radiation beam can be carried out using matrix techniques to manipulate the vectors (x, x') and (y, y') as one would do for the electron trajectories. However, as we have seen, there are significant diffraction effects in undulator action that are not accounted for by a geometrical optics analysis nor by the computer ray-tracing techniques that have been so valuable up to now in modeling beamline optical systems. In physical optics, we are obliged to work with the fields, so we represent the electric field at distance z from the mid-plane as E(x, y; z). We will also need the frequency-space representation of E, E(x', y', z), where we note that, for the small angles of interest to us, the angle variables (x', y') are proportional to the spatial frequencies [(sinx')lA, (siny')/A]. E and E are thus related by a Fourier transform. We now define the rms spatial and angular extent of the fields as

+00 f x2 IE(x)12 dx

~ (x2) = --=-:"--00 -­

fIE (x)12 dx

+00 f x'2IE(x')12 dx'

~(x'2) = -=-:=-00--­

f IE(x')12 dx'

(35)

As with any signal represented in the direct and frequency domains, the widths of the two representations are reciprocally related. In fact, the product of the widths has a minimum value that corresponds to a signal with minimum information content. Specifically, the rms widths that we have just defined are related (as shown, for example, by Bracewell, 1978) in the following way:

(36)

The minimum information signal, corresponding to the equals sign in Eq. (36), can be shown to be a Gaussian wave packet. Physically, Eq. (36) represents the fact that, if the width is restricted, the angle (i.e., the frequency) will increase because of diffraction. The minimum allowed value of the width-angle product corresponds to the single-mode beam we are seeking to define, and this, therefore, has the emittance £C, characteristic of a spatially coherent beam, given by

, A Ec = GrGr =-

4n (37)

Equation (37) is the same as Eq. (33), which was derived from the Gaussian-laser-mode representation. The rectangular function of equal area to a Gaussian has a width .J2ii G, so, assuming we are dealing with Gaussian-distributed beams, we find that the phase-space area of a single-mode (spatially coherent) beam is given by

334

(38)

The above results are derived from fundamental considerations and represent a physically correct measure of the size of the coherent phase space. However, in practical experiments, one usually needs to choose the amount of phase space to accept on the basis of a resolution-flux trade-off. Insufficient spatial coherence (accepting too much phase space) leads to a loss of resolution in a hologram, for example, while accepting too little phase space is equivalent to a loss of flux. A common compromise is exemplified by the case of illumination of a zone plate lens by a pinhole of diameter d at distance z. The complex coherence factor (Born and Wolf, 1980) of the pinhole source (taken to be incoherently illuminated) is a circular Airy function peaked at zero separation of the two test points. This function is of the same form as the amplitude distribution of the pinhole Faunhofer diffraction pattern that has a zero at a radius 1.22k/d. The bright region inside the zero is known as the Airy disk. To maintain a high degree of spatial coherence over the whole zone plate, it would be necessary to accept light only within a region near the central peak of the complex coherence factor. However, a compromise that causes only slight loss of resolution is to set the diameter of the zone plate equal to the radius of the Airy disk. This choice maximizes the so-called "resolution-luminosity" product of the system and is equivalent to accepting phase-space areas in x and y of (1.22}.,)2 instead of (Y2)2, roughly a six-fold flux gain. As an example of the consequences of these ideas, we show in Fig. 10 a graph of the spatially coherent fraction of the light from ALS undulators for both the single-mode and the half-the-Airy-disk definitions of coherent phase space. The main point, of course, is that undulators are capable of delivering enough coherent flux to do many interesting coherence experiments.

5.3. THEDEGENERACYFACTOR

We traditionally characterize the usefulness of an undulator by quoting its time-averaged spectral brightness B, which is the number of photons per unit phase-space volume per unit fractional bandwidth per unit time. However, a more fundamental quantity would be its degeneracy parameter Ow (Goodman, 1985). This dimensionless quantity is defined as the number of emitted photons per coherent phase-space volume per coherence time or the number of photons per mode. The coherent phase-space volume is (}.,12)2 and the coherence time is }.,2/(Li}" c), so Ow is given by

o =DB(~)2(J:.:...)(Li}")= DB}.,3 w 2 eli}.,}., 4e'

(39)

where D is the duty cycle of the storage ring. In practical units, this is

Ow = 8.33 X 10-25 DB(ph/mm2/mr2/0.l %Bw/s)}.,(A)3 (40)

It is significant that the bandwidth cancels out and we are left with a measure of the probability that two wave trains will overlap in the same wave mode. Since photons are bosons, Ow is allowed to be greater than unity; however, it is only with the advent of undulators on modem storage rings that values greater than unity have been achieved in the XUV spectral region. As an example, the ALS undulators will achieve Ow values greater than unity for wavelengths longer than about 50 A.

c o ~ 10-2

~ -+-' C Q) ..... Q)

-g 10-3 ()

10-4

, " -. ------ --.. -. --- -- ~ ----- ---------------- ~ -- ------------ ,- - -- -

Gaussian modes

Half Airy disk

-------------------~ ---------------------! ---------------------~ ----------------.\. ~ -, , ' , , , , ' , , , , , , , , , , , , , , , ' , , , , , , , , , , , , , , '

10-5L-~~~~--~~~~~~~~~~~~~

10°

Photon energy (eV)

335

Figure 10, Coherent fraction of the central cone radiation from ALS undulators for the two definitions of coherent phase space discussed in the text.

One of the phenomena that are understandable in terms of Ow is the bunching of photoelectron counts due to the stochastic variations of the classical electromagnetic field, This is expected to be observable with thermal light for which the intensity fluctuates in a chaotic way, but not with light from a good-quality laser for which the intensity is stable. Bunching is a separate effect from shot noise, which affects all types of light beams equally. Undulator radiation is produced as a coherent sum of the fields radiated by one electron, but an incoherent sum of the wave trains emitted by the population of electrons, The resulting intensity, therefore, has chaotic fluctuations like thermal light, but does not have the black-body spectral distribution. Accordingly, an undulator beam should be described as pseudo thermal light with' a high Dw indicating a high effective temperature. This combination of qualities is more unique than one might suppose. Based on the Planck thermal distribution function for a black body, one can show that the degeneracy parameter for thermal sources, even very hot ones like the sun, is much less than unity (Goodman, 1985). Moreover, it can be shown that Ow is equal to the ratio of the size of the photo count fluctuations due to the stochastic variations of the classical electric field to the size of those due to shot noise. Therefore, for sources with a very small value of Dw, the shot noise dominates and bunching is essentially not observable. Thus, even in the visible region, neither

336

lasers nor thermal sources produce easily observable photo count bunching under normal conditions. The only way to imitate a pseudo thermal source with strong bunching is to pass laser light (which also has a very high Sw) through a moving diffuser.

In view of the above conclusions, we expect the measured instantaneous and time-integrated intensity in an undulator beam to show chaotic behavior. Specifically, we expect that the probability-density function of the instantaneous intensity will be negative-exponential, while that of the intensity integrated over a finite time will be Gaussian. The time scale of these fluctuations would be on the order of the coherence time of the wave field, which is in the femtosecond region for cases of practical interest. These physical quantities would be constants for a well-stabilized laser beam. Thus, in spite of the practical similarities between undulator beams and laser beams based on their low phase-space volume, the physics of their emission processes and the statistical properties of their radiations are very different.

The degeneracy parameter has importance in other matters as well. For example, it determines the detectability of the intensity fluctuations of the classical field in an intensity-interferometer experiment (Gluskin et ai., 1992). This is a close parallel to its role in determining the degree of bunching. The conclusion appears to be that soft x-ray intensity-interferometer experiments will be quite feasible with undulator beams on third-generation storage rings, while only ultraviolet experiments could be considered at older facilities.

5.4. "DEPTH-OF-FIELD BROADENING" EFFECTS

Undulators and other sources of synchrotron radiation are essentially small transversely and very extended in the emission direction. With an intuition based on geometrical optics, one, therefore, expects that it will be impossible to make a perfect image of the source due to depth-of-field effects. This has been discussed by various authors, especially Green, 1977, and Coisson and Walker, 1985. As discussed by the latter, the effect can be described using the phase-space representation. An electron with coordinates (x, x') at z = 0 transforms to (x+x'z, x') at z = z. If, at this point, it emits a photon at an angle xe' to its trajectory, then the apparent emission point of the photon in the z = 0 plane is x - xe'z. If the trajectory was steered by an angle xs ' in traveling to z, then the apparent emission point would be x - (xe' + Xs ')z. The point to note is that this expression is independent of x'. This implies that depth-of-field broadening is not caused by the electron beam angular spread. Rather, it results from the emission angular spread or from steering of the beam, as in a wiggler, and is still present even for a zero-emittance beam. The calculation of the form of the depth-of-field-broadened source is rather cumbersome. Even for the case of a zero-emittance beam (treated by Coisson and Walker), the expression must be written in terms of the exponential integral and is infinite at its center point. The more realistic case, including a finite emittance but still within the geometrical optics approximation, is treated by Green, who represents the source by a new function ef(a, Y) (see Appendix 3). The function ef(a, Y) has a finite peak at the origin and, for long sources, has large non-Gaussian tails extending out to many sigmas of the original unbroadened source.

For the case of an undulator, the amount of steering is negligible and the possibility for depth­of-field effects rests on the angular spread of the emission from a single electron. However, we have already noted that such spreading of the one-electron pattern is a diffraction effect and its counterpart is a broadening of the source (to ~ A.mL /4n). Both effects are included in the representation of the source as a Gaussian laser mode. The diffraction picture thus includes essentially the same broadening effects that we discussed in the previous paragraph. We conclude that the "depth-of-field broadening" is simply the geometrical optics approximation of the diffraction picture of the single-electron pattern and its convolution with a realistic source

337

with finite emittance. Therefore, diffraction and depth-of-field broadening represent the same thing and should not be added in calculations.

On this basis, we can get some idea of what will happen when we try to image the one-electron undulator source. It will behave like any other diffraction-limited source, and we will not see evidence that the source had a great depth. As a consistency check, we compute the transverse and longitudinal resolutions (~t and Lit) to be expected from an imaging system at wavelength A. and numerical aperture NA = A. / L. This yields

(41)

Thus, roughly speaking, the resolution of the optical system would be such that it could not tell the difference between a point and an object of the size and shape of the undulator. In summary, we expect no harmful depth effects in imaging the undulator source.

5.5. PARTIAL COHERENCE EFFECTS IN UNDULATOR BEAMS

We have already noted that geometrical ray tracing is not adequate to represent all behaviors of an undulator source because of diffraction. Nevertheless, it is very desirable to have a way to model the performance of undulator beamlines with significant partial coherence effects, and such modeling would, naturally, start with the source. The calculation would involve a knowledge of the partial coherence properties of the source itself and of how to propagate partially coherent fields through space and through the optical components used in the bearnline. We discuss the source properties further below, but it is important to recognize that, although most of the these calculations are, in principle, straightforward applications of conventional coherence theory (Born and Wolf, 1980; Goodman, 1985), there is not much current interest in this type of problem in the visible optics community. Therefore, there is not a large body of literature to help us with solutions to specific cases. For example, even for the rather simple problem of diffraction by an open aperture with partially coherent illumination, we have found published solutions only for circular and slit-shaped apertures and only for sources consisting of an incoherently illuminated aperture of similar shape to the diffracting aperture. Thus, there is no counterpart in these types of Fourier optics problems to the highly developed art of ray tracing in geometrical optics, nor is there anything as simple as a ray to which an exact system response can be calculated.

This is not to say that no progress has been made. One of the difficulties of coherence-theory calculations is that integration over a large number of variables and a high degree of complication is often encountered. A major simplification of the problem for cases where the small-angle approximation applies has been achieved by Kim, 1986, 1989. This author has developed an extension to the normal coherence theory based on the use of the frequency-space representation of the mutual intensity (see Appendix 2) rather than the usual direct-space representation. The Fourier transform of the mutual intensity (called the "brightness" by Kim) is shown to be invariant with respect to propagation through free space and simple lenses. This means that representation of such propagation is very simple and consists of linear operations on the phase­space coordinates. This allows the brightness to be calculated anywhere without multiple integrals. It would take us too far away from our main subject to give a full presentation of this, but we do consider in the next section the coherence properties of the source itself.

We first recall that the undulator source consists of an incoherent superposition of many one­electron patterns, each of which is to be represented as a Gaussian laser mode with rms width and

338

angular width ar and ar' as given by Eq. (33). Therefore, throughout the source area, there is an rms coherence width ar with a complex coherence factor )1(Lh, .1y) of Gaussian form. Thus, apart from the Gaussian intensity distribution of the source, its field correlations are spatially stationary. We, therefore, consider the undulator to be a quasi homogeneous source (Goodman, 1985). The latter is defined as one for which the mutual intensity can be written as

(42)

The expressions we use in this section are all separable in x and y so, starting with Eq. (42), we give only the x part. Substituting the above Gaussian forms into this equation gives

( .1x .1x) [-x2 -.1x2 ] J12 x+- x-- =exp --+--2 '2 2ai 2aax '

(43)

where

1 1 1 -----+-a 2 - 4a2 a 2 '

L1x x r

and J12 is a function of the spatial variables (x, Lh). The propagation law for J12 is a standard result of coherence theory (Born and Wolf, 1980; Goodman, 1985) and involves a multiple integral over four variables altogether, including y and .1y. In general, this is difficult and can be avoided by using the brightness function defined by Kim, which is valid for many practically interesting cases. The brightness function is denoted by B(x, x';O). It is a function of both position and angle coordinates and is defined by

+~ , f ( .1x .1x) ·'.A __ ' B(x,x ;0)= C J12 x+T' x-T e-I~ d.1x , (44)

where k = 211111. and C is a constant. Using Eq. (43) in Eq. (44), we find

(45)

This forms the starting point for the simplified propagation and optical calculations that are enabled by knowledge of the brightness function. It is noteworthy that the brightness function used by Kim is not a physically measurable quantity, although several such quantities can be obtained from it. It is, therefore, necessary to pay special attention to the meaning of the brightness function as described in the published accounts (Kim, 1986, 1989) before using the results derived from it.

As an example of the use of the brightness function, we calculate the mutual intensity at a plane distance z downstream of the source. We begin by propagating the brightness a distance z using the transform (x ~ x - x'z, x' ~ x1. This gives

339

(46)

from which we obtain 112 via the transfonn that is the inverse ofEq. (44);

(47)

where

It is noteworthy that 112 is no longer spatially stationary (L1x part separable). For the special case x = 0, meaning that the two test points are disposed symmetrically about the axis, we see that 112 is a Gaussian with a width consisting of the quadratic sum of two tenns. The first tenn is equal to the van-Cittert-Zemike-theorem result for an incoherent Gaussian source of nns width Gx. The second tenn is a constant width of{2 times Gr. This shows that the van Cittert-Zemike theorem result is a good approximation when it predicts a large coherence width (»-.fi Gr) such as in the far field of small sources (the ALS, for example). On the other hand, at shorter distances from larger sources, the van Cittert-Zemike theorem predicts a very small coherence width, and the constant tenn then dominates. The failure of the van Cittert-Zemike theorem should not be surprising since the strong directionality of the undulator beam shows that the incoherent representation must break down eventually.

