advances in electronics and electron physics, volume 46

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS

VOLUME 46

CONTRIBUTORS TO THIS VOLUME

E. R. Chenette L. G. Christophorou J . M. Cowley J.-F. Delpech J.-C. Gauthier Allan Rosencwaig A. van der Ziel

Advances in Electronics and Electron Physics

EDITED BY

L. MARTON

Smithsonian Institution, Washington, D.C.

Associate Editor

CLAIRE MARTON

EDITORIAL BOARD

T. E. Allibone H. B. G. Casimir W. G. Dow A. Rose A. 0. C. Nier

E. R. Piore M. Ponte

L. P. Smith F. K. Willenbrock

VOLUME 46

1978

ACADEMIC PRESS New York San Francisco London A Subsidiary of Harcourt Brace Jovanovich, Publishers

COPYRIGHT @ 1978, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by ACADEMlC PRESS, INC. ( L O N D O N ) LTD. 24/28 Oval Road, London NW17DX

LIBRARY O F CONGRESS CATALOG CARD NUMBER: 49-7504

ISBN 0-12-014646-0

PRINTED IN THE UNITED STATES OF AMERICA

CONTENTS

CONTRIBUTORS TO VOLUME 46 . . . . . . . . . . . . . . . . FOREWORD . . . . . . . . . . . . . . . . . . . . . .

Electron Microdiffraction J . M . COWLEY

I . Introduction . . . . . . . . . . . . . . . . . . . . I1 . Theory of Imaging and Diffraction . . . . . . . . . . . . .

III . Diffraction Techniques . . . . . . . . . . . . . . . . IV . Operational Factors . . . . . . . . . . . . . . . . . V . Interpretation and Application . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . .

The Lifetimes of Metastable Negative Ions L . G . CHRISTOPHOROU

I . Introduction . . . . . . . . . . . . . . . . I1 . Experimental Methods . . . . . . . . . . . .

I11 . Metastable Atomic Negative Ions . . . . . . . . . IV . Extremely Short-Lived Metastable Molecular Negative Ions . V . Moderately Short-Lived Metastable Molecular Negative Ions .

VI . Long-Lived Parent Molecular Negative Ions Formed by Electron Capture in the Field of the Ground Electronic State (Nuclear-Excited Feshbach Resonances) . . . . . . .

VII . Long-Lived Parent Negative Ions Formed by Electron Capture in the Field of an Excited Electronic State [Electron-Excited Feshbach Resonances (Core-Excited Type I)] . . . . .

VIII . Long-Lived Metastable Fragment Negative Ions . . . . . IX . Autodetachment Lifetimes of Doubly Charged Negative Ions .

References . . . . . . . . . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

Time-Resolved Laser Fluorescence Spectroscopy for Atomic and Molecular Excited States:

Kinetic Studies J.-C. GAUTHIER A N D J.-F. DELPECH

Introduction . . . . . . . . . . . . . . . . . . . . I . Direct and Indirect Methods for Excited-State Kinetics Studies

I1 . Experimental Techniques for Pulsed-Laser Fluorescence Spectroscopy . . 111 . Methods of Data Analysis and Reduction . . . . . . . . . . . IV . Applications to Atomic and Molecular Physics . . . . . . . . . V . Recent Developments and Concluding Remarks . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . .

. . . .

V

vii ix

1 5

14 31 36 50

56 61 75 71 79

89

116 118 122 125

131 132 143 164 172 195 199

vi CONTENTS

Photoacoustic Spectroscopy ALLAN ROSENCWAIG

I . Introduction . . . . . . . . . . . . . I1 . The Early History of the Photoacoustic Effect .

111 . The Photoacoustic Effect in Gases . . . . . . IV . Theory of the Photoacoustic Effect in Solids . . . V . Theory of the Photoacoustic Effect in Liquids .

.

. VI . Experimental Methodology . . . . . . . .

VII . Photoacoustic Spectroscopy in Physics and Chemistry VIII . Photoacoustic Spectroscopy in Biology . . . .

IX . Photoacoustic Spectroscopy in Medicine . . . . X . Future Trends . . . . . . . . . . . .

References . . . . . . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

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

Noise in Solid State Devices A . V A N DER Z ~ E L AND E . R . CHENETTE

I . Introduction . . . . . . . . . . . . . . . . . . . . II . Sources of Noise . . . . . . . . . . . . . . . . . .

111 . Noise in Diodes . . . . . . . . . . . . . . . . . . IV . Noise in Transistors . . . . . . . . . . . . . . . . . V . Noise in JFETs and MOSFETs . . . . . . . . . . . . . . VI . Miscellaneous Solid-state Device Noise Problems . . . . . . . .

References . . . . . . . . . . . . . . . . . . . .

208 209 211 214 241 247 256 280 289 306 308

314 314 320 335 356 314 380

AUTHORINDEX . . . . . . . . . . . . . . . . . . . . . 385 SUBJECT INDEX . . . . . . . . . . . . . . . . . . . . . 398

CONTRIBUTORS TO VOLUME 46

Numbers in parentheses indicate the pages on which the authors’ contributions begin

E. R. CHENETTE, Electrical Engineering Department, University of Florida, Gainesville, Florida 3261 1 (313)

L. G. CHRISTOPHOROU, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 and The University of Tennessee, Knoxville, Ten- nessee 37916 (55)

J . M. COWLEY, Department of Physics, Arizona State University, Tempe, Arizona 85281 (1)

J.-F. DELPECH, Groupe d’Electronique dans les Gaz, lnstitut d’EIec- tronique Fondamentale, Facultk des Sciences, Universitk Paris-XI, Bitiment 220, 91405 Orsay, France (13 I )

J.-C. GAUTHIER, Groupe d’Electronique dans les Gaz, Institut d’Elec- tronique Fondamentale, FacultC des Sciences, UniversitC Paris-XI, Bitiment 220, 91405 Orsay, France (131)

ALLAN ROSENCWAIC, Lawrence Livermore Laboratory, University of California, Livermore, California 94550 (207)

A. VAN DER ZIEL, Electrical Engineering Department, University of Minnesota, Minneapolis, Minnesota 55455 (3 13)

vii

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FOREWORD

Twice before, in our 11th and 39th volumes, did we treat the subject of electron diffraction. The review by J . M. Cowley, which starts this volume, differs from the previous treatments by its viewpoint. Whereas in the earlier reviews the diffraction phenomenon was independent of the size of the sample, here the considerations focus on the methods and effects, when the sample is very small. This justifies the use of the word “microdiffraction” in describing the theory, the techniques, the operational factors, the interpretation and applications of this method, which is closely related to electron microscopy.

L. G . Christophorou’s review, entitled “The Lifetimes of Metastable Negative Ions, ” concentrates on molecular negative ions. Metastable atomic negative ions are covered in a short section in this review, but only so far as their role in the formation of the molecular species requires them. After a discussion of the experimental methods, the review distinguishes between the short-lived and the long-lived metastable molecular negative ions, and concludes with the autodetachment lifetimes of doubly charged negative ions.

Time-resolved laser fluorescence spectroscopy is a new subject for our volumes and is reviewed here by J.-C. Gauthier and J.-F. Delpech. The introduction of this review emphasizes how much the study of the transfer processes in atomic and molecular states has benefited from recent pulsed dye laser technology, from advances in fast photodetectors, from modern electronics, and from new data acquisition systems. After examining some direct and indirect methods for the kinetic studies of excited states, the main discussion centers on the experimental techniques for pulsed laser fluorescence spectroscopy, on the methods used for the interpretation of the data, and on the possible applications to atomic and molecular physics.

While “Photoacoustic Spectroscopy” seemes to be a brand new field of research, A. Rosencwaig points out in his review that the photoacoustic effect was discovered as early as 1880 by Alexander Graham Bell. To quote from the review: “The photoacoustic signal is a measure of the amount of energy absorbed by a system, that is dissipated through nonradiative or heat producing processes. ” After discussion of the theory of the photoacoustic effect in solids and in liquids, its techniques and applications in physics, chemistry, biology, and medicine are reviewed.

The last article in this volume was written by two previous contribu- ix

X FOREWORD

tors : A. van der Ziel (Volume 4) and E. R. Chenette (Volume 23). The subject of their present review, “Noise in Solid State Devices,” is closely related to the earlier reviews. The treatment of the subject is largely theoretical, without neglecting, however, comparisons with experiment, as well as some applications of the theory. The ample bibliography provides the necessary background for appreciation of the progress made in the various devices under discussion.

Following the pattern established for many years, we list here the reviews, and their authors, of forthcoming volumes:

High Injection in a Two-Dimensional Transistor The Gunn-Hilson Effect A Review of Applications of Superconductivity Digital Filters Physical Electronics and Modelling of MOS Devices

Thin-Film Electronics Technology Characterization of MOSFETs Operating in Weak

Electron Impact Processes Sonar Microchannel Electron Multipliers Electron Attachment and Detachment Electron-Beam-Controlled Lasers Amorphous Semiconductors Electron Beams in Microfabrication. I and I1 Design Automation of Digital Systems. 1 and I1

Magnetic Liquid Fluid Dynamics Fundamental Analysis of Electron-Atom Collision

Electronic Clocks and Watches Review of Hydromagnetic Shocks and Waves Beam Waveguides and Guided Propagation Recent Developments in Electron Beam Deflection

systems Seeing with Sound Wire Antennas Ion Beam Technology Applied to Electron Microscopy Microprocessors in Physics The Edelweiss System A Computational Critique of an Algorithm for the

Large Molecules in Space Recent Advances and Basic Studies of Photoemitters Application of the Glauber and Eikonal

Inversion

Processes

Enhancement of Bright-Field Electron Microscopy

Approximations to Atomic Collisions

W. L. Engl M. P. Shaw and H. Grubin W. B. Fowler S. A. White J. N. Churchill, T. W. Collins,

T. P. Brody and F. E. Holrnstrom

R. J. Van Overstraeten S. Chung F. N. Spiess R. F. Potter R. S. Berry Charles Cason H. Scher and G. Pfister P. R. Thomton W. G. Magnuson and Robert J.

R. E. Rosenweig Smith

H. Kleinpoppen A. Gnadinger A. Jaumotte & Hirsch L. Ronchi

E. F. Ritz, Jr. A. F. Brown P. A. Ramsdale J. Franks A. J. Davies J. Arsac

T. A. Welton M. & G. Winnewisser H. Timan

F. T. Chan,, W. Williamson, G. Foster, and M. Lieber

FOREWORD xi

Josephson Effect Electronics Signal Processing with CCDs and SAWS

Flicker Noise Present Stage of High Voltage Electron Microscopy Noise Fluctuations in Semiconductor Laser and LED Light

X-Ray Laser Research Ellipsometric Studies of Surfaces Medical Diagnosis by Nuclear Magnetism Energy Losses in Electron Microscopy The Impact of Integrated Electronics in Medicine Design Theory in Quadrupole Mass Spectrometry Ionic Photodetachment and Photodissociation Electron Interference Phenomena Electron Storage Rings Radiation Damage in Semiconductors Solid State Imaging Devices Particle Beam Fusion Resonant Multiphoton Processes Magnetic Reconnection Experiments Cyclotron Resonance Devices The Biological Effects of Microwaves Advances in Infrared Light Sources Heavy Doping Effects in Silicon Spectroscopy of Electrons from High-Energy Atomic

Solid Surfaces Analysis Surface Analysis Using Charged Particle Beams

Low-Energy Atomic Beam Spectroscopy Sputtering Infrared Detector Arrays Photovoltaic Effect Electron Irradiation Effect in MOS Systems

Sources

Collisions

Light Valve Technology High Power Lasers

Supplementary Volumes: Image Transmission Systems High-Voltage and High-Power Applications of Thyristors Applied Corpuscular Optics Acoustic Imaging with Electronic Circuits Microwave Field Effect Transistors

M. Nisenoff W. W. Brodersen and R. M.

White A. van der Ziel B . Jouffrey H. Melchior

Ch. Cason and M. Scully A. V. Rzhanov G. J. Bend B . Jouffrey J. D. Meindl P. Dawson T. M. Miller M. C. Li D. Trines N. D. Wilsey E. H. Snow A. J . Toepfer P. P. Lambropoulos P. J. Baum R. S. Symons and H. R. Jorg H. Frohlich Ch. Timmermann R. Van Overstracten

D. Berinyi M. H. Higatsberger F. P. Viehbock and F.

Riidenauer E. M. Horl and E. Semerad G. H. Wehner D. Long and W. Scott R. H. Bube J. N. Churchill, F. E.

Holmstrom, and T. W. Collins

J . Grinberg V . N. Smiley

W. K. Pratt G. Karady A. Septier H. F. H m u t h J. Frey

In the thirty years of publication of Advances in Electronics and Electrons Physics, we have enjoyed the wholehearted collaboration of the

xii FOREWORD

scientific community. To our innumerable friends and colleagues, we wish to reiterate our heartfelt thanks, both for the advice received and for their contributions.

L. MARTON C. MARTON

ADVANCES IN ELECTRONICS AND ELECTRON PHYSICS, VOL. 46

Electron Microdiffraction

J. M. COWLEY

Department of Physics Arizona Stare Universiiy

Tempe, Arizona

B. Variants on CBED . . . . . . . . . . . . . C. Selected-Area Electron Diffraction ( D. Incident-Beam Scanning

B. Contamination . . . .

A. Identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . ......................... 50

I. INTRODUCTION

The term “microdiffraction” has been coined, with more respect for convenience than grammatical elegance, to suggest the diffraction of radiation from a very small volume of material. The smallness of the sample examined is defined in relationship to current practice. For X rays the dimensions of the single-crystal samples normally used are of the order of 100-500 pm, but special microdiffraction techniques have allowed patterns

1 Copyright @ 1978 by Academic Press. Inc.

All rights ofreproduclion in any form reserved. ISBN 0-12-014646-0

2 J . M. COWLEY

to be obtained from regions a few tens of microns in diameter. The use of new high-intensity sources, including synchrotron radiation, may reduce this limit considerably. In electron diffraction the established practice is to obtain patterns from samples that are very thin (hundreds of A) in the incident beam direction but may be fractions of a millimeter in diameter in electron diffraction instruments or about 1 pm in diameter when the usual selected-area electron diffraction (SAED) mode is used in a 100 keV electron microscope. “Electron microdiffraction” thus refers now to the obtaining of diffraction patterns from regions much less than 1 pm in diameter and a few hundred angstroms in thickness or less.

In principle, the smallest sample that can give a detectable diffraction pattern is a single atom. Already the scattering from single heavy atoms has been collected to give clear images in the scanning transmission electron microscope (Crewe and Wall, 1970) and has been made to interfere coherently with the transmitted beam to give phase contrast images in the conventional transmission electron microscope (e.g., Iijima, 1976, 1977). Given sufficient specimen stability the number of electrons that may be scattered from a single heavy atom in an experimentally convenient time (say, 1 minute) using the electron sources now available is about lo1’, which is sufficient to define the intensity distribution within a diffraction pattern with reasonable accuracy. The scattering of this number of electrons from each atom of an assembly of atoms should allow the relative positions and scattering powers of the atoms to be determined. Thus, in principle, the basis exists for the atom-by-atom determination of the structure of matter.

In practice, of course, many difficulties of experimental technique, specimen stability, and interpretation of data prevent the attainment of this goal. However, the progress in recent years has been spectacular. Diffraction patterns have been recorded from specimen regions 20 A in diameter. The impact of this achievement on practical investigations in materials science has not yet been very great, partly because the techniques are in early stages of their development, but the potential of the method is obvious. A review of the history and current status of electron microdiffraction at this stage seems justified as a basis for discussions on the refinement of the experimental methods and the applications to the multitude of current problems involving structural variations over small distances in solids.

The possibilities for electron microdiffraction arise because of the strong interaction of electrons with matter. The scattering cross sections of atoms for electrons are approximately lo6 times those for X rays of the same energy. The small thicknesses of samples used for electron diffraction have always been both a limiting factor and the basis for the special advantages of the method. Transmission electron diffraction developed slowly because

ELECTRON MICRODIFFRACTION 3

of the experimental difficulties of preparing samples in the necessary thickness range of 10-1000 A. The advent of the electron microscope provided the means whereby the thin sample areas could be recognized and the techniques for the systematic correlation of diffraction effects with crystal morphology could be developed.

Reflection electron diffraction, both at low energies (LEED, 10-500 eV) and at high energies (RHEED, 10-100 keV) has been concerned with the scattering from the few top layers of atoms on crystal surfaces. Since with few exceptions the specimen areas used have been quite large (fractions of a square millimeter) we shall not be very concerned with these methods.

Early in the development of electron diffraction instruments a single long-focus lens was added to provide a sharply focused pattern on the photographic plate (Fig. la) and this geometry is still used in many cases to observe fine detail in the diffraction pattern. To improve the resolution in the diffraction pattern, it is now common to produce smaller effective source sizes by using one or more strong lenses to demagnify the actual electron source.

( b ) FIG. 1 . Electron optical arrangement for a diffraction instrument, [a) with the electron

beam focused on the screen or plate and (b) with the electron beam focused on the specimen.

For this configuration the beam diameter at the specimen level is defined by the limiting aperture and is usually several hundred microns. The very severe limitations on specimen thickness imply that only very rarely is it possible to obtain a nearly perfect single crystal covering a significant

4 J . M. COWLEY

fraction of such a large area. Hence this lens configuration gives excellent averaging over orientations for polycrystalline patterns or mosaic or bent single-crystal films, but does not allow any well-defined correlations of diffraction intensities with the thickness, structure, or perfection of single crystals. The basic information on the scattering of electrons by crystals and the detailed comparison of experimental observation with theory depended on the development of methods to obtain patterns from much smaller regions.

The first and most obvious means for decreasing the irradiated area of the specimen was to increase the power of the electron lens of Fig. l a to focus the beam on the specimen as in Fig. lb. The diffraction spots on the photographic plate were thereby spread out into shadow images of the limiting aperture, but a small amount of spreading is not important if we are interested only in the positions and intensities of the well-spaced pattern of spots given by single crystals. This idea was used, for example, in the electron diffraction instruments of von Ardenne et al. (1942), Hillier and Baker (1946), and Cowley and Rees (1953), in which reasonably sharp diffraction spots could be obtained from crystal regions about 1 pm in diameter.

The idea of focusing the electron beam on the specimen had been introduced much earlier, of course, by Kossel and Mollenstedt (1939), who used much larger angles of convergence to obtain spectacular convergent beam electron diffraction (CBED) patterns from small regions of thin mica crystals. This work and its interpretation by MacGillavry (1940) were of great significance in revealing the strong dynamical diffraction effects that have been the dominant factor in determining the limitations and the unique capabilities of electron microdiffraction.

The large-scale development of electron microscopy in the late 1940s and 1950s extended the range of electron diffraction work. The wealth of knowledge on the sizes, shapes, and imperfections of submicron crystalline regions provided a much better background for the interpretation of electron diffraction patterns. The development of the selected-area electron diffraction (SAED) techniques (Boersch, 1936) allowed the direct correlation of diffraction intensities with crystal morphology. The mutual advantages of electron diffraction and electron microscopy and their strong interdependence have made them inseparable to the extent that, except for the highly significant cases of exploratory research on diffraction phenomena, their combined use in a single instrument is standard practice for most materials science.

The SAED techniques has suffered from the restriction that, in a 100 keV electron microscope designed primarily for high-resolution imaging, the area from which the diffraction pattern can be obtained is limited by the spherical aberration of the objective lens to be greater than about 1 ,urn.


To overcome this limitation it was necessary to go to instruments having limited imaging capabilities. Goodman and Lehmpfuhl (1965) and their colleagues (Cockayne et al., 1967) developed the convergent beam diffraction technique with special instruments that allowed areas of 200 A diameter to be studied. Riecke (1962) developed instrumentation for “microbeam” microdiffraction. It is only with the recent production of high-resolution scanning transmission electron microscopy (STEM) instruments that the techniques have become available for the simultaneous observation of high-resolution images and diffraction patterns from extremely small regions. The diameter of the region giving the diffraction pattern may, in principle, be equal to the resolution limit of the microscope.

In the following pages, after a brief outline of the relevant theoretical considerations, we shall describe in more detail the various techniques available for electron microdiffraction and then give some account of the nature of the information that can be obtained, with examples drawn from the applications already made.

11. THEORY OF IMAGING AND DIFFRACTION

A. Focusing and Imaging

The aspects of electron optics relevant to our purposes may be discussed in terms of the familiar ideas developed for the physical optics of light (Cowley, 1975). We may confine ourselves to scalar wave theory, since polarization effects are negligible and the nonscalar effects of the magnetic fields of electron lenses on electron trajectories can be assumed, as a first approximation, to provide only a trivial image rotation.

The monochromatic wave from a point source is described by the wave function in a plane a t a distance R from the source by

$,(xy) = i(RA)-’ exp( - ikR) exp[ - i2n(x2 + y2)/RA] (1)

where we have used the small-angle approximation, which is valid for most cases of interest. For a more general source function $l(xy), the wave produced on this plane is

$ 2 ( X Y ) = $I(XY) * $,(w) (2)

where * indicates the convolution integral defined by f ( x ) * g(x) = J f ( X ) g ( x - X ) d x . For RI1 large compared with the maximum x2 + y 2 values, this reduces to the Fourier transform relationship of Fraunhofer

6 J . M. COWLEY

diffraction

Yz(uu) = Yl(uu) = Y$l(xy) = JJ$l(xy)exp[2zi(ux + uy)]dxdy

where

u = 2 sin 6JA z x/RI, u = 2 sin 6,JA x y/RL (3)

and 8, and 8, are half the angles made with the z axis in the x-z and y-z planes. If such a wave falls on a lens, it is modified by multiplication by the

transmission function of an ideal lens, which would exactly reproduce the object function, and by the function

A(u4 expCiX(u4-J (4)

where A(uu) is the aperture function, usually assumed to be unity within and zero outside an axially placed physical aperture, while ~ ( u u ) is the phase change produced in the wave relative to the axial beam path. Usually ~ ( u u ) is assumed to take the simplest possible form :

~ ( u u ) = z A h 2 + +nC,A3u4 ( 5 )

where A is the deviation from exact focus and C, the spherical aberration constant.

The wave function produced at a distance R,, near the focus of the lens, is then given by a second Fourier transform, Fraunhofer diffraction, operation as

= qw3 4A(uu)exp[ix(udl}

= $1 -x-, -y- * 9[Aexp(i~)] ( R", 3 or

W y ) = $I(% y) * CC(XY) + iS(XY)l (7)

where, in the final expression, we have referred the functions to the object dimensions by removing the magnification factors - R/R,, ignored terms of modules unity and trivial constant factors, and expressed the spread function in terms of its real and imaginary parts. The image intensity is then

4 X Y ) = I$l(XY) * {CbY) + iS(XY)}12

I(xy) = c2(xy) + s2(xy) = T(xy)

(8)

(9)

For a point source represented by a &function,

which defines the intensity spread function T(xy). If the object can be represented by an incoherently emitting region of intensity distribution


Io(xy), the image is then given by summing the intensities from all points of the object and so is written

I ( X Y ) = I O b Y ) * W Y ) (10)

Then T(xy) is the Fourier transform of the contrast transfer function of the lens for the incoherent imaging case,

T(xy) = F{A(uu)exp[i~(uu)] * A ( - u , -u)exp[-i~(-u, -u)]} (11)

With this background we may describe the essential imaging steps in the various forms of electron microscopy. For CTEM the important, resolution- determining step is the action of the objective lens to form the first magnified image of the electron wave transmitted through the object.

If i+b I(xy) is the wave at the exit surface of the specimen, the image intensity will be given by (8), magnified by the objective and subsequent lenses. In the practical case that the incident wave is not strictly monochromatic and the focal length of the objective lens varies slightly with time, the intensity distributions for the various wavelengths and focal lengths are added incoherently or, if the probabilities of occurrence for various values of A and f can be described by a distribution function D(A, f),

If the specimen is sufficiently thin in relation to the wavelength and the resolution, as discussed below, the wave function at the exit surface of the specimen may be described in terms of a transmission function q(xy), which multiplies the incident wave function :

i+b l (XY) = $ O ( X Y M X Y ) (13)

For many purposes it is a useful approximation to assume that $o(xy) is a plane wave of unit amplitude. The actual beam convergence in CTEM is usually of the order of lop3 rad. However, for an increasing number of applications, larger convergence angles are used, as in the case of high- resolution imaging with the specimen immersed in the objective lens field, in which case the forefield of the objective lens often acts as a short-focal- length condenser to give angles of convergence approaching lop2 rad. The incident-beam convergence then has important effects on the image resolution and contrast (Wade and Frank, 1977; O’Keefe and Sanders, 1975; Anstis and O’Keefe, 1976; O’Keefe and Anstis, 1978).

It is customary to consider the effects of beam convergence with the assumption that waves coming in different directions are incoherent so that the total image intensity is given by summing the image intensities given by plane waves for all angles of incidence. This would be appropriate

8 J . M. COWLEY

if the specimen were illuminated by an ideally incoherent source without intermediate lenses or apertures. It is a good approximation for the conventional system in which the aperture of the condenser lens is illuminated by a hot-filament electron gun. In this case an effectively incoherent source of several microns diameter gives a coherence width of the radiation at the condenser aperture (w x L/as, where as is the angle subtended by the source) of about 1 pm, which is very small compared with the usual dimensions of the condenser aperture, 50-200 pm.

However if, as in some of the newer CTEM instruments, a field emission gun is used, the effectively incoherent source size may be less than 50& so that the condenser aperture may be illuminated coherently. Then the amplitudes of the waves incident on the specimen at various angles must be added. The incident wave is approximated by c(xy) + is(xy), as in (7), where the form of the function is determined by the aperture and aberrations of the condenser lens system.

For high-resolution STEM instruments, the high brightness of a field emission gun is necessary in order that a sufficiently high current of electrons should be concentrated into the small probe, a few angstroms in diameter, which is scanned across the specimen. It is a good approximation to assume coherent illumination of the specimen by a wave described by (7) with a &function source. The functions c(xy) and/or s(xy) thus have sharp peaks of form depending on the objective lens aberrations and defocus, with radially diminishing oscillations (Cowley, 1976b).

For sufficiently thin specimens, the exit wave is given as in (13) by a transmission function

$ l ( X Y ) = d x - x, Y - Y ) [ C ( X Y ) + i S ( X Y ) l (14)

where X and Y are the coordinates of the center of the incident-beam probe on the specimen and we have assumed, for convenience, that the specimen rather than the incident beam is moved. On the distant plane of the detector, Fig. 2, the amplitude distribution will be given by the Fourier transform

FIG. 2. Scheme for scanning transmission electron microscopy.


of (14) as

yx, Y ( U ~ ) = Qxr(uu) * A(uv)exp[i~(u~)] (15)

where Q(u, u) = P q ( x y ) . The signal detected and used to form the magnified image is then given

by integrating the intensity Ixy(uu) = ~Yxy(uu)~2, multiplied by a detector sensitivity function W(uu):

ZOb(XY) = J I x Y ( u ~ ) W ( u ~ ) du du (16)

As in the case of CTEM, the effects of finite source size and variations of I or f are included by the incoherent integration of intensities over the corresponding variables.

The diffraction pattern (15) for a thin object will consist of a strong central spot surrounded by a weaker, wide-angle (to 10- rad) distribution of scattered electrons. The variety of image signals that may be produced by detection of all or part of the central spot (bright-field images) or all or part of the surrounding scattered radiation (dark-field images) has been described, for example, by Cowley (1976b).

An aspect of electron microscopy that is of importance in the present context is the relationship of the imaging process to the formation of the Fraunhofer diffraction pattern. In the case of CTEM we have seen that the angular distribution of the scattered waves from a specimen is described in terms of the Fourier transform (3). The Fraunhofer diffraction pattern is produced where each of the component plane waves leaving the specimen is condensed to one point of a two-dimensional spatial distribution. This happens on the back focal plane of the lens, where the intensity distribution is

I(uu) = p J ' , ( U U ) I * (17)

with u = x f l , v = y f l in the small-angle approximation. If the lenses following the objective are used to magnify this back-focal plane distribution rather than the image plane distribution, the Fraunhofer diffraction pattern of the object, rather than the magnified image, will appear on the final viewing screen or photographic plate (see Fig. 3).

The arrangement of the STEM instrument (Fig. 2) is essentially that of Fig. lb. A diffraction pattern is produced on the plane of the detector. The alternative configuration, Fig. la, can be provided by changing the lens excitations to produce a focused diffraction pattern but in this configuration a high-resolution image cannot be formed. It should be emphasized that for the convergent beam mode of Fig. lb, which is compatible with high- resolution imaging, the intensity distribution of the diffraction pattern

10 J . M. COWLEY

Specimen

Oblect I Y ~ lens

Back l o c a l p lane

Selected-area aperture

Intermediate lens

Prolector lens

Final

FIG. 3. Ray paths for a transmission electron microscope, following the specimen, as used to obtain a high magnification image (left) and to obtain a selected-area diffraction pattern fright).

given from Eq. (15) will be strongly dependent on the aperture and aberrations of the objective lens, including defocus.

B. Diffraction of Plane Waves

For a very thin sample, the effect on the incident wave may be represented by multiplication by a transmission function q(xy) that describes the changes in phase and amplitude. The predominant effect is the change of phase due to the potential distribution &r) of the scattering matter so that, in the so-called phase-object approximation with a plane incident wave, the transmitted wave is given by the transmission function

d X Y ) = exPC - W(xy) l (18)

where &y) = !4(r) dz, with the beam direction taken to be the z axis, and c = n/AE, where E is the accelerating voltage. Absorption effects may be included by assuming 4(xy) to be complex.

The one-to-one correspondence between amplitudes at points of the incident and transmitted waves, implied by use of the transmission function, assumes no lateral spreading of the perturbations of the wave, i.e., no appreciable effect of the Fresnel diffraction smearing represented by Eq. (1). Rough estimates suggest that the spreading of wave perturbations during


transmission through a thickness T is of the order of (TA)’j2, so that for 100 keV electrons the spreading is 1 A for T % 30 8, and 3 A for T % 250 A. Since the atom sizes and separations in projection are of the order of 1 A, the thickness limit for this approximation is normally taken as 20-30 A.

For even single heavy atoms the phase change represented by (18) may be quite large, exceeding n. However, for very light atoms the phase change is relatively small and one may approximate

q(xy) = 1 - ia4(xy)

so that the diffraction pattern given by Fourier transform is

Y(w) z Q(uu) z ~ ( u v ) - ~o@(uv) (19)

This weak-scattering or “kinematical” approximation may be improved by adding the Fresnel diffraction effects. For each scattering element the transmitted wave is convoluted by the Fresnel propagator, Eq. (1) with R = z - z,,, and the out-going wave is found by integrating over z to give a diffraction amplitude for a plate of uniform thickness T,

Y(uu) = d(uu) - ia s @(uu, z)exp[ -2i~iz[(uu)] dz (20)

where [(uu) is the distance of the Ewald sphere from the reciprocal lattice plane perpendicular to the incident beam.

In the case of a three-dimensionally periodic potential distribution, such as a thin crystal, the diffracted amplitude is expressed in terms of the structure amplitudes y,,k[, the Fourier coefficients of the Fourier series describing the potential, as

yhkl rx - io@hkl[(sin n[hklT)/nchkl] (21)

where [hk[ is the hkl excitation error, or the distance of the hkl reciprocal lattice point from the Ewald sphere (Cowley, 1975).

Since this kinematical approximation assumes that the scattered amplitudes are small compared with the incident beam amplitude, it is valid only for light atoms and for very thin samples. It fails for even thinner samples in the case of crystals viewed along the directions of principal axes, for which there is a progressive phase change due to rows of atoms aligned in the beam direction.

Because for very thin crystals the scattering is approximated by that of a two-dimensional phase grating, the diffraction patterns can contain a very large number of diffraction spots, regularly arranged in a close representation of a planar section of the reciprocal lattice. The small curvature of the Ewald sphere of reflection in reciprocal space corresponds to the small spreading of the electron wave in real space due to Fresnel diffraction.

12 J. M. COWLEY

For thicker crystals the factor in (21) involving the excitation error may be small for all but a small number of reciprocal lattice points so that only a few diffracted beams may be produced and for particular orientations a single diffracted beam may dominate the diffraction pattern. However, except as a rough first approximation, it is rarely possible to make the assumption, common in X-ray diffraction, that only one diffracted beam appears at a time (the “two-beam” case, when the direct transmitted beam is countec?).

C . Dynamical Scattering

The single-scattering kinematical approximation may fail for a sample one atom thick. For very thin samples the phase object approximation (18) without the weak scattering approximation (19) can often serve because it includes all multiple-scattering processes. This is evident from the power- series expansion of the exponential in (18), which on Fourier transforming gives

Q(uu) = 6(uu) - iaO(uu) - $020(uu) * O(uu) + &a3@ * (D * O + . . * where successive terms represent single, double, triple, . . . scattering.

In general the Fresnel diffraction effects must also be included to give a full “dynamical” theory of scattering. This has been done in various ways (see Cowley, 1975). A simple, straightforward approach due to Cowley and Moodie (1957a) considers the progressive phase change of the electron wave by successive thin slices of crystal perpendicular to the incident beam, with Fresnel diffraction of the wave between slices. In the limit that the thickness of the individual slices tends to zero this gives the same results as the classical formulation of the problem by Bethe (1928), who solved the wave equation for electrons in a periodic potential field. On the basis of the Cowley-Moodie approach a computing method has been developed (Goodman and Moodie, 1974; Cowley 1975) in which the progressive modifications of wave amplitudes by slices of crystal of finite thickness are calculated. This method has been used for calculating diffraction patterns and images for crystals giving thousands of diffracted beams simultaneously (O’Keefe, 1975) and for crystals containing defects or disorder giving rise to continuous distributions of diffuse scattering (Fields and Cowley, 1977; Spence, 1977). Computer programs based on the matrix formulation of the Bethe theory (Hirsch et nl., 1965) have also been applied for many-beam diffraction problems, although usually for smaller numbers of diffracted beams, and the differential equation formulation of Howie and Whelan (1961) has proved valuable for computing images of dislocations and other extended defects in two-beam and several-beam approximations.


D. Diffraction in Convergent Beams

The approximation of assuming the incident radiation to be a plane wave of unit amplitude is sufficient for many purposes in that for parallel- beam, focused-electron diffraction patterns and for many applications of CTEM the range of angles of incidence present is too small to give any appreciable variation of difrraction intensities. However, for convergent- beam electron diffraction (CBED), STEM, and some high-resolution CTEM applications, the variation of diffraction conditions within the range of incidence beam directions is important.

It has been customary in the past to calculate CBED patterns or CTEM and STEM images on the assumption of incoherence of the incident beam components: the intensities are calculated for each incident beam direction and are added, as mentioned in Section I1,A. The calculation of CBED patterns (Goodman and Lehmpfuhl, 1967) or of CTEM images (O’Keefe and Sanders, 1975) on this basis involves a large number of n-beam dynamical calculations in general, although for CTEM of very thin specimens an analytical modification of the contrast transfer function of the objective lens may provide a useful approximation, eliminating the need for multiple calculations (Wade and Frank, 1977; Anstis and O’Keefe, 1976; OKeefe and Anstis, 1978).

The same approach of making a separate n-beam dynamical calculation for each angle of incidence may be used for the case of coherent illumination, as when a field emission gun is used, except that then it is necessary to add the amplitudes, rather than the intensities of the patterns or images given for different incident beam directions. It appears likely that, because interference effects will cause a more rapid variation of amplitudes than intensities with angle of incidence, a finer sampling of incident beam directions will be necessary in the coherent case, with a corresponding increase in the amount of computation required.

An alternative method of calculation is possible for the STEM and CBED cases in which the diameter of the incident beam on the specimen is small. The incident-wave amplitude described by (7) is a localized, non- periodic function. The transmission of this wave through a crystal may be calculated by use of the method of periodic continuation. The single incident beam is replaced by a two-dimensionally periodic array of nonoverlapping beams spaced at multiples of the crystal periodicity. A single calculation, with a very large number of beams, is then made for a superlattice having the periodicity of the beams. The diffraction pattern will then correspond to that for a single incident beam, sampled at a finely spaced array of points. This method has been used by Spence (1977) to calculate CBED patterns from small perturbed regions of crystal.

14 J. M. COWLEY

It has been suggested (Cowley and Nielsen, 1975; Cowley and Jap, 1976) that in the case of coherent illumination, the interference effects near the crossover formed by a lens may lead to some surprising diffraction phenomena. The flow of energy in the beam is actually parallel to the axis near the focal point (Fig. 4a) and cannot be represented by the classical geometric optics picture of intersecting ray paths (Fig. 4b). Arguments or ideas based on the picture of Fig. 4b may be misleading. It has been shown by the calculations of Spence and Cowley (1978), for example, that the details of intensity distributions in a CBED pattern are strongly dependent on the focus of the objective lens although, for a perfect crystal pattern, this dependence appears only in the regions where the extended spots overlap.

fa)

- A- --\- /

( b ) FIG. 4. Ray paths (lines indicating the flow of energy) at the focus of a lens for (a) coherent

wave optics and (b) classical geometric optics.

111. DIFFRACTION TECHNIQUES

A. Convergent-Beam Electron Diffraction (CBED)

From the early work of Kossel and Mollenstedt (1939) it was obvious that CBED patterns from crystals gave more information than was contained in focused patterns for particular incident-beam directions. The variations of diffracted-beam intensities with the directions of incidence were clearly displayed in the complex distributions of intensity within the large circular disks formed by the diffraction spots from single crystals. The Kikuchi line patterns formed in the diffuse scattering outside the diffraction spots showed


intensity distributions that were obviously related but essentially different. A thorough understanding of the patterns came slowly.

The first stage in the interpretation of the patterns in terms of Bethe’s dynamical theory came from MaGillavry (1940), who showed that with the assumption of two-beam diffraction conditions the intensity of a reflection as a function of excitation error [,, and the thickness T of a plane- parallel crystal slab could be written

when w = &(h, th = 1/(2r~@~), and Qk is the h Fourier coefficient of the potential distribution of the crystal.

For the special case of reflection at the exact Bragg angle, Ch = 0, this gives the well-known “pendellosung solution” with 1, varying sinusoidally with thickness, with periodicity equal to C h , the “extinction distance”. The intensity of the transmitted beam, I , , which from the conservation of energy must be equal to 1 - 1, in the absence of absorption, likewise varies sinusoidally. As a consequence, both bright- and dark-field images of crystals show “thickness” fringes if the thickness in the beam direction varies.

For a crystal of constant thickness, the variation of intensity with angle of incidence, and so with [,,, given by (22) may be compared with the kinematical result, the square of (21). The kinematical result is illustrated in Fig. 5. Different angles of incidence of the incoming beam correspond to

FIG. 5. Diagram illustrating the formation of a convergent-beam diffraction pattern in the kinematical approximation; in reciprocal space and (below) in the real space of the diffraction pattern.

16 J. M. COWLEY

different positions of the Ewald sphere relative to the row of reciprocal lattice points and to different positions across the diameter of each of the circular diffraction spots. The intensity distributions across the spots therefore reflect the distributions of scattering power along the shape- transform extensions of the reciprocal lattice points, each having the form

For c h latge (w >> l), the two-beam approximation formula (22) approaches this form and the crystal thickness can be deduced from the periodicities of the fringes. For (h small, the intensity is dominated by the contribution of l@hl. By analysis of the patterns obtained by Kossel and Mollenstedt from mica, MacGillavry was able to deduce I@,,( values in excellent agreement with those calculated from the structure of mica as then understood from X-ray diffraction results. The measure of agreement achieved must be regarded as fortuitous since knowledge of the structure of mica has since been modified by further refinements and the validity of the two-beam approximation is questionable for the patterns used.

CBED patterns were obtained later by Ackerman (1948) from a wide variety of materials using an incident-beam diameter estimated to be 1 pm or less and by many subsequent authors. The interpretation of the details of such patterns in terms of two- or several-beam dynamical diffraction theory was made by Ackerman (1948), Fues and Wagner (1951), Hoerni (1950), and others, but the technique remained of rather limited academic interest until revived with the application of more advanced techniques and more quantitative interpretation by Goodman and Lehmpfuhl (1965, 1967). These authors initially used an electron microscope with the specimen placed near the back focal plane of the objective lens so that the strong lens gave a focused probe diameter of a few hundred angstroms at the specimen. Similar small probes, with improved operational convenience, were subsequently provided by a specially designed diffraction instrument (Cockayne et al., 1967) in the Melbourne laboratory.

With these instruments, patterns such as Figs. 6 and 16 were obtained from very small, perfect crystal regions under carefully controlled experimental conditions and with sufficient control of the relevant experimental parameters to allow detailed systematic comparisons of observed intensities with those computed using accurate multibeam dynamical diffraction theory. The valuable series of results in the refinement of crystal potential distributions and the determination of crystal symmetries by this group will be summarized in Section V.

In the last few years a further advance in CBED techniques has become possible with the introduction of field emission guns and STEM techniques. CBED patterns are readily obtained in both dedicated STEM instruments and scanning attachments for CTEM instruments, which increasingly use

sin2(nch T)/(dh)*.


FIG. 6. Convergent-beam electron diffraction pattern for a “systematic” row of reflections and some weak nonsystematic reflections. The center of the second-order spot is at the Bragg reflection condition.

field emission guns. A STEM instrument has electron optics ideally suited to CBED (Fig. 2) . If instead of scanning the incident beam across the specimen the beam is held stationary, a CBED pattern of a fixed small area is formed on the detector plane. With such instruments diffraction patterns have already been recorded from areas as small as 20 A in diameter (e.g., Brown et al., 1976). In principle, the radius of the selected area from which the CBED pattern is obtained may approach the resolution limit of the microscope, which is currently about 3 A.

For an ideal leiis having no aberrations and limited only by a circular aperture of radius u = uo, the beam intensity at the specimen is given by J f ( 2 7 ~ u ~ r ) / ( 2 n r ) ~ , where J , is the first-order Bessel function and r the radial coordinate. This has the well-known form of the Airy disk, a central maximum of radius r l = 0.61/u0 to the first zero of intensity. In terms of the angle M subtended by the aperture at the specimen, rl = 1.22,+. The central maximum is surrounded by concentric circles of intensity with intensity maxima decreasing approximately in proportion to r P 3 . Thus the incident intensity is by no means limited to a well-defined central spot. Integrated around the circumference of each concentric circle, the intensity falls off quite slowly, being proportional to r P 2 . Approximately 16% of the incident intensity lies outside the central maximum. This distribution and the even

18 J. M. COWLEY

slower decrease with r of the wave amplitude of the focused spot become increasingly relevant as the size of the spot is decreased. A probe with central maximum 5 A in diameter passing through a thin crystal will give a CBED pattern reflecting the periodicity of the lattice even if the unit cell dimensions are much greater than 5 A. Each diffraction spot will take the form of a sharply defined disk and, although the disks may overlap, the fact that they are in a periodic array will be readily discernible.

The diameter of the disk of intensity corresponding to each spot in the CBED pattern will be La, where L is the distance from specimen to detector screen or plate. Disks will just touch at the edges for planar spacings in a crystal specimen given by

d = 1/2sin9 = L/a (23)

If we assume incoherent STEM imaging, which is sometimes used as a first approximation for the dark-field STEM mode, the Rayleigh criterion gives a resolution for this case of

Ax = 1.221ia (24)

which is slightly larger than the planar spacing (23), so that in order to resolve a spacing d in the STEM image, it is necessary to use an objective aperture that will make the CBED spots overlap by about 20%.

These considerations must be modified by the presence of lens aberrations in practice. Spherical aberration will not affect the size of the diffraction spots, but it will modify the shape and size of the central maximum of amplitude or intensity at the specimen, it will modify the surrounding fringes, and it will modify the STEM resolution.

It is well known that for bright-field phase contrast imaging of very thin specimens there is an optimum aperture size of approximately u,,, = 1.5Cs- 1/4jZ-3/4 (see Cowley, 1975) giving a least resolvable distance of approximately

AX = 0.66C,"41314 (25)

on the assumption that Ax = 1/umax. For dark-field imaging it may be assumed as a first approximation that, because the square rather than the first order of the spread function is relevant, the resolution is improved by a factor of about 2l'* (the figure for gaussian spread functions) and the constant in (25) has been given various values around 0.4.

Thus with spherical aberration and coherent imaging a larger objective aperture may be used, giving larger spots and also better resolution (and it is usually assumed that the CBED spots will just touch for a d spacing just resolved). Hence it may be concluded that while STEM imaging can give information on spacings in the sample down to a certain dmi,, the CBED


pattern produced by the same incident beam contains information on spacings in the range of dmin and below as well as, in many cases, some information on spacings much larger than dmin.

The usual reservations concerning the interpretation of diffraction patterns must, of course, be made. Under kinematical scattering conditions the “phase problem” applies if the CBED spots do not overlap. The relative phases of the reflections are lost when the intensity is recorded and only an autocorrelation function, or Patterson function, can be deduced, rather than the actual potential function, which would specify relative atom positions. For the more usual dynamical scattering conditions some information on relative phases of diffracted beams is present in the diffraction pattern but the problem of deducing these phases is complicated and in general is susceptible only to trial-and-error methods of solution.

If the CBED spots do overlap and the incident beam has sufficient coherence, interference effects in the regions of overlap will, in principle, give information on the relative phases of the reflections (Nathan, 1976; Cowley and Jap, 1976).

B. Variants on CBED

1. Defocused CBED; Shadow Imaging

In the lens in a CBED instrument is defocused so that the small crossover is formed either before or after the specimen, the central spot of the CBED pattern will become a bright-field shadow image of the specimen. If the illuminated portion of the specimen contains a single crystal, each diffraction spot of the CBED pattern will become a dark-field shadow image showing the variation of diffraction intensity within the illuminated region (Fig. 7). For both the central beam and diffracted beams, the range of angles of incidence across the illuminated area will be the same as for a beam focused on the specimen. For a perfect, unbent, thin shgle-crystal plate of uniform thickness illuminated from an incoherent source, the only source of contrast in the spots would be the changes of diffraction intensity with incident-beam direction and the out-of-focus patterns would be exactly the same as the in-focus pattern. For other types of specimen, the contrast variations will

FIG. 7. Diagram suggesting the formation of bright- and dark-field shadow images in a defocused convergent-beam diffraction pattern with the beam crossover before the specimen.

20 J. M. COWLEY

depend on both the variation of diffraction angle and the variation of orientation and dimension of the lattice planes across the specimen area. In general this will give complicated images that may be difficult to interpret, but there are cases in which the variation of the structure is relatively simple and useful information may be derived.

For example, Fig. 8 is an out-of-focus CBED pattern of a thin crystal plate containing a dislocation. The variations of lattice plane orientation around the dislocation line are clearly indicated by the deflection of the dark diffraction contour in the bright-field image and the bright line in the dark-field image.

FIG. 8. Out-of-focus convergent-beam diffraction pattern from a graphite crystal containing a dislocation showing the perturbation of the bright- and dark-field extinction contours due to the local distortion of the crystal.

The use of the bright-field shadow images as a rapid and convenient method for associating information on specimen morphology with CBED patterns has recently been advocated by Dowel1 (1976). The resolution obtainable in such images is, in the incoherent approximation, comparable with that of STEM using the same lenses. It was shown by Cowley and Moodie (1957b) that in an ideal case the resolution could actually be better by a factor 2lI2 than for CTEM using the same lenses, but this does not represent an experimentally feasible imaging mode.

2. Wide-Angle CBED

If the angle of convergence of the incident beam is greatly increased by using a very large limiting aperture or none, the CBED pattern from a


single crystal changes its nature. The individual round diffraction spots become large, overlap, and merge to give a more-or-less uniform background on which are superimposed patterns of black and white lines, somewhat similar to the Kossel lines of X-ray diffraction or the Kikuchi lines formed from electrons diffusely scattered in thick crystals. Figure 9 shows such a pattern obtained from a thin crystal of silicon.

FIG. 9. Wide-angle convergent-beam diffraction pattern from a thin crystal of silicon.

The geometry of these patterns may be understood in terms of the geometry of the crystal lattice. Parallel sets of lines correspond to reflections from various orders of reflection from a given set of planes. The prominent line pairs come from low-index hkl reflections plus the corresponding &2 reflections. Patterns of this sort have been used by Goodman and Lehmpfuhl (1968), Cockayne et al. (1967), and others to give rapid and convenient determinations of crystal orientations, using the symmetry of the patterns to identify the principal axiaI directions in the crystal. The intensity distributions in these patterns are necessarily complicated and highly dependent on crystal thickness, orientation, and perfection as well as on the focus of

22 J. M. COWLEY

the incident beam, but interesting observations on this subject have been given by Fujimoto and Lehmpfuhl (1974).

The specimen area giving the wide-angle CBED patterns is necessarily greater than that for the CBED patterns discussed earlier, because the contribution to the electron beam diameter at focus, due to third-order spherical aberration alone, varies with a3. However, beam diameters of a few hundred angstroms appear to be feasible.

Smith and Cowley (1971) showed that for bent crystals the contrast shows striking variations with defocus. Pairs of strong black and white lines occur with a separation that is roughly proportional to defocus and to the curvature of the crystal in the direction at right angles to the lines. It was suggested that the separations of these line pairs could be used as a measure of crystal curvature. Also, since the defocused patterns took on some of the aspects of shadow images of extended regions of the crystal, perturbations of the lines by crystal defects, similar to those of Fig. 8, were observed and could conceivably be used to study local lattice strains. A further application of this type of wide angle pattern was suggested by Smith and Cowley (1975). The separations of line pairs in a pattern from a simple crystal of known structure such as silicon provide a convenient calibration of angles of electron beams leaving the specimen. This was useful, for example, in measuring detector angles in a STEM instrument for which the poorly defined power of electron lenses following the specimen made direct geometric measurements unreliable.

3. Grigson Scanning CBED

In the STEM instruments designed by Crewe and co-workers (Crewe and Wall, 1970) and the commercial and other instruments based on this design, no provision is made for the direct observation or recording of the diffraction pattern by use of two-dimensional detectors such as fluorescent screens or photographic plates. Instead, recourse is made to the method developed by Grigson (1965) for use in conventional electron diffraction (HEED) instruments. The diffraction pattern is scanned over a single detector of small aperture using deflection coils after the specimen, and the signal detected is used to modulate either the intensity or the y deflection of a cathode ray tube. (Figure 15 was obtained in this way.)

The advantage of this method is that a quantitative measure of the diffraction pattern intensity, in the form of an electronic signal, is provided in a form suitable for recording or display in a variety of ways. The main disadvantage is that it is very inefficient in its use of the electrons scattered by the specimen. Only a very small fraction of the scattered intensity is recorded at any one time so that, especially with the small detector sizes


needed to give reasonably high resolution in the diffraction pattern, the time to record the diffraction pattern is long and the radiation damage and contamination of the small specimen area illuminated may be severe.

Often in practice the diffraction patterns given when the STEM instrument is operated in a high-resolution imaging mode is too weak for convenient observation by the Grigson technique. Instead a more intense larger diameter beam (20-50A diameter) is formed by use of a weaker, preobjective lens to give a sharper diffraction pattern of higher intensity. The difficulty is then that the correspondence between the diffraction pattern and the image is less direct and more uncertain. A scheme involving a two- dimensional detector system, designed to overcome these difficulties, has been proposed by Cowley (1978).

C . Selected-Area Electron Difraction (SAED)

1. SAED in CTEM

The use of selected-area electron diffraction with CTEM is sufficiently well established and well known to require only a brief summary. The principle is illustrated in Fig. 3. In the imaging mode (Fig. 3a) an aperture placed in the image plane of the objective lens will select the part of the image coming from a very small area of the specimen. If the focal length of the intermediate lens is changed so that the diffraction pattern formed in the back-focal plane of the objective, rather than the image, is magnified on to the final viewing screen (as in Fig. 3b), then only those electrons diffracted by the selected area of the specimen will contribute to the diffraction pattern recorded.

For an aberration-free objective lens, the size of the selected area would depend only on the diameter of the selected-area aperture and the magnification of the objective lens and so could be made very small. However, the minimum size is severely restricted in practice by the spherical aberration of the lens. An analysis of the situation has been given, for example, by Hirsch et a/. (1965) and by Bowen and Hall (1975) and is illustrated in Fig. 10. For an object point on the axis, an electron beam diffracted at angle c1 ( x A/d, where d

FIG. 10. Diagram illustrating the limitation of specimen area from which a selected-area electron diffraction pattern can be obtained, due to spherical aberration.

24 J. M. COWLEY

is the lattice plane spacing) will be displaced from the corresponding image point by an amount MC,a3, where M is the magnification. Hence for a selected-area aperture of radius Ro the maximum diffraction angle included in the diffraction pattern will be amax = (MC,/R,)1i3. Alternatively, one can see that an electron beam scattered at an angle LY from a point that is off axis by a distance ro will pass through the on-axis point of the image, the center of the selected-area aperture, if roM = MC,a3.

Hence for a diffracted beam corresponding to a 0.5 I$ lattice spacing, for example, the uncertainty in the position of the diffracting region is of the order of CSct3 = 8C,A3, with 3, in angstroms, and the useful minimum size of the selected-area aperture corresponds to a region of diameter about 8000 I$ of the specimen for 100 keV electrons with C, = 2 mm.

These arguments give only a rough indication of the limitations of this method. It is usually assumed that the minimum useful selected area is about 1 pm for 100 keV electrons. However, it may be noted that the minimum dimensions of the selected area are strongly dependent on the electron wavelength. For electron microscopes operating at 0.5 or 1 MeV selected-area diffraction has been obtained from areas 500 A or less in diameter (Popov et al., 1960; Dupouy, 1976). In principle the diffraction pattern from a much smaller area of the specimen could be synthesized by choosing the appropriate parts of the intensity distributions of the diffraction patterns recorded for different amounts of defocus of the objective lens, but this procedure would be inconvenient and rarely feasible.

2. Microbeam Selected-Area Patterns

The limitation of SAED due to spherical aberration can be avoided if the selected area of the specimen is chosen not by an aperture in the image plane of the objective lens but by restricting the incident beam so that it illuminates only a very small region of the specimen. This is the method used to obtain CBED patterns from small areas as described above, but a number of microbeam methods have been devised with the aim of getting as close as possible to a “parallel-beam” sharp diffraction pattern from a small area.

Riecke (1962) produced a fine incident beam of small divergence by use of a triple condenser lens system. Two strong lenses were used to obtain a very small reduced image of the electron source and a third, long-focal-length lens served to image this on the specimen plane, giving an illuminated area approximately l00OA in diameter. Later (Riecke, 1962) he used a strong short-focal-length final condenser lens, actually the strong forefield of the objective lens, to produce incident beam spots in the specimen only a few hundred angstroms in diameter. The principle of this method is illustrated in Fig. 11.


FIG. 1 I . Use of a small crossover to produce a nominally parallel beam to give a microdiffraction pattern.

At one extreme, the beam defined by a small aperture could be focused on the specimen. This is CBED. At the other extreme, illustrated in Fig. 11, the electrons from a fine crossover could be focused by a short-focal-length lens to give a nominally parallel beam over a small area of the specimen. The appearance of parallel illumination is, however, an illusion derived from the geometric-optics diagram. The convergence angle of the beam on the specimen will be given by the ratio of the diameter of the crossover to the focal length of the lens.

The same basic limitation applies to both extreme cases and all intermediate degrees of focusing. The diameter of the specimen area illuminated is inversely proportional to the angle of convergence. The "paraIle1-beam" case differs from the usual CBED case only in that the diffraction spots, being images of a crossover, are diffuse maxima rather than sharply defined images of an aperture. Correspondingly the area of the specimen illuminated may be more sharply defined, e.g., with roughly a Gaussian shape, rather than the slowly decreasing [ J l ( x ) / x ] * form for the usual CBED case.

Thus the microbeam technique, used with a CTEM system to magnify the diffraction pattern and relate it to the image, can give results comparable to the CBED technique used in conjunction with STEM. For convenient use, however, it requires a specially designed condenser lens system not usually found in commercial electron microscopes.

26 J . M. COWLEY

D . Incident-Beam Scanning

In the diffraction modes we have so far considered, a fixed beam is incident on the specimen and the intensity of diffracted electrons is measured as a function of the angle of scattering. The same intensities will be observed if, with essentially the same geometry, the scattered-beam direction is fixed and the intensity is recorded as a function of the angle of incidence, i.e., with a fixed detector and a scanning system used to vary the incident beam direction systematically (Fig. 12).

Spec.

FIG. 12. Incident-beam scanning system to provide microdiffraction in a CTEM instrument.

This follows from application of the principle of reciprocity (Cowley, 1969a), which may be stated as follows: The amplitude (or intensity) of radiation at a point B due to a point source at A will be the same as the amplitude (or intensity) at the point A due to an equivalent point source at B. Applying this principle to all points of a finite incoherent source and a finite incoherent detector, we see that the intensity recorded with the detector in Fig. 12 will equal the intensity of the diffraction pattern produced by an incident beam from an incoherent source subtending the same angle p at the specimen.

It must be remembered, ofcourse, that the illumination in a CTEM instrument is not necessarily equivalent to illumination from a finite incoherent source subtending an angle equal to the angle of convergence of the incident beam. Especially if a field emission gun is used, the convergent incident beam may be almost completely coherent and the simple reciprocity relationship of Fig. 12 no longer applies.

In both STEM and CTEM instruments, deflector coil systems are usually present before the specimen and have been used to produce diffraction patterns by incident beam scanning.

In STEM instruments, a weak, long-focus lens instead of the strong objective lens is sometimes used to give a focused, high-resolution electron diffraction pattern from a relatively large area of the specimen, as in Fig. la. Use of the scanning coils and a fixed detector then allows this diffraction pattern to be displayed and recorded, using the scheme of Fig. 12.

The reciprocity relationship suggests that this arrangement is equivalent to the SAED method used in a CTEM instrument. The limitations are the same. We have seen that the effect of spherical aberration of the objective lens


on electron beams scattered through large angles puts a lower limit on the size of the selected area in the CTEM case. For the STEM configuration the effect of lens aberrations on beams incident at high angles will impose the same restrictions on the area from which the diffraction pattern is obtained. However, for the STEM case the effect of lens aberrations could be reduced by modulating the objective lens focal length in synchronism with the scan of the incident beam.

The use of incident-beam scanning with CTEM does not suffer from this limitation of the SAED method. The diffracted beam that is detected can be the axial beam, which is magnified to give an image of optimum resolution by the microscope lenses. The selected area of the specimen can be chosen from the high-magnification image seen on the final viewing screen. A small- detector aperture is used to select the specimen region and the intensity passing through the aperture is recorded as the direction of the incident beam on the specimen is varied.

This method was suggested by van Oostrum et af. (1973) and has been used by Geiss (1976) to obtain diffraction patterns from regions as small as about 30A in diameter. It has the advantage over the microprobe methods of microdiffraction in CTEM that it does not require a nonstandard illu- minating system. Also the heavy contamination and irradiation damage usually associated with a microbeam system are avoided because the normal broad-spot, near-parallel CTEM illumination can be used. Apart from the instrumental limitations of microscope instabilities and low signal strength, the method is limited only by the same fundamental factors as apply to the microbeam CTEM or STEM cases. In the absence of lens aberrations, the resolution obtainable in the diffraction pattern is inversely proportional to the diameter of the selected area of the specimen. The effect of spherical aberration of the objective lens is clearly seen in this case because it will limit the resolution of the final image and hence restrict the accuracy with which the selected area of the specimen can be defined.

E . Rejlection Microdiffraction

The methods described above for electron microdiffraction from thin specimens by transmission may, in principle, be applied equally well to the glancing-angle reflection electron diffraction from flat surfaces of bulk specimens (reflection high-energy electron diffraction, RHEED) or to the near-normal incidence diffraction of low-energy electrons (LEED).

While practical difficulties of electron optics have prevented any success with low-energy electrons, some limited results have been obtained for electron energies in the range 5-100 keV (Cowley et al., 1975; Nielsen and Cowley, 1976).

28 J . M. COWLEY

For 100 keV electrons the angles made by incident and diffracted electron beams with a flat crystal surface are usually around rad. An incident beam of circular cross section will therefore intersect the surface in a highly elongated ellipse, with major axis 100 times the minor axis. Hence the selected area giving the diffraction pattern will be 100 times as large as for the transmission case. In order to reduce this area one could consider reducing the electron energy so that the diffraction angles would be increased in proportion to A. However, for a probe-forming lens of the same focal length and C, value, the area of the minimum spot will be increased by a factor proportional to A3I2. Hence the minimum selected area possible will vary roughly with ,I1/’.

Similar considerations apply to the imaging of flat surfaces (Nielsen and Cowley, 1976). The resolution of the image formed by a scanning-microscope system will be approximately given by the probe diameter. As the accelerating voltage is increased and the diffraction angles decrease, the images formed by detecting the diffracted beams will become increasingly foreshortened and difficult to interpret even though the resolution in the one direction, perpendicular to the beam direction, will improve.

These conclusions apply, of course, only to surfaces that are very nearly planar. RHEED intensities from very small regions will be very strongly influenced by any waviness, steps, or projections on the surfaces. The diffraction patterns and images may be almost completely dominated by contributions from the tips of small surface irregularities.

In order to overcome the effects of this extreme sensitivity to surface morphology, the larger diffraction angles for lower-energy electrons are advisable. The energy range of 5-10 keV seems to offer the best compromise, although so far the beam sizes used for diffraction and diffraction imaging in this range have not been less than a few hundred angstroms (Cowley et al., 1975).

F. Optical Microdiflraction

The optical diffractometer has become a valuable accessory in electron microscope laboratories and has been widely used for the evaluation of imaging conditions by observation of the optical diffraction patterns obtained from selected regions of the photographic negatives of micrographs. The photographic plate is illuminated by parallel coherent light from a laser source and the diffraction pattern is observed and recorded at the back focal plane of a long-focus lens. For images of periodic objects, the optical diffraction pattern can be useful in revealing the presence and the nature of the periodicities in the image.

Since it is possible to obtain the optical diffraction pattern from particular areas of the photographic images of any size, the use of the optical diffracto-


meter has been regarded as an alternative to the use of electron microdiffraction techniques in the microscope. It has the great advantage that once the micrograph has been taken, it may be examined in detail before the areas for microdiffraction are carefully selected and the diffraction patterns can be obtained without further radiation damage or contamination of the specimen. The principal defect is that, except in special cases, the optical diffraction pattern does not have the same intensity distribution as the electron diffraction pattern, although the periodicities revealed are usually similar.

The optical diffraction pattern may give valuable information when used in conjunction with lattice fringe images to follow variations of periodicities in crystals corresponding to variations of composition or ordering, as in the cases of the spinodally decomposed Au-Ni alloy and the partially ordered Cu,Au alloy examined by Sinclair et a/. (1976). However, applications of this sort must be made with care. The fringes in the image, corresponding roughly in spacing, but not in position, with the planes of atoms in the crystal usually arise as a result of strong dynamical scattering. It is well known (Cowley, 1959; Hashimoto et al., 1961) that under simple two-beam diffraction conditions the fringe spacing may vary if there are changes in thickness or orientation of the crystal. The spacing may also vary in a bent crystal at places where other strong reflections are excited. The more general n-beam diffraction case is even more complicated.

The intensities of the optical diffraction pattern will resemble those of the electron diffraction pattern only under a very restricted set of conditions. For example, if the specimen is very thin and scatters weakly, the image intensity obtained in bright field at the Scherzer optimum defocus (Cowley, 1975) will be given approximately by

Ie(xY) = 1 + 2 g 4 ( ~ ~ ) * S(XY) (26)

where &xy) is the projection in the beam direction of the potential distribution of the object. If the amplitude of the light transmitted through the photographic plate is proportional to Ie(xy) , the optical diffraction pattern intensity will be given by I@(uv)~(uv)sinx(uv)1’, where @(uv) = 9 4 ( ( x y ) . This will be proportional to the electron diffraction intensity I@(uv)I’, only to the extent that the contrast transfer function of the lens, sinX(uv), is of constant amplitude and sign. However, even for this most favorable case, with optimum defocus, the optical diffraction intensities will be greatly reduced for small and large angles of scattering.

Should there be deviation from optimum focus, or any appreciable dynamical scattering, i.e., if second- or higher-order terms must be included in (26), or if the relationship of the light transmitted through the photographic plate to the electron image intensity should not be exactly linear, further perturbations of the image intensity will result. This may be illustrated by

30 J. M . COWLEY

reference to an idealized object with projected potential distribution

$(x) = A + 2B cos 27rx/a (27)

which, under the above assumptions would give electron and optical diffrac- patterns

I (u) = A’S(u) + B16(u - l/a) + B?S(u + l/a) (28)

i.e., a central beam and one diffracted beam on each side with amplitude B , modified by the contrast transfer function in the optical case. Unless o$(x) << 1, second- and higher-order terms will appear in (26), and (28) will be replaced by

I(u) = A’S(u) + 1 B,”S(u - n/a) (29) n

with appreciable intensities for several values of n. Even for an ideal optical diffraction arrangement the B,” coefficients will be modified in the optical diffraction pattern by the contrast transfer function of the electron microscope lens, which is in general complex, to give Bt # B,. If there is not a strictly linear relationship between the amplitude of the transmitted light and the electron image intensity, further higher-order reflections will be generated in the optical diffraction pattern, which will then be written

(30)

with Cn # B.’ # B,. Thus in each case there will be a central beam and more than one diffracted beam on each side and in general the optical diffracted beam intensities will not be the same as the electron diffraction intensities.

For greater crystal thicknesses or for different values of the lens defocus, the relative amplitudes and phases of the central and diffracted beams will vary. In particular, the crystal thickness or the defocus can be adjusted so that the diffracted beams are out of phase with the central beam so that instead of being given by (26) the intensity for the object (27) will be

I&) = A’S(u) + 1 C,2S(u - n/a) n

I ( x ) = A’ + 48’ cos ~ T C X / U = A’ + 2B2 + 28’ cos ~ T C X / U (31)

Then the optical diffraction pattern will contain no first-order diffraction spots nor any of the odd-order reflections of (30), whereas the electron diffraction pattern will contain strong first-order reflections and all odd- and even-order reflections of (29).

Thus the optical diffraction pattern may differ very markedly from the electron diffraction pattern in the relative intensities of the spots and also in the extent of the pattern.

However, if these limitations are fully appreciated and taken into consideration the optical diffraction patterns may provide a valuable means for


analyzing images even in cases where strong dynamical scattering is known to exist but the image is known to give an approximate nonlinear representation of the projected potential, as in the images of complex oxides obtained with atomic resolution (Cowley and Iijima, 1976). Arguments have been presented (Iijima and Cowley, 1978) to justify the interpretation of the local intensity variations of diffuse scattering in an optical diffraction pattern from an image of disordered material even though the overall intensity distribution of the pattern is very different from that of the electron diffraction pattern (see Fig. 13).

FIG. 13. (a) High-resolution electron microscope image of a disordered Nb,W oxide, (b) electron diffraction pattern from the same crystal, (c) optical diffraction pattern from the electron micrograph negative of (a).

IV. OPERATIONAL FACTORS

A. Radiation Damage

For many specimen materials the most important experimental limitation on the use of microdiffraction methods will undoubtedly be the radiation damage produced in the specimen by the incident electron beam. This

32 J. M. COWLEY

limitation is particularly severe for organic and biological materials but will be important for most substances other than the electrical conductors.

Some energy is transferred from the incident beam to collective excitations of the specimen crystals to give either correlated vibrations of the atoms (phonons) or collective oscillations of the nearly free electrons (plasmons), which decay rapidly, generating phonons. In either case the net result is a heating of the specimen.

Other processes of inelastic scattering of the incident electrons result in ionization, leading to the breaking or rearrangement of bonds in the case of molecular or homopolar crystals or the generation of localized defects in ionic compounds. In thin samples, molecular fragments or displaced atoms may be ejected or evaporated into the surrounding vacuum. In many cases, even for metals and semiconductors, the enhancement of diffusion processes under irradiation may result in phase changes, crystal growth, or annealing, or any process that brings the specimen more nearly into equilibrium with its environment.

A further source of radiation damage is the “knock-on” collisions of electrons with atoms, in which atoms are given sufficient energy to displace them from their lattice sites. Since the incident electron is relatively light, it must have very high energy in order to do this (e.g., 400 keV to displace copper atoms). Hence this type of damage is important only in high-voltage electron microscopy where it can be the predominant damage effect for metals for which ionization effects are relatively small.

The importance of radiation damage effects for various specimen materials has been extensively investigated and discussed in relationship to high- resolution electron microscopy (see, e.g., Glaeser 1974; Misell, 1977). The situation for microdiffraction may be judged by comparing the amount of intensity information required to define the diffraction pattern with sufficient accuracy with that needed to give a satisfactory representation of the image of the same area.

A commonly made, rather conservative assumption is that in order to adequately define the image intensity, it is necessary to detect lo4 electrons per picture element. A smaller number may suffice for dark-field images. In order to characterize the intensity distribution in an electron diffraction pattern with comparable accuracy, one would have to detect a similar number of electrons, one the average, at each of a large number of points, perhaps lo5 points if there is a large amount of fine detail in the pattern or as few as 10 points if one requires to know only the intensities of the stronger spots in the diffraction pattern of a single crystal of simple structure.

We may assume, as a reasonable compromise that one must detect a minimum of lo6 scattered electrons in order to define the diffraction pattern intensity distribution. Then the minimum area from which a diffraction


pattern can be obtained will have a diameter roughly 10 times the resolution limit for microscopy set by the radiation damage effects. Rough figures for this minimum diameter would then be, for example, l W A for 1-valine, 400 for polyethylene, and 30 A for phthalocyanine (Glaeser, 1974).

As an alternative indication, we may go directly to the observations on the decay ofcrystal diffraction patterns that were used as the primary evidence for radiation damage effects in many cases. The radiation dose required to substantially reduce the extent of the diffraction pattern of organic materials, has been found to vary from 0.25 electrons/A2 for hydrated biological specimens to 130 electrons/A2 for phthalocyanines (Misell, 1977). If we assume that lo6 electrons will be scattered from an incident beam of lo7 electrons, the rpinimum diameters of the area that can be selected for microdiffraction then vary from 1 pm for hydrated biological material to 300A for the phthalocyanines.

The use of microdiffraction from areas of 20A or less in diameter will therefore be restricted to a limited range of stable materials. These will mostly be electrical conductors for which ionization effects represent only transient perturbations of the cloud of free electrons.

For most specimens the heating by the incident beam will be a relatively minor contribution to the radiation damage effect. The increase in temperature of an electron microscope specimen may be as much as 100 or 200 degrees, although it is usually much less than this. In spite of the fact that higher current densities may be used for microdiffraction, the temperature rise may be no greater because the cooling of the specimen by radiation and conduction of heat becomes much more effective as the volume of the heated region is reduced.

B. Contamination

In the relatively poor vacuum systems of most commercial electron microscopes (about torr) the accretion of a layer of contamination on specimen surfaces is a well-known phenomenon, which may be a serious limitation to high-resolution microscopy. As the size of the incident beam decreases the importance of this effect increases. With an incident beam less than 100 A in diameter, a mound of contamination thousands of angstroms in height may grow on each side of a thin specimen in a few seconds, completely masking the diffraction from the specimen (Knox, 1976).

The contamination is believed to result from polymerization, under electron irradiation, of organic material in the system, coming from the pump oils, gasket materials, grease on the surfaces of structural components of the specimen stage, or from the specimen itself. It has been found by Isaacson et a/ . (1974) that contamination may occur even in the baked,

34 J. M . COWLEY

ultrahigh-vacuum system of a STEM instrument, presumably as a result of the introduction of mobile organic components of the specimen.

The accumulated evidence suggests strongly that the contamination deposit is built up mainly as a result of migration of the organic molecules across the surface of the specimen. As the molecules enter the intense electron beam they become ionized, polymerized, and partially graphitized. Their mean free path within the electron beam is of the order of hundreds or thousands of angstroms.

In normal high-resolution CTEM the incident beam is usually a few microns in diameter. As can be seen by subsequent observation at low magnification, a ring of heavy contamination may be formed around the boundaries of the beam. The central part of the incident beam is thus well guarded from contamination and a contamination rate as low as 1 &ec or better is commonly achieved.

If the incident beam is less than 1000 A in diameter the contamination will cover the entire beam area. The dimensions of the mounds of contamination formed may be determined by subsequently shadowing the specimen with a thin layer of evaporated metal (Fig. 14). The observations of Knox (1976) showed an almost linear initial increase in the volume of deposited material with time. The minimum diameter of the mound of contamination formed with an incident beam of 30 A diameter was about 150 A. For mounds 2 pm high formed in about 400 sec of irradiation, the diameter at the base was about 2000 A.

Obviously the further development of electron microdiffraction is possible only if contamination can be eliminated or, at the least, reduced by a very large factor. It is important that the amount of mobile organic material should be reduced to a very low level by use of a clean ultrahigh-vacuum system, by use of low-temperature specimen environments, and by removing

FIG. 14. Electron microscope image of contamination mounds formed by an electron beam of small diameter (less than 100 A) in increasing intervals of time, with metal shadowing to indicate the heights of the mounds (after Knox, 1976).


as much organic material as possible from the specimen grids and the specimen itself. Beyond this, various means have been suggested for preventing the migration of the organic material across the specimen surface.

The observation of the “guard-ring’’ effect of the wide incident beam in CTEM suggests that low-level irradiation of a large surrounding area of the specimen with an auxiliary electron beam would be effective. Since this could interfere with the observation of the diffraction pattern, the alternative of an intermittent irradiation of the wider specimen region is probably more convenient. Lehmpfuhl has found that if a large specimen area is irradiated briefly with a high-intensity electron beam, the center of the irradiated area will remain free of contamination for a period sufficient to allow series of measurements to be taken. This technique is obviously of limited value for radiation-sensitive specimens, although the level of background irradiation may be several orders of magnitude less than the irradiation of the specimen area being examined.

For specimens to be observed in clean ultrahigh vacuum conditions the most important source of contamination is the specimen itself and several means of pretreatment of the specimens have been reported as effective. Irradiation of the specimen with infrared radiation (Isaacson et al., 1974) apparently helps to clean it. Irradiation with ultraviolet light may be effective in fixing the molecules on the surface (Engel et al., 1977) and cooling the specimen reduces the mobility of the molecules. For heat-resistant specimens, baking at moderate temperatures may be very effective.

C. Instrumentaf Stability

It is well known that the performance of STEM instruments is very dependent on their mechanical and electrical stability. Any rapid or irregular fluctuation of the beam position relative to the specimen, due to movement of the filament, vibration of the column, or stray electric or magnetic fields, will degrade the resolution. In microdiffraction the size of the selected area and the precision with which its position can be correlated with an image depend even more strongly on these factors.