Before leaving this subject, we should point out that, for designing coherence experiments, it is essential to know the shape and extent of the function J12 [or its normalized fonn, the complex coherence factor J.l12 = 112/(111h2) 112)] at the location of optical components, microscope sam­ples, etc. This is usually calculated by an approximation, the main one being the van Cittert­Zemike theorem, in which (under suitable conditions) J.l12(L1x) is given by the Fourier transfonn of the source intensity distribution /(x). We show in Table I the nature of the available approxi­mations to help in judging when they can be safely used. The main point is that the coherence character of undulator sources varies, in practical cases, over the whole range from essentially coherent, to essentially incoherent, so that no simple approximation can cover every case.

6. Brightness: Compromises and Limitations

6.1. OPTIMUM CHOICE OF BETA FUNCTIONS

High brightness is one of the most desirable properties of undulators, and a great deal of effort is devoted to optimizing it. One question which arises is whether the f3 functions at the undulator location have a large effect on the brightness. When £ » /I., i.e., the source is far from diffraction-limited, the brightness is dominated by the electron beam emittance. Conversely, when £« /I., the source is extremely diffraction-limited, and the brightness is dominated by

340

TABLE 1. Methods to find the complex coherence factor downstream of an undulator source.

RMS Coherence

Assumed Source Width at the Character Source*

Coherent ""as (diffraction-limited)

Quasi homogeneous < as (general case)

Almost incoherent « as

Incoherent (electron- = 0 beam-limited)

Complex Coherence Factor Distance z

Downstream

Constant

J12/(l} 1122)112 from Eq. (47)

~[/(x)] ~LudL1x)]1 A z**

\

~[/(x)]1 10**

*The source is taken to have an rms width as. **/0 is the integrated flux, ~ represents the Fourier transform.

Method of Calculation

None

Brightness function

Generalized van Cittert-Zernike theorem (Goodman, 1985)

van Cittert-Zernike theorem

diffraction. In both cases, the brightness is relatively insensitive to {3, although there is a shallow minimum. On the other hand, when E - A, it is possible to suffer a major loss of brightness by a poor choice of {3. To see the effect of the {3 functions, consider the dimensions of a diffraction­limited x-ray beam. Its phase-space ellipse has semiaxes ar • a; while that of the electron beam has semiaxes (e.g.) aX. a;, and the two would have similar area because E - A. The optimum value of the {3 function would match the two ellipses by having a; - a; and ax - ar while the worst choice would mismatch them in the manner of a cross. In the latter case, the resulting photon phase-space area would be approximately a circle with the crossed ellipses inside it! To fmd the optimum {3, we set the ratio of the major to minor axes equal for the two ellipses

or (48)

leading to

(49)

In practice, this is a rather low but possible value for {3.

6.2. INTENSITY DISTRIBUTION NEAR THE CENTRAL CONE

It comes as a slight surprise to leam that there is somewhat of a shortage of central-cone radiation at the exact frequency of a harmonic even in the one-electron pattern. The angle-integrated flux per unit fractional bandwidth is actually twice as high at a frequency aJpeak = mIDI (O)(I-lImN) as it is at the exact harmonic frequency mIDI (0). This arises because the exact harmonic intensity on-axis can only receive contributions from the sinc functions in Eq. (22) centered on directions

341

at higher angles, whereas, the hollow cone of frequency OJpeak can receive contributions from beams at both higher and lower angles. Thus, there is a peak of intensity on the axis at mOJI (0) with an approximately Gaussian angular distribution, but a decidedly non-Gaussian, hollow-cone distribution at OJpeak. From a practical standpoint, OJpeak is better for flux while mOJI (0) is better for brightness. This is illustrated quantitatively for an ALS 5-cm-period undulator in Fig. 11.

lt is important to note that Eq. (33) is true for the frequency OJpeak. The corresponding equation at the exact harmonic frequency mOJI (0) (Kim, 1993) is

_ ~2A,mL '-mm _ -' _ Am (Jr - (Jr - - e - (JrUrr--

4n' 2L' 4n (50)

6.3. FAILURE OF THE FAR-RELDAPPROXIMATION (WALKER, 1988)

The far-field approximation is widely used to simplify the calculation of undulator spectral and angular distributions, allowing, in particular, their expression in closed form. It consists essentially of assuming that the observation direction is constant as the electron traverses the undulator, or that all parts of the undulator are at the same distance from the observer. However, there are many practical cases, including some at the ALS (which has especially long undulators), where the far-field approximation is not satisfied. To evaluate the effect, consider the situation depicted in Fig. 12. The observer angle changes from 81 to 82 as the electron traverses the undulator, and so, according to Eq. (12), the emission wavelength changes. The result is a "chirped" spectrum as shown in the figure. From Eq. (12), the change in wavelength LU is given by

(51)

From Fig. 12, we can also see that 81 = 8/(I+U2D) and 82 = 8/(l-U2D). Therefore, the spectral lines will be broadened by their own fractional width limN when 8 is given by

(52)

After so~uction, this leads to

(J~ a' {D(I_~). rf"L 4D2 (53)

If we take (J = (J; as a reasonable collection half angle, then the conclusion is that the spectral lines will be broadened significantly.

6.4. LIMITATIONS ON TWO-PHOTON EXPERIMENTS

There are two kinds of two-photon experiments that one might consider suited to undulator radiation. The first is a two-color experiment involving two coaxial undulators giving two different photon energies. This experiment might be imagined as a way to probe a short-lived

342

U5 3rd harmonic, Eo = 284 eV, tJ.E = 0

5X1017 5xl017

~ 4x 1 4x l017

;!.

0

'" 3x1 017 "0 01

~ "' (;j

2x1017 c % ~ a..

1x1 017

U5 3rd harmonic, Eo = 284 eV. tJ.E = -0.8 eV

5x 1017

~ 4x l017 en ~ 0 ~

Q 3x1 017 '" "0 01

E (;j

2x1017 (;j c 9 0 ~ 0-

1x l017

Figure 11 . One-electron intensity distributions near the axis for the ALS 5-cm-period undulator in the third harmonic. Curve (a) is for the exact harmonic energy (E) and curve (b) is for an energy £(1 - limN). which is the energy of the peak of the angle-integrated spectrum. ilE is the energy difference between (a) and (b). Note that the former has a peaked and the latter a hollow-cone shape.

343

F(w)

Figure 12. Illustration of the effect of the change in observer angle from one end of the undulator to the other and the resulting chirped spectrum, which becomes important when the far-field assumption is not satisfied.

intermediate state, but is not promising for the following reason. Considering that, for a single mode, we would have df2== 21C(J'j-, L1aYw == limN, and n(1 +K212)Fm(l() is approximately unity, then Eq. (27) shows that the number of radiated photons per incident electron is about a == 1/137. Therefore, the probability of getting two photons from two undulators is proportional to (1/137)2, which would give a very low rate.

In the second type of two-photon experiment, two nominally identical photons of energy E would do something that needed energy 2E. This experiment is much more promising because the probability of getting two photons in the same mode at the same times (from a single undulator) is equal to the degeneracy factor Ow, which, as discussed earlier, can be much larger than unity for some conditions. This type of experiment can be considered for samples witb sufficiently high interaction probability.

6.5. BENDING MAGNET BACKGROUND

An observer near an undulator axis will see radiation from both the upstream and downstream bending magnets. The nature of this radiation will vary from a spectrum characteristic of the bending magnet fringing fields at zero and small angles to that characteristic of the bending magnet full field at sufficiently large off-axis angles. As an example, we show in Fig. 13 the power density due to an ALS 3.65-cm undulator and that due to its upstream and downstream bending magnets as calculated by the POISSON magnetic field code. It is noteworthy that the two bending magnet beams are unequal and very much weaker than the undulator beam. This has important consequences for the operation of beam position monitors, although the situation is not as good as it seems because the monitors respond to photons in proportion to their photoelectron yield, not their energy. One can also see that the full bending magnet power density is not achieved until several milliradians off axis. Another feature with implications for beam position

344

1000~----------------------------------------.

100

-C\I

E C,) 10.0

~ -~ ·00 c: Q) "0 .... Q) ~ 1.0 o a..

0.1

'\

, Undulator total power

Central cone of fundamental

Downstream bend "'"

'\ Upstream bend

.'.-./ Full .' field

,.. value .... ..~

.. ." .. . .. ."

0.01 L-__ ----i ____ ~ ____ ~ ____ __'_. ____ __1_ ____ .L.._ ____ LJ

o 5 10 15 20 25 30 35 Horizontal off-axis distance (mm), 14 m from source

Figure 13. The power density distribution near the axis of an ALS 3.65-cm undulator showing the undulator total-power and central-cone distributions and the power density from the fringing fields of the upstream and downstream bending magnets using magnetic fields from POISSON.

345

stabilization is that the central cone is a narrow and relatively weak beam buried in a much wider and stronger power-density distribution; however, it is the broad power distribution that will be sensed by the beam position monitors.

6.6. IMPERFECfUNDULATORS: MEASUREMENT AND ANALYSIS OF DEFECTS

Until now, we have assumed that we were dealing with a perfect undulator. We now tum to assessing the effects of the inevitable imperfections of real undulators. The consequences of departures of the undulator magnetic field from its nominal form are illustrated qualitatively in Fig. 14. This figure shows a calculated electron trajectory for a realistic imperfect undulator field. Obviously, if one wants to obtain near-theoretical performance from the undulator, one must pay careful attention to the size of the field errors and their effects. In seeking to maintain good field quality, it is worth considering the consequences of failure. Electrons traversing even the most imperfect undulator still radiate, and the power must go somewhere. In the worst case, all coherent superposition of the one-electron signals from successive periods of the undulator is lost and the coherent sum is replaced by an incoherent one. In this case, each half-period of the undulator acts like a small bending magnet, and the resulting power output is equal to 2N times the output from each half-period. The spectrum then loses the undulator peaks and becomes smooth like the spectrum of a wiggler.

30

20

10

-10

-20

-30

Random Walk of Trajectory (uncorrected)

- 1-a envelope ...... undulator central cone

K=2 N=50 a= 0.005

o 10 20 30 40 50

Figure 14. Three sample orbits for a 50-period undulator with nns field errors of 0.5%. The orbit deviation is expressed in units of the amplitude of its ideal sinusoid.

346

The consequences of field errors fall into two main classes: (1) effects on the storage ring. and (2) effects on the radiated spectrum. We consider the first category in Section 6.7 and the second in Section 6.8. However. the prerequisite for any rational approach to these effects is an ability to measure the undulator fields accurately enough to compare the fields of real devices with their nominal values and the error tolerances derived from experience or calculation. The first two ALS undulators have been extensively measured. and the analysis of the measurements has been reported by Marks et al.. 1993B. We use the results of this work to illustrate the following material.

Like their counterparts in other laboratories. the ALS group has developed a magnet measurement facility for qualification of undulators (Marks et al.. 1993A). The measurement system consists of two primary elements. The first element is a moving stage with precise position measurement and control. This moving stage carries Hall probes capable of measuring Bx and By; it can map Bx and By throughout the three-dimensional region between the undulator poles with an accuracy of±O.5 Gauss. Bearing in mind that the undulator gap varies from 14 mm to 210 mm and that a single scan of the 4.5-m length of the undulator generates 2500 data points. one can see that a great many scans and a large quantity of data are involved in fully characterizing the undulator at a reasonable range of gaps. The second measurement system element is an integral coil used to measure the field integrals JBydz and JBxdz. where the z axis is the undulator axis. The coil is 550 x 1 cm2 in area and measures the field integrals with an accuracy of ±20 Gauss·cm.

Figure 15 shows an example of a spline fit to a data set. derived from a scan of the Hall probe measuring By as a function of z. This type of data can be analyzed using a variety of processing

(j) C/l ~ ro .9 cO

10000

5000

0

-5000

-10000

-200 -100 o Z(cm)

Figure 15. Measured By as a function of z for an ALS 5-cm-period undulator.

100 200

347

tools including tools to identify field peaks, truncate the data to eliminate end fields, least-squares fit the data to a set of harmonics, take the Fourier transform, half-period filter the data, calculate the optical phase errors, and calculate the expected radiation emission. We discuss some of these tools further below.

Given that the undulator structure is nominally a periodic function with a symmetry of the formj{z + Au/2) = -j{z), its field (without the non-periodic parts at the ends) should fit a cosine Fourier series with only odd harmonics:

Bh(Z)= LBmcos(mkuz+tf>m) , m=l, 3, 5, ... , m

(54)

where ku = 2n1Au. A nonlinear least-squares fit routine is used to fit Bh to the measured data with Bm and t/>m as fitting parameters. The rms value (Ie of the residual By-Bh between the fit and the measured data is then defined as the measure of the overall size of the field errors. It includes both local errors and global effects such as taper and sag.

Another interesting technique is the half-period filter, which is applied to the spline fit to the measured Bx or By data. This is defined, for example, by

(55)

It is implemented in the frequency domain by means of the convolution theorem. For any function that is exactly periodic with period Au and that has only odd harmonics, we can see that Fh(Z) will be zero. The output of the filter provides a measure of the field errors over a half­period range (i.e., local errors). Figure 16 shows the half-period-filtered output corresponding to half of a data set similar to the one in Fig. 15. One can see the small values representing local errors in the periodic part of the undulator and the large values representing the transition to a nonperiodic field at the end. Examination of the above equation also shows that the integral of Fh(Z) is equivalent to the integral over Bx or By, provided the limits of integration correspond to constant field regions. Therefore, this procedure also allows separation of the contributions to JBx d z, JBy dz into portions corresponding to the periodic and nonperiodic parts, a capability which is useful in correcting the field integrals. Integrals like JBy dz are important in considering the effect of the undulator on the electron beam as discussed in the following section.

6.7. IMPERFECT UNDULA TORS: EFFECT ON THE STORAGE RING

An undulator is generally short compared to a betatron wavelength, so the primary effect of the undulator magnetic fields on the electron beam is via their line integral through the device and its variation with horizontal and vertical position. We first note that in free space, B(x, y, z) satisfies the three-dimensional Laplace equation, a fact which follows from Maxwell's equations. Therefore, B(x, y) = iB(x, y, z)dz satisfies the two-dimensional Laplace equation as do its components, Bx and By. separately. The values of the line integrals of Bx and By can, therefore, be expressed as general solutions of Laplace's equation in polar coordinates (r, 9) as follows (Jackson, 1975):

m=~ m=~

JBydz= LamrmPm(cos9) and JBxdz= LbmrmPm(cos9) , (56) m=O m=O

348

u;-(/)

::J <1l .9 crt

3000

2500

2000

1500

1000

500

0

- 500

-1000

-1500 - 3000 -2000 -1000 o

Z(mm)

1000

Figure 16. Half-period filter output corresponding to a curve like that of Fig. 15.

2000 3000

where we have imposed the condition that lEy dz and lEx dz are finite at the origin. These are essentially Taylor-series representations of the functions lEy dz, fEx dz and are equivalent to two­dimensional multipole expansions of the integrated magnetic fields as shown in Table 2.

The radius of convergence of the series in Eq. (56) is equal to the magnetic half-gap. The area of validity includes the central region where the electron beam core is always located, but does not include some regions that lie outside the circle of convergence but still inside the dynamic aperture. The latter can contain scattered electrons that are not lost and are in the process of being returned back into the central region by radiation damping. If these particles get lost as a result of undulator magnetic field errors not represented by the multi pole expansion, then the beam lifetime will be reduced. The field integral variations that are represented by multi poles are described by the coefficients am and bm, which can be determined from the integral coil measurements. The size of the unwanted multipoles, as defined by these coefficients, can then be compared to a calculated tolerance value. Table 3 shows the tolerance values calculated for the ALS and the storage ring operational consequences expected for each type of unwanted multipole.

In general, it is difficult to correct the errors listed in Table 3 by means of adjustments to the accelerator optics because the errors change in value as the undulator gap is tuned. Consequently, the best strategy is to make the errors negligible for each undulator. Two exceptions to this are the horizontal and vertical dipole components for which there are both fixed corrections (by means of permanent magnet rotors) and tunable corrections (by means of the horizontal and vertical bump-coil systems). These correction mechanisms are already installed at the ALS and other third-generation storage rings to achieve beam stability. .

349

TABLE 2. Field integral multipole terms.

m Field Integral Tenn Multipole Character

0 J Bydz = ll() dipole

J Bydz=alx quadrupole

2 f Bydz = a; (2x2 - y2) sextupole

3 f Bydz = a; (2x 3 -3xy2) octupole

4 f Bydz = ag4 (5 x4 -30x2y2 +3) decapole

0 f Bxdz=/Jo skew dipole

1 f Bxdz=qx skew quadrupole

2 f Bxdz =;: (2x2 - y2) skew sextupole

3 f Bxdz = i (2x 3 - 3xy2) skew octupole

4 f Bxdz = b; (5x4 -30x2y2 +3) skew decapole

As an example of the magnitudes of the errors, we show in Table 3 the tolerances used at the ALS. The as-built undulators had values of the multipole tenns about two to three times larger than those in Table 2, so a correction system comprising several small, individually adjustable,

TABLE 3. Storage ring effects of undulator magnetic field integral errors.

Integrated Multipole Tenn

Horizontal (vertical) dipole

Quadrupole

Skew quadrupole

Sextupole

Skew sextupole

Octupole

Skew octupole

Tolerance Values Operational Consequences atALS

100 Gauss' em Vertical (horizontal) steering

100 Gauss Tune shift

100 Gauss Horizontal-to-vertical coupling, beam rotation

50 Gauss/em

20 Gauss / cm2

Amplitude-dependent tune shift; loss of dynamic aperture

Amplitude-dependent tune shift; loss of dynamic aperture

Higher-order tune resonances; loss of lifetime

Higher-order tune resonances; loss of lifetime

350

pennanent magnets (Hoyer, 1992) was installed at each end of the devices. By this means, the multipoles were brought within tolerance or within the range of adjustment in the case of the dipoles.

6.8. IMPERFECfUNDULATORS: EFFECfONTHESPECfRUM

The main purpose of an undulator is to deliver a spectrum as close as possible to theoretical. Thus, there is a need for a theoretical analysis of the errors that impair its ability to do this. Such an analysis has been provided by Kincaid, 1985. A primary conclusion of this study was that the perfonnance of the undulator source is degraded by random field errors and that the degradation increases like the square of the harmonic number. The peak value of the flux per unit solid angle (and hence also, via Eq. (27), the brightness) of the nth harmonic is degraded by a factor GL1 or F L1 as follows:

where

GL1 == e-30gp2

F L1 varies like (gp )-2

g~1

g~ 1

n

g=u2N3 , p=~ , 1+­

K2

(57)

and u is the nns field error. The regime g ~ 1 corresponds to a small degree of wandering of the orbit (Fig. 14) while g ~ 1 corresponds to a large one. Some of the conclusions of Kincaid's paper are summarized in Fig. 17, which shows the contours GL1 = 0.7 and FL1 = 0.7 on a log-log plot of g against p. The circular plotted points represent harmonics of actual undulators for which p and g are known. The fact that the points lie to the left of the contours shows that the predicted losses of intensity are less than 0.7. Note that all these examples are in the regime of small walking of the orbit, which implies a Gaussian dependence of the degradation factor on the size of the field errors.

The ALS group set a goal of achieving at least 70% of theoretical flux in the 5th harmonic, which, according to the above equation for GLI , implies that u must be less than 0.25%. There is some difficulty in finding a rigorous procedure for detennining a value for u from measured magnetic field data, but roughly speaking, we could identify it with ue. At the time the ALS undulators were started, the 0.25% value was about a factor of 2 beyond the state-of-the-art. Nevertheless, it was achieved, even at the minimum gap of 14 mm.

In view of the arguments following Eq. (16), it is also clear that another factor that can lead to degradation of the brightness is the electron beam angular spread resulting from its finite emittance. The emittance sets a limit to the quality of undulator fields beyond which further improvement to the undulator does not improve its perfonnance. We show an example of the effect of emittance in the following paragraph.

Once an undulator has been built, there is no longer a need to study it by means of a general theory. One can simply calculate the radiation output using the actual measured field of the device. This can be accomplished more easily than hitherto using the simplified radiation equation derived by Wang, 1993B [Eq. (21)]. As an example of the procedure, we show in

351

102

Performance F.d = 0.7 >30% loss

101 U5.0 0.25%

• • •

U10.0 0.65%

10°

SLVI 0.5% • • M ~

N b II 0)

10-1

SLX 0.35% • •

10-2 TOKO.1% • •

Gt. = 0.7

10-3

10-2 10-1

Figure 17. Curves of constant G.d and F.d on a plot of p against g as explained in the text. The circles represent the harmonics of various real undulators as follows: TOK means the transverse optical klystron undulator at Brookhaven, BL X means the Beamline 10 wiggler at Stanford, BL VI means the 54-pole wiggler at Stanford, UlO.O means a putative lO-cm-period undulator at the ALS, and U5.0 means an actual 5-cm-period undulator at the ALS.

352

Fig. 18 the flux per unit solid angle of an ALS 5-cm-period undulator operated at a magnetic gap of 23 mm (K = 2.13) for three cases: (1) ideal field and zero emittance, (2) actual field and zero emittance, and (3) actual field and actual emittance. The spectra are taken from Wang, 1993A. It is noteworthy that all the harmonics are reduced by both field error and emittance effects and that the size of the reduction increases rapidly with harmonic number as predicted by the theory. The fust, third, and fifth harmonics are all still large enough to be useful, consistent with the goals of the ALS undulator design and manufacturing program.

7. Undulator Performance Trade-OtIs: Discussion and Conclusions

7.1. UNDULATOR DESIGN STRATEGIES

To give an overview of the material presented so far, we consider some of the scaling laws that prevail under various conditions. Equation (25) shows that the on-axis intensity scales like N2,

while Eq. (27) shows that the central-cone flux scales like N. The brightness scaling, given by Eq. (33), requires a closer examination. The value and scaling of the l: 's depend on whether the beam size and angle are dominated by diffraction or by the electron beam dimensions. Four cases can be distinguished (Kim, 1989):

(1) O'x, O'y < < O'r and 0';, 0' Y < < 0'; (size and angle are both diffraction limited)

(58)

(2) O'x> O'y »O'r and 0';, 0' y « 0' ; (size is electron-beam limited and angle is diffraction limited)

(59)

(3) O'x, O'y « O'r and 0';, O'y» 0'; (not realistic in cases of interest to us), and

(4) O'x,O'y »O'r and 0';, O'y» 0'; (size and angle are both electron beam limited)

'!E m (60)

From these cases, we see that the brightness scales like '!Em (which scales like N) when the size and angle are either both diffraction limited or both electron-beam limited. On the other hand, if the size is electron-beam limited and the angle diffraction limited, then the brightness scales like N2. The latter is a common case of which there are several examples at the ALS.

The conclusion of this discussion for the undulator designer is to use the largest possible N for a given L (i.e., the smallest possible Au) consistent with achieving enough field to get to the longest wavelength desired. This implies that the gap has been reduced to the minimum consistent with storage ring operation. This same philosophy of using the minimum possible Au also maximizes the required K, which increases the total power output.

353

6x1017 (a)

5x1017

«

""'" e 4x1017 N "0 ~ .E ~ 0 3x1017

e Ul en c:

2x1017 0 0 .c a.

1x1017

o 200 400 600 800 1000 1200 1400 1600 1800 2000

Photon energy (eV)

4.5x1017 (b)

4x1017

« 3.5x1017

""'" e 3x1017 N

"0 ~ .E 2.5x1017 ~ 0

~ 2x1017 c: .8 1.5x1017 0 .c a.

1x1017

0.5x1017

0 I 1 1 ~ 1 J 1 o ~ ~ ~ B 1~1~1~1~1B~

Photon energy (eV)

Figure 18. On-axis flux per unit solid angle of an ALS 5-cm undulator at K = 2.13 (a) for the ideal field and zero emittance. (b) for the actual field and zero emittance. and (c) for the actual field and actual emittance.

354

3x1017 (c)

2.5x1017

« oo::t Q

2x1017 '" "C ~ E --~ 0

0 1.5x1017

--(/) en c: 0 (5 ..c:

1x1017 D..

0.5x1017

0 1 1 1 1 1 1 1 ,.\ ,J,. ,.I.

o 200 400 600 800 1000 1200 1400 1600 1800 2000

Photon energy (eV)

Figure 18. (continued).

As a final example to illustrate these principles, we show in Table 4 three ALS undulator schemes that provide 50-eV photons. Table 4 illustrates two important points: (1) it is possible to gain flux by producing low-energy photons at high K, but there is a significant price in higher power output, and (2) such unfavorable trade-offs generally happen when trying to generate photons at an energy well below that for which the storage ring is optimized.

TABLE 4. Various designs to generate 50-e V photons using an ALS undulator.

Existing 8-cm Existing 5-cm Bending Magnet Characteristic Low-K Device Device Device for Comparison

Undulator period (cm) 36 8 5 Number of poles 12 55 89 K value at 50 eV 1.2 3.0 3.9 Field at 50 e V (T) 0.036 0.40 0.83 Useful flux (usual units) 4.5 x 1014 3 x 1015 5 x 1015 9.2 x 1013

(per 10 rnr) Brightness (usual units) 5.5 x 1016 5 X 1017 9 X 1017 2 X 1014

Unwanted power low high higher Wiggler Ecrit (eV) not a good 600 1250

measure

355

Acknowledgements

The authors wish to acknowledge valuable conversations with K.-J. Kim and S. Marks. This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, of the U.S. Department of Energy, under contract DE-AC03-76SFOOO98.

Appendix 1

We want to prove that the K2 that should be used in Eq. (12) is LK~. We start with Eq. (10). m

To obtain f3z, we use Eq. (3) for f3 so f3z=~f32-f3; =~I-1/r2-f3; and Eq. (9) for f3x

(meluding ']1 the hrumoni,,) '0 p; = :' ( ~ Km 'in mk", r We then expond ,m'mk." = (1 -

cos2mkuz)12 and take the average over z. All the cosine and cross terms of the };2 then vanish,

and we are left with lii = ~ L K~. Substituting this in the square root and expanding by the 2r m

binomial theorem (lIr2« 1), we finally get liz =1-~(1+.!. LK~). When this is inserted 2r 2 m

in Eq. (10), we obtain Eq. (12), provided that we define K = ~~K~ , which completes the

proof. The equation for Be.trfollows from Eq. (13).

Appendix2 .

The mutual intensity J 12 is a measure of the spatial coherence of light at two transversely separated points, 1 and 2. It is assumed that the points are illuminated by quasi-monochromatic light and that the optical paths to the two points differ by an amount that is much less than the coherence length of the light. This implies that, in using J12, it is understood that there is full temporal coherence and only spatial coherence is to be considered. The mutual intensity J12 is then equal to the correlation function of the optical fields at the points 1 and 2.

Appendix 3

In Green, 1977, the author represents the finite-emittance, depth-broadened source as a function

ef(a, y), where a = y/(jy and tan Y = (j L 12, where L is the length of the source and (j is the rms (jy

opening angle of the radiation emission. The value of ef(a, Y) is defined by

Y {a2 } dt ef(a, Y) = f exp - -cos2 t --. 2 cost

o

+00

It is normalized by f ef(a, Y)da = -v'2i tan Y, has the

356

value ef(O, y) = -In . at a = 0, and asymptotically approaches Yexp( -a 12) for Y < 0.1. 1 (l+SinY) . 2 2 I-smY

Graphs and numerical tables are provided by Green for a comprehensive range of values.

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MIRRORS FOR SYNCHROTRON-RADIATION BEAMLINES

MALCOLM R. HOWELLS Advanced Light Source Lawrence Berkeley Laboratory Berkeley CA 94720, USA

ABSTRACT. We consider the role of mirrors in synchrotron-radiation beamlines and discuss the optical considerations involved in their design. We discuss toroidal, spherical, elliptical, and paraboloidal mirrors in detail with particular attention to their aberration properties. We give a treatment of the sine condition and describe its role in correcting the coma of axisymmetric systems. We show in detail how coma is inevitable in single-reflection, grazing-incidence systems but correctable in two-reflection systems such as those of the Wolter type. In an appendix, we give the theory of point aberrations of reflectors of a general shape and discuss the question of correct naming of aberrations. In particular, a strict definition of coma is required if attempts at correction are to be based on the sine condition.

1. Introduction

Mirrors are the standard way to manipulate radiation beams at synchrotron-radiation facilities. They are almost always used at grazing incidence, and with the increased sophistication of optical designs and increased power in the radiation beams, they have become an important and challenging branch of optical technology. It is becoming well known that there are important limits to what it is possible to manufacture, so that mirror technology is one of the major limits to the performance of a beamline.

In this paper, we consider the functions of mirrors, the shapes one can conceive, and the standard way to initiate the process of design based on a paraxial analysis. We consider quantitative geometrical descriptions of the important mirror shapes, both in an exact way and by using series expansions. The latter both simplify calculations and make it possible to identify the terms involved in approximating a surface with particular aberrations of the radiation beams reflected from the surface. We study the "sine condition" as a way to understand some of the special limits that apply to single grazing-incidence reflectors and to see the benefits of double-reflection schemes such as the Wolter telescope. Although the benefits of such aberration-canceling schemes are not normally necessaty for beamline mirror systems, the same principles apply to grating systems for which aberration canceling can have practical importance. In order to study the aberrations quantitatively, we give a table of some of the terms of the optical-path function expansion in the Appendix, together with a discussion of the naming conventions of the aberrations for systems with and wi~hout a symmetry axis. The names must be strictly defined if rules for aberration correction, such as the sine condition, are to be successfully used.

359

A.S. Schlachter and F.J. Wuillewnier (eds), New Directions in Research with Third-Generation Soft X-Ray Synchrotron Radiation Sources, 359-385. © 1994 Kluwer Academic Publishers.