In STEM a slow drift of the beam or specimen will result only in a slight distortion of the image. A vibration of the instrument or a ripple on the deflecting or focusing fields will likewise produce only an image distortion if it has a frequency that is a multiple of the frequency of one of the scans. However, for microdiffraction any drift or vibration will be significant if it produces an appreciable motion of the beam relative to the specimen within the time taken to record the pattern (see Brown et al., 1976). It follows that the requirements for instrumental stability and freedom from stray fields are extreme and that optimum performance will depend on the development of

36 J . M . COWLEY

recording systems of high efficiency to allow patterns to be recorded in the shortest possible time.

The effect of these instabilities on the diffraction pattern intensities will be an incoherent addition of intensities from different specimen areas. It will resemble the effect of using a larger incoherent eIectron source (Cowley, 1976b). In general the coherent diffraction effects that depend on the relative phases of the diffracted wave across the beam diameter will be reduced. Some of the information that is unique to electron microdiffraction will be lost.

V. INTERPRETATION AND APPLICATION

A. Identification

The most immediate and obvious application of microdiffraction is for the identification of the crystalline phases present in very small volumes of sample materials observed by electron microscopy. With the usual limitations of selected-area diffraction in CTEM, it is not possible to obtain clear evidence on the nature of small particles, precipitates in a matrix, or the components of a multiphase mixture when the sizes involved are less than a fraction of a micron. The possibility of obtaining a diffraction pattern from an area 30 A or less in diameter, identifiable in a microscope image, represents an important advance in technique.

Applications of microdiffraction for these purposes to date have been mostly exploratory, designed to test the technique rather than to apply it systematically to the solution of structural problems, but systematic applications are beginning to appear,

Geiss (1976) used his scanning technique with CTEM to observe diffraction patterns from single grains in an aluminum sample and from portions of single asbestos fibers. Carpenter et al. (1977) using the STEM attachment of a CTEM microscope obtained microdiffraction patterns from separate phases forming a lamellar microstructure in a'Cu -Ti alloy, and with similar apparatus de Diego et al. (1976) were able to sort out the diffraction patterns due to small precipitates in a Ta-N alloy.

The detection of local ordering to form an ordered superlattice within a microdomain of an alloy has been demonstrated by Dunlop and Porter (1976) for metal carbides using the STEM/CTEM combination and by Brown et al. (1976) for Cu-Pt alloys using a dedicated STEM instrument. Chevalier and Craven (1977) have made a systematic study of ordering in the Cu-Pt microdomains with a dedicated STEM instrument, making estimates of the degree of order within 40 A microdomains from the intensities of superlattice reflections (see Fig. 15).


FIG. 15. Microdiffraction patterns obtained with a STEM instrument from microdomains of a Cu-Pt alloy approximately 40 A in diameter, showing the (hhh) row of spots. In (a) the direction of ordering is perpendicular to the beam; in (b) the direction of ordering is parallel to the beam.

38 J. M . COWLEY

The difficulties of this method are those of normal selected-area diffraction, compounded by the difficulties inherent in the use of small regions barely resolved in an electron microscope image. A single diffraction pattern obtained in an arbitrary orientation is rarely sufficient to allow identification of a phase, even in the favorable case that there is so much known of the system in advance that a single feature of the diffraction pattern, such as the presence of superlattice reflections, is sufficient to give an unequivocal indication on the information required. For known phases, tilting the sample to a particular orientation will usually give the desired result. For unknown phases, a series of diffraction patterns in well-defined orientations may be required. With the microscopes now available it is often a very difficult and time-consuming task to obtain diffraction patterns from exactly the same, barely resolved region with the sample tilted through a series of widely different angles. It is easier to obtain diffraction patterns at random from a large number of different selected areas in random orientation if it can be assumed on the basis of independent evidence that all the selected areas have equivalent structure. There is a danger in this that one can always be tempted to assume on the basis of insufficient evidence that the one selected area giving a clear interpretable pattern is typical of all the regions that appear superficially to be similar in a micrograph.

B. Symmetry

1. General

Electron microdiffraction patterns contain a great amount of information concerning the structure of small regions of matter. As in the more familiar case of X-ray diffraction patterns, the systematic derivation of this information involves the interpretation of first the geometry and symmetry of the diffraction patterns and then the intensity distributions. The differences from X-ray diffraction become evident early in this process, since it has long been realized that the well-known rules for X-ray diffraction on the evidence for crystal symmetry in the diffraction patterns do not apply for electron diffraction because of the very strong dynamical diffraction effects.

The detailed exploration and understanding of the relationship of electron diffraction intensities to crystal structure and symmetry has been strongly dependent on the development of electron microdiffraction techniques. The diffraction patterns from larger areas of crystalline specimens are usually complicated by the averaging over a wide and undefined range of crystal thickness, orientation, and perfection, and the intensities vary strongly with all these factors. Except for a few cases of unusually perfect and uniformly thin crystal samples, it is possible to obtain sufficient control over


all relevant experimental variables only by choosing a very small area of perfect crystal for which the thickness is shown to be uniform.

The relationship of diffraction intensities to crystal symmetry differs from that for X-ray diffraction in several respects. The coherent multiple scattering (dynamical diffraction) renders the relationship more complicated but at the same time provides the possibility for deriving a greater amount of information.

In situations for which more than two beams have appreciable intensity, Friedel’s Law does not hold, i.e., 1, # 1, in the absence of a center of symmetry. It is relatively easy in electron diffraction to find orientations for which the intensities of reflections corresponding to reciprocal lattice points symmetri- cally placed with respect to the origin are clearly not equal even though in geometrically equivalent situations. Hence the ambiguity of space group determination and in the determination of the sense of a polar axis, which limits kinematical X-ray diffraction, does not apply. This is demonstrated clearly in the CBED pattern from CdS, Fig. 16 (Goodman and Lehmpfuhl, 1968).

In kinematical X-ray diffraction, the so-called forbidden reflections, which are systematically absent as a result of the presence of screw axes or glide planes in the structure, are essential guides for space group determination. There are many observations of electron diffraction patterns from thick crystals or crystal surfaces in which the forbidden reflections not only occur but appear to be the strongest reflections present. They can result from coherent or incoherent multiple scattering of electrons. If the reflection corresponding to the reciprocal lattice vectors h and g occur, the reflection h + g can be produced by double scattering.

In the coherent diffraction case, the wave amplitude for a reflection I), can contain contributions arising from waves I)h and $ , - h for combinations involving all h and also for combinations of three, four, or more wave amplitudes. The resulting intensity will depend on the relative phases of all these contributions. Under certain conditions as in crystal orientations of high symmetry, these amplitude contributions can cancel out systematically, giving zero intensities (Gjq5nnes and Moodie, 1965; Moodie, 1972).

2. Two-Dimensional Symmetry

The conditions under which systematic absences (forbidden reflections) occur in electron diffraction patterns can most readily be appreciated by recourse to the real-space argument of Cowley and Moodie (1959; also Cowley et al., 1961) based on the concept of an initially plane wave pro- gressing through successive layers of a crystal with scattering through small angles only. First we consider the approximation that the three-dimensional

40 J . M. COWLEY

FIG. 16. CBED pattern From a thin CdS crystal in the [2130] projection showing the absence of a center of symmetry, but the mirror symmetry about the [OOOl] axis (horizontal) is clear (Goodman and Lehmpfuhl, 1968).

distribution of atoms in the unit cells can be ignored and only the projection of the unit cell contents on some central plane is relevant. This is equivalent to the assumption that the only reflections occurring are those corresponding to a planar section of reciprocal space perpendicular to the incident beam.

The symmetry of the wave function for a planar incident wave after passing through one slice, one unit cell thick, will then be the symmetry of the transmission function exp[ - io+(xy)] , where +(xy) is the projection of the potential distribution in the unit cell. Propagation of this wave by Fresnel


diffraction to the next slice will not affect the symmetry since it is represented by convolution with a propagation function having cylindrical symmetry. If the potential distributions of the first and second slices coincide exactly when projected in the beam direction, the wave leaving the second slice will have exactly the same symmetry as that leaving the first. For a perfect crystal of any thickness, therefore, the wave at the exit face will have the same symmetry as that of the transmission function of the unit cell, projected in the beam direction. The systematic absences in the diffraction pattern will then be exactly those in the corresponding reciprocal lattice plane for the phase object approximation.

It should be noted that these absences are not necessarily those given by kinematical theory. The symmetry of exp[ - io+(xy)] is not that of +(xy) in general. This is illustrated in Fig. 17a, which represents the [ 1101 projection

I-4

Wave function

(b-1) ( b - 2 ) FIG. 17. (a) [110] projection of the structure of silicon or germanium. (b) Diagrams illus-

trating the difference of symmetry of the projection with Fresnel diffraction for a twofold screw axis with (1) the atoms at the same z coordinates and (2) with atoms at different z coordinates.

42 J. M. COWLEY

of silicon or germanium. Kinematically, for spherical atoms, both the 200 and 222 reflections are forbidden. The 200 is absent because the projections of interatomic vectors AB and BC on the [200] direction are equal. However the potential fields of A and B overlap more than those of B and C. The function &xy) is additive for the overlapping regions but exp[ - icrc$(xy)] is not.

Hence the projections of exp[ - io4(xy)] for the AB and BC pairs are not equivalent and the 200 reflection is not forbidden dynamically. On the other hand, this effect of overlap has less effect on the projections for the 222 reflection, which remains very weak, and no effect on the 110, which remains forbidden. These arguments are consistent with the results of n-beam com- putations made by Dessaux et al. (1977) using the matrix method and by O’Keefe (private communication) using multislice calculations.

With even a small tilt away from the exact axial orientation, the condition of exact superposition of the projections of slices will no longer be valid and the forbidden reflections may appear. This is clearly seen in the C3ED pattern (Fig. 18) obtained by Goodman and Lehmpfuhl (1968) from CdS. The odd-ordered 00.1 reflections have zero intensity at the centers of their disks, which correspond to the exact axial orientation. These reflections are kinematically forbidden by the twofold screw axis in the crystal (a glide-line

FIG. 18. CBED pattern from a CdS crystal near the [lOTO] zone axis. The odd-order 0001 reflections (center horizontal row) show black bands corresponding to the dynamically forbidden reflections.


symmetry of the planar group pg). The disks also have zero intensities along the 00.1 line since tilts in this direction do not destroy the symmetry element of the wave function corresponding to the screw axis in the crystal. However, tilts in other directions do destroy this symmetry relationship and the reflections gain appreciable intensity.

Gjplnnes and Moodie (1965) have demonstrated that a further condition may give dynamically forbidden reflections, namely, when the incident beam is at the Bragg angle for the kinematically forbidden reflection. This may be understood by considering that atoms separated by half the lattice plane spacing will be illuminated by waves that are n/2 out of phase. The modifications of the wave functions due to dynamical effects in the corresponding rows of atoms through the crystal will then be independent and equivalent, giving waves at the exit surface that will cancel out when combined with an additional 742 phase change in the Bragg reflection direction.

3. Three-Dimensional Symmetry

In the previous section we made the assumption that the symmetry of a wave passing through a slice of unit cell thickness will be determined by the projection of the unit cell in the beam direction. This is not strictly true. An atom near the top of the slice will produce a perturbation on the wave function at the bottom of the unit cell that is different from that due to an atom near the bottom, because the Fresnel diffraction spread of the wave from the atoms will be different. Hence, as suggested in Fig. 17b, the wave function symmetry will reflect the symmetry of a twofold screw axis only if the atoms related by this symmetry element lie in a plane perpendicular to the beam or if, as suggested by Gjplnnes and Moodie (1965), the “forbidden” reflection is at the Bragg angle. For a threefold screw axis, the symmetry of the wave function can never be exactly that of the unit cell projection.

These modifications of wave function symmetry due to the three-dimensional nature of the distribution of atoms will usually be small, especially for thin unit cell slices (10 A or less for 100 keV electrons) and the resulting nonzero reflections will be very weak and rarely discernible in principal orientations. They correspond in reciprocal space to the involvement of reflections associated with reciprocal lattice points not in the plane perpendicular to the electron beam. Calculations of the magnitude of their contributions have been made by Lynch (1971) for the favorable case of gold crystals in [ 11 11 orientation. By analogy with X-ray diffraction jargon, they are sometimes referred to as “upper layer-line” interactions. They can be made to contribute appreciably to the intensities of CBED patterns by a suitable choice of crystal orientation, usually well removed from a principal axis orientation (see Goodman and Secomb, 1977).

44 J. M. COWLEY

These three-dimensional diffraction effects introduce symmetry relationships quite different from those encountered in X-ray diffraction. A systematic evaluation of these relationships was made by Gj#nnes and Moodie (1965) and subsequently in more detail by Buxton et ul. (1976). Goodman (1975) has shown that, on the basis of these symmetry relationships, it is possible to develop a practical method for three-dimensional space group analysis without any of the ambiguities of kinematical X-ray diffraction. The most difficult determination is probably that of “handedness,” or the absolute configuration for noncentrosymmetric space groups. Goodman and Secomb (1977; Goodman and Johnson, 1977) showed that, with an appropriate choice of orientations, giving enhanced sensitivity to the three-dimensional dynamical diffraction effects, determinations of this sort can be made with a few observations plus some relatively simple calculations. In this way they confirmed the right handedness of a-quartz.

4. Crystal Symmetry

Under dynamical diffraction conditions it cannot be assumed that the diffraction pattern reflects the symmetry of the unit cell contents. The electron wave interacts with the whole crystal, and it is the symmetry of the crystal that is relevant. The previous paragraphs have referred to cases that are ideal both theoretically and experimentally, in which the symmetry of the crystal is determined predominantly by the symmetry of the unit cell contents, with perfect crystal structures and crystal boundaries that introduce no further symmetry restrictions. However, these are special cases. Other special cases have been observed for which the effects of crystal symmetries, which differ from the unit cell content symmetries, are pronounced.

The diffraction pattern is given by the Fraunhofer diffraction from the wave function near the exit face of the crystal. This wave function results from transmission through the whole crystal and retains the influences of all portions of the crystal including the entrance and exit faces and any faults encountered in between.

A striking illustration of this is given by the CBED patterns of graphite obtained by Johnson (1972), which show very pronounced threefold symmetry in place of the usual, kinematical sixfold symmetry obtained with the beam parallel to the c axis. The threefold symmetry is that of a crystal having one or more stacking faults in the basal plane. While a very high proportion of the unit cells in the crystal have the usually hexagonal graphite structure, the lateral displacement at a fault plane reduces the symmetry of the wave function.

An equally striking although more subtle example is given by Fig. 19, which shows a CBED pattern obtained by Goodman (1974) from a thin MgO platelet with faces parallel to 100 type planes, tilted so that the beam is in the


FIG. 19. CBED pattern from a MgO platelike crystal with (100) faces with the incident beam along the [ 1 1 11 direction.

[ 11 11 direction. If the entrance and exit surfaces had been perpendicular to the incident beam, the pattern would have shown the hexagonal symmetry of the crystal structure viewed in this direction. However, for the tilted crystal with the entrance and exit faces making large angles with the plane normal to the beam, the symmetry of the crystal as seen by the incident beam

46 J . M. COWLEY

cannot be hexagonal but it can, at most, have twofold symmetry. The diffraction pattern of Fig. 19 shows only the mirror-plane symmetry indicated.

C. Intensities: Structure Analysis

Following the determination of unit cell dimensions and crystal symmetry, the next stage in the quest for structural information is the interpretation of the observed intensities in terms of crystal structure. While the term “crystal structure’’ can refer to the atomic arrangement, or more precisely, the potential distribution, throughout the whole crystal, including surfaces and defects, we use it here in the more common, restricted sense, referring to the potential distribution within one unit cell of the idealized periodic structure. This is the equivalent of the electron density distribution determined by the processes of X-ray diffraction crystal structure analysis. The determination of crystal structure in practice usually consists of the determination of the magnitudes and relative phases of the Fourier coefficients of the potential distribution +(r) (the structure amplitudes Qh). It is the occurrence of strong dynamical diffraction effects that complicates the interpretation of electron microdiffraction intensities, making the derivation of structure amplitudes more difficult but at the same time making the method intrinsically more powerful. The “phase problem” of kinematical X-ray diffraction does not apply. Relative phases as well as magnitudes of the structure amplitudes can be derived from coherent diffraction effects within the crystals. Also it appears possible that, in favorable cases, the observation of coherent interferences between overlapping diffracted beam spots in CBED patterns may provide direct evidence on relative phases even under kinematical scattering conditions (Cowley and Jap, 1976; Nathan, 1976).

Historically, structure analyses of crystals based on electron diffraction patterns from small single-crystal regions have followed one or the other of two extreme courses (Cowley, 1967). On the one hand, rough indications of structure have been obtained using patterns from specimens of poorly defined morphology and degree of perfection on the basis that, if there is sufficient averaging of intensities over crystal thickness and orientation, an interpretation in terms of kinematical theory will give reasonable results provided that forbidden reflections or other gross effects of dynamical scattering are absent or may be ignored. The assumptions of this approach are most nearly justifiable for light-atom materials. Recent examples of such analyses include the studies on relatively large areas of organic crystals by Dorset (1976) and of small areas of Ko.z,WO,, +,,) using selected-area techniques by Goodman (1976b).

At the other extreme are the highly accurate studies carried out under conditions for which the experimental variables are well defined, a small


region of perfect crystal is selected, and the dynamical diffraction effects are not only taken into consideration but are utilized as an essential basis for the accuracy of the structure factor determinations. These methods have been applied to a limited number of materials for which suitable crystals are readily available, for which the dynamical diffraction calculations are not too complicated, and for which the accurate determination of structure factors is significant in relationship to theoretical results on the ionization and bonding of atoms in crystals.

Very few of these investigations have relied only on the measurement of relative intensities of different reflections. The recent work of Goodman (1976a) using CBED patterns from areas approximately 100 8, in diameter of graphite crystals approximately 120 A thick greatly extended the earlier CBED work of Hoerni and Weigle (1949). As well as showing that graphite crystals contain twinned or faulted regions and also areas having an ortho- rhombic structure, Goodman was able to select regions of perfect hexagonal structure and determine their thickness with an accuracy of better than one unit cell. By comparison of spot intensities with the results of detailed n-beam calculations he was able to refine the structure amplitudes of some of the inner reflections and attained an accuracy of about 1%.

The use of CBED patterns to give rocking curves that may be used for structure refinement is more thoroughly established. The variation of the intensities of reflections with angle of incidence of the electron beam, as mentioned in Section III,A, is strongly dependent on dynamical interactions and can be used for determination of structure amplitudes with high accuracy. The methods used and early results have been reviewed previously (Cowley, 1969b) and will not be described in detail here.

Early work on MgO (Goodman and Lehmpfuhl, 1967) used patterns such as Fig 6, due as closely as possible to a set of reflections lying on a line through the origin. The dynamical interactions between diffracted beams were thus limited to this so-called systematic set so that the dynamical theory calculations involved relatively few structure amplitudes. Later work by McMahon (1969) showed that neglect of all reflections except the systematic set could lead to errors that vary strongly with orientation but may be as small as 1% for orientations carefully chosen. Taking such effects into account, accuracies of perhaps p/, were attained for some of the inner MgO reflections.

Determinations approaching these in accuracy have also been made for germanium by Shishido and Tanaka (1976).

It seems probable that precision determinations of structure amplitudes by microdiffraction methods will continue to provide valuable data on particular problems of fundamental interest but will not be a popular pastime because of the considerable amount of care and effort involved. However,

48 J. M. COWLEY

as the techniques of microdiffraction and the programs for n-beam dynamical calculations become more widespread, there will be a considerable expansion in the variety and sophistication of structure analyses applied to an increasing number of problems in solid-state science.

D. Disordered Systems

Microdiffraction has obvious potential as a means for investigation of disordered systems, although the instrumental performance is not yet at the level where this technique can be exploited fully. The disorder of the structure may take the form of a disordered occupancy by different types of atoms (or vacancies) on the atom sites in an extended crystal or it may take the form of a disordering of the atom sites as in amorphous or heavily distorted crystalline materials.

In each case, the questions of current interest relate to the possible existence and nature of small, well-ordered microdomains having dimensions in the range 10-50 A. It is very difficult to make deductions regarding these microdomains from the evidence of the usual diffraction patterns, which represent averages over diffraction effects from a very large number of microdomains having a great variety of shapes, sizes, and orientations. The possibility of obtaining diffraction patterns from individual regions of 10- 20 A in diameter is obviously important.

For disordered alloys, considerable evidence regarding the presence of microdomains of an ordered superlattice structure has been obtained by inference from diffraction patterns and by direct observation in high- resolution electron microscopes, particularly by use of dark-field images obtained from the diffuse superlattice diffraction spots. Although there are serious difficulties in interpreting high-resolution dark-field images (Cowley, 1973), the evidence is clear in some cases for the existence of well-defined regions in which the ordered superlattice structure is well established in one of its possible orientations relative to the average disordered lattice. These are mostly cases of quenched alloys in which local ordering has progressed to the extent of forming relatively large microdomains, as in the case of the Cu-Pt alloys studied by Chevalier and Stobbs (1976), where dark-field images showed highly reflecting patches averaging about 40 A in diameter. Microdiffraction from these regions (Chevalier and Craven, 1977) showed them to give the superlattice reflections corresponding to one or other of the possible orientations of the ordered, low-temperature Cu-Pt structures (Fig. 14).

For the more purely short-range ordered alloys existing above the critical temperature for ordering, it has been concluded from X-ray diffraction evidence that a description in terms of microdomains can be made if


the average microdomain dimensions are 10-30 A (Moss, 1962; Gragg et al., 1971) and that local configurations of atoms are not necessarily those of the low-temperature ordered structures (Clapp, 1971). Microdiffraction from small regions will clearly add further light on these questions.

For amorphous materials the standard diffraction evidence on structure comes from the very diffuse ring patterns, from which it is possible to deduce radial distribution functions giving correlations in the distances between atoms and their near neighbors. From this it is very difficult to distinguish among the extreme structural models, the microcrystalline model that regards the amorphous material to be made up of small ordered regions 10-30 A in diameter, and the random-network model in which the deviation from an ordered structure is progressive and continuous, being governed by energetically possible deviations in local interatomic bonding. Observa- tions by high-resolution electron microscopy of sets of parallel fringes extending over distances of about 15 A in images of amorphous germanium (Rudee and Howie, 1972) appeared at first to favor the microcrystalline model, but it was later shown that the evidence is indecisive because of an imaging artifact (MacFarlane, 1975).

Microdiffraction patterns from thin amorphous germanium films have shown a mottled appearance in place of the smooth continuous halos given by larger specimen areas (Brown et al., 1976; Geiss, 1976). This is to be expected because the diffracting regions contained relatively small numbers of atoms in partially ordered arrays and there was little of the averaging over the large number of local arrays that takes place in the usual diffraction experiment. Clearly, microdiffraction patterns taken from even smaller regions can give valuable information on local atomic arrangements, even though difficulties will arise from the fact that the diffraction patterns will refer to two-dimensional projections of regions of films that are usually thicker than the lateral dimensions of the selected areas.

E. Microdifraction in Relationship to Other Techniques

It has been emphasized in this chapter that electron microdiffraction is strongly and inevitably related to high-resolution electron microscopy, either CTEM or STEM. The relationship extends to the common instrumentation, the essentially complementary nature of the observations, and the body of dynamical diffraction theory used for both.

Microdiffraction is also an essential component of the relatively recent concept of combining an array of techniques under the heading of “analytical electron microscopy.” It is seen that it is of enormous advantage, particularly in materials science, to combine in one instrument, or a few compatible instruments, the capabilities for imaging a specimen in bright

50 J. M. COWLEY

or dark field, for obtaining microdiffraction to determine the crystal structure, and for making a chemical analysis of small areas by means of either X-ray microprobe analysis or electron energy loss spectroscopy.

In modern microscopes, and particularly in those equipped with high- brightness field emission guns, it is possible to detect the characteristic X-ray emission of the elements contained in specimen regions as small as a few hundred angstroms in diameter. The minimum detectable mass of an element may correspond to fewer than lo6 atoms (Joy and Maher, 1977). By observing the energy losses of transmitted electrons due to the excitation of the inner-shell electrons of the specimen atoms, it is likewise possible to identify the elements in regions a few hundred angstroms in diameter (Silcox, 1977). This method is particularly suited to the detection of relatively light atoms for which the X-ray methods are less sensitive.

In this context microdiffraction has been used to date mostly in its simplest form as a method for phase identification, but more sophisticated applications will undoubtedly appear with time. It is possible to envisage complete chemical and crystallographic analyses being made on an unknown crystal having dimensions less than 100 A.

In further developments of the combination of techniques it may be possible to go beyond the practice of obtaining separate although related pieces of information by sequential application of the various methods. It has been suggested (Cowley, 1976a; Cowley and Jap, 1976) that STEM imaging and microdiffraction may be combined to enhance the power of both. The greatest amount of information concerning the specimen obtainable with the STEM technique will be derived by recording the diffraction pattern in the detector plane for each image point in the STEM image. Combination of diffraction information from successive image points may assist in the derivation of structural information on a much finer scale than the image resolution. Pattern recognition techniques based on the diffraction intensities may allow the more efficient detection of particular atomic groupings.

With these and the many other possibilities now appearing it is evident that after a relatively long history of sporadic use and limited accomplish- ments electron microdiffraction may well grow to a technique of major significance in the near future.

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Carpenter, R. W., Bentley, J., and Kenik, E. A. (1977). Scanning Electron Microsc. 1,411. Chevalier, J.-P., A. A., and Craven, A. J . (1977). Philos. Mag. [8] 38,67. Chevalier, J.-P., A. A,, and Stobbs, W. M. (1976). In “Electron Microscopy/l976” (D. Brandon,

Clapp, P. C. (1971). In “Critical Phenomena in Alloys, Magnets and Superconductors,”

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Cowley, J. M. (1959). Acta Crystallogr. 12, 367. Cowley, J. M. (1967). Prog. Muter. Sci. 13, 269. Cowley, J. M. (1969a). Appl. Phys. Lett. 15, 58. Cowley, J. M. (1969b). Acla Crystallogr., Sect. A 25, 129. Cowley, J. M. (1973). Acta Crystallogr., Sect. A 29, 537. Cowley, J. M. (1975). “Diffraction Physics.” North-Holland Publ., Amsterdam. Cowley, J. M. (1976a). Ultramicroscopy 1, 255. Cowley, J. M. (1976b). Ultramicroscopy 2, 3. Cowley, J. M. (1978). Proc. Int. Conf. High Voltage Electron Microscopy, Sth, 1977 (in press). Cowley, J. M., and Iijima, S. (1976). In “Electron Microscopy in Minerology” (H.-R. Wenk,

Cowley, J. M., and Jap, B. K. (1976). Scanning Electron Microsc. 1, 377. Cowley, J. M., and Moodie, A. F. (1957a), Acta Crystallogr. 10, 609. Cowley, J. M., and Moodie, A. F. (1957b). Proc. Phys. SOC., London, Ser. B70,505. Cowley, J. M., and Nielsen, P. E. H6jlund (1975). Ultramicroscopy 1, 145. Cowley, J. M., and Rees, A. L. G. (1953). J . Sci. Instrum. 30, 33. Cowley, J. M., Moodie, A. F., Miyake, S., Takagi, S., and Fujimoto, F. (1961) Acta Crystallogr.

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Hoerni, J. (1950). Helu. Phys. Acta 23, 587. Hoerni, J. , and Weigle, J. (1949). Nature (London) 164, 1088. Howie, A., and Whelan, M. J. (1961). Proc. R. Sac. London, Ser. A 263, 217. Iijima, S. (1976). Proc. 34th Ann. Electron Microsc. Soc. Am. Meet. p. 490. Iijima, S. (1977). Optik (Sruttgart) 47,437. Iijima, S. , and Cowley, J. M. (1978). Proc. Znt. Symp. Order Disord. Solids, 1977 (in press). Isaacson, M., Langmore, J., and Wall, J. (1974). Scanning Electron Microsc. 1, 19. Johnson, A. W. S. (1972). Acta Crystallogr., Sect. A 28, 89. Joy, D. C., and Maher, D. W. (1977). Scanning Electron Microsc. 1, 325. Knox, W. A. (1976). Ultramicroscopy 1, 175. Kossel, W., and Mollenstedt, G. (1939). Ann. Phys. (Leipzig) [5] 36, 113. Lynch, D. F. (1971). Acta Crystallogr., Sect. A 27, 399. MacFarlane, S. (1975). J. Phys. C 8, 2819. MacGillavry, C. H. (1940). Physica (Utrecht) 7, 329. McMahon, A. G. (1969). M.Sc. Thesis, University of Melbourne. Misell, D. L. (1977). J. Phys. D 10, 1085. Moodie, A. F. (1972). 2. Naturforsch., Teil A 27,437. Moss, S. C. (1962). D.S. Thesis, Massachusetts Institute of Technology, Cambridge. Nathan, R. (1976). In “Digital Processing of Biomedical Images, 1976” (K. Preston, Jr. and

Nielsen, P. E. H$jlund, and Cowley, J. M. (1976). Surface Sci. 54, 340. O’Keefe, M. A. (1975). Ph.D. Thesis, University of Melbourne. O’Keefe, M. A., and Anstis, G. R. (1978). In press. O’Keefe, M. A., and Sanders, J. V. (1975). Acra Crystallogr., Sect. A 31, 307. Popov, N. M., Kasatochkin, V. I., and Lukianovich, V. M. (1960). Dokl. Akad. Nauk SSSR

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53


The Lifetimes of Metastable Negative Ions'

L. G. CHRISTOPHOROU

Oak Ridge National Laboratory Oak Ridge, Tennessee

and The University of Tennessee

Knoxville, Tennessee

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Negative-Ion Resonances (Modes of Formation of Metastable Negative Ions) B. The Range of Variation of the Lifetimes of Metastable Negative Ions

11. Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . A. Electron Scattering and Dissociative Attachment Experiments . . . . . . . . . . . . . B. High-pressure Electron Attachment Experiments . . . . . . . . . . . . . . . . . . . . . . . . C. Time-of-Flight Mass Spectrometric a n Resonance Techniques

111. Metastable Atomic Negative Ions . . . . . . ..... IV. Extremely Short-Lived Metastable Molecular Negative Ions . . . V. Moderately Short-Lived Metastable Molecular Negative Ions . .

A. The Lifetime of O;* at Near-Thermal Energies . . . . . . . . . . . . . . . . . . . . . . . . . . B. The Lifetimes of SO;*, C,H;*, and C,H,Br-* at Thermal Energies . . . . . . .

VI. Long-Lived Parent Molecular Negative Ions Formed by Electron Capture in the Field of the Ground Electronic State (Nuclear-Excited Feshbach Resonances) . . A. The Formation of Long-Lived Parent Negative Ions and Their Cross Sections B. Variation of the Autodetachment Lifetime with Incident Electron Energy . . . C. A Theoretical Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. The Lifetimes of the Parent Negative Ions of Nitrobenzenes . . . . . . . E. Biological Significance . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VII. Long-Lived Parent Negative Ions Formed by Electron Cap Excited Electronic State [Electron-Excited Feshbach Resonances (Core-Excited Type I)] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

VIII. Long-Lived Metastable Fragment Negative Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . IX. Autodetachment Lifetimes of Doubly Charged Negative Ions . . . . . . . . . . . . .

A. Atoms . . . . B. Molecules . References . . . . . . . . . . .

56 56 60 61 62 63 65 15 11 19 80 85

89 89

100 102 111 115

116 118 122 122 124 125

' Research sponsored by the Energy Research and Development Administration under contract with Union Carbide Corporation.

55 Copyright @ 1978 by Academic Press, Inc.

All rights of reproduction in any form reserved. ISBN 0-12-014646-0

56 L. G . CHRISTOPHOROU

I. INTRODUCTION

In this chapter, recent work on the lifetimes of metastable’ molecular negative ions is synthesized, and a wider understanding is endeavored. Previous discussions are extended, and a coherent picture is attempted. Special emphasis is accorded to the lifetimes of long-lived parent negative ions and their relation to molecular structure.

A. Negative-Ion Resonances (Modes of Formation of Metastable Negative Ions)

The negative-ion resonances of atoms and molecules have been discussed by many authors (see, for example, Taylor et al., 1966; Bardsley and Mandl, 1968; Schulz, 1973; Christophorou et al., 1977a). It is nonetheless instructive to elaborate on these briefly for the purpose of clarifying and aiding the discussions that will follow.

A bound electron in an atom or a molecule is characterized by a wave function *,, which can be expressed as

II/m cc exp[ - iEm(t/h)] (1)

where h = 2nh is Planck‘s constant and Em the energy eigenvalue. The probability density I$,l’ in this case is not a function of time, and the state is stationary. Atoms or molecules in excited stationary states can decay by photon (not electron) emission.

A negative-ion resonance (NIR) is formed when the incident electron attaches itself to a neutral atom or molecule for times longer than the electron’s normal transit time through the molecule, and they are viewed as nonstationary states of the electron-atom (molecule) system. For a NIR state, the energy is complex, Em - iir,, and the probability density

l#ml ’ a ~ X P [ - r m ( t / h ) l (2)

is time dependent (r, is the associated autoionization width). Such resonances are nonstationary states of the electron-molecule system, and they thus decay by electron emission with a characteristic lifetime

z, = h/r,,, (3)

Since rm is (for molecular systems) a function of the internuclear separation R, either the average value f, of rm is used in Eq. (3) or the value re of rm

The term metastable is used to describe unstable (superexcited) negative ions irrespective of their lifetime. The term autodetachment is used for the autoionization of the metastable negative ion, although autodetachment and autoionization are used interchangeably in the text.

THE LIFETIMES OF METASTABLE NEGATIVE IONS 57

RADIAL DISTANCE OF INCIDENT ELECTRON, p INTERNUCLEAR DISTANCE

FIG. 1. Schematic illustrations of (a) shape and (b) nuclear-excited Feshbach resonances. The symbols 10) and IR) designate, respectively, the electronic ground state of the neutral species and the negative ion resonance (from Christophorou et d., 1977a).

at the most probable internuclear separation R e . We shall refer to t, as the autodetachment lifetime. It is thus seen that all negative ions in their very initial step of formation are in a superexcited state, i.e., they possess sufficient internal energy to revert back to a neutral molecule and an electron.

Basically, four types of NIRs can be distinguished, depending on the mechanism by which the electron is trapped: shape or single-particle resonances, core-excited resonances (type I),4 core-excited resonances (type 11),4 and nuclear-excited Feshbach resonances.

Shape resonances result when the incident electron is trapped in a potential well arising from the interaction between the incoming electron and the neutral molecule in its electronic ground state. The neutral molecule exerts an attractive force on the incident electron while the relative motion of the two bodies gives rise to a repulsive centrifugal force. The combined effect of the attractive and repulsive forces is an effective potential exhibiting a barrier as shown in Fig. la. A shape resonance arises when the incident electron is trapped in the attractive region by the barrier. Shape resonances usually lie at low (0 to -4 eV) energies and have lifetimes ranging from - 10- to 2 10- l o sec. A shape resonance lies above the neutral-molecule potential-energy curve (or surface), i.e., the electron affinity is negatiue, and it decays back into the target plus a free electron. The decay, being in effect

There exist, of course, other types of nonstationary states in addition to NIRs, such as doubly excited atoms(e.g., the 2s2 and 2s2p states of He). These latter, however, are nonstationary states of the neutral atom-not the negative ion-and they decay to an electron plus a positive ion rather than to an electron plus a neutral atom as the NIR states do.

Core-excited (type I) and core-excited (type 11) resonances are also called, respectively, electron-excited Feshbach and core-excited shape resonances.

58 L. G. CHRISTOPHOROU

electron penetration through the potential barrier, is analogous to the c1-

particle decay of radioactive nuclei. The width and height of the barrier are functions of the dominant partial waves responsible for the formation of the resonance and naturally affect the rate of decay and thus 7,. It is further possible that the shape resonance may decay via dissociative attachment, a process illustrated schematically in Fig. 2.

A X t e

FIG. 2. Schematic illustration of the process of dissociative electron attachment via a shape resonance. The asymptote, A + X-, in the R,,., direction is as a rule different from that, AX + e, in the p direction (from Christophorou et al., 1977a).

In core-excited (type I ) resonances, the incident electron is captured with the simultaneous excitation of one of the electrons of the target molecule, i.e., these resonances arise when the interaction potential between the excited target molecule and the incident electron is strong enough to support a bound state. They lie below the parent state (usually 0 to -0.5 eV below the energy of the corresponding excited neutral state) and decay via autodetachment or, if energetically possible, also via dissociative attachment. When decay into the parent state is not energetically possible, such resonances have long z, because, although they can decay into some nonparent states, this involves a change in configuration and is therefore not fast.

Core-excited (type ZI) resonances are similar to type I, but they lie above the parent state to which they can decay. Type I1 core-excited resonances are also called core-excited shape resonances since their mode of formation is the same as the shape resonances except that the effective potential now arises from the attractive interaction between the incident electron and an


excited electronic state of the target molecule rather than the ground state. Core-excited shape resonances have properties similar to those of the ground state. Then since the potential barrier is formed by the angular momentum of the electron we expect p-, d-, f-wave resonances, but not, as a rule, s-wave resonances. Shape resonances decay preferentially into their parent states-and they are thus shorter-lived than Feshbach resonances- unless they exist barely above an inelastic threshold, in which case they become long-lived because of a very thick barrier.

Nuclear-excited Feshbach resonances are similar to type I core-excited resonances in that the negative-ion state lies below the parent ground state exhibiting a positive electron affinity. Unless in vibrational levels v’ higher than the lowest vibrational level v = 0 of the parent neutral state (see Fig. lb) nuclear-excited Feshbach resonances cannot decay into the parent state. A similar situation may exist for type I core-excited resonances. In nuclear- excited Feshbach resonances the loss of energy by the captured electron is solely to vibrational modes of the molecule.

Resonances and their energies are often associated with one (or more) molecular orbitals of the target molecule. Although such a description of the resonances is an approximate one, viewing the incident electron as going into one’ of the unoccupied molecular orbitals of the target molecule is a rather convenient way of describing the resonances in terms of the electronic structure of the neutral molecule. Recently negative-ion states of atoms and small molecules have also been viewed (Read, 1977; Spence, 1977; Wang and Christophorou, 1977a,b) within the energetics of the genealogical sequence

core (“grandparent”) state 5 neutral (“parent”) state

-negative + e ion (“daughter”) state Wang and Christophorou (1977a,b), in particular, related the electron affinity (EA) of an atomic or molecular ground (or excited) electronic state to the ionization potential (IP) of the neutral state through

EA = [ 2 ( 2 - p)2 - 1]IP (4)

where 2 is the charge of the core state, and p a parameter that depends on the electronic configuration of the neutral state, the change in orbital size and its degree of penetration into the core in going from the neutral to the negative ion, and the extra electron-electron and extra electron-core interactions. The authors used the observed dependences of p on the electronic configurations and determined reasonable values of EA from a knowledge of I P for a number of atoms and small molecules.

Often a superposition of configurations is necessary to adequately describe the resonance.


In big polyatomic structures, other modes of electron trapping are possible, such as geometrical changes accompanying the electron collision and capture processes connected with specific portions of the molecule. There is little known about them at the present time.

B. The Range of Variation of the Lifetimes of Metastable Negative Ions

Let us denote by c,u the rate constant for capture of an electron of velocity u by a molecule AX and by z, the rate constant for release of the electron for the process

( 5 ) AX + e(a) a A X - * A AX(*) + e(')

where the (*) indicates that AX may be left with some internal energy and the (') that the electron may have different kinetic energy than the captured one. It has been pointed out by Christophorou (1971) that the cross section for the formation of AX-* depends strongly on the position of the resonance and varies by 10-11 orders of magnitude, from to 2 1 O - l 4 cm2. The lifetimes of metastable negative ions vary by an even larger factor; they range from sec, i.e., by more than 14 orders of magnitude. This is to be contrasted with radiative deexcitation lifetimes of excited neutrals, which are 2 10-8-10-9 sec, and to autoionization and preionization lifetimes of superexcited neutral atoms and molecules, which are as a rule6 c

Depending on the magnitude of z,, three classes of negative ions have been distinguished (Christophorou, 1971).'

(1) Extremely short-liued (10- 5 z, 5 10- l 2 sec): these are observed as resonances in electron scattering and/or in dissociative attachment studies.

(2) Moderately short-lived 5 z,5 sec): these can be stabilized in high (1 - - 1000 torr) and very high ( > 1000 torr) pressure swarm experiments.

sec): these can be conveniently studied with conventional time-of-flight mass spectrometers.

to

sec.

(3) Long-lived (z, >

Long-lived autoionizing states of atoms with lifetimes in the microsecond region have been reported [e.g., for the alkali atoms (Feldman and Novick, 1963)l.

' Both the lower and upper limits of C, quoted for each class are not stringent. Each class represents the approximate range of C, over which the three experimental methods-electron scattering, high-pressure swarm, and TOF mass spectrometry-can be successfully employed to determine T ~ .


Autodetachment is not the only mode of decay of AX-*. There are two other channels through which AX- * can be destructed:

9 AX'"' + e"' (64

% (6b)

AX- + energy (6c) where k, is the rate constant for autoionization of AX-* (indirect elastic or inelastic electron scattering), k,, the rate constant for dissociative attachment, and k,, the rate constant for stabilization of AX-* by radiation and/or by collision. The rate constant k , for radiative stabilization of AX-* is usually considered small, as can be seen from the Einstein coefficient for emission of vibrational energy (Herzberg, 1950):

AX + e A, A X - * A'') + X-'''

p1-0 = 3 2 n 3 D ~ , v ~ , / 3 c 3 h (7)

where Dol is the dipole matrix element of the 0 -, 1 transition, vo l the frequency of the transition, and c the speed of light. However, radiative stabilization can play a significant role in stabilizing AX-* when z, is large, when the probability of stabilization by collision is small, or when the excess energy of AX-* is large and radiative decay into many lower-lying states is possible. On the other hand, since the cross section for vibrational to translational energy transfer is small, the energy removed from AX-* in collisions with a stabilizing body S resides in the preponderance of collisions as internal energy in S. In this respect, energy levels (rotational and vibrational in almost all cases considered here), density of states of S, and degeneracy in the vibrational levels of S and AX-* affect the probability of stabilization p of AX-* upon collision with S.

If all three channels in process (6) are operative simultaneously, the lifetime z of AX-* is given by

5 = l/(ka + k,j, + k,,), 7 < 7, (8)

11. EXPERIMENTAL METHODS

As discussed in Section I, the particular experimental method appropriate for the measurement of z, (or z) depends on the size of the negative- ion lifetime itself. Lifetimes of extremely short-lived negative ions are obtained from electron scattering or from dissociative attachment studies, lifetimes of moderately short-lived negative ions from analyses of high- pressure electron attachment swarm data, and those of long-lived negative ions by using time-of-flight (TOF) mass spectrometers and/or ion cyclotron resonance (ICR) techniques. These methods will be discussed briefly.


A . Electron Scattering and Dissociative Attachment Experiments

NIRs decay by electron emission into many final states, and electron scattering and electron transmission experiments have been employed to detect such resonances by measuring the structure in the energy dependence of the excitation cross section of any state that lies energetically below or near the NIR. The search for structure in the various cross sections for elastic and inelastic electron scattering includes vibrational excitation and dissociative attachment for the case of molecules. From such studies, it has been possible to determine the autoionization width and thus z, through Eq. (3). Values of z, for a number of species determined this way are given in Sections I11 and IV [additional data on small molecules and atoms can be found in Schulz (1973)l. The magnitude of z, affects crucially the extent to which the transient anion is distorted from the neutral molecule’s equilibrium configuration. This in turn influences the distribution of vibrational levels of the ground electronic state of the neutral molecule to which the anion decays.

Dissociative electron attachment data for isotopic species can be used under certain assumptions to provide estimates of z,. To illustrate how such estimates can be made, let us refer to Fig. 3, where dissociative attachment to a diatomic or a “diatomic-like” molecule is illustrated. The attachment process is viewed as a Franck-Condon transition between the initial (neutral molecule AX plus the electron at rest at R + m) and the final (the negative ion state AX-*) state. Let us denote by oo the cross section for

= AUTOIONIZATION REGION

A t X + e

R4 Rc

INTERNUCLEAR SEPARATION, R

FIG. 3. Schematic potential energy diagram illustrating the dissociative electron attachment process for a diatomic or a “diatomic-like” molecule. l’€’,I2 and IYHl2 schematically represent, respectively, the squares of the zero-point vibrational wave functions for the deuterated and nondeuterated analogs.


formation of AX-* at an internuclear separation distance R,. Once formed, AX-* will either decay by autodetachment, a process possible for R 2 R, , or by dissociation into A + X- . Beyond R,, AX-* will dissociate with unit probability. For cases such as schematically illustrated in Fig. 3, the dissociative attachment cross section bda can be expressed as

(9) - aoe-(r. / t ,) b d a -

where Ts is the mean time taken by A and X - to separate from the point of formation at Re to the point at R = R , beyond which the negative-ion curve lies below that for the neutral molecule and thus autodetachment is no longer possible, i.e.,

u(R) is the velocity of separation of A and X- and is a function of R,Z,( = h/Ta) is the mean autodetachment lifetime, and r, is the mean autoionization width (r, and thus z, are functions of R).

Let us now assume that b d a has been measured for a diatomic molecule such as H2 and its deuterated analog (D2 or HD). If for two isotopic species no is assumed to be affected only by the square of the zero-point vibrational wave function ( c c M : / ~ , where M, is the reduced mass of A, X), Ts is assumed to be proportional to M,"', and 7, is assumed to be the same for the two isotopic species, then an estimate of z, can be obtained by applying Eq. (9) and by using the measured g d a for the two species. Schulz and Asundi (1967) applied the above procedure to the case of H2, HD, and D2 and obtained T,(~Z;) N 0.7 x lo-'' sec for the 3.75 eV resonance of H;*, HD-*, and D;*.

In a similar fashion, Christophorou et al. (1968) estimated the za of HCl-*, DCI-*, HBr-*, and DBr-* to be - 4 x sec using their measured dissociative attachment cross sections as a function of electron energy for these systems.

In view of the many assumptions intrinsic in these calculations and the often complex dependences of b d a on M , (Christophorou, 1971), this is not an accurate method, but it provides an order-of-magnitude estimate of za .

B. High-pressure Electron Attachment Experiments

Such experiments can yield, indirectly, lifetimes of moderately short- lived negative ions from appropriate modeling of the measured pressure dependences of the rate of electron attachment, the modeling itself depending on the assumed reaction schemes. They have been conducted with both unitary and binary gas mixtures. In the latter case, the electron-attaching


molecule under study is in very minute amounts compared with the number density of the stabilizing body S, whose pressure ranges from a few to - 60,000 torr (Christophorou, 1971, 1976). For a given ion AX-* the rate constant for the reaction

(10)

depends, often very strongly, on the nature of S . From such experiments and ensuing analyses, one finds the concentration [S] of S at which k , , [S ] = k,, i.e., the number density of S at which the rate for stabilization of AX-* via collisions with S is equal to the rate of autodetachment of AX-*. Once this “critical density” of S is found, k,, and thus k , can be obtained from k , , [ S ] = k,p, where k , is the collision rate of AX-* and S , and p is the probability of stabilization of AX-* at each AX-*$ encounter. The quantity p is I 1, and it can vary considerably depending on the nature of the AX-*,S pair. Usually k, can be estimated from the Langevin classical cross section for spiraling collisions, which is quite appropriate for such slow ion-molecule collisions. Thus one may write for the collision frequency v, between AX-* and S

AX-* + s A X - + s + energy

V, = voLNc (11)

where v is the relative velocity of AX-* and S (the product uoL is averaged over the distribution in velocity of both AX-* and S ) , N , is the number density at which k , , [ S ] = k,, oL (for a nonpolar’ molecule S with static polarizability a) is

and M , is the reduced mass of AX-* and S . Accordingly, we write

k , , [ S ] = v,p = 2aN,(e2a/M,)1/2p (13)

and since p I 1, t,(AX-*) 2 l /v , . Thus, although for certain ion-neutral pairs, p can be reasonably well assumed to be - 1, for certain others p << 1. Of course, p can be a function of the internal energy of AX-* and thus for a given AX-*$ pair a function of E. In this manner then, one obtains only lower limits to z,(AX-*), the major uncertainty originating from a lack of knowledge of p .

For a polar S, uL is rather difficult to obtain accurately (see, for example, Su and Bowers, 1973a,b).


Although high-pressure swarm experiments provide only an indirect method of determining .r(AX-*) and the lifetime value obtained may be influenced by the kind of the assumed electron attachment reaction schemes and the modeling of the observed density dependences of the attachment rate, the latter are measured absolutely over a wide range of mean electron energies and densities of S . This allows not only for a pretty good estimate of the magnitude of T as a function of ( E ) but also for a significant insight into the negative-ion process itself [see Section V and Christophorou (1971, 1976)l. Presently, ultrahigh-pressure experiments constitute the only method for determining lifetimes of metastable anions in the range - to - sec.

C. Time-&Flight Mass Spectrometric and Ion Cyclotron Resonance Techniques

Time-of-flight (TOF) mass spectrometers have been employed extensively in the direct determination of the lifetimes of metastable negative ions in the range - sec, while ion cyclotron resonance (ICR) techniques in the range 2 lop3 sec. Since a great deal of direct determinations of z, have been made using TOF mass spectrometers, a description of this type of measurements is indicated.

to -

1. Time-of-Flight Mass-Spectrometric Determinations of z,

The use of TOF mass spectrometers to measure autodetachment lifetimes of long-lived negative ions was first made by Edelson et al. (1962) for SF;*. Subsequently, extensive studies were made on many systems by the Oak Ridge researchers. TOF mass spectrometers offer two other advantages in addition to allowing a direct measurement of .ra : they permit an identification of the masses of the metastable ions and also a study of the energy dependences of both the metastable-ion yields and the autodetachment lifetimes.

A schematic diagram of the TOF mass spectrometer employed by the Oak Ridge groups is shown in Fig. 4. Electrons from a heated filament are pulsed (every - 100 psec and for -0.5 psec) by applying a positive pulse on grid l’, these are made quasi-monoenergetic (resolution -0.1 to -0.25 eV) by the retarding potential difference (RPD) grid assembly following grid 1’ and enter the collision chamber where they may be attached to molecules AX forming AX-*, and if not they are collected in the electron trap. About 0.5 psec after the injection of the electron pulse (i.e., -0.5 psec after grid 1‘ was made positive) a potential pulse of -100 to -150 V is applied on the backing plate, pushing AX-* toward the flight tube. Upon


COLLIMATING MAGNETS

ACCELERATION r

PUMPING SYSTEM

LINER I I

1

ELECTRON t- MULTIPLIER

1 HORIZONTAL DEFLECTION

PLATES

I- 1 ;-L I -

.-?

FIG. 4. Schematic diagram of the TOF mass spectrometer (dimensions not in scale) used by the Oak Ridge groups. Invariably the dimensions y , I , and L ranged from - 1.4 to 1.8 cm, 56 to 61.5 cm, and 146 to 150 cm, respectively. Also, either one or two ion focus grids were used.

entering the flight tube at x = 0 all ions are given the same kinetic energy T (variable from 1 to 6 keV) and travel the distance L to the detector under field-free conditions. From the time tAX it takes the ions to travel the distance L, their mass MAX is determined from

MAX = 2T(tm/L)2 (15)

MAX = MR(tA&d2 (16)

or, as is usually the practice, from

where MR and t R are, respectively, the known mass and the measured transit time of a reference ion. By changing the incident electron energy E the yield (current) Zm-* as a function of E is obtained. For most of the long-lived ions observed so far (Section VI), the negative-ion currents attained their maximum intensity at -0.0 eV, although in a few cases they were found to maximize at higher energies, still, however, below 1 eV. Three examples are shown in Fig. 5.

Let us now assume that at time t = 0, NT negative ions AX-* are injected into the flight tube of the TOF mass spectrometer and that at x = 0 they all attain a kinetic energy T. As these ions travel along the flight tube, some lose their electron and change into neutrals, but they still travel with the same velocity and reach the detector at x = L, where both neutrals and survived negative ions are detected with equal efficiency. Thus the intensity of the detector signal under these conditions gives a measure of NT. If now a "flat-top" potential is applied on the horizontal deflection plates [ f 500 V


FIG. c , ELECTRON ENERGY IEVI

The negative-ion current versus electron energy for (a) CzCI; (0) and SF; * ( 1, (b) p-C6H,N02NH;* (0) and SF;* (O), and (c) 2-CIoH,CHO-c(0) and SF,* (0). The data for (a) were taken from Johnson et al. (1977) and for (b) and (c) from Hadjiantoniou et al. (1973a,b). For (a)-(c) the SF;* current was normalized to the respective AX-* current at the peak.

with respect to the liner; Collins et al. (1970a)l located at x = 1 (see Fig. 4) and the measurement is repeated, the ions that have survived autodetachment in traveling the distance from x = 0 to x = I will be deflected and removed from the beam, and thus they will not reach the detector at x = L. Under these conditions, therefore, only the neutrals N o pass through the plates and are detected. The quantity N T - N o gives the number N - of AX-* that have not decayed within the time t, that takes the ions to travel the distance between the acceleration grid (x = 0) and the deflection plates (at x = I ) . If it is now assumed that the autodetachment of the AX-* ions follows the exponential decay law

N - = N T exp( - t,/z,) (17)

z, can be determined from a measurement of N-/NT and t , . Usually, N - / N T is measured for a number of values oft, (the Iatter being varied by changing the kinetic energy of AX-*, i.e., by changing the liner voltage), and z, is then obtained from the inverse slope of

-InN-/NT vs. t , (18)

which should be a straight line passing through the origin for a single decay process. Examples of this procedure are shown in Fig. 6, At times, the negative-ion signal may be too weak to make such plots, and in these cases, 7, is determined by averaging many values of z, calculated from individual


1.2 -

‘0 2 4 6 8 U I2 Y llME ff fLlOll1,tly-l

1.2 -

1.0 -

0.0 -

‘ 0 2 4 6 0 0 I2 I4 16 TIME W FLIGYT. IIpW

O J -

0.a -

TIME DT FLIWT. I U

FIG. 6. Plot of -In(N-/N’) against time offlight c for 12 NO,-containing benzene derivatives (discussed in Section VI). The measurements were made at the energy (-0.0 eV) of the maximum of the respective negative-ion resonance [based on the work of Johnson et al. (1975)l.

measurements of N - / N T and t l . Separation of the neutral- and negative-ion components of the “ion packet” can be achieved not only by deflection but also by retardation. For an outlay and description of electronics and for further details, see Collins et al. (1970a) and Hadjiantoniou et al. (1973~).

In these experiments, the various researchers carefully considered and eliminated a number of possible sources of error that can affect the lifetime measurements such as

(1) destruction of A X - * by processes other than autodetachment (e.g., collisional detachment, collision-induced dissociation),

(2) collisional detachment and charge transfer in the flight tube, (3) detachment by grids in the path of the ion beam, (4) discrimination due to stray fields and focusing by TOF ion optics, and (5 ) differences in detector sensitivity for neutrals and negative ions.

TOF mass-spectrometric studies also allowed detection of the variation of z, with incident electron energy E. Although this important finding was discovered through these techniques, the measured variation of z, with E is less pronounced (see Fig. 7 and discussion in Section VI) than it actually is


350

300

250

- ::

3 200

I

W

w

A

z

W

-

0

? 450

3

8 z

100

50

0 0 0.2 0.4 0.6 0.8 q.0

ELECTRON ENERGY (eV)

FIG. 7. Experimental and calculated lifetimes for 1,4-CIoH60;* as a function of electron energy normalized at the lowest energy point (see text] (from Christophorou et al., 1973b).

mainly because of the large width of the electron pulse (a measure of which can be estimated from the data in Fig. 5 ) especially in those cases where the RPD is not applied due to weak signals (a measure of the electron pulse width in this case can be seen from the ion current versus E data in Fig. 8a). In spite of the large width of the electron pulse, it has been possible for this molecular anion to obtain data such as shown in Fig. 8b, which dramatize the effect of incident electron energy on 7,. Had the electron beam been truly monoenergetic, fa(&) would be a very much stronger function of E although because of the sharpness of G,(E) always a small fraction of the electrons in the pulse lead to the observed AX-*. This explains why in some cases, such as SF;*, the 7, as measured in TOF mass spectrometers was not found to vary with E, although such a variation is expected. In this connection, a combination of TOF mass spectrometers and high-resolution electron spectrometers would be a desirable improvement, but problems with beam intensity may arise.

70

I I M

L. G . CHFUSTOPHOROU

225 -

200 -

- 8 g 5

- 6

- 5 5

G e - 4

- 3

150

I I

i

I I

50 - I i

0 0.2 0.4 0.6 0.8 1.0 1.2 ELECTRON ENERGY lev1

0.5 rV, ra= 35 pmc

0.3 eV. ra= 99 pmc

4- 0 y r, .I87 prec

0 2 4 6 8 (0 12 14 16 I , TIME-OF-FLIGHT IpseCI

0. 1

0 2 4 6 8 (0 12 14 16 I , TIME-OF-FLIGHT IpseCI

FIG. 8. (a) Dependence of the negative-ion lifetime (0) and the negative ion current ( x ) for the m-C6H,N0,CF; * metastable ion on the incident-electron energy. (b) Determination of T~ for m-C,H,NO,CF;* at incident-electron energies equal to -0.0, 0.3, 0.5, and 0.7 eV (from Johnson et al., 1975).

When z, is not observed to vary with E, for all practical purposes the measured lifetimes correspond to E 2: 0 eV, since the measurements were, as a rule, made at the peak of the negative-ion resonance. When, however, z, decreases with E, it is difficult to assign a specific value of z, to a specific value of E. [It is not clear, for example, what significance should be attached to the lifetime data at apparent electron energies <O.O eV (broken parts in Fig. 8a.)] The lifetimes for those ions whose z, varied with E (Table VII, Section VI) are those at the peak of the associated negative-ion resonance.

It should finally be noted that because z, is a strong function of E , the measured values of z, are sensitive to the source conditions, especially


to the temperature of the collision chamber, which is, as a rule, higher ( - 60-70°C) than ambient due to heating by the filament. This may account in part for some of the observed differences in the values of z, of certain negative ions (e.g., SF6, see Table VII, Section VI) obtained by the various groups of investigators, all of whom used TOF mass spectrometers. Since z, is sensitive to temperature, it would be desirable to measure the z, for vibrationally “cold” molecules, a rather difficult task.

2. Ion Cyclotron Resonance Determination of z,

The ICR technique was first applied to the study of long-lived negative ions (SF, * and c-C,F,*) by Henis and Mabie (1970). This study concluded that the z, of SF;* is 2 500 psec, i.e., very much longer than TOF mass spectrometric determinations (25 to - 68 psec). Subsequent ICR studies by Odom et al. (1974,1975) and by Foster and Beauchamp (1975) also found the lifetime of SF; * to range from - 100 psec to > 2 msec, depending on experimental conditions. Let us now elaborate briefly on the ICR .technique as it has been employed by Odom et al. (1975) for the measurement of z, (SF;*).

A beam of low-energy electrons is introduced into the ICR cell at position 1 (Fig. 9a) by pulsing a grid “on” for times between 50 and 100 psec. These electrons produce SF, * and SF; and also inelastically scattered electrons, which drift through the ICR cell with a characteristic velocity ud = E/B, where E is the magnitude of the static electric field applied via the drift plates, and B is the magnetic field strength. Since v, is mass independent, all charged particles (ions and electrons) take the same time z,,, (residence time or TOF) to travel from position 1 (position of incident electron beam) to position 2 (position of total ion current collection at the end of the cell). The charged particles in the ICR cell oscillate in a plane parallel to the magnetic field with a characteristic frequency

where VT is the potential applied to the trapping plates, d is the distance between the trapping plates, and m and q are, respectively, the mass and charge of the particle. If a radio frequency (RF) electric field is applied to one of the trapping plates at the trapping plate frequency of a particular ion, this ion absorbs energy from the resonant electric field, its oscillation amplitude increases, and the ion is ejected, striking one of the trapping plates and being neutralized.

Let us now assume that at some time tdelay (< ere,), after the primary electron beam has been modulated “off,” an initial burst of RF of time duration r2j is applied to the trapping plates of one side of the cell and ejects all

72

( e l I I I I I I I I I

r‘e’ I I

e- BEAM PULSE -

~, ‘delay

A A V V ”

L. G . CHRISTOPHOROU

TIME (psec)

9

7

< 4-

5

3

i I 1 1 1

i

0 I I 1 I

0.2 0.6 1 .o lei I msec)

FIG. 9. (a) Basic pulse scheme used by Odom et a/. (1975) to measure the lifetime of SF;* with an ICR technique. (b) Semilog plots of the normalized SF;* ion current against tei for delay times of 0.1 (a), 0.4 (m), and 1.2 msec (A). The primary electron pulse “on” time was 50 p e c (from Odom et a/., 1975). (c) SF;* “lifetimes” as a function of tej for delay times of 0.1 (a), 0.4 (m), and 1.2 msec (A) (see text and Odom et a/., 1975).

free electrons from the cell. The ion current collected after this ejection of unbound electrons gives the ion current I , at time to = tdelay + t2j. If now I , is the total ion current at time t = rdelay + tej > to (the duration of the RF pulse tel > tZj), and the autodetachment process is assumed to follow first-order kinetics, then

ln(Z,/I,) = - kat (20)


where k, is the autodetachment rate constant. From Eq. (20), we see that k, can be obtained from a measurement of the current ratio I J I 0 as a function of tej , the duration of the electron ejection pulse.

Odom ef al. (1975) studied thermal electron attachment to SFs, which under the normal operating conditions of ICR experiments can be quite complex as is indicated by the reaction scheme

e(t.,) + SF, Td ’

SF;* I SF‘,“ + e(c,)

)/ (21)

SF, + h v SF; + F SF, + S“’

In (21), O,(E;) is the attachment cross section for an electron of energy ci and an SF6 molecule without excess vibrational energy, a r ) ( ~ ~ ) is the attachment cross section for an electron of energy ef and an SF:) molecule with possible vibrational energy (*), and .ra- k,, kd, and k,[S] are, respectively, the rate constants for autodetachment, radiative stabilization, autodissociation, and collisional stabilization of SF;*. Odom et a/. (1975) argued that under their experimental conditions the probability of collisional and radiative stabilization of SF, * is small’ and that the decomposition of SF, * to form SF; + F is negligible. Also, they argued that the electron ejection R F does not affect the transmission of the SF;* ions through the cell and that the probability of a neutral molecule capturing an electron within the time the electrons are ejected from the ICR cell is small.

In Fig. 9b, typical decay curves obtained by Odom et al. for SF:* are presented, showing that the ln(ZI/Zo) versus tej plots are not linear as predicted by Eq. (20), but substantially curved. This would indicate that the SF;* ion under their experimental conditions is not characterized by a single k , . Odom et al. evaluated the slopes of tangent lines to the experimental curves at various points along the curves in Fig. 9b and deduced the “apparent lifetimes” shown in Fig. 9c. It is seen that the autodetachment lifetimes of SF;* determined in this manner depend on tej and are longer for a fixed value of te, the longer the [delay is; they range from -0.25 msec (tdelay =

0.1 msec) to -2 .5 msec (tdelay = 1.2 msec). As will be discussed in Section VI, the lifetime of a metastable polyatomic

negative ion is a strong function of its total internal energy. In both the TOF and IRC types of experiments, SF;* ions can be formed in a number of internal energy states. The longer lifetimes observed for SF;” (and some

’ These assumptions, however, are not valid for longer observation times ( - 10 msec), for which the radiative process could be important. Evidence for this has been obtained by Foster and Beauchamp (1975), who studied SF;* at longer observation times than Odom et al. (1974, 1975).

74 L. G . CHFUSTOPHOROU

other ions such as c-C4Fi*) in ICR experiments compared with those measured for the same species in TOF mass spectrometers and the nonexponential decay of SF;* observed in the ICR experiments could be rationalized in terms of formation of the AX-* ions in a number of autodetaching states with a spectrum of lifetimes that depends on the internal energy of AX-*. The most obvious mechanism of forming ions with different amounts of internal energy is by capture of electrons with a different distribution of kinetic energy and/or capture of electrons by neutral molecules with varying amounts of internal energy. This becomes even more important when (as is the case for most AX-* under discussion) the attachment cross section is very large at -0.0 eV and decreases sharply with increasing energy above zero.

3. Rationalizing the Diflerences between the Autodetachment Lifetimes as Measured in TOF Mass-Spectrometric and ICR Studies

The differences between the autodetachment lifetimes as measured using TOF mass spectrometric and ICR techniques can be rationalized by considering the different conditions under which the lifetime measurements were made. Of these, two seem to be the most significant: (1) differences in the electron energy distribution and (2) differences in the observation times.

In connection with the first condition, in TOF mass spectrometers the metastable ions (say SF; *) are formed directly by the incident-electron beam, which crosses the collision chamber only once. This beam may contain only a small fraction of subthermal electrons. The lifetime measured in TOF mass spectrometers is therefore indicative of the internal energy states of the ions that are formed with the initial electron energy distribution. On the other hand, in ICR experiments the scattered electrons can make many oscillations (passes) before capture occurs and the ions may be formed by capture of electrons with very much lower energy than in the TOF experiment. These slower electrons can be formed at the turning point of the oscillation and also by repetitive attachment-detachment processes. In either case SF;* ions formed in the ICR experiment could have a lower average internal energy and thus a longer lifetime than in TOF studies. Accordingly, Henis and Mabie (1970) attributed the longer lifetime they measured for SF,* in an ICR study to the capture of electrons with a lower kinetic energy distribution than in the TOF studies.

In connection with the second condition, TOF mass spectrometers are characterized by a short time delay (5 1 pec) between the formation of SF; * and the ensuing TOF analysis and a small time window (tens of microseconds for SF;*). In contrast, in ICR experiments there exists a long delay (time


window - 1 msec) between the initial electron injection and the detection of SF,*. Thus in the two techniques a different portion of the lifetime distribution is sampled; the autodetachment process that occurs in tens of microseconds would be the one most readily observed in TOF mass spectrometers, while those processes that occurred in the order of milliseconds would be the ones observed in the ICR study. That is, due to the long observation times employed in the ICR experiments, only long-lived species are observed, the short-lived ones having decayed. In this regard, and contrary to Henis and Mabie's explanation, Odom et al. maintained that in both TOF and ICR experiments SF;* ions are formed with the same lifetime distribution (i.e., both experiments are characterized with the same distribution of electron energies), but the measured lifetimes are different in the two kinds of experiments because of different observation times."

From the preceding discussion, we may conclude that the longer autodetachment lifetimes in the ICR experiments, the exponential decay observed in TOF mass spectrometric studies, and the nonexponential decay observed in ICR experiments at long observation times, can be rationalized by assuming that the metastable ions are formed in a number of autodetaching states (which are most likely different in the two experiments) and that the measured lifetime depends on the experimental time of observation (which is very much longer for the ICR studies).

111. METASTABLE ATOMIC NEGATIVE IONS

In contrast to the case of molecules, resonances in atoms are mostly associated with excited states (core-excited). NIRs associated with ground- state atoms have been established for the alkalis. Although in this chapter the discussion is concerned with metastable molecular negative ions, in this section we note that even for monatomic molecules (atoms) we can distinguish extremely short-lived (z, 5 10- l 2 sec), moderately short-lived (10- l 2 5 z, 5

sec) metastable negative ions. Since the long lifetimes of atomic negative ions cannot be attributed to vibrational redistribution of the ion's excess energy (as we shall discuss in Section VI for the long-lived polyatomic negative ions), the long lifetimes of metastable atomic negative ions must be due to specific electronic (orbital) configurations that inhibit fast electron autoejection. This mechanism can also be operative in certain polyatomic molecular negative ions (see Sections VI and VII).

l o These authors also showed that the autodetachment process in an ICR experiment can follow an exponential decay if the observation time is short enough so that a narrow region of the lifetime distribution is sampled, an observation that is consistent with TOF mass spectrometric studies.

sec), and long-lived (z, F


Lifetimes of some extremely and some moderately short-lived metastable negative ions are listed in Table I. As an example of long-lived atomic negative ions, we give in Table I1 the lifetimes for the He-* ( 1 . ~ 2 ~ 2 ~ ) metastable negative ions. Although the nuclear field in the helium atom is strongly attenuated by the atomic electrons (closed 1s shell) at distances corresponding to shells with principal quantum numbers n > 1 and the He- ion is thought not to exist in the ls22p('S) ground state, the He- in the( ls2~2p)'P,~, ,3129

configurations exists and is quite long-lived. Recently, Hiraoka et al. (1977) presented experimental evidence for the existence of long-lived (z, > 10- sec) N-* (ID) [and possibly N-* ( ' S ) ] anions for which a situation similar to that for He-* ('P) is expected.

A number of other long-lived metastable excited-state atomic negative ions have been predicted semiempirically, observed in mass spectrometers (thus their z, 2 sec), or studied by field ionization and laser photo-

TABLE I

EXPERIMENTAL LIFETIMES OF SOME EXTREMELY

SOME MODERATELY SHORT-LIVED ATOMIC NEGATIVE ION STATES'

SHORT-LIVED AND THEORETICAL LIFETIMES OF

Metastable Position of ion NIR (eV) Ta (set)

H- * ('S) 9.56 15.3 x 1 0 - 1 5 c

('D) 10.128 90 x 1 0 - 1 5 d

He-*(%) 19.3 82 x 1 0 - 1 5 '

H -*('S) 10.151 3.5 x lo-"/

(W 9.738 117.5 x

10.150 3.2 x ('P) 10.179 2.9 x lo -"/ ('P) 10.177 1.5 x

Determined from the reported autoionization widths through Eq. (3).

All these resonances are for the hydrogen atom below n = 2. See Schulz (1973) for theoretical estimates of rm for these and other NIR states in hydrogen below n = 2.

' McGowan (1967); experimental.

'Golden and Zecca (1970, 1971), Gibson and

Burke (1968); theory, close-coupling calcu-

@ Burke (1968); theory, close-coupling calcu-

Sanche and Burrow (1972); experimental.

Dolder (1969); experimental.

lation.

lation plus correlation.


TABLE I1

LIFETIMES OF THE He-* (ls2s2.p) METASTABLE NEGATIVE ION^

Metastable Position of ion resonance (eV) 5, (set)

19.75 > 10-5b 1.7 x 10-3'

4.55 x 10-4d (3.45 f 0.9) x 10-4'

(5.0 2) x 10-4f

(1 kO.2) x (1.82 0.27) x 10-58*h

(1.6 0.4) x lo-'/ (1.1 k 0.5) x

_ _ _ ~ ~~ ~~ ~

The electron binding energy of He-* (ls2s2p) was reported to be 0.067 e'J (Estberg and LaBahn, 1970) and 0.080 f 0.002 eV (Brehm et al., 1967) in a laser photodetachment experiment.

Sweetrnan (1960); experimental. Pietenpol (1961); theory. Estberg and LaBahn (1970); theory.

Novick and Weinflash (1971): experimental. This value is low; it appears that it represents a mean lifetime

for all three 4P9 states of He-. According to Blau et al. (1970), this is due to the short drift-path lengths used in the experiment of Nicholas et al. (1968).

' Blau et al. (1970); experimental.

Nicholas et al. (1968); experimental.

detachment electron-spectroscopy techniques. These include (Hotop and Lineberger, 1975) C-* (2D), Si-* ('D, 'P), Ge-* ('D?), Sn-* (2D?), Pb-* (2D?), Be-* (4P?), Mg-* (4P?), Ca-* ("P?), Sr-* (4P?), and Ba-* ("P?). The autodetachment lifetimes of these have not yet been determined. Currently, there is very little quantitative information on the states of such ions.

IV. EXTREMELY SHORT-LIVED METASTABLE MOLECULAR NEGATIVE IONS

The lifetimes of such metastable negative ions are determined principally from the linewidths of NIRs studied in electron-scattering experiments. Theoretical estimates have also been made for a few simple diatomics. Examples of extremely short-lived metastable molecular negative ions are given in Table 111.

It is worth noting the differences in the lifetimes of the shape and core- excited Feshbach resonances of the isoelectronic molecules N2 and CO


TABLE 111

EXAMPLES OF EXTREMELY SHORT-LIVED METASTABLE MOLECULAR NEGATIVE IONS

Metastable Maximum of NIR ion (eV)

3.75b 11.40 11.40 11.43 13.63 2.3

11.48 -2 10.04

-0 to 1.5 -0.8' - 0.3b

2.3 3 to - 4

Shape resonance. ' Peak of dissociative attachment resonance. ' Determined from the observed isotope effect in the dissociative attachment cross section.

' Schulz and Asundi (1967); dissociative attachment. The autoionization width is strongly dependent on the internuclear separation R.

Comer and Read (1971), Joyez et al. (1973), Schulz (1973); electron scattering. Ehrhardt and Weingartshofer (1969), Weingartshofer et al. (1970); electron scattering. This lifetime is for r = 0.13 eV at the equilibrium distance (2.3 au) of N2-; r = 0.8 eV

Krauss and Mies (1970); theory. Ehrhardt et al. (1968); electron scattering.

Ir Core-excited Feshbach resonance. ' Schulz (1973); electron scattering.

" 9.4 x lo-" sec for the first vibrational peak at -0.2 eV. ' Spence and Schulz (1971); electron scattering. Ir Possible assignment based on discussion by Rohr and Linder (1976).

at the N2 equilibrium distance (2.0 au) (Krauss and Mies, 1970).

Sanche and Schulz (1971); electron transmission.

Christophorou et al. (1968); dissociative attachment. Burrow and Sanche (1972); electron scattering. ' This is the second shape resonance of N 2 0 ; the first one lies at -0.0 eV (Azria et al.,

1975). The lifetime corresponds to a 0.7 eV width at the center of the Franck-Condon region (Dube and Herzenberg, 1975).

Dub6 and Herzenberg (1975); theory.

since they show the validity of earlier statements as to the factors affecting za. Thus in the CO ('n) shape resonance, the trapped electron (1-3 eV) tunnels through a p-wave barrier, while in the N2 ('ll,) shape resonance the trapped electron (-2.3 eV) tunnels through a d-wave barrier. Since the


p-wave barrier is not as high as the d-wave barrier, z, is expected to be shorter for the CO ('n) than for the N, (211,) shape resonance. This is clearly seen in Fig. 10 where the energy dependence of the resonant excitation of the v = 1 state of N,, CO, and H, (Ehrhardt et a/., 1968) are shown; for N;* and CO-*, 5, is sufficiently long for vibrational structure to be resolved. Similarly, the NO shape resonance should contain a p-wave component in the partial wave, while the 0, shape resonance a d-wave component. The lower barrier height associated with a p-wave dominated electron escape in the case of NO agd the higher barrier height associated with a d-wave dominated electron escape in the case of O2 accounts for the shorter (- sec) lifetime of the former and for the longer ( - lo-'' sec; see Section V,1) lifetime of the latter. ir--

V ' l - . . . . . . I . . . . , . . . .