360

2. Mirrors in Synchrotron-Radiation Beamlines

We show in Table 1 the main functions of mirrors in beamlines and address the important question of whether the optical quality of the mirror limits the spectral resolution of the beamline. Horizontal deflection of the beam is necessary to achieve separation of branch lines derived from the same bending-magnet port. At higher photon energies (smaller grazing angles), such separation can be difficult to achieve, and one must resort to a separation in the distance of the experiments from the source. On the other hand, undulator beams are difficult to split between simultaneous users and are often time-shared. The switching of the beam between the users then involves a mirror or mirrors that can be moved under computer control. This is easier to do nearer the source, such as in switching between monochromators. After the exit slit of a monochromator, the separation achievable with grazing reflections is too small for large experiments like surface-science stations, and the only recourse is to place the experiments on a rotating platform centered at a beamline bellows to act as a "knuckle."

The use of mirrors as energy filters has been practiced since the earliest days of synchrotron radiation-research and has been analyzed, for example, by Rehn (1985). Roughly speaki~ the mirror reflects efficiently only for grazing angles smaller than the critical angle '1/28, where (j is the difference from unity of the real part of the refractive index of the mirror coating. The cutoff energy varies by about a factor of 2-2.5 between the

TABLE 1. Functions of beamline mirrors.

Type of mirror Resolution Function (typical) determining? Applications

Deflection High power, often No Separation of branches from a flat port

Energy Filtration Any No Low-pass energy filter, order suppression

Power Absorption High power, often No Rejection of power at flat unwanted ·photon energies

Condensation Spherical, toroidal, or No Source to entrance slit (high ellipsoidal power), exit slit to sample (low

power)

Collimation High power, Yes Plane grating and crystal spherical, toroidal, or monochromators, e.g., to paraboloidal match the beam angular

spread to the rocking curve width of a given crystal

Microprobe Low power, Sets spatial Microscopy, fluorescence Formation Kirkpatrick -Baez resolution microanalysis

pair, or ellipsoid

Focusing Low power, spherical, Yes for grating, Plane grating and crystal toroidal, or no for crystal monochromators paraboloidal

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most and the least reflective coatings. The ability of mirrors to carry out this crude filtering is often important for suppressing unwanted high-order diffracted beams from grating monochromators.

In cases where significant power is carried by x rays with energy above the intended range of operation, it is usual to absorb such x rays in the first mirror at grazing incidence. This approach allows subsequent components, which may have a higher grazing angle and may be resolution-determining (with correspondingly tighter tolerances), to operate at lower power load.

Many beamlines have condensing mirrors that either deliver the beam from the source into the entrance slit of a monochromator or relay the beam from the exit slit to the sample. The first type of mirror is normally high power and the second, low power. The surface tolerances are set by the sizes of the object and image in each case, while failure to meet the tolerances leads to a loss of flux and/or spatial resolution but not to a loss of spectral resolution.

The natural vertical opening angle of the synchrotron radiation from the bending magnets of a typical high-energy storage ring is about 5-10 times larger than the rocking curve width of commonly used crystals. Thus there is a motivation to collimate the radiation. Plane diffraction gratings can also profit from a collimation mirror that leads to a wavelength­independent focal position at infinity. Such mirrors are resolution determining.

Finally mirrors may be used for focusing. When they focus the light from a grating to an exit slit, they affect the spectral resolution. When they are used as concentrators for microscopy or microanalysis, they determine the spatial resolution. The most critical cases of the latter occur when the grazing-incidence "forgiveness factor" is not in effect and the (often-multilayer-coated) mirrors are used at normal incidence.

3. Paraxial Design: Coddington's Equations

The first step in designing a beamline is the same as for any optical system: paraxial design. This is a preliminary design process that only considers behavior that is second order in the optical path function, that is, focusing effects in the tangential and (separately) in the sagittal plane. Third- and higher-order effects (aberrations) are neglected at this stage. One manifestation of this level of approximation is that the curvatures at every point of an optical surface are approximated by the two principal curvatures at the mirror center (pole). That is, surfaces that have a curvature that varies with position, such as ellipsoids, are effectively approximated as toroids. Even though we shall consider aberrations and the exact shapes of the surfaces later, the paraxial properties, focal lengths, image positions, and so on that we calculate during the paraxial analysis will remain valid. The equations that govern the focusing behavior of a toroid, known as Coddington's equations, thus assume a special importance in beamline design. They are

1 1 2 -+-=---r r' Rcosa

1 1 2cosa -+-=---r r' p

(1)

where r is the object distance, r' the image distance, R the major axis, and p the minor axis of the toroid. Thus we see that the tangential and sagittal focal lengths, It and Is are given by

1 It =-Rcosa 2

I =.!.--L.. s 2 cosa '

(2)

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and if one desires that!t =!s (stigmatic image), then evidently r/R = cos2a. These equations can be proved by a geometrical argument (Longhurst, 1962) or by setting F20c) and F020

equal to zero (see Eq. Al and Table AI).

4. Geometrical Descriptions of Mirror Surfaces

Another useful way to approximate a mirror surface is to express it as a two-dimensional Maclaurin's series with respect to a coordinate system whose y-z plane is the tangent plane at the mirror pole and whose x axis is the normal at the pole. That is

x= I,aijyizj . ij

The coefficients in this series, the ail s, are used as descriptors of the mirror surface shape in the aberration series given in the Appendix, which is therefore universal for all mirror shapes. These coefficients are associated with particular point aberrations, and a study of the Maclaurin's series can provide some insight into the nature of the distortions to be expected in wave fronts reflected from a mirror. We give in Tables 2 and 3, the aij's for the ellipsoid of revolution and the bicycle-tire toroid which is an arc of the minor radius (p ) rotated about a point at distance R. From these, one can get the ail s of the paraboloid of revolution and

TABLE 2. Ellipsoid ofrevolution* Oi/S.

rx 0 1 2 3 4

0 0 0 r+r' o C 4rr'cos2a) 4rr'cosa aJ sm2a+ 2 2

(r + r')

1 0 0 SinaC I) 0 * a02-- ---2 r r'

2 cosa(r + r') o 3 . 2 [3 4rr' ( cot2a)] 0 *

4rr' ai12sm 2a - - --- 1---2 (r+ r,)2 2

3 SinaC I) 0 * 0 *

a20-2- ;-7 4

[5Sin2ac Iy I] 0 * 0 * a20 --- --- +-

16 r r' 4rr'

*Paraboloid of revolution Oijs are obtained from those of the ellipse by setting r'~ co for a collimating parabola or r ~ co for a focusing one, the rays always traveling to the right.

TABLE 3. Bicycle-tire toroid* aij's

Ix 0 1 2 3 4

0 0 0 1 0 1 -2p 8p3

1 0 0 0 0 *

2 1 0 1 0 * -2R 4R2p

3 0 0 * 0 *

4 1 0 * 0 * 8R3

*Apple core toroId aij's are the same as those gIven In thIS table except for the replacement R2 p -+ Rp 2 in a22.

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a point at distance R. From these, one can get the aij's of the paraboloid of revolution and the apple-core toroid (an arc of radius R rotated about its chord at maximum distance p ) as explained in the table footnotes. Those of the corresponding cylinders are obtained by setting j = 0 and those of the sphere by setting R = p.

As an example of how useful this representation is, consider the case of an elliptical cylinder mirror such as one might get by bending techniques. The height x of the surface does not depend on z, so we may write it

(3)

Hence the curvature is given by

(4)

Suppose the segment of the ellipse is chosen to demagnify the object (Fig. 1). We will then have r'< r, so that a30 will be negative and the linear term in Eq. (4) will represent a curvature that diminishes with increasing y as it should (Fig. 1). In order to produce an unaberrated image of the axial object point, that is, a circular wave front in the image space, the ellipse needs an a30 term of the proper value as given in Table 2. If the mirror were circular instead of elliptical, a30 would be zero (according to Table 3), which is larger than the correct (ellipse) value and according to Eq. (3) would cause the wave front to lead the reference sphere when y is positive and lag behind it when y is negative. Alternatively one can also see from Eq. (4) that the change in a30 represents an error in the mirror curvature (and hence a similar error in the reflected wave front) that varies linearly with the position, y, in the aperture. This type of twisting of the wave front (which we call aperture defect) moves the outgoing ray in the same direction for all points in the mirror aperture, so that the effect on the image of a point or a line is to produce an asymmetry of the delivered image.

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y

x

~-----------ae----------~

I+-------a------------+-I

Figure 1. Ellipse geometry and notation.

Similar arguments can elucidate other types of image defect due to other departures of the mirror from the ideal ellipsoid of revolution. It is important to recognize that incorrect aij's lead to point aberrations (that is, those that do not depend on the z coordinate). Such aberrations can, in principle, always be corrected by altering the shape of the single reflecting surface we are discussing, while those that do depend on z cannot.

5. Toroidal Mirrors

The surface of the bicycle-tire toroid is the easiest aspheric one to fabricate with good figure and finish because it is possible to move the lap in a pseudo-random motion while still maintaining contact with the mirror surface at all points. Thus for toroids (but not for conics), one can use a large lap, which is a great advantage. This leads us to investigate the image quality that can be achieved by using a bicycle-tire toroid mirror, which from now on we will simply call a toroid.

The steep sagittal curvature of the grazing-incidence toroid typically leads to a curvature of the tangential line image, which is the one we would normally like to use to deliver light into a monochromator entrance slit. We will start by evaluating this effect. We are looking for a "L\y' = kL\z'2 " type of relationship in the image plane [where (L\y', L\z') are the coordinates of the ray intersection point relative to the Gaussian image point as origin]. We know from applying Eq. (A5) to the astigmatism term (which dominates in determining L\z') that

&'=lr'(S+S') , (5)

so we need those terms that give a L\y' proportional to L\z'2, after we take the derivative with respect to w and substitute for I from Eq. (5). There are three such terms as discussed by Welford (1965): F102, F111, and F120. Thus,

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(6)

Note that none of these terms depends on z (only on z'), so that this is stilI a point aberration. We consider the case in which the source is a point in the symmetry plane (z = 0) and there is some astigmatism. This leads to a type of line curvature named astigmatic curvature by Beutler (1945). There is another type that arises when the source has a finite extent in the z direction (for example, a slit). The latter type comes from F102, with z '" 0 and was named enveloping curvature by Beutler. In practice, both may be present and would be combined as shown in Fig. 2 (Welford, 1965). Returning to Eqs. (5) and (6) and the Appendix, and taking F102 and F111 from (Noda, 1974), we now have

L1y/ = z,2 tana [§.._ S' + 2(S+S') (s+s'f] c 2(S+S'fr' r r' r' (7)

Using the above equation for Is, this becomes

) (a)

(b)

(c)

Figure 2. The two types of line curvature. (a) shows the one due to the finite length of the entrance slit, called "enveloping curvature" by Beutler, and (b) shows that of the astigmatic focal line due a point source at the entrance slit, called "astigmatic curvature" by Beutler. In practice the two are combined into a shape formed by displacing either curve along the other as shown in (c). The two effects can have either the same or opposite signs.

366

~ , = 12 tan a [3M + 1- 2r'] Ylc 4fs fs'

(8)

where M (= rlr' ) is the magnification. This a useful equation for calculating the depth (sag) of the curved line image in the general case where some astigmatism is present (is "# ft). An important special case is the stigmatic image (is =ft). Usingfs= (lIr+lIr') -I, Eq. (8) becomes

(9)

It is noteworthy that the undesirable broadening Liy'lc, passes through zero for particular choices of the conjugates in both Eq. (8) and Eq. (9). When there is a stigmatic image, the line-curvature aberration is evidently zero for unity magnification. This is sensible intuitively since the mirror surface needed for aberration less imaging of a point at unity magnification is an ellipsoid with the mirror pole in the symmetric position in the Y-Z plane (Fig. 1). Now the curvatures of an ellipsoidal mirror vary with position on the surface but take stationary (minimum) values in the Y-Z plane. Therefore, a toroid, which has two constant curvatures, is a better approximation there than elsewhere. The advantage of unity magnification goes further than eliminating line curvature. It also eliminates the aperture defect (F 300

aberration), as we discuss below. The good-quality image provided by a toroid at unity magnification has several practical applications in beamline design.

To see the usefulness of Eq. (8) let us define separate magnifications and image distances in the tangential and sagittal planes. Thus M t = r't1r and Ms = r'slr. Equation (8) now tells us that the curvature of the tangential focal line will be zero provided that

(10)

so that there is always an Ms that gives zero line curvature. This equation is potentially useful also. It is also significant that the line curvature has opposite signs on opposite sides of the magnification value where it passes through zero.

We can derive a useful rule of thumb from Eq. (9) that helps in thinking about possibilities for using toroids in practical situations. Suppose that the radii are chosen for the image to be stigmatic, and the source point is at infinity (M = 0). This is the worst case for using a toroid, being the furthest from unity magnification. At grazing incidence (sina =1), Eq. (9) reduces to

(11)

which is the sag of the toroid in the minor radius direction. This gives a useful "worst-case" feeling for the amount of line curvature to be expected and shows that, in general, the time when a toroid will work best is when it is acting as a weak lens.

We tum now to the next-largest toroid aberration after the line curvature, namely, aperture defect. The ray aberration LiY'300 is given by

(12)

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which leads, in the general case to

(13)

Specializing to the case of a toroid for which a30 = 0 and a20 = 1I2R and assuming we are at the tangential focus so that we can apply Coddington's equations, this reduces to

(14)

or in the common case of grazing incidence (sina =1) and M«l, we get another useful rule of thumb

(15)

which applies equally to spherical and toroidal mirrors since the sagittal properties are not involved. If w is the mirror half-width, then this equation gives the aberration of the marginal ray and is thus a rather pessimistic estimator of image degradation. One can see, for example, that half the rays have aberrations less than one-quarter that of the marginal ray. The aperture defect is the dominant aberration in spherical condensing mirrors such as one would use on an undulator beamline with a monochromator that has an entrance slit. As we noted above, it vanishes at unity magnification or, more generally, it is zero on the Rowland circle.

6. Spherical Mirrors: Kirkpatrick-Baez Systems

The theory given so far for toroidal mirrors applies to spherical ones in the special case p = R. In view of this, a stigmatic image is not obtainable using a spherical mirror unless a = O. In fact, when spherical mirrors are used at extreme grazing angles (a few degrees or less), the astigmatism is essentially complete. That is, the mirror achieves almost no focusing in the sagittal plane and the rays continue to diverge (or converge) at the same angle as before .. At first sight, this appears to be a disadvantage, but in fact it is very useful because one can use a second spherical mirror focusing in the sagittal plane of the first to achieve a focused image in two dimensions as shown in Fig. 3. Such a scheme has less aberration than a single stigmatic toroid at the same grazing angle and is easier to make with good tolerances. It was used in a magnifying microscope configuration in the 1940s and 1950s (Kirkpatrick, 1948), but is now more often used in a demagnifying geometry to form a microprobe (Underwood, 1988) or to condense a beam into a monochromator entrance slit.

Just as toroids are easier than conics to fabricate, so spheres are easier than toroids. The easiest surface of all is a sphere with a loose tolerance on the radius, which is in many ways even easier than a flat. From a fabrication point of view, a flat is like a sphere with a particular value of the radius. The use of spherical optics is now established as the way to get the best figure and finish accuracy. It is often the only way to get optics with the tolerances required to deliver x rays at modern synchrotron radiation facilities without degrading the optical quality of the beam.

The lowest-order aberration is a familiar one from grating theory where it is known that the image of an erect object is formed in a plane steeply inclined to the outgoing principal ray.

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/ /

/

/ /

" Object

/ /

/ /

/' /'

I"

/

Image Jt /

/ /

/

Figure 3. The layout of the two spherical mirrors in the Kirkpatrick-Baez microscope with their tangential planes perpendicular.

One example of such an inclined focal plane is the Rowland circle. This defect, known as obliquity of field, is not represented in the listing in the Appendix because it is a purely in­plane field aberration and there is no field coordinate in the symmetry plane in the Noda description. Nonetheless, the behavior can still be described by considering the focusing condition F200 = 0 and noting that the tangent of the angle e between the focal plane and the principle ray is given by

tan e = r,aal = tan aG ar'r M +1

where aG is the grazing angle of incidence. The angle e is seen to be always smaller than the grazing angle: obviously a very unfavorable condition for imaging. The difficulty in avoiding the loss of performance caused by this defect was partly responsible for the decline in popularity of the Kirkpatrick-Baez microscope as a device for imaging extended object fields. However, if one only wishes to image a small distant object such as a slit or the synchrotron source, then the defect becomes tolerable and the spherical mirror is very useful. Without the steep curvature of the toroid we now have p large and therefore Is large by Eq. (2) and the line image becomes essentially straight as one would expect [see Eq. (8)] (Hogrefe, et aI., 1986). The largest aberration for long-radius spherical mirrors is therefore the aperture defect, which we have already discussed and which is typically the main limit to performance for the spherical condensing mirrors that are now quite widely used. Another limit is the difficulty of making mirrors larger than about one meter long. Current capability does extend up to 1.5 meters but with some worsening in cost, weight, and tolerances for

369

figure and finish. The consequence of this size limit is that the horizontal collection angles of Kirkpatrick-Baez systems used on bending magnets are rather severely restricted when the photon energy is above a few-hundred eV. However, such systems are well suited to use with undulator beams.

7. Ellipse- and Parabola-Shaped Mirrors

Mirrors shaped as elliptical cylinders and as ellipsoids of revolution are both of interest and have been used on beamlines. For example, both have been used as focusing mirrors for the SX700 plane-grating monochromator (Petersen, 1982; Nyholm, 1986). They are obvious choices for many applications as the two-dimensional and three-dimensional surfaces of exact point-to-point imaging. Paraboloidal cylinders and paraboloids of revolution also have obvious applications and, as we have seen, can often be regarded as a special case of an ellipse with one conjugate equal to infinity. However, it often turns out that these cases are less useful than they might appear. Firstly, the ideal imaging property only applies to the axial object point, and other points are imaged poorly. This is not a fatal disadvantage, and we discuss it further in the section on the sine condition. Secondly, if one puts a full-size lap in contact with any of these surfaces, then the only possible motion of the lap relative to the surface without losing contact is a linear motion in a single direction for the cylinders and rotation in a single direction for the surfaces of revolution. This is not sufficient for good polishing, so one must have recourse to zone polishing using a small or a flexible lap. Such an approach gives much worse errors in figure and finish and with greater effort and cost than using a large lap. The result is that it is hard to get good-quality mirrors in this category. The ellipsoidal mirrors in the SX700s have always been the limiting component of those systems, and their manufacturing tolerances determined the achievable spectral resolution. The plane-grating monochromators at the National Synchrotron Light Source were similarly limited in resolution by the fabrication tolerances of their parabolic mirrors. The most promising strategy for obtaining an ellipse of high accuracy, in this author's opinion, is to use bending, which is only applicable for an elliptical cylinder but does allow the surface to be manufactured as a flat. Diamond turning is an acceptable way to generate the surfaces of revolution, but the polishing problem described above still remains.

First consider an elliptical mirror whose action is defined by the object and image conjugates rand r' and the included angle 2a (see Fig. 1). The equation of the ellipse is X2/a2 + y2/b2 = 1 where a and b are the semi-major and semi-minor axes, respectively. The ellipse parameters a, b and the eccentricity e can be expressed in terms of the user-specified quantities r, r' and a by means of the focus-directrix definition of the shape of the ellipse and the geometry of Fig. 2:

2a = r+r' (2ae)2 = r2 +r'2 -2rr'cos2a

b2 =a2(I-e2) .

The coordinates of the pole of the mirror are

~ Xo =±a~l-tt

1': _ rr'sin2a 0- 2ae '

(16)

(17)

370

where the square root is +, 0, or -, depending on whether r is greater than, equal to, or less than r'. The tangential radius of curvature Rp at the pole of the mirror is given by

(rr,)3/2 Rp=--- ,

ab

and the angle 0 between the tangent at the mirror pole and the x axis is given by

(18)

(19)

For a parabola, defined by the focal length r and included angle 2a, the equation is y2 = 4aoX, where ao is the semi-latus rectum. The latter is given by

2 aO = rcas a

while the pole of the mirror is the point

xo =ao tan 2 a Yo =2aotana

Rp is given by

R=~ P cos3 a '

(20)

(21)

(22)

and the angle 0 between the tangent at the mirror pole and the x axis is cot -1 a. Rp is useful for paraxial design, and 0 for making coordinate transforms between the X-Y and x-y systems.

The applications of conic mirrors that one encounters in synchrotron practice are usually quite unsophisticated, and one has little need to understand the geometry in a serious way. For applications in which a high-resolution image is required, as opposed to reproduction of a simple shape, this situation changes and one needs to understand the behavior of the wave fronts in a more complete way. In such cases, the reader is referred to one of the treatments in the literature that deal with conic and similar mirrors at a deeper level (Brueggemann, 1968; Cornbleet, 1984; Korsch, 1991).

8. The Sine Condition and Coma in Axisymmetric Grazing-Incidence Mirrors

8.1. GENERAL ARGUMENTS

We turn now to the role of the sine condition in determining the aberrations of grazing incidence systems with an axis of symmetry. The sine condition (Abbe, 1879; Welford, 1962, 1976) states that for all rays one must have

sin!/> !/>p --=-sin!/>' !/>~

(23)

371

where t/J and t/J' are the angles of the inward and outward rays to the symmetry axis and t/Jp and t/J'p are the same thing in the paraxial regime. Satisfying the condition guarantees that a system free of spherical aberration is also free of coma. The coma involved in this theorem is strictly the aberration of the axisymmetric system, which depends linearly on the field angle measured from the symmetry axis and is defined, for example, by Born and Wolf (1980). As pointed out by Underwood (1992), the aberrations of grazing-incidence systems with only a plane of symmetry (which have been loosely called coma by some workers) cannot be corrected by obeying the sine condition or any derivative of it. The proper naming of aberrations is discussed further in the Appendix. Here we only emphasize again that since the aberrations of greatest interest to us-line curvature and aperture defect-are not really coma, they are not corrected by obeying the sine condition.

Of course, there are grazing-incidence systems that do have a symmetry axis. Such systems are widely used in x-ray telescopes. The main ideas on which they are based were first described by Wolter in a landmark paper in 1952, long before the technology needed to implement the ideas effectively became available. We give a distillation of these ideas in what follows and discuss the possibility of applications to synchrotron-radiation systems.

The first important idea can be expressed as follows. Given object and image points 0 and I lying on the axis and distant u and v, respectively, from the center A of the system (Fig. 4), construct a new point, B, such that A and B are harmonic conjugates with respect to 0 and I. That means that B divides 01 externally in the same ratio that A divides it internally. Draw a sphere with AB as diameter. It can be shown that the sine condition is equivalent to the requirement that the locus of the intersection points P of the inward rays from 0 and the corresponding outward rays to I should be the sphere AB. This locus, called the

.",.,...-------- .........

/' " ./ .............

Optical / / '- Principal surface ...... "'\

~/ \ _pi \

\ \

II Ie I B I I I

: \ :: / ~----U----"I' I I I

~V~ I I \ I /

\ /

" / " / ...... ./ ...... ./

....... /'

........... --------"...; Figure 4. The geometrical form of the sine condition. Let A and B be harmonic conjugates with respect to the object and image points 0 and I. Draw a circle on AB as diameter. The requirement that the locus of the ray intersection point P be the circle AB is equivalent to the sine condition.

372

"Knieflache" by Wolter, is usually translated "principal surface" in English. From the point of view of grazing-incidence systems, which generally have small deflection angles, the plane through A normal to the axis will be a reasonable approximation to the sphere.

Suppose now that we have a segment of a single grazing-incidence ellipsoid of revolution with object and image points at the foci of the ellipse (Fig. 5a). We immediately see that the principal surface in this case is the surface of the mirror itself, and it is roughly perpendicular to the spherical surface that it would need to be to obey the sine condition (Underwood, 1978). It is obvious from this that any single-reflection grazing-incidence mirror violates the sine condition grossly and can never be coma-corrected. On the other hand, consider a different segment of the ellipse (Fig. 5b) again operating with the object and image points at the foci but this time in normal incidence. In this case, the principal surface is still the mirror itself, but now it closely approximates the spherical surface required to satisfy the sine condition. A high degree of coma correction is therefore expected. One way to achieve a similarly high degree of coma correction in a grazing-incidence system is to use a double reflection as in the Wolter telescope and microscope systems (Fig. 5c). Variants of these have been widely used in recent years, particularly as x-ray telescopes. In the next section, we give plausibility arguments showing how coma is produced and how the double-reflection principle can be used to correct it.

8.2. CALCULATION OF THE COMA CIRCLE

First consider a paraboloid of revolution being used to focus parallel light (Fig. 6) and consider the image formed by a thin ring of reflecting surface PIP2P3. For the rays that enter parallel to the axis, shown as thin lines, the image is perfect and the rays unite at the focus F. Thus, there is no spherical aberration. For the rays (shown dashed) that enter at a small angle 8 below the axis-parallel ones, the reflected rays from PI and P3 will both be deflected downward and will meet the focal plane at F2 and F3, which are both below F. In the projection, in which the ray is seen through P2, it appears undeflected and arrives at F2, which is above F. From the geometry of the figure we can see that

FF _ -r'sin8 1-

cos( lfI+ 8) FF _ -r'sin8

3-cos( lfI- 8)

FF2 = r'tan8 , (24)

where r' = PIF and the angle 2lf1 = PIFP3. If we now take 8 « lfI, FI and F3 become the same, and both have

while F2has

-r'8 y=-- ,

cos lfI

y = r'8 .

(25)

(26)

By considering other reflection points on the ring PIP2P3, we can see that, for the rays inclined to the axis by 8, the image will be a circle centered on F, and, from (25) and (26), the radius of the circle Rc must be given by

(a)

(b)

(c)

I I I I I ,

Actual principal surface

, '--Desired

principal

____ ----------------------s_u_rf_aces~ ,

:... !>

I ~ Principal surface "~ _'1--

373

Figure 5. (a) Single-reflection imaging geometry in which the principal surface is the mirror itself, which is roughly perpendicular to the principal surface (shown dashed) needed to satisfy the sine condition. (b) Another single-reflection geometry but now at near normal incidence; the mirror and the desired principal surface almost coincide and thus lead to much higher quality imaging. (c) Double-reflection Wolter system in which the principal surface is a much better approximation to the desired surface than in (a).

RC=T10(1+COS lfl ) . 2 cos lfI

(27)

This has been an approximate treatment (Howells, 1980). Wolter's more accurate one reveals that the radius of the circle is actually given by

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y

rsin ~ ------

I I I I I I I Ip

I I I I I I I I

2

IP3

-------,.-----~

x -----------r

~

------r-----~

Figure 6. Definition of the points F, Fl, Fl', F2, F3 used in explaining the origin of coma in the single­reflection, grazing-incidence mirror with a symmetry axis.

R = r'tan8(I+cOSljl) , c 2 cos ljI

(28)

and that the center is shifted off axis by s where

s = _ r'tan~ (1- COS'JI) . 2 cos'JI

(29)

Now consider a grazing-incidence ellipse in the same geometry but with the object point at a finite distance r from PlP2P3 and displaced a distance L1 from the axis so that 8 = ,1/r. Since ljI is small for grazing-incidence systems, we have (l + cosljl)/cosljl"" 2. Consequently, Rc "" (r'/r)L1 or L1 times the magnification. Moreover, (l - cos ljI)/cos ljI "" 0 so s "" O. The conclusion is that the image of an off-axis point is a circle centered on the axis with a radius such as to pass through the paraxial image point.

That this aberration is really Seidel coma can be seen from the fact that it varies linearly with the field angle 8 and also because each circular zone of the aperture contributes a circular aberration figure in the focal plane. Furthermore, each time the reflecting point runs once round the ring PlP2P3, the ray traces out the circle twice, which is again characteristic of coma (Welford, 1962).

It is also clear from the above treatment that the normal-incidence conic has the expected well-corrected coma. For this case, the value of ljIis roughly 1800 leading to (I + cosljl)/cosljI "" O.

375

With this insight into the dominant field-angle-dependent aberration of the single grazing­incidence mirror, we can understand the well-known "bow-tie" shaped image that used to be troublesome at synchrotron radiation facilities before low-emittance electron beams became widespread. Consider a unity-magnification mirror comprising a segment of an ellipsoid of revolution that subtends a maximum angle !2 at the axis. The image of an off-axis point will be an arc of the image circle of angle 2!2 passing through the paraxial image of the object point. When the object is extended in one direction much more than the other, as synchrotron radiation sources often are, the result is a "bow-tie" image as explained in Fig. 7. Note that this behavior also follows the theory closely for a unity-magnification toroid, which does have a type of symmetry axis with the center points of the object and image lying on it. However, as the magnification departs from unity, the behavior initially continues roughly similarly, but the symmetry of the system has been broken and the aberrations are no longer strictly Seidel coma. Although there is no sudden change, the behavior becomes significantly different for magnifications far from unity.

Actual "image"

ro-t --

I ------I -_ .-----------II II \I ,---------::::= \ --\ --,----

Gaussian image

c:::-------------- I -- --_ I -- / --

Figure 7. Explanation of how coma leads to a "bow-tie" image when a mirror with an axisymmetric (or nearly axisymmetric) shape is implemented over a segment of angular width Q = 16.5°.

376

9. Mirror Pairs in Wolter Geometry

We now return to the analysis of the focusing system shown in Fig. 6 with a view to elucidating the principle of the Wolter double-reflection system. Suppose that the ring PlP2P3 contains the joint between the two reflectors and imagine the rays to be reflected twice, just in front of and just behind the joint. Considering the same three rays, we find that P2F2 behaves as before, while the rays through Pl and P3 are now deflected upward by 8 instead of downward and arrive at Fl' as shown in Fig. 6. The rays through Pl and P3 now have

r'8 y=+-- ,

cos 1fI (30)

while the ray through P2 continues to have y "" r'8, so that the aberration circle, in the case of a Wolter system, has a radius Rw given by

RW=r'8(I-cOS lfl ) . 2 cos lfI

(31)

Equation (31) for Rw has the factor (l - cos lfI)/cOS lfI "" 0 for a grazing-incidence system, whereas the corresponding factor in Eq. (28) for Rc was approximately equal to 2. Thus the introduction of the double reflection brings about a large reduction in the aberration and allows one to design grazing-incidence systems with image quality similar to that of the normal-incidence conic. This is in accord with expectations based on the sine condition and the principal surface for the double-reflection system as shown in Fig. 5.