1.0 2.0 3.0 b.0 10 2 0 3.0 1? 0 2.0 4.0 6.0 m.0- collision energy C e V 3

FIG. 10. Comparison of the energy dependences of the resonant (via shape resonances; see Table 111) excitation of one v = 1 vibrational quantum of N2, CO, and Hf. It is seen that as 5, becomes shorter in the order N,, CO, and H2, the vibrational structure is progressively washed out, and it disappears entirely for H2. Ehrhardt et al. (1968) reported rN2- = 0.15 eV, Tco- Y 0.4 eV, and rHi u 3 eV (from Ehrhardt et al., 1968).

The lifetime of the core-excited state 'XC,. of N;* at 11.48 eV is (Table 111) about two orders of magnitude longer than the lifetime of the core-excited state 'C+ of the isoektionic molecule CO, although compound-state formation in N, and CO is expected to be similar. The much shorter lifetime of the 'Z' resonance of CO most probably reflects the opening up of an additional channel of decay of the resonance, namely, that of dissociative attachment. The 'X' resonance state of CO can partially decay into 0- ('P) + C (3P), but no similar dissociative attachment process has been observed for N,. Thus, the natural width of the 'ZC,. state in N, would be small, and 5, long, since this state decays predominantly to the ground state of the molecule.

V. MODERATELY SHORT-LIVED METASTABLE MOLECULAR NEGATIVE IONS

In this section of the lifetimes of O;*, SO;*, C6H;*, and C2H5Br-* determined recently from analyses of high-pressure electron swarm studies are discussed.


A. The Lifetime of 0; * at Near-Thermal Energies

Vibrationally excited negative ionic states of the type

e + o,(x3q; v = 0) --+ o;*(x*n,: v’) . (22)

where v and v’ are the vibrational quantum numbers of 0, and O;, respectively, form at thermal and near-thermal (5 1 eV) energies. Electron scattering experiments allowing observation of the electrons autodetached via the process

O;*(X2n,: v’) + 02(x31; ; v) + e (23)

can yield an estimate of the “natural” widths of the various v’ states of O;* and thus estimates of the lifetimes of these states. In this manner Linder and Schmidt (1971) estimated T,{O;*(X~II,; v’ = 9)) = 1.3 x lo-’, sec.

If the transient O,* is embedded in a dense gaseous medium, it can be stabilized by collision with another body S, viz.,

O;* + S 0; + S + energy. (24)

From the pressure dependence of the rate constant for reaction (24), it is possible to evaluate the lifetime of 0; *. Earlier efforts to determine ~ ~ ( 0 ; *) via Eq. (24) seem to have been permeated by errors in the energy dependence and the magnitude of the rate constant for (22). Recently, Goans and Christophorou (1974) studied electron attachment to 0, in mixtures with dense gases, and they deduced a lifetime for O;* (X211,; v’ = 4) equal to -. 2 x 10- l 2 sec. In view of the substantial difference between this value and those calculated theoretically (see later this section), it is considered necessary to discuss the results of Goans and Christophorou and their corresponding analysis, especially on 0, in C,H4 (ethylene).

The rate of attachment of electrons to 0, in C2H4 shown in Fig. 11 has been found to be consistent with the reaction scheme

e + 0, -L o;* (254

o;* A o2 + e (25b)

(254 o;* t C , H ~ --h 0; + C ~ H ~ + energy

which predicts

1 k , 1 - +-- 1

(aw)O kl k l k 3 PC2H4

In Eq. (26), (aw)o is the measured attachment rate for Po, + 0 torr, and PkZH4 is the ethylene pressure corrected for compressibility. The experimental data on (crw),, versus Pk2H4 in Fig. 11 are plotted in Fig. 12 in the manner suggested


7

3

f - 0

Q g o o t ' " I ' I ' " ' I 0 4 8 12 16 20 24 28

P;~~CARRIER-GAS PRESSURE (to3 Tow)

FIG. 11. Attachment rate (aw)o for O2 in N2 (0) and C2H4 (0) as a function of the carrier- gas pressure. The data plotted are for E / P & values equal to 0.03 V cm-' torr-' for N2 and 0.1 V cm-' ton- ' for C2H4. These E/PZ9* values correspond to a mean electron energy of -0.05 eV. (aw), are the measured attachment rates for Po, + 0 torr. The carrier-gas pressures were corrected for compressibility (from Goans and Christophorou, 1974).

FIG. 1 0.1 V cm-

1.0

0.9

0.8

0.7 - b 5 0.6

'9 0.5

9

% 0.4

u lo

- 5

0.3

0.2

0.1

0 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

I I P ; ~ ~ ( I O - ~ T ~ ~ ~ ' )

2. I /(aw), as a function of 1/P298 for 0, in C2H4. The data plotted are torr-' ((6) 1 0 . 0 5 eV) (from Goans and Christophorou, 1974).

for E/P',, =


by Eq. (26). The data plotted are for ( E ) = 0.05 eV and are consistent with Eq. (26). From least-squares fits to six such plots in the range 0.05 - 0.064 eV, Goans and Christophorou (1974) obtained k , = 2.33 x lo7 sec- torr- and k 2 / k 3 = 10,700 torr. These two quantities are quite important, since the former yields an estimate of the rate of attachment of thermal electrons to O2 at a density corresponding to that of liquid C2H4 and the latter can be used, as described in Section II,B, to determine z,(O; *). The quantity k 2 / k 3 gives the medium pressure at which the rate of autodetachment of 0; * is equal to the rate of stabilization of O;* via collisions with C2H4. Through Eq. (14), Goans and Christophorou (1974) obtained 7,(O; *) = ( k 2 ) - = l/v,p, using for N , the number density at 10,700 torr and for tx the static polarizability of CzH4. It can be argued" that p 2: 1 for each 0;*/C2H4 collision, in which case one obtains ~ ~ ( 0 ; " ) N 2 x 10- l 2 sec. This value is in agreement with an earlier estimate made by Christophorou (1972a) based on electron attachment to O2 in high pressures of N2, but it is considerably shorter than earlier determinations deduced from low-pressure (of the order of a few torr) swarm experiments (Prasad, 1966; Chanin et al., 1962). The lifetime of 2 psec is considered to be representative of the 0; * (v' = 4) since the attachment rates in Fig. 12 are for ( E ) N 0.05 eV, and the electron-scattering experiments position (Table IV) the 0; * (v' = 4) level of 0; at 0.082 eV above the v = 0 of the neutral 02.

The theoretical estimates of z,{O;*(X*II,; v' = 4)) are summarized in Table V and are very much larger than the experimental values of Goans and Christophorou (1974). For this reason a discussion of the theoretical work and a further scrutiny of the swarm analysis is in order.

Because the extra electron in 0; is in a ng orbital, it has as its lowest component an 1 = 2 wave for which the penetrability of the barrier varies as E " ~ . Herzenberg (1969) made use of this fact and extrapolated from higher energies to v' = 4, for which he assumed an energy-integrated capture cross section equal to 2: 10- l 8 cm2 eV and obtained an autoionization width of -2 x eV, which yields a lifetime of - 3 x lO-"sec. This value is a rough estimate, dependent on the theoretical assumptions made and the assumed energy integrated cross section.

" As stated in Section lLB, p depends on AX-* and the stabilizing body S, especially its vibrational energies, its density of vibrational states, degeneracy of vibrational states of AX-* and S, and perhaps also the internal energy of AX-* and therefore the energy of the captured electron. When S is a polyatomic molecule or when S = AX (in which case the levels of S are nearly degenerate with those of AX-*), p may indeed be close to unity. This seems to be the case for the O;*/O, collision pair (Chanin et al., 1962; Herzenberg, 1969). Since the three-body coefficients for e + O2 + 0, -, 0; + O2 and e + O2 + CIH4 -+ 0; + C2H4 are not too dissimilar (Goans and Christophorou, 1974; Christophorou, 1978), it would seem that the assumption that p for the O;*/C2H4 pair is close to unity is not unreasonable (see Christophorou, 1978).

THE LIFETIMES OF METASTABLE NEGATIVE IONS

TABLE IV

THEORETICAL ESTIMATES OF THE LIFETIMES OF

0; *(XTl,, v’ = 4, . . . , 10)

83

VI 4 5 6 7 8 9 10 Ref. ~~~

T (lo-’’ sec) 88 9.2 3.6 1.9 1.3 0.9 0.7 a 165 18.3 5.5 2.5 1.4 0.9 0.6 b

E (eV) 0.082 0.207 0.330 0.450 0.569 0.686 0.801 c

a Parlant and Fiquet-Fayard (1976); theoretical analysis based on the electron scattering

Koike and Watanabe (1973); configuration interaction calculation [values based on

Linder and Schmidt (1971); electron scattering data [based on their data they estimated for

data of Linder and Schmidt (1971).

electron scattering data of Linder and Schmidt (1971)l.

the lifetime of O;* (X2n, , v’ = 9), a value equal to 1.3 x lo-’’ sec].

TABLE V

THE LIFETIME OF

o;* (XZrI,, v‘ = 4)”

Lifetime sec) Method

330 Theoryb 165 Theory‘ 12 Theoryd 88 Theory“

2 Experimentf

’ Located at 0.082 eV above the 0, (X’ZZ,, v = 0) state.

Herzenberg (1969). Koike and Watanabe (1973). Koike (1973, 1975). Parlant and Fiquet-Fayard

Goans and Christophorou (1976).

(1974).

Koike and Watanabe (1973) performed a configuration interaction calculation and also predicted that the autodetachment width should vary as ~ ~ ’ ~ ( 2 , cc E - ~ ’ ~ ) . Based on the experimental data of Linder and Schmidt (1971) on the vibrational excitation of O2 and their theoretical analysis, they calculated Jo,(E) ds = 0.012~: eV = 3.4 x 10- l 9 cm2 eV for the v’ = 4 vibrational level of O;*, and a lifetime for this state of 165 x sec. It is


believed that this lifetime value is too long due to an underestimate of J O , ( E ) ~ E . Koike and Watanabe calculated lifetimes for higher (to v' = 10) vibrational levels of 0; *; these are listed in Table IV. In a similar calculation Koike (1973, 1975) found T,(O;*, v' = 4) = 72 x lo-'' sec.

The most recent theoretical estimate of -c,(O;*, v' = 4), equal to 88 x lo-'' sec, is by Parlant and Fiquet-Fayard (1976), who again based their calculations on the experiments of Linder and Schmidt (1971). They reported lifetimes for O;* in V' = 4 , . . . , 10 states that are compared with the values of Koike and Watanabe (1973) in Table IV.

All of the above theoretical estimates were naturally made on isolated O;*. Although it may be argued that the large differences between the experimental and the theoretical values reflect in part the effects of the medium on the 0; * lifetime in the case of swarm experiments, it is believed that the theoretical values for z,(O;*, v' = 4) are too long mainly due to a large uncertainty in the cross sections used to determine the autodetachment widths. The cross sections used seem to be too low, as is indicated below.

We have treated the data of Goans and Christophorou (1974) on 0, in C2H4" in the manner suggested by Eq. (26) for six values of the mean electron energy ( E ) in the range 0.045-0.085 eV. In this manner kl((E)) was obtained, which is interpreted as the absolute rate constant for process (25a) as a function of ( E ) . These data (solid circles in Fig. 13) were used to unfold the monoenergetic attachment rate constant (i.e., the rate constant that would be measured had all the electrons in the electron swarm had the same kinetic energy) kl(&) through

(27) kl((E)) = som kl(Elf(E, (&)Id&

In the unfolding procedure, we took for f ( ~ , ( E ) ) the known distribution functions for N,(Christophorou, 1971). The use of these distributions has been made for two reasons: first, because f ( ~ , ( E ) ) for C2H4 are not known, and second, because for the low values of (8) (50.085 eV) under consideration, f(~, ( E ) ) for C2H4 and N, are not expected to be too dissimilar. A knowledge of kl(&) allowed13 determination of the cross section c,(E) for reaction (25a), which is shown in Fig. 13. The cross section function peaks at -0.13 eV, i.e., somewhat higher than the peak associated with v' = 4 in the electron-scattering data at 0.082 eV. Admittedly, this cross section is an approximate one with a large uncertainty at the peak. However, the energy- integrated cross section loa(&) dE = 1.3 x 10- '' cm2 eV should be reasonably

l 2 These data apparently do not indicate, as those on O2 in N, do, a significant effect of the CzH4 medium on the 0; * potential energy curve.

l 3 The attachment cross section o,(E) and the monoenergetic attachment rate constant (in sec-' torr-') are related by UJE) = k , ( ~ ) / N ~ ( 2 / m ) " ' e ' ' ~ , where N o is the number of attaching gas molecules (0,) per cm3 per torr ( T = 298"K), and m is the electron mass.


(2) [eV)

0 0.4 0.2

0 0. 4 0.2

f (eV)

FIG. 13. kl((&)) against (E) and o&) against for 0, in C2HL (see text).

accurate (see Christophorou, 1978). It is much larger than the values used in the theoretical calculations cited in this section. If one scales down the theoretical estimates for z,(O;*, v' = 4) by an appropriate factor to account for the differences in the energy-integrated cross section found from the analysis just mentioned and those used in the theoretical calculations, the theoretical and the experimental values are brought to within the range of confidence limits. Thus, the value 165 x lo-'' sec of Koike and Watanabe would be 4.3 x IO-l'sec and the 72 x 10-l2sec of Koike would be 1.9 x

sec, both in comfortable agreement with the experimental value of -2 x lo-l'sec.

B. T h e Lqetimes of SO;*, C6H;*, and C,H,Br-* at Thermal Energies

High-pressure electron attachment studies have allowed determination of lower limits to the lifetimes of the metastable negative ions SO;* and C6H;* (Table VI). It is worth noting that Rademacher et aE. (1975) found that SO;* in Nz gas is stabilized by a process other than collision, which they attributed to radiative stabilization. In spite of the short lifetime of SO; *, radiative stabilization may become important at low pressures, especially if one considers the fact that since EA(SOZ) = 1.097 eV there would be a number of vibrational states to which SO;* can decay radiatively. It is furthermore worth noting that the lifetime of C,H;* was found by Christophorou and Goans (1974) to decrease with increasing E, as can be seen

86

-- ' b 8 -

L. G. CHRISTOPHOROU

-

TABLE VI

LIFETIMES' OF MODERATELY SHORT-LIVED PARENT MOLECULAR NEGATIVE IONS

Metastable Energy Electron negative Lifetime range affinity

ion (10-'2sec) (ev) (eV 1 Comments

' D P - - 6-

5 ;

0 ; * b 2' Thermal (298°K) O M d From data on 0, in

so; * 200' Thermal (298°K) 1.097@ From data on SO2 in

C,H, * 1-0.2h 0.04-0.18 20.0, to' From data on CsH6 in

C2H, (750-17,OOO torrye

C,H, (200-15,OOO torr)/

N, (400-15,ooO torr)h

1 * ~ <0.0.063eV

" 0 . <a>= 0.087 RV * <.>~0.109a~ /?' *

I / .

' These must be considered as lower limits since they were determined under the assumption

' For the O;* (X21T,, v' = 4) (see text). that p = 1.

Goans and Christophorou (1974). Celotta et al. (1972).

' A similar value has also been obtained from data on O2 in N2 (300-25,000 torr)

' Rademacher et a/. (1975). @ Celotta et a/. (1974). * Christophorou and Goans (1974). ' Christophorou (1976), Christophorou et al. (1977a).

(Christophorou, 1972a).

from the aw versus PN2 data in Fig. 14, which show that the attachment rate initially increases linearly with PN,, but as P,, increases further it shows a less than linear dependence on P,,. The higher the value of ( E ) , the higher the range of N2 pressures over which the rate varies linearly with PN2. Christophorou and Goans (1974) found that the data in Fig. 14 are consistent

I-

L i : ; 2 - - - 6

I / O 1' ,/' . ' z$/yk<:, ,/ ~ < o = O . l 8 1 e V

1, ,',.",,- -. : ,,!;?=:,If-*-**-'- :&Z c +

0.' ' I ' I ' ' ' I ' ' 1


10".

- : Id 2 co

16'*0

with a reaction scheme similar to that for O2 in C2H4 [Eq. (25)] on the basis of which they obtained t , / p versus ( E ) for C,H;* (Fig. 15). Since p I 1, the value for z,(C,H;*) given in Table VI is a lower limit. The decrease in z,/p with E is consistent with that observed for long-lived negative ions (Section VI).

1 ....,__

005 010 015 020 025

Additionally, Goans and Christophorou (1975) observed that the rate for electron attachment to C,H,Br below - 3 eV in mixtures with N, (for pressures to 25,000 torr) and with Ar (for pressures to 42,500 torr) depended on the medium's density. They analyzed the observed density and energy dependences of the attachment rates on the basis of the reaction scheme

9 C2H, + Br-

e + C,H,Br C,HsBr-* C2HsBr'*' + e(.) (28) * [sl C2H,Br- + S + energy

This model assumes that only one state of C,H,Br- is involved in the electron capture process below - 3 eV, and it predicts that

(29) k d k 2 + kstns)

k2 + k 3 + k,,ns (aw)o =

where S refers to either N, or Ar and n, is the number density of S, which is proportional to P,. In the limit where P , + 0, Eq. (29) reduces to

(aw),, 0 = k,k, / (kz + k 3 ) (30)


(aw)o, is interpreted as the rate for dissociative electron attachment, which requires no stabilization but is in competition with autodetachment and stabilization of C2H5Br-* by collision. Both the pressure-dependent and the pressure-independent components of the attachment rate peaked at -0.75 eV.

In Fig. 16, the ratios k2/k3 and k2/(k2 + k3) are presented as functions of ( E ) . Dissociation of C2H5Br-* into C2H, + Br- becomes increasingly more probable compared with autodetachment with increasing ( E ) . Estima- tion of k2, k,, and the lifetime z = 1/(k2 + k3) of C2H5Br-* requires an explicit knowledge of k,,. If we assume that the stabilization of C2H5Br-* is collisional and consider the stabilization rate constant for C2H5Br-* and S collisions to be k,, = 211(e2c(/M,)'i2 p = 5.5 x lo-'' cm3 sec-', we obtain (for p = 1) z N lo-" sec. Since p is likely to be < 1, z is > lo-" sec. Goans and Christophorou found k2 to increase with ( E ) (0.2 to -1.1 eV) and to exceed k3 at energies 2 0.7 eV. Thus, on the basis of this analysis, the lifetime of C2H5Br-* is determined predominantly by autodetachment below and by dissociation above the resonance energy (-0.75 eV).

5 1.0

4 0.8

3 0.6 *"

< + d 2 .

2 0.4

+ 0.2

0 0 0 0.25 0.50 0.75 4.00

{r). MEAN ELECTRON ENERGY lev)

16. Dependence ofthe ratios k2/k3 and kZ/(k2 + k3) for C,H,Br-* on ( E ) . The data Points A A are for mixtures ofCzH,Br with Nz, and the data points 0, 0 are for mixtures ofC2H5Br with Ar (from Goans and Christophorou, 1975).

Finally, it is noted that although the lifetime of C6H;* lies in the picosecond range, the lifetime of C6F;* lies in the microsecond range (Section VI). This reflects primarily the increased stability of C,F; due to the fluorine substitution, which is manifested by the large positive electron affinity of C6F6 and the large cross section for C6F; formation (Section VI and Table VII). It is expected-and experiments are in progress at the author's laboratory to investigate this-that the lifetimes of parent ions of multiply substituted fluorobenzenes will cover the range from to -lo-, sec, depending on the number of fluorines around the benzene periphery.


VI. LONG-LIVED PARENT MOLECULAR NEGATIVE IONS FORMED BY ELECTRON CAPTURE IN THE FIELD OF THE GROUND ELECTRONIC

STATE (NUCLEAR-EXCITED FESHBACH RESONANCES)

A. The Formation of Long-Lived Parent Negative Ions and Their Cross Sections

A variety of polyatomic structures have been found-principally by using TOF mass spectrometric and electron-swarm techniques-to capture thermal and epithermal electrons in the gas phase via a nuclear-excited Feshbach resonance mechanism and to form long-lived (7, > sec) parent negative ions. The existing data on these systems are summarized in Table VII. Structures whose parent negative ions are long-lived include benzene derivatives with highly electron withdrawing substituents such as -NOz, - C N ,

-CHO; -CN-(po1y)substituted organic molecules [e.g., C,(CN),] ; a variety of perfluorinated organic compounds with 71- or with only o-orbitals (e.g., C,F,, c-C4F8, C6FI0), and other multiply halogenated molecules for which dissociative attachment is either not energetically possible or not too fast (e.g., CzCI4); C-0-containing organic molecules (see Table VII,4); strained molecules (e.g., azulene, cyclooctatetraene) ; higher aromatic hydrocarbons (e.g., anthracene, 1,2-benzanthracene); and miscellaneous organic and inorganic polyatomics (Table VII,6,7). Hadjiantoniou et al. (1973b) also reached the conclusion that organic molecules containing the groups -COCO--, -COCH(OH)--, -COOH, and =CHCHO capture thermal electrons and form long-lived parent negative ions.

As for benzene, substitution of an H atom on the periphery of the naphthalene molecule by the electron withdrawing -CHO group increases the molecule’s electron affinity and electron attachment cross section, and consequently the lifetime of the parent negative ion. This is seen from a comparison of 2-naphthaldehyde and naphthalene. The thermal electron attachment rate, t,, and EA for the former far exceed those for the latter (Christophorou, 1971). The observation of long-lived parent negative ions for benzonitrile, perfluorinated benzonitrile, and tetracyanoethylene at thermal energies can similarly be attributed to the large electron-withdrawing ability of the CN group. It is noted, in this connection, that for a number of cyanocarbons Farragher and Page (1967) observed that the EA of the molecule increases as the number of the CN substituents increases, in a fashion similar to the increase observed for benzene derivatives with increasing number of fluorine (halogen in general) atoms on the benzene periphery. Perfluorination increases the EA of a molecule for both aromatic and aliphatic hydrocarbons and has a profound effect on z, as is indicated by the

... n

v,

Y

n

t- U

d

t 3

n

09

r- U

N

4

n nnn n

nn -nn

'm

'n

n 'm

"m"m

' n

'w'b

o'n

N

m

00 m

m

(u

03

m m

m

ua

a Y

UU

u

ua

N

r

-m

o r

-r-b

0

0-

0

-*

$22 rn

23-

u Y

UU

UU

Y u

uu

(u

*m

- -6- 4

NI

-c

L- (urn

m N

r

nh

lb

**

* I

L7

0

u, z, 5

u U

* I n

4 z, 0

U

3

* I n

z 0

u, z, 5

u

U

90

n

t- U

n

-1 3

U

IA

2

2 a .~ n

P’

U

* _I

n

I- Y

m

13 N

it I l-7

* I

* I

it I

n

u 0

0

* I *

* I T;:

s s

0

Lr,

u, u,

u, u,

a z,

z, z,

2, z,

X

0

n

n

0

0

0

0

U

U

YY

5

E

u

Eu

91

TABLE VII (Continued)

Compound

Thermal attachment Parent negative Lifetimeb rate Electron affinity

ion sec) (set-' ton-') (eV)

Bromopentafluorobenzene

Pentafluorobenzaldehyde Octafluorotoluene

Cinnamaldehyde

4. C=O-containing organic structures Glyoxal

Acrolein Acetic acid

Hexafluoroacetone W N Biacetyl

2.3-Pentanedione

Adipic acid

Cinnamaldehyde Pentafluorobenzaldehyde 0-; rn-; p-Nitrobenzaldehyde

0-; rn-; p-Nitrobenzoic acid 0-; rn-; p-Nitroanisole

Bend Benzoin

l&Naphthoquinone

1-Naphthaldehyde

2-Naphthaldehyde

[(CHO)zl-* 2.5 [21]

[CH,CHCHO] - * [CH3COOH]-* 326 f 30 [22]

38 f 3 [22]

[(CF3)2COl -* -60 [23]; 65 & 10 [24] 1.2 x lo9 [25] 20.4 [ll]; 0.6 [20] [(CH,CO)J -* 12 [11

[CH,COCOC,H,]-* 7 f 0.3 [22]

[COOH(CH,),COOH] - * See VII,3

See VII,3 See VIIJ

See VIIJ See VII,l

[C6H5CHOHCOC6H5] -* [C~H,COCOC~HS]-*

[1,4-G OH&]-* [l-CloH&HO]-*

[2-C ,,H,CHO]-*

30 f 3 [22]

12

36 395; 205; 41

44; 338; 142

189; 310; 196 - 90 [22]'

10 f 1 [22]

350 [26y

15 & 0.5 [22]

7.6 0.3 [22]

3 x lo9 [25]; 1.7 x lo9 [25]

1.9 x 109 ~ 2 6 3 1.71 [7]; 1.78 [7]; 0.7 [20] 0.68 [12]; 0.71 [18]

0.63 [I21

n

r- N

m

d

Y

n

r-

N

U

z N m

m

*

n

r- N

Y

Al

x I

.- 8 P E

m

u

2

2 E m

8

%

.- h

c

m

P

n

N

N

Y

z +I N

m

W

* I n

5

Y

3 s .- - c

.- 6

93

TABLE VII (Continued)

Compound

Thermal attachment Parent negative Lifetimeb rate Electron affinity

ion (lo+ sec) (sec- torn-') (eV)

Acenaphthylene

Fluoranthene

6. Perjuorinated organic structures Peduoro n-butane Peduoro n-pentane

Peduoro n-hexane Perfluorobutene-2

Perfluorobut yne-2

Peduorocyciobutane

\D P

Perlluorocyclohexane

Perlluoromethylcyclohexane

Perfluorocyclobutene [C4F61 - * Perfluorocyclopentene [CP~I-* Perfluorocyclohexene CC6Fd'

Hexduorobenzene See VII,3 Octduorotoluene See VII,3

I. Miscellaneous polyatomics Sulfur tetrafluoride W4I - * Sulfur hexafluoride [SF61 - *

88 [33]

189 [333'

12.7 [41; 151

34.3 [l5]

43.9 [l5] 30.6 [ 151

16.3 [l5]

14.8 [l5; 441; 12 [13]; - 200 [45]'

450 [13]; 236 [IS] 793 [13]; 757 [l5]

6.9 [13]; 11.2 [l5]

50 [13]; 26.2 [15]

113 [13]; 106 [Is]

0.6 [38]; 0.9 [40]; 0.9 [39]

3.1 x lo5 [42]

1.6 x lo9 [43]

1.6 x lo9 [54]

3.6 x lo8 [17]; 3.9 x 10' [43]; 6.8 x 10' [55]

3.2 x 109 ~463; 1.3 x 109 [ITJ

4.6 x lo9 [43]

3.8 x lo9 [I71 1.0 x 10" [17]; 2 1.4 k 0.3 [5]

1.13 x 10" [54]

16.3 [3] 1.2 [3]

25 [l]; 25-35 [8; 9; 22; 231; 68 [3]; - 500 [45] ;' Very long (> msec) [42]'

8.8 x lo9 [47]; 20.6 +_ 0.1 [S]; 7.2 x lo9 [43] 0.65 rt 0.2 [48]

4.1 x lo7 [17] 3.0 f 0.2 [SO] Selenium hexafluoride [SeF,] - * 9 Tungsten hexduoride cwF61- * 28 f 4 [49] 4.5 f 0.2 [SO]; 3.7 [57] Rhenium hexafluoride [ReF,] - * 20.3 [51] 2 3.89 [48] Uranium hexduoride CUF,I - * h; Very long (> msec) [S2If3’ 23.61 [52]

Nickelocene [(C5H5)2Nil -* > 100 [53]‘ Cobaltocene [(C,H,)*Col- * -25 [533’

References: [1] Compton et a/. (1966b), [2] Naff et al. (1971), [3] Harland and Thynne (1971), [4] Chaney and Christophorou (1970), [S] Lifshitz et al. (1973), [6] Henglein and Muccini (1959), [7] Chen and Wentworth (1975), [PI Johnson et a/. (1975), [9] Hadjiantoniou et a/. (1973a), [lo] Batley and Lyons (1962), [ l l] Christophorou et a/. (1973a), [12] Wentworth et al. (1975), [13] Naff et a/. (1968), [14] J. P. Johnson and L. G. Christophorou (unpublished results, 1975),[1qThynne (1972),[16]Gant and Christophorou (1976),[17]Davis ef al. (1973),[18] Wentworth and Chen (1967),[19] Johnson ef a/. (1977), [20] Briegleb (1964), [21] Compton and Bouby (1967), [22] Hadjiantoniou et a/. (1973b), [23] Collins et al. (1970a), [24] Naff (1971), [25] Bouby et a/. (1965), [26] Collins et ai. (1970b), [27] Cooper and Compton (1973), [28] Compton et a/. (1974), [29] Lyons and Palmer (1973), [30] Page and Goode (1969), [31] Houk and Munchausen (1976), [23] Thynne and Harland (1973a), [33] Frey et a/. (1973), [34] Wentworth and Ristau (1969), [35] Chaney et a/. (1970), [36] Wentworth et al. (1966), [37] Roberts and Warren (1971), [38] Michl(1969), [39] Chaudhuri et al. (1967), [40] Jannes and Pottemas (1971), [41] Harland and Thynne (1973b), [42] Fessenden and Bansal(1970), [43] Bansal and Fessenden (1973), [44] Harland and Thynne (1973a), [45] Henis and Mabie (1970), [46] Mahan and Young (1966), [47] Christophorou et a/. (1971b), [48] Cooper et al. (1975), [49] Thynne and Harland (1973a), [50] Compton and Cooper (1974), [Sl] Stockdale et al. (1970), [52] Beauchamp (1976), [53] Begum and Compton (1973). [54] Christophorou et a/. (1977c), [SS] Christophorou (1976), [56] Foster and Beauchamp (1975), [57] Dispert and Lacmann (1977).

Lifetime values listed are for -0.0 eV unless otherwise indicated. Lifetime found experimentally to decrease with increasing E ; the value recorded is for the energy at which the u,(E) peaks. This lifetime value is for E u 0.6 eV, where the u,(E) peaks. The lifetime has been found to decrease with E. The NIR has a maximum at 0.7 eV for benzii and at 0.2 eV for fluoranthene. The lifetime listed is for

0.0 eV. ’ Measured using the ion cyclotron resonance technique. @ Stockdale et a/. (1970) suggested that r,(SeF;*) and r,(TeF;*) are < lo-” sec, if they capture electrons at all. See this paper for a discussion of the

* Stockdale et a/. (1970) failed to detect this ion in a TOF mass spectrometer. possible implication of a Jahn-Teller effect in explaining the differences in the electron attachment properties of SF;*, ReF;*, WF;*, and TeF;*.

Beauchamp(l976) concluded that the behavior of UF;* is similar to that observed in their ICR experiments for SF;*, where an infinite lifetime against

j The lifetime listed is for 0.0 eV, T~ decreases with increasing E. In addition to the 0.0 eV process, there is evidence for an additional one at -0.5 eV (at

’ The lifetime listed is for thermal energies. 7* decreases with increasing E. In addition to the thermal process there is evidence for an additional one at

autodetachment for SF; * was indicated (Foster and Beauchamp, 1975).

- 1 eV,

- 1 eV for which T~ decreases from -60 psec at 0.5 eV to - 20 psec at 1.5 eV.

N 15 psec).


data in Tables VII and VIII. For monosubstituted benzenes for which the substituent X is an electron donor, and correspondingly the EA of the parent molecule is < 0 eV, perfluorination substantially increases t, since it lowers the negative-ion state making EA > 0 eV. When, however, X is an electron acceptor and the EA of the parent molecule is > 0 eV, perfluorination has a less pronounced effect on t, (Table VIII).

TABLE VIII

LIFETIME OF PARENT NEGATIVE IONS OF MONOSUBSTITUTED BENZENES

EFFECT OF PERFLUORINATION ON THE

Lifetime (sec)

x [CsH5 - XI-" [C,F5 - XI-"

F < 10-12 12 x CI < 1 0 - 1 2 17.6 x Br <lo- '* 21 x CN 5 x 10-6 17 x CHO < 10-6 36 x

Especially significant is the systematic study of NO,-containing benzene derivatives of Johnson et al. (19751, which suggests that all such compounds capture thermal electrons very efficiently and form long-lived parent negative ions, unless a dissociative electron attachment process (which is usually very fast) destroys the ion in times < sec. There is strong evidence that in the long-lived aromatic parent negative ions the electron is maintained by the entire molecule in a delocalized n orbital. This is supported by the observed successive lowering (increase in EA) of the lowest negative-ion state(s) of fluorinated benzenes with increasing number of F atoms around the benzene ring, resulting in a more stable ion with increasing number of F atoms: C6H6 [ - 1.4 eV, Compton et al. (1966a)l; C6H5F [ - 1.27 eV, Christophorou et al. (1974)l; 1,3-C6H4Fz [-0.6 eV, Naff et al. (1968)l;

et al. (1968)l; C6F6 [ + 1.80 eV, Lifshitz et al. (1973)].14 The drastic effects of the position and the nature of the substituents on the z, of the nitrobenzene anions provides further support (see Section V1,D).

It is thus seen that in these aromatic anions the capture of a slow electron by the molecule depends on the availability of vacant orbitals belonging to the entire molecule (delocalized). Christophorou et al. (1974) demonstrated

1,3,5-C6HsF3 [-0.3 eV, Naff et d. (1968)I; 1,2,3,4-C6HzF, [ -0.0 eV, Naff

l 4 Recent electron-scattering data on substituted fluorobenzenes [Frazier et al. (1978)l indicate that the lowest negative-ion state of C6F6 may not be a H state.


that for monosubstituted benzenes with substituents COOH, CHO, or NO,, both of the two lowest NIRs associated with the two lowest virtual n orbitals (n4, n5) are located below those (degenerate) of benzene. This, again, indicates the ability of electron-withdrawing substituents to lower the positions of NIRs. Actually (Table VII) the lowest NIR of nitrobenzene should be located below the energy level of the neutral molecule in its ground electronic, vibrational, and rotational state; this is exemplified by the positive value of the EA for this molecule. It might be possible then that when two or more strongly electron-withdrawing substituents are attached to the benzene ring or when the molecule has a large positive EA that the second (higher-lying) NIR state also lies below the energy level of the neutral molecule in its ground electronic, vibrational, and rotational level. This could allow parent negative ions to form via two distinct nuclear-excited Feshbach resonant states that may both be long-lived. Such systems may form long-lived doubly charged negative ions in the gas phase.

In nonaromatic structures, empty orbitals of specific groups can provide localization of the attached electron on only a part of the polyatomic molecule. Such may, for example, be the case for aliphatic hydrocarbons containing >C-0 or -COOH groups.

For all observed long-lived parent negative ions that are formed via a nuclear-excited Feshbach resonance mechanism, EA is positive ( > 0 eV), the number of vibrational degrees of freedom N is quite large, and the Tlectron attachment cross section G,(E) is substantial. A positive EA is necessary to facilitate strong binding for the attached electron and a large N is required to provide an effective distribution of the unstable ion’s excess energy among the many degrees of freedom to delay autodetachment. The data in Table VII indicate that z, is crucially affected by the details of the molecular structure and by the nature of the functional groups present in the molecule. Unfavorable Franck-Condon factors between the initial (unstable ion) and final (neutral molecule and electron) states can allow considerable time to elapse before the ion’s excess energy is concentrated in a mode that ejects the electron. Also, electronic state and symmetry selection rules as well as distinct structural characteristics such as the molecular geometry and the geometrical changes that often occur concomitantly with electron capture may result in the formation of long-lived negative-ion states.

For most of the long-lived parent negative ions studied so far (Table VII) the attachment cross sections have two distinct characteristics: (1) they are large (very often 2 100 A’) and (2) they decrease sharply with increasing energy above thermal. Actually in some cases (Christophorou, 1972b; Christodoulides and Christophorou, 1971 ; Christophorou et al., 1971b) the cross sections from thermal energy to less than - 0.5 eV decrease as CY with y values between 1 and 1.4 depending on the compound, which implies that


c,(E) maximizes at subthermal energies. At thermal and epithermal energies the cross sections for many such compounds are close to the maximum s- wave capture cross section, nit’, where 3, = 2n;Z is the electron de Broglie wavelength. An example is shown in Fig. 17. The maximum attachment rate was determined as a function of ( E ) from

using the known (see Christophorou, 1971) distributions fN2 in Nz (Nlorr is the number of attaching gas molecules per cm3 per torr at T = 298°K). The swarm-unfolded (Christophorou et al., 1971a) electron attachment cross section is shown in the inset of Fig. 17, where it is compared with the maximum (nit2) s-wave capture cross section over a range of electron energies. If these experimental cross sections are used to determine the thermal attachment rate via

0 0.2 0.4 0.6 0 . B 4 MEAN ELECTRON ENERGY, <<>(&‘I

FIG. 17. Electron attachment rate as a function of ( E ) and swarm-unfolded electron attachment cross section as a function of E for C,Fl0 (perfluorocyclohexene) (from Chris- tophorou et al., 1977c), compared with maximum s-wave capture rate and cross section (see text).


wherefM(e) is a maxwellian function (T = 298"K), it is found that = 1.13 x 10" sec-' torr-', which is, to the author's knowledge, the largest attachment rate measured to date. The lifetime of C6F;$ at thermal energies is - 1.1 x

To further illustrate the precipitous decline of c,(E) above $kT for most long-lived parent negative ions, we present in Fig. 18 the recent data of Gant and Christophorou (1976) on the electron attachment rates for SF6 and C6F6. The swarm-unfolded attachment cross sections are shown in the inset of Fig. 18. These two examples, as well as that in Fig. 17, have been chosen for an additional reason : They show that such "zero-energy" electron attachment processes are, as a rule, accompanied by higher-lying NIRs, leading to either dissociative attachment or to shorter-lived metastable ions. Thus, for SF6 a second resonance is evident at 0.32 eV [in a swarm-beam study Christophorou et al. (1971b) found the peak of this resonance at 0.37 eV], which is mainly due fo SF; produced by dissociative attachment. For C6F6 a second resonance is seen at 0.73 eV, which, as the one at thermal energies, is due to C6F; *. The lifetime of C6F; * produced via the 0.73 eV resonance is, however, shorter [ lo-* I t, < sec (Gant and Christophorou, 1976)l compared with the long-lived (1.2 x lo-' sec; Table VII,3) one at -0.0 eV.

sec (Table VI1,6).

3 (s>.MEAN ELECTRON ENERGY Iw1

FIG. 18. Electron attachment rate M W ( ( E ) ) as a function of (E) for C6F6 and SFs. The solid and open data points are, respectively, for the carrier gases N2 and Ar (from Gant and Christophorou, 1976). The inset is the swarm-unfolded electron attachment cross sections.


It is emphasized that the existence of higher-lying negative ion states is the rule rather than the exception. In cases of closely spaced and relatively long- lived negative-ion states, the possibility exists of intramolecular radiationless transitions from the upper- to lower-lying negative-ion states, in a similar fashion to intramolecular radiationless transitions in excited polyatomic neutral molecules.

In experiments using quasimonoenergetic electron beams, the yield of long-lived parent negative ions at thermal and near-thermal energies shows up as a sharp resonance centered at - 0.0 eV, whose shape is identical to that of SF;* and instrumental (see Fig. 5). In some cases, however, such resonances were found in TOF mass-spectrometric studies to peak at energies above thermal (still below - 1 eV). One such example is shown in Fig. 19 for C6H&OCOC6H; * (benzil- *). This behavior may be associated with geometrical changes concomitant with electron capture, the energy requirement being similar to that for an “activation process.” Changes in geometry upon electron capture and the need for an activation energy to bring about these changes have been reported for a number of molecules [N,O (Ferguson et al., 1967; Chaney and Christophorou, 1969); C o t (Krauss and Neumann, 1972; Cooper and Compton, 1972); cyclooctatetraene (Wentworth and Ristau, 1969)l. Similarly, Frey et al. (1973) attributed the peaking of the fluoranthene long-lived negative ion resonance at 0.2 eV (rather than at 0.0 eV) to the strain energy (of -0.5 eV) for fluoranthene.

W

- I20 f W

I l l , , , , , , , l l l j l l l . l I I , , , , 0 0 0.5 1.0 15 20

ELECTRON ENERGY (eVI

FIG. 19. Negative-ion current (0) in arbitrary units and parent negative ion lifetime (0) as a function of electron energy for C6H&OCOC6H5 taken without RPD (from Christophorou et al., 1973b).

B. Variation of the Autodetachment Lifetime with Incident Electron Energy

For the ion C6H5COCOC6H;* whose yield as a function of incident electron energy E maximizes at an energy (0.7 eV) well above thermal, T, decreases greatly (Fig. 19) with increasing E and is large to electron energies


well in excess of thermal. The dependence oft , on E, however, is not unique to these cases. Collins et al. (1970b) first observed experimentally the decrease oft, with increasing E. Their first such observation was for the “zero-peaking’’ resonance of 1,Cnaphthoquinone (Fig. 7). Subsequently, similar observations were made for o-nitrophenol (o-C6H,N020H) and tetracyanoethylene (C,(CN),) by Christophorou et al. 1973b), for m-nitro-ol,ol,a-trifluorotoluene (m-C6H4No2CF3) (Fig. 8), m-nitroacetophenone (rn-C6H4N02COCH3), and m-nitrobenzonitrile (m-C6H,No2CN) (see Fig. 20) by Johnson et al. (1975), for maleic anhydride (C4H203) and phthalic unhydride (C8H403) by Cooper and Compton (1973), and for cobaltocene (CloHloCo) and nickelocene (CloHloNi) by Begum and Compton (1973).

The decrease of t, with E is a general phenomenon. The larger the internal energy of the metastable ion, the larger is the rate of autodetachment. The failure of the experiment to show the decrease in z, with E in the preponderance of the cases studied so far (Table VII) can be attributed to the large

80

60

40

- i ; 20

g 6 0

s

J

’ [&OH]-

O L L L L L L L L L L

400

300

200

400

0

o 0.4 o e 0 0 4 0 8 E. ELECTRON ENERGY ( e V )

FIG. 20. Negative-ion current (0) in arbitrary units and parent negative ion lifetime (0) as a function of electron energy for C2(CN4) and o-C6H4N020H (Christophorou et a/., 1973b) and m-C6H4N02CN and m-C6H4N02COCH3 (Johnson ez al., 1975), taken without RPD.


widths of the electron beam pulses employed and to the precipitous decline of a,(€) above thermal energy, so that even though the mean energy of the electron pulse is increased, experimentally one observes predominantly ions produced by capture of electrons whose energy coincides with the position of the cross section maximum.

C . A Theoretical Treatment

The autodetachment process is a radiationless transition from a discrete state into a continuum of states. It is thus closely linked to other processes of this kind such as autoionization, preionization, and predissociation, and falls within the domain of unimolecular reaction-rate theory.

From the data in Table VII and the discussion in the preceding sections, it is clear that complex polyatomic molecules can possess large attachment cross sections and at the same time the respective ions have long autodetachment lifetimes. The large cross sections and long lifetimes can be reconciled by assuming that the transition from the neutral molecule to the unstable negative ion is strongly allowed, but the excess energy (comprised primarily of the molecule's electron affinity and the kinetic energy of the captured electron) is distributed among the ion's many internal degrees of freedom. The latter assumption explicitly implies that intramolecular relaxation in the ion is fast and that all forms of energy input are equivalent. The large cross sections, therefore, can result from the large-transition probability to the negative-ion state, and the long lifetimes can result from the time delay required for the metastable negative ion to return to a configuration that will lead to autodetachment.

Using the above hypothesis, Compton et al. (1966b) considered the reaction

(33) ( A B . . . CD)-* A AB . . , CD + e

where uaa is the rate of the forward (capture), and za-' is the rate of the backward (autodetachment) reaction. The two rates are related through the principle of detailed balance by

ra- 1 '

7, = (p-/po)(l/vaa) (34)

where p- is the density of states of the negative ion, po the density of states of the electron plus the molecule, and u the velocity of the incident electron. Compton et al. assumed that the molecule is left in its ground state after electron ejection, and they thus used for po the density of states of the free electron pe, given by

pe = m2v/n2h3 (35)


If we now view the molecule as a set of weakly coupled harmonic oscillators and assume that the ion’s internal energy is shared equally with the system’s N (= 3n - 6, for a nonlinear molecule with n atoms) vibrational degrees of freedom, the continuous-energy density expression of Rabinovitch and Diesen (1959)” can be used for p - :

In Eq. (36), E, = 4 hvi is the “zero-point” energy; is the internal energy of the metastable ion in excess of E,, i.e., the sum of the kinetic energy of the incident electron, the adiabatic electron affinity EA of the molecule, and the vibrational energy of the molecule above the zero-point energy prior to electron attachment (here assumed to be zero); TfN) is a gamma function of N ; n;=, hvi is the product of all vibrational energies hvi (assumed to be the same for the ion and the neutral molecule); (1 - pw’) is a correction factor described by Whitten and Rabinovitch (1963), which is to be evaluated at each value of The quantity /3 is a frequency dispersion parameter given by

N - 1 (v’) P=(7)w (37)

where (v’) is the mean square of the vibrational frequencies and (v)’ is the square of the mean frequencies; o’ varies with E‘ ( = E ~ / E , ) as shown in Fig. 21.

l 5 For other expressions of p - and for a general discussion on unimolecular reactions and decompositions, see Robinson and Holbrook (1972) and Forst (1973).


Combining Eqs. (34)-(36), we have

(38) 2 2 ~ 3 LET + (1 - P ~ ’ ) & J N - ~ 1

z, = ___ m2u r(N) ny=l hvi uo,(u)

which relates z, to o,(u), EA, N , and v i . Equation (38) has been employed to provide an estimate of EA from a

measurement of z, and o,(u) and a knowledge of N and v i a In Table IX, EA values are listed for some polyatomic molecules estimated in this manner. They are seen to be only in modest agreement with those obtained by other methods.

TABLE IX

ESTIMATES OF EA USING EQ. (38); COMPARISON WITH LITERATURE VALUES

Electron affinity (eV)

Molecule Using Eq. (38)” Literature ~ ~ ~~~~~~

Sulfur hexafluoride (SF,) 2 1.06 20.54;b 20.6;‘ 0.75;d 1.49‘ Biacetyl (CH,CO), 20.41 0.6’ Nitrobenzene (C,H,NO,) 20.53 0.5;’ 20.7;‘ < 1.1;@ 1.19;h 1.34h 1 ,4-Naphthoquinone (C, oH,O,) 2 0.60 0.7;’ 1.6: 1.71;h 1.78” Azulene (ClOH8) 2 0.46 0.5;’0.62;’ 0.65;k 0.66;‘ 0.86d

~~ ~

’ Chaney and Christophorou (1970).

‘ Lifshitz et a/ . (1973).

Henglein and Muccini (1959). Chen and Wentworth (1975). Compton and Cooper (1974).

Roberts and Warren (1971). Kay and Page (1964). Briegleb (1964).

’ Kunii and Kuroda (1968). j Wentworth et al. (1966). Ir Jannes and Pottemas (1971). ’ Becker and Chen (1966).

The above treatment has been extended by Christophorou et al. (1973b) in an effort to explain the observed large dependences of z, on E. They considered the reaction

7A ’ e ( ~ , ) + AX(t;, + E,) a AX-*(E, + c: + c, + EA1

(a)

AX“’(&, + E, + (6’ - (39)

(c1

where E, is the translational energy of the neutral molecule and the negative ion (assumed to be zero in the ensuing analysis), E~ and ef are, respectively, the energies of the incident and autodetached electrons, E:( =4 Ey=, hvi) is the zero-point energy of the ion, vt is the ion’s ith fundamental vibrational


frequency, and the rest of the symbols were defined earlier. We note that the attachment cross section (the one actually measured experimentally) for (a) -+ (b) is g,,(q), while the attachment cross section for (c) + (b) is o ~ ) ( E ~ ) .

Equation (34), which relates the transitions between (b) and (c), can now be expressed as

0 - 1 - P

P 7, - 7 u , Q ( Of)

where po is the number of states per unit energy per unit volume for the final products AX"' and e(Ef), and p- is the number of states per unit energy for the negative ion AX-*. In the treatment of Christophorou et al. (1973b), it was assumed that uft$)(uf) = uiga(ui) and vi = vi. The density of states of the negative ion p- was taken as in Eq. (36). The density of states of the neutral molecule to which the metastable negative ion decays pM and the density of states of the products (molecule + e) to which the metastable negative ion decays po were assumed to be given, respectively, by

In the above equations, 1 - Po is a correction factor for the neutral molecule analogous to 1 - Po' for the negative ion discussed earlier; o can be determined from Fig. 21, where the variable E' is now taken to be (q - Ef)/E:. It should be observed that the density of states p o was taken equal to the density of states of the ejected electron ( =m2u,/n2h3) times the number of molecular states N'. From Eqs. (36) and (40)-(42), we have (Christophorou eta!., 1973b)

where d, u, and E are 1 - Po', 1 - Pw, and *mu2, respectively.

and divide Eq. (43) by (ci + U E ~ ) ( ~ + ' I 2 ) , we have If we assume a two fold degeneracy for the negative ion and multiply

aa(Ei)&y m z,' = -(&i + UEZ)(N+ lI2)z

[ti + EA + U ' E ~ ] ~ - ' n2h3 (44)

1 06 L. G . CHRISTOPHOROU

where

I = joA X"'(1 - XIN-' dX, A = gi/(ci + UE,), X = &,MEi + a&,)

The quantity I can be evaluated numerically for any value of A.

we have If we now remove the assumptions that o,(q) = ur)(uf) and v: = v i ,

In deriving the above expressions, it was assumed that the molecules were initially in their ground vibrational states, that the density of states of the neutral molecule and the negative ion can be represented by similar expressions, and that the excess energy in both the negative ion and the neutral molecule is shared equally among the available vibrational degrees of freedom. This last assumption will be scrutinized later in this section.

In spite of the many well-recognized (Compton et al., 1966b; Klots, 1967; Christophorou et al., 1973b; Johnson et al., 1975) limitations of the above treatment, Eq. (45) is important: it relates z,(E), o,(E), N , hvi, hv;, EA, and also the energies ei and 8, of the incoming and outgoing electrons. It allows a qualitative and at times a semiquantitative understanding of long- lived negative ions as well as of the energy dependence of z, on ci.

Expression (44) has been applied by Christophorou et al. (1973b) to the case of 1,Cnaphthoquinone for which the pertinent quantities are known. The calculated z, as a function of E~ has been shown earlier in Fig. 7. The calculated values of T~ were normalized to the experimental data of the lowest energy point.

To illustrate the dependence of z, on EA, N , and hvi, we now refer to the recent work of Christophorou et al. (1977b) on C6F6. For C6F6, N = 30, EA = + 1.8 eV (Lifshitz et al., 1973), and u , ( ~ ~ ) is known (Gant and Chris- tophorou, 1976). The 20 fundamental frequencies reported by Steele and Whiffen (1959) for C6F6 were used by Christophorou et al. (1977b) to calculate /3 and E,, with the 10 degenerate levels assigned to those degenerate in benzene. Using the aforementioned values of N , EA, o,(ci), and hvi, Christophorou et al. calculated the negative-ion lifetime zCa,(ci) for C6F; * under the assumption that v; = vi and o:)(E,) = oa(zi). Their result is shown in Fig. 22 by the broken curve. The theoretical dependence of z, on EA can also be seen from the data in Fig. 22, where T ~ , , ( E ~ ) is presented for values of EA between 0.5 and 2.5 eV. It is seen that z, decreases with decreasing EA.


1

6, ISVI

FIG. 22. Calculated negative-ion lifetime T ~ ~ , for C,F;* as a function of the incident- electron energy zi for several values of the electron affinity EA of CsF,. The number of vibrational degrees of freedom was taken equal to 30 (see text) (from Christophorou et al., 1977b).

The theoretical dependence of z, (E~) on N is shown in Fig. 23. zca1(q) was calculated for EA = + 1.8 eV, v f = vi, Q ( E ~ ) = g,(q), but N was substituted by N’ in the first term of Eq. (45), where N’ is an effective number of degrees of freedom (< 30) sharing the ion’s excess energy. The negative ion’s zero-point energy EL was multiplied by WIN, and p was as given in Eq. (37). It is seen from the data in Fig. 23 that a decrease in N’ results in shorter -cCa,(ci).

It is observed (Fig. 22) that at thermal energies the value of rCal(q) is several orders of magnitude larger than the 12 psec measured in TOF mass- spectrometric experiments at these energies (Table VII). A reduction in EA lowers T,, but only for EA = 0.5 eV is the value zCa1 at thermal energies comparable with the experimental value texp. Such a degree of error in EA seems improbable. An explanation of this discrepancy is provided by the data in Fig. 23, where it is seen that at thermal energy (-0.04 eV) z,,, N zeXp for N’ 2: 9.

To facilitate a more direct comparison between the calculated and the measured lifetimes, Christophorou et a/ . (1977b) integrated zcal(ci) over an electron energy distribution function F(ti) , which is characteristic of the


0 0.2 0.4 0.6 0.8 1.0 6, (eVl

FIG. 23. Calculated negative-ion lifetime T,., for C,F;* as a function of the incident- electron energy ci for several values of the effective number of degrees of freedom N'. The electron affinity of C,F, was taken equal to 1.8 eV (see text) (from Christophorou et al., 1977b).

electron beam in the TOF mass-spectrometric studies, viz.,

JOm Teal (Ei)oa(Ei)F(Ei) dEi (Tcal(&i)) = (46) J: oa(Ei)F(Ei)dEi *

In Eq. (46) ~ ~ ~ ~ ( 6 ~ ) are the functions in Fig. 23, oa(ti) is the attachment cross section for the lowest (near-thermal energy) resonance in Fig. 18, and F(EJ is the instrumental shape of the SF;* resonance as measured without application of the retarding potential difference technique in the TOF studies. The best agreement between ( T ~ & ~ ) ) and the experimental value was again obtained for an N' value between 8 and 9.

The above result cannot be attributed to a difference between vi and v f . For a large molecule such as C6F6, the vibrational frequencies vf of the negative ion are not expected to differ appreciably from the v i of the neutral molecule. To investigate, however, the effect of any such possible differences between v; and vi on the value of N' determined above, Christophorou et al. (1977b) took v; = fvi, where f is a constant, and repeated the lifetime calculations giving f a number of values between 0.5 and 1.1. The results of


these calculations are shown in Fig. 24. Differences between vi and vi of as high as 20% do not affect the optimum value (8-9) of N determined for f = 1. Furthermore, small changes in na(ci) could not alter the above conclusions.

I 3 6 9 12 15 18 2( 24 27

N’

FIG. 24. Integrated negative-ion lifetime for C6F;* as a function of the effective number of degrees of freedom N’ when the fundamental frequencies of the negative ion are a fraction f of those of the neutral molecule. The horizontal broken line is the value of T,,~ (from Chris- tophorou et a/., 1977b).

It thus seems that an agreement between the calculated and the measured lifetimes of C6F; * can reasonably be achieved by a considerable reduction in the effective number of degrees of freedom, which share the ion’s excess energy. The consistent optimal value of 8-9 for N’ leads to the conclusion that only about 30% of the possible vibrational modes are involved in sharing the energy of the C,F;* ion in the lowest state configuration. A combination of factors such as a slight reduction in EA and in vf would raise the best value of N’ but not enough to alter this conclusion.

The findings of Christophorou et al. (1977b) are supported by studies of excess energy partitioning in dissociative attachment processes (see, e.g., DeCorpo et al., 1971; Harland and Franklin, 1974; Franklin, 1976) where it is found that only a fraction [0.35-0.57; DeCorpo et al. (1971)l of the available vibrational modes are utilized in the absorption of the ion’s


energy. They are also consistent with electron-scattering experiments on benzene, which belongs to the same symmetry group (D6h) as C6F6. Wong and Schulz (1975) observed that only a few (five) vibrational modes corresponding to stretching and deformation of the carbon ring are strongly excited by electrons in the region ( - 1.2 eV) of the lowest shape resonance. The excitation of only a limited number of vibrations was explained (Wong and Schulz, 1975) on the basis of selection rules, derived from symmetry considerations and some postulates. If the same selection rules were applicable to the lowest electron attachment resonance in C6F6, one would expect N to be much less than N , as is indicated by the lifetime calculations of Christophorou et al. (1977b). The value of N’ may depend on the symmetry of the state into which the electron is captured.

The preceding discussion, confined as it is only to C6F6, indicates that the assumption that all available vibrational modes share equally the ion’s excess energy (i.e., N = N ) may not be valid. If for a series of similar molecules it were possible to keep the main parameters in Eq. (44) approximately

I I I I

20 30 40 50 60 70

FIG. 25. 7, as a function of N . (0) 1, Perfluoro n-butane (12.7, 36); 2, perfiuoro n-pentane (34.3,45); 3, perfluoro n-hexane (43.9,54). (a) 4, PerRuorobutene-2 (30.6,30); 5, perfluorobutyne- 2 (16.3, 24). (U) 6, Perfluorocyclobutane ( - 13.4, 30); 7, perfluorocyclohexane (-343, 48); 8, perfluoromethylcyclohexane (- 775, 57). (A) 9, Perfluorocyclobutene (9, 24); 10, perfluorocyclopentene (38, 33); 1 1 , perfhorocyclohexene ( 1 10, 42). (0) 12, Hexafluorobenzene (12, 30). (0) 13, Chloropentafluorobenzene (17.6, 30); 14, bromopentafluorobenzene (21, 30); 15, cyanofluorobenzene (17,33); 16, pentafluorobenzaldehyde (36,36); 17, octafluorotoluene ( - 25, 39). (m) 18, Cyclooctatetraene (6.4,42). (A) 19, Azulene (7,48); 20, acenaphthylene (88, 54); 21, fluoranthene (189, 72). The first number in parentheses is the value of 7. and the second the value of N .

N


the same except for N , the theoretical model discussed in this section would predict that t, increases with N . Actually, under the above assumptions the model would roughly predict that

lnt, cc N (47)

Indeed, some authors (e.g., Naff et al., 1968) claimed to have observed such a quantitative dependence of t, on N . In Fig. 25 In T, is plotted as a function of N for different groups of molecules. For each particular group t, increases with N . However, the increase of In t, with N is not linear as predicted by Eq. (47); there is a great deal of variation between the various groups, and within a particular group of molecules the dependence of t, on N seems to be leveling off as N increases. The bending over of the In t, versus N functions may actually reflect the fact that as the molecules get bigger, G,(E) gets larger, and thus In cata versus N might be a more appropriate function. The limited data on 0, (see Table VII) do not allow a test of this latter relationship but lend support to it. Considering the variation of EA, c,, and vi from one molecule to another, relation (47) could at best be an approximate one even for a specific group of molecules.

The simple model discussed in this section, although accounting qualitatively and at times semiquantitatively for the experimental data, is beset by a number of shortcomings, as can be seen from the discussion. The need for a more sophisticated theoretical treatment is quite apparent. See a recent discussion by Ivanov and Ponomarev (1977).

D. The Lifetimes of the Parent Negative Ions of Nitrobenzenes

In this section we shall focus attention on the results of the systematic study of Christophorou and co-workers on nitrobenzenes, which allow an insight into the details of electron attachment and autodetachment processes of polyatomic molecules. Christophorou and co-workers studied over 40 nitrobenzenes containing, in addition to the NO2 group another substituent X around the benzene periphery, and investigated the effects of the nature and the position of X on t,. The main findings and conclusions of these studies are as follows:

(a) NO,-containing benzene derivatives capture thermal and epithermal electrons and form long-lived nuclear-excited Feshbach resonant states (unless dissociative attachment destroys the ion at times < sec). The cross sections for these attain their maximum value at thermal energies and decrease rapidly thereafter with increasing energy (see Fig. 5).

(b) The lifetimes T~ depend on the electron affinity EA and the ionization potential I , of the parent molecule. If c,(~) and vi for these compounds are


OCH,. I I I 1 I I

not too dissimilar, from Eq. (44) we would expect

(48)

Since all za were measured under identical experimental conditions and temperature, and since all B,(E) peak at thermal energies and decrease rapidly with increasing energy above thermal, the preponderance of the parent negative ions of each compound is formed by capturing thermal electrons. Since, moreover, these compounds are similar in structure, the zero-point vibrational energies of the neutrals and negative ions should not be too dissimilar, the internal energy of the latter in excess of EL being determined principally by the magnitude of EA. Thus if it is assumed that for the negative ions of nitrobenzenes in Table VII,l, vi (and v; ) and B,(E) are not too dissimilar, the observed differences in z, for these ions can be attributed to differences in the EA of the respective neutral molecules. Electron affinities for the nitrobenzenes are virtually nonexistent. For some of these compounds, however, polarographic half-wave potentials Eli,, which should be proportional to EA (see, for example, Matsen, 1956), are available, and in Fig. 26 T : / ( ~ - ' ) is plotted as a function of Eljz. The experimental results are in reasonable agreement with the predicted dependence of z, on EA [Eq. (48)J. For a given value of z, tends to be longer for the meta- than for the para-isomers.

1/(N- 11 cc EA z a

1.18 I I 1 I I I I 1

1.16


In Fig. 27, 7, is plotted as a function of I , , the parent molecule ionization potential, for meta- and para-nitrobenzenes. It is seen that when I, < I, (IH is the ionization potential of nitrobenzene), z, is basically unchanged. When, however, I, > I,, a sharp increase in z, is observed. This behavior is related to the electron donor-acceptor properties of the substituent X. When X is an electron donor, T~ is basically independent of the value of I , ; when X is an electron acceptor, z, increases greatly with increasing I,. The data in Figs. 26 and 27 become clearer if one considers the effect the nature of X has on both EA and I, [see (c) below]. Both of these observations and the following discussion add support to the arguments presented in Section VI,A that the electron is captured into a delocalized 71 orbital.

600

500

400

t 100

I I 1 I

NO2 <

CN

CN 0

8.5 90 9.5 40.0 (0 5 Iu (eV)

FIG. 27. Lifetime of parent negative ion as a function of the parent neutral molecule’s ionization potential I , for a number of meta- and para-substituted nitrobenzenes. The lifetime data are from Table VII, and the ionization potential data are from Christophorou et a / . (1977a): (*) C6H5NO2 (29, 10.1). (0) Meta-isomers: rn-C6H,NOzF (28, 9.93); rn-C6H4N0,CI (47,9.92); m-C6H4N02Br (21. 9.82); twC6H4NO2CN (315, 10.29); m-C6H4N02NH2 (21, 8.76); m- C 6 H 4 ( N 0 ~ ) 2 (537, 10.62); m-C6H4N02CH3 (19, 9.48); m-C6H4N020CH3 (52, 9.09) (0) Para-isomers: p-C6H,NOZF (10, 10); p-C6H4NoZCl (14. 9.98); p-C6H4N02Br (10, 9.76); P-C~H~NOZOH (14, 8.58); P - C ~ H ~ N O ~ C N (205, 10.23); P - C ~ H ~ N O ~ N H ~ (15, 8.63); p- C6H4(N0J2 (421, 10.63); p-ChH4NO2CH3 (14, 9.63); p-C6H4N020CH3 (10, 9.04). The first number in parentheses refers to the value of T& and the second to the value of I,.


(c) The lifetimes of the parent negative ions of disubstituted nitrobenzenes depend strongly on the n-electron donor-acceptor properties of the substituent groups (NOz and X): strong electron-withdrawing substituents X (NOz is a strong electron-withdrawing group as well) greatly increase z,, while z, is little affected by n-electron donating substituents.

CNDO-2 molecular orbital calculations by Johnson et al. (1975) on nitrobenzenes with the two substituents NOz and X in the para-position to avoid intramolecular complexing between them have shown than when X is an electron acceptor, the magnitude of z, correlates with the amount of n-electron charge withdrawn from the ring, while z, is little affected by the amount of n-electron charge donated to the ring by X when X is an electron donor. This is shown in Fig. 28a, where z, is plotted as a function of the ring n-electron charge, in electron units. A charge of six corresponds to benzene. If the ring n-electron charge for the neutral molecule is subtracted from that for the negative ion, the net increase in ring n-electron charge can be estimated for the parent negative ion. Johnson et a!. (1975) found that the ring n-electron charge for the 12 parent negative ions they studied (see Fig. 28b) increased by 0.2-0.3 electron units when X is an electron donor and by 0.4-0.7 electron units when X is an electron acceptor. This large change in the latter case is due to the greater Coulomb attraction between the attached electron and the positively charged ring, resulting in a longer z, (see Fig. 28b). It suggests that the extent of intramolecular charge

Ix) c fj

X [$I

1 CZO M OW w

0 I 1 Nd$$,F,

5 9 4 196 5 9 0 S O 0 6 0 2 604 606 0 01 0 2 03 0 4 0 5 0 6 0 7 R l N G 7 -ELECTRON CHARGE INCREASE IN RING .- ELECTRON CHARGE

FIG. 28. (a) T~ versus ring n-electron charge for the neutral p-disubstituted NO,-containing benzene derivatives. The compounds are identified by the second substituent X and nitrobenzene by H. (b) T, versus increase in ring n-electron charge for the parent negative ions ofp-disubstituted NO,-containing benzene derivatives. The compounds are identified by the second substituent X and nitrobenzene by H (from Johnson et al., 1975).


transfer between the substituent and the bezene ring can be inferred from the measured values of z,.

(d) The lifetime of the parent anions of nitrobenzenes is not only affected by the nature of X but also by the relative position of X with respect to NO,. For a given X, the z, of the meta-isomer is longer than the z, of the corresponding para-isomer. This may, in part, be due to small changes in EA. Although the t, of most ortho-isomers are shorter than for the corresponding meta-isomers (and close to those of the respective para-isomers), for certain ortho-substituted nitrobenzenes (o-nitrophenol, o-nitroaniline, and o-nitrobenzaldehyde) the lifetimes are much longer compared with the meta- and para-isomers. This most significant finding was attributed (Hadjiantoniou et a/., 1973a) to an intramolecular interaction (complexing) between the substituents (X and NO,) at the ortho-position, i.e., to hydrogen bonding between the 0 atom of NO, and the H atom of X (=OH, NH,, CHO). Such a high sensitivity of z, on the interaction of X and NO, suggests that it might be possible to infer intramolecular complexing between X and NO, from a measurement of z,.

E. Biological Signijicance

In closing this section, it might be appropriate to indicate the biological significance of such basic physical studies of biologically related structures. First, they help elucidate the mechanisms and significance of electron transport in certain biological reactions where the existence of electron transfer has long been recognized. Second, t, can be used as a probe to identify electrophilic sites in bigger biostructures by identifying electrophores in structurally similar smaller compounds. Certain of the electrophores identified in this way, such as the carbonyl group (TC-0) of the aldehydes and ketones, are believed (see, for example, Szent-Gyorgyi, 1968) to be the main electron acceptor groups in the cell. Most carbonyl compounds react readily with hydrated electrons (Hart and Anbar, 1970). Recently, also, Bakale et at. (1976; also private communication, 1976) measured the rate constants for electron attachment to a number of o-, m-, and p-nitrobenzenes in nonpolar liquids for which Christophorou and co-workers measured 7, in gases. They found that the rate constants are very large (-5 x lo', M-' sec-' in c-hexane at 20°C) and are affected very little by X and its relative position to NO,. Although this may be taken to add weight to the assumption made earlier that G,(E) for the nitrobenzenes discussed in Section VI,D are roughly the same, it should be kept in mind that in a liquid the rate constants for electron attachment are strong functions of the liquid itself. Christophorou (1976) has stressed that processes and properties of


molecules involving charges (charge-separated states) are the ones most dramatically influenced by the medium, both by its density and by its nature. Third, z, can be used as a probe to infer intramolecular charge transfer and molecular complexing in big organic structures. Fourth, the identification of long-lived transient species carrying an extra electron and a considerable amount of internal energy is in itself quite significant, since such species should be highly reactive.

VII. LONG-LIVED PARENT NEGATIVE IONS FORMED BY ELECTRON CAPTURE IN THE FIELD OF AN EXCITED ELECTRONIC STATE

[ELECTRON-EXCITED FESHBACH RESONANCES (CORE-EXCITED TYPE I)]

Short-lived electron-excited Feshbach resonances (core-excited type I; see (Section 1,A) are abundant for both atoms and molecules. Long-lived negative ions in the field of an excited state, however, are rare; He-* (ls2s2p) ‘Pj (Table 11) is a distinct such example. As noted earlier, specific electronic configurations can lead to long-lived negative-ion states.

Christophorou et al. (1969) and Collins et al. (1970b) presented the first experimental evidence of long-lived polyatomic parent negative ions formed by electron capture in the field of an excited electronic state. In Fig. 29a, we reproduce their results on the rate of electron attachment as a function of mean electron energy for p-benzoquinone (p-BQ). Although the ground-state electron affinity of p-BQ is > O eV [0.77 eV, Briegleb (1964); 1.37 eV, Farragher and Page (1966); 1.83 eV (or 1.98 eV), Chen and Went- worth (1975); 1.89 eV, Cooper et al. (1975); 2.0 eV, Batley and Lyons (1962); 2.08 eV, Kunii and Kuroda (1968)], this molecule behaves in a strikingly different manner than other similar polyatomics discussed in Section VI. The attachment rate (Fig. 29a) and the attachment cross section (Fig. 29b) do not have their maximum at -0.0 eV, but rather at -2.0 eV, and their magnitudes at -2.0 eV are more than two orders of magnitude larger than at thermal energies. The z, of p-BQ-* at -2.0 eV is long, and it decreases with E across the resonance (Fig. 29c). The lifetimes in Fig. 29c were measured without RPD, and the dependence of z, on E is therefore not as pronounced as it would have been had the RPD technique (or, generally, a narrower electron beam) been employed. Cooper et al. (1975) repeated Collins et al.’s (1970b) measurements on p-BQ applying the RPD technique. They confirmed their findings and, as expected, found a sharper decrease oft, with E.

They also determined the average position of the p-BQ-* resonance to be 1.4 2 0.1 eV rather than 2.1 eV as found in the swarm-beam analysis of Christophorou et al. (1969). Collins et al. (1970b) did not observe p-BQ-* ions at thermal energies in their TOF mass-spectrometric study, although their

FIG. 29. (a) Attachment rate as a function of mean electron energy for p-benzoquinone in ethylene, nitrogen, and argon carrier gases as indicated. (b) Electron attachment cross section as a function of electron energy (established by the swarm-beam technique) for p-benzoquinone. (c) Parent negative-ion lifetime for p-benzoquinone (measured without RPD) as a function of electron energy [based on the work of Christophorou et al. (1969) and Collins et a/. (1970b)l.

3

q 4

i?

4

. 0


swarm work showed a weak capture process at thermal energies (see Fig. 29a). This would imply that the z, of p-BQ-* at thermal energies is < sec. Thus for p-BQ, both o,(E) and z, for the -2.0 eV process are orders of magnitude larger than at thermal energies.

The -2.0 eV p-BQ-* resonance has been attributed by Christophorou et al. (1969) to electron capture in the field of the lowest triplet state T I of p-BQ, resulting from an n + n* transition at -2.31 eV, concomitant with electron capture. The energy difference 2.31 - 2.1 = 0.21 eV (or 2.31 - 1.4 = 0.91 eV if we use the 1.4 eV value of Cooper et al. (1975) for the position of the p-BQ-* resonance) suggests that the electron affinity EAT, of p- benzoquinone in TI lies between 0.2 and 0.9 eV.

The above observations are not only of physical but also of biological significance. Such molecular species carrying an extra unit of electron charge and electronic excitation energy can be highly reactive. This richness in negative charge and internal energy is especially significant for systems such as the quinones, whose role in electron transport in living systems has long been recognized (see, for example, Wolstenholme and O’Connor, 1961 ; Morton, 1965).

Similarly, Chaney et al. ( 1 970) associated an electron attachment process they observed in azulene at -0.35 eV above thermal, with electron attachment in the field of the first triplet state of azulene. But since parent ions due to this process were not observed in a TOF mass spectrometer by these workers, the T , must be sec. Interestingly also Aberth et al. (1975) reported a mass-spectrometric observation of long-lived ( T , > 10- ’ sec) HD-*, D;*, H;*, H2D-*, HD;*, and D;* ions. The authors suggested that the large cross sections (-6 x lo-’’ cm2) and long lifetimes of these ions indicate that they may be formed in a quartet electronic state analogous to He-* (ls2s2p) 4Pj. Schnitzer and Anbar (1976a) observed isotope effects in the formation of partially or totally deuterated analogs of H i and H; ions, on the basis of which they suggested a hydrogen abstraction reaction for the formation of H;* and a hydrogen transfer reaction involving an excited H: in the formation of H;*.

VIII. LONG-LIVED METASTABLE FRAGMENT NEGATIVE IONS

Fragment negative ions produced in dissociative electron attachment processes can be metastable and subject to autodetachment. Their lifetimes can be long enough to permit their study in TOF mass spectrometers. A number of such Iong-lived metastable fragment anions have been observed (Table X) by Cooper and Compton (1973, 1974) and Compton et al. (1974) in collisions of electrons and fast cesium atoms with a number of anhydrides. Their z, ranges from a few microseconds to milliseconds and, as for the


TABLE X

LONG-LIVED METASTABLE-FRAGMENT NEGATIVE IONS

Energy of Metastable- maximum T.

Molecule fragment ion intensity (ev) ( sec)

Maleic anhydride (C4H,0,)

0

Succinic anhydride (C4H403)

Glutaric anhydride (C5H,0,)

0 Qo

cis-1,2-Cyclobutane dicarboxylic anhydride (C6H603)

0

Hexafluoro glutaric anhydride (C5F603)

COT* co; * C2H2CO-* C2H2CO; * c,o;*

c o y * COT* C2H4CO;*

CSHsCO; * CZH4CO;*

or c,o;*

co; *

or ChH6CO; *

C4H20; *

C2F4CO; * CzF4CO-* CzF;*

C,F,CO;* CjFsCO-* C3Fa*

3.1 2.8 2.3 0.2

1.7 1 .1

0.9

1 .o

2.0

1.4

0.0 0.0 0.5

0.0 0.0 0.4

62 & lo'.* 60 + 5'rd 58 f 5c,d 40 f 5',d

1 1 7 ~ d . e

71 f 26+ Yd

145c.d.e

84Wd.'

45 + 5 4

45 f 5C.d

27y.d.e

2 500OC*'*f 124'J

1 4 ' ~ ~

2 1260'.'.f 80C.J

760'9J

Using the Cs collisional ionization technique. Compton et a/. (1974).

Cooper and Compton (1973). T, has been observed to decrease with increasing incident electron energy; the

Cooper and Compton (1974).