Although this discussion of the Wolter double-reflection principle contributes to our understanding of grazing-incidence mirror systems, it does not provide a blueprint for a new generation of improved beamline mirrors. The kind of high-quality image provided by a Wolter system is useful in imaging systems such as x-ray telescopes and may eventually be useful in x-ray microscopes. However, beamline mirrors are generally condensers and the fact that detail features within the object (which is usually the synchrotron source or a slit) are not accurately reproduced in the image is unimportant provided the overall size of the image is not significantly enlarged. The only useful improvement one would get by using a Wolter system as a condenser would be to eliminate the bow-tie effect, but as Fig. 7 shows, the gain in the flux that could pass through a slit would hardly make up for the losses of the mirror itself and would scarcely repay the investment needed for an extra aspheric mirror and all its accompanying systems.

One might ask whether there could be a role for the Wolter system as a collimating or focusing system for a monochromator. This is a different case and the higher-quality "lens" would have certain advantages. For example, even a perfect paraboloid mixes the horizontal and vertical divergences of the beam from a bending magnet, and this is a disadvantage in illuminating crystal monochromators which could probably be avoided by using the Wolter system. However, even when such advantages are taken into account, it is hard to imagine the high cost of a Wolter system of sufficient optical quality being considered acceptable for a beamline component. Moreover, the aberrations of focusing and collimating optics in grating monochromators must be combined with those of the grating, and the use of a better "lens" is an oversimplification of what is needed. The conclusion is that Wolter optics probably do not have a role in beamline systems for the time being. This is not to say that mirror pairs in general are not useful. Indeed, there are already several examples existing and proposed, and we can understand their operation in terms of the principles described above.

377

10. Mirror Pairs in General

There have been several studies carried out to identify the best way to combine the action of two mirrors or a mirror and a grating (Pouey, 1981, 1983; Aspnes, 1982; Hunter, 1981; Chrisp, 1983). Figure 8 shows several configurations involving two identical toroids, some of which tend to obey the sine condition (I/J' increases when I/J increases). Others radically disobey it (I/J' decreases when I/J increases). The dominant point aberrations F300 or F120 will not be improved by obeying the sine condition, but they can still be made to cancel. We will analyze this possibility in terms of wave-front errors. Even for the field aberrations, we expect a high degree of correction to be achieved only if there is an exact or approximate symmetry axis and the sine condition has an exact meaning. First consider the toroid in Fig. 8a and take F300 as an example. Based on the fact that the circular curvature of the mirror is too weak on the upstream side and too strong on the downstream (compared to the ideal paraboloid), we would expect the wave front emerging from the first mirror to be

(a)

.==-- -=-- ----=== •

~_""""'_ ... ;;;;;:;;;:;::oo_ •

Figure 8. Four possible ways of combining two toroidal mirrors in pairs: (a) and (c) violate the sine condition grossly because the angle to the axis decreases at the outgoing side when it increases on the ingoing side; (b) and (d) satisfy the sine condition in this sense. On the other hand, (a) and (c) are configured for approximate cancellation of point aberrations depending on an odd power of w (aperture defect and line curvature), while (b) and (d) are not.

378

twisted, as described earlier, by an amount proportional to w 3 and to be leading the reference sphere above the principal ray and lagging it below. If we apply the same argument to the second mirror, we see that wave-front errors of opposite sign will be introduced by the second reflection, provided its direction is as shown in Fig. 8a. Note that this is the direction that strongly disobeys the sine condition. Similar arguments show that the wave-front errors corresponding to line curvature (which also depends on an odd power of w) will also be self­canceling for the same mirror configuration (Fig. 8a). That the configuration in Fig. 8a indeed gives a better image of the axial point than the inverse configuration (shown in Fig 8b) can be verified by ray tracing for particular geometries. It is noteworthy that these types of arguments depend on the aberrations being small enough and the mirrors close enough together that each ray is reflected at aperture coordinates (w. I ) with substantially the same magnitudes (although maybe not the same signs) in each mirror. This is a condition that may not be met in real beamlines.

There are some similar comparisons between pairs of mirrors that do and do not obey the sine condition reported by Aspnes (1982) and Hunter (1981). One has to be careful in interpreting the results given by Aspnes because the toroids used had magnification values limited to unity or infinity. The choice of unity has special consequences because the point aberrations, line curvature and aperture defect, which would normally be dominant, happen to vanish at that value. As we have seen, the point and field aberrations must be considered separately and their relative importance depends on the source size and aperture size. Roughly speaking, the worst that can happen due to a field aberration is an extreme bow-tie effect that enlarges the short dimension of the synchrotron source or slit to equal the long one. On the other hand, the potential damage due to point aberrations is unlimited. For the small source sizes of modern storage rings, the best system design for a condenser will normally be one that corrects the point aberrations.

This discussion does not exhaust the possibilities of two-mirror systems. Readers wishing to explore further can refer to the paper by Namioka et al. (1983).

One can obviously design mirror pairs analogous to those in Fig. 8 using conics. Naturally these have no point aberrations, whichever way round they are, since they are the ideal point­imaging surfaces. However, it is important to recognize that manufacturing tolerances are likely to be larger for a conic than a toroid and will often more than outweigh the aberration advantage of conics.

Acknowledgements

This work was supported by the Director, Office of Energy Research, Office of Basic Energy Sciences, of the U.S. Department of Energy, under contract DE-AC03-76SF00098.

379

Appendix. The Optical Path Function Expansion for Gratings and Mirrors

According to Noda et al. (1974), the diffraction-grating optical, path function is given by

(AI)

We have included the 102 term explicitly, whereas Noda et al. included it implicitly in the 100 term. F is the actual path length AB, and wand 1 are defined in Fig. AI. In Fijk. i, j, and k are the powers of w, I, and z (or z' ) in the series expansion of F, each term of which represents a particular geometrical optical aberration. The terms go up to fourth order (i + j + k ~ 4) and are exactly those given by Noda et al. Those that have j + k = odd have been omitted, being equal to zero by symmetry, and terms that have i = j = a are omitted because they do not represent aberrations. For the study of mirrors, we can use a still more restricted subgroup of the expressions for the F ijk'S. First we reject the parts that represent the possibility that the grating is a holographic recording. This leads to

(A2)

z

Figure AI. Coordinate system used discuss the optical path function analysis (after Noda, 1974).

380

where Eijk is the expression given in Table AI. For a beamline mirror, it is often sufficient to consider a point source located in the symmetry plane of the mirror. This means z = z' = 0, k = O. We also know that from the law of reflection that f3 = -a, so that (A2) becomes

FijO = Eijo(a,r,O)+Eijo(-a,r',O) . (A3)

The function EijO(a, r, 0) is therefore tabulated in Table A-I, which uses the notations

cos2 a T = --- - 2a20 cos a

r

I S = - - 2a02 cos a

r (A4)

S ' and T ' are also defined and are the same as Sand T except that r is replaced by r'. The aij

parameters have been discussed in detail for some important surfaces in the main text. The terms in Eq. (AI) have some similarities with the terms in the aberration expansion of

an axisymmetric optical system. The aperture coordinates wand I, for example, are basically the same as the aperture coordinates of an axisymmetric system. However, since our system has only a plane of symmetry rather than an axis, the notion of field coordinates is completely different. In fact, the conventional field angle or field coordinate, which would be measured from the axis, no longer exists in the absence of an axis and so aberrations that depend on it, such as Seidel coma, have no analog in the present study. One should not make the mistake of regarding z or z' as directly analogous to the "axisymmetric" type of field

TABLE AI. Values of EijO(a,r,O).

IX 0 1 2 3 4

0 0 0 S 0 4a52 _S2 8~cosa ,

1 -sina 0 Ssina 0 * --- 2a12 cosa , 2 T 0 4l1zoa02 - TS - 2a12 sin 2a 0 * + ,

2Ssin2 a 4a22 cosa ,2

3 Tsina 0 * 0 * --- 2a30 cosa , 4 4aio - T2 - 4a30 sin 2a 0 * 0 *

, 4Tsin2a

8a40cosa + ,2

381

coordinate. To see the error of this, consider the case in which both types of system are specialized to two dimensions so that only their tangential planes are considered. The axisymmetrical system still has a field coordinate measured from the axis, but for the plane­symmetric system, the restriction to the tangential (symmetry) plane implies z = z' = 01 The same error might lead us to look in Eq. (AI) for terms like w 3z or wl2z to find the coma terms. But these terms are both symmetry-forbidden. There is no coma in the usual sense, and none of our coordinates w, I, or z (or z ') can be identified as a conventional field coordinate.

In light of the above, it is perhaps unfortunate that a tradition has grown up in the synchrotron-radiation community of giving traditional names to the aberration terms in Eq. (AI) based on only partial similarities to the corresponding aberrations of the axisymmetric system. For example, the F120 and F300 aberrations, which have the same dependence on the aperture coordinates as conventional coma, have been referred to as "coma," even though there could be no similarity in their dependence on the field coordinates, as explained above. The present author has been among those guilty of this. It has been pointed out by Underwood (1992) that this can lead to important errors in the treatment of coma (see the section on the sine condition). Therefore we propose to continue to use traditional names only when the analogy is fairly complete. The following system of names is proposed:

F 100 Grating equation F 102 Line curvature FOil Law of reflection in the sagittal plane F 200 Tangential defocus F020 Astigmatism (sagittal defocus) F 300 Aperture defect F 120 Line curvature F III Line curvature F 400 Spherical aberration F220 220 aberration F 040 Higher-order astigmatism F 202 202 aberration F022 022 aberration F03l 031 aberration F2ll 211 aberration

Of course, the treatment given so far does not enable one to calculate the most interesting thing, which is the extent of degradation (blurring) of the image that will be caused by any particular aberration. We now proceed to address that issue in the geometrical optics approximation. For each term of the aberration series, we calculate the ray aberrations (displacements from the paraxial image point), which in our notation are known as L1y' and .dz':

8y~.0 =~(OF) IJ cosa ow ijO '

(AS)

and

382

&1.;0 = r'( a F) a 1 ijO

The total ray aberration in the Liy' (Liz' ) direction is the algebraic sum of the LiY'ijO's (Liz'jjO's ). This means that partial or total cancellation of aberrations (known as "aberration balancing" when done deliberately) is possible and is sometimes useful.

(A6) ij ij

Equations (AS) are central to the geometrical theory of aberrations. Proofs are provided, for example, by Welford (1962) and Born and Wolf (1980). Neither of these authors includes the case of grazing incidence, which differs from the standard case of axial symmetry by the factor lIcosa in the first of Eqs. (AS). This arises simply from the coordinate change involved in rotating the exit pupil so that it can be perpendicular to the outgoing principal ray. For the slow systems involved in grazing-incidence optics, Eqs. (AS) give an excellent approximation. For very fast systems of f-l and faster, more complex expressions are needed as provided, for example, by McKinney and Palmer (1987).

As an example of the application of Eqs. (AS) and (A6), we show some ray traces in Fig. A3. The rays traced are shown in Fig. A2 and are done in double precision, so all aberrations up to very high order are accurately represented. To illustrate the action of Eqs. (AS) in determining the pattern of ray intersections in the receiving plane, we try to explain the general features of the ray traces in terms of the lowest-order aberrations. In the general case, we approximate Ax' and Liy' as

I I I

2 D 0 + X D 0 + X - D 0 + x

E D 0 + X D 0 + X 0 D 0 + X -c:: D 0 + X

0 0 D 0 + X :;:::: D 0 + IC

'en D 0 + IC

0 D 0 + IC

D- D 0 + IC D 0 + IC D 0 + IC

-2 D 0 + IC

I I I

-10 0 10 Position (em)

Figure A2. Layout of the 5 x 15 ray points and their plot labels used in the ray trace shown in Figure A3. This figure shows the pattern of intersections of the rays with the tangent plane at the pole of the toroidal mirror. The rays travel from left to right.

383

1.0 (a) (b)

o.

0.5

E .s E .s

c: 0.0 0

~ 0 a.

c: 0 .2 .~ 0 a.

-0.5 .... -1

-1.0 -2~~~~~~~~~~

-1.5 -1.0 -0.5 0.0 -2.0 -1.5 -1.0 -0.5 0.0 Position (mm) Position (mm)

(c)

5 5

E .s E .s

c: 0 0

~ 0 a.

c: 0 0

"" 'Cij 0 a.

-5 -5

-5 o 5 -0.3 -0.2 -0.1 0.0 Position (mm) Position (mm)

Figure A3. Ray traces of the images of a point source produced by a toroidal mirror under various conditions. The 5 x 15 pattern of incoming rays and their plot symbols are shown in Figure A2. The parameters of the system for the image shown in (a) are: r = 10 m, r' = 2 m, a = 88°, R = 95.55123612 m, p = 0.1163316560 m. The mirror area (tangential x sagittal), measured in its tangent plane, is 300 x 40 mm2. Rand p are calculated to give a stigmatic image, and we follow the standard practice of using extreme precision for calculated numbers input to the ray-trace code. Figure (a) is the image in the focal plane for the above system. Figure (b) shows the image from the same system but in a plane 15 cm downstream of the focus and with a 5 x 31 ray pattern. In Figure (c) the system is the same as in (a) except that the value of p has been increased to the value (0.1744974840 m) given by Eq. (10) to demonstrate that the line image indeed becomes straight as predicted. Figure (d) is the same as (c) but with an expanded transverse scale so that the residual aberrations can be seen.

384

(A7)

and

, r' [I 2 3 2] L1y = -- -Fj201 + F200W+-F300W cosa 2 2 (A8)

Notice we are ignoring the second and third terms in Eq. (6) in our representation of the line­curvature aberration. This is valid provided we are dealing with a steeply curved toroid (p« R). Substituting for t from (A7) into (A8) we get

(A9)

Equation (A9) shows some of the characteristics we see in the ray traces. For example, when F020 = F200 = 0 representing a stigmatic focus, which is the condition prevailing in Fig. A3(a), Eq. (A9) predicts a family of parabolas, each with a semi-latus rectum that increases with increasing wand a Liy'-directed shift proportional to w2. Most of these features can be seen in Fig. A3(a). On the other hand, when F 020 = w = 0 (the plus signs in Fig. A2), Eq. (A7) predicts that Liz' = O. The line traced out by the plus signs does show this behavior at low t, but at high t, some higher-order aberrations give an increase in Liz'. Study of the rate at which Liz' increases with t (measured in plus-sign intervals) shows it to be an F040 effect, i.e., Liz' proportional to [3. Figure A3(b) shows more aberrations because now F020, F200 *" 0, but it continues to show, basically, a family of parabolas. In the symmetry plane (l = Liz = 0), we have two effects determining Liy', a w effect (defocus) and a w2 effect (aperture defect). Starting from the plus sign at (0, 0), we can see that the sizes and directions of the shifts along the Liy' axis are intelligible on this basis. Turning to Fig. A3(c), we see a large astigmatism (long focal line) and no defocus, and we see clearly that application of Eq. (10) in choosing p correctly delivers a straight focal line. Examination of the expanded diagram in Fig. A3(d) reveals that the only horizontal shift along the Liy' axis is the w2 one (aperture defect), as it should be, since we are now in focus. Moreover, we see another aberration that gives a slight line curvature with equal sizes but opposite signs for positive and negative values of w: the hallmark of the F220 effect.

385

References

Abbe, E., "Ueber die Bedingungen des Aplanatismas der Linsensysteme," Sitz. Ienaisch. Ges. Med. Naturwissen 13 (Sitz. Ber. VIII-2), 129-142 (1879).

Aspnes, D.E., "Imaging Performance of Mirror Pairs for Grazing Incidence Applications: a Comparison," Appl. Opt. 21, 2642-2646 (1982).

Beutler, H.G., "The Theory of the Concave Grating," I. Opt. Soc. Am. 35, 311-350 (1945). Born, M., and E. Wolf, Principles of Optics (Pergamon Press, Oxford, 1980). Brueggemann, H.P., Conic Mirrors (Focal Press, London, 1968). Chrisp, M.P., "Aberrations of Holographic Toroidal Grating Systems," Appl. Opt. 22, 1508-

1518 (1983). Cornbleet, S., Microwave and Optical Ray Geometry (John Wiley, Chichester, 1984). Haber, H., "The Torus Grating," J. Opt. Soc. Am. 40, 153-165 (1950). Hogrefe, H., M.R Howells, and E. Hoyer, "Application of Spherical Gratings in Synchrotron

Radiation Spectroscopy," Proc. SPIE 733, 274-285 (1986). Howells, M.R., "Beamline Design for Synchrotron Spectroscopy in the VUV," Appl. Opt.,

19, 4027-4034 (1980). Hunter, W.R., "Aberrations of Grazing Incidence Systems and Their Reduction or

Toleration," Proc. SPIE, 315, 19-29 (1981). Kirkpatrick, P., and A.V. Baez, "Formation of Optical Images by X-Rays," I. Opt. Soc. Am.

38, 776-774 (1948). Korsch, D., Reflective Optics (Academic, Boston, 1991). Longhurst, RS., Geometrical and Physical Optics (Longmans, London, 1962). McKinney, W.R., and C. Palmer, "Numerical Design Method for Aberration-Reduced

Concave Grating Spectrometers," Appl. Opt. 26, 3108-3018 (1987). Namioka, T., "Design Studies of Mirror-Grating Systems for Use with an Electron Storage

Ring Source at the Photon Factory," Nucl. Instrum. Methods 208, 215-222 (1983). Noda, H., T. Namioka, and M. Seya, "Geometrical Theory of the Grating," I. Opt. Soc. Am.

64, 1031-10366 (1974). Nyholm, R., S. Svenson, and I. Nordgren, "A Soft X-ray Monochromator for the MAX

Synchrotron Facility," Nucl. Instrum. Methods A246, 267-271 (1986). Petersen, H., "The Plane Grating and Elliptical Mirror: a New Optical Configuration for

Monochromators," Opt. Commun. 40, 402-406 (1982). Pouey, M.R, M.R Howells, and P.Z. Takacs, "Visible Ultra Violet Optical Design of Toroidal

Mirror-Toroidal Grating Combinations," Proc. SPIE 315, 37-43 (1981). Pouey, M., Howells, M.R., and Takacs, P.Z., "Optical Design of Grazing Incidence Toroidal

Grating Monochromator," Nucl. Instrum. Methods 195, 223-232 (1983). Rehn, V., "Optics for Insertion-Device Beamlines," Proc. SPIE 582, 238-250 (1985). Underwood, I., "X-ray Optics," American Scientist 66, 476-486 (1978). Underwood, I.H. (private communication, 1992). Underwood, I.H., A.C. Thomson, Y. Wu, and RD. Giauque, "X-ray Microprobe Using

Multilayer Mirrors," Nucl. Instrum. Methods A266, 296-302 (1988). Welford, W., "Aberration Theory of Gratings and Grating Mountings," Progress in Optics,

edited by E. Wolf, 4, 241-280 (1965). Welford, W., "Aplanatism and Isoplanatism," Progress in Optics, edited by E. Wolf, 13, 268-

292 (1976). Welford, W.A., Geometrical Optics (North Holland, Amsterdam, 1962). Wolter, H., "Spiegel system Streifenden Einfalls als Abbildende Optiken flir

Rontgenstrahlen," Ann. Phys. 10, 94-114 (1952).

INDEX

(e, 2e) experiments 164-166 (e, 3e) experiments 177 (e, e' Auger) experiments 171 (hv, 2e) experiments 177 (1+1) double-resonance experiments, purpose of 131 (1 + 1) double-resonance photodissociation 130 3d transition metals, interaction energies 223-224

A

Absorption edges, definition of 3, 4 Absorption spectroscopy (see Photoabsorption spectroscopy and X-ray absorption spectroscopy) ACO (Orsay) synchrotron radiation facility 60,68 Advanced Light Source (ALS) 1,5, 11, 17, 72,91,211,213,299,310,341,342,343, 344,350,351,352

on-axis flux per unit solid angle 353-354

time structure of 136 Advanced Photon Source (APS) 91,213 Aladdin synchrotron radiation facility 73, 74,301,304 Alignment of atoms 103-106

by laser excitation 107 Alignment of photoions and Auger electrons 54 Alkaline earths

MBPT calculations 86 random phase approximation with

exchange (RP AE) 86 random phase approximation,

relativistic (RRPA) 86 ALS (see Advanced Light Source) Angular distribution,

of fluorescence radiation 108-112

387

of photoelectrons 115-124 of photoelectrons emitted from inner s

shells 44 of photoelectrons from rare gases 62-

64,80-81 Anisotropic molecules 19 APECS spectra 177-178 Argon

(e, e' Auger) experiments 171 double-ZEKE coincidence spectrum

76 PEPICO experiments 174-175 photoelectron cross section of (3s) 63 photoelectron spectra 63, 74 threshold PEPICO spectra 175 two-electron excitation in 82-83

Atomic planes in crystals, distance between 2 Atomic structure

reasons for investigating 23 study with synchrotron radiation 23-46

Atomic vapors, photoionization 85-89 Atoms

alignment of 103-106, 107 electron structure of 23-46 measurements of 2

Auger angular distribution, measurement in rare gases 173 Auger decay of vacancies 30 Auger spectra 65,77,177 Autoionization resonances 65, 118

B

Barium fluoride core-core transitions in 291 mapping of occupied and unoccupied

states 290-292 Barium

photoionization cross sections of 86, 89

388

photoionization spectrum of 66 Beamlines

mirrors 357-383 optics 357-383 U4 Dragon (NSLS) 203, 259 U5U (NSLS) 206, 211 use for spin-polarized photoemission

studies 206-211 x-ray microprobe 15

Bending magnets 343 characterization of radiation 12

Benzene, Cls~1t* excitation in hydrogenated and deuterated 263 BESSY synchrotron radiation facility 16, 91, 114, 124,91

time structure of 136 Binding energy 3 Bonn synchrotron radiation facility 56 Born approximation (distorted wave) 173 Boron, K soft x-ray emission spectra 287 Bremsstrahlung 41 Brightness (see Spectral brightness) Bromine

C

photodissociation of 143-144, 149-150

photoelectron spectra for atom 150 photoionization of atomic 149-150 photoionization spectrum of molecule

145 potential curves of molecule 146

Calcium ions, electron spectra of photoexcited 21,49 Calcium, photoemission spectra 307-308 Carbon monoxide

absorption spectra 261 fluorescence spectrum 199 Rydberg states 262

Chemical bonds, measurement of 2 Chemical shifts, separation of 13 Chemical states of atoms, distinguishing between 5 Circularly polarized synchrotron radiation 19, 124,203-219

experimental application (ColPd multilayers) 246-248

undulators for producing 217

use for photoionization 292-293 use in soft x-ray photoabsorption

spectroscopy 268-277 CO++, TPEPICO spectroscopy 174-175 ColPd multilayers, magnetic circular dichroism study of 246-248 Cobalt

magnetic circular dichroism study of 246-248

x-ray absorption spectra 225 x-ray magnetic circular dichroism 223

Coddington's equations 361 Coherence of radiation from undulators 11, 14,331-339 Coincidence experiments 51

continuous vs. pulsed sources 185-185 dynamical correlations 167-170 electron collision 161-173 electron correlation and time correlated

161-163 electron scattering and photoionization

163-164 on rare gases 75 inner-shell ionization 171-173 internal state correlations 166-167 photoionization 173-178

Coincidence spectrometers 174,178-184 optimization of 181-183

Coma 370-375 Cooper minimum 80 Copper

APECS spectrum 177-178 emission spectra of 195 L3 satellite intensity vs. excitation

energy 195 x-ray absorption spectra 225

Core-core transitions, in barium fluoride 291 Core-excited ionic state 49 Core-hole

lifetime 256 relaxation processes 199 effect on dichroism intensity 245-246

Core-ionized ionic states 50 Correlation effects 48, 54-55

in two-electron transitions 64-65 Cu(OOI), holographic reconstruction of 22 Cylindrical mirror analyzers 48,59,77, 78

D

Daresbury Advanced Photon Source (DAPS) 211 Daresbury synchrotron radiation facility 56, 59,60,78,90 DCI (Orsay) synchrotron radiation facility 60 Decay processes

influence of excited electron 40-41 multiatomic states 39 satellites 38-40 semi-Auger 38 two-vacancy states 38-40 vacancy 37-40

Deflection parameter 10 DESY (Hamburg) synchrotron radiation facility 56

time structure of 136 Detection methods, for pump-probe experiments 137 Diamond, K emission spectra of 197 Dichroism 19,200 (see also magnetic circular dichroism)

band-structure model 241-245 calculation of effect in one-electron

model 228-229 calculation of effect in Stoner

model 232-235 concepts and theory for 3D transition

metal atoms 221-250 effect of core hole 245-246 two-step model 236-238

Diffraction gratings, optical path function expansion 379-384 Dirac-Fock calculations, for rare gases 64,81 Dirac-Slater calculations, for rare gases 64 Direct double photoionization 176-177 DORIS (Hamburg) synchrotron radiation facility 60, 77, 194 Double photoionization 33, 44

phenomena in rare gases 84 threshold experiments 173-178

Double vacancy states 32 Double ZEKE coincidence spectroscopy 76 Dragon bearnline 203 Dragon monochromator 258, 264 Dysprosium, photoion yield spectrum 88

E

Electromagnetic spectrum 1, 2 Electron beam, effect of finite emittance 326-330

389

Electron correlations 51,54--55,64--65, 103 in ionization and related coincidence

techniques 161-188 creation of satellite levels by 44 effects in photoionization 23-46 manifestations of 24--26

Electron-electron (e-e) correlations continuum final state 162 internal state correlations 161

Electron-electron coincidence spectroscopy 77 Electron spectroscopy 50 Electronic bonding at buried interfaces 288 Electronic structure of atoms 23-46 Electrons

acceleration to relativistic state 5 binding energy of 3 principles of radiation emission by

relativistic 316-317 Electrostatic analyzers, use in pump-probe experiments 137 ELETIRA 1,91,299,310 Equivalent core-hole model 259 Ethylene, Cls~1t* excitation in hydrogenated and deuterated 263 European Synchrotron Radiation Facilities (ESRF) 91 Excited molecules, photoionization of 156-157 Extended x-ray absorption fine structure (EXAFS) 251

F

Faraday effect 221-223 Ferromagnets, magnetic circular dichroism 211-215 FIR lasers 155, 156 Fluorescence

angular distribution of 108-112 polarization of 108-112

Fluorescence spectroscopy 51, 193 with third-generation synchrotron

radiation sources 281-297

390

Flux (see Photon flux) Franck-Condon effect 255 Franck-Condon principle 259,263 Frascati synchrotron radiation facility 56 Free-electron lasers 133, 134, 135, 155, 156 Fresnel zone plates 303, 305 Fullerenes 43

G

GaAs spin-polarized electron source 203 Gadolinium iron garnet

absorption spectra 214-216 magnetic circular dichroism spectra

214-216 General Electric synchrotron 47 Giant magnetic resistance 209 Giant resonance 24-25 Glasgow synchrotron radiation facility 56, 59,60

H

Halogen molecules, photodissociation of 143-150 Hamiltonian 252 Harmonics of radiation from undulators 11 Hartree-Fock calculations 24,61 HCN, photodissociation of 152 Helium

absorption coefficient of 48, 56-57 double ionization cross section 84 n = 2 satellite 167-169 photoelectron spectrum 64, 68-69 photoionization cross section 69 two-electron excitation in 81-83 variation ofHe+ (n = 2)lHe+ (n =1)

branching ratio with photon energy 65

High-Tc superconductors, x-ray absorption spectroscopy 265-267 History of research with synchrotron radiation 47--49

advances in the production and use 66-74

electron synchrotrons and photoabsorption experiments 56-59

first electron storage rings and photoionization experiments 59-66

new storage rings and new experiments 74-90

History of synchrotron radiation facilities 3 Holography 15,17,20,22 Hydrogen, dissociation study with PIPICO technique 176

I

Inner-shell photoionization 171-173, 177-178 Insertion devices 67,71-74

capable of delivering linearly and circularly polarized light 125

definition of 10 deflection parameter 10

Interaction energies, for 3d transition metals 223-224 Interaction Hamiltonian 252 Interferometry 18 Iodine

ion-yield spectrum 148 photoabsorption spectrum 144 photodissociation of 143, 149 photoelectron spectra 147, 148 photoionization of 144-149 potential curves of molecule 145

Ion spectroscopy 51 Iron

K

angle-resolved photoemission experiments 207-209

magnetic circular dichroism cross section 214,215,223

total x-ray absorption cross section 214,215

x-ray absorption spectra 225

Kerr effect 221-223 Kirkpatrick-Baez systems 367-369 Krypton, two-electron excitation in 83

L

L-edge x-ray absorption 224-229 Lanthanum, photoion yield spectrum 88 Lasers

far-infrared (FIR) 155, 156 free-electron 155, 156 in two-color experiments 129, 133-135 temporal characteristics 134 use in pump-probe experiments 294

Lebedev Institute, Moscow, synchrotron 47 Light (nature of) 1, 2 Linear accelerator (linac) 5 Lithium

M

photoabsorption cross section 50 photoelectron spectrum 51 photoionization cross section of 86 photoionization processes in 50 resonance decay 120 resonance energies and assignments

122 resonant excitations 118 two-photon ionization of 112-115

Magnetic circular dichroism 211-218,268 (see also Dichroism)

and orbital magnetization 271 applications 246-248 band-structure model 241-245 calculation of effect in one-electron

model 228-229 calculation of effect in Stoner

model 232-235 concepts and theory for 3D transition

metal atoms 221-250 effect of core hole 245-246 two-step model 236-238

Magnetic multilayers 209-210 Magnets, for storage rings 9 Many-body effects 54 Many-body theories 55,61,69

diagrams 26-42 Many-electron effects 103 Mass spectrometry, as a detection method for pump-probe experiments 137 MAX II (Lund) 91 MAXIMUM scanning photoemission spectromicroscope 304-306 MBPT calculations, for alkaline earths 86 Metastable states of atoms 43 Microimaging, of neuron netw0rk 309

of spatial distribution of specific elements 308-309

391

of subcellular structures 307,308-309 Microscopes (see also Spectromicroscopes)

x-ray secondary emission microscope (XSEM) 303,309

Microscopy 299 (see also Spectromicroscopy and X-ray microscopy) Mirrors in synchrotron radiation bearnlines 359-385

functions of 358-359 Axisymmetric grazing-incidence 370-

375 coma 370-375 ellipse-shaped 369-370 geometrical description of

surfaces 362-364 optical path function expansion 379-

384 pairs 375-378 parabola-shaped 369-370 paraxial design 361 sine condition 370-375 spherical 367-369 toroidal 364-367 Wolter geometry 375-376