' Dissociative electron attachment study.

value recorded is at the cross section maximum.


parent anions, it should depend on the anion's internal energy. Indeed, t, has been found to decrease with E for a number of fragment anions (Table X). The parent negative ions of some of the molecules in Table X have been observed, and their lifetimes have been given earlier in Table VII. For some, however (e.g., hexafluoroglutaric anhydride with EA 2 1.5 i- 0.2 eV), no parent ions were observed in electron impact studies, although such anions were detected in a reaction

Cs,,,, + M d M - + Cs' (49)

This can be attributed to the fact that the M- in Eq. (49) can be formed in its ground vibrational state, while in electron impact studies M-* has initially substantial internal energy ( 2 EA), which can cause fast autodetachment and/or autodissociation. This is consistent with abundant earlier observations (see Christophorou, 1971) of negative ions in ion-molecule reaction processes that normally are not observed in direct electron impact studies. They all relate to the fact that in neutral-atom or ion collisions lower-lying states of M - are reached for which M- is stable or less unstable with respect to autodetachment and/or autodissociation than in direct electron impact reactions. Consistent with this is the finding of Compton et ,al. (1974) that in cesium collisional ionization experiments a number of parent and fragment anions are much longer-lived than under direct electron impact.

Three other observations on these systems are worth noting:

(1) Dissociative electron attachment to cyclic anhydrides produces abundantly RCO; ions (i.e., neutral CO is ejected from the molecule), and the RCO; * ion undergoes autodissociation following autodetachment.

(2) Metastable CO; * ions are produced from molecules containing "bent" carboxylic groups in stereospecific configurations similar to that of co;.

(3) The t, of CO; * depends on the parent neutral molecule.

In connection with (l), it is interesting to note that the RCOz radicals seem to exist only in their anionic form, the attached electron providing the binding of R and CO, units.

In connection with (2) , according to Walsh rules COz with 16 valence electrons should be linear, but CO; with 17 valence electrons should be bent in the ground state. The CO; bond angle is - 134", and the CO bond distance is 1.25 A, i.e., - 10% larger than for COz . According to Cooper and Compton (1973) the minimum of the potential energy surface of the lowest state 'A1 of COT is -0.5 eV above that of the ground state 'Al of COz, i.e., EA ((20,) = -0.5 eV. Ab initio SCF calculations by Krauss and Newmann (1972) showed that the minimum in the potential energy surface for the 'A, state of CO; lies below that for the ground state 'A, of COz for a bond


a 5

angle of 134", in agreement with the measured value of the 0-C-0 angle measured by Ovenall and Whiffen (1961) for CO; in solids, but above the minimum for CO, at 180".

The gross differences in the molecular configurations of CO; and CO, would inhibit the direct formation of CO; in its ground state. Electron impact studies of CO, at low pressures failed to produce CO; ions. Pre- liminary recent work on C 0 2 in high pressure (> 10,000 torr) of N2 (D. L. McCorkle, L. G. Christophorou, and M. Forys, unpublished results 1977) indicates some weak electron attachment to CO, at thermal and epithermal energies. Consistent with this finding is the earlier observation (Lehning, 1968) that the mobility of thermal and epithermal electrons in pure C 0 2 decreases significantly with increasing CO, pressure and approaches the mobility of COY (see a discussion in Christophorou, 1975).

In Fig. 30a, CT,(E) for the production of COY* from succinic anhydride and maleic anhydride are shown. The lifetime of COY* from succinic anhydride is 26 * 5 psec and from maleic anhydride is 60 f 5 psec. The long

SF;*

n I

to

(bl 0.6 - I

0 .5 -

0.4 -

n

2 4 6 6 10 (2 14 16 TIME of FLIGHT 1pr.c)

FIG. 30. (a) Cross section for production ofCO;* as a function of incident electron energy from succinic anhydride (I) and maleic anhydride (11). The energy scale was calibrated with the Kr 10.0 eV energy loss peak. (b) -In N - / N T versus time of flight for CO; * produced by dissociative electron attachment in succinic and maleic anhydrides (from Cooper and Compton, 1972).


lifetime of CO; * has been attributed to poor Franck-Condon overlap between the potential energy surfaces of the bent (134") CO;* and the linear

In connection with (3), the observed two values for the z, of COT* have been attributed by Cooper and Compton (1972, 1973) to different Franck- Condon overlap between different vibrational states of CO; * and the vibrational states of C 0 2 . The shorter value of z, could originate from CO;* primarily in a higher (v' = 1) vibrational level and the longer value of z, from CO; * primarily in the v' = 0 vibrational level.

Finally, it is of interest to note the recent results of a TOF mass-spectrometric study of 2-C4F8 (perfluoro-2-butene) by I. Sauers, L. G. Christo- phorou, and J. G. Carter, unpublished (1977). In addition to the formation of the parent anion C,F;* with z, 'u 11 psec at -0.0 eV, these indicated the formation of a variety of long-lived metastable fragment anions when slow electrons collided with 2-C4F8, namely, C,F;* (t, = 7 psec at -0.0 eV), C4F;* (z, 2: 18 psec at -0.0 eV), and C,F;* (z, = 70 psec at -4.2 eV).

(180") COZ.

Ix. AUTODETACHMENT LIFETIMES OF

DOUBLY CHARGED NEGATIVE IONS

A. Atoms

In recent years there have been a number of reports on short- and long- lived doubly charged atomic negative ions in free space. Dinegative ions should be stable if the additional binding energy due to the second electron exceeds the Coulombic repulsion of the mononegative ion. Atomic dinegative ions are listed in Table XI. In spite of the fact that the existence of some of these is still uncertain, long-lived doubly charged negative ions are of fundamental importance from both the basic and the applied point of view. It is, for example, of great interest to identify the atomic configurations that allow for the formation of such negative-ion states and the processes that lead to their production, and to consider whether beam intensities can be high enough to seriously consider their use in high-energy accelerators. They constitute a challenge for both theory and experiment.

The doubly charged H2- ion is an extreme case of the three-electron isoelectronic series, H2-, He-, Li, Be', B2+, etc. The extremely short-lived HZ- ion was first observed by Walton et al. (1970, 1971) as a resonance in e + H- scattering. These workers associated HZ- with the structure they observed at 14.2 eV in the cross section for the electron detachment process e + H- -+ H + 2e. A higher-lying (at - 17.2 eV) resonance in the e + H- scattering cross section was later observed by Peart and Dolder (1973) and was similarly ascribed to H2-. Theoretical calculations by Taylor and

TABLE XI

LIFETIMES OF DOUBLY CHARGED METASTABLE ATOMIC NEGATIVE IONS

& L Ion" (eV) (set) Comments

H2- b 14.2 - 10-15 '

4 10- 1 5 d 17.2 & 0.35

5.38 -

- F2 -

- 5 x 10-16'

(2.3 f. 0.4) x 10- *

- 10-4hJ

-lo-' to 10-6'

?'

H2- associated with structure observed at 14.2 eV in the cross section for the reaction e + H- + H + 2e

H2- associated with structure observed at 17.2 eV in the cross section for the electron detachment process e + H- + H +2e

Theory Inferred indirectly from ex-

periments using a tandem mass spectrometer with a hollow cathode duoplasmatron ion source (see discussion in text)

An electron source and a mass spectrometer were employed

A Penning ionization source and a 60" magnetic sector for mass analysis were used

An electron source and a coincidence mass spectrometer using quadrupole mass filters for analysis were employed

See text for possible configurations. Contrary to these experimental observations, Levy-Leblond (1971) argued that a system

Walton et al. (1970, 1971). Peart and Dolder (1973). Herrick and Stillinger (1975). For the ejection of one electron only. Schnitzer and Anbar (1976b,c). See discussion in text. Stuckey and Kiser (1966).

j Baumann et al. (1971). Ir Ahnell and Koski (1973).

consisting of one proton and three electrons has no stable bound state.

Bethge (1974).


Thomas (1972) and by Thomas (1974) support the existence of these resonances and predict short autodetachment lifetimes, comparable to the experimental values. Taylor and Thomas (1972) attributed the 14.2 eV resonance to the ( 2 ~ ) ~ 2 p ( ~ P ~ ) configuration; the higher lying one (at 17.2 eV) most probably has largely the configuration ( ~ P ) ~ [ ~ P O ; ’Do] (Taylor and Thomas, 1972; Thomas, 1974).

More recently Schnitzer and Anbar (1976b,c) reported the existence of H2- and D2- with a much longer lifetime (2.3 x low8 sec). This was inferred indirectly from experiments using a tandem mass spectrometer with a hollow-cathode duoplasmatron ion source. According to Schnitzer and Anbar, this longer-lived state probably has a different electronic configuration than those at 14.2 and 17.2 eV and resembles the long-lived autoionizing states of He- (BIau et al. 1970; Estberg and LaBahn, 1970) and Li (Feldman and Novick, 1963; Pietenpol, 1961). Durup (1976) pointed out that Schnitzer and Anbar’s observation requires H2- to be metastable with respect to both H - + e and H + 2e and proposed that it is in a ~P,[~SO] state. How- ever, the result of Schnitzer and Anbar has been questioned by Aberth (1976) and Vestal (1976).

sec were reported by Stuckey and Kiser (1966). These findings were questioned by Fremlin (1966) and could not be duplicated by Chupka et al. (1975). However, Baumann et al. (1971) verified Stuckey and Kiser’s findings and reported additional (TeZ-, Bi2-, and I2 -) long-lived dinegative ions. Similarly, Ahnell and Koski (1973) reported F2-, and Bethge (1974) long-lived 02-, Pz- , F2-, ClZ-, As2-, BrZ-, SbZ-, Te2-, 1’-, and Biz- (see Table XI) ions. Such long-lived dinegative species are in contrast to H2- (at 14.2 and 17.2 eV) for which z, ‘v lo-’’ sec. According to Schulz (1973), U. Fano suggests that a plausible explanation for these long-lived atomic dinegative ions could be a sextet of the type 3p4(,P)4s4pZ for C1- and a configuration 2 ~ ~ ( ~ S ) 3 s 3 p ’ for 0’-. The energy of these levels is not known.

02-, FZ-, C12-, and Br2- ions with lifetimes of -

B. Molecules

Dinegative ions should be stable in the gas phase if the additional binding energy due to the second electron is greater than or equal to the Coulomb repulsion in the ion. Large molecules for which Coulomb repulsion could be substantially diminished might then be expected to form long-lived doubly charged anions. Reports of doubly charged parent anions of complex molecules have appeared recently. Dougherty (1969) reported a doubly charged anion for benzo [cd] pyrene-6-one, and Bowie and Stapleton (1976) reported similar results for nitrobenzoic acid, cyanobenzoic acid, and p-NO2-C,H4-(CHZ),-CO2R with R = H or CH,. For the last series of


molecules, Bowie and Stapleton found that the most abundant dinegative ions occur when n = 3 or 4. Bowie and Stapleton also presented evidence that the parent molecular dianions M2- they observed are formed via electron attachment to the mononegative ion, M -, viz.,

M- +- e - M 2 - (50) Although no lifetimes have been mentioned for these dianions in the above reports, the fact that they were detected with mass spectrometers implies that they are long-lived (0, 2 lop6 sec).

The observation of dianions for nitrobenzenes containing one (or more) electron withdrawing groups is consistent with the discussion in Section VI,A. We suggested there that when the EA of an organic molecule is highly positive, as in the case of nitrobenzenes with one or more additional electron withdrawing substituents, two negative-ion states may lie below the energy level of the neutral molecule in its lowest electronic, vibrational, and rotational state, allowing for two long-lived nuclear-excited Feshbach resonances. Such and similar compounds, therefore, are expected to be candidates for parent doubly charged negative ions.

Additionally, Bowie and Stapleton (1976) observed a number of doubly charged fragment anions such as (M-H)2- from organic acids, NO:- from nitrobenzenes, and (M-CH3)Z - from p-nitroanisole and p-nitrobenzylmethyl ether. As discussed in the previous section, C12-, BrZ-, and 1’- ions were detected as fragment ions from suitable alkyl halides (e.g., CH3X [X = C1, Br, I] and CHX, [X = C1, Br]) and CC14, which themselves have not been found to form parent dinegative ions.

It is stressed that our discussion is about gas-phase anions. In liquids and solutions, multiply charged negative ions can be stabilized via solvation since solvation adds extra stability, and many such ions have been reported. It is noted also that even in the gas phase, hydration of ions (e.g., 0-) could lead to species able to support more than one electron.

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Time-Resolved Laser Fluorescence Spectroscopy for Atomic and Molecular Excited States : Kinetic Studies

J.-C. GAUTHIER AND J.-F. DELPECH

Groupe CEElectronique dans les Gaz Institut CEElectronique Fondamentale*

FacultP des Sciences UniuersitP Paris-XI

Orsay, France

Introduction ..................................... I. Direct and Indirect Methods for Excited-State Kinetics Studies . . . . . .

A. Stationary Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Time-Resolved Methods . . . . . . . . .....................

ectroscopy . . . . . . . . . . . A. Dye Lasers : Modes of Operation and Performance ........................ B. Photon Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Time-Resolved Measurement of Fluores D. Possible Systematic Errors . . . . . . . . . . .

A. The Master Equation of Level Populations B. Multicomponent Exponential Decay An

D. Least-Squares Simulation of Experimental Data .......................... E. PressureFits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV. Applications to Atomic and Molecular Physics A. Atomic Lifetimes: Radiative Decay,

Transfer ......................... B. Relaxation of Molecular Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V. Recent Developments and Concluding Remarks .............................. References . . . . . . . . . . . . . . . . . . . . . . . . . .... . . .

11. Experimental Techniques for Pulsed-L

111. Methods of Data Analysis and Reduction . . . . . . . . . . . . . . . . . . ..............

C. Least-Squares Fitting of the Coefficients of the Master Equation . . . . . . . . . . . .

. . . . . . . . . .

131 132

133 138 143 144 150 154 163 164 164 166 168 170 171 172

172 186 195 199

INTRODUCTION

Radiative deexcitation and collisionally induced excitation transfer processes involving various atomic and molecular states are of great fundamental and practical interest; they play a major role in our understanding of such systems as gas lasers, stellar atmospheres, molecular formation in astrophysics, gaseous plasmas up to thermonuclear conditions, and ionosphere physics. In these systems, atoms or molecules are being continually excited

* Laboratory affiliated to CNRS


All rights of reproduction in any form reserved. ISBN 0-12-0146(6-0

132 J.-C. GAUTHIER AND J.-F. DELPECH

and deexcited by absorption and emission of radiation (radiative transfer) and by mutual collisions. The experimental study of the cross sections for the various collisional processes, as well as of the radiative lifetimes, is also of great importance for testing theoretical models, which have become both very powerful and very sophisticated over the last few years.

The study of such reactions in the gas phase at or near thermal energies has received a fresh impetus with the recent development of pulsed-dye laser technology coupled with the progress of fast photodetectors and with modern electronics and data acquisition systems. With these new tools, transient phenomena involved in the interaction of light with matter can be studied with high temporal and spectral resolution. It thus has become possible to inject energy into a well-defined state of an atomic and molecular system and to follow its relaxation within internal and external degrees of freedom. One can thus study, among others, radiative decay of excited atomic, ionic, and molecular states, internal conversions among the various internal degrees of freedom of excited molecules, and chemical reactivity dynamics.

This review is essentially concerned with the use of pulsed-laser fluorescence spectroscopy for the study of excited-state kinetics. In Section I, however, we give a very brief review of the main other direct and indirect methods that have been used for such studies. Sections I1 and 111 are devoted to the experimental aspects of time-resolved laser fluorescence spectroscopy : dye lasers and photon detectors, electronics and data acquisition systems, deconvolution of the time response of the detection systems, and finally, the important-and often somewhat vexing-problem of extracting from the raw experimental data the kinetic coefficients for deexcitation and/ or transfer between various excited states. Sections IV and V are devoted to an examination of the application of these laser techniques to atomic and molecular physics.

Laser fluorescence spectroscopy is a fast-growing subject. While we have tried to include all the papers that had been published by July 1977 and that demonstrate in a significant way the originality and reliability of these techniques, there are doubtless important papers that have escaped our notice: we apologize for their omission.

I. DIRECT AND INDIRECT METHODS FOR EXCITED-STATE KINETICS STUDIES

An ideal method for investigating the kinetics of collisional and radiative energy transfer processes between atomic or molecular excited states should fulfil several requirements:

(1) It should be applicable to all kinds of atoms and molecules, neutral or charged, over a broad range of wavelengths and times.

PULSED-LASER FLUORESCENCE SPECTROSCOPY 133

(2) For fast, efficient measurements, excitation and detection efficiencies should be high.

( 3 ) For unambiguous data interpretation, excitation and detection should be highly selective; in some cases, sub-Doppler resolution will be necessary (i.e., the excitation and/or the detection should be free from Doppler broadening).

Needless to say, no practical method can simultaneously meet all these requirements; while time-resolved laser fluorescence spectroscopy is often the most precise and convenient method, and will be discussed in Sections 11-V, this section is devoted to a rapid survey of other methods that have been used in the past and that are still of great value in various situations. These methods can be broadly classified in two categories, depending on whether temporal resolution is used or not.

A. Stationary Methods

These methods are characterized by the fact that measurements of atomic and molecular level populations are carried out without temporal resolution. They are indirect methods, as the decay rate of the state is not directly determined.

The p h ~ ~ shft method (Duschinsky, 1932,1933) has been used to measure atomic and molecular lifetimes. An optically allowed transition is excited with intensity-modulated light: the fluorescent light is modulated at the same frequency w but is shifted in phase relative to the exciting light because of the finite lifetime of the emitting state. The phase shift A q is related to the lifetime z by the relation

tan A 9 = wz (1)

Choosing the best modulation frequency (Acp should be about 45") makes this method very sensitive (lifetimes down to a fraction of lo-' sec can be measured). The excitation is done using either a broad-band light source (Chutjian et al., 1967) or a pulsed electron beam (Smith, 1970). Electron excitation introduces the ability to excite a wide range of upper levels of neutral or multiply ionized species. However, both excitation schemes are particularly susceptible to errors originating in radiative cascade, since the excitation step is largely nonselective: much detail is thus lost. Use of a monoenergetic electron beam at threshold energy or of monochromatic light sources generally overcomes these shortcomings. The latter possibility has been recently considered by Armstrong and Feneuille (1975), taking advantage of the performances of single-mode tunable CW dye lasers. Doppler effects being suppressed by the use of an atomic beam, an experiment can be performed in which an essentially monochromatic (but


amplitude-modulated) light source excites the atoms or molecules. They find that the phase shift is no longer independent of the strength of excitation and that the dependence of A q on the level lifetime changes at higher values of the excitation strength. Monochromatic light excitation was first reported by Baumgartner and his co-workers (1970) who studied the radiative lifetimes of alkali molecules with the phase shift method, using CW argon ion laser lines for excitation (see Fig. 1). Comparison of lifetime measurements of the excited molecular states with lifetimes of corresponding states in the separated atoms gives information about transition probability variations with internuclear separation.

Often employed before 1970, the phase shift method is of minor interest now, compared to highly selective optical methods involving CW or pulsed tunable dye lasers.

supcnonio Rohctor

Mcillatitinp qwrtz

I

A2

L

Ahupha+ B Light

----*--. = w Absorption

COU

FIG. I . Typical experimental system used in phase shift lifetime measurements. The high- frequency amplitude modulation of the laser beam is provided by a supersonic modulator: beam enlargement is necessary to obtain a good modulation phase stability. From Baumgartner et al. (1970).


With the level-crossing technique (Franken, 1961), the natural width (and hence the lifetime) of two crossing levels is determined from the change of the fluorescence light intensity and polarization at the crossing point. Zeeman sublevels of different hypedine structure (hfs) states in a magnetic field or Stark levels in an electric field may be used as crossing levels. To illustrate the potentiality of the method, let us focus the discussion on 'P,!, states of alkali atoms (Schmieder et al., 1970). When the magnetic field IS zero, the 2P3,z level is split into four hfs sublevels of total angular momentum F = I + $, I + 3, I - $, I - 3, where I is the nuclear spin. As the field is increased from zero, the F sublevels are split into their magnetic substates, some of which may cross others. If the condition (Am1 I 2 is satisfied, where Am is the difference in magnetic quantum numbers of the crossing levels, it is possible to detect the crossing by its effect on the scattered light (an increase or a decrease, depending on light polarizations during excitation and detection). The shape of the intensity vs. field strength curve depends on the hfs constants and on the lifetime of the observed levels. A special use of this technique is the study of the magnetic depolarization of atomic resonance fluorescence, the so-called Hanle effect. At zero field, the Zeeman levels of an excited atom are degenerate and may be excited coherently by an uni- directional beam of polarized light. If the excited state is unperturbed, it radiates to the ground state with a characteristic decay time (its natural lifetime), and the fluorescence radiation is observed to have the same polarization as the exciting beam. When a magnetic field is applied along an arbitrary direction to this excited state, the degeneracy between magnetic sublevels is removed and the coherence is partially or totally destroyed, resulting in a depolarized resonance fluorescence light. The lifetime z of the excited state is then deduced from the width of the zero-field resonance, knowing the magnetic moment of the excited-state Zeeman levels.

The Hanle effect and the more general level-crossing method have been widely used to determine atomic lifetimes (Budick, 1967) but can also be extended to molecules (Zare, 1966). However, the best application resides in the determination of atomic fine structure (fs) and hfs of excited states. The Stark effect in excited states has been also very precisely investigated using this technique with parallel electric and magnetic fields. Pure electric-field level crossings, which are possible when the uniform electric fields attainable in the laboratory are comparable to the hfs separation (Khadjavi et al., 1968), have also been obtained.

A more promising technique to determine excitation transfer cross sections is based on excited-state anticrossings (Eck et al., 1963). When a small interaction, for example, of the spin-orbit type, couples two magnetic substates, the levels involved are prevented from crossing; the levels instead repel each other (anticrossing) and the wavefunctions of the two substates


interchange their identities as the magnetic field is swept through resonance. When the anticrossing involves, for example, singlet and triplet states that may have, at zero field, very different populations due (for example, in a discharge) to quite different excitation cross sections, the wavefunctions at the anticrossing field are completely mixed and populations are equalized. This results in a large change in the ratio of the populations. Anticrossing spectroscopy has been recently used to determine collisional angular momentum transfer cross sections in the Rydberg states of helium by Freund et al. (1978).

However, these methods, which involve magnetic resonance, seem to be more suitable for precise atomic and molecular fs and hfs determination than for kinetic parameters determination. They are dependent on measurements that can only be related indirectly to these parameters through some model of the excited sublevel array. Although direct methods in which the natural decay time of the observed level is determined have the highest inherent accuracy, individual transition probabilities cannot generally be determined by such techniques alone, and indirect methods like those previously described are sometimes found to be useful.

This is particularly the case for the steady-state collision-induced juores- cence method. This highly selective technique allows the measurement of lifetimes and collisional cross sections in atoms and molecules. It is essentially a perturbation method in which an assembly of excited levels in thermal or collisional-radiative equilibrium are disturbed by steady-state or quasi- steady-state monochromatic light, possibly produced by a laser. Only one level is perturbed by the highly selective excitation and the resulting variations in neighboring level populations are probed by usual fluorescence techniques. Cross sections for excitation transfer, for quenching by collisions with parent or foreign atoms, and radiative transition probabilities are deduced using the techniques described in Section I11 to solve the time- independent rate equations. The method has been widely applied to study inelastic collisions involving excited alkali atoms in collision with noble gas atoms, at thermal energies. One of the atoms most studied is certainly cesium; in their comprehensive work, Cuvellier and co-workers (1975) have used a flashlamp-pumped dye laser for excitation. Transfer of excitation between the fs levels of the first doublet of the alkali metals have also been widely investigated with techniques analogous to that described by Siara et al. (1974) in the 7’P doublet of cesium. Rydberg states of potassium have also been studied (Gounand et al., 1976a) with this technique.

A slightly different method has been devised by Bakos and Szigeti (1968) and by Wellenstein and Robertson (1972) to study the collisional relaxation processes for the n = 3,4 states of helium. Direct pumping of the singlet levels from the ground state is not practicable because the corresponding


resonance radiation falls in the VUV spectral range and pumping the triplet is impossible owing to spin selection rules.

Use is made of a discharge to populate the 23S and 2's metastable levels in a first excitation step. A helium-pumping lamp, provided with suitable filters, is subsequently used to populate the n = 3,4 levels. All these levels are in collisional-radiative equilibrium, and the principle of the technique consists in measuring the small disturbance of the equilibrium when one level is slightly perturbed by optical radiation. To increase the signal to noise ratio, use is made of synchronous detection techniques at frequencies very much smaller than the collision frequencies to be measured, so that a steady- state condition is preserved. Coupled with mass spectroscopic detection, the Wellenstein and Robertson investigation provided the first direct measurement of the cross sections for associative ionization of the n = 3 states of helium. Measurements on the n = 4 states of helium, using a perturbation method similar to Wellenstein's but with a laser instead of a lamp for excitation, have been made by Abrams and Wolga (1967).

The collision-induced steady-state fluorescence method has also opened an elegant possibility for the study of inelastic collisions in molecular excited states. The molecular species to be studied is excited optically with monochromatic light, preferably from a CW laser, into a single rotation-vibration level. The fluorescence from this level is observed spectroscopically using conventional techniques. If the experiment is performed in the presence of a foreign gas, the steady-state fluorescence spectrum contains additional weak lines, which result from inelastic collisions of the excited molecules with the gas atoms during the molecular excitation lifetime. A typical laser-excited fluorescence spectrum of NaLi is shown in Fig. 2. When a single collision occurs during the lifetime of the excited state, the dependence of the intensity on pressure is linear and cross sections can be obtained directly from the measured intensity of the collision-induced fluorescence. Relative cross sections measured in the Na, (B'II,) molecule (Bergmann and Demtroder, 1972a) and in the NaLi (In) molecule (Ottinger, 1973) are found to be markedly different depending on whether the collision-induced transition originates from the upper or from the lower component of the A doublet of the Il state. Such propensity rules were also observed by Ottinger and his co-workers (1970) in the three isotopic forms of the Liz molecule. These effects were further discussed by Klar (1973) and, using the scattering theory, by Green and Zare (1975). Absolute cross sections for rotational and vibrational transitions in the Na, molecule have also been investigated as a function of the collision partner by Bergmann and Demtroder (1971,1972b) using the laser fluorescence technique.

It has recently been proposed to use the collision-induced fluorescence technique (Gersten, 1973) to obtain information concerning the energy

138 J.-C. GAUTHIER AND 1.-F. DELPECH

I

PI2 p., 50

Na Li ('TT I v '=?) - (X 'Z ' ~ " - 1 ) ~ Collision gas Neon FIG. 2. Typical laser-excited fluorescence spectrum of NaLi, prepared by the laser in the

J = 28 level with unidentified v'. Collision-induced rotational transition cross sections are inferred from the relative strengths of the numerous rotational lines. From Ottinger (1973).

curves and dipole matrix element as a function of internuclear separation in atomic gases. This technique has the unique advantage of allowing a direct measurement of the dependence of the electronic transition moment on internuclear separation; this information is not readily obtainable by other techniques. It has been applied recently to the case of Na dimers by Callender and his co-workers (1974).

B. Time-Resolved Methods

Direct methods, where the light emission of the levels of interest is analyzed, should be used whenever possible, as indirect methods are liable to many pitfalls.


Under the simplest experimental conditions, the exponential decay rate of the light fluorescence originating from the excited levels is directly related to the radiative probability of the observed transition and, at higher pressures, to the various collisional cross sections with ground-state atoms.

Usually, photon excitation is restricted to neutral species, except where special arrangements, such as a pulsed discharge, are provided to produce a ionized species that may be studied during the period before recombination has too severely reduced its concentration. The most serious limitation of photon excitation is the limited range of accessible upper states. Beam-foil excitation and electron excitation introduce the ability to excite a wide range of levels in neutral, singly, or multiply ionized species. The first method has poor selectivity, which may result in strong cascading and/or line identification problems (Dufay, 1970). In the second method, we have already pointed out that threshold energy electrons are essential to reduce cascading. As both these methods have unique possibilities for the investigation of highly excited states, they will be reviewed very briefly.

Beam-fail spectroscopy was developed in 1966 by Bashkin and his co- workers (1966). Basically, singly charged positive ions of the elements of interest are accelerated by a Van de Graaf accelerator to energies ranging from 10 keV to 500 MeV. These ions are excited when the beam undergoes collisions on passing through a thin foil (generally made of carbon 5-10 pg/cm2 thick). Lifetimes of the ions in excited states are measured by time of flight techniques (see Fig. 3). The decay curves are analyzed with a least- squares fitting computer routine that is able to fit multiexponential functions

FIG. 3. Lifetime measuring apparatus in beam-foil spectroscopy. The insert shows the arrangement used to determine the spectrum. From Smith (1970).


to the experimental points (see Section 111). In order to deduce the lifetime from the decay curve, several corrections are necessary, so that the lifetimes can only be known to 20% (Smith, 1970). Although the excitation technique is highly nonselective and thus has to be corrected for cascading (this can be done by varying the excitation energy), beam-foil spectroscopy has been able to provide lifetime data of great value (Wiese, 1970; Martinson, 1973) on neutral, singly, and doubly ionized atoms. Recent developments in experimental techniques (Church and Liu, 1973; Masterson and Stoner, 1973) using quantum beats and photon coincidences permit the measurement of lines of selected levels without the necessity of correcting for cascades into those levels. Other arrangements combining foil and laser excitation (Harde and Guthohrlein, 1974) have been devised to obtain cascade-free lifetimes. The main disadvantage of this extremely useful method is that molecules cannot be studied, since molecular ionic beams dissociate in the foil; in addition, only one point of the decay curve can at present be obtained for each position of the detector, thus considerably increasing the data acquisition time.

Pulsed electron excitation was used as early as 1932 (Lees, 1932; Lees and Skinner, 1932) to study the excitation function of helium. As in the case of beam-foil excitation, the use of electron pulses of high current densities coupled with direct observation of the decay curve is normally nonselective and is also prone to interpretation difficulties. Experimental techniques for the direct measurement of excited-state lifetimes in atomic and molecular gases down to a few nanoseconds have now been devised by Bennett and co-workers (1965) and Bridgett et al. (1970). They use essentially monoenergetic electron beams at fairly low current density (0.1 A/cm2) at energies slightly above the excitation threshold of the state of interest. Space charge effects and cascading from unwanted excited states are thus made negligible with these improvements. Typical applications of this technique to the 2p53p configuration of neon (Bennett and Kindlmann, 1966) and to helium n = 3 states in collision with other rare gas atoms (Kubota et al., 1975) are representative of its potential usefulness. A severe shortcoming of the method is the possible heating of the gas in the observation region by the electron beam if high current densities are used. Very mild excitation must be used if one wants to measure excitation cross 'sections at temperatures close to 300°K; in that case, the severe reduction in available excited-level populations reduces the optical signal to a level where single-photon counting techniques must be used, resulting in a drastic increase of the experimental acquisition time. Apart from these difficulties, the technique is very attractive, as shown by the recent work of Thompson (1974) and Thompson and Fowler (1975), who used a pulsed inverted triode (Johnson and Fowler,


1970) to study lifetimes of 49 levels of helium; they found results in good overall agreement with accepted values (Wiese et al., 1966). Other versions of this technique, like the high-frequency deflection technique (Erman, 1975), have been proposed as refinements of the conventional multichannel delayed-coincidence technique with periodic electron excitation.

In the electron-photon delayed coincidence method introduced by Imhof and Read (1969), delayed coincidences between inelastically scattered electrons and the subsequent decay photons are recorded. This method is an adaptation of standard techniques used in nuclear physics. Its main advantage is that cascade complications can often be eliminated to yield lifetime measurements that are free from systematic errors. The state of interest is excited with a monoenergetic electron beam at an impact energy well in excess of threshold. Inelastic electrons scattered into a small solid angle in the forward direction are energy analyzed before being detected by a channel electron multiplier. The excited state of interest can thus be selected by tuning the energy analyzer to the energy loss corresponding to the state excitation energy. The lifetime is then measured by observing delayed coincidences between these electrons and the corresponding fluorescence photons. The resolving power for discriminating against cascade is typically 0.05 eV, a value largely sufficient for most atomic (and some molecular) states of interest.

In neutral atoms, the electron-photon coincidence method has been used to study mercury (King et al., 1975a) and cadmium (CvejanoviE et al., 1976). However, its most interesting feature is to allow measurements of radiative lifetimes of ionic excited states without the cascading problems inherent to beam-foil excitation. When ionic species are to be studied, the incident electron simultaneously ionizes and excites the target atom or molecule into the desired ionic excited state. Now, however, there are two inelastic electrons: the scattered electron and the ejected electron, which share the available energy between them (use is made of electrons with energies in excess of threshold). Investigation of the energy distribution between electrons shows that, away from threshold, it is probable that one of the electrons will take most of the energy, leaving the other with much less. In order to maximize the detection efficiency of the higher-energy electron, the energy resolution of the analyzer is broadened (typical values are 2 V FWHM). This very promising technique has been applied to atomic ionic states like Cd I1 (Shaw et al., 1975) and to molecular ionic species like N:, CO', and CO: (Smith et al., 1975).

A much more direct and precise version of this technique has also been devised. Often called the photon-photon delayed coincidence method (Fig. 4), it uses a discharge or an electron beam to prepare the states. Photons arising

142

Hypodrrmic ntedle

J . 4 . GAUTHIER AND 1.-F. DELPECH

Electron optics

\l

Fused silica lens

-1 optical filters

L

Y 0 - & -

from transitions from higher states to the state of interest are selected in one channel while photons emitted as a result of the decay of the state of interest to lower energy states are detected in a second channel (see Fig. 4). Delayed coincidences are observed between the initiating and decay photons and the resulting time spectrum is used to measure the lifetime of the intermediate state (Camhy-Val and Dumont, 1970). This purely optical method has been applied to levels of mercury and argon I1 (Camhy-Val et al., 1970; Mohamed et al., 1976) and its applicability to molecular excited states has been proved in H2 by King and his co-workers (1975b) and Eland et al. (1976). These methods are characterized by very long acquisition times (typically of the order of 50 hours or greater). However, owing to the high selectivity in both photon channels, which discriminates completely against cascading effects from higher excited states, coincidence methods should be substantially more accurate than the other techniques; typical claimed accuracies are well below 5%. These techniques have been comprehensively reviewed recently by Imhof and Read (1977).

- $ E

FIG. 4. The experimental system typically used in the photon-photon delayed-coincidence method. The gas is fed by the hypodermic needle, in front of the exciting electron gun, perpen- dicularly to the plane of the figure. From King et a/. ( I 975b].


11. EXPERIMENTAL TECHNIQUES FOR

PULSED-LASER FLUORESCENCE SPECTROSCOPY

A typical time-resolved fluorescence experiment is shown in Fig. 5. Such an experiment comprises typically three parts: the light source (in this case, a dye laser), the interaction region (in this case, a plasma cell at a pressure of a few torrs), and a fluorescence measurement system.

In the interaction region, atoms must initially be prepared in certain well- defined quantum states before interacting with the laser beam. This initial state may be the ground state, in some particularly simple situations (for example, when studying some properties of alkali atoms). Excited states that are not optically connected to the ground state (metastable states) are more difficult to prepare, as well as states that lie so high in energy that they cannot be reached from the ground state by using presently available lasers (as in helium and the other rare gases). Nonselective excitation is then the only possible way to sufficiently populate these states. Pulsed afterglow techniques are particularly useful in this connection (Delpech et al., 1975). In afterglows, the metastable or first resonance states are principally formed during the discharge, and to a lesser extent during the afterglow; their concentration is comparatively large (densities of 1010-1012 cm- are typical) and they can be used very conveniently as initial states from which it is possible to populate efficiently by optical pumping higher-lying excited

microwave diagnostics - linewidth

1 1 1 L monitor

referenci monoch

DYE res. 7 i 1 meter -1 PMT

monoc hromator res. 0.5A LASER

I - - k u,, I u) ,, sampling WA 16 channels

head convertor + analyzer

FIG. 5. Simplified schematic diagram of the system used by the Gaseous Electronics Group

- - - at Orsay. Parts of the nanosecond time resolution sampling head are described in Fig. 11.

144 J.-C. GAUTHLER AND J.-F. DELPECH

states. Beam-foil nonselective excitation may also be used to populate long- lived or metastable ionic species; a laser pulse is then needed to populate the desired upper level.

A. Dye Lasers: Modes of Operation and Performance

Among the many tunable sources of coherent light that have been developed up to now (Kuhl and Schmidt, 1974), organic dye lasers offer a unique combination of desirable properties and are thus particularly attractive. Specific areas where tunable coherent radiation may be expected to play a significant role (Yardley, 1975) have been recently discussed by Mooradian et al. (1976). Organic-dye lasers have been described in detail in a number of excellent review articles since their discovery in 1966 (Lempicki and Samelson, 1966; Bass et al., 1971; Schafer, 1973). However, for convenience, we summarize here some of their properties and modes of operation, inasmuch as they are relevant to our purpose.

Dye lasers comprise two essential elements : the active medium, which exhibits gain at optical frequencies and which provides a source of optical energy, and the cavity, which determines the tuning characteristics of the radiation source.

1. Optical Properties of Organic Dyes in Solution

Dye materials can absorb and emit in the ultraviolet and near infrared as well as in the visible. The spectroscopic properties of these materials are determined by their physical and chemical structures and by their interaction

WAVENUMBER (rO3Chl-’)

FIG. 6 . Singlet absorption and fluorescence spectra of PBD in toluene as a function of wavenumber (27 x lo3 cm- ‘ = 3704 A).


with the solvent. Most dye solutions have spectral properties similar to those of PBD [2-phenyl-5 (4-biphenyl)- 1,3,4-oxadiazole] in toluene (Fig. 6 ; strictly speaking, PBD is not a dye but a scintillator). The widths of the principal absorption and emission bands are usually of the order of a few thousand cm-' and the fluorescence peak occurs at a longer wavelength than the principal absorption peak. This displacement is easily understood with the help of the energy level diagram shown in Fig. 7.

{- s-s - fluorescence

s-s - absorption

- - - I I I -& - iT2 - - - I I

1-1 - Internal conversion

T-S Phorphorrncrnce

(Slow)

I I I I

~~

Contigurotton coordinate

FIG. 7. Schematic energy level diagram for a dye molecule. Transitions relevant for dye laser action are shown: singlet and triplet levels are respectively labeled S and T. From Bass er al. (1971).


KMTILLAIW -

COUMARINE

BRILLIANT -

XANTHENE

OXAZINE

C YA NINE

WES

SULFAFLAVINE

The events involved in the absorption and reemission of light begin and end with the molecule in its ground-state equilibrium configuration. After absorbing one photon of light (produced, for example, by the pump laser), the molecule is excited on the electronic level S, and relaxes nonradiatively into the low-lying vibrational sublevels of S1 in times on the order of 10- ‘I-

lo-’’ sec (Ricard, 1975). Since the radiative lifetime of S, is usually of the order of s, the broad-band fluorescence spectrum corresponds to transitions from the lower edge of S, to the ground state and is thus shifted to the red.

An important figure of merit of a particular dye is the fluorescence quantum efficiency, i.e., the ratio of the number of emitted photons to the number of absorbed photons; this number usually ranges from 0.01 to nearly unity.

There are several processes that compete with the fluorescence of the dye molecule and therefore reduce the efficiency of laser action. The most important parasitic processes are usually nonradiative internal conversion from level S, to So and nonradiative intersystem crossing from level S, to the metastable level T I , followed by slow triplet-singlet phosphorescence. Ab- sorptions from singlet and triplet states to higher-lying levels also result in further losses in dye efficiency; however, these processes are less important with short pumping pulses (e.g., with N2, KrF, or Nd:YAG lasers).

Quantitative analysis of dye laser action is difficult, as it requires a precise knowledge of molecular deexcitation parameters (a review was written by Bass et al. in 1971). Numerical solutions of the rate equations governing pulsed laser operation in dyes have been proposed (as an example, see Pappalardo e,t a/., 1972); more recently, high-gain systems have been investigated under strong pumping conditions to study amplified spontaneous emission in more details (Ganiel et al., 1975).

Dye chemistry is outside the scope of this chapter: important dye families and their respective tuning range are shown in Fig. 8. Drexhage (1973) gives a compilation of laser dyes and Basting et al. (1974) give a list of 73 new laser dyes having a maximum fluorescence wavelength ranging from 3750 to 6700 A. A classified bibliography on work on dye lasers before 1972 has been compiled by Magyar (1974).

m - - - m - I . I . 1


2. Pumping Techniques

Dye laser performances are closely dependent on pumping techniques. Essentially three broad classes of pulsed-dye lasers are of interest in time- resolved fluorescence spectroscopy; the corresponding typical wavelength ranges are given in Fig. 9.

FIG. 9. Approximate wavelength coverage for the different pumping light sources (“Fast flashlamp” means rise times on the order of 100 nsec). From Walther (1973).

(1) Flashlamp-pumped dye lasers deliver a typical output energy of 10 mJ (much larger energies are possible), but the pulse duration is relatively long (300-500 nsec is typical) and repetition rates tend to be low (although repetition rates up to 100 Hz can be reached). While they have been used successfully in some special cases (as we shall see in Section IV) they are usually not very well suited to time-resolved fluorescence spectroscopy.

(2) Frequency-doubled ruby- and Nd:YAG-pumped lasers are mostly used to generate near IR radiation and their performances are strongly dependent on the nature of the dye-solvent combination used.

(3) We shall only describe here the nitrogen-laser-pumped dye laser, which is by far the most useful device in the near UV and visible spectral range for atomic and molecular studies requiring an efficient pulsed-light source of short duration (a few nanoseconds), high repetition rate (usually 50-100 pps, but rates up to 1000 pps are possible), and adequate output energy (up to several hundred microjoules).

The nitrogen laser oscillates at 3371 A. It has attained increasing importance as a pumping-light source for dye lasers because of its convenient wavelength for dye absorption, short risetime, and high repetition rate. Very simple systems can be designed (Basting et al., 1972) with peak powers in excess of 1 MW and active lengths as short as 30 cm. Nitrogen lasers de- livering peak powers in the 0.1-1 MW range are relatively inexpensive, both in initial investment and in operation expense.


A high-performance dye laser has been designed by Hansch (1972) that is particularly suited to being pumped by a nitrogen laser, and this design is now widely used with only minor variations. Its basic components are shown in Fig. 10. A dye cell is excited by the focused nitrogen-pumped light and the resulting superradiant narrow beam is reflected by a grating that forms one end of the laser cavity. The beam is expanded in an inverted telescope to ensure appropriate operation of the grating; a plane output coupling mirror is mounted at the other end of the cavity. Beam expansion has the advantage of greatly improving the resolution, which may be further increased with a tilted Fabry-Perot etalon inserted between the telescope and the grating.

ECHELLE GRATING

FABRY-PEROT ETALON

} TELESCOPE I

DYE CELL

POLARIZER

ri Y IRROR

FIG. 10. Basic components of a narrow-band tunable dye laser. From Hansch (1972).

Bandwidths of 300 MHz, which are substantially narrower than most atomic Doppler widths, have been achieved with a grating-Fabry-Perot combination. Poststretching of the output pulses is possible with an external spherical Fabry-Perot; with this setup, Hansch et al. (1971) achieved a linewidth of 7 MHz. Much improved tuning techniques have been proposed independently by Wallenstein and Hansch (1974) and Flach et al. (1974). They used the simultaneous scanning of the wavelength-selective elements of the cavity (grating and Fabry-Perot etalon) and of an optical external con- focal filter interferometer by changing the gas pressure in an enclosing chamber. Tuning over continuous ranges of 150 GHz with a bandwidth of 25 MHz was achieved with excellent linearity. A convenient narrow-bandwidth oscillator-amplifier has been described by Lawler et al. (1976).

Depending on the context, linewidths are evaluated in wavenumber units AT, in frequency units Av, or in wavelength units A l ; the practical units are


respectively cm - ’, GHz, or A. The following simple numerical relations among these quantities are useful:

AT= 3.34 x l o w 2 Av, A V = 30.0 AT, A 2 = 10-’;L2 AT (2)

Thus a linewidth of lo-‘ cm- ’ corresponds to a width of 3 GHz in frequency units, and of 2.5 x lo-’ A in wavelength units at 5000w.

When working in the high-resolution mode with linewidths below 10- A, large losses in efficiency and power have to be accepted. A considerable increase of output power can be obtained by combining a gas-pressure-tuned dye laser oscillator followed by a multistage dye laser amplifier, pumped by the same nitrogen laser with suitable delays. With such a device, peak powers of 50 kW in the visible at linewidths down to 6 x A have been achieved (Wallenstein and Hansch, 1975).

Spectral fluctuations are observed in the output of a pulsed dye laser; they have their origin in the statistical quantum noise of the laser medium. Intensity stabilization of dye laser radiation is possible by saturated amplification. This is done by using a partly saturated high-gain traveling-wave laser amplifier with a nonsaturable absorber distributed in the amplifying medium. The feasibility of the proposed method was demonstrated by Curry et al. (1973), who found that output powers in the kilowatt range could be generated with this scheme, with negligible dependence on input power.

A very promising technique for pumping dye lasers is to use rare-gas- halide lasers, which have much higher “wall plug” efficiencies than N2 lasers and are able to,pump liquid scintillators in the near UV range. The Blumlein driven KrF laser oscillating near 2500 A (Sutton et al., 1976a; Godard and Vannier, 1976) is a good candidate to pump such new scintillator dyes. Laser action has been obtained in solutions of paraterphenyl with a broad-band optical conversion efficiency of about 30% in the 3350-3650 A range (Sutton and Capelle, 1976b; Godard and de Witte, 1976).

The practical overall efficiency of laser-pumped dye lasers is still well below 1% and their usefulness would be considerably improved if it could be enhanced. This may be accomplished by direct electrical excitation, which requires operation of the dyes in the vapor phase. Successful laser operation of a N,-laser-pumped, solvent-free gaseous-phase dye has been reported (Steyer and Schafer, 1974). However, fluorescence studies on electron-beam excited mixtures of POPOP dye vapor in various buffer gases (Marowsky er al., 1976) indicate that important problems are still to be solved before the realization of an electrically excited dye vapor laser.

3. Special Use of Dye Lasers

Simultaneous multiple-wavelength operation of a pulsed tunable dye laser has been demonstrated (Friesem et al., 1973; Lotem and Lynch, 1975).


The system utilizes either gratings in cascade or a small wedge inserted in front of the wavelength selector to form two independent cavities. Applica- tions of double-wavelength-output pulsed dye lasers are numerous in nonlinear optics. Three pulsed UV frequencies can be obtained in a KDP crystal by generating the second harmonic and the sum frequency of the two output beams; difference frequencies can also be generated for tunable pulsed infrared applications.

Frequency doubling of single-wavelength dye lasers in ADP and KDP crystals was achieved soon after the development of operational organic-dye lasers (Bradley et al., 1971). Lithium formiate monohydride crystals have a conversion efficiency of 2% for input powers in excess of 50 kW; they allow (Dunning et al., 1973) the generation of harmonic radiation at wavelengths 150 A below those attainable using a refrigerated ADP crystal. The range between 2300 and 3000 bl is thus fairly easily covered with peak powers in the kilowatt range by using several dyes at fundamental wavelengths between 4600 and 6000 A.

Efficient frequency conversion requires phase-matching between the fundamental wave and its second harmonic. Noncritical index matching may be obtained in ADP by temperature tuning between -120 and +14o"C. However, owing to its higher scan velocity and its broader spectral range, orientation tuning is almost always preferred in practice. The proper tilting angles of the dye laser tuning elements and of the nonlinear crystals are different nonlinear functions of the wavelength. An ingenious mechanical coupling of the two motions servocontrolled by the spatial position of the generated UV beam has been proposed and successfully tested by Kuhl and Spitschan (1975).

A new class of ultrafast transient processes has become accessible to experimental investigations with the development of ultrashort optical laser pulses (Fabelinskii, 1971; Shank and Ippen, 1973). A review of experimental techniques for the generation of frequency-tunable picosecond pulses from passively mode-locked dye lasers has been presented by Bradley (1974). One or two photon traveling-wave excitations of superradiant emission in rhodamine 6G has also been used (Rubinov et al., 1975) to generate 1 psec dye laser pulses. Relaxation processes in excited molecular states and vibrational relaxation in liquids are under current investigation with these ultrashort pulsed dye lasers (Mooradian et al., 1976).

B. Photon Detectors

Most of the modern photodetectors used in the detection of time-resolved fluorescence are photoelectronic. From the near UV to the near IR spectral range, the most efficient detectors are photomultipliers and photodiodes.


Outside this range, particularly in the middle and far infrared regions, photoconductive detectors are most sensitive (Stockmann 1975).

Silicon photodiodes are widely used for monitoring the output of pulsed lasers because of their high quantum efficiency ( > 50%), high speed of response ( ~ 0 . 5 nsec), constant sensitivity, and low dark current. Their uses for measuring the power and energy of short laser pulses have been discussed by Edwards and Jefferies (1973). Germanium avalanche photodiodes have been operated as photon counters in delayed coincidence measurements (Fichtner and Hacker, 1976) at photon energies below 1 eV, where no photomultipliers are available. At energies greater than 1 eV, photomultipliers (PM) are definitely the most efficient and sensitive detectors when used as photon counters in the nanosecond resolution mode. Table I gives the most significant performances of available photomultiplier tubes that have been widely used in time-resolved fluorescence experiments. The data have been taken from application notes of the manufacturers and agree with our own observations.

The time resolution of a photomultiplier is essentially limited by (1) transit time spread due to different path lengths, (2) transit time spread due to different initial electron velocities, and (3) transit time between the last dynode and anode. When careful optimizations of operating conditions for a minimum transit time are performed (Leskovar et al., 1976), fluorescence lifetimes as short as 90 psec can be measured. Reducing the number of dynodes used in a 1P28 photomultiplier, Beck (1976) obtained an output signal rise time of 360 psec. Normal operation of photomultipliers provides time resolution usually of the order of 2 nsec (see Table I).

Photomultipliers for the detection of low light levels can be used in two basic modes of operation: charge integration, in which the anode current integrates the individual photon pulses, and digital mode, in which the pulses corresponding to individual photoelectrons are counted. The digital technique is superior in practice to charge integration at very low light levels because it eliminates dark-current components originating at places other than the photocathode. In the selection of a photomultiplier for use in photon counting, several important parameters must be considered. Let I, be the rate of photon arrival on the photocathode: the photons can be assumed to be emitted at random time intervals so that Poisson statistics is obeyed

where P(n, z) is the probability that n photons strike the photocathode in the time interval z. Let q be the photocathode quantum efficiency and I , the rate of single-electron dark pulses originating from the photocathode.

TABLE I

PERFORMANCE OF TYPICAL PM TUBES USED IN TIME-RESOLVED FLUORESCENCE DETECTION

Spectral range Risetime Quantum efficiency (%) Anode dark current PM Type (A) Gain (nsec) (wavelength, A) (4

RCA IP21 RCA 885ob RCA 4501, RCA 8575 RCA 4818 Hamamatsu R446 UR RCA C31024b RCA C31034 Philips 56 AVP

3200-5600 2900-5200 3000-5200 2500-5200 1850-9000 3000-5500 2400-9000 3000-5200

3 x 106 1.4 x 10'

5 x 106 1.3 x 107

1.4 x 107 > 4 x 105

3 x 107 6 x lo5

1.6 2 2 1.6 1.6 1.5" 2.5 2

13 (3800) 31 (3850) 31 (3850) 17 (3600) 14 (3000) 20 (4ooo) 14 (5500) 19 (4000)

10-9

1 0 - ~ 2 x 10-9 5 x 10-8

10-8 3 10-9 5 x 10-7

6 x lo-'@

a This PM typically has a single-electron anode pulse risetime of 800 nsec. These photomultipliers have been extensively studied by Leskovar et al. (1976).


The variance of the measured number of photoelectrons during the time T is given by (Robben, 1971)

c2 = Ipr]r + (4)

so that the signal-to-noise ratio (SNR) of the measurement is given by

It is clearly apparent from the above expression that in a photon-counting experiment where the photon flux and the time T of data acquisition are given, the SNR will be maximized with a photomultiplier tube having the highest available quantum efficiency and the lowest dark noise. In this limit, the SNR becomes

SNR S (Ipy/r)1’2 (6)

To minimize thermoionic dark-pulse count, the photocathode area should be no larger than necessary for light collection: this is the case for the RCA C 31034 tube. Tubes should have as low a dark noise as possible so that I d << Z,q. For the majority of the tubes listed in Table I where the anode dark current is around A, this corresponds, however, to counting rates greater than lo3 counts per second or one count every millisecond, and the contribution of the dark noise to the SNR is thus negligible in many fluorescence experiments having a good time resolution.

The RCA 8850 and RCA 4501 (8575) photomultiplier tubes have a particularly high quantum efficiency (around 30%) in the blue part of the spectrum. When used over the whole visible spectrum, all the photomultipliers listed are almost equally suitable, with quantum efficiencies above 10% in the 3500-7000 A range. If high photon fluxes are available in the experiment, the photomultipliers may be used in the analog mode of operation: tubes with side-on photocathodes (RCA 8850,4501 and HAM R446UR) having current amplification factors (or gain) greater than lo7 are particularly suitable for these applications.

Saturation of photomultipliers is a possible source of error in measuring fluorescent decay times (Jones and Calloway, 1973). Saturation can occur by exceeding the maximum linear output current, which decreases with increased light pulse duration. The operating stability of a photomultiplier depends on the magnitude of the average anode current. Sensitivity changes (fatigue) are a direct function of large currents imposed for great lengths of time. Output linearity and fatigue problems have been investigated on many types of photomultipliers, and methods for reducing errors from this source have been proposed (Coates, 1975; Fenster et al., 1973).


In fluorescence measurements, particularly at wavelengths equal or close to that of the exciting source, it is necessary to measure a weak, time- varying signal immediately after an intense pulse of light. The response of the photomultiplier to the strong pulse often introduces artifacts that interfere with these measurements. The simplest way to avoid this problem is to gate the photomultiplier on for the duration of the measurement. At moderate switching speeds, this can be simply done by pulsing the first dynode in such a way as to cause its potential to become positive with respect to the cathode potential (see, as an example, Yamashita, 1974). However, this technique is not suitable for risetimes below 20 nsec and photomultipliers containing a focus electrode should be used for gating in the nanosecond time scale (Jameson and Martin, 1975). When overloading of the photomultiplier is not a problem but when the output has to be pulsed, high-speed gates of a design similar to that of Albach and Meyer (1973) can be used. In the analog mode, the high-speed, unity-gain buffer (National Semiconductor Corp. LH0033) can be conveniently utilized as a matching amplifier between the load of the photomultiplier (which to a certain extent determines the PM time resolution) and the subsequent electronics, with a typical 50 L2 input impedance.

C. Time-Resolved Measurement of Fluorescence Signals

As already noted, photomultipliers have two basic modes of operation; signal treatment techniques are accordingly rather different.

1. Analog Techniques

The charge integration technique, in which the anode current integrates the individual photoelectron pulses, requires reasonable emission intensity. The resistive load at the anode determines the time resolution of the fluorescence experiment. In the nanosecond time scale, matched loads must be used so that individual photoelectron pulses constitute the analog signal. Overlapping of photoelectron pulses is not a problem-contrary to the digital technique-and apart from quantum efficiency, the most important multiplier characteristic is the current amplification factor, or gain. The time-dependent fluorescence signal can be directly amplified, displayed on an oscilloscope, or sampled by gating either the detector (see Section II,B) or the resulting output signal. The circuit shown in Fig. l l a can be used for gating the output signal of a photomultiplier (Gauthier et al., 1976a). The sampling device is a Hewlett-Packard HP 5082 matched quad in a suitably polarized bridge configuration. The bridge is opened for a very short time by a subnanosecond pulse (input P) and acts as a transmission


33

-b. FIG. 11. Schemes of an analog sampling gate (a) and its associated pulse generator (b) with

nanosecond time resolution. Resistor values are in ohms and capacitances in picofarads. Suitable polarization is applied on A and B inputs to provide adequate output offset and symmetry control of the gate.

gate for the PM signal applied on input S . The high-input impedance LM 308 operational amplifier acts as a low-frequency active filter (integrator) to collect the net charge that crosses the bridge during its aperture time. After further amplification by the LM 307 operational amplifier, the voltage at the output is proportional to the value of the signal during the gate sampling pulse, i.e., at the selected time delay after the laser pulse. Various integration techniques can then be used for interfacing with the data acquisition system.

The subnanosecond pulser shown in Fig. l l b is both inexpensive and efficient; it makes use of the avalanche properties of ordinary transistors (in this case, readily available Motorola 2N3904). The synchronization pulse applied at input A is shaped by a monostable integrated circuit; the delay between the input signal and the output signal is finely adjusted on a subnanosecond time scale by the variable polarization applied to the


base of the avalanche transistor by the LM 307 operational amplifier. The output pulse lasts about 1 nsec and the delay may be adjusted from 0 to 200 nsec, not including the fixed delay of the monostable circuit, with a jitter below 50 psec in a laboratory environment.

Time resolutions of 500 psec are readily obtained with this gate and its associated pulser, with an input dynamic range of 10 mV to 2 V and on/off ratios exceeding lo4. Only one sample can be taken for each laser shot with such a system; when the pulsed laser operates at a maximum repetition rate of a few pulses per second, data acquisition may thus be very lengthy. This can be overcome by coupling in parallel several gates with different time delays.

The boxcar averager is also an useful analog system for recurrent signals; it suffers from the same drawback of making poor use of the information contained in the photomultiplier output signal. Basically, it incorporates a sampler similar to the one described above, plus a simplified data acquisition system that varies the sampling delay slowly and integrates the sampled signal over a large number of shots. If the entire waveform is to be recovered, the boxcar gate should be scanned across the desired time interval at a rate compatible with the integration time constant.

As the output of pulsed dye lasers are subject to 5-10% amplitude fluctuations under normal operation, it may be convenient to monitor continuously the fluctuating laser output for appropriate normalization. Instead of measuring directly the output light level of the laser, it is much better, whenever possible, to monitor the time integrated fluorescence intensity. As this intensity is generally found to be proportional to the laser power at constant wavelength, monitoring of the laser output power and wavelength stability is thus made possible.

2. Digital Techniques

In the digital mode of operation, there is one output pulse for each photoelectron leaving the photocathode. These pulses are shaped by a preamplifier before entering a pulse-amplitude discriminator (see, for example, Giittinger et al., 1976).

This ensures that only pulses with amplitudes larger than a predetermined value and with the proper risetime pass through the signal-processing circuits. Time jitter errors introduced in discriminators have been discussed by Janney (1976).

Time-resolved fluorescence measurements using digital methods began with the delayed-coincidence pulse counting methods (for a review, see Bennett et a/., 1965). In this technique, periodic excitation laser pulses (as detected by a photodiode) are applied to the start input of a time-to-height converter, causing a capacitance to begin charging linearly with time. This


voltage is reset to zero after reaching a predetermined value, except if a photon pulse is detected before the end of the integration period; a stop pulse is then applied to the circuit and causes it to cease charging. Extremely good time resolutions can thus be attained. If a multichannel height analyzer is then used, one obtains a multichannel delayed-coincidence analyzer. If the photomultiplier detects more than one fluorescence photon after a single laser shot, i.e., if there is photon “pile-up,” only the first photon will be recorded because only one fluorescence photon stop pulse can be processed per excitation start pulse. To avoid this problem, only very low light levels can be used when working with this technique (which is often called the single-photon time correlation technique) so that very long acquisition times are necessary at low laser repetition frequencies. Several techniques have, however, been proposed (Coates, 1968; Davies and King, 1970; Hartig et al., 1976) to avoid this problem, which has also been the object of some theoretical work (Bader et al., 1972; Donohue e f al., 1972).

The multichannel photon counter avoids the necessity of having on the average much less than one photon per cycle by counting all of the detected photons in a number of channels, which are opened sequentially following a trigger pulse. Collins et al. (1972) used counters having only one binary stage per channel in their 100 MHz photon-logging electronics. Contrary to multi- scaling systems, this device requires only that light intensities be such that the probability remains small for two photons to be detected in the same time interval. A similar device has been reported by Berger (1973) but the performances of both systems were, at that time, limited by the fact that high- speed emitter-coupled logic (ECL) was not fully developed. A 512 channel photon counter with a time resolution of 10 nsec per channel has been described by Lawton et al. (1976); an instrument following the same prin- ciples but having a resolution time of 3.5 nsec was built and used for helium Rydberg state studies by Boulmer ef al. (1977a).

3. Optimization of Temporal Response and Deconvolution

It is very important to approximate as well as possible the true decay that would be obtained by using an infinitely short excitation pulse together with a detection system having an infinite time resolution, if significant time- resolved fluorescence studies are to be made. The response function of the apparatus introduces a distortion in the measured signal; the finite width of the excitation laser pulse is also important in determining the shape of the luminescent response. The true decay function G(t) can in principle be extracted from the experimental data by solving the integral equation


where D(t) is the observed decay curve and Z(t) the measured instrumental response function. This equation has often been used for fluorescence decay curve deconvolution with the simplifying assumption that the undistorted decay function is either a single exponential or the sum or difference of two exponentials (Bennett, 1960; Helman, 1971).

An alternative method, deconvolution by convolution (illustrated in Fig. 12) has been reviewed by Ware et al. (1973): it involves only the usually reasonable assumption that the exact functional form of the undistorted response of the fluorescent system under investigation is well approximated by a sum of exponentials, namely,

where the a,, can be either negative or positive and the Y k are a priori selected constants that span a suitable range. Deconvolution using Eq. (7) is then reduced to the least-squares resolution of a set of n simultaneous linear

FIG. 12. Typical example of deconvolution using the exponential series method of Ware et a/. (1973). Curve A is the measured instrumental response curve. Curve B (points) is the measured fluorescence decay curve and curve B (line) is the result of the convolution of curve A with the “best fit” decay time of 2.84 nsec. Time scale is 0.248 nsec per channel. From Lewis et al. (1973).


equations where only the a k are varied. The Y k are arbitrary, and have no physical significance; there is no simple relationship between them and the relaxation times of the system. A set of 6-15 Y k is usually found to be sufficient.

This particular approach appears to have some advantages over Fourier and Laplace transform methods of resolution of Eq. (7): the required integra- tions span only the range of the actual data in hand. Misbehavior is generally smooth, easily recognized, and readily distinguished from the photophysics of the system. Unfortunately, it is not suited for small computers because of the large number of numerical integrals involved. For many applications it is however possible to restrict the number of terms in the series of exponentials and to come within the limitations of small computers by careful programming.

Fluorescence decay curves are generally obtained in digital form with a limited number of experimental points. In addition, the sampling interval in the photon detection and timing apparatus is much smaller than the fastest fluorescence decay involved in the experiment. The convolution integral [Eq. (7)] may then be replaced by the discrete sum (Wiener, 1964)

N

D ( N ) = 1 I ( N - k)G(k) k = O

(9)

where D ( N ) is the decay function (before deconvolution) measured at times t = N At, where At is the sampling interval; it is, of course, required that D(t) and Z( t ) be measured under strictly identical experimental conditions, with the same sampling interval. Deconvolution is then simply done by noting that Eq. (9) may be written in matrix form

D = I . G (10)

and thus

G = C . D with C = l - '

where I is a triangular matrix with band structure. These expressions are particularly suitable for fast evaluation on small computers, as they involve only sums of products of measured data points. To be specific, from Eq. (9) we obtain

N

G ( N ) = 1 C(N - k)D(k + 1) k = O

with


The deconvolution coefficients C are readily deduced from the measured response function Z ( t ) of the system [with I(0) = 01 :

C(1) = 1/1(1)

The C(i) can be seen to be ordered in ascending powers of the 1(2)/1(1) ratio. Thus if 42) > Z(l), the C(i) coefficients increase with their order i and have alternate signs. At large values of i, the C(i) coefficients have an oscillatory behavior (two consecutive coefficients have same absolute value and opposite signs) with a characteristic frequency fl2, where f is the inverse of the sampling interval. Statistical noise components at frequencies close to 1 2 are thus amplified and the deconvolution process, while manageable, is not well behaved. A large amount of data (in fact, all the data from the initial time on) has to be handled in that case. It is thus advisable to choose a sampling interval such that 42) < Z(1); the C(i) coefficients then decrease rapidly to zero and only a limited number of data points have to be entered in the evaluation of Eq. (1 1). More detailed discussions and bibliographies have been given by Rollett and Higgs (1962), Ioup and Thomas (1967), and Hill and Ioup (1976).

For meaningful deconvolution, it is obviously essential to measure accurately the laser time profile L(t), as distorted by the detection system. The overall temporal response of the system must be optimized in order to assure the validity of the convolution integral [or the sum; see Eq. (9)] as a reasonable representation of the instrument output. These problems are particularly critical when one attempts to measure lifetimes that are short compared to the decay time of the laser (Lewis et al., 1973). One must then take into account the wavelength dependence of the response function of the system due to the laser and photomultiplier wavelength-dependent characteristics (Wahl et al., 1974); when using pulsed lamps, possible drifts may be associated with the sequential measurement of Z(t) and D ( t ) (Hazan et al., 1974).

In any case, whether or not deconvolution is used, it may be advisable to reduce the experimental noise contained in the discrete data by using smoothing algorithms. However, care should be taken not to unduly degrade the underlying information. Least-squares smoothing algorithms have been described by Savitzky and Golay (1964). If carefully used, they introduce only a negligible distorsion into the original data. A smoothing algorithm acts as a low-pass filter, and its cutoff frequency should be larger than the largest characteristic frequency of the system under consideration, which is, within a factor of order of unity, the reciprocal of its shortest relaxation time.


General problems in digital filtering are discussed by Hamming (1962) (see also Gold and Rader, 1969). Errors introduced by smoothing algorithms and the optimum use of these techniques have been recently discussed by Wertheim (1975) and by Giannelli and Altamura (1976).

4. Relation between the Pumping Source and the Number of Fluorescence Photons

When the phase relaxation time is very much shorter than the laser pumping time, as is usually the case in fluorescence experiments (McIlrath and Carlsten, 1973; Sharp and Goldwasser, 1976), the rate equation coupling two levels of an atom reduces to the equation for simple two-body collisions between atoms and photons (cgs units are used throughout):

d N , I dt L'

~ = - J-+; L(v - v,)S(v - V S ) ( B O l N 0 -

where the subscripts 0 and 1 denote the ground (or initial) state and the excited state, respectively; I is the specific laser intensity (ergs sec-' cm-2) integrated over its spectral distribution L ( v - vL); S(v - vs) is the spectral profile of the line connecting the two atomic states; B,, and B,, are the absorption and stimulated emission Einstein coefficients; and R l o is the total deexcitation rate coefficient of level 1 (including all processes, whether radiative or collisional).

We assume now that the laser and atomic line profiles are both well represented by gaussian distributions with full widths at half-maximum (FWHM) AL and A s , and with identical center frequencies, i.e., vL = vs = v. Then, at the end of a laser pulse of duration 5,

if we assume that spontaneous and stimulated deexcitation both remain negligible during the laser pulse (the situation is only slightly more complicated if this is not the case).

and the absorption oscillator strength fol are related by

Einstein's absorption coefficient Bo

The laser photon flux @, defined as the total number of emitted photons per laser pulse divided by the surface of the beam, is related to I by

@ = I.r/hv (17)


and we obtain at the end of the laser pulse (see, for example, McIlrath, 1969)

e2

mc N, = -@folNo x 2

Numerically

@fo 1 No Nl = 2.49 x lo-' (A; + A;)"'

with linewidths A in hertz. Consider now a typical situation:

(1) A frequency-doubled laser pulse at 2700 8, with a peak power of 100 W during 2 nsec (FWHM), with a spectral FWHM of 0.015 and a beam diameter of 0.2 cm. The flux is thus 8.66 x lo1' photons cm-2, and the linewidth is AL = 6.2 GHz.

(2) The line to be pumped at 2700 8, is Doppler broadened (As = 4.1 GHz) with an absorption oscillator strength fol =

(3) The number density of atoms in the initial state is N o = 10" cm-3; the observed volume is I/ = 0.25 cm3; the optical system comprises a monochromator open to fll0, i.e., with a solid angle 0 = 10-2/4n, a mirror system with a magnification factor G = 5, and an overall efficiency q =

The equivalent solid angle of observation is then

I L aeq. = __ x R 4G2

and the total number of photons detected per unit time is

W = VJ"lA100eq. (19)

where A I o is the spontaneous emission coefficient of the line under observation. If A l o = lo6 sec-',

N = 1.8 x lo7 photons sec-'

or one detected photon every 55 nsec, well within the capacity of any reasonable data acquisition system. This number is typical, despite the rather low photon flux cm-2) used in the calculation. Of course, in an actual experiment, only a fraction of the sublevels of the excited state will be connected to the ground state by, for example, linearly polarized light. Because of this, the ratio of excited-state atoms to ground-state atoms may also depend on the polarization of the laser light and on the kinetics of the relaxation between sublevels.


D. Possible Systematic Errors

Despite its general reliability, there are several possible experimental error sources that may be associated with the time-resolved laser fluorescence method and that should be carefully avoided. Systematic errors and uncertainties fall into two categories, according to whether they arise from the optical system and electronics, or from the medium under study itself.

Time scale calibration uncertainties and improper use of photomultiplier tubes are the most common errors affecting the electronics. Accurate calibration of the time scale can be made by recording with the complete apparatus the light emitted by a source of known pulse shape, by measuring the time delay between two light pulses having different calibrated path lengths to the detector, or for larger time delays by using standard oscillators. Typical accuracies of better than 1% can be achieved with these methods.

When rather long radiative lifetimes are to be measured, care should be taken to ensure that the signal decay is not distorted by the diffusion of excited atoms or molecules out of the observation volume. Such an effect depends upon the size of the observation volume. It would be most pronounced at low pressures and for highly excited (Rydberg) states where radiative and collisional rates are comparatively small, and would then result in an overestimation of the decay rate. As a check, decay rates may be measured with an aperture in the detection system, which reduces the solid angle of observation to a fraction of that normally employed: measured values must be identical to those obtained with the nominal solid angle.

Of a completely different nature are the errors arising from parasitic phenomena during the process of interaction of pulsed light with the atoms or molecules under study. Coherent resonant propagation, resulting from the cooperative interaction (superradiance) of a large number of excited two-level systems, has been discussed by Arecchi and Courtens (1970) and observed by Gross et al. (1976) and Wallenstein (1976). These effects are important at higher laser powers and higher excited state population densities: they strongly shorten the observed excited state decay rate. Threshold conditions for observation and laser intensity variations of superradiance have been discussed by Haroche and his co-workers (Gross et al., 1976). Such effects are easily tested by simply recording the measured decay rate as a function of exciting laser power.

Light from pulsed dye lasers generally exhibits a large degree of linear polarization. Many experimental observations are made at right angles to the exciting beam with a detection system not involving polarization analysis (Deech et al., 1975). However, optical systems using a monochromator to isolate fluorescence light are not insensitive to polarization, depending on


the relative orientation of the grating grooves and of the fluorescence polarization direction. Errors that can be introduced into fluorescence measurements by polarization effects have been discussed comprehensively by Cehelnik et al. (1975).

Radiation trapping is also known to introduce errors in absolute intensity measurements when radiating atoms are embedded in their parent gas at sufficiently high pressures (Holstein, 1947). In time-resolved fluorescence experiments, the apparent lifetime may be greater than the actual lifetime because of radiation trapping. Correction factors have been recently (Holt, 1976) computed for spherical and cylindrical geometry. To first order, resonance trapping perturbation is shown to depend on the magnitude of the quantity k,R, where ko is the absorption coefficient at line center and R a characteristic radial dimension of the measuring cell.

111. METHODS OF DATA ANALYSIS AND REDUCTION

A. The Master Equation of Level Populations

The study of relaxation kinetics of laser-excited atomic or molecular levels has led to the development of a number of analysis methods pertinent to each specific problem. Basically, the populations of the N coupled levels form the components n, of a vector n(t) that describes completely at time t the system under investigation. The rate equations governing the time evolution of the population vector n, with a selective laser pumping term L(t) affecting only one level (labeled p), are written in matrix form

(20) dnldt = R - n + 6,L(t)

where R is the reaction rate matrix (negative definite) and the pth basis vector 6, has all its components equal to zero except the pth, which is equal to unity. Equations of this form have been integrated by a number of authors (Matsen and Franklin, 1950; Brau, 1967; Snider, 1976) in connection with chemical reaction kinetics (law of mass action), particularly dissociation and recombination of molecules possessing rotational, vibrational, and electronic degrees of freedom.

Numerical integration of Eq. (20) is not always straightforward. There is sometimes no convergence problem when first-order time derivatives are evaluated by the finite-differences method. As a rule of thumb, the time step of the calculation should be about one-tenth the shortest time-constant relevant to the particular situation. However, in many cases, this leads to problems, because the system includes both fast and slow reactions: the fast


reactions control the stability of the integration method, while the truncation error is determined by the slow components. Special methods for these "stiff" problems have been described by Gear (1971a,b; see also Morrison, 1962; Greenspan et al., 1965). Furthermore, in most cases, many components of the reaction rate matrix are either zero or negligible for physical reasons. The matrix is then said to be sparse, and it may be advisable to use this fact in numerical techniques. A detailed survey of sparse matrix research has been given by Duff (1977).

The rate coefficients R,,, which are the elements of the reaction matrix R, usually have a simple physical interpretation. Diagonal elements (i.e., where 1 = m) are the total deexcitation rates per unit time of level 1. These loss terms are negative and include both radiative and collisional contributions :

Rtt = - ri + 1 K t m + Kf) (21) ( m , m Z f

where r, is the total radiative loss rate given by the sum of the transition probabilities for all optically allowed transitions from level 1, K,, is the rate for excitation transfer to the level m, and K , is the reaction rate coefficient for processes involving reactions where the product atom or molecule is different from the reactant 1-excited atom. A typical example of such reactions is the associative ionization process

M * + M + M , f + e (22)

which has been studied experimentally by Wellenstein and Robertson (1972) in helium.

Off-diagonal elements (i.e., 1 # m) are the I to m transfer rates (these transfer terms are positive). Radiative contributions to excitation transfer are in many cases completely negligible. Energy differences between the levels of interest are generally small as well as the corresponding oscillator strengths for allowed transitions. However, there is a possibility of cooperative effects (superradiant transitions) when the laser pumping power is high enough (Gross et al., 1976). These effects may be avoided by working at low laser energies; when they are not negligible, the apparent deexcitation and transfer rates become dependent on laser energy.

The solution of Eq. (20) is easily obtained, at least formally, in terms of the eigenvalues of the reaction matrix R, provided the problem is linear. If

R = T - T-' (23)

where 5 is the diagonalized form of matrix R and T is the usual diagonaliza- tion matrix, we may write

n(t) = T - expht T-' - no (24)

166 J.-C. GAUTHIER AND I.-F. DELPECH

where no is a vector determined by the initial conditions. The eigenvalues d are negative and exp St is a diagonal matrix of the form

The origin of times has been chosen at the end of the laser pulse, i.e., when t ( t ) is zero again. If meaningful results are to be obtained, the FWHM of the laser pulse should not exceed the shortest relaxation time to be measured. Using Eqs. (24) and (25), each of the N elements ni of the population vector n may thus be written

N

ni( t ) = 1 clijexpLjt j = 1

and is a linear combination of N exponential decay curves. At later times, the time evolution of the level populations is governed by the largest exponential time constant (i.e., the smallest d j ) and all levels remain in a collisional- radiative quasi-equilibrium, which in turn relaxes to thermal equilibrium.