Molecular ions, photodissociation of 139-141, 155 Molecules, photodissociation of 131-133, 139-150 Monochromators 133

crystal 257 Dragon 258, 264 FLIPPER I 194 high-resolution soft x-ray 257-258 on U5U beamline (NSLS) 211,212,

213 toroidal grating 48 toroidal grazing-incidence (TOM) 66,

67-69 Multichannel devices 184, 194 Multicolor experiments 292-296, 282 Multilayer coatings 301 Multilayers 209-210

ColPd 246

392

N

National Synchrotron Light Source (NSLS) 15,16,17,194,203,211,258,259, 282,303,351 Natural emission angle 3, 5 NBS l80-MeV synchrotron 48,56 Near-edge x-ray absorption fine structure (NEXAFS) 222 Near-threshold processes 41-42 Negative-ion photodetachment 44 Neon, triple ionization in 84 Neurobiology, as application for spectromicroscopy 306-309 Neurons

photoemission spectra of cell bodies 308

microimage of network 309 Nickel

L3 emission spectra 198 magnetic circular dichroism

in 223,269-271 x-ray absorption spectra 225

Nitrogen

o

K emission spectra of 197 K-shell photoabsorption spectrum 259 photoelectron spectrum 153 Rydberg states 262

Open-shell atoms 85-89 photoionization 87

Optics (see X-ray optics) Oxygen

p

emission spectra of 196 K emission spectra 198 TPEPICO spectroscopy 176

Phonon relaxation 287 Photoabsorption spectroscopy 14,48,56-59

experimental setup 47 Photodetachment, in negative ions 44 Photodissociation 129-160

molecular 131-133 of diatomic molecules 132 of halogen molecules 143-150

of core-excited molecules 154-155 ofHCN 152 oflarge molecules 141-143 of molecular ions 139-141,155 of poly atomic molecules 150-154 of small molecules 139-141 of s-tetrazine 132, 151-154

Photoelectron spectroscopy 103-127 as a detection method for pump-probe

experiments 137 Photoelectron-photoion coincidence (PEPICO) experiments 174 Photoelectron-photoion-photoion triple­coincidence (PEPIPICO) spectroscopy 178 Photoelectrons, angular distribution of 115-124 Photoemission

history of experiments 300 lateral resolution in experiments 300 spin-polarized 206-205

Photoemission spectromicroscopy 299-313 (see also Spectromicroscopy)

programs in 303 two modes of 301-303

Photoion-photoion coincidence (PIPICO) experiments, study of hydrogen dissociation 176 Photoionization

Auger decay of 30 by polarized light 122 coincidence experiments 173-178 cross sections 51-53 direct double 176-177 double 44, 76 double-vacancy states in 32 effects of electron correlations 23-46 inner-shell, double 177-178 near inner or intermediate shell

threshold 44 of atomic iodine 144-149 of atomic vapors 85-89 of atoms 23-46,47-127,59-66, 104-

127 of bromine atom 149-150 ofCa+ 49 of excited atoms 44, 78, 89 of excited molecules 156-157 of ions 47-127 of ions (singly charged) 78-79

of ions (positively charged) 90 of ions (experimental setup) 48 of lithium 50,51 of metastable states of atoms 43 of multiatomic fonnations 43,45 of open-shell atoms 87 ofradicals 131 of sodium atoms 90 of state-prepared atoms 292 rearrangement effects in 29-32 relativistic 45 resonant effects 65-66 single-electron events in rare gases 79-

81 subshell 60-62 threshold effects 65-66 vacancies in 30

Photoions, alignment of 54 Photon energy, vs. photon emission 6 Photon flux 20

estimating for undulator radiation 330--331

from second- and third-generation sources 213,332

Photon-in, photon-out studies 282, 283-292 of B203 285-288 of silicon 284-285

Polarization, of fluorescence radiation 108-112 Polarized light

introduction to 221 photoionization by 122 use in atomic alignment 107

Polarized synchrotron radiation 19, 199, 200 (see also Circularly polarized synchrotron radiation) Polyatomic molecules, photodissociation 150--154 Post-collision interactions 24, 65-66

in rare gases 84-85 Potassium

L2,3 photoabsorption spectrum 264-265

photoemission spectra 307-308 Predissociation of molecules 140 Pulsed light sources, use for study of dynamical phenomena 293

393

Pulsed nature of synchrotron radiation 10, 17, 18 Pump-probe experiments 129-160,292-296 (See also Two-color experiments)

apparatus for 295

Q

cwoperation 136--137 detection methods in 137-138 synchronization of sources 138

Quasi-equilibrium theory 141

R

Radicals, photoionization of 129, 131 Radio frequency (rf) cavities 7 Random phase approximation 24

generalized, with exchange (GRPAE) 30,32,33

relativistic (RRPA) 55,61 relativistic (RRPA), for alkaline earths

86 relativistic (RRPA), for rare gases 62-

64,81 with exchange (RPAE) 24,26--32,33-

34,55,60,61 with exchange (RP AE), for alkaline

earths 86 with exchange (RP AE), for rare gases

62-64,80--81 Rare earth metals

dysprosium 88 lanthanum 88

Rare gases, angular distribution of photoelectrons 62-64, 80--81

argon 61,63,74,76,171,174-175 Auger angular distribution 173 coincidence experiments in 75 Dirac-Fock calculations 64, 81 Dirac-Slater calculations 64 double-ionization phenomena 84 helium 48,56--57,64,65,68-69, 167-

169 random phase approximation,

relativistic (RRPA) 64,81 random phase approximation, with

exchange (RPAE) 62-64,80,81

394

single-electron photoionization 79-81 triple ionization in 84 two-electron processes in 81-85 xenon 57,58,60,61,62,63,77,79,

168,170 Rare-earth metals, photoabsorption spectra 87 Rearrangement effects in photoionization 29-32 Relativistic effects 54 Relativistic, time-dependent, local density approximation (RTDLDA) 81 Relaxation processes, core-hole 199 Resolution (see Spatial resolution or Spectral resolution) Resonance decay, in lithium 120 Resonance energies and assignments for lithium 122 Resonance-enhanced multi-photon ionization (1+1 REMPI) 129 Resonant effects in photoionization 65-66 Resonant excitations, in lithium 118 Resonant ionization 168 RRKM statistical theory 141 Rydberg states 260, 262

s

Satellites, 32-35, 167 decay 38-40 formation 35-37 separation from emission spectra by

selective excitation 194-196 Schwarzschild objectives 304 Secondary particles 41-42 Self-consistent field (SCF) approximation 24 Silicon

chemical states of 5 emission as a function of photon

excitation energy 284 L2,3 soft x-ray emission spectra 286,

288-289 photon-in, photon-out studies 284-285

Silver, study of multilayers 210 Sine condition 370-375 Sodium

photoabsorption cross section of 58-59

photoelectron spectra of photoionized atoms 90

photoemission spectra 307-308 photoioniiation of 91

Soft x rays, photon energy of 3 Soft x rays, utility of for microscopy 1, 3 Soft x-ray emission spectra 190-192

copper 195 diamond K 197 nickel L3 198 nitrogen K 197 oxygen 196 oxygen K 198 titanium L 197 YBa2Cu30 7- x 195 zinc 196

Soft x-ray emission spectroscopy (SXE) 285 (See also X-ray emission spectroscopy)

grazing-incidence spectrometer for 193 history of 189-190 instrumentation for 192-194 polarization-resolved 200

Soft x-ray emission selectively excited 194-197 surface and bulk probing 197-199

Soft x-ray photoabsorption measurements in gases 259-264 representative studies 259-267

SOLEIL, time structure of 136 Solids, absorption spectroscopy 264-267 Spatial resolution 13 Spectral brightness 13

as a function of photon energy 7 benefits of 315 challenge of exploiting in third­

generation synchrotron radiation sources 299

compromises and limitations 339-352 definition 3 estimating for undulator radiation 330-

331 of synchrotron radiation facilities 8,

332 Spectral resolution 13 Spectrometers

coincidence 178-184 electron-ion coincidence 174

for soft x-ray emission spectroscopy 193

Spectromicroscopes, MAXIMUM 304-306 (see also Microscopes) Spectromicroscopy 299-313 (see also Microscopy and Photoemission spectromicroscopy)

future of 310-311 in neurobiology 306-309 programs in 303 use in differentiating chemical states of

atoms 5 Spherical mirrors 367-369 Spin analysis 203-219 Spin-orbit interaction in the d shell 238-241 Spin polarization in photoemission from solids, from magnetic effects 203-205 Spin polarized photoemission, at NSLS U5U Beamline 206-211 Spot size 125 Stanford Synchrotron Radiation Laboratory (SSRL) 16,20,60,64,246,351 Stokes parameters 108 Stokes shift 287 Storage ring

description 7 magnets in 9 rf cavities in 7

Subcellular structures 308 Super ACO 48,49,72,73

time structure of 136 Superconductors 265-267 SURF II synchrotron radiation facility 60 SXES spectra of organic molecules 199 Synchronization of sources in pump-probe experiments 138 Synchrotron radiation facilities

ACO (Orsay) 60, 68 Aladdin 73, 74, 301, 304 ALS 1,5,11,17,72,91,136,211,

213,299,310,341,342,343,344, 350,351,352,354

APS 91,213 BESSY 16, 114, 124 BESSY II 91 Bonn 56 brightness 8, 331 characterization of 1-22 comparison of flux 213

Cornell University 47 Daresbury 56, 59, 60, 78, 90, 211 DCI (Orsay) 60 DESY (Hamburg) 56 DORIS (Hamburg) 60,77,194 ELETIRA 1,91,299,310 ESRF 91 evolution of 6,66-74 first generation 70 flux 213, 332 Frascati 56 General Electric synchrotron 47 Glasgow 56, 59, 60 history of 3 Lebedev Institute, Moscow 47 MAX II 91 NBS synchrotron 48, 56

395

NSLS 15,16,17,194,203,211,258, 259,282,303,351

parasitic use 56 salient parameters for third-generation

low-energy 8 second generation 70-71 spectral brightness of 8 SSRL 16,20,60,64,246,351 Super ACO 48,49, 72, 73 third generation 91, 157 time structure data 136 use for photoabsorption experiments

56-59 use for photoionization experiments

59-66 SURF II (Washington, D.C.) 60 TANTALUS I (Univ. of Wisconsin)

60 Synchrotron radiation

circularly polarized 19, 124,203-219, 246,268-277,292-293

definition 3 from undulator vs. bending magnet

source 12 history of research with 47-49 in two-color experiments 130, 133-

136 natural emission angle 3, 5 photon flux 20,213 source characteristics 133-136 special characteristics of 13

396

T

time structure 10,17,18,133-136, 184,292-296

TANTALUS I synchrotron radiation facility 60 s-Tetrazine

energy diagram 152 photodissociation of 132, 151-154 photoelectron spectrum 153

Thin films (magnetic) 209-210 Third-generation synchrotron radiation facilities 1-22

capabilities and limits 157 Three-body photodissociation 150 Threshold double photoionization 174-176 Threshold effects in photoionization 65-66 Threshold electron spectroscopy, as a detection method for pump-probe experiments 137 Threshold photoelectron-photoelectron coincidence (TPEPECO) 76 Threshold photoelectron-photoion coincidence (TPEPICO) 76, 174

spectra for argon 175 study of molecular oxygen 175

Time compression 316 Time structure

of lasers 134, 135 of synchrotron radiation 133-136

Time-of-flight electron analyzers 77 Time-of-flight mass spectrometry, as a detection method for pump-probe experiments 137-138 Time-to-amplitude converter (TAC) 179 Titanium, L emission spectra of 197 Toroidal mirrors 364-367 Triple ionization in rare gases 84 Triple-reflection polarizer 217 Tunability

of synchrotron radiation 14 undulators 11

Two-color experiments 89,90 (See also Pump-probe experiments)

in molecules 129-160 on aligned atoms 103-127 source characteristics 133-136 time resolution of detectors in 135

Two-electron excited states 48 Two-electron processes 75

in rare gases 81-85 Two-electron transitions 64-65

limitations on use of undulators for 340-342

with lithium 112-115

U

Undulators 67,91 and two-photon experiments 341-343 as source of high-brightness

synchrotron radiation 301 basic description 317-320 calculating spectral and angular

distribution of radiation from 324-329

central cone 321-322 characteristics of radiation 324-331 characterization of radiation 12 coherence of light from 11,14,331-

339 compromises and limitations of spectral

brightness 339-352 crossed-field 19 definition of 10 degeneracy factor 334-336 depth-of-field broadening 336-337 description of 71-74 design strategies 352-354 diffraction limits 321-322 effects of imperfections on spectrum

350-352 effects of imperfections on storage ring

347-350 electron orbit 345 electron trajectory 10 far-field approximation 341 flux and brightness estimates 330-332 for producing circularly polarized

synchrotron radiation 217 fundamental equation of action 320-

321 gap 73,211 history 315 intensity distribution 326-328, 342 measurement and analysis of defects

345-347

modes of beam 333-334 partial coherence effects in beams

337-339 photon flux 213 photon flux vs. gap width 73 polarization of radiation from 19 power density distribution 344 practical realization of 322-324 properties of radiation 315-358 radiation used in two-photon

experiment 114 tunability 11 U5U (NSLS) 203,211,212,213

University of Wisconsin synchrotron radiation facility 69

v

Vacancies

w

creation of 35-37 decay of 30,37-41 decay of inner 190 double-vacancy states 32-35 near and subthreshold formation and

decay 40-41

Wigglers 67, 72 definition of 10 elliptical 19

x

X-ray absorption decay of 196 L-edge 224-229

X-ray absorption spectroscopy 14 and x-ray circular dichroism 251-279 experimental setup 47 measurements in solids 264-267

X-ray absorption spectrum, broadening mechanisms 255-256

397

X-ray emission spectroscopy 198-202 (See also Soft x-ray emission spectroscopy) X-ray fluorescence 281-297

excitation scheme 283 X-ray magnetic linear dichroism 222 X-ray microprobe 15 X-ray microscopy 1 (see also Microscopy and Spectromicroscopy)

use in imaging chromosome 4 X-ray optics

fabrication tolerances 13 interferometer for testing 18 requirements for 11

Xenon

y

angular distribution of photoelectrons 63,81

Auger energy vs. photon energy 85 Auger spectrum 77 N4,5 (e, e' Auger) spectra 171-172 photoabsorption cross section of 57, 58 photoionization cross section of (4d)

60, 79-80 photoionization cross section of (5s)

61,62 resonant ionization 168, 170 two-electron excitation in 83

YBa2Cu30 7- x, emission spectra of 195 Ytterbium atoms, photoionization of 122

z

Zinc, emission spectra of 196