Experimental systems usually yield as raw experimental data the absolute populations ni( t ) of each level at predetermined times (equidistant or not) after the beginning of the laser pulse; the problem is to extract from the data as much information as possible on the kinetic coefficients, i.e., to go back from the observation of the n,(t) to a reasonable approximation of part or all of the matrix R. The analytic forms of Eq. (20) and of its solution, Eq. (26), suggest three methods of data analysis, which will be described below :

(1) multicomponent exponential decay analysis, (2) a least-squares fit of the coefficients of the master equation, using

measured values of the populations and of their time derivatives, (3) a least-squares fit to the experimental data through a numerical

resolution of the system of first-order differential equations using trial values of the unknown rate coefficients.

B. Multicomponent Exponential Decay Analysis

The analysis of multicomponent exponential decay function is a fre- quently encountered problem in many different branches of experimental science (Isenberg and Dyson, 1969; Laiken and Printz, 1970). The method of non linear least squares (Bevington, 1969; Johnson and Schuster, 1974) is the most widely used and is probably the best method to solve that problem,


provided a good enough initial estimate of the 2N parameters (a i j , ,Ij) is available and the number N of exponentials, i.e., the number of relevant independent levels, is known. Otherwise, the iterative process, starting from the biased initial estimates, may well converge to a local minimum of the x 2 hypersurface or not converge at all ( x 2 designates, as usual, the suitably normalized sum of the squares of the errors). The most common method used to resolve a decay curve into its components is the graphical approach. The data are plotted on semilog paper and time constant estimates are found by a repeated substraction of straight lines. The method may be refined by employing a least-squares technique to fit the straight lines. In many practical situations, it gives a few reasonably good initial trial values for the nonlinear least-squares analysis, and should thus always be used first.

Considerably more refined approaches were developed by Gardner et al. (1959) using a Fourier transform solution and more recently by Provencher (1976a,b). These methods automatically give N and the best estimates of (ai j , A j ) together with their estimated standard errors. These methods are typically restricted to N < 5. The initial data must be extremely accurate if more than two or three exponentials are to be separated. In most cases, the accuracy required is far beyond that usually available.

Even when the accuracy of the data is sufficient, the problem of relating the eigenvalues Aj to the excitation transfer rate coefficients R,, is far from simple. As an illustration, let us consider the simplest case, where there are only two independent levels (the low-lying alkali doublets for example). Possible processes that can take place when a single level of a doublet system is excited are shown schematically in Fig. 13. We suppose that RIo and R20 are the radiative transition probabilities to lower-lying states labeled 0. R and R , , are the upward and downward excitation transfer rates between levels 1 and 2. The rate of change of the upper level populations nl and 11,

R20

0 1 FIG. 13. A simple model for doublet excitation transfer. R , , and R,, are the doublet

deexcitation coefficients [radiative plus collisional) while R,, and R l z are the intradoublet mixing reaction rate coefficients. The ground level is labeled 0; 1 and 2 refer to the doublet I e v e I s .


may be written

dn,/dt = - (RI2 + Rlo)n, + R2,n2

dn2P = R12n1 - (R2l + R20)nt

(27)

(28)

and from detailed balance considerations, AE being the energy difference between levels 1 and 2,

Rl2IR2, = (92/91)exp(-AE/W = P

n2(t) = a2,exp(-Llt) + a2,exp(-12t)

I1 + 22 = Rzo + RIo + (1 + ~ ) R 2 1

(29)

(30)

If level 1 is pumped and fluorescence is observed on level 2, the solution is

with

(31) &1”2 = R10R20 + (RIO + PR20)R2,

Clearly, in this particularly simple case, the transition probabilities R l o and R,, and the pressure-dependent transfer rate R,, can be deduced from the decay rates I, and I,, if measurements are possible at least at two different pressures, and if there are good reasons to believe that Eqs. (27) and (28) give a good description of the process under consideration. This is not often the case in a practical situation; when there are more than two levels, the extraction of exponential time constants is more and more difficult and there is no direct and simple relationship between the measured decay rates and the unknown rate coefficients.

There are fortunately particular situations ofgreat practical interest where simple exponential decay analysis yields useful and reliable results. The laser-pumped level p is often the only level that is substantially populated at the end of the laser pulse; its time evolution at early times is then only governed by excitation transfer processes to the other levels. The initial decay slope of level p, which is easily measured on a semilog plot, is then proportional to its total deexcitation rate. If the experimental conditions are such that it is possible to pump different levels by tuning the laser to different transitions, detailed information about the state-to-state transition rates may thus be obtained. These methods are illustrated in the recent papers by Gauthier et al. (1976a) and Delpech et a/. (1977b) on the rotational relaxation of ’He,, ’He4He, and 4He, excited molecules.

C. Least-Squares Fitting of the CoefJicients of the Master Equation

While very useful as a first step, multicomponent exponential decay analysis is usually not sufficient; the possibility of obtaining biased results is particularly serious with exponentials, because they are severely non-


orthogonal. A grossly incorrect solution may thus reproduce the data well enough to be accepted, while having no basis in the physics of the phenomenon.

A much more accurate method is possible, if the populations of all relevant levels have been measured as a function of time. Consider for example a three-level linear system that decays freely after the initial laser pulse,

dn,/dt = R l l n l + R 1 2 n 2 + R I 3 n 3

(32) dn3Jdt = R 3 1 n 1 + R32n2 + R33n3

and suppose now, to be specific, that n l , n 2 , dn2/dt , and n3 are known to a good accuracy at a sufficient number of times between t = 0 and the time it takes for the three levels to reach quasi-equilibrium and to decay at the same rate, which corresponds in a linear system to the inverse of the smallest eigenvalue. The problem of finding R , , , R 2 2 , and R Z 3 reduces now to a problem of multiple linear regression (Snedecor and Cochran, 1957; Bevington, 1969), i.e., of finding the coefficients R that fit best, in a least-squares sense, dn2/dt to the measured functions n , , n 2 , and n3 (in practice, it may be more efficient to normalize all populations to the population of the laser-pumped level, thus reducing by one the number of variables, as well as reducing the range spanned by each variable).

dn2/dt = R z l n , + R2,n2 + R23n3

In the more general case of a N-level system, N dni

- ( t ) = c Rijnj ( t ) dt j = 1

( 3 3 )

after substitution of the measured values of dni/At, nj( t ) obtained under given experimental conditions at the M times t k after the beginning of the laser pulse, we obtain a set of M linear equations with the N unknowns R i j . Provided, as can always be the case, that M > N (Margulies, 1968), this is easily solved by a least-squares method yielding the “best” values for the Rij and the corresponding estimates of the statistical error bars. The principle of the method is to chose the R i j that will minimize the sum x 2 of the squares of the distances between the measured dni/dt and its fit with the measured n j . As

(34)

the N values of R i j (remember that i is fixed andj varies from 1 to N, i included) are determined from equations of the type d(x2) /dRik = 0, so that

170 J . -C. GAUTHIER AND J .-F. DELPECH

Writing Eq. (35) for the N values of k, one obtains a system of N linear equations with N unknowns, which is then solved by ordinary determinant methods. We can estimate the uncertainties in the coefficients Ri j by numerical methods. The standard deviation squared 0; that characterizes the uncertainty in the determination of a given coefficient Rij is the sum of the squares of the products of the standard deviation of each data point om by the weight of this data point on the determination of Ri j :

The resolution of Eq. ( 3 9 , with the full set of coefficients R,,, sometimes yields error bars that are comparable to or larger than the coefficients themselves. This is particularly true when N 2 3 and when the original data are noisy on some levels. Negligible transfer rate coefficients are responsible of such effects. This introduces additional statistical fluctuations on the remaining significant coefficients, and it is desirable to use as much a priori information as possible to reduce the number of degrees of freedom of the system in order to improve accuracy.

There is of course no general law to impose selection rules or other constraints in the analysis of an experimental situation. The only way is to try to apply physical judgment when reviewing the results of different least- squares resolutions with particular constraints. A good standard procedure, as in any multiple linear regression, is to start with as few “explicative” levels as possible, i.e., to limit the sum on the right-hand side of Eq. (33) to the smallest possible number of levels (for example, the level under consideration and the laser-pumped level), and to add explicative levels one at a time, using physical insight. The procedure should stop as soon as the xz begins to increase. This is very conveniently done on a small computer programmed in a reasonably interactive mode, where the experimenter may add or delete at will explicative levels. This method was used by Gauthier et al. (1976b) in the study of the temperature dependence of the collisional relaxation processes for the n = 3 triplet states of helium.

D. Least-Squares Simulation of Experimental Data

Least-squares fitting of the coefficients of the master equation is very efficient when the number of relevant levels is not too large, and when the populations of all relevant levels can be measured simultaneously with sufficient accuracy. This is not always the case: some lines and thus some levels are difficult to monitor either because their wavelength is so close to that of the excitation laser that scattered light is sufficient to overload the


photomultiplier tube, or because their wavelength lies outside the range of availability of rapid and accurate detectors. On the other hand, much a priori information about the transition rates may already be known, and it is desirable to use it so that the number of unknown variables to be fitted may be reduced to a practicable level.

In this case, the most useful method is certainly the numerical resolution of Eq. (20), using digital computer integration methods (Nordsieck, 1962; Gear, 1967, 1971a,b) with parameters adjusted by hand to minimize the sum of the squares of the distances between experimental and computed points. The number of measured level populations may of course be much smaller than the number of levels used in the simulation. For example, Shaw- and Webster (1976) have measured the populations of two levels only while using six levels in their model calculation of the excitation transfer processes in the n = 4 levels of helium.

The numerical problems associated with the resolution of Eq. (20) have been discussed in Section II1,A. Since the use of the least-squares fitting procedure amounts to a search along the x2 hypersurface rather than to an exact analytical solution, there is no analytic form for the uncertainties in the final values of the parameters. Some idea of the errors involved can, however, be qualitatively determined by varying each parameter around its optimum value so that the simulated time evolution of the level populations remains within the error bars of the original data. Large, undetected systematic errors remain possible, however, if the initial guess of the functional form of the rate coefficients is poor.

Use of this method is thus open to the criticism that there may be more than one model that adequately describes the experimental facts. Con- sequently, it should never be used alone, but initial inputs to the model calculations should be obtained using different methods, such as the graphical methods already described in Section II1,B. While it can never be absolute, confidence in the method increases considerably when a unique set of rate coefficients and constants is found under very different experimental situations and leads to the precise reproduction of the experimental results (when for example the pressure or the temperature are varied on a wide range). This was done by Gauthier et al. (1976a,b) and Delpech et al. (1977b).

E. Pressure Fits

Once the diagonal (radiative and collisional deexcitation rates) and off-diagonal (excitation transfer rates) elements of the reaction matrix have been determined, the decay rate of a given level as a function of pressure must be analyzed to sort out the contributions from the various relaxation


processes. The observed decay rate is generally fitted to functions of the tY Pe

(37)

where p is the gas pressure. It should be noted that in experiments where the quenching of a level belonging to a specific atomic species is studied using various foreign gases, both the pressure of the gas under study and of the perturbers must be varied to discriminate between homonuclear and heteronuclear effects. If electrons are present, as in experiments using discharges as a primary excitation step to the wanted levels, the electron density (and if possible the electron temperature) should be varied independently of the gas pressure (Gauthier et al., 1976b) to take into account the electron contribution.

Provided enough experimental points are available, the parameters C1, C,, and C 3 can be easily deduced from Eq. (37) by simple least-squares analysis. They may then be identified with care, provided the relevant levels have been taken into account, to the total radiative (or, more generally, collisionless) decay rate (Cl ) , to the two-body collisional deexcitation rate (C,) for reactions of the type

R = C 1 + C2p + C3p2

M(P) + M(0) -+ W q ) + M(0) (38)

where p and q are different sublevels of the system (q # p ) , and finally to three-body reactions (C3), which lead to products belonging to a different system [as in Eq. (22)].

IV. APPLICATIONS TO ATOMIC AND MOLECULAR PHYSICS

A. Atomic Lifetimes: Radiative Decay, Cross Sections for Quenching and for Transfer

This section is devoted to a summary of results on the kinetics of excited states of atoms obtained by time-resolved laser fluorescence spectroscopy. Both lower-lying (resonance) states and levels with energies close to the ionization limit have been thus investigated. Much information on lower excited states has already been obtained-and is still obtained-by the use of the classical methods described in Section I, but as already noted, the use of time-resolved laser fluorescence techniques has led to considerable improvements: selective excitation, freedom from cascading, etc. Highly excited Rydberg states of atoms (Stebbings, 1976) behave rather differently from atoms in their ground or lower excited states, and their study would


be extremely difficult without modern pulsed-laser fluorescence techniques. Radiative and collisional transfer experiments are also reviewed below.

1. Helium

The lowest excited level of helium that is radiatively connected to the ground state is the He (2lP) level, which lies 21.22 eV above the ground state. Thus, it cannot be reached from the ground state with practical dye lasers. Large concentrations of metastable singlet and triplet states do, however, exist in helium afterglow discharges (Delpech et al., 1975) and can also be produced in atomic beams, and all n > 2 excited states, including the continuum, are easily reached from them with simple experimental techniques using a dye laser pulse.

Collins et al. (1972) were the first to use the full possibilities of time- resolved laser fluorescence decay measurements. They applied this technique to the study of the rate coefficients for the reactions

He (53P) + He (1's) + He (53D) + He (1's)

He (53P) + e --+ He (53D) + e

He (53P) + He (1%) + He: + e

(394

(39b)

(394

Metastable concentrations on the order of 10" cmF3 were obtained in a flowing afterglow. Metastable states, (in this case the 23S state) were subsequently pumped to the reacting state by intense sec-') pulsed radiation at 2945 A produced by a frequency-doubled flashlamp-pumped dye laser. An external KDP crystal was used for frequency doubling. The delayed fluorescence from the optically pumped populations was analyzed by a 100 MHz photon-logging system having 512 channels, with a time resolution of 10 nsec. With this system, the rate coefficient for reaction (39c) was found to be 8 x cm3 sec-'. Upper limits on reactions (39a) and (39b) were found to be about 8 x and 8 x cm3 sec-', respectively.

Hurst et al. have developed a photoionization method for complete conversion of a quantum-selected population to ionization, making possible sensitive and absolute measurement of the selected populations in a gas (see also Section V). They have applied this technique to a study of the kinetics of He (2lS) states, in good agreement with an energy pathway model (Payne e l al., 1975a,b).

Nanosecond time-resolved absorption spectroscopy has been recently used by Lawler et al. (1977) to study the populations of the 2'P, 23P, 2lS, and 23S levels in a high-pressure (50-3000 torr), fast (a few nanoseconds) helium discharge. They report results significantly different from those of

photons A-


Payne et al. (1975a,b), although the interpretation of their data is substantially more difficult owing to fast transient phenomena and comparatively high electron densities.

Preliminary experimental results of the selective excitation by a pulsed dye laser of the triplet n = 3 states of helium were presented by Moy et a!. (1975). Transfer cross sections between the 33P and 33D states were found to be in reasonable agreement with the values quoted by Wellenstein and Robertson (1972). This work was considerably extended, using the technique developed by Gauthier and co-workers (1976a), to the study of the temperature dependence of the collisional relaxation processes of the n = 3 states (Gauthier et al., 1976b). A typical temporal evolution of the n = 3 helium states populations is shown in Fig. 14. Basically, in a first step, metastable concentrations in excess of 10' cm- are obtained in the stationary afterglow of a ultrapure helium discharge at pressures of a few torrs. A short- pulse nitrogen-pumped dye laser (Moy, 1976) tuned to the 3888.6 A, 23S-33P transition is then used to transfer a small fraction (0.5%) of the 2% population to the 33P level. Collisional depolarization among the 33P sublevels was complete, so that fluorescence light was totally unpolarized. The SNR

tlm (Ild FIG. 14. Fluorescence light intensities of all n = 3 triplet levels of helium after selective

excitation of the 'P level by a pulsed dye laser. Experimental conditions: [He] = 4.6 x 10'' an-', To = 3WK, [el = 2.9 x 10" ~ r n - ~ , i", = 600°K. Statistical error bars are representative of data reproducibility.


of fluorescent light intensity is not a problem in this type of experiment: the photomultiplier tube that records fluorescent light was used in an analog mode and its output amplitude was converted into digital form with 1 nsec resolution. The corresponding data were processed on-line after each laser shot by a 16-channel data acquisition system. The coefficients of the master equation were then fitted in the least-squares sense, as described in Section II17C, to obtain the kinetic coefficients pertaining to the three n = 3 triplet states of helium. Their radiative lifetimes, two-body quenching rates, and excitation transfer rates were finally deduced from the variation of decay rates with helium pressure at constant temperature. To obtain elevated neutral temperatures, a small oven was placed around the afterglow tube so that the temperature variation of the collisional cross sections could be derived. The experimental results were compared with the recent theoretical study of Cohen (1976); agreement between theory and experiment is satisfactory. Moreover, a detailed discussion of the experimental results brings useful complementary information on the intermolecular potentials and on the interstate couplings pertaining to He (n = 3)-He (1's) collisions.

Attempts have also been made to study the n = 4 levels of helium with time-resolved laser excitation. The situation is substantially complicated by the fact that triplet-singlet interconversion is observed between D and F states (Abrams and Wolga, 1967, and references therein): the corresponding number of levels to be studied simultaneously is doubled. Catherinot and his co-workers (1976) have used a glow discharge as a first excitation step for populating 2lS and 23S metastables; 2'S-3'P and 2IP-4'D transitions were then pumped with a pulsed dye laser. Population transfers to the 3'D, 33D, and 33P levels were observed when pumping the 3'P level and to the 4'P, 43D7 and 3'P levels when pumping the 4'D level. A cross section of 5 x cm2 was estimated for the reaction

He (4'D) + He (1's) + H e (43D) + He (1's) in reasonable agreement with previously published values. Only apparent decay rates were recorded in this particular experiment and no attempt was made to interpret the observed transfers on the basis of a time-dependent collisional and radiative model.

Shaw and Webster (1976) have published a more quantitative study of n = 4 excitation transfer processes in helium. The 4'D level in a helium flowing afterglow is populated by two-step excitation: the 2'P level is first populated by resonance radiation trapping and is then pumped with a pulsed dye laser tuned to 4922 A. The fluorescence originating from the 4'P and 43D levels is accurately measured with a 10 nsec multichannel photon counter (Lawton et al., 1976). The data are then fitted to a simple simulation model including excitation transfer and associative ionization, using the method


outlined in Section II1,D. The results compare favorably with the few available previous measurements: the quoted error limits are f30% for the measured cross sections. However, only two excited states populations were monitored in this experiment. A higher degree of confidence in the results might have been reached by monitoring the four remaining levels, but 4ls3F states are difficult to observe with nanosecond resolution, as their emission lines fall in the 2 pm range. On the other hand, no theoretical calculations are currently available for the n = 4 transfer cross sections in helium. Cohen's approach is not simply transposable to these states since the L - S coupling scheme for the F states becomes invalid when the principal quantum number is larger than 3.

The collisional kinetics of higher excited Rydberg states of helium (n 2 10) can also be studied with the laser fluorescence method. Delpech and his co-workers (Boulmer et al., 1977a) have reported detailed spectroscopic measurements of light emission and level populations of highly excited states in helium. The experimental data were consistent with a collisional-radiative theory where electron-impact-induced transitions between bound Rydberg states were described in terms of Mansbach and Keck (1969) collisional rates. Excitation transfers between atomic excited states by collisions with ground-state helium atoms were also found to be important in the stabilization stage of collisional-radiative recombination of Hef ions. Photoionization studies of high-lying (n 2 11) Rydberg states by 10.6 pm photons helped to confirm that neutrals and electrons were both involved in the collisional stabilization of the recombining electron.

Various mechanisms may be invoked for excitation transfer between highly excited atomic states. In addition to neutral collisional transfers, one may, for example, envision curve crossings between diabatic He, states and the He: ion potential curve near its dissociation limit. A detailed study of these individual mechanisms is now under progress at Orsay. Use is made of the techniques developed for the study of the n = 3 states of helium. In the afterglow period of a pulsed discharge in helium, metastable z3S states are pumped to n3P states (with n around 10) with a frequency-doubled pulsed tunable dye laser. Peak powers over 1 kW around 2676 h; are obtained with a temperature-tuned ADP crystal. Fluorescent light originating from all levels with principal quantum number n, 3 I n I 15, is detected by a standard monochromator-photomultiplier combination and monitored by a 3.5 nsec multichannel photon counter. Excited levels with high principal quantum number may be considered to be energy degenerate with respect to the orbital angular momentum quantum number 1. Energy differences between different 1 sublevels are very much smaller than An = 1 energy jumps. This greatly simplifies the data analysis of excited-le'vel populations by leading to a large reduction of the number of equations to be solved simultaneously.


Preliminary experiments relating the observed apparent lifetimes to electron and neutral densities give results in reasonable agreement with a transient level population model using Mansbach and Keck state-to-state transfer rates (Delpech et a/., 1977a).

Energy transfers from excited helium atoms to other ground-state rare gases have been investigated recently by Nayfeh and his colleagues ( 1976). Time-resolved VUV emission from 2l S helium states decaying through collisional-induced emission was used to monitor this population in foreign- gas quenching collisions. Results of these studies are of current interest in UV laser applications, e.g., He-Cd and He-N, lasers.

Pulsed selective-excitation spectroscopy has also been applied to plasma diagnostics (Burrel and Kunze, 1972) in helium discharges. In a plasma where electron-atom collision rates dominate over atom-atom and radiative decay rates, the decay time of the intensified spontaneous emission arising from the upper level of the laser-pumped transition should be a sensitive function of the electron density n, (Measures, 1968). Calibration functions relating the decay time to n, and the electron temperature T, can be obtained in a controlled plasma, where all parameters are measured by standard methods.

2. Other Rare Gases

Time-resolved collisional and radiative relaxation experiments in the heavier rare gases are rather scarce, despite the fact that a very large number of relaxation phenomena play a role and are of great interest in these gases. For the time being, there are only a few theoretical estimates of the corresponding collision cross sections. However, specific experiments have been performed to explain or even to achieve laser action in these gases (or in mixtures involving these gases). Estimates of the lifetimes, quenching rates, and collisional transfer cross sections of the atomic levels involved in gas laser action are thus available.

Arrathoon and Sealer (1971) have studied the radiative lifetimes of the 3s2 and 2p4 (Paschen notation) states of neon I. Inelastic-destruction cross sections of these states for collisions with neutral helium atoms were also evaluated. Intense nanosecond bursts of laser radiation at 6328 A obtained by mode locking in a He-Ne laser cavity were used for the excitation. Fluorescent light decay was recorded on a sampling oscilloscope; stability was such that there was no appreciable long-term drift error over a typical 20 min data-averaging period.

The lifetime of the 4p2D,,, level of argon I1 was also measured using mode-locked optical pulses from an argon-ion laser; shorter fall time pulses were necessary since lifetimes in argon I1 are generally much shorter than in neon I. Results were in good agreement with those determined from previous


pulsed-electron-beam experiments. The methods used by these authors are, however, limited to currently available continuous gas lasers and, due to the almost total lack of tunability of gas laser transitions, to the study of radiative and collisional processes in their parent gas.

Metastable states of rare gas atoms are known to play a role as highly efficient energy sources in chemical reactions. Metastable argon deexcitation by molecular nitrogen has been extensively studied (Setser et al., 1970; Nguyen et al., 1974) and laser action has been achieved in such mixtures. Direct pumping of metastable levels from the ground state is not currently possible with lasers but pulsed radiolysis with high-energy electron beams (Le CalvC and Bourkne, 1973) has been used to study, with time-resolved fluorescence detection, the collisional transfer rates between argon and nitrogen. More precise, rotationally resolved excitation transfer rates have been obtained by Dunning et al. (1975) by depopulating strongly, with a pulsed dye laser, one of the metastable levels of argon and following with time analysis the excitation transfer processes among the rotational sublevels of u' I 5, C3n, levels of Nz .

Quantum states of high principal quantum number (Rydberg states) are known to be of great importance in the collisional-radiative recombination processes in cold pure-gas plasmas (Stevefelt et al., 1975). Studies of xenon atoms in high Rydberg states have been made by Stebbings and his co- workers (1975) with laser-excited beams. A beam of xenon atoms, containing a small fraction of atoms in metastable 3P0 and 3P2 states produced by electron impact, is intersected by the output beam of a pulsed dye laser tuned to excite transitions from the 3P, metastable state to levels that lie just below the first ionization limit Xe' ('P3,Z). Two techniques were used to detect the high Rydberg atoms: photoionization by the iesonant pulse itself and field

FIG. 15. Rydberg atoms production in a typical beam-laser experiment. A beam of xenon metastable atoms is intersected by the output beam of a pulsed laser. Rydberg atoms are detected by photoionization or field ionization. From Stebbings et al. (1975).

PULSED-LASER FLUORESCENCE SPECTROSCOPY I79

ionization for levels very close to the continuum. Lifetimes were determined by recording the number of high Rydberg atoms in a given state as a function of the time delay between the laser pulse and the application of the ionizing field. The field ionization properties and radiative lifetimes of these highly excited atoms were shown to be nearly hydrogenic. The beam laser apparatus is shown in Fig. 15.

3. Alkali Atoms

Oscillator strengths and hence radiative lifetimes of alkali atoms have been known for a long time (Heavens, 1961) from calculations based on the Bates-Damgaard method (1949). However, it is only quite recently that accurate experimental results have become available, with the progress of time-resolved spectroscopy.

In cesium, a comprehensive study by Deech and other investigators (1977) of radiative lifetimes and depopulation cross sections has been presented. The atoms were prepared in a two-step process: the 6*P level is reached by excitation with a cesium lamp, and a pulsed tunable dye laser is then used to reach various S and D states. Measurements of the time-resolved fluorescence of these atoms are in good agreement with previous measurements using a similar technique (Pace and Atkinson, 1975). Results extend to n2S, ,2 and n2D,,, states of cesium ( n = 8 to 14) over a range of vapor densities covering the onset of collisional depopulation. Lifetimes are found to vary as (n*)2.62 for D,,, states and as (n*)4 for depopulation cross sections, where n* is the effective quantum number, in good agreement with elementary theoretical considerations.

Some of these results have been partially confirmed by a simultaneous study by Marek (1977), who directly produced n = 7 to 9 states by pumping the ground state with a pulsed laser. Excitation transfers between fs doublet states of cesium induced by collisions with cesium atoms have been studied by Pace and Atkinson (1974b) using sensitized fluorescence measurements with a pulsed dye laser (Pace and Atkinson, 1974~). Their results confirm the importance of the fs energy splitting AE. Figure 16 shows the relationship between the values of several 3/2 + 1/2 excitation transfer cross sections for the different alkalis as a function of A E - '. Except in the case of sodium, the graph indicates an essentially linear relation; Franck's rule is thus obeyed.

There is much current interest in theoretical studies of the cross sections for the excitation transfer between excited states of alkali atoms, induced by collisions with ground-state atoms of the same kind or, more simply, by rare gas atoms. Either a quantum treatment (Mies, 1973) or a semiclassical treatment (Gaussorgues et al., 1975) may be applied to cross-sectional calculations. Experimentally, the Cs-Xe couple is currently of much interest because of its


1 L L k t 1 1 L L I I I 1 , , , , I 5 0 5 0 m 500 too0

VAE ( e V P FIG. 16. Intradoublet transfer cross section vs the inverse of the doublet energy separation

for all the measured alkali-alkali mixing collisions. The straight line is a best fit, excluding the Na (3P,,, + 3P,,,) transition. From Pace and Atkinson (1974b). Reproduced by permission of the National Research Council of Canada from Can. J. Phys. 52. 1641-1647 (1974).

potential for exciplex laser action. The 7P,/z, 7P3iz cesium doublet was first studied (Kielkopf, 1975) at low xenon partial pressures (below 150 torr). Lifetime measurements were in very good agreement with previous experimental works but transfer cross sections were found to be rather small, around 1.5 x 10- l 7 cm2, A more detailed work by Marek and Niemax (1976) gives the lifetimes of many levels and the qualitative variations of the observed lifetimes with xenon pressures up to 600 torr. Under the influence of collisions with xenon gas atoms, the lifetimes of the 7, 8, and 9 2P1,z and zP3,z levels are shortened. Recent experiments at much higher pressures (Gauthier et al., 1976c) have shown the importance of cesium and xenon densities in the kinetics of formation and decay of Cs-Xe excited molecules formed by irradiating with a short pulsed dye laser a Cs-Xe mixture (xenon density ranging from 0.9 to 6.9 amagats). Mixing cross sections for the 7P levels were found to be much smaller in the high pressure region ( > 760 torr) than at very low xenon pressures (Siara et al., 1974) but a very rapid variation of this cross section with cell temperature was reported, indicative of an energy threshold of the cross section at thermal energies.

In sodium, the first experimental study of radiative lifetimes using pulsed laser excitation was made by Erdmann et al. (1972) and later by Gornik and other investigators (1973a). They employed stepwise excitation involving two pulsed dye lasers, the first pumping either D, or D, lines and the other


pumping the 3’P --f 42D transition. A 20-channel fast transient analyzer recorded the fluorescence decay light. A schematic drawing of the apparatus is shown in Fig. 17. Results are in excellent agreement with previous sodium lifetime determinations using electron beam excitation. A comprehensive study of radiative lifetimes in sodium has been presented by Gallagher et al. (1975a). Two nitrogen-pumped dye lasers were used, according to the scheme developed by Gornik et ul. (1973a), to selectively pump the n = 5-13, s, and d states of sodium. Results indicate that the quantum defect theory, which predicts that the lifetime of a state should be proportional to ( r ~ * ) ~ , can be used to calculate with confidence the lifetimes of the Rydberg levels in sodium. Moreover, the lifetimes were found to be in good agreement with a Coulomb approximation calculation. The effects of collisions of rare gas atoms with sodium atoms in high-lying s and d states have also been studied (Gallagher et ul., 1975b). A lengthening of the fluorescence decay times of the nd levels of sodium in collision with argon, helium, and neon atoms was observed and interpreted as a collisional mixing of the initially excited nd levels with the higher angular momentum substates of same n; the observed lifetime is thus in fact the lifetime of the states with 1 2 2. The cross sections

PDPll dye laser

Trigger

dye laser

Monochrornator n

a L

0

I

1

recorder

Trigger

FIG. 17. A beam laser experiment with two-photon excitation. This technique is useful for populating D states in atoms with S ground state. From Gornik et a/. (1973a).


for the process, which are of the order of several thousand A2, appear to increase as the geometrical cross section of the excited atom. Using improved experimental capability, these authors (Gallagher et al., 1977) have refined and extended their initial measurements. In a recent theoretical work, Olson (1977) gives I mixing cross sections for helium, neon, and argon based on their scattering length and polarizabilities. Despite some simplifications, this treatment yields results in good agreement with experimental values.

Quenching of sodium atoms in various 2p states by iodine molecules has been reported by Bersohn and Horwitz (1975) with direct VUV pulsed excitation and time-resolved fluorescence detection. The main goal of these studies was to investigate cross sections of gas phase reactions between good electron donors like alkali atoms and good electron acceptors like halogen atoms. Quenching cross sections obtained for Na(42P) and Na(5’P) show the expected decrease from those of the first excited states.

In rubidium, low-lying levels like 62P1,, and 62P3,2 states have been studied by Pace and Atkinson (1974a). Cross sections for excitation transfer between fs levels were found to be consistent with the empirical relationship between the magnitude of the cross sections and the fs splitting that has previously been established for the alkalis. Highly excited (12 I n I 22) P states have been investigated more recently. Natural lifetimes (Gounand et al., 1976b) were found to be in substantial disagreement (about 50%) with Coulomb approximation calculations. Furthermore, cross sections for the collisional depopulation of nP states (12 I n s 22) do not vary with principal quantum number for n 2 14, contrary to previous hypotheses (Gounand et al., 1977).

Mies (1973) has pointed out that the velocity dependence of intramultiplet mixing in alkali atoms should be studied experimentally to check our theoretical understanding of the processes involved. Molecular beam studies of collisions of laser-excited atoms have been initiated by Anderson and co-workers (1976); K (42P1j2) atoms in collision with ground-state helium atoms were first studied (see Fig. 18). The pumping 7699 A radiation was obtained from an optical parametric oscillator pumped by a doubled Nd:YAG laser. The system generates 75 pulses/sec of about 100 nsec duration. Fluorescence emission from K (42P3,2) atoms was observed with a 128- channel photon-counting system with microsecond resolution. The velocity dependence of intramultiplet mixing among the 4P levels has been measured (Fig. 19) using crossed optically excited alkali and helium beams over the relative velocity range u = 1.3-3.4 km/sec. The cross section appeared to fit a linear function u(u) = A(v - v,,) over that range with a typical value of 76 A2 at u = 2 km/sec. Only relative cross sections are available with this technique, and normalization to thermal average cross sections of intramultiplet mixing (Krause, 1966) is necessary.


FIG. 18. Schematic drawing (top view) of an experimental setup used to study the velocity dependence of intramultiplet mixing in alkalis. The multiplet excitation is provided by a laser beam reflected vertically through the particle beam interaction region by a polarization- conserving prism (V). From Anderson et al. (1976).

ENERGY IN KCALIMOLE 0.2 0.5 1.0 2.0 3.0 5.0 r I 1 I I I

0 0 10 20 30

RELATIVE VELOCITY IN lo4 CMISEC

FIG. 19. Intramultiplet mixing cross sections of potassium (4P) in collision with helium (1s) as a function of relative beam velocities. Data points: experimental results of Anderson et al.; curve: modified Nikitin theory. From Anderson et ul. (1976).

Applications of time-resolved laser fluorescence techniques to plasma diagnostics have been already mentioned. This diagnostic method was proposed by Measures (1968) and has been applied in a potassium plasma by Measures and Rodrigo (1972) and Rodrigo and Measures (1973) using a


thermally tuned Q-switched ruby laser. However, careful calibration experiments are necessary and should be performed under well-diagnosed plasma conditions in order to determine empirically the dependence of the decay time on electron density and temperature. The present diagnostic technique is thus of limited usefulness in standard plasma experiments where only average values of the plasma parameters are needed. However, when spatial resolution is needed, this new tool is of considerable interest since laser irradiation and fluorescence detection can be made to define a small volume, thus probing locally the characteristic parameters of the electron gas.

4. Miscellaneous Atoms

While many results begin to emerge for rare gases and alkali atoms, this is not always the case for many other atoms of interest in atomic and molecular physics. Accurate potential surface calculations for testing theoretical models of chemical and physical quenching of electronically excited atoms are difficult for large-2 atoms. However, atoms of astrophysical interest or atoms that present great chemical activity or are potential candidates for excimer laser action have been studied with the techniques of laser fluorescence.

Transition probabilities for the emission lines from the metastable 3P0, 1 , 2

states in magnesium are of considerable interest in astrophysics: its cosmic abundance is only one-tenth that of carbon atoms. There is a large discrepancy between theoretical and experimental determinations of the 3P1 lifetime: values range from 0.5 to 4.2 msec. Magnesium atoms in the 3P1 state can be prepared (Wright et al., 1974) by an intense pulse of 4571 8, radiation from a pulsed dye laser. This corresponds to the intercombination line 3p3P1 + 3s1S0. The time variation of the number of atoms in the 2P, state is monitored by observing the metastable absorption of the 3P1 + 3S1, 51 73 8, magnesium resonance radiation. A first measurement (Wright et al., 1974) gave a lifetime of 2.2 -t 0.2 msec, but quenching of the metastable state by contaminants in the sample was invoked to explain the discrepancy between theory and experiment. A remeasurement of 3P1 lifetime, avoiding the contamination problem, was performed by Furcinitti et al. (1975). They found a lifetime of45 f 0.5 msec, in better agreement with theoretical results. Besides, cross sections for the quenching of the 3PJ states by several polyatomic gases were determined experimentally (Blickensderfer et al., 1975). Information about short-lived molecular states selectively sensitized by excited atoms was obtained by the pulsed dye laser method. Lifetime measurements of the 4s5s3Sl level of calcium by use of a pulsed dye laser have also been reported (Gornick et al., 1973b).

The reactivity of excited atomic states may also be studied with this technique. Reactions of 0 (ID) atoms with NzO, H20, CH2, and H2 are


particularly important in the high atmosphere, in connection with the earth's ozone shield. Such deactivation studies on 0 (ID) atoms have been made by Davidson and his co-workers (1976) with excited oxygen atoms produced by the dissociation of ozone via a frequency quadrupled Nd-YAG laser as a photolytic source. Fluorescence on the 0 ('D) + 0 (3P) emission observed with time resolution provides a way to measure the rate for relaxation of 0 (ID) atoms in collision with various diatomic and polyatomic gases present in the upper atmosphere.

Along the same lines, excimer studies of Hg, molecules were made by Siara and Krause (1975). They investigated by the method of optical delayed coincidences the formation and decay of Hg, molecules produced in Hg-Ar mixtures irradiated with pulsed 2537 A resonance radiation. Mixing cross sections between excited states of Hg, dimers induced in the Hg,-Ar collisions were found to be of the same order of magnitude as the corresponding cross section for Hg-N2 collisions.

Measurements of excited-state lifetimes in ionic species is also possible using the time-resolved fluorescence technique. The lifetimes of the 7S,,, , 6D5,2, and 6D312 levels of singly ionized barium have been measured by Havey ef al. (1977). The lifetime of each level was determined by direct fast- oscilloscope observation of the decay of fluorescence emitted from the excited level, following excitation by two pulsed dye laser beams.

Absolute population densities can be measured by time-resolved absorption techniques involving the detection of the fluorescence on a transition originating from the level under investigation. Local measurements can be made because the viewing volume is restricted to the area where the laser beam and the observation region overlap. This was applied to the n = 2 level of atomic hydrogen by Bergstedt and other investigators (1975), who monitored the time-resolved Ha-Balmer transition (n = 2 to n = 3) and deduced the population of the lower level and depopulation rates of the upper level.

Note finally that optical pumping techniques coupled to electric quadrupole or magnetic-resonance state selection make it possible to prepare atoms in definite magnetic fine or hyperfine sublevels and molecules in particular magnetic sublevels of a rotational state. Anisotropies in interatomic and intramolecular potential can be studied by the measurements of individual cross sections between magnetic sublevels. Optical-pumping techniques (Happer, 1972) have been widely used in spectroscopy to determine quantities such as fs and hfs of atoms. Polarized-light excitation and detection is a choice tool to perform such experiments (Haroche et al., 1973). These techniques have recently been used in time-resolved laser fluorescence (Deech et a/., 1975) of cesium atoms and iodine molecules (McCaffery et al., 1976) to investigate lifetimes and collisional cross sections of individual magnetic sublevels. Quantum interpretations of fully oriented rotationally inelastic


atom-diatomic molecule collisions have been developed by Alexander et al. (1977): the potential use of laser-induced fluorescence detection and a possible experimental arrangement are discussed. Recently proposed (Wieman and Hansch, 1976; Teets et al., 1976) atomic and molecular polarization spectroscopy schemes may also prove useful for studying collisional processes if the molecular dynamic information contained in probing laser beams is fully extracted with time-resolved detection,

B. Relaxation of Molecular Energies

State-selected fluorescence excitation of simple molecules and free radicals in the gas phase by tunable pulsed dye lasers has been employed to investigate radiative lifetimes and collisional deexcitation rates of individual rotational-vibrational levels of electronically excited states. This technique has also been applied to study reactive and nonreactive inelastic collisional processes in molecular beam scattering studies (Zare and Dagdigian, 1974). Measurements of the individual rates at which the quantum states of the reactants evolve into the quantum states of the products during a single reactive encounter are very useful in gaining an understanding of a large number of chemical phenomena. By coupling time-resolved laser-induced fluorescence detection of products with selective pulsed-laser excitation of reagents, it is now possible to obtain detailed information on simple bi- molecular gas-phase exchange reactions.

Lifetime studies are equally important since they can be used to determine transition moments, transition probabilities, oscillator strengths, and branching ratios among the numerous optical properties of molecules.

Measurements of the transition moment and of its variation with internuclear distance for electronic transitions in diatomic molecules provide a sensitive check on the accuracy of the wavefunctions used for the calculation of molecular parameters. Phase shift methods or absolute intensity measurements were previously used to determine such quantities. Time-resolved fluorescence techniques have recently led to a considerable increase in experimental versatility and accuracy.

1. Lifetimes and Quenching Cross Sections of Electronically Excited States

Alkali and alkali-hydride molecules are well suited for comparison between theory and experiment since they are to some extent hydrogenlike. The radiative lifetime of LiH (A'C') states was determined by pulsed-laser excitation in the ( 5 , O ) band (Wine and Melton, 1976). Time-resolved fluorescence from single rotational levels was followed as a function of rotational


quantum number. The radiative lifetimes were found to be independent of J . The averaged collision-free lifetime was in excellent agreement with a multi- configuration self-consistent field calculation by Docken and Hinze (1972). The radiative lifetime of the B’n, state of K, has been measured (Lemont et al., 1977) using the time-correlated single-photon-counting technique with a mode-locked H e N e laser excitation source. Radiative lifetime and quenching cross sections with potassium atoms were found to be in good agreement with experimental and theoretical work performed by Tango and Zare (1970).

The Na, molecule has been extensively studied both theoretically and experimentally. The A’C, state (Ducas et al., 1976) and the B’n, state (Demtroder et al., 1976) lifetimes have been determined experimentally using, respectively, a nitrogen-pumped dye laser and a mode-locked argon ion laser. Both experiments used the single-photon-counting technique to record fluorescent light intensities on various u”, J” rovibrational levels. Results are in excellent agreement with recent ab initio calculations by Stevens et af. (1977).

The lifetime problem of excited NO, states has been a puzzling question for many years (Donnelly and Kaufman, 1977, and references therein). Fluorescence spectroscopy with time resolution helped progress toward the understanding of excited NO, properties. Sackett and Yardley (1970) used a flashlamp-pumped dye laser to monitor the radiative lifetime as a function of wavelength. Sakurai and Capelle (1970) were the first to use a nitrogen- pumped dye laser to excite the NO, molecule in the 4220-6040 A range. No attempt was made to resolve the excited electronic state but the lifetime was found to be roughly independent of the excitation wavelength. Narrow-band excitation experiments (Haas et al., 1975) on the ,B, states of NO, revealed that these states exhibit long collisionless lifetimes of the order of 33 p e c (in agreement with Sakurai and Capelle’s results), in contrast with previous experiments and expectations of short lifetimes based on integrated absorption coefficient measurements. The technique of selective excitation of single hyperfine levels in electronically excited NO, states was simultaneously used (Paech et a/,, 1975) to determine directly the lifetimes of such levels under collision-free conditions. The spectral width of the frequency-stabilized argon ion laser was 1 MHz, and NO, pressures in the beam were less than torr. The CW laser output was modulated to form rectangular pulses with 100 nsec risetime. The laser beam was perpendicular to a well-collimated beam of NO, molecules. The spectral widths of the absorption profiles were reduced to 1% of their Doppler width at thermal equilibrium, allowing the resolution of the hfs. Fluorescence intensities were then followed with conventional photon-counting techniques in the microsecond range. Experi- mental results indicate that vibronic states, which are populated by nonradiative transitions from the initially excited state, exhibit fluorescence


decays with lifetimes around 30 pec, in good agreement with Schwartz and Senum (1975).

Comparison with other measurements indicates that the initially excited state has ,B1 characters, while the fluorescence is mainly emitted from ,B2 states. These results have been confirmed and extended by Donnelly and Kaufman (1977), who excited directly the ,B2 levels with a frequency- doubled Nd:YAG laser pumping a rhodamine 6 G dye laser.

In pure halogens, selected vibrational levels from v’ = 0 - 25 of the B3n& state of 1, (Sakurai et al., 1971) and from v’ = 1 - 31 of the B3n& state of Br, (Capelle et al., 1971) have been studied and their lifetimes and self-quenching cross sections have been accurately determined. A simple, sensitive method of detecting coherent multiphoton transitions by observation of subsequent photoionization has been described by Dalby et al. (1977) and has been applied to a sttrdy of molecular iodine.

Time-resolved laser-induced fluorescence from NH, ( ,A,) states has also been observed following excitation of the radical in its ground state by means of a pulsed dye laser (Halpern et al., 1975). Analogous experiments were performed on OH and OD (German, 1975).

Large polyatomic molecules have also been investigated in single vibronic states. This is the case for naphtalene (Boesl et al., 1976) and glyoxal (Beyer et al., 1975a,b). In the latter case, deuterium effects on the energy transfer rate constants (Zittel and Lineberger, 1977) were found to be nonexistent. Isotope- independent collisional rates were also found in the e3n, excited state of the He, molecule by Delpech and his co-workers (1977b).

The main advantage of fluorescence lifetime measurements with pulsed- laser excitation is in avoiding the effects of cascading during the lifetime of the excited state. The use of this technique should be highly preferred when selective excitation is an essential step of the investigation. However, there is only a limited number of molecular excited states that may be studied with this technique by pumping directly from the ground state. Additional excitation steps may be necessary to study highly excited states or molecular ionic states. In the latter case, this was done by J4rgensen and Wrensen (1975) using the technique of laser-excited, fast-molecular-ion beams. The collision processes are spectroscopically studied within a gas cell (beam gas method), and under certain assumptions, the molecular lifetimes can be evaluated. Fluorescence intensities are measured as a function of position along the beam in the cell so that, in the single-collision regime, the lifetimes may be determined from semilogarithmic plots of intensities vs distance. Excited states of CH, CO’, CO:, and CS: have been investigated with this method.

Many electronic transitions of diatomic molecules fall in the VUV range (A < 2000 A) where pulsed and intense laser sources are still scarce. Although it does not fall in the class of laser excitation, the use of synchrotron radiation


from electron storage rings should be mentioned here (Lindqvist et al., 1974; Lopez-Delgado et al., 1974). Its radiation characteristics are unique: high intensity, low beam divergence, pulsed and recurrent in nature.

2. State-to-State Chemical Reaction Rates

Molecular spectroscopy may be applied to the direct determination of the internal states of the reaction products by tunable laser excitation and fluorescence detection. This technique has been shown to be a sensitive detector of individual rotational-vibrational product states in molecular- beam reactive-scattering studies (Zare and Dagdigian, 1974). A typical experimental system (see Fig. 20) consists generally of three basic parts : (1) the molecular beam apparatus for carrying out the reaction under single- collision conditions, (2) the product detection system, which probes the excited-state distribution of the products using laser-induced fluorescence, and (3) the reactant preparation system (in this case a HF laser), which can excite selectively one of the reactants to the desired state prior to the reaction.

I PROWCT DETECTION SYSTEM

FIG. 20. Schematic of an experimental system used in lifetime-separated molecular reaction studies, showing the molecular-beam apparatus (cutaway drawing), the product detection system, and the reactant preparation system. From Pruett and Zare (1975).

This type of experimental system can be used to record simple excitation spectra of products as a function of reactant internal state. The fluorescence decay is then integrated and the detection laser is tuned over the product excited states under investigation. In this way, state-to-state reaction rates for the reaction Ba + HF ( u = 0, 1) + BaF ( u = 0, 1,2) + H have been studied (Pruett and Zare, 1976). Apart from the detection of product states in different electronic and vibrational levels (rotational levels are also accessible), the additional use of time resolution of fluorescence signals allows the measurement of radiative lifetimes and quenching rates of product molecules. When


very different lifetimes are involved in the excited states of the product molecule, lifetime-separated spectroscopy may be used to detect a weak emission band overlapping a strong emission band having a much shorter lifetime. This has been done by Pruett and Zare (1975) in the Ba + CO, 3

BqO + CO reaction, where the weak, long lifetime A' I I -P X fluorescence was separated from the strong, short lifetime A'C + X fluorescence of BaO. Pure lifetime measurements in LiH and NaH in their A'Z' excited state (Dagdigian, 1976) have been reported using a supersonic alkali oven coupled to an H, cell. Reactions of the type Na + H, -+ NaH + H were analyzed by probing NaH and LiH molecules by a pulsed tunable laser coupled to time-resolved fluorescence detection. Results for LiH were in good agreement with the theoretical predictions of Docken and Hinze (1972) and the experimental results of Wine and Melton (1976), who used a completely different method (see above).

3. Vibrational Excitation Transfer

Molecular studies of vibrational energy transfer have been limited up to now to vibrational ground-state levels. Vibrational excitation energies of most diatomic or triatomic molecules fall in the range from 100 to lo00 cm- so that direct optical excitation of these states requires IR light sources or lasers. Nonselective and indirect techniques have been used for a number of years, based, for example, on the optic-acoustic effect: when a polyatomic gas is exposed to intensity-modulated infrared radiation, the energy is absorbed on vibrational levels before being redistributed through inelastic collisions to the other degrees of freedom; pressure variations induced by the sound wave so generated are detected and related to the time const.ants of the collisional exchanges. Recent developments of this technique, often called the spectrophone method, are reviewed, for example, by Huetz-Aubert and Lepoutre (1974); its application to rotational relaxation will be discussed below. Such methods are generally highly nonselective in the excitation process. They are also quite indirect since vibrational decay rates are inferred from phase and amplitude measurements of the sound wave.

Pulsed-laser excitation and time-resolved detection of the subsequent fluorescence were first used by Hocker et al. (1966) in vibrational relaxation measurements in CO,. Of course, when molecular gas lasers ( C o t , HCN, HF, HzO, CO, etc.) are used in the frequency range involved in vibrational transitions, wavelength coincidences will severely restrict the number of molecules that can be studied. However, nonlinear mixing of two laser beams can be used to generate reliably narrow-band tunable infrared light. Much work remains to be done to develop this technique (Hansch, 1973).

Vibrational relaxation of HC1 (u = 1) by C1 atoms at room temperature has been measured by MacDonald et al. (1975). Flowing HCI was excited to


its first vibrational level by a pulsed HCl chemical laser and C1 atoms were generated by a microwave discharge in C1,. Fluorescence on the 1-0 branch was monitored by a Ge: Au detector cooled to 77°K: the response time of the system was shorter than 5 psec. Results of the excitation transfer rate from u” = 1 to u” = 0 were in good agreement with previous results (Craig and Moore, 1971). Vibrational-vibrational energy transfer in ground-state SOz molecules was studied by Siebert and Flynn (1975). They used a Q-switched COz laser oscillating on a single rotational transition of the 9.2 pm band to excite selectively the v1 (symmetric stretch) mode of SO2. Fluorescence was detected on the vz (antisymmetric stretch) mode with a fast 1R detector. Similar experiments have also been reported in CH3Cl (Grabiner and Flynn, 1974).

Vibrational deactivation of HF (v = 1-3) by H atoms has been recently studied (Bott and Heidner, 1977) using laser-induced fluorescence in a discharge flow tube in which H atoms were produced by a microwave discharge. A small fraction of the injected HF ( u = 0) was pumped first to HF ( u = 1) and subsequently to HF ( u = 2 and u = 3) by the multiline output from a pulsed (TEA)HF laser. Photomultipliers were used to detect the time variation of the fluorescence on the various vibrational states. Results for v = 1 and 2 are in reasonable agreement with Monte Carlo classical trajectory calculations by Wilkins (1972) but u = 3 rates are too fast to be explained by this theory. This is related to the techniques used to construct the potential energy surfaces needed for theoretical predictions of upper-level reaction rates.

Considerably less work has been so far reported on the vibrational relaxation in electronically excited states. Molecules that can be excited in the available range of present pulsed dye lasers may be studied from the point of view of their vibrational properties by pumping, for example, a (0-1) band and monitoring the (0-0) time-resolved fluorescence spectrum. In this way, the u’ = 1 to u’ = 0 vibrational relaxation can be obtained. This has been done by Hogan and Davies (1975) on the OH (A2X+, u’ = 1) state using a frequency-doubled pulsed dye laser at 2821 A.

4. Rotational Energy Transfer

Rotational transitions in molecular collisions have been studied by a number of authors (for a review, see La Budde and Bernstein, 1971). Until recently the experimental methods used to study the collisional processes involving rotational states were mostly limited to the measurement of ultrasonic dispersion (for a recent experiment, see Kistemaker and de Vries, 1975). Ultrasonic dispersion experiments yield the collision-induced transition probability averaged over all the initial and final levels. To reduce the number of initial rotational levels so that the process will be described by a single


relaxation time, the temperature has to be lowered to the cryogenic range (for an example, see Prangsma et al., 1970) Other methods have been developed using molecular beams, optical fluorescence, and double resonance, some of which have been discussed above (for a review, see Oka, 1973).

The development of tunable dye lasers has made possible the selective excitation of one rotational level by pumping only one line in the desired rotational band. The population of the perturbed level and ofmany rotational levels subsequently populated by collisional energy transfers may then be followed, on the time scale of the collisional processes, by fluorescence light measurements. The first application of this technique was reported by Collins and Johnson (1972), who examined the rotational relaxation of the 4He, (e31T,) state by pumping the a3C: (u” = 0, K” = 7) to e3n, (u’ = 0, K‘ = 8) rotational transition with a pulsed tunable dye laser. He, (a3X:) metastable states were produced in a flowing-helium afterglow illuminated by a pulsed flashlamp-pumped dye laser. The excitation transfer rate coefficient between K’ = 8 and K‘ = 7 was measured to be 2.4 x lo-” cm3 sec-’, a fairly large rate, which was assumed to be due to complex reaction chains involving other electronic states energetically lying within a few kT as intermediaries (Collins and Johnson, 1976). A more complete study of electronic and rotational energy relaxation in molecular helium (Gauthier et al., 1976a) was undertaken, using a nitrogen-laser-pumped dye laser for excitation. Fluorescence intensities were processed on-line after each laser shot by a 16-channel data acquisition system with a 1 nsec time resolution. The relaxation of electronic excitation energy and the energy redistribution among rotational sublevels were particularly studied. The results show that in fact the AK = 1 excitation transfer rates were ten times faster than measured by Collins and Johnson (1972). While collisions with AK = -t 1 account for more than 60% of total rotational transfer, it was necessary to include a substantial probability of multiquantum rotational transitions in order to explain the observed results (see Fig. 21).

To gain more insight into the details of rotational relaxation mechanisms, further investigations (Delpech et al., 1977b) were undertaken in 3He and in an equimolar mixture of 3He and 4He. The total rotational relaxation rates of a given K rotational sublevel of 3He, on 3He and of 3He4He on 3He and 4He were found to be very nearly the same and also the same as those measured in pure 4He,. These measurements show that the difference in symmetry plays a negligible role. A similar conclusion has been reached in N, experiments by Kistemaker et al. (1970). Moreover, we have shown that reactive collisions play a substantial role in the rotational energy relaxation of He, (e31T,) molecules: about 16% of the total energy transfer from a given rotational level occurs through the exchange of helium nuclei. Results of individual state-to-state excitation transfer rates were found to be in good


FIG. 21. Rotational energy relaxation in molecular helium. Reduced rotational line intensities as a function of rotational energy and of time after peak laser power at a helium pressure of 35.1 torr. The laser is pumping the J = 6 level. Dots and triangles are experimental data and the curves represent the best fits through the reaction rate equations using the exponential gap law of Polanyi and Woodall (1972).

agreement with the simple one-parameter Polanyi-Woodall empirical law (Polanyi and Woodall, 1972) in the three isotopic forms of He,.

The iodine molecule has been also studied to some extent in a more general investigation of excited-state lifetimes and predissociation by selective excitation (Lehmann, 1976). Systematic measurements of lifetimes of many individual vibrational-rotational levels with a pulsed dye laser having a spectral width of 1 GHz enabled Broyer and his co-workers (1975) to demonstrate clearly the J ( J + 1) dependence of the inverse of the lifetime due to gyroscopic predissociation.

Two additional techniques have been derived from the direct laser excitation method. The first is the double-optical-resonance method in which a pulsed laser is used to populate specific rotational energy levels (pump laser) and a second laser is used to probe the population in this or in any other J level of the molecule by absorption measurements. The probe laser may be continuous and thus follows the population change with time after the pulse. Rotational relaxation effects have been observed in HF with this technique (Hinchen and Hobbs, 1976). Rotational transfer rates were found to be three orders of magnitude faster than I/-T relaxation and to vary with AJ. The


data for rotational relaxation have also been compared to a simple model based on the Polanyi-Woodall exponential gap law, and the agreement is also good, as in the case of helium described before.

In the second technique, relaxation measurements can be made in the ground state by laser-induced depopulation (Feinberg et al., 1977). A pump pulsed laser is applied to strongly depopulate one v”, J” level of the ground state and a delayed pulsed laser is used to probe the repopulation of the ground state by collisional rotational relaxation. In order to probe states whose populations do not change substantially, a contrast enhancement technique such as polarization spectroscopy (Wieman and Hansch, 1976) may be necessary.

With electric quadrupole or magnetic resonance state selection, it is possible to prepare beams of neutral diatomic molecules in particular magnetic sublevels of a given rotational level. The determination of cross sections for rotationally inelastic transitions between individual magnetic sublevels thus becomes possible. Such cross sections are useful probes of the anisotropy

- 70.0

- 60.0

- 50.0

-40.0

- SDQ

- 200

- loo

8 conserving model

MJ conwving modd 1

1 . . , . , , . a 4 4 I 10 11 14 14 11 30 ‘12 14 14 18 50 51 54 56 58 40

AJ’( from J‘=1*)

FIG. 22. Selection rules affecting the m, states of rotational levels. Theoretical circular polarization ratios as a function of change in excited-state rotational quantum number AJ. Upper curve, angular momentum orientation conservation model; lower curve, m, conserving model. From Jeyes et a!. (1977).


of the intramolecular potential. Recent experimental results (Jeyes et al., 1977) have shown the m, properties of inelastic rotational relaxation rates. This was done by exciting one particular rovibrational level of iodine with circularly polarized laser light and measuring the circular polarization of rotationally resolved levels as a function of AJ, the rotational momentum transfer during the collision. Figure 22 shows the results, which indicate that a selection rule [Am, = 0 or Am, = 4 (McCaffery, private communication)] is apparent in rotationally inelastic transfer.

Along the same lines, rotational alignment in inelastic collisions has been investigated (Kato et al., 1976) in the iodine molecule and used to assign the rotational spectrum of the B3n& state. Recent theoretical studies (Alexander and Dagdigian, 1977) suggest that simultaneous detection of time-resolved fluorescence line intensities and polarizations would be extremely useful to monitor the occurrence of rotationally inelastic collisions and the resulting m, distribution.

v. RECENT DEVELOPMENTS AND CONCLUDING REMARKS

We shall conclude this review by describing a few recent experiments using pulsed dye laser excitation where time-resolved detection or excitation is an essential step of the experimental process. This section does not purport to be exhaustive. The applications of pulsed dye lasers are numerous; studies in this field are rapidly growing and cover many areas of science and technology. The authors only wish to point out some applications of time- resolved fluorescence studies with lasers that, in their opinion, may be of interest in the near future.

Among these, the time-resolved method for detecting a single atom of a specified type in a well-defined volume of space is particularly fascinating. Using a CW dye laser, it has been possible recently (Fairbank et al., 1975) to detect by resonance fluorescence as few as 100 sodium atoms/cm3. This method can be applied to detect many other atoms, as well as many molecules, ions, and radicals. A list of 87 atoms where resonance fluorescence from the ground state can be used with available techniques has been given by Fairbank et al. with estimated sensibilities ranging from 10’ to lo4 atoms/ cm3, depending on the oscillator strength of the transition. When using pulsed lasers, the sensitivity decreases to about lo6 atoms/cm3, because of a reduced duty cycle.

The demonstration of single-atom detection by resonance ionization spectroscopy has been given very recently by Hurst et al. (1977a). While CW tunable lasers are particularly appropriate to sample a steady-state concentration of atoms, single-atom detection requires time-resolved excitation


so that pulsed lasers are necessary. All atoms within the volume delimited by the laser beam are ionized by two-photon processes and resulting electrons are detected in a gas proportional counter. Details of the resonance ionization process are shown in Fig. 23. The laser is tuned to promote the atom to an excited state lying more than one-half of the way to the ionization continuum and a second photon from the pulsed laser will complete the resonance ionization processes. Competing processes, namely, radiative and collisional processes to otherwise inaccessible state 2 or chemical reactions out of state 1, are also sketched in Fig. 23.

FIG. 23. Schematic of the two-photon resonance ionization process. Radiation is included to a directly inaccessible state 2, with E~ + hv i E, , whereas for state 1, E , + hv < e c . The shaded low-energy zone represents chemical products, which are created out of reaction with the intermediate state 1. From Hurst et al. (1977b).

Although the method has only been used up to now on cesium atoms, extension of this technique to detect about one-half the elements with three- photon resonance ionization using commercially available lasers has been considered (Hurst et al., 1977b). Statistical density fluctuations of a small number of atoms in a well-defined volume of space, which previously could only be deduced from such phenomena as Brownian motion or Rayleigh light scattering, can now be detected at the single-atom level. Many applications, like purification of atomic species, have been proposed Letokhov, 1976) using resonance ionization spectroscopy. Single-atom detectors could also become very useful devices for analytical chemistry and should have many environmental applications. Derived methods may serve as a new way to study chemical kinetics, and particularly combustion processes.

Similar methods have been proposed and used for one-molecule detection. Using a simple experimental arrangement, Grossman et al. (1977b) have photodissociated with a laser pulse every CsI molecule in a small volume; a second pulsed laser detected each Cs atom through resonance ionization spectroscopy. Absolute cross sections for photodissociation of CsI as a function of wavelength were thus obtained with excellent accuracy and resolution. This technique was also used to measure the diffusion of cesium


atoms in argon and the reaction of cesium with O2 in argon gas (Grossman et a/. 1977a).

Resonance ionization techniques can also be applied to the laser isotope separation of uranium. Such separation processes involve three basic steps, as do many fluorescence measurements on atomic and molecular species. The first, preparation of ground-state uranium atoms and the third, removing the ions from the separation region, fall outside the scope of this review. The second step requires a two-photon excitation and ionization scheme of the type shown in Fig. 23. This step has been extensively studied in the comprehensive work of Sargent Janes'et al. (1976); the experimental system is shown in Fig. 24. The exciter section comprises a nitrogen-pumped dye laser

BEAM CWBINEI!

POWER IONIZER 1 PULSE I - SUPPLY TRIGGER BEAM OUTPUT

EXCITER

I I I I I I I I I I I I I I I I I I

1 I I I I I I I I

ETALON WITH PIEZOELECTRIC SPACER I I I

I I

CYLINDRICAL LENS I I I - ECHELLE GRATING I

AMPLIFIER

2%- FOCAL LENGTH LENS

MASTER OSCILLATOR I

- ETALOW

I

L ----------------------- ---A FIG. 24. Two-photon excitation of uranium vapor: schematic diagram of the tunable dye

laser, master oscillator, and power amplifier combination. The echelle grating and the Fabry- Perot etalon narrow the laser line-width down to 0.02 A. From Sargent Janes et al. (1976).


oscillator, an external etalon, and a nitrogen-pumped dye amplifier operating at 4266.275 A for 235U (and 4266.324 A for 238U) with a resolution of 0.02 A. The ionizer section was either simply a nitrogen laser operating at 3371 or another nitrogen-pumped dye laser system for detailed spectroscopic studies of the photoionization process. The ions produced were detected and analyzed by a mass spectrometer; the results demonstrated 50% enrichment of U235/U238. Excited-state lifetimes were measured by observing yields as a function of the delay between the exciter and ionizer laser pulses. The two- step photoionization cross section was also measured as a function of the wavelength of the ionizing laser.

Rydberg states in atomic uranium have been investigated by three-step laser excitation and detected by photoionization by an intense, pulsed-CO, laser (Solarz et al., 1976). By delaying the infrared ionizing pulse and thus discriminating against the shorter-lived valence states, Rydberg levels with principal quantum number exceeding 60 were preferentially detected (see Fig. 25). In this way, Rydberg progressions were precisely assigned and a more accurate value of the ionization potential was derived. The use of

1.0

- v1 0.5 Y

c

2,

0

Y

.r

.r

f o

E x c i t a t i o n energy, E (cm")

FIG. 25. Detection of Rydberg states of uranium using time-resolved stepwise laser excitation and photoionization. Simplification of the Rydberg states spectrum achieved by going from (a) short time delay between excitation and photoionization to (b) long time delay. From Solarz et al. (1976).


time-resolved spectroscopy is possible for the high-lying states of uranium : investigation of the ionization rates, oscillator strengths, and radiative and collisional lifetimes are currently under progress.

Considerable progress has been possible in the study of Rydberg states of atoms or molecules with the use of time-resolved fluorescence and ionization techniques. Apart from their fundamental interest, Rydberg states may be attractive for producing tunable coherent sources in the infrared and the far-infrared region. Lau and his co-workers (1976) have shown how alkali Rydberg states could be optically pumped to generate coherent infrared radiation; their analysis is based on previous accurate experimental and theoretical data on these states. Wavelengths near 16, 12, 8.6, and 7.7 pm, which are of interest for isotope separation, were found to be potentially attainable with milliwatt power levels; such systems seem capable of both pulsed- and continuous-wave excitation. In fact, 16 pm laser operation on the 626 cm-’ transition between the 6D,/, --f ’PI/, states in potassium vapor has been recently demonstrated by Grischkowsky et al. (1977). This laser features an extremely high gain, a favorable branching ratio, and a laser- enhanced cascade process.

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Photoacoustic Spectroscopy

ALLAN ROSENCWAIG

Lawrence Livermore Laboratory University of California Livermore, California

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111. The Photoacoustic Effect in Gases

A. Historical Theories

. . . . . . . . . I . . . . . . . . . . . . . . . .

C. Special Cases . . . . . . . . . . . . . . . . . . . . . . . . . . D. General Case E. Nonhomogene F. Experimental Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Conclusions on the Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V. Theory of the Photoacoustic E A. Gas-Microphone Coupling B. Liquid-Microphone Coupli C. Experimental Verification

VI. Experimental Methodology . , . . . , . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Radiation Sources . . . . . . , . , . , . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . B. Experimental Chamber C. Data .4cqu i s i ' ; u .~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . D. Commercial Photoacoustic Spectrometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

A. Inorganic Insulators . . . . . . . . . . . . . . . .

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

VII. Photoacoustic Spectroscopy in Physics and

B. Inorganic and Organic Semiconductors C. Metals. . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . D. Liquid Crystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Catalysis and Chemical Reaction Studies F. Surface Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Deexcitation Studies . , . . . . . , . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Conclusions

VIII. Photoacoustic Spectroscopy in Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Hemoproteins , . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Plant Matter . . . . . . . . . . . . . . . . . . . . . . .

IX. Photoacoustic Spectrosco ................................... A. Bacterial Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Human Eye Lenses ..................................................

. . . . . . . . . . .

B. Drugs in Tissues . . . . . . . . . . . . . . . . .

208 209 211 214 215 216 225 22 I 233 235 240 241 241 242 244 247 248 249 255 256 256 251 258 264 264 266 270 213 280 280 28 1 283 289 289 289 291



208 ALLAN ROSENCWAIG

D. TissueStudies ...................................................... 293 E. The in V i m Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306

X. FutureTrends . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308

I. INTRODUCTION

Spectroscopy is the science devoted to the study of the interaction of energy with matter. As such, it is a science encompassing many techniques and practiced by large numbers of scientists of many disciplines. The oldest form of spectroscopy is optical spectroscopy, where the energy is in the form of photons with wavelengths ranging from less than 1 A in the x-ray region, to 100 pm (lo6 A) in the far-infrared region. In spite of its long history, optical spectroscopy is still the most useful and active spectroscopic field, in part, because it is a nondestructive and often nonperturbing means for investigating the properties of materials.

Conventional optical spectroscopy techniques tend to fall into two major categories ( I ) . In the first category, one studies the optical photons that are transmitted through the material under study. In the second category, one studies the light that is scattered or rejected from the material. Almost all conventional optical techniques are variations of these two basic methods, and as such they are distinguished not only by the fact that optical photons constitute the incident energy beam, but also by the fact that the data are obtained by detecting these photons after their interaction with the matter or material under investigation.

There are, however, a great many substances, both Qrganic and inorganic, that are not readily amenable to the conventional traiismission or reflection forms of optical spectroscopy. These materials are usually highly light scattering, such as amorphous solids, powders, gels, smears, and suspensions, or else optically opaque with dimensions that far exceed the penetration depth of the incident photons. Over the years, several techniques have been developed to permit optical investigation of such materials, the most common of which are diffuse reflectance (2), attenuated total reflection (ATR) or internal reflection spectroscopy (3), and Raman scattering (4). All of these techniques are very useful, but they all suffer from similar limitations, to wit, each is applicable to only a small category of materials, each is useful over only a small wavelength range, and the data obtained are often difficult to interpret.

During the past few years another optical technique has been developed to study those materials that are unsuitable for conventional transmission or reflection methodologies (5). The technique, called photoacoustic spec-

PHOTOACOUSTIC SPECTROSCOPY 209

troscopy (PAS), is distinguished from the conventional techniques chiefly by the fact that even though the incident energy is in the form of optical photons, the interaction of these photons with the material under investigation is studied not through subsequent detection and analysis of the photons, but rather through a direct measure of the energy absorbed by the material due to its interaction with the photon beam.

Although more will be said about experimental methodology later in this chapter, a brief description of this technique at this point might be appropriate. In PAS of solids, the sample to be studied is placed inside a closed cell containing a gas, such as air, and a sensitive microphone. The sample is then illuminated with chopped monochromatic light. The analog signal from the microphone is processed by a tuned amplifier whose output is recorded as a function of the wavelength of the incident light. In this way, photoacoustic spectra are obtained, which are found to correspond to the true optical absorption spectra of the samples.

One of the principal advantages of PAS is that it enables one to obtain spectra, similar to optical absorption spectra, on any type of solid or semi- solid material, whether crystalline, powder, amorphous, smear, or gel. This capability is based on the fact that only the absorbed light is converted to sound. Scattered light, which presents such a serious problem when dealing with many solid materials by conventional spectroscopic techniques, presents no serious difficulties in PAS. Furthermore, it has been found experimentally that good optical absorption data can be obtained with the photoacoustic technique on materials that are completely opaque to transmitted light. PAS has already found important applications in the research and analysis of inorganic, organic, and biological materials (5- 10). Furthermore, it appears to have a strong potential as a tool in surface studies and deexcitation studies ( I O J Z ) . These several applications and potential uses will be explored in Sections VII-IX.

11. THE EARLY HISTORY OF THE PHOTOACOUSTIC EFFECT

Although photoacoustic spectroscopy made its official debut as recently as 1973 (7), the concept on which it is based is quite old. An analogous technique, commonly referred to as optoacoustic spectroscopy, has been used for many years in the study of optical absorption phenomena in gaseous systems. The photoacoustic effect is now generally accepted as applying to the investigation of nongaseous matter,while the word optoacoustic is still used in many circles for gas studies. The change of name from optoacoustic to photoacoustic was instituted by this author to reduce confusion with the


acoustooptic effect, in which light interacts with acoustic or elastic waves in a crystal.

The photoacoustic effect in both nongaseous and gaseous matter was first reported in 1880, when Alexander Graham Bell gave an account to the American Association for the Advancement of Science of his work on the photophone (12). In this report, he briefly reported the accidental discovery of what we now term the photoacoustic or optoacoustic effect in solids.

In brief, Bell’s photophone consisted of a voice-activated mirror, a selenium cell, and an electrical telephone receiver. A beam of sunlight was intensity modulated at a particular point by means of the voice-activated mirror. This intensity-modulated beam was focused onto a selenium cell whose electrical resistance varied with the intensity of light falling on it. The selenium cell was incorporated in a conventional telephone circuit, and thus a voice-modulated beam of light resulted in electrically reproduced telephonic speech.

While experimenting with the photophone; Bell discovered that at times it was possible to obtain an audible signal directly, that is, in a nonelectrical fashion. This phenomenon occurred if the beam of sunlight was rapidly interrupted, as with a rotating slotted disk, and then focused on solid substances that were in the form of thin diaphragms connected to a hearing tube.

In a publication in 1881 (13), Bell described in detail his further work on the photoacoustic effect. He found that the need for diaphragms was un- necessary. In particular, he found that if solid matter were placed inside a closed glass tube to which a.hearing tube was attached, a quite audible signal could be detected if the material in the tube were then illuminated with a rapidly interrupted beam of sunlight. He noted that “the loudest signals are produced from substances in a loose, porous, spongy condition, and from those that have the darkest or most absorbent colours” (13, p. 515).

In a series of definitive experiments, Bell (13) demonstrated that the photoacoustic effect in solids was dependent upon the absorption of light, and that the strength of the acoustic signal was in turn dependent on how strongly the incident light was absorbed by the material in the cell. He concluded that “the nature of the rays that produce sonorous effects in different substances depends upon the nature of the substances that are exposed to the beam, and that the sounds are in every case due to those rays of the spectrum that are absorbed by the body.”

In addition to studying the photoacoustic effect with solids, Bell and his associate Sumner Tainter also studied the effect in liquids and gases (13). They observed that only weak signals were produced when the cell was filled with a light-absorbing liquid, but that quite strong signals were obtained when the cell was filled with light-absorbing gases. Photoacoustic experiments on gases were also performed by John Tyndall (14) and Wilhelm

PHOTOACOUSTIC SPECTROSCOPY 21 1

Roentgen (15), who had heard about Bell’s discovery of the previous year. They found, as did Bell, that the photoacoustic effect in light-absorbing gases could be readily observed for many gaseous substances.

111. THE PHOTOACOUSTIC EFFECT IN GASES

The series of papers by Bell, Tyndall, and Roentgen in 1881 reported experiments in which optical radiation from the sun or a mercury arc was chopped or intensity modulated by passing it through a rotating slotted disk, and then directed into a closed chamber containing a gas. This gas was either colored and thereby could absorb light in the visible region of the spectrum, or else was a colorless but infrared-absorbing gas. The absorbed radiation produced pressure fluctuations in the gas that were detected as audible sound, having the same frequency as the chopping or modulating frequency, through a hearing tube attached to the gas-containing chamber. The acoustic pressure was correctly interpreted,as arising from the transformation of the absorbed light energy into translational or kinetic energy of the gas molecules.

After this initial flurry of interest, and several subsequent papers by other authors, experimentation with the photoacoustic effect both with solids and gases apparently ceased, and was not revived until nearly 50 years later, when the microphone became available to record the photoacoustic signal. Working in 1938 at the State Optical Institute, Leningrad, Viengerov began using the photoacoustic effect to study infrared light absorption in gases and to evaluate concentrations of gaseous species in gas mixtures (16). His light sources were black-body infrared sources, such as Nernst glow bars, and he employed an electrostatic microphonic arrangement whereby he measured the voltage change between charged capacitive microphone diaphragms. Viengerov was able to measure COz concentrations in Nz down to -0.2% by volume. Measurements of lower concentrations were limited both by the low sensitivity of his microphone, and by background absorption of the incident radiation by the cell windows and walls, that is, by the presence of an unwanted photoacoustic effect in solids.

A year later, Pfund (17) described another gas analyzer system used at Johns Hopkins Hospital for measuring concentrations of CO and C 0 2 . Pfund‘s experiments are of additional interest because, instead of observing pressure changes, he measured the corresponding changes in gas temperature directly, using a thermopile shielded from direct radiation. This instrument had a sensitivity comparable to Viengerov’s initial apparatus.

A major improvement in the sensitivity of gas concentration analysis occurred in 1943 when Luft described a commercial automatically recording


gas analyzer that employed two photoacoustic cells in a differential design (18). One cell contained the gas mixture to be analyzed, while the other contained the gas mixture without the particular species of interest. In this instrument, the microphone output was proportional to the pressure difference between the two cells. Luft’s differential analyzer had two major improvements over Viengerov’s original design. It minimized the signal due to background absorption in the cell windows and walls, since the same background signal was obtained in both cells. It also permitted the analysis of gas mixtures containing more than two species. Gas analyzers based on Luft’s design became commercially available in 1946. These instruments had a sensitivity that allowed the measurement of CO, in N, down to a few parts per million (ppm), as compared to Viengerov’s sensitivity of a few parts per thousand.

In addition to the gas analyzer, a variation of Luft’s design was also used to construct an infrared gas spectrometer, called the spectrophone (19). This instrument utilized infrared light sources such as Nernst glow bars, an infrared dispersive monochromator to obtain monochromatic radiation, and a differential photoacoustic cell design to obtain infrared absorption spectra of gases and vapors.

It was recognized that the photoacoustic signal is a measure of the amount of energy absorbed by a system, such as a gas, that is dissipated through nonradiative or heat-producing processes. As such, the photoacoustic effect can be used to study this ubiquitous channel of energy level deexcitation. In 1946, Gorelik (20) first proposed that measurement of the phase of the photoacoustic signal could be used to investigate the rate of energy transfer between the vibrational and the translational degrees of freedom of gas molecules. When a sample of gas in a photoacoustic cell is irradiated by photons, which it absorbs, the absorbed energy is used to excite a vibrational or vibrational- rotational energy state if the irradiation is in the infrared, or to excite an electronic state if the irradiation is in the visible or ultraviolet regions of the electromagnetic spectrum. After a time delay determined by the rate of energy transfer through interatomic collisions, this excited state will deexcite with a transfer of energy to the translational modes of the gas molecules, causing the gas to heat up by appearing as increased kinetic energy of the gas molecules. If the irradiation time is long compared to the time required for the energy transfer, then essentially all of the absorbed energy will appear as heat energy, and the resultant pressure fluctuations will be in phase with the periodic incident radiation. If the frequency at which the incident light is chopped or modulated is high enough such that the irradiation time is less than the time required for energy transfer to take place, then not all of the absorbed energy will appear as periodic heat, and the phase of the maximum pressure fluctuation will be noticeably different than that of the incident


radiation. Thus a study of the phase of the acoustic pressure signal as a function of modulation frequency will give information about the rate of intermolecular energy transfer. Gorelik‘s proposal was successfully put into practice by Slobodskaya in 1948 (24, and the use of the photoacoustic effect to study vibrational lifetimes of gaseous molecules has now become a well- established technique.

If we consider a gas with a simple two-level system, a ground state Eo and an excited state El, one can find (20,221 that the amplitude q and phase 4 of the photoacoustic signal are given by

4 = 7c/2 - tan-’wz

where A and C, are constants dependent on the intensity of the incident light beam, the beam and cell geometries, and the thermal properties of the gas; w is the radial frequency at which the beam is intensity modulated; and z is the relaxation time constant for the energy transfer that deexcites the system from state E , to state E , .

For further details about the early photoacoustic work with gases, the reader should consult the review articles by Delaney (23) and Read (24).

Between 1950 and 1970 the photoacoustic gas analyzer employing a conventional light source gave way to the more sensitive gas chromatography technique. Similarly, the spectrophone gave way to the more versatile infrared spectrophotometer. The photoacoustic effect was henceforth primarily used to study vibrational lifetimes and other aspects of radiationless deexcitation in gases. The advent of the laser provided, in the 1970s, a major new impetus to gaseous PAS. One of the reasons for this impetus is the fact that a photoacoustic signal arises solely from molecules that absorb the incident radiation, and thus this signal is linearly proportional to the strength of the incoming radiation. High-power lasers therefore permit the analysis of very low concentration species in a gaseous mixture. The recent revival of gaseous photoacoustic measurements can be traced to the work of L. B. Kreuzer in 1971 (25,26). With the use of a high-power tunable infrared laser, low-noise microphones, and phase-sensitive lock-in amplifiers, Kreuzer demonstrated that it was possible to detect and identify gaseous species in concentrations as low as a few parts per billion (ppb) (25-28). The highly collimated laser beam also makes possible the use of multipass resonant systems (29) that increase the effective photoacoustic signal from the gas while minimizing the background signal due to photoacoustic absorption in the cell windows and walls. This capability of detecting gas concentrations in the parts per billion range has become extremely useful not only for atmospheric pollution


monitoring (30), but also in studies of stratospheric chemistry by means of a balloon-borne photoacoustic system (31).

In the field of gas spectroscopy, lasers permit the study of the detailed shape of gas absorption lines, since laser linewidths are much narrower than the Doppler and collision-broadened absorption lines of a gas. Coupled with the fact that much lower absorptions can be studied with the photoacoustic effect than with transmission spectroscopy, where the transmitted flux can constitute an immense background signal, laser PAS allows for the study of the detailed absorption lineshapes of molecules for all levels excited by the laser beam, even for those with very low absorption cross sections (32).

The study of radiationless deexcitation processes in gaseous systems with the photoacoustic effect has also been pursued actively in the past few years. In a series of very pretty experiments, M. B. Robin and his co-workers have used the new optical and electronic techniques to further the investigations of both kinetic and photochemical effects in gases (33-37). A study of the phase of the photoacoustic signal as a function of incident photon wavelength irradiating a sample of biacetyl gave useful information about the relative efficiencies of the radiative and nonradiative channels of deexcitation. By a similar technique these investigators were able to study intersystem energy transfer in gas mixtures of pyridine and biacetyl and of oxygen and biacetyl. Robin and Keubler have also studied the photochemical decomposition of aromatic ketones with PAS through a study of phase as a function of wavelength. For further details about these interesting experiments the reader is referred to the publications of Robin and his co-workers.

Although the above discussion is short, it is clear that the field of gaseous PAS has a very interesting past and an active present. In particular, as tunable lasers expand further into the ultraviolet and infrared regions of the elctro- magnetic spectrum, the applications described above will be expanded and new applications will be developed.

Iv. THEORY OF THE PHOTOACOUSTIC EFFECT IN SOLIDS

As was pointed out in Sections I1 and 111, the photoacoustic effect was originally discovered in solids, and then subsequently in gases and liquids. Nevertheless, the original investigators concentrated almost entirely on the gaseous phenomenon, no doubt because it was the easiest to understand. It is most intriguing that in spite of the strong rebirth of the photoacoustic effect in 1938, this rebirth was apparently limited to gases. It was not until a few years ago, some 90 years since Bell’s original discovery, that the photoacoustic effect in solids and its potential applications were “rediscovered.”


How, in spite of the many years of work with gases since 1938, solid-state PAS remained forgotten is a puzzling mystery.

Since 1973, however, the photoacoustic effect with solids has strongly reemerged, with the consequent development of a modern spectroscopic technique that appears to be extremely useful for the investigation of nongaseous materials.

A. Historical Theories

Alexander Graham Bell’s experiments on the photoacoustic effect with solids, liquids, and gases, reported in 1880 and 1881 (12,13), generated considerable interest at that time. However, the phenomenon was evidently regarded as a curiosity of no great functional or scientific value and was quickly forgotten. Fifty years later, the photoacoustic effect was reborn for the study of gases, and the theory of this effect in gases has since been thoroughly expounded. Unlike the situation for gases, a satisfactory explanation for the photoacoustic effect in solids has been developed only recently, although several attempts to do so have been made both in the last century as well as in this one.

In attempting to account for the audible signal obtained from his experiments with dark spongy solids, such as lampblack, Bell hypothesized that:

When a beam of sunlight falls upon the mass, the particles of lampblack are heated, and consequently expand, causing a contraction of the air-spaces or pores among them. Under these circumstances a pulse of air should be expelled, just as we would squeeze out water from a sponge. The force with which the air is expelled must be greatly increased by the expansion of the air itself, due to contact with the heated particles of lampblack. When the light is cut off, the converse process takes place. The lampblack particles cool and contract, thus enlarging the air-spaces among them, and the enclosed air also becomes cool. Under these circumstances a partial vacuum should be formed among the particles, and the outside air would then be absorbed, as water is by a sponge when the pressure of the hand is removed. I imagine that in some such manner as this a wave of condensation is started in the atmosphere each time a beam of sunlight falls upon the lampblack, and a wave of rarefraction is originated when the light is cut off (13, pp. 515-516).

In the case when the illuminated solid is in the form of a thin flexible membrane or disk, Bell supported the theory of Lord Rayleigh (38), who concluded that the primary source for the photoacoustic signal was the mechanical vibration of the disk resulting from uneven heating of the disk when struck by the beam of sunlight.

Mercadier (39), who also experimented with the effect at this time, suggested that the sound is due to “vibrating movement determined by the alternate heating and cooling produced by the intermittent radiations, principally in the gaseous layer adhering to the solid surface hit by these radiations” (39, p. 410).


In my opinion, Preece (40) suggested a mechanism that comes closest to the modern theory. He wrote that the photoacoustic effect ‘‘is purely an effect of radiant heat, and it is essentially one due to the changes of volume in vapours or gases produced by the degradation and absorption of this heat in a confined space.” He further noted that “Cigars, chips of wood, smoke, or any absorbent surfaces placed inside a closed transparent vessel will, by first absorbing and then radiating heat rays to the confined gas, emit sonorous vibrations” (40, p. 51 8).

We have found in our laboratory, from experiments in which we first thoroughly evacuated the photoacoustic cell and then refilled it with non- adsorbing noble gases, and from experiments with two-dimensional solids and other materials with weak surface adsorption properties, that adsorbed gases do not play a significant role in the production of the acoustic signal. Furthermore, it can be readily shown that thermal expansion and contraction of the solid, and in fact any thermally induced mechanical vibrations of the solid, are generally too small in magnitude to account for the observed acoustic signal. From both experimental and theoretical considerations, we have concluded that the primary source of the acoustic signal in the photoacoustic cell arises from the periodic heat flow from the solid to the surrounding gas as the solid is cyclically heated by the chopped light. We are thus in agreement with the conjectures of Mercadier and Preece some 90 years ago.

Neither Mercadier nor Preece attempted to mathematically develop their hypotheses into an actual theory. It appears that the first attempt at this was by J. G. Parker in 1973 (41). Parker was performing experiments with the photoacoustic effect in gases, and in the course of these experiments noticed that a small but measurable signal was being produced in his gas cell, even when the gas within the cell was completely transparent to the incoming light. He determined that this signal was due to the absorption of light within the quartz windows of his cell, and analyzed his results in terms of a periodic heat flow from the windows to the enclosed gas. In his analysis, however, Parker had to make the rather unrealistic assumption that an anomalously large fraction of his incident light was being absorbed within a very small surface layer of his quartz windows in order to explain his results. The reader is referred to his paper for further details.

B. Present Theory

In the following pages we shall present a theory (42,43) for the photoacoustic effect in solids (RG theory) that assumes, as Mercadier, Preece, and Parker did, that the primary source of acoustic signal is due to a periodic


heat flow from the solid to the surrounding gas, as the solid is cyclically heated by the absorption of the chopped light. This theory is able to account, in a natural direct fashion, for all of the observed effects in photoacoustic spectroscopy of solids.

1. Heat Flow Equations

It is known that any light absorbed by a solid is converted, in part or in whole, into heat by nonradiative deexcitation processes within the solid. We formulate here a one-dimensional model of the heat flow in the cell resulting from this absorbed light energy. Consider a simple cylindrical cell as shown in Fig. 1. The cell has a diameter D and length L. We assume that the length L is small compared to the wavelength of the acoustic signal and the microphone (not shown) will detect the average pressure produced in the cell. The sample is considered to be in the form of a disk having diameter D and thickness 1. The sample is mounted so that its front surface is exposed to the gas (air) within the cell and its back surface is against a poor thermal conductor of thickness &. The length 1, of the gas column in the cell is then given by 1, = L - 1 - lb. We further assume that the gas and backing materials are not light absorbing.

FIG. 1. Cross-sectional view of a simple cylindrical photoacoustic cell. From Rosencwaig and Gersho (43).

We define the following parameters:

k j : thermal conductivity of material j (cal/cm-sec-"C) pj: density of materialj (gm/cm3) C j : specific heat of material j (cal/gm-"C)

pi = k ./ .C .. thermal diffusivity of material j (cm2/sec)'

pj = l/aj: thermal diffusion length of materialj (cm) aj = (a/$)/': thermal diffusion coefficient of materialj (cm- ')

where j can take the subscripts s, g, and b for the solid, gas, and backing material, respectively, and w denotes the chopping frequency of the incident light beam in radians per second. Table I lists the above photoacoustic parameters for various substances.

TABLE I

Thermal diffusion Density p Specific heat C Thermal conductivity k Thermal diffusivity length at 100 Hz,

Substance (gm/cm3) (d/gm-"C) (cal/cm-sec-"C) j = k/pC (cm2/sec) = (2j/w)'" (cm)

Aluminum 2.7 0.216 4.8 x lo-' 0.82 5.1 x lo-' Stainless steel 7.5 0.12 3.3 x 3.7 x 10-2 1.1 x Brass 8.5 0.089 2.6 x lo-' 0.34 3.3 x 10-2 KCI crystal 2.0 0.21 2.2 x 10-2 5.2 x lo-' 1.3 x lo-' Crown glass 2.6 0.16 2.5 x 10-3 6.0 x 10-3 4.4 x 10-3 Quartz 2.66 0.188 2.2 x 10-3 4.4 x 10-3 3.7 10-3 Rubber 1.12 0.35 3.7 x 10-4 9.4 x 10-4 1.7 x 1 0 - 3

Polyethylene 0.92 0.55 5 x 10-4 9.9 x 10-4 1.8 x 10-3 Water 1.00 1.00 1.4 x 10-3 1.4 x 10-3 2.1 10-3 Ethyl alcohol 0.79 0.60 4.2 x 10-4 8.9 x 10-4 1.7 x 10-3 Chloroform 1.53 0.23 2.9 x 10-4 8.4 x 10-4 1.6 x 10-3 Air 1.29 x 10-3 0.24 5.7 x 10-5 0.19 2.5 x lo-' Helium 1.80 x 10-4 1.25 3.4 x 10-4 1.52 7.0 x lo-'


We assume a sinusoidally chopped monochromatic light source with wavelength 1 incident on the solid with intensity

1 = $lo(l + cosot)

where I , is the incident monochromatic light flux (W/cm2). Let ct denote the optical absorption coefficient (cm- ') of the solid sample for the wavelength 1. The heat density produced at any point x due to light absorbed at this point in the solid is then given by

+aI,eux(I + cos or)

where x takes on negative values since the solid extends from x = 0 to x = - I with the light incident at x = 0. Note also from Fig. 1 that the air column extends from x = 0 to x = 1, and the backing from x = - 1 to x = - ( 1 + &).

The thermal diffusion equation in the solid taking into account the distributed heat source can be written as

with A = ctIoqf2ks

where 4, is the temperature, and v] the efficiency at which the absorbed light at wavelength 1 is converted to heat by the nonradiative deexcitation processes. In this chapter we shall assume v] = 1, a reasonable assumption for most solids at room temperature. For the backing and the gas, the heat diffusion equations are given by

for - ~ - I , I x I -1 a24 1 a4 ax2 - fib at

for O I X I I , a24 i a4 ax2 - p, at

(4)

(5)

A more exact treatment for the gas can be derived using equations of fluid dynamics as has been done subsequently by others (44,45). However, the approximate treatment given here gives results consistent with the more exact method, and has the advantage of presenting a simpler physical description of the process. The real part of the complex-valued solution 4(x, t ) of Eqs. (3)-(5) is the solution of physical interest and represents the temperature in the cell relative to ambient temperature as a function of position and time. Thus the actual temperature field in the cell is given by

T(x, t ) = Re[$(x, t)J + CD

where Re denotes the "real part of" and @ is the ambient (room) temperature.


To completely specify the solution of Eqs. (3)-(5), the appropriate boundary conditions are obtained from the requirement of temperature and heat flux continuity at the boundaries x = 0 and x = - 1, and from the constraint that the temperature at the cell walls x = 1, and x = -(I + l,,) is at ambient. The latter constraint is a reasonable assumption for metallic cell walls but in any case does not affect the ultimate solution for the acoustic pressure.

Finally, we note that we have assumed the dimensions of the cell are small enough to ignore convective heat flow in the gas at steady-state conditions.

2. Temperature Distribution in the Cell

be written as The general solution for &x, t ) in the cell, neglecting transients, can

4(x, t ) = el + e2x + de"" + ( Ueusx + Ve-OEX - Ee"")e'"' for - 1 I x I 0 (1 -x/1,)00+8exp( - o , x + i ~ t ) for O I X I l ,

(6)

where W, U , V , E, and 8 are complex-valued constants; el, e2, d, W,, and 8, are real-valued constants; and oj = (1 + i)uj with uj = ( 0 / 2 j ~ ) l / ~ . In particular, it should be noted that 8 and W represent the complex amplitudes of the periodic temperatures at the sample-gas boundary (x = 0) and the sample-backing boundary (x = - I), respectively. The dc solution in the backing and gas already make use of the assumption that the temperature (relative to ambient) is zero at the ends of the cell. The quantities W, and O0 denote the dc component of the temperature (relative to ambient) at the sample surfaces x = -1 and x = 0, respectively. The quantities E and d, determined by the forcing function in (31, are given by

1

d = -A /u2 (74

E = A/(u2 - 05) = ~ 1 , / 2 k , ( ~ ~ ~ - 05) (7b)

In the general solution (6) we have omitted the growing exponential component of the solutions to the gas and backing material, because for all frequencies w of interest the thermal diffusion length is small compared to the length of the material in both the gas and the backing. That is, pb << 1b and p, << 1, (p, - 0.02 cm for air when w = 630 rad/sec), and hence the sinusoidal components of these solutions are sufficiently damped so that they are effectively zero at the cell walls. Therefore, the growing exponential


components of the solutions would have coefficients that are essentially zero in order to satisfy the temperature constraint at the cell walls.

The temperature and flux continuity conditions at the sample surfaces are explicitly given by

k " ( 0 , a4 t ) = ks L ( 0 , a4 t )

k b -( a 4 b - I , t ) = k, "( 84 - 1, t )

ax ax

ax ax

where the subscripts s, b, and g identify the solution to (6) for the temperature in the solid, backing, and gas, respectively. These constraints apply separately to the dc component and the sinusoidal component of the solution. From (8), we obtain for the dc components of the solution

Bo = e , + d

Wo = el - e21 + de-"

k - $ B o = kse2 + k,ad

Equations (9) determine the coefficients e l , e 2 , d, W,, and Bo for the time- independent (dc) component of the solution. Applying (8) to the sinusoidal component of the solution yields

B = U + V - E ( 104

(lob)

(10c)

( 104

W = e-"s'U + &'.'I/ - e-alE

- k,a,O = k,o,U - k,o, V - k,uE

kb(TbW = k,a,e-"q'U - k,o,euslV - k,ae-aiE

These quantities, together with the expression for E in Eq. (7b), determine the coefficients U , V , W , and 8. Hence the solutions to (9) and (10) allow us to evaluate the temperature distribution (6) in the cell in terms of the optical, thermal, and geometric parameters of the system. The explicit solution for 8, the complex amplitude of the periodic temperature at the


solid-gas boundary (x = 0) is given by

( r - l)(b + l)eus' - ( r + l)(b - l)e-usl + 2(b - r)e-" (g + 1)(b + l)eus' - (9 - l)(b - l)e-us'

e =

where

b = k,U,,/k,a,

9 = k,a,/ksas

r = (1 - i)cc/2a,

and as stated earlier, as = (1 + i)as. Thus, Eq. (11) can be evaluated for specific parameter values, yielding a complex number whose real and imaginary parts O1 and O2 respectively determine the in-phase and quadrature components of the periodic temperature variation at the surface x = 0 of the sample. Specifically, the actual temperature at x = 0 is given by

T ( 0 , t ) = ~ + + o + 8 1 c o s o t - 8 2 s i n w t

where 0 is the ambient temperature at the cell walls and O,, is the increase in temperature due to the steady-state component of the absorbed heat.

3. Production of the Acoustic Signal

As stated in Section I, it is our contention that the main source of the acoustic signal arises from the periodic heat flow from the solid to the surrounding gas. The periodic diffusion process produces a periodic temperature variation in the gas as given by the sinusoidal (ac) component of the solution (6),

4ac(~ , t ) = 6 exp( - agx + iot) (15)

Taking the real part of (15), we see that the actual physical temperature variation in the gas is

T,,(x, t ) = e-'Sr[O1 cos(ot - a,x) - e2 sin(ot - a,x)] (16)

where 61 and 6 2 are the real and imaginary parts of 6, as given by (11). As can be seen in Fig. 2, the time-dependent component of the temperature in the gas attenuates rapidly to zero with increasing distance from the surface of the solid. At a distance of only 24a, = 2xpg, where pg is the thermal diffusion length, the periodic temperature variation in the gas is effectively fully damped out. Thus we can define a boundary layer, as shown in Fig. 1, whose thickness is 2npg ( ~ 0 . 1 cm at 4271 = 100 Hz) and maintain to a good approximation that only this thickness of gas is capable of responding thermally to the periodic temperature at the surface of the sample.


FIG. 2. Spatial distribution of the time-dependent temperature within the gas layer adjacent to the solid interface. From Rosencwaig and Gersho (43).

The spatially averaged temperature of the gas within this boundary layer as a function of time can be determined by evaluating

From (1 5 ) this gives

using the approximation e - 2 n << 1. Because of the periodic heating of the boundary layer, this layer of gas

expands and contracts periodically and thus can be thought of as acting as an acoustic piston on the rest of the gas column, producing an acoustic pressure signal that travels through the entire gas column. A similar argument has been used successfully to account for the acoustic signal produced when a conductor in the form of a thin flat sheet is periodically heated by an ac electrical current (46).

The displacement of this gas piston due to the periodic heating can be simply estimated by using the ideal gas law,

where we have set the average dc temperature of this gas boundary layer equal to the dc temperature at the solid surface, To = 0 + O0. Equation (18)


is a reasonable approximation to the actual displacement of the layer since 27rpg is only -0.1 cm for w / 2 n = 100 Hz and even smaller for higher frequencies.

If we assume that the rest of the gas responds to the action of this piston adiabatically, then the acoustic pressure in the cell due to the displacement of this gas piston is derived from the adiabatic gas law P V y = const, where P is the pressure, V the gas volume in the cell, and y the ratio of the specific heats. Thus the incremental pressure is

YPO YPO VO 4

8P( t ) = -6V = - -dx(t)

where Po and Vo are the ambient pressure and volume, respectively, and - 6V is the incremental volume. Then from ( lQ

where

Thus the actual physical pressure variation AP(t ) is given by the real part of 6 p ( t ) as

(21)

(22)

where Q1 and Q2 are the real and imaginary parts of Q, and q and -$ are the magnitude and phase of Q, i.e.,

Q = Q1 + iQ2 = qe-i*

Ap(t ) = Q1 cos(ot - 4 4 ) - Q2 sin(ot - 744)

Ap(t) = qcos(wt - $ - 4 4 )

or

Thus Q specifies the complex envelope of the sinusoidal pressure variation. Combining (1 1 ) and (20) we get the explicit formula

( r - l ) (b + l)eus' - ( r + l)(b - l)e-us' + 2(b - r)e-"l (g + l)(b + l)eus' - (g - l ) (b - l)e-"" [

where b = kbab/k,a,, g = kgag/ksas, r = (1 - i)a/2aS, and as = ( 1 + i)a, as previously defined. At ordinary temperatures, T o x CD so that the dc components of the temperature distribution need not be evaluated. Thus Eq. (23) may be evaluated for the magnitude and phase of the acoustic pressure wave produced in the cell by the photoacoustic effect.


C . Special Cases

The full expression for M ( t ) is somewhat difficult to interpret because of the complicated expression for Q as given by (23). However, physical insight may be gained by examining special cases where the expression for Q becomes relatively simple. We group these cases according to the optical opaqueness of the solids as determined by the relation of the optical absorption length, p, = l / a , to the thickness 1 of the solid. For each category of optical opaqueness, we then consider three cases according to the relative magnitude of the thermal diffusion length ps, as compared to the physical length 1 and the optical absorption length pa. For all of the cases evaluated below, we make use of the reasonable assumption that g < b and that b 1, i.e., kgUg < kbab and kbab - k,a,.

The six cases are illustrated in Fig. 3. It is convenient to define

Y = yPolo/2J21gT, (24)

which always appears in the expression for Q as a constant factor. We also define the optical pathlength as y, = l /a .

FIG. 3. Schematic representation of special cases for the photoacoustic effect in solids. From Rosencwaig and Gersho (43).

1. Optically Transparent Solids (pa > I )

In these cases, the light is absorbed throughout the length of the sample, and some light is transmitted through the sample.

Case la: Thermally Thin Solids ( p s >> I ; p, > p,) 1 - al, e faS ' z 1, and \r\ > 1 in (23). We then obtain

Here we set e-"


The acoustic signal is thus proportional to al, and since pb/a, is proportional to l/w, the acoustic signal has a w - dependence. For this thermally thin case of ps >> 1, the thermal properties of the backing material come into play in the expression for Q.

Case Ib: Thermally Thin Solids (ps > 1; p, < pa) Here we set e-"' z 1 - al, e*OS' E 1 f osl, and Irl c 1 in (23). We then obtain

[(a' + 2a:) + i(a' - 2a:)l 'v ctN

Q = - 4ksa,a,3b

The acoustic signal is again proportional to al, varies as w - l , and depends on the thermal properties of the backing material. Equation (26) is identical to Eq. (25).

Case lc: Thermally Thick Solids (ps c 1; p, << pa) In Eq. (23) we set e-aI N- - 1 - al, e-un' N- O, and Irl << 1. The acoustic signal then becomes

Here the signal is now proportional to a k rather than al. That is, only the light absorbed within the first thermal diffusion length ps contributes to the signal, in spite of the fact that light is being absorbed throughout the length 1 of the solid. Also since ps < I, the thermal properties of the backing material present in Eq. (26) are replaced by those of the solid. The frequency dependence of Q in Eq. (27) varies as w - ~ ' ~ .

Cases la, lb, and lc for the so-called optically transparent sample demonstrate a unique capability of PAS, to wit, the capability of obtaining a depth profile of optical absorption within a sample. That is, by starting at a high chopping frequency we shall obtain optical absorption information from only a layer of material near the surface of the solid. For materials with low thermal diffusivity this layer can be as small as 0 . 1 ~ at chopping frequencies of 10,000-100,O00 Hz. Then by decreasing the chopping frequency, we increase the thermal diffusion length and obtain optical absorption data further within the material, until at - 5 Hz we can obtain data down to 10-100 pm for materials with low thermal diffusivities, and up to 1-10 mm for materials with high thermal diffusivities.

This capability for doing depth profile analysis is quite unique and opens up exciting possibilities in studying layered and amorphous materials, and for determining overlay and thin film thicknesses.

2. Optically Opaque Solids (p, << 1 )

compared to 1, and essentially no light is transmitted. In these cases, most of the light is being absorbed within a distance small


Case 2a: Thermally Thin Solids (ps >> 1 ; p s >> pa) In Eq. (23) we set e-a‘ % - 0, e*usJ z 1, and Irl >> 1. We then obtain

In this case, we have photoacoustic “opaqueness” as well as optical opaqueness, in the sense that our acoustic signal is independent of a. This would be the case of a very black absorber such as carbon-black. The signal is quite strong (it is l/al times as strong as that in case la), depends on the thermal properties of the backing material, and varies as cow’.

Case 2b: Thermally Thick Solids ( p s < I ; ps > pu) In Eq. (23) we set E 0, e-us’ z 0, and Irl > 1. We obtain - a1

(a - 2a, - ia) E - Y

2a,ask,a QE-

Equation (27) is analogous to Eq. (26) but the thermal parameters of the backing are now replaced by those of the solid. Again the acoustic signal is independent of a and varies as co-’.

Case 2c: Thermally Thick Solids (,us << I ; ps < pJ We set eVut z 0, e - ~ J N - 0; and )rI < 1 in Eq. (23). We obtain

- iaY 4a,a,3kS

Q = - (2as - a + ia) z -

This is a very interesting and important case. Optically we are dealing with a very opaque solid (al>> 1). However, as long as ups < 1 (i.e., ps < l/a), this solid is not photoacoustically opaque since, as in case lc, only the light absorbed within the first thermal diffusion length ps will contribute to the acoustic signal. Thus even though this solid is optically opaque, the photoacoustic signal will be proportional to ups. As in case lc, the signal is also dependent on the thermal properties of the solid and varies as wV3/ ’ .

D. General Case

As has been shown in the RG theory developed above, the three sample parameters that play the major role in determining the photoacoustic signal are the sample thickness 1, the optical absorption length pu, and the thermal diffusion length ps. In most experimental situations the values of the relevant parameters are often outside the range of the special cases described above. For these situations, it is imperative to use the exact expression developed in the RG theory. To illustrate the use of the exact expression we present in this section computer-generated plots that give the change in magnitude and

228 ALLAN ROSENCWAIQ

phase of the photoacoustic signal as a function of the chopping frequency f and also as a function of a normalized length L , which we shall define later (47 ) .

1. Dependence on Modulation Frequency

Referring to Eqs. (12)-(14), and noting that k , << k, while a, is usually of the same order of magnitude as a, for air at room temperature and pressure, we can therefore ignore g since g << 1. Also when the density and specific heat of the backing material are not greatly different than that of the sample, we can set

(31)

Furthermore,

Thus Eq. (23) can be rewritten as

where

is a frequency-independent term. It should be noted that the sample-backing combination can also be

regarded as a two-layer or thin-film-substrate sample with the bottom layer or substrate not absorbing any of the incident radiation.

In Figs. 4-6 we show the computer-generated plots derived from Eq. (32) for the magnitude q and phase tj of the photoacoustic pressure Q as a function of chopping frequency f = 27101. In Fig. 4 we consider the optical case of a more or less “transparent” sample or first layer, that is, one in which the optical pathlength p, = l/a >> I (e.g., pa = 101); in Fig. 5 we have the case of an “absorbing” sample or layer, that is, where p, N 1; and in Fig. 6 we have the case of an “opaque” sample or layer, where pa << 1 (e.g., pa = 0.11). In each of Figs. 4-6 we also consider three cases for the parameter b, which, as defined by Eq. (31), can be considered as the root of the ratio of the thermal conductivities for the backing or second layer relative to the sample or first layer. The calculations were performed for b = 0.1, 1.0, and 10.0, that is, for k, /k , = 1, and lo2. To generate Figs. 4-6 we have taken the values 1 N 50 prh, k, N cal/cm-sec-°C, p, N 2 gm/cm3, and C, z 0.2 cal/gm-”C.


The values for k, , p,, and C, are typical of reasonably low thermal conductive solids.

We can see from the log-log plots of the relative magnitude q in Figs. 4-6 that q varies as f for low frequencies and as f - 3 / 2 for high frequencies. The change in slope occurs fairly abruptly in the 100-Hz region for the transparent and absorbing samples and much more gradually in the 1-10- kHz region for the opaque sample. The effect of b on the frequency dependence of q is minimal, being apparent only at low frequencies where the thermal diffusion length is large enough to encompass the backing or second layer. The effect of the backing or second layer on the frequency dependence of the magnitude of the photoacoustic signal is overshadowed, even at low frequencies, by the stronger frequency dependence of the signal generation process within the sample and at the sample-gas interface.

However, the phase of the photoacoustic signal is, as we can see, quite sensitive to the presence of a boundary in such a two-layer system. We note that there is a phase change of 45" as we proceed from low to high frequencies. The shape of the $ vs f curve is dependent on both the optical properties of the sample or front layer (a) and on the relative thermal conductivities of the

FIG. 4. Photoacoustic magnitude q (a) and phase I// (b) vs. chopping frequency f for a transparent sample having pa = 101. From Rosencwaig (47).

b - 1.0 -- b-10.0

lo-.' ' ' ' ' ' ' 85 -

10-1 I 10 Id 103 104 lo6 F n q m s y IHzI

FIG. 5. Photoacoustic magnitude y (a) and phase $ (b) vs. chopping frequency f for an absorbing sample having pe 5 1. From Rosencwaig (47).

1c

Fnqwnsv iHzI

FIG. 6. Photoacoustic magnitude y (a) and phase $ (b) vs. chopping opaque sample having pa = 0.11. From Rosencwaig (47).

frequency f for an


two layers (b). The change in phase is fairly abrupt for the transparent and absorbing samples and occurs in the 1 -100-Hz region. It is more gradual for the opaque sample and occurs mainly in the 0.1-50-kHz region. We note that for all of the optical cases, the II/ vs f curves are sensitive to b only at frequencies I 100 Hz. The explanation for this is that only at low frequencies is the thermal diffusion length in the sample or first layer large enough to extend into the backing or second layer as well. For transparent and absorbing samples, the major change in $ occurs at frequencies where the thermal diffusion length becomes comparable to the thickness I of the sample or first layer. For opaque samples, the major change in $ occurs at frequencies where the thermal diffusion length becomes comparable to the absorption pathlength of the sample or first layer. Therefore, the presence of the second layer is much less of an influence for the opaque sample than for the transparent and absorbing samples.

2. Dependence on Normalized Length

To illustrate more clearly how the photoacoustic signal depends on the magnitude of the thermal diffusion length relative to either the geometrical thickness or to the optical pathlength of the sample, we show in Figs. 7-9

Normlirsd 1

FIG. 7 . Photoacoustic magnitude g (a) and phase $ (b) vs. normalized length L = l/ps for a transparent sample having pm = 101. From Rosencwaig (47).

Normalized L

FIG. 8. Photoacoustic magnitude q (a) and phase (1 (b) vs normalized length L = l /ps for an absorbing sample having p. x 1. From Rosencwaig (47) .

- b- 1.0: --- - b-10.0 : (-I- l o - ' ~

10-12 -

tod b - 0.1 : - b- 1.0 : ---

l o - ' ~ b-10.0 : (-I-

Normalized L

FIG. 9. Photoacoustic magnitude q [a) and phase I) (b) vs normalized length L = pJps for an opaque sample having p., = 0.11. From Rosencwaig (47) .


the dependence of q and I+$ on the normalized length L. We define L = l /ps for the case of the transparent and absorbing samples and L = p a / p s for the case of the opaque sample. We have plotted these curves over six decades of L (equivalent to 12 decades in frequency space) to illustrate that the phase undergoes the 45" change only in the region L z 1.

As in Figs. 4-6, the magnitude q varies rapidly with L , with a dependence of L - 2 for L < 1, and of L - 3 for L > 1. The phase is independent of b for the transparent and absorbing cases when L > 5, that is, when the thermal diffusion length is much smaller than the thickness of the sample or first layer. For the opaque case, L = pa/ps = 0.1 l / p s , and thus is independent of b for L > 0.5. In the region L x 1, the thermal diffusion length becomes comparable to either the thickness I (transparent and absorbing cases), or to the optical absorption length pa (opaque case), and at this point the phase undergoes a 45" change. The slope of the curve in this region and in the region L < 1 is quite sensitive to the thermal properties of the second layer for the transparent and absorbing cases, but not sensitive for the opaque case, since at L x 1, the thermal diffusion length is already much smaller than I in the opaque case.

It is interesting to note that the phase $ is not a monotonic function of L for the case b > 1, that is, for the case when the backing or second layer has a significantly higher thermal conductivity (or k p C product) than the sample or first layer.

E. Nonhomogeneous Samples

Thus far we have dealt only with the photoacoustic effect in homogeneous solid materials. There is, however, considerable interest in applying the photoacoustic effect to layered and nonhomogeneous samples. Layered systems of interest include thin-film electronic materials, samples coated with paints or polymeric films, and multilayered materials such as photographic film. In addition, there are many problems of practical interest in which the sample is thermally homogeneous and the optical absorption varies continuously with depth from the surface. These problems include the characterization of doped semiconducting material, of laser windows whose surfaces have different absorption properties than the bulk, and of biological tissues.

The analysis of Section IV,D applies not only to a homogeneous sample in thermal contact with a different backing material, but also to a two-layer system in which the two layers may have both different thermal and optical properties, and in particular to the case where the lower layer or substrate does not absorb the incident radiation. For example, we can determine the thickness of a film on such a substrate quite accurately if we know something


about the film optical absorption coefficient a, and its thermal conductivity relative to the substrate (b). Knowing the approximate values of M and b, one can determine the film thickness to within 0.1% for a reasonably strong photoacoustic signal (>0.1 mV at the microphone), if one is working with a transparent or absorbing film. Furthermore, by measuring the phase II/ as a function of chopping frequency we can, using, e.g., Figs. 4a, 5a, and 6a, estimate the values of b, and from that obtain values for either the thermal conductivity k or density p if we know the thickness of the film 1. We can also estimate the optical absorption coefficient a at the wavelength of the incident radiation. The capability of evaluating b from the experimental II/ vs. f curves might be of particular interest in monitoring impurity concentrations and structural imperfections in a film, since both of these can alter the thermal conductivity and thus be reflected in the II/ vs. f curves through changes in b.

Although the treatment presented in Section IV,D is a model only for the case of a nonabsorbing substrate (the substrate, however, can be a light- scatterer), the RG theory can be formulated as well for the more general case of two or more layers, all of which might absorb the incident radiation but have different thermal as well as optical properties. The unique capability of PAS to perform depth-profile analysis can then be used to full advantage in the study of multilayered and nonhomogeneous substances.

Afromowitz, Yeh, and Yee have considered the case of a thermally homogeneous, but optically nonhomogeneous system (48). They found that the temperature at the solid-gas interface given in Eq. (1 1) can be represented by the expression

where s2 = iw/P andS(s ) is the single-sided Laplace transfer of H(x) , which H ( x ) is related to the absorption coefficient a(x) by the relation

where R is the reflection coefficient, lo the incident intensity, and ro the efficiency of the nonradiative transition (usually assumed to be 1). By measuring the photoacoustic signal as a function of chopping frequency w, one obtains O(w) and thus one obtains X(s) from Eq. (33). Knowing X(s), one can then invert to obtain H(x) , and then using Eq. (34), one can derive the spatially dependent absorption coefficient a&). In Fig. 10 is shown a test of this treatment. The solid line represents an absorption coefficient that decreases linearly with increasing depth below the surface until the point x =

6, below which the absorption coefficient is a constant value. Simulated photoacoustic data were generated at five chopping frequencies correspond-


I I

FIG. 10. Simulated photoacoustic data were calculated at five chopping frequencies for the spatially varying optical-absorption function shown (solid line). These data were inverted by the procedure described in the text, and the absorption function shown by the points was deduced. From Afromowitz el al. (48).

0 6 26 Dmance #"to smple. x

ing to thermal diffusion lengths having ps = 0.16,0.336,6,3.336, and 106. The points plotted in Fig. 10 represent the results of the data-inversion procedure described by Afromowitz et al. (48). The reasonably good fit to the actual a(x) indicates that this method is quite promising for depth-profile analysis of materials by means of the photoacoustic effect.

Aamodt et al., (44) and Bennett and Forman (45) have also analyzed the production of a photoacoustic signal from solid matter. These authors have used linearized hydrodynamic equations to describe the effects on the gas of the periodic heat flow at the solid-gas interface. In their derivation the acoustic piston effect described in Section IV,B,3 occurs in the form of an acoustic stress term. Bennett and Forman have applied their analysis to the problem of optical absorption in windows used in high-power laser systems. This is a particularly interesting problem in that there appears to be evidence that laser windows may have a quite different optical absorption coefficient at the surface compared to the value of the bulk material. In particular, the optical absorption coefficient at the surface may be considerably higher than the bulk value due to the impregnation of impurities by the polishing processes. Bennett and Forman have suggested that the photoacoustic effect may be used with considerable success to tackle this problem of a nonhomogeneous optical property and to obtain values for both the surface and bulk optical absorption coefficients (49).

F. Experimental Verijication

In this section we consider some of the predictions of the theory of the photoacoustic effect in solids and how these predictions have been borne out by experiment.


One of the most obvious and important predictions of the theory is that the photoacoustic signal is always linearly proportional to the power of the incident photon beam, and that this dependence holds for any sample or cell geometry. This prediction has been found to be fully accurate. In Section IV,C we showed that when the thermal diffusion length in the sample is greater than the optical absorption pathlength (cases 2a and 2b), the photoacoustic signal is independent of the optical absorption coefficient of the sample. In that case, therefore, the only term in Eq. (28) or (29) that is dependent on the wavelength of the incident radiation is the light source intensity I. . Thus, it is clear that the photoacoustic spectrum in the case of a photoacoustically opaque sample (ps > p,) is simply the power spectrum of the light source. This is verified in Fig. 11, where we show the PAS spectrum of a porous carbon-black sample and also a power spectrum of the same source, as seen by the photoacoustic cell, taken with a silicon diode power meter (7). In the wavelength region /2 > 400nm where the silicon diode power meter has a flat wavelength response, we see a one-to-one correspondence between the PAS spectrum and the power spectrum. Unlike the silicon diode power meter, however, a PAS cell containing a porous carbon-black (e.g., a loose wad of completely burned cotton, or a thick layer of carbon-black particles obtained from the incomplete combustion of acetylene) acts as a true light trap with a flat response at all wavelengths. In fact, it is clear from Fig. 11 that one can readily construct a power meter based on the photoacoustic principle that would have a greater wavelength range than other power meters, while maintaining high sensitivity and a fairly large dynamic range. The same power meter could be used from the x-ray range to the far infrared, requiring

CARBON-BLICK

101

ZOO 400 600 800

NANOMETERS

FIG. 11. (a) The photoacoustic spectrum of carbon-black, (b) the power spectrum of the xenon lamp using a silicon diode power meter. From Rosencwaig (7).


only a change of entrance window to permit appropriate transmission of the desired photons into the PAS cell.

The saturation of the photoacoustic signal that is theoretically expected to occur in an optically opaque material when the thermal diffusion length becomes larger than the optical pathlength has been demonstrated by McClelland and Kniseley (50) using fixed chopping frequencies and a variable a by working with aqueous solutions of methylene blue dye. Their results are shown in Fig. 12, where we can see that essentially full saturation is reached for a -2000 cm-', when the chopping frequency is 50 Hz, and for a - 30,000 cm-', when the chopping frequency is 1800 Hz.

lo-" 3 ' , I , - , ' ' ' j " ' , ' ' ' l ' , , , ' , ' ' ' l , " , ' ' ' ' j , ' U J 10-2 100 10' 102 '9 103 '3 104 1 05

1 5 0 ~ 1 1 llamnd ".Em-'

FIG. 12. Photoacoustic signal for methylene blue dye in water. The solid lines represent experimental data. while the dashed lines indicate the theoretically calculated signal. From McClelland and Kniseley (50).

The dynamic range for the absorption coefficient a that can be measured with the photoacoustic effect can be quite large. For example, Monahan and Nolle have shown that a PAS signal will vary significantly with changes in the absorption coefficient a of powdered amorphous As$, from less than 10 cm-' to more than lo4 cm- at a fixed chopping frequency of 510 Hz (51). If the chopping frequency were increased to 5000 Hz, one could measure an a for this material of over lo5 cm-'. At the low end, we can readily measure absorption coefficients in the range of lo-' cm-l or less if care is taken to eliminate background signals due to window and wall absorptions, and to account for diffuse reflection and absorption in powdered samples. Absorp- tion coefficients as low as cm-' have been measured in nonpowdered solids such as intact alkali fluoride crystals by Hordvik and Schlossberg, who used a variation of the photoacoustic technique (52). These authors measured the elastic strains that are produced in a crystalline solid when illuminated with high-power periodic laser light by bonding a piezoelectric transducer directly to the material under study. The sensitivity of this technique is limited only by the amount of radiation scattered directly onto the transducer by impurities and inhomogeneities in the material. Hordvik and Schlossberg estimate that in low light-scattering materials absorption


coefficients as low as cm-’ could be measured with laser powers of about 1 W. The total dynamic range for the measurement of a thus appears to be from to lo5 cm- ’, a range considerably greater than that available with current spectrophotometers.

As the theory of Section IV indicates, the photoacoustic effect is primarily dependent on the relationship between three “length” parameters of the sample: the thickness 1, the optical absorption length pu = l /a, and the thermal diffusion length ps.

In the case of a strongly absorbing material such as fine carbon-black particles 10-3-10-4 cm in diameter, we are dealing with the situation where pm < 1 since a is of the order of lo6 cm-’. At the same time, the thermal diffusion length ,us > 1 for chopping frequencies in the range 50-5000 Hz. In Fig. 13 the log-log plot of the experimental photoacoustic signal with frequency clearly shows the w - behavior predicted by the theory (47).

5’01------ 2.0,

CHOPPING FREOUENCY I H z )

FIG. 13. A log-log plot ofthe photoacoustic signal for carbon-black vs. chopping frequency showing the o- ’ dependence. From Rosencwaig (53).

The theory also predicts that for an opaque material (pu < I ) , the PAS signal will vary as 0-l when ps > pa and as w-3/2 when ps < pa. We have confirmed this prediction with an experiment on a 0.1-cm-thick disk of GaP (53). At a wavelength of 524 nm, o! - 25 cm- ’ for GaP at room temperature. Thus by varying the chopping frequency from 50 to 2000 Hz, the theory predicts that we will move from a region where the dependence is primarily w - l to one where it is o - ~ / ’ . The experimental results shown in Fig. 14 verify this predicted frequency dependence.


1 En Im 2m YY) iwo 2000

Chopping Irqumcv. Hz

FIG. 14. A log-log plot of the photoacoustic signal for a 0.1-cm-thick GaP disk at 524 nm vs. chopping frequency, showing a frequency dependence that varies from close to 0;' at low frequencies to w - 3'2 at high frequencies. From Rosencwaig (53).

Wetsel and McDonald have also shown this dependence in a photoacoustic study of aqueous solutions of phenol red sodium salt (54). In fact, knowing the thermal properties of their sample (mainly water), they were able to obtain values for the absorption coefficient M accurate to within 10% from an analysis of the frequency dependence of the photoacoustic signal. These authors also noted a deviation from the theory at very low chopping frequencies. This deviation can be accounted for by the fact that at very low frequencies the assumption made in the theory that the gas column in the PAS cell is always much larger than the gaseous thermal diffusion length is, of course, no longer valid. More will be said about this later.

There are several other aspects of the theory that have been experimentally verified. For example, the theory predicts that if a sample with ,us > 1 is firmly embedded in a backing of reasonable thermal mass, the photoacoustic signal will be proportional to l(lb/kb, where l(lb is the thermal diffusion length and kb the thermal conductivity of the backing material. However, if we are dealing with fibers, or thin films supported only on their sides, that is,


in situations where the backing is the gas itself, then the photoacoustic signal is proportional to p g / k g , where the subscript g denotes gas parameters. Since p/k for a gas is usually appreciably greater than p / k for a solid, the photoacoustic signal in the second case should be considerably greater than in the first case. We have verified this prediction with experiments in which sus- pended carbon-black samples, such as burned cotton, gave a signal ten times greater than samples of carbon-black particles embedded in double-sided tape mounted on aluminum.

G. Conclusions on the Theory

The theory of the photoacoustic effect in nongaseous materials appears to be quite well developed at this point, at least for the case of homogeneous samples. The problems of layered and nonhomogeneous materials are now receiving at tent ion.

From a study of the theory it is clear that one can use the photoacoustic effect to perform various kinds of measurements on the sample. It is obvious that one can utilize a photoacoustic spectrometer to obtain optical absorption data on any and all types of materials. The data may be qualitative when parameters such as the thermal diffusivity or the geometric dimensions of the sample are not known, and fully quantitative when they are known. In addition, one can, by changing the chopping or modulation frequency, obtain a depth-profile analysis of the optical properties of a material. At high chopping frequencies, information about the sample near the surface is obtained, while at low chopping frequencies the data come from deeper within the sample. This is a feature unique to the photoacoustic technique. Another unique capability lies in the ability to obtain optical absorption data on completely opaque materials, provided that one can operate at a chopping frequency high enough so that the thermal diffusion length is smaller than the optical pathlength. The bulk of the present work in PAS is concerned with those types of experiments done to determine the optical absorption properties of materials. Such experiments have been most fruitful, yielding valuable spectroscopic data on inorganic, organic, and biological systems, data that could not be readily obtained by more conventional techniques.

There are, in addition, two other classes of experiments that are of considerable interest and for which the photoacoustic technique is uniquely suited. It is possible to obtain information about the thermal conductivity of a material, by measuring the thermal diffusion length ps through knowledge of 1 and 01. The thermal conductivity is an important physical parameter that is often very difficult to measure, particularly on powders, amorphous materials, and biological samples. Finally, one can measure 1, through a


knowledge of o[ and ps, or by keeping ct constant and changing ps in a controlled fashion, or vice versa. Such an experiment can be extremely useful for measuring the thickness of thin films on substrates that possess different optical or thermal properties than the films. Measurements of the thermal and geometric parameters of materials could well become a most important practical application of the photoacoustic effect in industry.

v. THEORY OF THE PHOTOACOUSTIC EFFECT IN LIQUIDS

There has been considerable interest in the possibility of using the photoacoustic technique for investigating optical absorption processes in liquids. This interest is stimulated by two problems that cannot be effectively managed by conventional spectrophotometry, the accurate measurement of a weakly absorbing solution, and analysis of highly light-scattering liquid systems such as suspensions. It is at present quite difficult to measure absorption coefficients in solutions where the absorption coefficient is much less than

cm- '. It is also most difficult to obtain reliable absorption spectra when dealing with highly light-scattering liquid suspensions. The problem of dealing with low-absorption solutions has been partially resolved by the use of fluorescence techniques, and in fact it has been shown that laser fluorescence detection combined with chromatographic separation can yield a detection limit of - 7.5 x 10' molecules/cm3 of rhodamine 6G, or an effective absorption coefficient of .V lo-' cm-' (55). However, in trace analysis the analytical procedure often involves the use of a nonfluorescent, highly absorbing indicator for the substance to be measured. Here fluorimetry is not possible, and conventional spectrophotometric analysis permits a sensitivity of only 10- 3-10-4 cm- '. The problem of highly light-scattering solutions is not at all well resolved at present.

A. Gas-Microphone Coupling

There are two ways by which a photoacoustic signal can be obtained from an illuminated solution. In the first method, the solution is itself a sample, much as a solid material, in a conventional photoacoustic cell. The periodic heat generated within the solution by optical absorption processes diffuses to the liquid-gas interface, perturbs the gas within a boundary layer or so of the interface, and creates an acoustic disturbance detected by a conventional gas microphone. The theory for this effect is identical to that developed in Section IV, with the appropriate liquid parameters inserted where the sample parameters are needed. Table I lists for example, some of these parameters,


such as the thermal diffusivity and heat conductivity, for water and other liquids.

Several investigators have studied optical absorption in solutions by this technique (50,54,56). In general, these investigators have found that photoacoustic signals are readily detectable from solutions with optical absorption coefficients as low as 0.1 cm-'.

B. L iquid-Microphone Coupling

In order to achieve a higher sensitivity for photoacoustic measurements in liquids, some researchers have investigated the possibility of using a piezoelectric transducer that couples directly to the thermoelastic waves produced in the liquid. Condenser-type gas microphones are not suitable for measurements of pressure changes in liquids since they are sensitive primarily to the extent of motion of the microphone membrane. Since the amplitude of motion in an elastic medium is given by

AX = Ap/2npv

where Ap is the pressure fluctuation, p the density of the fluid, and v the frequency of the pressure fluctuation, it is readily seen that Ax for liquids is - smaller than in gases for the same pressure fluctuation.

Thermoelastic waves generated in liquids by the absorption of electromagnetic energy are well known from Bell's original work [13]. Laser sources for the electromagnetic radiation were first applied by R. M. White (57) and later by others using pulsed lasers (58,59). More recently CW laser beams have been used with quite good results (60).

The analysis given below makes use of potential functions for both the liquid and a piezoelectric ceramic transducer assumed to be a cylinder as in Fig. 15, with appropriate continuity conditions between the two media

FIG. 15. The cylindrical piezoelectric transducer used in a liquid photoacoustic cell. From Kohanzadeh et al. (60).


(60). The strain calculated in the cylinder yields the voltage induced across it through the piezoelectric coefficients. We shall neglect the coupling between thermal and acoustical effects, since the coupling term is usually negligibly small. Also the thermal effects die out rapidly in the liquid and can usually be neglected in a typical experiment of this sort.

We treat the liquid as an inviscid fluid and the ceramic cylinder of Fig. 15 as an isotropic elastic solid. Thermal energy is supplied by absorption of a collimated optical beam, directed along the z axis and is given analytically by the expression

(35)

where a is the optical absorption coefficient of the liquid at the wavelength of the photon radiation, w the chopping frequency, and l ( r ) the function describing the intensity radial dependence of the optical beam. If we assume that the beam is of constant intensity across the diameter of the ceramic cylinder, and if for the sake of simplicity we assume a << 1, then

H(r, z, t ) = al,(r)e-azfiWt

H(r , z, t ) N H(t ) = aloeiWt (36)

Following Nowacki (64, the displacement u, stress tensor 0, and temperature T in the solid are expressed in terms of a scalar potential Q, and a vector potential 'Y. In the fluid, the corresponding quantities are expressed in terms of a scalar potential only. The vector potential for the solid indicates the presence of shearing due to reflections at the cylindrical boundaries. Formulas for u, 0, and f are

u = vo + VXY

The scalar and vector potentials satisfy the equations

[(v - ; ;)( v2 - $31 CD = --& PH

The material constants p, p , cl, c 2 , P, and C are different for the solid and the fluid. For the solid, p, p, cl, and c2 denote shear modulus, density, longitudinal velocity, and shear velocity, respectively, and for the fluid p = c2 = 0 and c1 is simply the velocity of sound. The constant C is the


specific heat, while p and x are defined as

p = 3Ka~/pct , x = U/pc (39)

where K is the bulk modulus, aT the linear expansion coefficient, and IC

the thermal conductivity. The potentials that satisfy Eqs. (38a) and (38b) will have the form

@(t) = Aeio', w(t) = Beiwte (40)

where e is the azimuthal unit vector. For the cylindrical configuration considered here, the pressure in the ceramic will be given by

p = -grr = p J @ = po2Aeia' (41)

Solving Eq. (38a) we find that

A = (3KaT/p2CU3)d0

Thus the pressure p is given by

p = ( 3 K a T / p ~ C ) ~ l o (43)

In Table 11 we list the values for K, tlT, C , and p for a number of typical liquids. Also we show the product 3KuT/poC for a chopping frequency of 100 Hz. We note that water is not the best liquid for these types of measurements, while chloroform and carbon tetrachloride appear quite good.

It is of interest to compare Eq. (43) with the analogous equation for the photoacoustic effect in solids. The analogous situation would be one in which the material is the gas itself and we are dealing with case 1 in Section IV,C :

We note that Eqs. (43) and (44) are very similar. In both situations the signal varies as w-l and linearly with the absorbed energy al,. Both expressions also vary as (pC)- ' , where the parameters are for the gas in the case of Eq. (44) and for the liquid in the case of Eq. (43).

C. Experimental Verification

There has as yet been very little work reported on the photoacoustic effect in liquids, where a liquid-coupling piezoelectric transducer or crystal microphone was used. Nevertheless, not only is the technique feasible but as Lahmann et al. (62) have shown, it is highly practical, especially in the case of a weakly absorbing solution.

TABLE I1

Density p Specific heat C Linear expansion aT Bulk modulus K ~ K ~ T I P C Liquid (gm/cm3) (cal/gm-"C) ( T - 1 (dynes/cm2) (dynes-cm/cal)

Water LOO0 1.00 6.7 x 10-5 1.9 x 104 3.8 Ethyl alcohol 0.192 0.60 3.7 x 10-4 1.0 x 104 23.3 Glycerine 1.26 0.54 1.7 x 10-4 4.5 x lo4 33.7 Chloroform 1.527 0.23 4.2 x 10-4 9.9 x 103 35.5 Carbon tetrachloride 1.550 0.20 5.1 x 10-4 9.5 103 46.9


In Fig. 16 we show a schematic of their experimental arrangement. In a photoacoustic experiment on a weakly absorbing species in a solvent, there is often a strong background signal from absorption in the solvent itself. To overcome this problem, Lahmann et al. separated the output of an argon ion laser operating in the "all lines mode" by the prism P1 into its spectral components. A perforated plate S transmits the two most powerful beams at 488 and 514 nm and screens off all the others. The two beams are reunited with the lens L and the second prism P2, and the recombined beam is directed into the sample cell. A chopping wheel, situated after the lens L, interrupts the two beams, such that the laser beam striking the sample cell has a slightly varying intensity at twice the chopping frequency but with a switching wavelength. The laser was operated at 700 mW power and chopped at 700 Hz.

FIG. 16. Schematic diagram for the experimental arrangement used in the liquid photoacoustic experiment described in the text. From Lahmann et al. (62).

This dual-wavelength operation enables a substantial reduction of the background signal, which is considerable even for solvents having absorption coefficients as low as cm-'. By working with solvents that have a weak and fairly flat absorption profile in the wavelength region used, one can reduce the background signal down to a residual absorption coefficient in the neighborhood of 10- cm- '. This then permits the detection of a solute concentration so low that the net absorption coefficient differential between the wavelength regions used is only cm-'. In the experiment of Lahmann et al., a detection limit of about 0.08 ng/cm3 of /?-carotene (2.2 x cm-') was found. They also obtained a detection limit of 15 ng/cm3 for selenium in chloroform corresponding to an absorption coefficient of 3.5 x cm-'.

Thus the photoacoustic technique can be used to extend the sensitivity of absorption measurements of solutions to absorption coefficients as low as cm- ' and even higher sensitivities can be obtained at lower chopping

or


frequencies and higher laser powers. Although still not as sensitive as the fluorimetric method, the photoacoustic technique can be used for any absorbing compound and is not limited to those possessing fluorescence.

VI. EXPERIMENTAL METHODOLOGY

In PAS of solids, the sample is placed inside a specially designed closed cell containing air (or other suitable gas) and a sensitive microphone. The sample is then illuminated with chopped monochromatic light as shown in Fig. 17. If the sample absorbs any of the energy incident on it, some energy level in the sample has thus been excited, and this energy level must subsequently deexcite. The most common mode of deexcitation is through the nonradiative or heating mode, and thus the periodic optical excitation of the sample results in a periodic heat flow from the sample to the surrounding gas. This in turn results in a periodic pressure oscillation within the cell, which is detected by the microphone as an acoustic signal. In obtaining a photoacoustic spectrum, one records the microphone signal as a function of the wavelength of light incident on the sample. As we showed in Section IV, the strength of the acoustic signal is proportional to the amount of light absorbed by the sample, and thus there is a close correspondence between a photoacoustic spectrum and a conventional optical absorption spectrum. Furthermore, since only absorbed light can produce an acoustic signal, scattered light, which often presents a serious problem in conventional spectroscopy, presents no serious problem in PAS.

n

FIG. 17. Block diagram of a single-beam photoacoustic spectrometer. From Rosencwaig (53).

As with other forms of spectroscopy, a photoacoustic spectrometer is composed of three main parts: a source of incoming radiation, the experimental chamber, and the data acquisition system.


A. Radiation Sources

The most common and most economical sources of optical radiation in the ultraviolet, visible, and infrared regions are provided by conventional light generators. These are the arc lamp for the UV-visible, the incandescent lamp for the visible and near infrared, and the glow bar for the mid- to far- infrared regions. Since all three light sources provide a broadband optical radiation, they must be used in conjunction with suitable monochromators. From Section IV, we know that the signal :noise ratio in photoacoustic spectroscopy increases linearly with the light intensity impinging on the sample. Thus, it is advantageous to use an intense light source and a high light throughout (i.e., low f-number) monochromator.

The light source that we have commonly used with our photoacoustic spectrometer is a 1000-W Hanovia xenon arc lamp (Model 961-C). The monochromator used was an J3.5 1/4-m Jarrell-Ash Ebert monochromator (Model 82-410) equipped with two gratings, one blazed at 300 nm for use in the UV, and the other blazed at 500 nm for use in the visible. This monochromator was generally used with 2-mm slits, giving a resolution of - 3 nm in the UV and -6 nm in the visible, a resolution that is adequate for most nongaseous samples. With the 2-mm slits, the average power of the chopped beam at 550 nm was -2 mW/cmz as measured with a silicon diode radiometer.

Since conventional light sources generally operate in a continuous mode, a light chopper, usually electromechanical in nature, must be used. This chopper can be located before or after the monochromator. In the theory of the photoacoustic effect in solids, we have shown that optical absorption spectra on completely opaque sample can be obtained, and that absorption vs. depth studies can be performed, provided one is able to adjust the chopping frequency. Thus, for optimum versatility, a variable-speed chopper is to be recommended. We have often used the variable-speed electromechanical chopper produced by Princeton Applied Research (Model 192).

Alternatively, the lamp intensity can be modulated directly through modulation of the lamp power supply. This is possible with only a few lamps, such as the Varian VIX xenon lamps, which can operate at relatively low power levels because of their built-in parabolic reflectors. Electronic modulation has considerable advantages over electromechanical modulation since it can provide fairly good modulation (up to 80%) over a modulation frequency range of 0.1 Hz to 10 kHz without acoustic noise or vibration. However, the current electronically modulated lamp systems still suffer from considerable arc jitter and low usable lifetimes. The electromechanical choppers, on the other hand, are used with lamps running in a stable long- life DC mode. However, electromechanical choppers can cover only the


range 5 Hz to 5 kHz, and introduce considerable acoustic noise and vibration, particularly at the higher frequencies.

Another source of optical radiation that can be used in PAS of solids is the laser. A laser requires no monochromator and if operated in a pulsed mode would also require no chopper. In the visible wavelength region, dye lasers provide an intense highly monochromatic light readily tunable over a fairly large wavelength range. Dye lasers can also be used with reasonable intensity in the uv region with the aid of frequency-doubling crystals. Scann- ing with dye lasers in the uv region is, however, tedious at present. In the infrared there are as yet no continuously tunable lasers that cover a wide spectral range, although if the experiment can be performed over a narrow wavelength region, then a discrete infrared laser (e.g., the CO and C 0 2 lasers) or a tunable spin-flip Raman laser can be used to great advantage to provide intense, highly monochromatic radiation.

B. Experimental Chamber

The experimental chamber is the section containing the photoacoustic cell or cells, and all the required optics. The actual design of this chamber will vary, depending on whether one is using a single-beam system employing only one photoacoustic cell or a double-beam system containing two cells, with appropriate beam-splitting optics. The photoacoustic cell will generally incorporate a suitable microphone with its preamplifier. Both a conventional condenser microphone with external biasing and an electret microphone with internal self-biasing provided from a charged electret foil are good microphones to use.

Some criteria governing the actual design of the photoacoustic cell are

(a) acoustic isolation from the outside world, (b) minimization of extraneous photoacoustic signal arising from the

interaction of the light beam with the walls, windows, and the microphone in the cell,

(c) microphone configuration, (d) means for maximizing the acoustic signal within the cell, (e) the requirements set by the samples to be studied and the type of

Let us now consider the above criteria in more detail.

(a) We have found that the problem of acoustic isolation is not particularly serious providing one uses lock-in detection methods for analyzing the microphone signal. One should, of course use chopping frequencies different from those present in the acoustic and vibrational spectrum of the

experiments to be performed.


environment. In addition, the cell should be designed with good acoustic seals and with walls of sufficient thickness to form a good acoustic barrier. Some reasonable precautions to isolate the photoacoustic cell from room vibrations should also be taken.

(b) To minimize any photoacoustic signal that may arise from the interaction of the light beam with the walls and windows of the cell, one should employ windows as optically transparent as possible for the wavelength region of interest, and construct the body of the cell out of polished aluminum or stainless steel. Although the aluminum or stainless steel walls will absorb some of the incident and scattered radiation, the resultant photoacoustic signal will be quite weak as long as the thermal mass of these walls is large. A large thermal mass results in a small temperature rise at the surface and thus a small photoacoustic signal. In addition, one should keep all inside surfaces clean to minimize photoacoustic signals from surface contaminants. One should also design the cell so as to minimize the amount of scattered light that can reach the microphone diaphragm.

(c) Various microphone configurations can be used. We have found that both cylindrical microphones (Fig. 18) and flat microphones (Fig. 19) can be readily used. Cylindrical microphones have the advantage of being easy to construct and have a large surface area, thereby increasing their sensitivity. A disadvantage is that they do not usually possess a flat frequency response over a large acoustic range. This can be troublesome if one is planning to do experiments at different chopping frequencies. Flat microphones are commercially available, are quite sensitive when of reasonable size ($in. diameter or larger) and good quality, and possess a flat frequency response over a wide acoustic range. A good example of such a microphone is the General Radio Electret Microphone, Model 1961-9601.

The microphone chamber used for the cell depicted in Fig. 18 was 1 in. in diameter and 5 in. long. The electret foil was 0.001 in. Teflon@ sheet with

L -

ELECTRET

FIG. 18. A simple photoacoustic cell employing a cylindrical electret microphone. From Rosencwaig (53).


FIG. 19. A simple photoacoustic cell employing a Rat microphone. From Rosencwaig (53).

a lo00 A aluminum coating on one side. The charge density was - 10 nC/ cm’. (These foils were supplied by J. E. West of Bell Laboratories.) The cylindrical chamber was machined to have a large number of closely spaced holes, and small circumferential groves of -0.001 in. high on the outside wall upon which the electret foil rested. Because of the rather excessive volume of the cell, the signal away from mechanical resonance of the electret foil was quite low. At resonance however (-400 Hz), the signal was quite high, giving an output of - 1 mV at 500 nm when a carbon-black absorber was used in the cell. The Q at resonance was measured to be -30.

The electret microphone used in the cell of Fig. 19 was made at Bell Laboratories according to the design specified by Sessler and West (63). These microphones have a rated sensitivity of - 1 mV/pbar, with a flat frequency response from 50 Hz to 15 KHz.

For both the cylindrical and flat microphone, we used an Ithaco low- noise 40-dB preamplifier (Model 144). The signal from the preamplifier was then fed into a Princeton Applied Research lock-in amplifier (Model 129A). The cell shown in Fig. 19 had a volume of - 5 cm3. With a carbon-black absorber, the signal at 100 Hz and 550 nm for the incident light was - 1 mV. The signa1:noise ratio for the carbon-black absorber was - 1000: 1. A study of the noise indicated that it was mainly due to electronic noise, primarily from the microphone-preamplifier system. This noise level is several orders of magnitude greater than the photoacoustic theoretical limit imposed by Brownian motion (25).

(d) Since the signal in a photoacoustic cell used for solid samples varies inversely with the gas volume as shown in Eq. (23), one should attempt to minimize the gas volume. However, one must take care not to minimize this volume to the point that the acoustic signal produced at the sample suffers appreciable dissipation to the cell window and walls before reaching the microphone.

The distance between the sample and the cell window should always be greater than the thermal diffusion length of the gas, since as shown in


Section IV it is this boundary layer of gas that acts as an acoustic piston generating the signal in the cell. When the column of gas in front of the sample becomes comparable to the gaseous thermal diffusion length, then Eq. (6) has to be modified to include the growing exponential term in the expression for the region 0 < x < 1,. If one performs the subsequent calculations with this modification, one obtains an expression for the photoacoustic signal that increases monotonically as the ratio of lg /pg decreases toward 1, and then decreases monotonically as IJp, decreases below 1. This effect has been observed by Wetsel and McDonald (54).

This dependence also has been demonstrated by Aamodt et al. (44). They showed that when the gas column in front of the sample is greater than the gaseous thermal diffusion length p,, the PAS signal strength increases with decreasing gas column length l,, reaching a maximum at x = l , /p , 7 1. For cells shorter than the gaseous thermal diffusion length, the PAS signal decreases with decreasing length and is dependent on the thermal properties of the entrance window.

Aamodt et al. performed experiments at various chopping frequencies as a function of gas column length 1, for both helium and nitrogen gases. Their results are summarized in Figs. 20 and 21, where universal curves are plotted

Helium filled cell experimental

20

m 1

ThDoratical 0 = 0 027 ---- B - 0 30 P o 9 - 0 1 6 0 . 1

0.01 0.1 1 .o 10

x -q L. ml/2.54

FIG. 20. Universal experimental curves for PAS magnitude and phase vs. normalized gas length X = ls/pgfor helium. From Aamodt ei al. (44).

253

x =Giq L. cml2.54

FIG. 21, Universal experimental curves for PAS magnitude and phase vs. normalized gas length X = IJpg for N,. From Aamodt et a / . (44 )

using X = l g /pg and q’ = wq1l2, where q is the magnitude of the PAS signal and X the ratio of the gas column length to the gaseous thermal diffusion length. These curves also show the dependence of the phase angle $ as a function of X . From these curves, we note that the normalized PAS signal q‘ peaks at X x 1.4, and that the phase begins to decrease rapidly as X decreases below 1.4.

Thus it is clear that one should keep I , > pLs for all chopping frequencies of interest. Since pg K 1/0”~ we must consider the lowest chopping frequencies that we are likely to employ. For air at room temperature and pressure, the thermal diffusion length is - 0.06 cm at a chopping frequency of 10 Hz. A reasonable value for 1, would then be about 0.1 cm.

In the design of a suitable photoacoustic cell, one must take thermoviscous damping into account as well, since this could be a source of significant signal dissipation to the cell boundaries. Thermoviscous damping results in a e - t x damping, where E is a damping coefficient given by Kinder and Frey (64)


where d is the closest dimension between cell boundaries, as in a passageway, u the sound velocity, w the frequency, po the gas density, and re an effective viscosity that is dependent on both the ordinary viscosity and the thermal conductivity of the gas. Again, for air at room temperature and pressure, one finds negligible thermoviscous damping at 100 Hz as long as d > 0.01 cm. It should be noted, however, that whereas the thermal diffusion length varies as 1/0’/~, the thermoviscous damping coefficient varies as wl”. Thus, while the thermal diffusion length is the predominant parameter at low frequencies, the thermoviscous term is predominant at high frequencies. A cell that is designed to be used over a wide range of frequencies, while capable of handling a number of different gases, should then have a minimum distance between the sample and window, and minimum passageway dimensions, of 1-2 mm.

One can further enhance the acoustic signal in the photoacoustic cell by a number of means. For example, if one must work at only one frequency, one can take advantage of the nonflat frequency response of a cylindrical microphone and work at the frequency of its peak sensitivity. Another way is by making use of an acoustically resonant cell section with a length equal to n1/2, where n is an integer and I the acoustical wavelength. In order to achieve such a resonance effect without making the cell gas volume too large, one can construct a cell similar to that shown in Fig. 22. Here the sample is placed in a nonresonant section of the cell, with a resonant section of reasonably small cross-sectional area joining the sample section with the microphone. One can, in addition, place a suitable diaphragm between the nonresonant and resonant sections, and between the resonant section and the microphone, thereby allowing, for example, different gases or different gas pressures to be used in the separate sections. In the use of resonant cell designs, however, one limits oneself to a fixed chopping frequency.

FIG. 22. A photoacoustic cell with an acoustically resonant section. From Rosencwaig (53).

Other methods to enhance the acoustic signal include the use of gases with a higher thermal conductivity, and the use of both higher gas pressures and lower gas temperatures. The theory shows that the photoacoustic signal


varies in most cases as k:I2PAI2/TO, where k, is the gas thermal conductivity, Po the pressure, and To the temperature of the gas. All these methods will increase the photoacoustic signal without limiting the choice of chopping frequencies.

(e) How closely one adheres to the above criteria will, of course, depend on the type of sample used (powder, smear, liquid, etc.), its size, and the type of experiment one wishes to perform (low temperature, high temperature, etc.).

C. Data Acquisition

The tasks of acquiring, storing, and displaying the data can be performed in many ways. However, certain basic procedures should be followed. For example, the signal from the microphone preamplifier should be processed by an amplifier tuned to the chopping frequency in order to maximize signal :noise. If phase as well as signal amplitude is desired, or if very weak signals are to be measured, then a phase-sensitive lock-in amplifier should be used.

For a single-beam spectrometer, provisions must generally be made to remove from the photoacoustic spectrum any spectral structure due to the lamp, monochromator, and optics of the system. This normalization can be conveniently done by digitizing the analog signal from the tuned amplifier, and then performing a point-by-point normalization (i.e., division) with either a power meter reading or a previously recorded photoacoustic spectrum obtained with a black absorber.

In a double-beam spectrometer, normalization can be performed in analog real-time fashion by dividing the analog output from the tuned amplifier processing the sample signal, with the output derived from a reference signal. This reference output may be from a power meter or from a second photoacoustic cell containing a black absorber.

With regard to the storage and display of the data, there are of course many possible schemes, ranging from the relatively inexpensive chart recorder to the sophisticated minicomputer.

Although a double-beam spectrometer has been constructed, most of the experiments reported to date have been performed on a single-beam spectrometer. The data are often stored in digital fashion in either a multichannel analyzer or a minicomputer, and normalization is done using a previously recorded carbon-black spectrum. We have found such a system to be quite adequate, since lamps such as the Hanovia lamp used are fairly stable and show only a gradual change in spectral output with time. Updating the carbon-black spectrum once a month is usually sufficient. Furthermore, the digital data acquisition system permits one to perform various data


analyses and plotting routines, and to do multiscan experiments when dealing with very weak signals.

D. Commercial Photoacoustic Spectrometers

At this time, there are two commercial photoacoustic spectrometers available. The first is manufactured by Gilford Instrument Laboratories of Oberlin, Ohio, and the second by Princeton Applied Research of Princeton, New Jersey. Both spectrometers are versatile, general-purpose instruments using conventional xenon irridation sources.

The Gilford instrument (R-1500 PAS SPECTROMETER) is a double- beam instrument employing two photoacoustic cells. It can be used with high efficiency from about 200 nm in the UV to about 3000 nm in the near infrared since it utilizes four gratings. The use of two photoacoustic cells allows this instrument to be used for both conventional normalized PAS spectroscopy, in which porous carbon-black is used in the reference cell, and also for differential spectrometry, in which samples are placed in both cells. Furthermore, the Gilford instrument comes with either an electromechanical chopping system or an electronic lamp mudulation system.

The Princeton Applied Research (PAR) instrument (Model 6000) is a single-beam instrument employing only one cell. Normalization is accomplished by diverting a small fraction of the monochromatized beam into a conventional power meter. Although this method is quite suitable for normalization in the visible wavelength region, it is probably inadequate in the UV since most wide-band conventional radiometers do not have a truly flat wavelength response over the entire wavelength region of 200-3000 nm. The PAR instrument employs only one grating at this time, limiting its useful wavelength range to 350-800 nm. The PAR instrument can only be obtained at present with an electromechanical chopping system, and unlike the Gilford instrument only a few discrete chopping frequencies are available.

In general, although both instruments are quite good and competitive in price, the Gilford R-1500 appears to be the more versatile and superior spectrometer.

VII. PHOTOACOUSTIC SPECTROSCOPY IN PHYSICS AND CHEMISTRY

In spite of the fact that PAS of solids has been actively pursued at this time for only a few years, it has already shown itself to be a very useful research and analytical tool in the fields of physics and chemistry. In general, the applications of PAS in these fields can be divided into three main areas: bulk optical, surface, and deexcitation studies.


A. Inorganic Insulators

When the surface of a solid material is not highly reflective, PAS will provide optical data about the bulk material itself. The PAS technique can thus be used to study insulator, semiconductor, and even metallic systems that cannot be studied readily by conventional absorption or reflection techniques, e.g., substances that are in the form of powers, are amorphous, or for some reason are difficult to prepare for reflection studies.

In the case of insulators, photoacoustic spectra give direct information about the optical absorption bands in the material. This is illustrated in Fig. 23. Spectrum (a) shows the normalized PAS spectrum of some Cr203 powder in the 200-1000 nm region. Spectrum (b) is an optical absorption spectrum obtained by McClure (65) on a 4-pm thick Cr,O, crystal. Spectrum (c) is a diffuse reflectance spectrum of Cr203 powder obtained by Tandon and Gupta (66). The two crystal field bands of the Cr3+ ion at 600 and 460 nm are almost as clearly resolved in the photoacoustic spectrum of the Cr203

t.

c 12-

I I 100 400 700 1000

NANOMETERS

FIG. 23. (a) The normalized photoacoustic spectrum of Cr,03 powder at 300°K. (b) The optical transmission spectrum of a 4.4-pm thick crystal of Cr,O, at 300°K. (c) The diffuse reflectance spectrum of Cr,O, powder at 300°K. From Rosencwaig (7)


powder as they are in the absorption spectrum of the Cr,03 crystal, and they are much better resolved in the PAS spectrum than in the diffuse reflectance spectrum.

Figure 24 shows the photoacoustic spectrum of another inorganic insulator, CoFz. Again, this material was in the form of a highly light- scattering powder. We note that both the crystal field bands of the Co2+ as well as the charge-transfer band are clearly visible.

CoFl

Gharge- transfer

band /

Crystal field IweIs f o r d + in

octahedral environment

Crystal field IweIs f o r d + in

octahedral environment

I I I

2.0 3.0 4.0 GV

D

FIG. 24. The photoacoustic spectrum of CoF, powder at 300°K showing both crystal-field and charge-transfer bands.

B. Inorganic and Organic Semiconductors

In the case of semiconductors, both direct and indirect band transitions can be observed as shown in Figs. 25 and 26. In Fig. 25, results are shown for three direct-band semiconductors, all in powder form. The band edge as measured by the position of the knee in the PAS spectra agrees very well with the values recorded in the literature and given in parentheses (67). Figure 26 shows the PAS spectrum of the indirect-band semiconductor Gap, also in

> t z W t

z W > - t u

I

I I I I

2.0 3.0 4 .O 5.0 I FV

0

FIG. 25. Photoacoustic spectra of three direct-band semiconductors in powder form at 300°K. The band-gaps derived from these spectra are shown and compared to the values derived from specular reflectance measurements (in parentheses). From Rosencwaig (5,9)

UNDOPED GOP

r 1.0 2.0 3 0 4 0 5.0

PHOTON ENERGY ev

3

FIG. 26. Photoacoustic spectrum of the indirect-band semiconductor GaP at 300°K. The shallower slope is a result of the indirect-band transitions. From Rosencwaig (5).


the form of a coarse powder. Here again, the band edge value for the forbidden direct-band transition obtained from the PAS measurement agrees remark- ably well with the literature value (68). The close agreement is surprising since one would expect a lower value for the knee due to opacity effects, arising from the strong indirect-band transitions in this semiconductor. It is these indirect-band transitions that give rise to the more gradual slope of the absorption edge seen in Fig. 26.

Several points concerning these spectra should be made: (1) The PAS technique gives the correct spectrum for both direct (CdS) and indirect (Gap) bandgap semiconductors. In fact, Somoano has shown that for GaP both the indirect-band gap transition at 562 nm and the direct-band gap transition at 470 nm are revealed (69). (2) The samples can consist of powders taken off the shelf without further purification, or single crystals that are ground up to reduce the reflectivity and increase the surface area. (3) The sample mass need only be a few milligrams. (4) Each spectrum is obtained in only a few minutes. These points emphasize the ease and convenience with which data of importance for the optical and electrical properties of semiconductors may be obtained without the requirements of high-purification or high- vacuum techniques. In addition, one can differentiate between different crystalline phases of the same material, such as the rutile and anatase phases of TiOz (70). By utilizing PAS at low temperatures, it should be possible to detect excitonic structure as well as the occurrence of phase transitions. The above semiconductors are also of interest for use as electrodes in photo- electrochemical cells to convert solar energy into electrical and chemical energy. Thus, PAS provides a convenient tool for the physicist or chemist to quickly screen, characterize, and correlate the optical properties of numerous semiconductors for use in new systems.

Organic and organometallic semiconductors may also be investigated using PAS. These materials are of interest in the field ofquasi-one-dimensional (1D) conductors. The compounds consist of weakly coupled chains of molecular units and may exhibit very high electrical conductivity parallel to the chains. The compounds may be metallic if the atoms or molecules are uniformly spaced along the chain. However, quasi-one-dimensional metallic chains are fundamentally unstable with respect to certain lattice distortions (Peierls’ instability). Thus, in some compounds the chains will distort in such a manner that the atoms or molecules form dimers, trimers, tetramers, etc. This distortion, or transition, can occur at temperature above or below room temperature. The compounds with distorted chains are semiconductors since the nonuniform spacing of the molecules along the chain opens up a gap at the Fermi level in the electronic energy spectrum. Thus, the electrical and magnetic properties are crucially dependent upon the chain structure. The compounds are usually darkly colored powders obtained as precipitates from


chemical reactions. Single crystals, which are difficult to obtain, are often in the form of very small needles (a few millimeters long) making conventional optical studies very difficult.

An example of a study of 1D compounds is illustrated in Figs. 27 and 28. This is a photoacoustic study (71) of a series of iridium carbonyl compounds that contain either the semiconducting linear chains of square-planar ci~-[1r(CO)~X,] or the nonchain complex [Ir(CO),X,]-, where X is chlorine or bromine. The photoacoustic spectra seen in Fig. 27 show three absorption bands below 650 nm at 2.3, 2.9, and - 3.4 eV. These bands are assigned as metal-to-ligand charge-transfer transitions from the a( yz) and b(xz) metal orbitals to the predominantly ligand CO b(sl*, 6p,) orbital. At wavelengths longer than 650 nm, the spectrum of the semiconducting materials rises strongly toward the infrared region. This rise is the high-energy end of a

4

800 f0 200 400 600 nrn

FIG. 27. Photoacoustic spectra of “as made” samples of (a) Ko,981r(C0)2CI,,,, 0.2 CH,COCH,; (b) Ko.,,Ir(C0)2CI,~0.5 H 2 0 ; (c) (TTF),.6,1r(CO)2CI,; (d) Cs, ,,Ir(CO),Br2; (e) (C,H,),As[lr(C0)2CId. From Rosencwaig et al. (7/).


200 400 600 800 f01 nm

FIG. 28. Photoacoustic spectra of crushed samples of (a) Ko,981r(CO)2C12,42; (b) (TTF)o,,lIr(C0)2C12; (c) Cso,611r(CO)2Br2. These spectra show the increased infrared absorption due to smaller chain lengths. From Rosencwaig et al. (71).

broad absorption band extending from -0.1 to 2 eV, as subsequently was shown by conventional infrared transmission spectroscopy. This band has been assigned as a transition from the 5d,2 band to b(n*, 6pJ. As can be seen in Fig. 27, the nonchain complex (e) does not have this infrared band. In addition, all of the observed linear-chain transitions seen in Fig. 27 are considerably red-shifted with respect to the corresponding transitions in nonchain [Ir(CO)2C12] -. This red shift is attributed to interactions along the chain that raise the energy of a( yz) , b(xz), and a@’), while lowering the energy of b(n*, 6p,).

The results shown in Fig. 28 indicate that the width and energy maximum of the infrared transition depend upon the chain length, becoming broader and shifting to higher energy as the chains are shortened by the process of crushing the samples. Observation of this effect gives us a valuable means for determining chain lengths, and thus of correlating chain lengths with observed electronic transport properties.

Another example of the use of PAS in the study of organometallic compounds is illustrated in Fig. 29, which shows the PAS spectrum of an interesting rhodium “bridged” dimer compound (72), in which the rhodium atoms


I 1 I I

I I I 700 600 500 400 301

A (nm)

FIG. 29. Photoacoustic spectra of several rhodium organometallic compounds: (a) Rho- dium "bridged dimer, (b) Rhodium (phenylisocyanate), tetraphenylborate, [Rh(+-CN),]B+,. The absorption band at 575 nm in (a) and (b) is associated with the formation of dimers, (c) Rhodium (vinylisocyanate), perchlorate, [Rh(VI-CN),]CIO,. From Somoano (69).

are physically bound into a dimeric structure by bridging ligands. The bridged dimers are not part of a chain structure in this compound. However, the interesting feature is the presence of the strong absorption band at 575 nm, which is due to the dimeric structure. A similar band is found in the solution spectrum. Figure 29, shows PAS spectra of two rhodium chain compounds. The rhodium atoms are square-planar coordinated by isocyanate ligands (R-CN) and stack to form rhodium chains, in contrast to the rhodium bridged dimer. Nevertheless, the PAS spectra of rhodium (phenylisocyanate), tetraphenylborate (Fig. 29b) reveals the same strong rhodium-rhodium dimer band at 575 nm, indicating a dimeric chain structure. The room temperature electrical conductivity of this material is quite low (cRT = lo-'' Q-') as would be expected for the nonuniform (dimeric) spacing along the chain. In contrast, the PAS spectrum of rhodium (vinylisocyanate), perchlorate (Fig. 29c) does not reveal any sign of the dimer, suggesting that this material contains chains of uniformly spaced rhodium atoms. Indeed,


the electrical conducitivity of this compound (oRT z 2 Q-'-crn-') is the highest known for any rhodium chain complex (73). All of the structural information deduced from the PAS spectra of Fig. 29 has been fully confirmed by single-crystal x-ray diffraction studies (73).

In summary, PAS provides a very simple technique for gaining insight into the structural and electrical properties of quasi-1D conductors without having to grow single crystals or perform x-ray measurements. In this way, many new potentially conducting compounds may be screened without the expenditure of excessive time and funds.

In addition to the above experiments, PAS can also be used at cryogenic temperatures and at higher resolution (e.g., with the use of lasers) to study excitonic and other fine structure in crystalline, powder, or amorphous semiconductors, and thus to investigate in these materials the effects of impurities, dopants, and electromagnetic fields.

C. Metals

PAS may be used to study metals if the reflectivity is first diminished by grinding or through the use of powders. Because of the very large absorption coefficients of metals, the absorption spectrum cannot be obtained by PAS. However, the presence of the metal (e.g., on a surface) may be detected by its loss of reflectance, which may result from interband transitions, a plasma resonance, etc. Figure 30 shows the PAS spectra of copper and silver powder. The sharp structure in the PAS spectra as indicated by the arrows occurs at the point of reflection loss and can serve as a signature for the presence of a particular metal. The structure in these PAS spectra also emphasizes the care one must take in the choice of materials for use in the construction of a PAS cell (see Section VI).

As in the case of semiconductors, a major advantage of PAS over conventional reflection spectroscopy is that highly reflecting surfaces are not needed and in fact are undesirable for bulk studies. Furthermore, the need for ultraclean surfaces and thus the need for the elaborate high-vacuum equipment, so necessary for conventional reflection studies, is not present for photoacoustic spectroscopy.

D. Liquid Crystals

Another class of materials of considerable interest to both physicists and chemists is the organic liquid crystals (74). Because the solid, smectic, and nematic states of liquid crystals are highly light-scattering, optical spectro- copy on these states cannot be readily performed. Nevertheless, optical data on these states may be very useful in providing information about the inter-


t c_

z w c

z w Ag POWDER

\

0 ev

FIG. 30. Photoacoustic spectra of copper and silver powders. The arrows indicate positions of known structure in the reflection loss spectra of these metals. From Rosencwaig ( 5 )

molecular interactions that play so important a role in determining the unsual properties of these materials.

In Fig. 31a we show data from a photoacoustic study (9) of a class of liquid crystals. Since all members of this class are composed of molecules with essentially the same benzylidene aniline center, but with different heads and tails, the general features of the UV PAS spectra are quite similar. How- ever, there is a clear shift in the position of the primary absorption edge as we proceed from MBBA to CBOOA. This shift is likely due to the different heads and tails, and thus is probably a measure of the different intermolecular interactions in these crystals. A correlation between the spectral data and the intermolecular interactions is clearly seen in Fig. 31b, where we have plotted the temperatures at which these materials crystallize and at which they go into their isotropic liquid state along the y axis, and the value 1 / ~ , where E is the position of the primary absorption edge, along the x axis. Thus, photoacoustic data on these liquid crystals can be used to obtain information about the intermolecular interactions in these crystals, and therefore enhance our understanding of the unusual properties of these compounds.


LIQUID CRYSTALS (a 1 h-[@ C H m N G1-t

400- ( b l

360

' 340-

320-

300 -

280

- Y

-

028 050 032 034 036 I/E lev-')

FIG. 31. A study of five liquid crystals with a benzylidine aniline center section. Their UV PAS spectra are shown in (a). In (b) the temperatures at which these crystals undergo transitions to their isotropic form, and the crystal form, are plotted as functions of the reciprocal absorption-edge energies. From Rosencwaig (9).

E. Catalysis and Chemical Reaction Studies

A very important area where PAS is beginning to be applied is in the study of catalysis and chemical reactions.

In Fig. 32a we show data obtained from an experiment on the inorganic insulator system CoMoO,. This experiment was performed in hope of obtaining further understanding of the hydrodesulfurization catalytic action of CoMoO, supported on alumina (75). Both the high-temperature P-CoMoO, and low-temperature cr-CoMoO, phases are available only as fine precipitates, and thus their optical spectra are not readily obtainable by conventional techniques. The photoacoustic spectra shown in Fig. 32a indicate that the p-CoMoO, has a charge-transfer band similar to that seen in the parent Moo3, while the charge-transfer band in the a-CoMoO, has shifted noticeably to lower energy. In Fig. 32b we have studied in more detail


nm (0)

I I

400 500 600 70 nm j b)

FIG. 32. A photoacoustic study of cobalt molybdate. (a) Spectra of the low-temperature alpha and the high-temperature beta forms of CoMoO,. (b) Spectra of the Coz+ crystal-field bands on hydrated cobalt sulfate (octahedral cubic), the two forms of cobalt molybdate, and cobalt aluminate (tetrahedral). From Rosencwaig (9).

the crystal-field bands of the Co2+ ions in CoSO,.H,O (octahedral coordination) in p- and a-CoMoO,, and in CoAl,O, (tetrahedral coordination). These PAS spectra indicate that the Cot+ ions in both the p- and CI-COMOO~ are in a distorted octahedral coordination, and that there is no significant difference in the Co2+ d electron configuration between the two CoMoO, phases. This then implies that the known difference in catalytic activity between the a and /? phases (the a phase is catalytically more active than the p phase) cannot be attributed solely to differences in the localized 3d election configuration.

However, Fig. 32a does provide another possible explanation for this difference. We note that the charge-transfer band edge of the a-CoMoO, is at a considerably lower energy than that of the P-CoMoO,. If this were the only significant difference between the two phases, then on these data alone we would predict that the a-CoMoO, would be catalytically more active than the P-CoMoO,, since it requires less energy to excite electrons into the “mobile” charge-transfer state in a-CoMo0, than in fi-CoMoO,. Since this prediction is in agreement with experimental data, this photoacoustic study indicates that the possibility of catalytic differences arising from differences in ligand-electron configuration should be investigated further.


Another example in the field of catalysis involves the investigation of the reaction of transition metal complexes with polymeric ligands to form anchored catalysts, which have been used to catalyze hydrogenation and hydroformylation of olefins. PAS has been used to investigate the electronic structure of these metal-polymer complexes in order to elucidate chemical processes and structure-reactivity relationships. An example of a PAS study of a model catalytic system involving a well-characterized compound, tungsten hexacarbonyl W(CO),, is shown in Fig. 33.

The PAS spectrum of W(CO), is shown both before and after photochemical reaction with polyvinylpyridine (69). Figure 33a shows the PAS spectrum of W(CO), and reveals the singlet-triplet ligand field transition at 350 nm as well as the corresponding singlet-singlet transition at 310 nm (76). Figure 33b shows the PAS spectrum of W(CO)5L, where the ligand L is NH3. This compound is formed upon irradiation of W(CO), in the presence of the ligand and is shown to reveal the effect of ligand substitution in W(CO), . The characteristic singlet-triplet and singlet-singlet transitions at 457 and 416 nm, respectively, of W(CO),NH3 are observed in the form of a broad absorption band from 375 to 475 nm. Figure 33c shows the PAS

I I I I

FIG. 33. Photoacoustic spectra of tungsten hexacarbonyl, W(CO)6 complexes. (a) W(CO),; (b) W(C0)5NH3; (c) W(C0); polyvinylpyridine. The complexes in (b) and (c) are photoproducts. From Somoano (69).


spectrum of the photoproduct of W(CO)6 and polyvinylpyridine and clearly reveals the W(CO),-pyridine absorption band similar to that seen in Fig. 33b. The polymer-anchored W(CO)5 species has been observed to catalyze olefin isomerization and hydrogenation. Thus, PAS may be used to study and characterize solid-state photoproducts where conventional optical and structural (ie., x-ray) techniques would be totally inadequate.

An example of a less well understood, but potentially useful polymer- anchored catalyst involves the transition metal cluster complex, tetrarhodium dodecacarbonyl, Rh4(CO),z. Figure 34 shows the photoacoustic spectra of (a) Rh,(CO)lz, (b) the photoproduct of the reaction of Rh4(C0),* solution with polyvinylpyridine, and (c) the thermal product [i.e., the same as (b) but

FIG. 34. Photoacoustic spectra of tetrarhodium dodecacarbonyl, Rh,(CO),, , polymer complexes. (a) Rh,(C0),2, (b) the photoproduct of the reaction of Rh4(C0)12 solution with polyvinylpyridine, and (c) same as (b) but not irradiated with UV light (i.e., a thermal product). From Somoano (69).


not exposed to UV radiation] (69). Both the polymer-metal complexes (b) and (c) are found to be hydroformylation catalysts, but the phototriggered catalyst (b) is considerably more active. This extra activity is associated with the absorption band (shoulder) at 600 nm in Fig. 34b since the less active thermal catalyst lacks this absorption. Upon prolong exposure to air, the phototriggered catalyst decomposes as indicated by the loss of the 600 nm absorption. Its PAS spectrum then resembles that of the thermal catalyst and exhibits correspondingly less catalytic activity (77).

Another use of PAS in this field has been to detect the degree of reduction of metals reacted on polymeric substrates. By using PAS to monitor the reaction of, say, chloroplatinate (K,PtCl,) with polyvinylpyridine, one may easily tell when the chloroplatinate has been fully reduced to platinum metal. The use of PAS in catalysis studies is just starting, but there is every indication that it will prove to be a valuable tool.

PAS has also been used for the study of chemically treated controlled- pore glasses. These materials are growing in importance in the fields of chemical synthesis and analysis. Functional groups such as metal chelation reagents, acid-base indicators, and enzymes can be attached to the surface of the glass and used in a variety of chemical reactions. However, techniques for definitive characterization of the attached functional groups are rather primitive and ineffective. The photoacoustic technique has been used to determine the presence of these groups and to monitor subsequent chemical reactions and identify photochemical degradation processes.

It is quite clear that the applications of PAS in the fields of catalysis and chemical reactions has significant potential. For example, it is possible to construct a reaction chamber that is also a photoacoustic cell. Even in the presence of an on-going exothermic reaction, PAS spectra can be obtained at a solid-gas interface, and thus dynamic studies of catalysis and other chemical reactions can be performed. Similar studies could also be conducted on solid-liquid interfaces using a liquid photoacoustic cell.

F . Surface Studies

PAS can be used to great advantage in the study of adsorbed and chemisorbed molecular species and compounds on the surface of metals, semiconductors, and even insulators. Such studies can be performed at any wavelength, provided the substrate one is dealing with is either nonabsorbing or highly reflecting at this wavelength. Under either of these conditions, the PAS experiments will give the optical absorption spectra of the adsorbed or chemisorbed compounds.

Our first indication of the feasibility for using PAS for surface studies came from an experiment performed with thin-layer chromatography (TLC)


(10). TLC is a widely used and highly effective technique for the separation of mixtures into their constituent components (78). This technique is of considerable importance in the chemical, biological, and medical fields. The identification of the TLC-separated compounds directly on the TLC plates can, however, be a fairly difficult procedure, particularly if reagent chemistry is inappropriate. Conventional spectroscopic techniques are unsuitable because of the opacity and light-scattering properties of the silica gel adsorbent on the TLC plates. PAS offers a simple and highly sensitive means for performing nondestructive compound identification directly on the TLC plates.

In Fig. 35a we show the PAS spectra taken on five different compounds that were separated and developed on TLC plates. These PAS spectra were taken directly on the plates themselves and were run in the ultraviolet region

PHOTOACOUSTIC SPECTRA ON TLC PLATES

Od - ,r

0.8 -

I

a 0 8 - $

[ 08:

0.8 - f "

0.4 -

UV ABSORPTION SPECTRA - FIG. 35. A photoacoustic study on thin-layer chromatography. In (a) are shown the UV

PAS spectra of five compounds, p-nitroaniline (I), benzylidene-acetone (11), salicylaldehyde (111), 1-tetralone (IV), and fluorenone (V). These spectra were taken directly on the thin-layer chromatography plates. In (b) are shown the UV absorption spectra of the same five compounds in solution. From Rosencwaig and Hall (10).

272

1.0

0.6

2 5 0.8- n U

L z U W ? 5 0.4

1

0.6

0.2-

ALLAN ROSENCWAIG

-

-

- (,

-

of 200-400 nm. The five compounds starting from the top of Fig. 35 are p-nitroaniline, benzylidene acetone, salicylaldehyde, 1-tetralone, and fluorenone. In Fig. 35b we show, for the sake of comparison, the published UV absorption spectra of these compounds in solution (79). The strong similarity between the photoacoustic spectra and the optical absorption spectra permits a rapid and unambiguous identification of the compounds.

To test the sensitivity of the PAS technique in this application we performed the experiment shown in Fig. 36. Here we have the PAS spectra taken directly on a TLC plate on which benzylidene acetone spots of different concentrations were developed. The spectra were taken on spots developed from starting drops containing - 10, 1, and 0.1 pg of benzylidene acetone. The spectra were taken on the developed spots, and these can be expected to have an even smaller amount of material than the starting drops. Never- theless, we note that even for the case of 0.1 pg, the main absorption band of the benzylidene acetone is visible. Knowing the amount of material present

I

xy) 1111 300

nm

i, m

FIG. 36. Photoacoustic spectra on spots of benzylidene acetone on a TLC plate. The spots were developed from starting drops containing 9.49, 1.11, and 0.095 pg of benzylidene acetone. From Rosencwaig and Hall (10).


in this spot ( < 1 pg), the size of the spot (-0.3 cm2 in area), and the molecular weight of the compound (146), we estimate that we have roughly no more than one monolayer of the compound in this spot.

This experiment thus indicates the possibility that under certain conditions (low optical absorption or high reflectivity) of the substrate, the PAS technique may well be sensitive enough to detect and identify a monolayer of adsorbed or chemisorbed compound. Other experiments on both metallic and nonmetallic surfaces have shown that monolayer detactability is achiev- able. The use of PAS for such surface studies, particularly if used with high- resolution light sources such as tunable dye or infrared lasers, can lead to fundamental understandings of surface oxidation and reduction processes under a variety of conditions, and also to further knowledge about catalytic activity on solid surfaces. Other surface studies would include PAS studies of organic compounds and inorganic oxides purposely deposited on the surfaces of metals, semiconductors, and polymers for purposes of passivation. Such studies would yield data about the structure, valence, complexing, etc., of the deposited compound, information that is at present very difficult to obtain nondestructively.

An example of the use of PAS in surface passivation studies of metals (9) is shown in Fig. 37. Spectrum (a) is of a copper surface that has been treated with benzotriazole, a known passivating agent. Spectrum (bj is of an identical but untreated copper surface. Spectrum (a) differs markedly from spectrum (b) only in the wavelength region below 300 nm. Spectrum (c) is the difference spectrum produced by subtracting spectrum (b) from spectrum (a). Spectrum (c) thus represents the spectrum of the chemisorbed monolayer of benzotriazole. Spectrum (d) is a PAS spectrum of benzotriazole powder. We note that spectrum (c) is quite different than spectrum (d), thus indicating that the chemisorbed benzotriazole has undergone significant structural change. It may be possible to establish, from such spectra, what changes have occurred to the benzotriazole upon chemisorption onto the copper surface, and thus to understand how this compound passivates the surface of copper metal.

G. Deexcitation Studies

The photoacoustic effect measures the nonradiative deexcitation processes that occur in a system after it has been optically excited, or in more general terms, excited by any electromagnetic radiation, or in fact by any means whatsoever. This selective sensitivity of the PAS technique to the lionradiative deexcitation channel can be used to great advantage in the study of fluorescent (or phosphorescent) materials, and in the study of photosensitive substances.


FIG. 37. A PAS surface study. Spectrum (a) is a photoacoustic spectrum of a highly polished copper surface treated with benzotriazole (BTA). Spectrum (b) is of an untreated surface. Spectrum (c) is the difference spectrum, showing the absorption bands of the monolayer of BTA. Spectrum (d) is of bulk benzotriazole powder. From Rosencwaig (5).

1. Fluorescent Studies

When an optically excited energy level decays via fluorescence or phosphorescence, then little or no acoustic signal will be produced in the photoacoustic cell. This is illustrated in Fig. 38, where we consider the case of the fluorescent solid Ho,O, (5,9). Several of the trivalent rare earth ions, such as Ho3+, have strongly fluorescent energy levels, that is, levels that tend to deexcite through the emission of a photon, rather than through phoiion or heat excitation. The upper PAS spectrum is of Hoz03 powder containing cobalt and fluorine impurities. All of the lines present in this spectrum correspond


1.0 4 2 .o 3.0 4.0 5.0

0 V

FIG. 38. Photoacoustic study of a fluorescent material. Cobalt and fluorine impurities quench the natural fluorescence of the Ho3+ ions in holmium oxide here studied by PAS. The upper spectrum is of doped, and the lower spectrum of undoped, holmium oxide. The fluorescent levels are marked with dots. From Rosencwaig (9).

to known Ho3+ energy levels, whose positions are designated by the bars below. The dots indicate which of these levels are normally fluorescent. In this material the fluorescence is highly quenched by the presence of the cobalt and fluorine impurities, and thus both the fluorescent and nonfluorescent lines appear in the PAS spectrum. The lower spectrum is of pure Ho,O,. Here all of the fluorescent levels have a greatly diminished relative intensity, since these levels are now deexciting through the emission of a photon rather than through heating of the solid.

A more illustrative example of the potential of photoacoustic spectroscopy in such studies has been reported by Merkle and Powell (11). They have used PAS to study the radiationless decay processes between the excited states of Eu2+ ions in KCl crystals.

Figure 39a shows the absorption spectrum of KCl:Eu2+ at room temperature. The two strong, broad absorption bands are attributed to transitions from the lowest Stark component of the *S,,,(4f') ground state to the eB and tZg components of the 4f65d configuration, with the former being at higher energy. The structure on these bands (which is more easily observed

276

A

ALLAhr ROSENCWAIG

t 1 3WO 4ow

h IBI

FIG. 39. (a) Optical absorption (solid line) and excitation spectra monitored at 4350 A flashed line) for KCI:Eu*+ at 300°K. (b) Photoacoustic spectra for two different phase shifts; solid line is 0" phase, dashed line is 45" phase. From Merkle and Powell (21).

at low temperatures) (80 -82) is due to the electrostatic interaction between the d and f electrons and spin-orbit interaction of the latter. The splitting of the bands indicates the cubic crystal field strength for the d electron is about lODq = 12000 cm- ', whereas the strength of the electrostatic and spin-orbit effects is found from the fine structure to be on the order of -5000 cm-'. These are dipole allowed transitions and the radiative decay times for the reciprocal emission transitions from these excited states can be calculated (83) to be on the order of 1 psec for both the t2g and eB excited states.

The fluorescence spectrum at room temperature consists of a broad band peaked at approximately 23800 cm- ', which represents a Stokes shift of - 6000 cm- ' from the lowest energy absorption band (84). The fluorescence decay time at 12 K is found to be about 1.3 psec and the decay pattern is observed to be purely exponential with no measurable rise time. These results


are essentially independent of the wavelength of excitation. The measured fluorescence decay time at low temperatures is close to the predicted radiative decay time.

Figure 39a also shows the excitation spectrum at room temperature. Both absorption bands appear but their relative intensities are quite different. Excitation in the high-energy band leads to a smaller amount of radiative emission than excitation in the low-energy band.

The obvious conclusion to be drawn from these optical spectroscopy results is that the e, level has two different types of decay channels. The dominant one is total radiationless relaxation to the ground state and the secondary one is a multiphonon transition to the t,, level, which fluoresces to the ground state. Both of these processes must take place on a time scale much faster than the -1 psec predicted radiative decay time since no radiative emission is detected from this level.

PAS provides a method for directly monitoring the amount of energy dissipated through radiationless transitions. Figure 39b shows the results of PAS measurements made on this system. The experimental setup is similar to that described in Section VI.

Both absorption bands appear in the PAS results. The phase angle 8 at which the signal is maximum is related to the lifetime of the state through the expression z = tan 8/27cv,, where v, is the chopping frequency. The signal from the low-energy band is maximum for approximately zero shift in phase from that of the exciting light, which indicates that the decay time of the level is much less than 100 ysec. This is consistent with the measured fluorescence decay time. The peak intensity of the PAS signal in the high-energy band occurs at a phase shift of -35", which implies that the lifetime of the state is of the order of milliseconds. It is obvious that this cannot be the radiationless decay time of the e, level since the radiative decay time is much faster and no radiation is observed from this level.

The model energy level diagram shown in Fig. 40 is proposed to explain the apparent discrepancy between the PAS and optical spectroscopy results. For simplicity the electrostatic and spin-orbit splittings of the 4f 9 d levels are not shown and only one manifold of excited 4f7 levels is pictured. The photoacoustic signal at phase angle 8 is simply the sum over all the relaxation transitions that generate heat.

For the transition model shown in Fig. 40 Merkle and Powell estimate that the t,, PAS signal arises from two sources: those electrons undergoing excited relaxation within the t,, band and those undergoing radiationless and vibronic relaxation to the ground state. The e, PAS signal is somewhat more complicated. It has a contribution due to all of the electrons relaxing within the e, band. Another contribution comes from those electrons that relax to the tZg band and subsequently to the ground state. A third contribution arises


FIG. 40. Proposed model for excited state relaxation of E d + in KCI. From Merkle and Powell (11).

from those electrons that relax from the e, band to the 4f energy level and thence relax radiationlessly to the ground state.

There are thus two excited + ground state radiationless transition times. The one that goes from the t2, to ground (or e, 4 tZg 3 ground) is much faster than 0.1 psec since no radiative emission is detected from this level. The PAS signal that arises from these transitions are at 0" phase shift. The other main relaxation channel is the eg 4 4f7 + ground state. Absorption, fluorescence, and excitation spectra along with lifetime data can be used to predict the observed relative PAS intensity ratios Ieg / I t2s for different values of the phase angle. This analysis gives a value of the phase angle for the 4f7 -, ground transition of about 66", in reasonable agreement with experiment. This value implies a 4f lifetime of 3.6 msec.

The transition model predicted by Merkle and Powell, although rather qualitative, provides an explanation for the apparent discrepancies between optical and photoacoustic data. It thus appears that the dynamics of excited- state relaxation in KCl : Eu2+ are more complicated than previously thought, and that PAS techniques provide a method for elucidating some of the


characteristics of the relaxation processes that cannot be observed by conventional optical means.

The above experiments indicate the type of investigations that can be performed on fluorescent and phosphorescent materials with the photoacoustic technique. A combination of conventional fluorescence spectroscopy and PAS can provide data about both the radiative and nonradiative deexcitation processes within these solids. That is, the complete deexcitation process within these compounds can now be readily studied for the first time. By performing both fluorescence spectroscopy and PAS as a function of temperature and compound composition, one can determine in a straightforward manner how these two variables affect the efficiencies and rates for the two deexcitation processes. Furthermore, since PAS gives phase as well as amplitude information, one can study exciton processes (random walk, energy level lifetimes, etc.) in these materials as a function of temperature and dopant concentration.

2. Photochemical Studies

Another channel of deexcitation for absorbed light energy in some compounds is through photochemistry. PAS offers a unique tool for the study of photochemical processes in solids. An illustration of this is shown in Fig. 41, where we have performed PAS experiments on the photosensitive material

200 400 600 800 nrn

FIG. 41. Photoacoustic study of a photosensitive material. The Cooper blue was dark- adapted before the lower spectrum was made. The middle spectrum was run afterward, and then the upper spectrum, with only a few minutes separating the runs. From Rosencwaig (9).


Cooper blue (2,2-dimethy1-4-phenyl-6-p-nitrophenyl-l, 3-diazabicyc10[3.1.0] hex-3-ene) (9). This compound is colorless in the dark but turns a strong blue when exposed to light of short wavelength. The bottom spectrum of Fig. 41 is the PAS spectrum of dark-adapted Cooper blue. There is substantial UV absorption but little visible absorption. The middle spectrum was obtained immediately after the first and is quite different, showing two strong absorption bands in the visible. These are the bands that give Cooper blue its blue color. These bands arise from a photochemical change in Cooper blue wherein some of the photons absorbed in the short wavelength region have been utilized to break a ring in the Cooper blue molecule and thus create a new compound. The upper spectrum, run immediately after the middle spectrum, shows yet further changes, reflecting further photochemical and even photoinduced thermochemical processes.

Not only can one readily see the effects of photochemistry by means of PAS, but one can also establish the activation spectrum for the photochemical process directly, by simply comparing the PAS spectrum with a conventional absorption spectrum. Information about the activation spectrum of photosensitive materials is at present quite difficult to obtain by other means. In addition, one can obtain, from the phase measurements of the photoacoustic signal, data about photochemical reaction rates, and can even distinguish between true photochemical events and photoinduced thermochemical events. Photoacoustic studies on photosensitive materials will not only provide valuable basic information about the physical and chemical processes in these materials, but can also be of great benefit in the understanding of technologically important compounds such as photoresists, and in the study of photoinduced physical and chemical changes in polymers, plastics, and pigments.

H. Conclusions

Spectroscopy is the prime occupation of most physicists and chemists. It is therefore not surprising that the photoacoustic effect has been used already in a wide variety of physical and chemical studies, in which it has proved its worth. It will undoubtedly become a familiar tool in physical and chemical laboratories in the near future, not only to extend the power of spectroscopic analysis to hitherto unmanageable materials, but also to be used with all materials for its own truly unique capabilities.

VIII. PHOTOACOUSTIC SPECTROSCOPY IN BIOLOGY

One of the most promising areas for the use of the photoacoustic method is in the study of biological systems, for here most of the materials to be


studied are often in a form that makes them difficult if not impossible to study by any other optical technique. Although many biological materials occur naturally in a soluble state, many others are membrane bound or part of insoluble bone or tissue structure. These materials are insoluble and function biologically within a more or less solid matrix. Optical data on these materials are usually difficult to obtain by conventional techniques, since they are generally not in a suitable state for conventional transmission spectroscopy, and if solubilized are often significantly altered. PAS, through its capability of providing optical data on intact biological matter, even on matter that is optically opaque, holds great promise as both a research and diagnostic tool in biology and medicine.

A. Hemoproteins

To illustrate the capabilities of PAS in biology we consider an experiment on cytochrome-C (a), with the results shown in Fig. 42. The hemoprotein cytochrome-C plays an essential role in cellular respiration and has been extensively studied (85). Since it is readily soluble in water, its optical absorption spectrum is well known. Figure 42a shows the optical absorption

CVTOCHROYE -i lOULOUS SOLUIION

CVTOCMROME-C POWDER

L REDUCE0

I 1 400 600

nm

( b l

FIG. 42. A study of cytochrome-C. (a) The optical absorption spectra of oxidized and reduced cytochrome-C in aqueous solution. (b) The photoacoustic spectra of oxidized and reduced cytochrome-C in solid (powder) form. From Rosencwaig (8). Copyright 1973 by the American Association for the Advancement of Science.


spectrum of the oxidized and reduced forms of cytochrome-C in aqueous solution obtained with a conventional spectrophotometer. Figure 42b shows the PAS spectra of oxidized and reduced cytochrome-C in lyophilized or powder form.

We note that the photoacoustic spectra are qualitatively very similar to the optical absorption spectra. In particular, all of the differences between

FIG. 43. Photoacoustic spectra of smears of whole blood, red blood cells, and hemoglobin. All three spectra clearly show the band structure of oxyhemoglobin. From Rosencwaig (8). Copyright 1973 by the American Association for the Advancement of Science.


the oxidized and reduced forms visible in the absorption spectra are also visible in the photoacoustic spectra. This experiment indicates that it is possible with PAS to obtain spectra on biological material in solid form that are comparable to those obtained in solution. This capability becomes extremely important when the compound to be studied is insoluble.

The oxygen-carrying protein of red blood cells, hemoglobin (86) also displays a strong and characteristic heme absorption spectrum. Conventional absorption spectroscopy on whole blood, even when diluted in a suitable buffer, does not produce satisfactory results. The inadequacy of the conventional technique arises from the strong light-scattering properties of whole blood, due primarily to the presence of the other protein and lipid material in the plasma and the red blood cells. In the conventional process, one first extracts the hemoglobin from the whole blood by centrifuge techniques and then obtains an absorption spectrum of an aqueous solution of the extracted hemoglobin. The PAS spectrum of a smear of whole blood in Fig. 43 exhibits the characteristic spectrum of oxyhemoglobin as clearly as in the PAS spectrum of the red blood cells and even of the extracted hemoglobin itself (8). The presence of the other protein and lipid material in whole blood, which causes so much difficulty in conventional spectroscopy from light scattering, creates no problem in PAS. Thus, it is now possible to study the hemoglobin directly in whole blood, that is, in situ, without resort to extraction procedures.

B. Plant Matter

The unique capabilities of PAS enable it to be used to obtain optical absorption data on even more complex biological systems such as green leaf and other plant matter. A PAS spectrum on an intact green leaf in Fig. 44 clearly shows all of the optical characteristics of the chloroplasts in the leaf matter-the Soret band at 420 nm, the carotenoid band structure between 450 and 550 nm, and the chlorophyll band between 600 and 700 nm. PAS can thus be used to readily study intact and even living plant matter, and thus obtain valuable information about normal and abnormal plant processes and pathology.

Since we can study intact plant matter, it is clear that PAS can be used as a quick and efficient screening tool in natural products chemistry (5). Using the PAS spectrum obtained from only a few milligrams of a natural source, such as a plant, animal, or microorganism, one can readily determine the types and relative concentrations of secondary metabolites present, that is, the chemical by-products of the organism’s metabolism, compounds, which may prove to have important physiological or biomedicinal value. At present such information can be arrived at only after laborious extraction and analysis procedures, requiring many grams of the natural source.

FIG. 45.

I I I I 200 400 600 800

FIG. 44. A photoacoustic spectrum of an intact green leaf.

nrn

E

w 200 /I 2 WRVILLEAE \=

200 400 600 800 NANOMETERS

Photoacoustic spectra obtained on a few milligrams of dried marine algae.


We have illustrated the applications of PAS in natural products chemistry with a study on marine algae, a promising source of new and unusual natural products (87). The photoacoustic spectra shown in Fig. 45 were taken on the algae after they had been air-dried, using less than 10 mg of material in each case (5). Although there is a considerable amount of spectral detail in all the PAS spectra, we were mainly interested in the region around 320 nm, a region where the sought-after aromatic or highly conjugated secondary metabolites could be expected to have an absorption band. The PAS spectra indicated that only two of the algae would yield any quantity of these aromatic compounds. Conventional extraction and analysis procedures performed by S. S . Hall and his colleagues at Rutgers University have fully confirmed the photoacoustic data. It is clear therefore that PAS can provide, in minutes, information about the compounds present in complex biological systems, and that it is capable of doing this nondestructively, requiring only milligrams of materia1 and no special sample preparation.

Another illustration of the use of PAS in biology is in the study of marine phytoplankton (88) illustrated in Figs. 46 and 47.

-4 T

s

3 e ' 3

9 2

G c, CZ

P

i2

>

1

0

500 600 700 000 WAVELENGTH f n m l

FIG. 46. Photoacoustic spectra of representative marine phytoplankton. From Ortner and Rosencwaig (88).


100 em 100

nm

FIG. 47. Absorption spectra of representative marine phytoplankton. From Ortner and Rosencwaig (88).

Biochemical characteristics of the algae are being increasingly considered by phytoplankton taxonomists (89). Although complications abound (90), photosynthetic pigment composition appears to be a phylogenetically conservative feature (91). Further, since photosynthetic carbon fixation by microalgae supports the ocean's food chain, the specific photochemical mechanisms by which these cells absorb and transfer light energy are of evident ecological import. Three methods of pigment analysis have been widely employed: chromatography, fluorometry, and spectrophotometry. In general, the resolution and sensitivity of analyses of intact cells has been considerably lower than analysis using solutions of extracted pigment samples (91). PAS can, however, be effectively used for analysis of intact cells.

In PAS the signal depends not only on the amount of absorbed optical energy, but also on fluorescent reemission, photochemical transfer, phosphorescence, and thermal deexcitation. Since the PAS signal is a function of thermal deexcitation alone, a PAS spectrum would be closely analogous to an absorption spectrum if fluorescence, photochemical transfer, and phosphorescence were either small or constant fractions of the totallight absorbed.

In this study the spectra of five phylogenetically disparate algal species were determined in replicate by PAS and by conventional spectophotometric


analysis. Monospecific cultures were obtained of Coccolitus huxleyii, Thalas- siosira pseudonana, Dunaliella tertiolecta, Pyraminmonas sp., and Platymonas sp. Cells from these cultures were then collected on Whatman glass fiber filters and then examined by both PAS and spectrophotometric methods.

In comparing Figs. 46 and 47, it appears that the PAS spectra have more detail than do the absorption spectra. All three chlorophycaen exhibit distinctly separable chlorophyll b peaks at 662 nm (Fig. 46,3-5). Chlorophyll c peaks at 645 nm are clear in both the bacillariophycaen and the hap- tophycaen (Fig. 46, 1-2). Coccolitus and Thalassiosira exhibit similar but unidentified peaks at 595-600 nm (Fig. 46,l-2). These features are relatively more distinct in the PAS spectra (compare Fig. 46 with Fig. 47). Platymonas appears to possess a structured chlorophyll a peak at 683-687 nm (Fig. 46,5). Coccolithus and Pyramimonas have sharp unidentified peaks at 518 nm (Fig. 46, 2-31. These features are not observed in the absorption spectra (Fig. 47). On the other hand, the prominent, but unidentified 620 nm absorption peak exhibited by Dunaliella (Fig. 47,4) is not readily discernible in the PAS spectrum (Fig. 46,4). Overall, PAS spectra roughly parallel absorption spectra. This result is not unexpected since, according to Sauer and Park (92), even physically ioslated chloroplast fractions, whose photochemical transfer pathways have been chemically blocked, have relatively low fluorescence yields. This result implies that thermal deexcitation rather than fluorescence is preferred when photochemistry is inhibited. Yet, although the absolute energetic shunt represented by fluorescence is rather small, it is suggestive that the fluorescent efficiencies of the different chlorophylls, at least in acetone, are very different (93). If this were true in intracellular pigment systems, it would account for some of the observed differences in the PAS and the absorption spectra. For example, if it were true in whole cells, as it is in acetone, that the fluorescent quantum efficiency of chlorophyll a was much greater than the fluorescent quantum efficiency of chlorophyll b (.024 versus 0.09 (93)), the PAS signal of the larger chlorophyll a peak would be relatively smaller resulting in greater separation from the overlapping chlorophyll b peak. Accurate knowledge of the fluorescent activation spectra of intracellular pigments might also explain the wavelength shifts, relative to spectrophotometry, of the principle chlorophyll peaks identified by PAS. It is equally plausible that the observed spectral differences result from differential heat transfer by the different pigment molecules since they have very different intracellular microenvironments.

These experiments illustrate the usefulness of the photoacoustic technique in the study of intact biological matter. The study on marine phytoplankton also demonstrates that the PAS method may be a more sensitive tool for discriminating between quite similar specimens than conventional absorption spectra. This added sensitivity is due to the fact that the photoacoustic


I.., ...* ---- ........_.__ ^_- c-c-

I I l l I 1

signal is dependent not only on the absorption characteristics of the specimen, but also on its deexcitation mechanisms (fluorescence and phosphorescence) and also on its thermal properties, whereas absorption spectra depend only on the absorption characteristics.

Besides obtaining spectral information on bulk biological matter, one can also use PAS to investigate the macroscopic structure of the sample. This can be done in several ways. For exampie, in Fig. 48 we illustrate the PAS spectrum obtained on a piece of apple peel for two chopping frequencies. At the higher frequency, the PAS signal arises solely from the waxy layer at the surface. This layer exhibits mainly a UV absorption due to its protein matter. At the lower frequency, the thermal diffusion length extends below the waxy layer and we get a PAS signal from the biological material beneath the waxy layer, such as the carotenoids and chlorophyll compounds in the peel itself. By changing the chopping or modulation frequency in acontinuous fashion, one is thus able to perform a depth-profile analysis of biological specimens. Another method for obtaining depth-profile information is to record the PAS spectrum at different phase angles. Kirkbright and his co- workers have performed this experiment on spinach leaf (70). They were able


to demonstrate the existence of the wax layer on the leaf by recording the in-phase and out-of-phase PAS spectra.

Although there are, at present, few photoacoustic studies in the field of biology compared to physics and chemistry, these studies definitely will be increasing at a fast rate in the near future.

IX. PHOTOACOUSTIC SPECTROSCOPY IN MEDICINE

Perhaps the most exciting area for photoacoustic studies lies in the field of medicine. One can use PAS to obtain optical data on medical specimens that are not amenable to conventional study because of excessive light- scattering; one can study specimens that are completely opaque to transmitted radiation; and one can perform depth-profile analysis by phase or frequency adjustments. Furthermore as we shall show below, it is possible to construct a photoacoustic cell that can be used to perform all of the above measurements in an in uiuo mode.

A. Bacterial Studies

An example of the use of PAS in medical studies is in the identification of bacterial states. Although conventional light-scattering methods can be used to monitor bacterial growth on various substrates, this very light- scattering makes it most difficult to obtain spectroscopic data on bacteria and thus to identify them. However, bacterial samples are quite suitable for study by PAS. Thus the PAS spectrum of Bacillus subtilis var. niger, a common airbone bacterium, contains a strong absorption band at 410 nm when this bacterium is in its spore state (69). This band is absent when the bacterium is in its vegetative state. Thus, PAS enables the detection and discrimination of various bacterial states and allows one to monitor and detect bacteria in various stages of development.

B. Drugs in Tissues

Another medical use for PAS is in the study of animal and human tissues, both hard tissues such as teeth and bone, and soft tissues such as skin and muscle.

An example of a PAS experiment on soft tissue (5) is illustrated in Fig. 49, where photoacoustic spectra of guinea pig epidermis are shown. The top spectrum is of an epidermis that has been treated with a 2% in ethanol solution of tetrachlorosalicylanilide or TCSA. This compound is known


WITH TCSA

C 0 N T R 0 L

- DIFFERENCE SPECTRUM - TCSA-CONTROL

250 3K) 450 SW 6! nrn

FIG. 49. A study of guinea pig epidermis. The upper spectrum is of a guinea pig epidermis that has been treated with tetrachlorosalicylanilide (TCSA); the middle spectrum is of untreated epidermis; and the bottom spectrum is the difference spectrum showing the absorption bands of the TCSA compound within the epidermis. From Rosencwaig (5).

to be a highly effective antibacterial agent, but unfortunately also causes photosensitivity and other skin problems (94). Why it has these side effects is not completely understood. The central spectrum is of control guinea pig epidermis. The bottom spectrum is the difference spectrum found by subtracting the control spectrum from the TCSA spectrum. The difference spectrum is thus the absorption spectrum of TCSA bound within the epidermis, the first such spectrum obtained by any technique. From this spectrum we can now establish the state of the TCSA compound in situ, that is, when it is incorporated into the skin, and thus learn more of its action on and within the skin under various conditions.

Another example of such a study is illustrated in Fig. 50. Here PAS spectra have been taken on human epidermal tissue (stratum corneum) in the 200-400 nm region. The bare stratum corneum is shown in (a), stratum corneum on which a commercial sunscreen has been applied is shown in (b), and (c) shows the spectrum of the sunscreen on the stratum corneum as obtained by differential means on a Gilford R-1500 PAS spectrometer with a bare stratum corneum sample in the reference cell.

These examples illustrate the potential usefulness of PAS in the study of natural, medical, and cosmetic compounds present in and on human tissue.


250 300 350 4w

Wavelength, nm

FIG. 50. A study on sunscreen. (a) A PAS spectrum of outer epidermal tissue (stratum corneum) on which a commercial sunscreen has been applied. (b) Untreated stratum corneum. (c) The difference spectrum, giving the spectrum of the sunscreen in situ, that is, on and within the epidermal tissue.

C . Human Eye Lenses

In Fig. 51 we show PAS spectra taken on intact human eye lenses. These experiments were conducted to study the processes by which human eye cataracts are formed. Little is known about this ancient disease, except that it is probably due to a photooxidative process in which the tryptophan and tyrosine residues in the protein matter of the lens form complexed compounds that are either highly light scattering (cortical cataracts) or else are colored (brunescent or nuclear cataracts) (95). Cataract studies are severly hampered by the fact that few spectroscopic investigations can be conducted on the intact lenses (96). In general these lenses must be solubilized, and except under the strongest solubilizing agents, only about one-half


FIG. 51. A study on intact human eye lenses. (a) The PAS spectrum of an intact normal human eye lens; (b) the PAS spectrum of an intact human eye lens with a pronounced brunescent cataract. Note that (b) indicates that the cataract results in increased optical absorption in both the ultraviolet and infrared regions of the spectrum.

of the lens material goes into solution and is suitable for study (97). The stronger solubilizing agents do too much damage to the inherent protein to be useful.

The PAS spectrum shown in Fig. 51 were obtained from intact human eye lenses (98); (b) is of a lens that has a brunescent cataract, while (a) is of a normal lens. Both spectra exhibit the characteristic peak at 280 nm due mainly to absorption by the tryptophan and tyrosine protein residues. The cataractous lens shows a much broader 280 band than does the normal lens. This is in agreement with the theory that the cataract is a result of conjugated tryptophan and tyrosine compounds (95). In addition, however, the PAS spectra indicate that the cataractous lens also exhibits an increased infrared absorption. We do not have at this time an explanation for this new-found feature. It is interesting to note, however, that these spectra indicate that cataractous formation does not only impair vision in the blue end of the spectrum, but actually impairs vision throughout the visible region, with broad absorption bands moving in from both the ultraviolet and the infrared regions of the optical spectrum.


D. Tissue Studies

In beginning our photoacoustic studies on human tissues, we hoped that we might be able to use the PAS technique to study abnormal and pathological tissues and thereby obtain new information about diseases such as cancer, psoriasis, etc. However, no such study can be definitive unless adequate baseline data are available on the spectra of normal tissue. Since such data are not available from conventional sources because of light-scattering and opacity effects, it became imperative that we obtain these data using the photoacoustic effect. We therefore began with a study of epidermal tissue, and in particular of the stratum corneum. The experiment below is discussed in detail, not only because of the interesting results obtained, but also because of the potential importance of PAS in medicinal studies on human tissues.

The stratum corneum, the outermost layer of mammalian epidermis, is composed of highly organized units of flat horny cells stacked in vertical interdigitated columns. It is the end result of a specific form of epithelial cell differentiation consisting of synthetic and degradative processes (keratinization), which ultimately form the complex semipermeable and protective matrix consisting of bipolar lipid layers, carbohydrates, proteins, and several other chemical moieties (99). The major protein of stratum corneum is a-keratin, which in mammalian epidermis, wool, and other biological systems consists of two or three separate a helices wound around each other in a coiled-coil configuration (100). Comprehensive treatments of the structure and biochemistry of the stratum corneum have been published recently (100-102).

The stratum corneum, a translucent membrane in the visible region, is a highly effective light scatterer especially in the ultraviolet region, and thus cannot be studied readily by conventional optical absorption techniques. Some investigators have attempted to reduce the scattering problem by treating the stratum corneum with fluids of matching refractive index (103,104). A more popular procedure is to solubilize the sample and then study the resulting optically clear solution. However, this approach is not suitable for studying the stratum corneum, in that (1) the stratum corneum is chemically resistant to complete solubilization because of the strongly cohesive nature of the keratin matrix, and (2) the question of whether the properties measured of the solution are exactly the same as those of the unsolubilized sample is the subject of considerable investigation and debate (105), not only for this system, but for other solubilized systems as well.

To characterize properly the physicochemical properties of the strateum corneum, it is necessary to determine its water content quantitatively. This is a most difficult measurement to perform in v i m and most of the successful


techniques are performed in uitro. The most common method is gravimetric, in which any loss of weight upon heating of the sample is attributed to water. Since the photoacoustic effect is sensitive to the presence of water, through its dependence on the thermal diffusivity of the sample (Section IV), we have been able to apply PAS for moisturization (hydration) studies, related to the role of water in the stratum corneum (106).

PAS experiments on intact, excised newborn rat stratum corneum have also been performed, as a function of the post-partum age of the rats, with rather interesting results (107).

Stratum corneum specimens from newborn rats used in these investigations were obtained 24 hours post partum for the hydration studies, and from 0 to 60 hours post partum for the maturation studies. The intact stratum corneum was harvested using Vinson’s procedure (108). The excised stratum corneum membranes were typically 1 cm2 in area and 12-15 pm thick.

For the hydration study, the separated stratum corneum was placed in a dry box for 48 hours, and subsequently cut into 12 samples, with pairs of samples equilibrated in closed constant humidity chambers of 11, 33, 52, 75,86, and 93% relative humidity over saturated salt solutions for 24 hours. From each pair, one sample was used for a gravimetric determination of water content (mg of H20/mg of dry stratum corneum), while the other was placed in the photoacoustic cell and its photoacoustic signal measured at 285 nm.

For the maturation studies, the intact stratum corneum was kept at ambient conditions for at least two days, and then run in the photoacoustic spectrometer. Several sets of maturation data were taken, each set using the stratum corneum of rats from the same litter, but with a different postpartum age per rat.

a nm

FIG. 52. Photoacoustic and optical absorption spectra of a poly-l-glutamic acid membrane in the ultraviolet. From Rosencwaig and Pines (107).


1.0

0.5

0

Figure 52 shows the photoacoustic and optical absorption spectra of a poly-l-glutamic acid membrane in the ultraviolet region. The close similarity seen here between the photoacoustic and the optical absorption spectra was also obtained for the case of the collagen membrane and for optically clear tape. These results as well as previous work on biological systems (5,8,9) clearly indicate the suitability of PAS to obtain optical absorption data on translucent or turbid biological membranes.

In Fig. 53, we show a typical photoacoustic spectrum, in the 220-450 nm region, of a translucent stratum corneum from a newborn rat. The photoacoustic spectrum in Fig. 53, is quite clear in spite of excessive light scattering, with very little background signal and no appreciable noise up to - 220 nm, in contrast to the results obtained with transmission spectroscopy using an integrating sphere (109). The so-called protein band in the region of 280 nm is clearly seen. The apparent flatness of the photoacoustic spectrum below 225 nm is due to the saturation effect wherein the optical absorption length pa becomes much smaller than the thermal diffusion length ps (Section IV). This occurs because below 225 nm the combined absorption coefficient

-

-

I I I - 200 250 300 350 400 4

of all chromophores

2 v) 3 0 V

w

I-

Y

a z 4 a

0

nm

FIG. 53. Photoacoustic spectrum of newborn rat stratum corneum (-- 12 pm thick). From Rosencwaig and Pines (107).

Of the commonly occurring amino acids, only trypotophan, tyrosine, phenylalanine, and cystine possess a characteristic absorption band in the 230-330 nm region. Tryptophan and tyrosine have by far the greatest absorption coefficient in this region. Amino acid analysis on the proteins in the newborn rat stratum corneum gave a consistent value of - 2 tyrosine molx, and - 0.2 tryptophan mol using both alkaline and acid hydrolysis.


Solid

A man (Tyd 286 nm X man ITry) 282 nm r ltyrl 42 nm r (try) 88 nm

Tryptophan

Solution

278 nm 282 nm 38 nm 60 nm

I I 1 1 I

2 w 250 300 360 400 450

nm

FIG. 54. Photoacoustic spectra of tryptophan and tyrosine powder. From Rosencwaig and Pines (107).

Using these values, and knowing that tryptophan has an absorption coefficient roughly three times greater than tyrosine, we estimate that the 280 nm band seen in Fig. 53 is -70% tyrosine, -20% tryptophan, and - 10% cystine and other indolic and phenolic chromophores.

Figure 54 shows that the photoacoustic spectra of solid tryptophan and tyrosine are also quite similar to their solution absorption spectra. The photoacoustic spectra, obtained on powders, exhibit a red shift and a small amount of line boradening relative to the solution spectra, features that are often seen when the spectrum of a compound in its solid state is compared with that in solution.

Figure 55 shows the photoacoustic signal at 285 nm as a function of the water content present in the stratum corneum, as determined gravimetrically.

04 0 0. i oh 0:s 0.8

lMgH$/Mg DRY S.C.1

I

FIG. 55. Photoacoustic signal strength at 285 nm as a function of water content for newborn rat stratum corneum. From Rosencwaig and Pines (106,107).


Identical results were obtained at other spectral wavelengths. There is first very little change in the photoacoustic signal, then a fairly rapid and then a slower decrease, with the curve apparently approaching a limiting value at high water content.

In Fig. 56, we show a series of photoacoustic spectra obtained from newborn rat stratum corneum membranes during the initial 60 hour maturation period. The times shown represent the post-partum age of the rats at the time of sacrifice. We note that the 280 nm band undergoes a major change, particularly in the 10-30 hour post-partum period. Identical results were obtained on all sets of rat litters studied, irrespective of the period between the time of harvesting of the stratum corneum and the time the photoacoustic spectra were actually obtained.

I I AGE AFTER BIRTH

0 nm

FIG. 56. Photoacoustic spectra of a series of newborn rat stratum corneum during the post-partum maturation period. From Rosencwaig and Pines (107).

Figure 57 shows the change in peak position of this 280 nm band as a function of post-partum age, and Fig. 58 shows the change in bandwidth and spectral area under this band. Little or no change is observed during the first 10 hours, or after 30 hours post partum, while a substantial change occurs between 10-30 hours post partum. The band peak begins at -284 nm, which is close to the value found for tyrosine in a solid matrix, and then increases to 289 nm. The most marked changes are the increases in bandwidth from 38 nm for the + hour old rat, to a value of - 78 nm for the 30 hour old rat. Normal values are 42 nm for tyrosine in the solid state and 38 nm in solution, with corresponding values of 66 and 60 nm for tryptophan.


I I

28s-

m-

[ 281-

z 4

2 s - A MAX ltvrl * 278 lwluclonl

A MAXltvI - 282 lsolutianl

= 286 (solid1

- 2(n lurlidl 286-

284-

zB?* 0 lb 20 30 ;o M i

AGE AFTER BIRTH lhrl

FIG. 57. Maximum position of the 280 nm band in newborn rat stratum corneum as a function of post-partum age. From Rosencwaig and Pines (107).

' 3.0

' 2.5

3 -2.0 c

E -1.5 5

r -1.0

304 o r o z o 3 ~ a m 6

AOE AFTER llRTn Ihrl

FIG. 58. Bandwidth and total spectral area of the 280 nm band in newborn rat stratum corneum as a function of post-partum age. From Rosencwaig and Pines (107).


The area under this band almost trebles, indicating that the band has broadened considerably and that its total absorption or oscillator strength has increased as well.

1. Hydration Study

The determination of water content in biological tissues would be most useful if it could be performed in uivo in a simple and direct fashion. PAS is ideal for achieving this purpose, since an open-ended photoacoustic cell can readily be sealed acoustically against a mammalian body, limb, or organ. Furthermore, the dependence of the photoacoustic signal on the thermal properties of the sample makes the technique sensitive to the presence of water.

The shape of the curve shown in Fig. 55 can be readily understood in terms of the photoacoustic thcory of Section IV. For a dry sample of stratum corneum, the thermal diffusion length ps is roughly equal to the actual thickness of the sample ( - 12 pm). Conditioning the stratum corneum to less than -60% relative humidity (0.15 mg H20) results in only small changes in the cohesive forces maintaining the stratum corneum’s rigidity (110). Moreover, differential scanning calorimetry (1 11) as well as nuclear magnetic resonance studies show that as water is initially added to the stratum corneum, it goes into “bound” sites and in particular into the matrix micropores. Since this “bound water” is fairly tightly bound to specific sites, and minimally effects the matrix rigidity (110), its effect on the total specific heat and thermal conductivity of the stratum corneum is minor in this hydration region. However, its effect on the density of the stratum corneum is not dependent on its binding characteristics, and thus the density will increase steadily from the dry state value of -1.0 g/cm3. In this humidity range, therefore, the net effect of the “bound water” is to decrease the thermal diffusivity (b = k/pC), thereby decreasing the thermal diffusion length ,us = (28 /0) ’ /~ .

In Section IV we showed that the photoacoustic signal is not noticeably affected by changes in the thermal properties of the stratum corneum until the thermal diffusion length ps becomes smaller than the sample thickness 1. This is apparently the case in our stratum corneum experiment until we reach -0.15 mg H20. When < I , then the photoacoustic signal will be given by Eq. (27), which holds when p, < 1 and ,us << pa. We know from the work of Everett et al. (109), from the known composition of stratum corneum, and from our own experiments that pa > 1 in the 280 nm region. Thus, the conditions for Eq. (27) are met, and the photoacoustic signal will be given by


where pg is the thermal diffusion length, y the specific heat ratio, and Po and To the ambient pressure and temperature, all pertaining to the gas in the cell. Zo is the light intensity and 1, is the gas column length in the cell. Rewriting Eq. (46) and substituting pz = 2p/w we can set

2 2 1 Q=--

( I ) p c (47)

where Z represents all the nonsample parameters, and where p and C are the density and specific heat of the stratum corneum membrane. The photoacoustic signal is now independent of the heat conductivity of the membrane, which is known to increase with water content (112).

As we add more water (>0.15 mg H,O), the cohesive forces are progressively reduced and the matrix is plasticized, becoming increasingly more extensible and softer (110). The additional sorbed water behaves more like bulk or "free" water (111,112). We can thus expect that the specific heat of the hydrated stratum corneum matrix will increase from a value close to 0.3 cal/ gm"C (113), to one close to water itself (1 cal/gm "C). At the same time, the density p continues to increase until all the micropores are filled, and then begins to level off as it approaches the limiting value of 1.32 g/cm3 for highly hydrated stratum corneum (114). Therefore, in the hydration region where both p and C are increasing fairly rapidly (0.15 < water < 0.5 mg H,O), the photoacoustic signal will decrease fairly rapidly. At even higher humidities (water content >0.5 mg H,O), the increases in p and C become much more gradual and thus the photoacoustic signal begins to level off as seen in Fig. 55.

It is apparent from the hydration study, that from the use of the theory and some preliminary calibration, one should be readily able to determine water content in biological systems, not only in vitro but also in uiuo.

2. Maturation Study

Rothman (115) proposed a theory to the effect that before and/or during the keratinization process of stratum corneum, hydrolysis takes place and some amino acids are incorporated into keratin to form a more highly cross- linked and insoluble keratin matrix, while other amino acids remain free in the cell walls or cellular debris. A preponderance of sulfur-containing amino acids are incorporated into keratin during the keratinization process.

Although most of the published work relating to a-keratin involves wool, the a-keratin of the stratum corneum is quite similar to that in wool (100). Pauling and Corey (116) have shown that fibrous keratin in wool is pre- dominately a-helical in character, and Crick (11 7) has shown that the separate a-helical strands combine to form two stranded coiled-coil structures in


which the two a-helices coil around each other. These coiled-coil filaments are arranged in ring-core tubules, which are held together by a nonfibrous high-sulfur keratin protein matrix. Thousands of these tubules, or micro- fibrils, are then arranged in a close-packed array, bound together by the keratin protein matrix (100).

It is known how a-helices are formed and stabilized (116), and that the strong filament-matrix bonding is primarily due to disulfide bonds (99). What is not well understood is how the coiled-coil formations of the a-helical keratin filaments are stabilized, and as yet there has been no success in making synthetic a-helical polypeptide chains adopt a coiled-coil conformation (118).

Mammalian stratum corneum obviously serves quite different roles in the pre- and postpartum periods. Major and rapid biochemical and structural changes can be expected during the initial postpartum maturation period when the stratum corneum matrix undergoes alteration to develop its so- called barrier functions, and to adapt to its new and strikingly different environment.

The 280 nm band is primarily due to absorption of the UV radiation by the phenolic (tyrosyl) and indolic (tryptophanyl) chromophores, and in the case of the newborn rat stratum corneum, primarily by the phenolic tyrosine residues. The changes that we see from our photoacoustic spectra in Fig. 58 therefore reflect changes of the tyrosine, and to some extent tryptophan, residues, and to changes in their local environment.

Because of excessive light scattering, reliable UV absorption data on intact proteinaceous membranes has been unavailable up to now. However, there are voluminous conventional UV absorption data on model compounds and proteins in solution. We have made use of these solution data, as well as stratum corneum chemistry, to consider the most likely conventional mechanisms (listed below) to account for our observed spectral changes. We have made the reasonable assumption that the initial maturation period is similar to the final stages of keratinization.

(1) An increase in the polarizability of the local environment of the nonpolar amino acids will produce a small red shift of the 280 nm band (119). Such an increase in local polarizability can result from an increase in the sulfur-containing amino acid cystine during the keratinization (maturation) process.

(2) The less-well-developed and somewhat looser keratin matrix structure of the newborn rat becomes more cohesive and tighter as it matures. The indolic and phenolic residues may then find themselves coming closer to negatively charged groups, such as the carboxylate groups. This change in the Iocal environment can alter the absorption characteristics of both tyrosine and tryptophan residues (119).


(3) Donovan (120) and Beaven and Holiday (121) have suggested that several electronic transitions comprise the 280 nm absorption of indole (tryptophanyl) residues, and that one of these transitions may be very sensitive to the local environment. This mechanism is not too promising since our tryptophan content is so low.

(4) Another possible explanation lies in the formation of new hydrogen bonds such as tyrosyl-peptide carboxyl bonds and tryptophyl-carboxyl bonds (121).

(5) Charge-transfer complexes can often result in both a red shift and some observable line broadening (120). In most charge-transfer processes in proteins, the “donor” is a side-chain aromatic chromophore, such as tyrosine, and the “acceptor” a compound containing a substituted benzene ring, a situation that can well be present in a maturing stratum corneum matrix.

Although all of the above mechanisms can occur during the initial maturation period, none of them individually or collectively appears able to account for the excessive line broadening and the threefold increase in spectral area of the 280 nm band that we have observed. Certainly, major changes in the tyrosine absorption can occur in a highly alkaline solution (124, but not in the pH environment present in biological tissue.

In fact, as Donovan (120) points out, most environmental and protein conformational changes produce such subtle spectral changes of the 280 nm band that difference spectroscopy is often the only means for detecting these changes.

We have also found that our spectral changes in the 10-30 hour postpartum period appear to be correlated to the occurrence of a new protein conformation. This has been indicated by elasticity and differential scanning calorimetry measurements. In the elasticity study it was found that the stratum corneum membranes of rats younger than 10 hours displayed a monotonic change of elasticity with temperature, while the older rats showed an abrupt change in elastic properties at - 70°C. Similarly, the differential scanning calorimetry showed that the stratum corneum membranes of the older rates (age > 10 hours) displayed a new “protein melting” transition at 7PC, while the younger rats displayed no such transition. It thus appears that the newborn rat stratum corneum undergoes a structural change resulting in a new or modified protein conformation during the 10-30 hour postpartum period.

Since the conventional mechanisms (1 -5) appear inadequate to completely account for our results, we have considered some other possibilities :

(6) Major spectral changes can be expected to occur if the amino acid composition of the stratum corneum matrix changes substantially during


the first 30 postpartum hours. This time period, however, appears much too short for such a change. Moreover, our amino acid analyses showed no noticeable change in tyrosine or tryptophan content during this maturation period.

(7) Another possible explanation could be the metabolic synthesis of a strongly absorbing chromophore, such as melanin, during the maturation period. Melanin cannot be the chromophore, since we are dealing with albino rats. Furthermore, in Fig. 59, we show the difference spectrum between a $ and 30 hour postpartum stratum corneum membrane, noting that this difference spectrum does not appear to have the spectral features that one would expect from a typical UV-absorbing chromophore. Thus, this explanation does not appear too promising, although it cannot be ruled out completely.

ZM 3& 360 yw) 4 $ 2 0 0 m

DIFFERENCE SPECTRUM 30 HR. .1/4 HR.

L

-5

J 200 260 350 400 1

Jm nm

FIG. 59. Top: Photoacoustic spectra of newborn rat stratum corneum from rats and 30 hour postpartum age. Bottom: Difference spectrum between the $ and 30 hour spectra. From Rosencwaig and Pines (107).

(8) Another mechanism lies in the possibility that the existing tyrosine residues undergo a major molecular modification, probably by enzymatic action. This explanation appears attractive, because it is well known that a molecular alteration of tyrosine residues within a protein can dramatically


change both the shape and the area of the 280 nm band (122,123). In particular, the action of tyrosinase on tyrosine-containing proteins can alter the 280 nm band in a manner quite similar to what we have seen in our photoacoustic spectra of Fig. 56. This is shown very clearly in Fig. 3 of Yasunobu et al. (123) and reproduced here as Fig. 60.

WAVE LENGTH lnml

FIG. 60. The oxidation of hypertensin I by tyrosinase: absorption spectra taken at 0, 30, 60, 120, and 1200 minutes after addition of the tyrosinase enzyme. From Yasunobu et al. (123).

The primary action of tyrosinase on a tyrosine residue bound in a protein is to oxidize it, usually by enzymatically changing the tyrosine to a hydroxyl- tyrosine or DOPA-type molecule. If we hypothesize that an oxidation or hydroxylation of the tyrosine residues occurs in the stratum corneum by some form of enzymatic action (probably by an enzyme other than tyrosinase since we are dealing with albino rats) during the 10-30 hour postpartum period, then not only do we have a mechanism for explaining our photoacoustic data, but we also have a mechanism for accounting for the new protein conformational change.

In order to illustrate this latter possibility, we consider the known amino acid sequence for a single unwound a-helical keratin protein of wool as depicted in Fig. 5.4 of Fraser et al. (100) and shown in Fig. 61. The tyrosine residue is located within the nonpolar region of the proposed inner core of the coiled-coil conformation, and is shown as being hydrogen-bonded within


POSITIVELY CHARGED

NEGATIVELY 0 CHARGED

O H Y D R O W W ~ ~ C

( ~ I H Y D R O G E N . BONDING

0 4 1 6 2 6 3 POSITION IN SEQUENCE.

FIG. 61. Diagrammatic radial projection of the resi ies in one of the a-helices of a two- stranded coiled-coil keratin filament in wool. From Fraser et al. (100).

its own helix to a neighboring serine. Thus, this tyrosine cannot participate in any interchain bonding.

However, if this tyrosine is now modified to a hydroxyl-tyrosine or DOPA structure, then various interchain hydrogen bonds become immediately available. For example, the hydroxyl-tyrosine could readily hydrogen- bond to a lysine or arginine or even another hydroxyl-tyrosine, all on a neighboring helix. Such interchain hydrogen-bonding could play a crucial role in stabilizing the coiled-coil structure of the cc-keratin in the stratum corneum.

Although we have as yet no direct evidence for the appearance of a modified tyrosine, such as a hydroxyl-tyrosine, during the maturation or keratinization period, this hypothesis has some attractive features in its analogy with the situation in collagen, the structural protein in muscle tissue. It has recently been determined (124,125) that as collagen is being formed, some of the proline residues are enzymatically modified to hydroxyproline, and that the extra hydroxyl group of the hydroxyproline contributes significantly to the stability of the triple helix of collagen through additional hydrogen bonds. We feel that the possibility of an enzymatic molecular change in the tyrosine of a postpartum stratum corneum merits serious investigation.


3. Conclusion

The stratum corneum is a highly complex filament-matrix system with hydrogen bonds, sulfur bonds, salt linkages, electrostatic interactions, and other covalent and noncovalent bonds contributing to its cohesive nature. The study of this complex system is difficult and cannot be pursued by any one method alone. Nevertheless, PAS offers a new and exciting technique for increasing our knowledge of both the fully developed tissue in uitro and in uiuo, and of the maturing tissue. Our new-found capability of being able to characterize epidermal tissue noninvasively and in uiuo, and the possibility that we might learn more about the keratinization and postpartum maturation process by the use of PAS and other techniques, can play an important role in our understanding of moisturization, substantivity, percutaneous absorption, and normal and diseased states of mammalian tissues.

E. The in Viuo Cell

We have mentioned several times in this section the possibility of performing in uiuo studies with a photoacoustic instrument. The concept for such an instrument is straightforward, and an in oivo PAS instrument has now been built (126).

One end of such a cell is open. This end is sealed against the specimen to be studied. The seal may be accomplished either by pressure against the specimen or by creating a small negative pressure differential between the outside and inside of the cell. The optical radiation can be introduced into the cell by means of an optical fiber bundle or by appropriately placed mirrors. With an in uiuo apparatus, one should operate at chopping frequencies remote from the usual frequencies associated with bodily move- ments and functions. Nevertheless, it is to be expected that the noise background will probably be higher with this instrument than with the conventional PAS spectrometer. To overcome this difficulty it might prove advisable to increase the incident light intensity or to use more sophisticated data acquisition techniques, such as multiscanning and other averaging procedures. Considering the potential of such in uiuo studies, these additional procedures should well be worthwhile.

X. FUTURE TRENDS

PAS of solids and liquids is still in its formative stages, yet its potential both as a research and analytical tool appears almost boundless. Until the development of this technique, many materials, both natural and synthetic,


could not be readily investigated by conventional optical methodologies, since these materials occur in the form of powders or amorphous solids, or as smears, gels, oils, suspensions, and so on. With PAS, optical absorption data on virtually any solid and liquid material can now be obtained.

In this chapter we have reviewed experiments with the photoacoustic effect in the fields of physics, chemistry, biology, and medicine. In all of these fields we have done no more than indicate some of the possible applications of this technique in these fields.

As PAS becomes better known, investigators in many different fields and with many different problems and orientations will adapt this technique to their own uses. Furthermore, since this is a spectroscopic technique that is not detector limited, it will surely be extended into the far ultraviolet and infrared regions of the optical spectrum, and quite possibly into other regions of the electromagnetic spectrum as well.

In the near future, it is quite likely that PAS will become a common and useful analytical and research tool in many scientific laboratories. Its ease of operation and versatility can only increase its areas of applications. Examples of some new areas are

(1) Low-temperature studies of organic and inorganic compounds. Modification of a PAS cell for low-temperature work should be quite straightforward.

(2) Single-crystal studies, using polarized light, of strongly colored materials that cannot be readily examined by conventional optical techniques.

(3) The use of transform techniques, e.g., Fourier or Hadamard transforms to improve the signal to noise ratio when the source intensity becomes weak as in the infrared.

(4) In the field of catalysis, the characterization of heterogeneous metal oxides and oxide mixtures may prove to be an important application. Similarly, PAS may prove advantageous in the monitoring of surface reactions of metallic catalysts such as platinum and platinum-rhodium combinations, which are used in automotive exhaust systems and hydrogenation and hydroformylation reactions.

(5 ) In the field of biology PAS will be used to study intact biological systems both in the laboratory and out in the field, providing data that now can be obtained only after extensive wet chemical procedures.

(6) In medicine PAS offers the opportunity of extending the exact science of noninvasive spectral analysis to intact medical substances such as tissues, with the possibility that by such noninvasive techniques new light might be shed on the diseases that afflict mankind.

The next few years promise to be an exciting period of growth for the rediscovered science of photoacoustic spectroscopy.


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Noise in Solid State Devices

A. VAN DER ZIEL

Electrical Engineering Departmeni University of Minnesota Minneapolis, Minnesota

AND

E. R. CHENETTE

Electrical Engineering Department University of Florida Gainesville, Florida

I. Introduction . . . . . . . . 11. Sources of Noise . . . . .

A. Thermal Noise . . . .................................... B. Shot Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . C. Generation-Recombination (g-r) Noise

A. Noise in Schottky Barrier B. Shot Noise in p-n Junction Diodes , . C. Generation-Recombination in the Space Charge Region of a p-n Junction . . . D. Avalanche Multiplication in p-n Juncti E. Applications . . . . . . . . . . . . , . . . . . . . . .

IV. Noise in Transistors ............................................... A. Shot Noise in Tr . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. Equivalent Circui ............................................... C. Recombination i itter Space Charge Region . . . . . . . . . . . . . . . . . . . . . . . D Comparison of Theory and Experiment . . . . . . , , , . , . . . . . . , . . . . . . . . . . . ,

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

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

. . . . . . . , 111. Noise in Diodes . . . . . . . . . . ........................

A. The Klaassen-Prins Schematic and Its Applications

C. Other Noise Source D Comparison with E

B. Examples . _ . . .... ........................

E. Noise Figure of FE ........................

D. Switching Devices . . . . . . . . . References . . . . . . . . . . . . . . . . . . . ...................................

314 314 314 317 319 320 320 32 1 328 331 335 335 335 343 347 347 352 356 357 358 364 365 369 374 374 375 376 379 380



314 A. VAN DER ZIEL AND E. R. CHENETTE

I. INTRODUCTION

The purpose of this chapter is to present a survey of noise in solid-state devices. To that end a discussion of the various noise sources is given in Section 11, and this is applied to p-n junction diodes, Schottky barrier diodes, tunnel diodes, Josephson junctions, bipolar transistors, junction field effect transistors (JFETs), metal-oxide-semiconductor field effect transistors (MOSFETs), etc. in subsequent sections. (For earlier review papers see the list of references.)

11. SOURCES OF NOISE

The most important sources of noise in solid-state devices are thermal noise, shot noise, generation-recombination (g-r) noise, and flicker noise. The latter deserves a separate discussion and therefore will not be dealt with here.

A. Thermal Noise (1-3)

1. Nyquist’s Theorem

According to Nyquist’s theorem the noise in a frequency interval Aj for any resistance R or conductance G kept at temperature T, can be represented by an emf [S,(f)Af]”2 in series with R or by a current generator [Si(f) Af]1/2 in parallel to G, respectively, where

S”(f) = 4kTRP(f), Sdf) = 4kTGP(f) ( 1 )

P ( f ) = (hf/kT)(+ + Cexp(hf/kT) - 11-’> (la)

The term i(hfkT) represents the effect of the zero point fluctuations; it was omitted in Nyquist’s original paper (1). A detailed wave mechanical calculation shows, however, that it must be taken into account (2). See also Oliver’s

This follows from Nyquist’s original derivation. Two resistors R, having an available noise lower Pa, and kept at the temperature T, are connected by a very long lossless transmission line of length L and characteristic impedance R. Let u be the velocity of propagation along the line; then the average energy present in the line is 2Pa,L/u. If now suddenly both ends of the line are short- circuited, standing waves of natural frequencies n(u/2L) (n = 1,2, . . .) occur on the line; the number of natural frequencies in the frequency interval Af

and

paper (3).

NOISE IN SOLID STATE DEVICES 315

.a

is (2L/u)Af. If E is the average energy in each mode of vibration, the trapped energy is also equal to (2L/u) Af E. Hence

P,, = EAf (1b)

(W

but according to quantum theory

E = hf (3 + [exp(hf/kT) - l]-'}

and it is obvious that the zero point energy must be included. Substituting (lc) into (1 b) yields the result referred by (1).

The quantum correction factor p ( f ) is shown in Fig. 1. It is unity at lower frequencies and increases slowly with increasing frequency at the highest frequencies; for room temperature the transition lies in the infrared range and at cryogenic temperatures in the high kMHz range. The factor p ( f ) is about equal to 3(hflkT) for hflkT > 5; in that case practically all the noise comes from the zero point fluctuation term.

I I I I

O r

2. Thermal Noise as Velocity Fluctuation Noise

One can also look at thermal noise as velocity fluctuation noise of the individual carriers (4,5). Let a carrier in a semiconductor have a velocity u,(t), and let the average velocity T, # 0. We then introduce the velocity fluctuation Au,(t) = u,(t) - i j x , and apply the Wiener-Khintchine theory to it. This yields

/ . ,

/

0

I I 1 I

SAvx( f ) = 2 Jym Au,(t) Aux(t + s) exp(jos) ds = 4 Re@)


where

D = som Av,(t) Au,(t + s) exp( -jws) ds

is the diffusion constant and Re stands for “real part of.” The low-frequency value of D, which is obtained by putting o = 0, is real and has the value Do; at infrared frequencies D becomes complex.

We now prove that our definition of D makes sense for low frequencies. The distance Ax traveled in time t is

AX = sof Av,(u) du

or

E? = Ji JiAt~,(u) Au,(w) du dw = Jd du J:.” Au,(u) Av,(u + S) ds

= ji du j-mm Au,(u) Av,(u + s) ds = 2D0t

(3)

for sufficiently long t, which is the well-known Einstein formula. We next consider a small piece of semiconductor of cross-sectional area A

and length Ax, with two ohmic end contacts. An electron moving with a velocity u&) in the sample gives rise to a current i(t) in the external circuit, such that

i(t) = qu,(t)/Ax, or Ai(t) = qAv,(t)/Ax (4)

so that spectrum of the fluctuation Ai(t) is

If n is the carrier density, then there are nA Ax carriers in the sample and each gives an independent contribution to the total current I(t). Hence

S i ( f ) = 4(q2/Ax2) Re(D)nAAx = 4q2nRe(D)A/Ax (6)

Because of the occurrence of the diffusion constant in this expression, the noise is sometimes called difusion noise; we see, however, that velocity fluctuation noise is its cause.

Expression (6) holds independent of whether or not Tx = 0, and whether or not the velocity distribution in v, is maxwellian; these conditions simply reflect themselves in the value of D. Therefore, the equations remain correct under hot electron conditions, i.e., when D becomes field-dependent, or when the velocity distribution becomes nonmaxwellian, or both.

At equilibrium this corresponds to thermal noise, of course. At low frequencies, where p ( f ) = 1, D = Do, and ,u = p0, the Einstein relation

4Do = kTPo (7)


holds. Substituting into (6) yields

Si ( f) = 4kT(qn~d/Ax) (64

This corresponds to thermal noise of the conductance G = qnp,A/Ax of the sample.

Bearing in mind that at equilibrium the high-frequency noise is also thermal noise, we can evaluate a high-frequency correction to the Einstein relation (7). Since at those frequencies the mobility p becomes complex, the conductance G of the sample must be written

G = qn Re(p)A/Ax (8)

(8a)

where Re again stands for “real part of.” Hence

Si(f) = 4kTGdf) = 4kT[qn Re(~)A/Ax]p(f)

Comparing this with (6) yields the following extension of the low-frequency Einstein relation (7):

4 ReP) = kT Re(p)p(f) (9)

For an extension of this theorem see van Vliet and van der Ziel(6).

B. Shot Noise

It is well known that shot noise in saturated thermionic diodes, carrying a current I , can be expressed by a current generator [ S , ( f ) A f ] ” ’ in parallel to the diode, where

si(f) = 241 (10)

for relatively low frequencies. When transit times become significant, the right-hand side of (10) must be multiplied by a transit time factor, the magnitude of which depends on the geometry and on the potential distribution in the diode. This is called Schottky’s theorem.

In some devices, such as p-n junctions, Schottky barrier diodes, and tunnel diodes, the current Z consists of two currents II and I , flowing in opposite directions, so that I = I1 - I , . Both currents show full shot noise, and hence

si(f) = + 1 2 ) = 2qllI1 + IZ)/(~I - 1 2 ) ( 104

This expression may again be modified at high frequencies by transit time effects or it may happen that new noise terms must be added at those frequencies.

Even if transit time effects are unimportant, the expression for S i ( f ) can become frequency dependent, because the quantum correction factor p ( f ) for thermal noise also applies to shot noise.


This is most easily seen for a very heavily doped Schottky barrier diode at zero bias. The width of the space charge region is then so narrow that transit time effects are unimportant and current flow occurs by “thermionic emission over the barrier.” At zero bias, equal currents I. flow in opposite directions, each fluctuating independently and showing the same spectral intensity SIo( f). The high-frequency conductance of the diode under this condition is go = qZo/kT. Since this is an equilibrium situation, go should show thermal noise, that is,

si(f) = 4kTgop(f) = W0p(f) = 2SIo(f) or SI,(f) = 2qIOp(f)

(11) as had to be proved.

As van der Ziel(7) has shown, the quantum correction also holds for the incremental emission current A1” due to electrons emitted into the space charge region with an initial energy between Vo and Vo + AVO volts; it is then possible to incorporate transit time effects. Again, the quantum correction in the shot noise is needed in order to obtain the quantum correction in the resulting thermal noise term.

We next consider another very fast device-the tunnel diode. At an applied voltage V, let currents I , and I, flow across the junction in opposite direction, and let these currents show shot noise with a correction factor q(f). We shall then demonstrate that 4(f) = p(f).

The proof is as follows. Since

&,(f) = W,q(f), SI,(f) = 2qI2q(f) (12)

Sdf) = 2q(li + 12)df) = 2dq(f)(Ii -k Iz)/(11 - 12) (12a)

It may now be shown with the help of wave mechanics that

(11 + 12)/(Ii - 12) = ~0th(qV/2kT)

SAf) = 4W~/V‘)qff)

(13)

Now for 1x1 << 1, cothx ‘v l/x. Hence for IqV/2kTI << 1

If we now let V go to zero, then, since the characteristic is continuous at V = 0, I/V approaches the low-frequency conductance go = (dI/dV), . Hence

&(f) = 4kTg0df) (14)

But since this is a thermal equilibrium situation, S,(f) must show thermal noise. By comparing with (1) we see that q(f) = p(f) as had to be proved. (For further discussion see a recent paper by van der Ziel(8).

We have hereby demonstrated for three particular cases that shot noise shows the quantum correction factor p(f) at high frequencies for which the factor deviates from unity.


C. Generation-Recombination (g-r) Noise (9)

For semiconductor material one encounters noise due to generation and recombination of carriers. It shows up as fluctuations in the resistance R of the sample, which can be detected, in turn, by applying a voltage V to the sample and measuring the fluctuating current.

For simple cases the noise can be described by one fluctuating number N of carriers, either electrons or holes. This is true for noise due to traps, deep-lying donors or acceptors, Shockley-Read-Hall centers when there is a predominant lifetime, etc. The general equation for the fluctuation in N is then

(15) dN/dt = g ( N ) - r ( N ) + Ag(t) - Ar(t)

where g ( N ) and r (N) , describing the generation and recombination rates of carriers, respectively, are known functions of N , and Ag(t ) and Ar(t) describe the randomness in these rates. Putting N = N o + A N , where N o is the equilibrium number of carriers, and neglecting higher-order terms in AN yields the equilibrium condition g(N,) = r(No), and the linearized Langevin equation

+ Ag(t) - Ar(t) d A N A N

dt z -- --

where

Making a Fourier analysis and bearing in mind that Ag(t) and Ar(t) show shot noise, i.e.,

S&f) = W) = M N o ) = W N O ) (16b)

yields the spectrum

Since

Eq. (1 7) may be written


This equation is valid for all g-r noise sources describable by fluctuations in a single variable N .

Now we switch to current fluctuations by observing that a fluctuation AN in N gives rise to a fluctuation AZ in the current Z such that

The current noise thus depends on 1’. When heating effects occur due to the current, N o and/or S,(f) may depend on Z and S , ( f ) may deviate from an Z2 dependence at large currents.

A similar relationship occurs for the Shockley-Read-Hall centers in the space charge region of a back-biased p-n junction. These centers alternately emit an electron and a hole, and hence the centers exhibit a fluctuating charge. Let nT be the density of centers; then the number in a volume element AV is nT AV. Iff; is the probability that the center is occupied by an electron and 1 - J ; the probability that it is occupied by a hole, then the average number L\N, of trapped electrons in the volume element AV is n,AV’ and the fluctuation 6 AN, has a mean square value

(19)

The reader will recognize this as a case of partition noise. It should be obvious that the spectrum S, , , ( f ) of 6 A N , is of the form (17b).

In many cases the time constant z is governed by an activation energy E,, and may be written as

6m = nTAVS,(l - S,) = m(1 - A )

z = z0 exp(qE,/kT) (20)

For a more general discussion of g-r noise involving more complicated situations, see van Vliet and Fassett (10).

111. NOISE IN DIODES

A. Noise in Schottky Barrier Diodes

Since the characteristic of a Schottky barrier diode is of the form

Z = Z,[exp(qV/kT) - 11 (21)

where lo may be a slow function of the applied voltage V, and the two currents I , exp(qV/kT) and - I , fluctuate independently and show full shot noise, the spectrum of the current fluctuations may be written

SAf) = 2dIoexP(qV/kT) + 101 = 2 d I + 210) (22)

NOISE IN SOLID STATE DEVICES 32 1

Now, since the input conductances are

g = dI/dV = q( l + I0)/kT, go = dl/dVI,=o = qIo/kT (22a)

Eq. (22) may be written

S,(f) = 2 w 9 + 90) (23)

corresponding to half-thermal noise of the conductance g if g >> go. Since the device is a majority carrier device and since the current flow through the barrier layer is by diffusion, except for heavily doped substrates (see below), this expression should be valid well into the microwave region.

However, two obvious corrections must be made. (1) We must introduce the device capacitance Cd in parallel to the resistance rd = 1/g. (2) One must bear in mind that the series resistance r, of the bulk regions shows thermal noise. As a consequence the full equivalent circuit of the Schottky barrier diode is as shown in Fig. 2.

FIG. 2. Small-signal equivalent circuit of Schottky barrier diode.

At high doping levels (lo’ 8-1019/cm3) the current flow is by “thermionic emission over the barrier” and the time constant is the travel time through the space charge region. As van der Ziel has shown (7,11), this transit time problem can be treated theoretically. It has been conjectured that Schottky barrier diode detectors and mixers can be built for the 10-pm wavelength range and above (1 1 ).

At still higher doping levels ( N > 10’ 9 /~m3) the current flow through the barrier is by tunneling. Since this is a very fast process, the infrared frequency response should be even better than in the thermionic case. However, this problem has not yet been solved.

B. Shot Noise in p-n Junction Diodes

1. Injection over the Barrier. Corpuscular Model

At lower current levels part of the current flow is by recombination in the space charge region; this problem is discussed in Section 111,~. The


remaining current is by “emission over the barrier.” At higher forward bias it should predominate. We discuss this problem for a pf-n diode in which nearly all current is carried by holes.

The characteristic is again of the form

I = I,[exp(qV/kT) - 11 (24)

The currents I , exp(qV/kT) and - I , fluctuate independently and each shows full shot noise. Therefore, at low frequencies the theory of Section II1,A holds and

whereas the low-frequency conductance is

Wf) = 2qU + 210)

go = dZ/dV = q(Z + Io)/kT

(25)

(244 At high frequencies diffusion and recombination effects in the n region

must be taken into account explicitly. If the width of the n region is a few times the diffusion length of holes in the n region (long diode), the admittance Y of the diode becomes

Y = go(l + jozP)’l2 = go + go[(l + j ~ z , ) ’ / ~ - 13 = g + jb (26)

where zp is the lifetime of holes in the n region and 2 2 112 112 9 = SOL9 + 3(1 + 0 zp) 1

The noise at high frequencies is found to be

S,(f) = 2q(I + 210) + 4kT(g - go) = 4kTg - 2qI (27)

that is, the currents I + I , and - I , can be considered as showing full shot noise at all frequencies and the incremental high-frequency conductance g - go shows thermal noise (12,12u).

This is most easily seen from a corpuscular model of current flow (12~) . Here three independent groups of holes are considered :

(1) Holes injected from the p+ into the n-region and recombining there. This gives rise to the current I + I , and to the conductance go for all frequencies of practical interest. Since the crossing of the space charge region is a very fast event, the currents consist of a series of independent random pulses of very short duration carrying the charge q. Therefore, the noise I + I, should be full shot noise for all frequencies of practical interest. The pulse rate is modulated practically instantaneously by the applied voltage V ; hence these pulses give a contribution go to the admittance for all frequencies of practical interest.

(2) Holes generated in the n region and returning to the p region. This gives rise to a current - I , consisting of a series of independent random


pulses of very short duration, the rate of which is independent of V. The npise of - I , should therefore be full shot noise for all frequencies of practical interest.

(3) Holes injected from the p+ region into the n region and returning to the p+ region before being recombined. They give rise to a series of independent random pulse pairs, each of very short duration, carrying a charge + q and - q, respectively, with a random time delay z between the two pulses, and the rate of these pulse pairs is modulated instantaneously by the applied voltage V. This gives rise to an admittance Y - go. A detailed discussion shows that the noise associated with this group is thermal noise of the incremental conductance g - go.

We come back to this method in Section III,B,3.

2. The Collxtive Approach (I2)

In the collective approach one describes the noise problems by the generation-recombination processes and the diffusion processes in the n region.

For the solution of the one-dimensional problem of a p+-n diode with volume recombination only, one can make use of a transmission line model. Let the transmission line have a voltage V and a current I , a series resistance for unit length R, a conductance for unit length G, and a capacitance for unit length C. Let the junction have a junction current i , an excess hole concentration p’, and a cross-sectional area A. Then p’ corresponds to V, i , corresponds to I , R corresponds to l/qD,A, G corresponds to qA/t , , and C corresponds to qA. The characteristic impedance of the transmission line corresponds to

Z o = ~ i / ~ / [ q D i / ~ A ” ~ ( l + j ~ z , ) ’ / ~ ]

and the propagation constant of the transmission line corresponds to y = (1 + joz,)”2/(D,z,)’12, where zp is the hole lifetime. Since this transmission line problem can be solved by standard techniques, the noise can be evaluated as soon as the noise sources are known.

In a volume element AV = A Ax in the n region, having a hole concentration p , a hole current qp AV/z , disappears by volume recombination and a hole current qpnAV/zp appears by volume generation; here pn is the equilibrium hole density and 0, the hole lifetime. For strongly asymmetric Shockley-Read-Hall centers (the common case) these currents show full shot noise, and so the volume generation-recombination processes can be represented by a current generator (Ai;)’/2 for the volume element AV, where

- A? = 2q(qP W z , ) Af + 2dqPn W z , ) Af (28)


The diffusion noise in any volume element AV = A Ax due to diffusion - in the x direction can be represented by the current generator where

- Ai& = 4q2D,p(A/Ax) Af (29)

representing the spectrum of the x component of the fluctuation in the diffusion current [see Eq. (6)].

Van der Ziel (12) used a noise source (dp,z)1’2, describing the carrier density fluctuations Ap in the volume element AT/, of the form (in our present notation)

- Ap: = (4p AXIDPA) Af (30)

At that time Eq. (30) was only a conjecture, introduced in order to obtain the right amount of noise out of the diode. For that reason this equation was recently questioned (13). However, it is easily seen from the transmission line model that the series resistance RAx of the section Ax corresponds to Ax/(qD,A). Therefore

- Ap: = (Ax/qD,A)’ = (4p Ax/D,A) Af

in agreement with Eq. (30). The objections to (30) are therefore not well founded.

Equations (28) and (30) are valid when p << n for the n region; this is called the case of low injection. When p and n become comparable, the diffusion noise source (30) becomes (4)

A& = 4q2Da[pn/(p + n)l(A/Ax)Af

Ap: = a ( A x / q D a A ) 2 = (2p Ax/D,A) Af

(31)

(314

where D, is the ambipolar diffusion constant, corresponding to -

for p N n [see Ref. (14)] . If Dn and D, are the electron and hole diffusion constants and p il: n, Da may be written

Da = 2DnDJDn + Dp) (31b)

At high injection and strongly asymmetric recombination centers the volume recombination source may be written

- Aifx = 2q2pAVAf//z, (32)

where zp is somewhat different from the value for the low-level injection case.

We now apply the transmission line analogy to a long junction in a one-dimensional model. We calculate the current at the junction for the device high-frequency short-circuited; this means that in our analogy we


short-circuit the input of the transmission line (Fig. 3). This yields for the junction current

Ai = Airx exp( - yx) + (Apx/.Zo) exp( - yx) (33)

Putting y o = a. = ~/(D,T$/~, y = a + jp , and*

a = a. Re(1 + j w ~ ~ ) l / ' = aog/go, p ( x ) = p ( 0 ) exp( - yox)

where g is the high-frequency diode conductance and go its low-frequency value, and P at the junction is obtained by taking the mean square value of (33) and integrating over the device length.

1 x A FIG. 3. Transmission line analogy as used to calculate the noise in a one-dimensional

junction diode.

Applying this to the low-injection case, this yields, if the equilibrium hole concentration pn is neglected

- i2 = 2qI Af - 4 k T ( g - go) Af = 4 k T g Af - 2qI Af

= 2 d A f (29/90 - 1) (33d

corresponding to half-thermal noise of go at low frequencies and to full thermal noise of g at high frequencies. If the effect of pn is taken into account, Eq. (27) results.

We now turn to the high-injection case (19) for a long junction (19~) . The transmission line model is valid, but D, must be replaced by D, every- where and the factor 4 in (30) must be replaced by 2 in (31a); otherwise the previous expressions remain valid. But there is a complication; because of the contribution of the field in the n region, the current at the junction is now

i, = - W,A(dP/dX) IO = ( 2 ~ , / D , ) [ - 9 A ~ , ( d P / d X ) l o l (34)

* At high injection there is now a region where the high injection condition p ( x ) > N , is not satisfied. If p ( 0 ) >> N , , this gives only a small error.

326 A. VAN DER ZIEL AND E. R . CHENETTE

where 2Dp/Da = (p, + pP)/pn. As a consequence,

Ai = (2op/0a)CAi,xexp(-yx) + (A~x /zo) exp(-yx)] (344

at high injection. Furthermore we must take into account that the transmission line model evaluates the spectrum of -qqdDa(dp/dx)),, so that the final result must be multiplied by (2Dp/Da)2 to obtain the spectrum of the noise at the junction. Hence for a p+-n junction, since p ( x ) = po(0) exp( -sox), we have for the dc current(14).

and for the noise, after some manipulations,

where g/go is given by (26a). In the same way for an n+-p junction,

Hence in an n+-p junction the ‘noise is larger than full shot noise at low frequencies. These results were derived earlier by more sophisticated methods

Since the true junction admittance is go = eI/kT, at low frequencies (15-18).

where B = (p, + 4,)/2pn for a p+-n and B = (pn + pp)/2pp for an n+-p diode.

Next we take into account the effects in the bulk region (18). There are several terms involved here. In the first place there is a high-level diffusion effect that changes the high-frequency admittance Y = g + jb to Y’ = (g +jb)(l 2 m)-’, where m = (p, - pp)/(p, + ,up); the plus sign holds for a p+-n and the minus sign for an n+-p diode (19). As a consequence the high-frequency noise emf e in series with the junction has a spectrum

We have here made use of the fact that lY’12 = gi(l + w2z@(l & m)-2 , so that C = B(l i- m)2, corresponding to C = 2pn/(pn + pp) for the p+-n and to C = 24,/(pn + pp) for the n+-p diode.

We now turn to other effects in the n region. There is the dc resistance R, of the region. In series with it is a modulation impedance Zmd caused by the current modulation of R, ; at high frequencies Zmd is complex. The


total bulk impedance z b s = R, + Zmod. The noise associated with z b s is correlated with the junction noise discussed earlier; formally we can introduce a noise current generator [SIbs(f)Af]132 in parallel to Zbs; this is best treated empirically. In addition, the real part of the bulk series impedance Zb, shows thermal noise, that is, S,,,(f) = 4kT Re(&,). The total noise equivalent circuit thus becomes as in Fig. 4.

FIG. 4. Complete equivalent noise circuit of a one-dimensional p-n junction diode including bulk resistance and bulk modulation effects.

For very short junctions [w,, << (D, .c , ) ' /~] the situation becomes somewhat simpler. We refer to papers by Van Vliet and Min (18,20) and to Section IV of this chapter for details.

If one now drops the assumption of the one-dimensional geometry, one finds that the final expressions (33a), ( 3 3 , and (35a) remain fully correct, but the evaluation of the integrals is much more complicated (15-17,21).

3. Comparison of the Collective and the Corpuscular Approaches

At first sight both approaches seem quite different; the corpuscular approach looks at what goes on at the junction and the collective approach looks at what goes on in the n region. However, one can surround the n region by a closed surface, integrate over the volume of that closed surface, and apply Gauss' theorem (9). One thereby transforms the volume integral into a surface integral and thereby finds a one-to-one correspondence between what goes on inside the n region with what goes on at the junction. However, this should not be overinterpreted. Gauss' theorem merely transforms to the noise current (i2)1/2 at the junction; it does not justify the splitting of P into two terms with physical meaning,

We now assume p >> p n , so that group 2 is insignificant. There is nothing wrong then with the idea of splitting the carriers into groups 1 and 3. But

328 A . VAN DER ZIEL A N D E. R . CHENETTE

there is no guarantee that the pulses in group 1 or in group 3 are independent of each other, nor is there any guarantee that group 1 as a whole is not correlated with group 3.

The important feature of the low-level injection case seems to be that one can proceed as if all correlations were absent. However, the high-level injection case serves as a warning that this is not always true.

C. Generation-Recombination in the Space Charge Region of a p-n Junction (22)

For back-biased and slightly forward-biased silicon diodes the current flow is by generation and by recombination of carriers in the space charge region, respectively. At sufficiently large back bias a Shockley-Read-Hall center alternately generates an electron and a hole, and at sufficiently large forward bias the center alternately captures an electron and a hole and so removes an electron-hole pair in a two-step process. Between these extremes, both types of processes occur. All these processes generate noise.

These noise processes are important for three reasons:

(1) The noise for a back-biased diode is important for photodiode

(2) The noise for an unbiased diode is important for photovoltaic opera-

(3) The noise for forward-biased diodes is important for transistor

operation of the junction diode.

tion of the junction diode.

noise caused by recombination in the emitter space charge region.

We start with discussing the device characteristic. According to the Shockley-Read-Hall theory of the centers, the net recombination rate R , is

where zpo and zn0 are appropriate lifetimes, n, and p1 are the carrier concentrations that would occur when the Fermi-level would be at the trap level, n, the intrinsic carrier concentration and n,pl = n:. The first term on the right-hand side of (37) represents the recombination rate R(x) and the second the generation rate G(x). Therefore, the net current I is

where I, = Ji qAR(x)dx, I, = Ji qAG(x)dx (37a) I = IR - I,,

Since pn = nf exp(qV/kT), except at high injection

IR(V) = ZG(V)exp(qV/kT), Z = IG(V)[exp(qV/kT) - 13 (37b)


For strong back bias, exp(qV/kT) is negligible and

(38) I = - I G - - -qAn:d/(nizpo + plzno)

since n and p are negligible for most of the space charge region.

zero bias is For zero bias I = 0 and I , = I , . The low-frequencies conductance near

9 = dI/dV = (dI , /dV) [exp(qV/kT) - 11 - [(ql~(T/)/kT)] exp(qV/kT) (39)

SO that at zero bias (V = 0), where IG(0) = I , ,

9 = 90 = SIo/kT (394

Obviously go should show thermal noise, that is,

S~(f)lv=o = 4kTgo = f 2410 (40)

corresponding to full shot noise of the equal and opposite currents I , . For moderate forward bias, I,( V ) << ZK(V) and

I = I,( V ) = IG( V ) exp(q V/kT) = I ; exp(q V/mkT) (41)

where m is between 1 and 2. For a center at midband in a symmetrical junction m = 2, but often m 1: 1.5. Again, the full characteristic is given by Eq. (37b).

We now turn to the noise in greater detail. Let a center be located at x, let the boundary between the space charge region and the p region be at x = 0, and let the space charge region have a width d. An electron generated at x and traveling to the n region transfers a charge Qn(x) in the external circuit and a hole generated at x and traveling to the p region transfers a charge Qp(x) in the external circuit. For an electron coming from the n region and being captured at x the transferred charge is -Qn(x) and for a hole coming from the p region and being captured at x the transferred charge is - Qp(x). According to Lauritzen (22)*

Qn(x) = 4(d - x)/4 QJx) = W/dt Qn(X) + Qp(x) = 4 (42)

where 4 is the electron charge. Van Vliet verified this accurately (25). At low frequencies Qn and Q, represented fully correlated pulses, and

hence their effects must be added'linearly, so that the factor (Q, + Q,)' = q2 appears in the noise expressions. At sufficiently high frequencies they

* Equation (42) follows from Ramo's theorem (23) when the space charge due to ionized donors and acceptors is neglected. If the effect of this space charge is included, one obtains the wrong expressions for Qn and Qp, and hence the wrong expressions for the noise (24).


represent independent pulses and hence their effects must be added qua- dratically, so that the factor Q~(x) + Q~(x ) 5 q2 appears in the noise expressions. As a consequence the high-frequency noise is somewhat smaller than the low-frequency noise in most cases.

We now follow a simplified discussion given by van Vliet and van der Ziel (26). According to (38) we have for a back-biased diode with strongly asymmetric centers (the normal case) and at low frequencies

For high frequencies we have instead

as is found by substituting (42). This agrees with experiment (27). At zero bias R = G, and hence at low frequencies

For moderate forward bias and low frequencies, SR(0) < 2R because of a low-frequency smoothing effect.* At high frequencies S,(f) = 2R but q2 must be replaced by Q ~ ( x ) + Q~(x). Therefore, at low frequencies

whereas at high frequencies

The noise at moderate forward bias may therefore be written

SIR = <(m) * 2qIR (46)

Lauritzen (22) and van Vliet (25) calculated <(O) = 0.75; Wade and van der Ziel measured t(0) for transistors at high injection and low temperature,' and found <(O) > 0.75 and slightly dependent on current (28).

* The smoothing effect occurs because subsequent pulse pairs are somewhat correlated as low frequencies; at high frequencies they are uncorrelated.

' The recombination current I, varies as n, and the injection current I, varies as n', where n, decreases strongly with decreasing temperature. Therefore, the ratio I,/l, at a given bias varies as l/n,, so that the recombination effect becomes much more pronounced at low temperatures.


D. Avalanche Multiplication in p-n Junction Diodes

In sufficiently back-biased p-n junction diodes, breakdown can occur in which the current must be limited by the external circuit. There are two types of breakdown possible:

(a) True avalanche breakdown. Here, at a certain critical voltage, known as the breakdown voltage, avalanching due to collision multiplication results in a self-sustained discharge current that must be limited by the external circuit.

(b) True Zener breakdown. Here, at a certain voltage Zener emission of electrons from ionized acceptors or from the valence band by tunnel effect sets in. This results in a very rapid increase of current with voltage, helped somewhat by avalanche multiplication of the generated carriers; this breakdown is not as abrupt as in the previous case.

In silicon diodes the first effect is dominant in diodes that have breakdown voltages above about 14 V, whereas the second effect is dominant in diodes that have breakdown voltages below 2 V. In between, junctions can conduct heavily as a result of the combined effects of tunneling and impact ionization multiplication. We discuss first the situation for true avalanche breakdown and then for Zener breakdown associated with avalanching.

1. Avalanche Multiplication Resulting in True Avalanche Breakdown

First, let the electrons and the holes have equal ionizing power. Let I , be the dark current before multiplication and let on the average p hole-electron pairs be generated when an individual hole-electron pair traverses the space charge region. Then, for p < 1 the multiplied current is

(47)

Hence multiplication by a factor M occurs that results in true breakdown when p reaches the value unity. Since p increases with increasing back voltage, there is a well-defined breakdown voltage V, in this case.

Now the noise. The spectral intensity of the primary current is full shot noise of the current I , . Since the secondary, tertiary, . . . hole-electron pairs are generated independently and at random, the currents P I , , p z I , , . . . also show full shot noise. All these noise contributions are now subject to further multiplication; that is, all the noise contributions must be added and multiplied by M 2 . Consequently,

Io(1 + p + p z + . . .) = Z O / ( l - p ) = I,M

resulting in a large amount of noise for large M .


If the electrons and holes have unequal ionization coefficients ci and b, respectively, the ratio k = f l /a is found to be practically independent of the applied voltage. Expressed in terms of k and M , McIntyre (29) obtained the following results:

(1) p+-n diode (all current carried by holes)

S J ~ ) = 2 q z , ~ 3 1 + - - [ k "(M"- J] (2) n+-p diode (all current carried by electrons)

(49)

The first result is transformed into the second by replacing k by l/k. The first expression is quite large for k << 1 and quite small for k >> 1, so

that the lowest noise is obtained if the holes have the largest ionizing power. The second expression is quite large for k >> 1 and quite small for k << 1, so that the lowest noise is obtained if the electrons have the largest ionizing power.

These predictions have been well verified by experiment. In germanium the holes have the largest ionizing power (k > l), whereas in silicon the electrons have by far the largest ionizing power (k << 1). For GaAs and GaP one finds k N 1.

2. Zener Breakdown with Associated Avalanching (30)

In the case of a junction with a very narrow space charge region (SCR) the concept of a single-current carrier being involved in a large number of ionizing collisions is no longer valid. Instead, experimental results show much better agreement with a model that allows only a very few (say one to three) ionizing collisions per crossing of the SCR. Primary electrons are emitted by tunneling, and the multiplication process is dominated by the influence of threshold energies for ionization.

The multiplication threshold model can be explained with the help of Fig. 5 , which shows triangular approximation to the electric field as a function of position in the vicinity of an idealized reverse-biased abrupt junction. Also shown is a sketch of the energy bands as a function of position. Since the tunneling probability is strongly dependent on the electric field, most of the tunneling current originates at x = 0. At x = x1 the tunneling electrons become free carriers, absorbing energy corresponding to the width of the band gap, with minor variations corresponding to the simultaneous emission or absorption of phonons. The electron continues toward x2, acquiring energy from the electric field. At x = x2 the electron has acquired the energy required for an ionizing collision.


P-tyPe , n - type

I ' Z I

d l ,,- ionizations occur I

I I

1d2_+Er eV I I

I I I I I

I

I I

I I I I I I

I 2142 ev I I I I

I

FIG. 5. The energy threshold model for Zener avalanche breakdown in a diode with a narrow space charge region. The electron tunneling at x = 0 becomes a free electron at x1 and has a probability a1 of producing an ionizing collision at x 2 .

We now outline the calculation of the multiplication noise resulting from

The total number of carriers N collected at the diode terminals in 1 sec no more than one ionization per carrier transit across the diode SCR.

is given by

where no is the number of electrons tunneling in one second (giving rise to It) and n, is the number of hole-electron pairs produced in one second on the ith carrier transit across the diode SCR.

If E and 5 represent the average ionization probabilities of electrons and holes, respectively, HI = EA0, Ez = 6- nl - - o, A - - -- anz - - -' a Gi0, and so on, then

(52)

N = no + pli (51)

N = no(l + E + E 6 + Ziz6 + . . .)


102 - -

The modified multiplication factor M* = (M - l), given by

M* = (Zr/Zt) - 1 = (N/no) - 1

becomes

M* = a(l + 6)/(1 - &) (53)

Since I , = qN, the spectral density of the low-frequency current fluctuations can be obtained by calculating

Sir = 2qz var N

The result can be shown to be

10-2

1 [ 1 + 3a + 3 s + a26 sir = 2q4 (1 - &)2

TRANSIT)

1 . . . . I . . . .

If 7i = 6 [Eq. (53)] becomes M* = a/(1 - a)

and Eq. (54) becomes

Sir = 2ql,(l + 3M* + 2M*’)

(54)


To compare this theory with experimental results note that Eq. (54a) is of the form

sir = 2 d t P + f(M*)l (55)

Figure 6 shows f ( M * ) as a function of M* for a typical narrow SCR diode. For further details we refer to the quoted paper (30).

E. Applications

1. Schottky Barrier Diode Detectors and Mixers (9,30)

In Schottky barrier diode detectors (31) carrying a bias current I, the noise is shot noise of the currents (I + I,) and - I o , where 1, is the saturation current. When the substrate is heavily doped n-type (Nd N 10i9/cm3) detection, albeit with an impaired rectification efficiency due to transit time effects, should be possible down to 10 pm wavelength (32). Rectification has, in fact, been observed down to 40 pm (33).

In Schottky barrier diode mixers the shot noise of the device current produces if . primary noise and high-frequency primary noise; the latter is converted into if . secondary noise in the mixing process. These two if . noise sources are correlated and must be added properly. We refer to the literature for details (9).

In principle, mixing should be possible down to 10 pm in heavily doped samples (32). In practice, mixing has been achieved down to 70 pm (33).

2. Metal-Oxide-Metal Detectors and Mixers (31,34)

Metal-oxide-metal diodes consist of a layer of metal and a metal point contact separated by a 10-20 A thick oxide layer. Because of the short distances involved, and since tunneling is a very fast process, these devices work as detectors and mixers below 10 pm (34). The devices suffer from instability, however; this would be a handicap in communication systems.

IV. NOISE IN TRANSISTORS

A. Shot Noise in Transistors

1. Early Work on Transistor Noise

It was recognized by Montgomery and Clark (35) in 1952 that the low- frequency noise behavior in transistors could be expressed by a shot noise emf e,, in series with the emitter junction resistance rco, by shot noise of the


collector saturated current Ico and by thermal noise of the base resistance rb. They obtained good agreement with experiment at low emitter currents but there was a significant discrepancy at higher currents. Van der Ziel (36) suggested in 1954 that the current distribution between base and collector gave rise to partition noise; when this additional noise source was taken into account good agreement was obtained at all currents.

The equations governing this approach are as follows. Let a current I , + I , , be injected from the emitter into the base, and a current I E E be collected by the emitter from the base, then the emitter noise emf has a mean square value

E + ~ I E E ] Af E + IEE

- ef = 2kTr,,-, ['I

according to Section 111. The collector noise has a mean square value

(57) -3 iL = 2q[(IE + lEE)aF(l - aF) + lCO] Af

Here the first term is the partition noise for the injection current (IE + IEE), aF is the dc current amplification factor, and Ico is the collector saturation current. Van der Ziel neglected the part I,, of the first term; usually this gives only a negligible error.

The thermal noise of the base resistance r, is represented by a noise emf eb where -

e: = 4k TrbAf (58 )

The equivalent circuit thus obtained is shown in Fig. 7a. It is most appropriate in common base connection.

Giacoletto (37) suggested that the noise could also be represented by a current generator ib between base and emitter and a current generator i, between collector and emitter. Here (Fig. 7b)

- - it = 2qIB Af, if = 2qIc Af (59)

whereas the cross correlation was assumed to be zero. In addition, the base resistance t-b has thermal noise. This is correct at higher currents; at lower currents remains correct, but (38)

(60) - -

i t = 2q(IB + ~ I E E + 2Icc) Af ibi, = - 2qIccAf

where IEE was defined before and I,, is defined as the collector current due to minority carriers generated in the base and collected by the collector. For silicon devices operating at currents between 10 pA and 10 mA the terms IEE and Icc are so small that they can be neglected.

What is missing in this approach is that the noise should be expressed in terms of the fundamental noise sources operating in the base, that the results


FIG. 7. (a) Oldest equivalent noise circuit of common base connection for low frequencies (Montgomery-Clark, van der Ziel). (b) Oldest equivalent noise circuit of common emitter connection at low frequencies (Giacoletto). Note that i: and i, are difjerent current generators.

should be extended to high frequencies, and that the approach should be extended to high injection, for which the basic assumptions underlying Eqs. (56)-(59) are not valid. This leads to the low-injection collective approach (12), the high-injection level collective approach (39,39a) and the corpuscular approach (124, respectively.

2. The Collective Approach at Low Injection

We assume here that the devices are p-n-p devices, that all recombination is volume recombination, that the geometry is one dimensional, that all current is carried by holes, and that effects due to the equilibrium hole concentration in the base can be neglected. Min and van Vliet (39a) have solved the case of arbitrary geometry.

The base can then be represented by a transmission line. If the emitter- base and the collector-base junctions are short-circuited for high frequencies,


"AiW I X w-x T

the line is short-circuited at both ends and the noise currents at the junctions are the short-circuit currents of the transmission line, respectively (Fig. 8). The basic noise sources Airx and Apx are then given by Eqs. (28) and (30), respectively, and the characteristic impedance Z o and the propagation constant y of the line are the same as in Section 111.

The hole distribution in the base region is

p(x) = po(0)sinh yo(w - x)/sinh yow (61)

where po(0) is the hole concentration at the emitter side of the base ( x = 0) and w is the width of the base region.

In modern transistors recombination in the base region is a very rare event, so that it can be neglected. The emitter hole current IEp is then equal to the collector hole current Icp, and the base hole current I , is practically zero. The noise term Airx can now be neglected and, since yow << 1,

In our transmission line analogy we thus have

(63) Ap, sinh y(w - x ) Ap, sinhyx Z o sinhyw ' Z o sinhyw

Ai,, = ~ Aicp = -

Here i,, flows into the emitter and i,, out of the collector. Therefore, by integrating over the width w of the base region,

T - 2q2A Af sinh y(w - x ) i,,, = A& = ___ * 2 -

Z P /i /2 J: 1 sinh yw

= 4kTg,, Af - 2qIE, Af (64)

sinh yx ZP J: lsinhywl

g=m=-----.2 2q2A Af - p(x) ~ ax = 2qI,hf


2 sinh y*(w - x ) sinh yx d x

2q2 A Af

ZP 21t1 sip(x) lsinhywI2 i:picp = Aizp Aicp = ~ .

= 2kTyce Af = 2kTap yep Af

Here

I e p = Icp = qADpPdO)/w (67)

Yep = gep + = gepo(yw/tanh V) (68)

where gepo = qlEP/kT, gep is the real part of yep, and gepO is its low-frequency value. The transfer admittance is, if gceO = qlcp/kT,

YW

sinh yw Yce = gceo ~ - - u p Yep

where ap = l/cosh yw. The final results remain correct when recombination in the base region is taken into account; the only difference is that I,, # Icp.

In this approximation aF = IEp/ICp would be unity. In fact, there is an additional emitter current IEn due to electron injection from the base into the emitter, so that the base current I , = I,, and aF = (Icp + IE,-,)/ICp. We come back to that problem in Section IV,A75.

It is convenient to rewrite Eq. (64). Bearing in mind that gepO = qlEp/kT, we may put

*2 lep = 2qzEpAf(2gep/gep0 -

We need this result in Section IV7A,5.

3. The Collective Approach at High Injection (39,39a)

We can now follow the same approach as in the high-injection diode. The diffusion noise source is given by Eq. (31a), p(x) is given by Eq. (62), Zo and y contain the ambipolar diffusion constant D,, and ref. (19)

(70) 2 0 Apx sinh yx

ep D, Z , sinhyw ’ cp D, Z , sinhyw Ai 2 0 Ap sinh y(w - x) Ai = P X .

The latter expressions take into account the effect of the field in the base region. The transmission line model now calculates the current - qAD, dp/dx and its spectral intensity. Consequently,

sinh y(w - x ) . I dx 2 0 , 2q2AAf y

= (K) ~ 1 ~ 1 2 J: p ( x ) ( sinh yw


sinh y*(w - x) sinh yx dx

20, '2q'AAf y ' = (K) T l G I !: (sinh ywl'

- - 2kTy,, Af 2Pn

Here = Icp = (2Dp/D,)qAD,p,(0)/w, and

d c , YW y,, = __ 7 kT sinh yw

qlEp Yw yep = kT iiiiijGyw'

(73)

(74)

gep is the real part of the input admittance yep, and yce is the high-frequency transfer admittance. We have here made use of the fact that the integrals are the same as in (64)-(66) and that D,/D, = (p,, + pp)/2p,,.

Because of the use of (62) there is a small part of the base for which the condition p N n is not satisfied; at very high injection this gives only a small error.

For n-p-n transistors the same result holds except that (p,, + pp)/2p,, must be replaced by (p,, + pp)/2k. The noise at high injection is therefore somewhat smaller than at low injection for p-n-p silicon transistors, whereas for n-p-n transistors the reverse is true. If we drop for a moment the subscripts p, we may thus write

- i,2 = B(4kTge Af - 2qI,Af) (75) - i,' = B2q1,Af

izi, = B2kTyC, Af - (76)

(77)

where B = 1 at low injection for both the p-n-p and the n-p-n transistor, B = (p,, + pp)/2pn for the p-n-p and B = (p,, + pp)/2& for the n-p-n transistor.

We mentioned that our method makes a small error, because a part of the base is not at high injection. This means that the transition from B = 1 to B # 1 sets in at somewhat higher injection and goes somewhat more slowly than would otherwise be anticipated.

While the result holds for a one-dimensional transistor, it is easily seen that it must be independent of geometry. In the first place, at high injection the diffusion noise spectrum is reduced by a factor 2, the field in the base region multiplies the noise by a factor (2DP/DJ2, and the current by a factor


2Dp/Da; all these effects are independent of geometry. The net result is therefore that the low-injection noise is multiplied by the factor Dp/D, for a p-n-p transistor. A similar argument holds for the n-p-n transistor.

4. The Corpuscular Approach (12a,38)

To simplify the discussion we consider a p-n-p transistor in which all current is carried by holes, recombination in the base is negligible, and the effects due to the equilibrium hole concentration in the base are negligible. The hole can then be split into two groups: (1) holes injected by the emitter and collected by the collector, (2) holes injected by the emitter and returning to it.

At low injection these groups give independent random pulses. Hence, group (1) gives a contribution 2qlEpAf to z, a contribution 2qZ Af to i:p, and a low-frequency contribution 2qZcpAf = 2kTgce0 Af to izpicp. Since the signal and the noise transfer are affected by the random diffusion time of the carriers through the base in the same manner, gceO must be replaced by y,, at high frequencies. Finally, group (2) gives a contribution 4kT(g,, - gepO)Af to c. Adding all terms, we obtain Eqs. (64)-(66).

At high injection the method cannot be applied directly since the pulses are no longer independent. However, it is possible to interpret the high- injection noise as a multiplication of the low-injection noise by the factor 4(2Op/Da), where the factor 4 is a smoothing factor and the factor 2Dp/D, is an enhancement factor caused by the field in the base.

5 . EfSects of the Znjected Electron Emitter Current in

C p

p-n-p Transistors (39)

When electrons are injected into the emitter of a p-n-p transistor, the base current ZB equals the emitter electron current ZEn. Two cases must now be considered :

In that case po(0) c Nd, and hence the electron concentration n, at the edge of the emitter region is

a. Low injection into the base.

ne = Ndexp[-q(vdif - vEB)/kT] (78)

where hi, is the diffusion potential of the junction; this varies aSexp(qI/,B/kT). Consequently the injected electron current is

I E n = I E n s exp(qvEB/kT) (784

b. High injection into the base. In that case po(0) > Nd and since po(0) = pn exp(qT/,B/kT), where pn is the equilibrium hole concentration in the n region,

% = P n exp(qvEB/kT)exp[ -q(hif - (79)


which varies as exp(2qVE$kT). Consequently the injected electron current is

I E n = I h n s exp(2qVEB/kT) (794

Since the acceptor concentration N , in the emitter region is so large, we generally have n, << N , , so that the electron current I,, is always a low injection current.

The dc current amplification factor is now ciF = Zcp/(Ic, + IEn) and the low-frequency ac current amplification factor is

ciO = (azCp/a VEB)/(azCp/a VEB + arEn/avEB)

@O = ciF(zCp f zEn)/(zCp -k 2zEn) < aF

(80)

(804

(81)

This is equal to ciF at low injection into the base, but at high injection

We now turn to the noise. To that end we apply Eq. (64a) and write

= 2qzEn Af(2gen/gen0 - l)

This holds for both low and high injection into the base, the only difference being that in the former case genO = qlE,/kT and in the latter case genO = 2ql~,/kT.* Equation (81) holds independent of whether the emitter region is a long region (we =- L,,) or a short region (we < Lne); here we is the width of the emitter region and L,, the electron diffusion length of the electrons in the emitter region.

We are now able to evaluate the base noise. The electrons give a contribution to z; the holes give no contribution at low frequencies but give a significant contribution at high frequencies that can be calculated as follows. Since ibp = ie, - if,,

(82) 7 - T ibp - (i,, - icp)(i:p - ic*,) = i,, + - 2Re(i:,i,,)

This is equal to - izp = B4kT[ge, - Re( Y,,)] Af (8W

Here B = 1 at low injection and B = (p, + pn)/2pn for very high injection. Consequently

= 2qzEa Af(2gen/gen0 - l) + B4kT[gep - Re(xe)] Af (83)

A similar expression can be derived for an n-p-n transistor, the only difference being that the subscripts n and p must be interchanged.

* The electron contribution to the emitter admittance is


B. Equivalent Circuits

Evaluation of the emf e, and the Current Generator i (38,39)

We now introduce the parameters (Fig. 9)

(84) . .

ee = ieZe, I = I , - Cti,

where c1 is the high-frequencies current amplification factor, and 2, the total junction impedance. The advantage is that we hereby transform to a simpler equivalent circuit consisting of an emf ee in series with the emitter junction and a current generator i in parallel with the collector junction.

FIG. 9. The general noise equivalent circuit of a junction transistor (BJT). Note that the use of i, and i , as current generators has made it necessary to define the controlling current as i: and the dependent current generator as at;, where e, and i are defined in the text.

For a p-n-p transistor, we have

uF = I C / ( I C f IEn) Or IEn / IC = (1 - @F)/OIF (85)

(85a)

(85b)

- g e p o or gen~- 1-a0 c10 = d l C / d vEB - dI , /dbB + dIEn/dvEB g e p o + gene g e p o “0

a = ~ c e / ( Yep + Yen)

To evaluate Z , one must take high-injection diffusion effects in the base region into account (19). In analogy with the diode case, Z , = l/ye, where

Ye = Yen + Y e p M 1 + m) = g e + jbe (86)


for a p-n-p transistor, and

Ye = Yep + yenM1 - m) = ge + jbe (8W

for an n-p-n transistor, where m = (,un - pp)/(pn + ,up). The discussion so far neglects the impedance of the base region proper,

consisting of the dc resistance R, and the modulation impedance Zmod in series (39a). We come back to this in a moment.

We now rewrite

- i,' = B2qZcAf

= B2Taye Af = B2qIc Af (aye/gepo)

Next we write

At low frequencies, bearing in mind that (1 - aF) << 1 and genO << gene, whereas

(plus sign for p-n-p, minus sign for n-p-n transistors)

Ze = reO = (kT/qId(l zk m) (91)

1 - e,Z = B(l k m)2kTre0Af = 2kTrsiAf or rsi = Zre0 (90a)

since B(l rt rn) = 1. Finally,

reO = (kT/qzE) * 2pn/(pn + pp) (914

re0 = (kT/qIE) 2pp/(pn + pp) (91b)

for a p-n-p transistor, whereas for an n-p-n transistor

Equation (90a) corresponds to the low-injection expression, except that reO has to be replaced by its high-injection value given by Eqs. (91a) and (91b). The resistance reo is quite small at high injection, and it is doubtful that it can be measured accurately. Any high-level injection effect on reO might thus easily esca e notice. 4 Next we evaluate i . We write


At higher frequencies we can substitute

wheref, is the alpha cut-off frequency of the transistor; we can also substitute for gcn/geno. For low frequencies we have gen/genO = 1 and

(924 -5- i - 2 e a ~ l ~ AfB( 1 - go)' -k 1 - QF) where I, = IC/aF. The first term in (92a) is usually so small that it can be neglected; in that case P is not affected by high-injection effects.

Experimentally, no high-injection effects on ? have been found, and the magnitude of is found to be as Eq. (92a) indicates (do), so this is reasonable agreement.

We now compare these results with those obtained by Wade and van der Ziel(41). There is, of course, a significant difference in that they do not take high-injection effects for the hole current into account. Moreover, they do not take high-frequency effects in the (electron) emitter admittance yen into account.

Equating 7 = 2qZeqAf, and compare the results, they found at low frequencies

obtaining good agreement with experiment. This is not a very substantial difference with (92a), since (1 - up)' and (ao - aF)' are very small. At high frequencies they found

I, , = IE[aF(l - aF) + (QF - ~ 0 ) ' l (92b)

whereas Eq. (92) yields, for gen N geno,

The difference is mainly in the factor B, and it should only show up at very high injection.

Finally - e,*i = i:(ic - ai,)Z,*

At higher frequencies, neglecting small terms, this reduces to the well-known formula (12,39a)

- e,*i = 2kTa[ - 1 + Z~pjgepO] Af (944


We have here made use of the fact that Z,, = Zepj(l + m), gepO = qlE/kT, and B(l + rn) = 1, where ZePj is the true junction impedance, equal to l/gepo at low frequencies. We note that Eq. (94a) does not contain the factor B, that is, 8 does not show high-injection effects at high frequencies.

A similar result holds for n-p-n transistors; only the subscript p must be replaced by the subscript n.

The low frequency value for a p-n-p transistor, taking all terms into account, is

N 2kT[(1 - No) - (1 f m)a,( 1 - U F ) / ~ F ] (94b)

To discuss this expression we introduce a correlation impedance (38) Z,,, = a / P . We then obtain at low frequencies

where reO is given by Eq. (91a). This is never very large, for the two terms partly cancel each other at any injection level. This does not agree with experiments, for reasons to be discussed below.

We must now correct an omission made earlier, where we ignored conductivity modulation effects in the base region. If we take these effects into account, we have a dc longitudinal resistance Rs in the base region that is modulated by the ac current and so produces a net bulk base impedance z b s .

At high frequencies Z,, is more complicated, but at lower frequencies it may be expressed as (20)

where the minus sign holds for the p-n-p and the plus sign for the n-p-n transistor. Since the junction impedance and the high-injection diffusion effect combined give an impedance

zbs = (1 rn)(kT/gzE) (96)

reO = (l * m)(kT/qlE)

the total low-frequency input impedance is approximately

(ze)in = reO + zbs = 2kT/q1E (964

for both the n-p-n and the p-n-p transistor. Thermal noise is associated with the real part of the bulk series impedance

zbs : s V , , ( f ) = 4kT (97)

In addition, due to the flow of current through Z,, , there is an additional current generator [SlbB(f) Af]”’ in parallel to the impedance z b , that is


- correlated with the junction noise. This additional noise term affects both e,' and 8, and so the correlation impedance Z,,,. Min and van Vliet (39a) have used this to explain the relatively large correlation impedance observed by'Tong and van der Ziel(40) at relatively low frequencies; we refer to their paper ( 3 9 4 for details.

C. Recombination in the Emitter Space Charge Region

The theory given for the recombination in the emitter space charge region is identical with the one given for the diode in Section II1,C. For a p-n-p transistor, we therefore have - i,' = B*2qzEp Af(2gep/ge.p0 - l) + 2qzEn Af(2gen/gen0 - l) + 2qIR (98)

whereas 2 and pi, are not affected. Here the first term is due to holes injected into the base region, the second

term to injection of electrons into the emitter region, and the third term to recombination in the emitter space charge region. The currents associated with these effects are lep, I,,, and IR, respectively. The function t ( w ) is defined in Section II1,C.

It is not difficult to modify the expressions for 2, F, and e*i accordingly. In Section 111 we have refered already to Wade and van der Ziel's (28)

measurement of ( (w) . For a more detailed account of recombination in the emitter space charge region see Wade et al. (42) and Section IV,D.

D. Comparison of Theory and Experiment

In this section we give a brief summary of an extensive experimental study of the noise in bipolar junction transistors (42,43). Bias conditions were varied over a wide range to assure measurements at both low and high injection levels; the ambient temperature was controlled as a parameter and ranged from 60 to 300°K. At the lower temperatures g-r processes in the emitter-base SCR and in the base region became important. It will be seen that the experimental results are well explained by a theory that was an earlier version of what is presently in Sections IV,A and IV,B, in which the noise reduction factor ((0) was assumed to be unity.

The experiment was basically that which has long been advocated by van der Ziel and his group: 7 was measured directly at the output terminals of the transistor operating in common-base connection with the input ac open-circuited; e,' and e,*i are determined indirectly from the measurements of the equivalent input noise emf, measured at a function of source impedance. From the point of view of this study, where the emphasis is on the use of


noise measurement to understand the physical processes of the transistor, the measurement at the output may be considered the more fundamental experiment. Comparison between theory and experiment requires only information about the bias currents and the dc and ac current gain (including the frequency dependence). There would be little point in comparing theory and experiment at the input if there were not good agreement at the output.

The theoretical expression for the noise referred to the input depends on the small signal equivalent circuit. The common base physical-T model used here has the merit that reo can be accurately calculated, and measurements of the input resistance can be used to determine the base resistance. An important point in the results summarized below is that below about 100°K there is evidence of an additional noise source in the base region, which can be represented as an additional base noise emf. It can be explained by g-r processes, which become significant at the lowest temperatures.

That the experimental results are in essential agreement with the theory is shown clearly by Fig. 10. The data points are the results of measurements; the curves were calculated from the expression

Here aF is the dc alpha and a0 the low-frequency ac alpha, and the noise reduction factor < is assumed to be unity. Emitter-base SCR recombination was not important even at the lowest temperature.

A slightly different story is told by Fig. 11. Again the theoretical curves were obtained using Eq. (99) with 5 = 1. A value of ( N 0.8 will explain the

-THEORETICAL

EXPERMENTAL

10- - I02 103 I 04

EMITTER CURRENT I€ (+A)

FIG. 10. IEQ as a function of IE with temperature as a parameter. Theory and experiment agree over the entire range.


- LEQVS.tE

4

s - 20. LOW-FREQUENCY THEORY

10

4

- I . . . . I 1

20 40’ 00 400 lo00 EMITTER CURRENT ( u A I

FIG. 1 1 . I,, as a function ofl, with temperature as a parameter. A value of < = 0.8 in Eq. (99) will explain the discrepancy between theory and experiment.

small discrepancy shown. This is consistent with the discussion presented earlier.

With this evidence of the basic agreement between theory and experiment we turn to the studies of the input noise emf. Here it is convenient to write the expression for the spectral density of the noise referred to the input in terms of the equivalent noise resistance:

Rk = rb’b + rnb + rSi + gsi(RS + rb,, + re + rCoJ2 ( 1 00)

Here R, is the real part of the source impedance that was maintained at room temperature, rb’b the extrinsic base resistance, rnb the noise resistance of the g-r noise source in the base, rsi tke noise resistance of the uncorrelated part of 2, gsi the noise conductance of i2 referred to the input, re the real part of the emitter junction impedance, and r,,, the real part of the correlation impedance. The theoretical expressions for these various terms are

kT a. Ieo = - -

qlE aF

( 1 OOC)

( 1 OOd)

A typical set of curves of Rk as a function of R, with lE as a parameter is shown in Fig. 12. Figure 13 shows the temperature dependence of R; for several different operating points. The noise shows a broad minimum in the

DEVICE: 2N4062 NO. I2 lo? TEMP: D0.K /

SOURCE RESISTANCE,R' (0)

FIG. 12. Noise resistance as a function of R, and I , as a parameter at T = 100°K.

5 600 - RN' VS. T 214062 NO. 12

d

E 8100 # A L

60 100 150 200 250 300

TEMPERATURE PK) FIG. 13. Noise resistance as a function of temperature with I , as a parameter.

R i V S . IE 2N4062 NO.12

6oo --- t 1s 7 70K

20 40 100 200 400 I000

FIG. 14. Comparison of theoretical and experimental values of noise resistance as a function of emitter current at T = 77°K. The discrepancy is attributed to a g-r noise source in the base region.

EMITTER CURRENT 1~ (PA)


4 s 80 c

70 80 90 00 110 120 TEMPERATURE m)

FIG. 15. Excess (g-r) noise resistance as a function of temperature for several operating currents. These curves were obtained from the differences between theoretical and experimental values of noise resistance as shown in data similar to Fig. 14.

vicinity of 150°K. Figure 14 shows a comparison of theoretical and experimental curves of R; as a function of I,. r,b was assumed to be zero in determining the theoretical curve. The difference between these two curves can be interpreted as being evidence of a g-r noise source in the base region; it leads directly to.the value of rnb. Figure 15 is a plot of this excess low noise resistance rnb as a function of temperature. Further study at this temperature dependence shows that the activation energy of the noise is 0.09 eV, corresponding to twice the activation energy for phosphorous donors in silicon ; this is the value expected theoretically. Another calculation shows that this excess noise emf should depend linearly on the function KI&$b. That this is so is demonstrated by the display shown in Fig. 16.

RNB vs. IB2R&2 100 -

~ = 5 0 0 k &

20 *

C -

0 * . * ' I - 4 6 lo5 2 4 6 O4

FIG. 16. Excess (g-r) noise resistance as a function of the parameter I&&,. The linear

Ig2Rgg2 ( A 2 - l l 2 )

dependence is predicted by the theory.


E. Applications

1 . The Noise Figure of BJT Amplijers (9,38)

The noise figure of a common-base amplifier can be calculated with the help of the physical-T equivalent circuit shown in Fig. 9 after adding the source impedance Z, and its thermal noise (4kTR, Af)’”. The details of the calculation are available in many references (9).

When the same common-base physical-T equivalent circuit is used to calculate the noise figure of a common-emitter amplifier an expression exactly the same as in common base can be obtained. That result, using the circuit of Fig. 17 is

However, here a‘ and i’ include a modification caused by the coupling capacitance c b c between the input and the output. These modified terms are a’ = (a + jwCbcZe) and i’ = i, - a’i, = i, - i,(a + jf&,,z,). For a device like the 2N5829, which will be used as an example in the discussion that follows, Cb, N 0.5 pF at an operating point with 1, N 1 mA, vcE N 10 v, SO that d b c & 5 O.la for f S 1.2 GHZ. Hence, the effect of c b c can be neglected to first approximation in many cases. The noise figure can then be written as

where i = i, - ai, and e = i ,Ze. Many calculations [see for example, Nielsen (441 proceed from this point with the assumption that e and i are completely uncorrelated. In many cases this is acceptable. However, a more general

ai’e

II Cbc Y n

i 8 s

FIG. 17. The common-base physical T noise-equivalent circuit redrawn for use in calculating the noise performance of a common-emitter amplifier. The effect of the capacitance C,, is discussed in the text.


expression results from allowing for correlation of these two noise sources. A convenient procedure is to divide e into two parts e = e' + e", where e' is fully correlated with i and e" is uncorrelated with i. One can then define the correlation impedance

Making these substitions yields the expression

The theoretical expressions for rsi , gsi, and Z,,, are

rsi = $12(ge - g e 0 ) + (q/kT)(zE + zEE)]lze12 (1044

gsi = P/[laI24kT Af] (104b)

(104c)

The low-frequency asymptotic theoretical values of these expressions were used in Section IV,D [Eqs. (10la-d)]. The correlation between e and i can only become significant at high frequencies or at low frequencies, when differences in the ac and dc current gains become important.

As can be seen from Eq. (104), a tuned-noise figure may be obtained when the source reactance X, is adjusted so that

zcar = [ - + 2z:ge0 - Z,*(q/kT)(zE + 21EE)]/2gsi

xs + x, + x,,, = 0 (105)

That tuned-noise figure has the value

Yb'b + rsi gsi F,=l+- + -IR, + re + rb + rc,,12

Rs Rs

This expression is of the form F = A + B/Rs + CR, , which has a minimum value of Fmin = A + 2(BC)'I2 when

R, = (Rs)opt = (B/C)l i2 (107)

Here

A = 1 + 2gsi(re + rb + r,,,) (107a)

(107b)

C = Ssi (107c)

= rb'b + rsi f gsi(re + rb + rcor)2

Hence 2 1 / 2

Fmin = + 2gsi(re + rb'b + reor) + 2[gsi(rb'b + rsi) + g,i2(re + rb'b + rcor) ] ( 108)

354

when

A. VAN DER ZIEL AND E. R . CHENETTE

To expedite the rest of the calculation the parameters G,(f) and G , ( f ) are introduced (45):

Gl(f! = (re + rcor)/reO G 2 ( f ) = 2rsibeO (110)

where f h = fa( 1 - and approximately

Then Fmin may be written as

F m i n = 1 + [1 - a F + ( f P ’ 3 [ G , ( f ) + (rb/reO)l + “1 - aF + (fl!)21[G2(f) + (2rbbeo)l

(112) 2 1/2 + (l - @‘F + (f!)2)[Gl(f) + rbheO1 1 Theoretical expressions for Gl( f ) and G,(f) can be obtained by making the appropriate substitutions. The results are

(1 10a)

The evaluation of IZe12/r,?o and re/reo follows from (19)

G,(f) and G 2 ( f ) are tabulated in Table I as functions offl! for (1 - a) N 0.01. To demonstrate the use of the table consider the example of the type

2N5829 transistor considered earlier. (Assume rb N 100 R and an operating point with I, N 1 mA.) The minimum noise figure at the frequency where f’ = 0.1 1: 100 MHz is calculated to be Fmin = 1.5, which corresponds to about 1.8 dB.

This is in good agreement with the information shown on the contours of constant noise figure in the Zc-Rs plane as shown in Fig. 18 (46). When this type of information is available on a device sheet there is little need for making the detailed calculations as just outlined, except for comparing theory and experiment.


~

0.00 1.000 1.Ooo 0.05 0.892 0.910 0.10 0.730 0.775 0.15 0.624 0.687 0.20 0.563 0.636 0.25 0.526 0.605 0.30 0.501 0.585 0.40 0.468 0.557 0.50 0.445 0.537 0.60 0.425 0.521 0.70 0.407 0.505 0.80 0.390 0.491 0.90 0.374 0.478 1.00 0.359 0.465 1.10 0.345 0.454 1.20 0.332 0.443 1.30 0.321 0.433 1.40 0.310 0.424

a2 a3 05 a7 1.0 u) 30 50 zo 10

Ic, COLLECTOR CURRENT (mA dc)

FIG. 18. Contours of constant noise figure in the Rs-Ic plane for a typical p- high-frequency transistor. The value calculated with Eq. (113) agrees well with (Courtesy Motorola Semiconductor Products, Inc.)

-n-p silicon these data.


2. Noise Performance of Low-Frequency Integrated BJT Amplifiers

Recent improvements in materials and careful attention to design problems has resulted in a new generation of integrated amplifiers with noise performance that is competitive with the best that can be achieved with careful design using selected discrete devices.

A serious and fundamental problem with direct-coupled integrated amplifiers is that, for low source impedances, the noise of the two input transistors contributes equally to the apparent input noise. The apparent input noise power is at least twice that which would be obtained from a single transistor operating under the same condition. As pessimistic as this sounds, Fig. 19 shows that the situation is not too grim when well-designed low-noise BJTs with low rb'b, high fl, and careful processing to minimize llf noise and popcorn noise, are used in the input stage.

-mn 1 I

I02 d FREWENCY (HI)

FIG. 19. The input noise emf of a low-noise integrated amplifier using BJT devices. This design is optimized for use with low to medium source impedances. Source noise is subtracted. (Courtesy of Harris Semiconductor, Melbourne, Florida.)

V. NOISE IN JFETs AND MOSFETs

In JFETs and MOSFETs one has in the first place thermal noise in the conducting channel. There is also g-r noise in the space charge regions. In JFETs one has in addition shot noise of the gate current (the gate-channel


junction is a back-biased p-n junction). Furthermore, especially at lower temperatures, in JFETs there is g-r noise in the channel due to deep-lying traps or to donors (in an n-type channel) or to acceptors (in a p-type channel).

All these mechanisms, except the shot noise mechanism, can be treated by a general schematic (47).

A . The Klaassen-Prim Schematic and Its Applications (47)

When capacitive leakage effects to the gate are neglected, the current equation for a JFET or MOSFET may be writtten as

Z(t) = g ( V ) d V / d x + h(x, t ) (1 14)

where g( V ) is the conductance for unit length at x , V the potential at the point x taken with respect to the source, and h(x, t) a random source function describing the distributed noise in the channel.

We now split the parameters into dc and ac parts, that is, we write Z(t) = I. + Al( t ) and V = V, + AV(x, t), develop into a Taylor series, and neglect higher-order terms. This yields

I , = g(Vo)dVo/dx, AI( t ) = d [ g ( V , ) A V ] / d x + h(x, t ) (115)

where AV(x, t ) = 0 at x = 0 and x = L, L is the device length, and V, is the drain voltage. Integrating (1 15) yields

Z,L = so'" g( V,) dVo, AZ(t)L = soL h(x, t ) dx (116)

Making a Fourier analysis for 0 < t I T and introducing Fourier amplitudes in and h,(x) we can define a current spectrum S r ( f ) for AI(t) and a cross spectrum sh(& x', f ) for h(x, t),

S , ( f ) = lim 2Tini,*, Sh(x, x ' , f ) = lim 2Th,(x)h,*(x') (117) T - r m T-. m

where the asterisk denotes the complex conjugate. Then

The fluctuating sources h(u, t ) at x and x' are uncorrelated, and so Sh(x, x', f ) is a delta function in x' - x . We thus write

sh(x, x', f ) = F(x', f ) s ( x ' - x ) (118a)

or, integrating with respect to x' and replacing x by u


Often one knows the value S , ( x , f ) for a small section Ax centered around x. Hence, applying (1 19) for the section Ax

That is, if we know S,(x, f) we have solved the problem.

B. Examples

1. Thermal Noise in the Conducting Channel

then the thermal noise of a section Ax around x can be represented as If the channel has a conductance g(Vo) for unit length at the point x,

S,(x, f) = 4kTg( b ) / A x or F(x, f) = 4kTg( Vo) (121)

so that

and is found by substituting dx = g(V,)dVo/Zo, introducing Vo as a new variable and substituting I, from (115). When the field in the channel is so large that the mobility p becomes field dependent, T must be replaced by the electron temperature T,, which increases with increasing field. When T, is independent of the field, T = To, where To is the lattice temperature.

Equation (122) holds when the transistor is not saturated, i.e., when Vd < h0, where V,, is so defined that g ( K 0 ) = 0. Beyond that point V, must be replaced by KO. In the case of saturation we may write, if T is field independent, or T = T o ,

where gmax = ard/i?< is the transconductance at saturation and V, is the gate voltage. We now apply this to various cases.

a. The JFET with an n-type channel (48-50). If 2a is the height of the open channel, w its width, and o0 its conductivity, then g ( Vo) is

g(V0) = 2a0wa[l - (- v, + V,i, + VO)l ' z /Vg] ( 124)


where The device saturates at the drain when g(&) = 0 or

is the diffusion potential of the junction and V,, a device constant.

v, = v,, + v, - v,, We find upon integration

a = )(1 + 3y,”’)/(1 + 2y,’l2) ( 126)

where y, = ( - % + Vdif)/VOO, so that a varies between 4 (for y , = 0) and + (for y , = 1). Usually the value of a lies close to 3 , except when source resistance effects are important (49).

GaAs FETs have Schottky barrier gates and a somewhat different geometry. As a consequence some modifications in the theory are necessary. A recent review paper gives details (51).

b. The MOSFET with low-conductivity substrate (52). For this case

dvo) = pwCox(v, - v, - V,) (127)

where w is the width of the channel, ,u the mobility of the carriers, Cox the oxide capacitance per unit area, and V, the turn-on voltage of the channel. The dzvice saturates if

v, = v,, = v, - V, (128)

Evaluating c( one finds a = 4 throughout. Experiments give a N 4 in some units, but in other units a was a factor 2-4 too large (533). There is no satisfactory explanation for this behavior at present.

If the substrate has a higher conductivity and voltage V, is applied to it, then a space charge region develops between the channel and substrate. Since that space charge (due to ionized acceptors for an n-type channel on a p-type substrate) is fixed, only part of the induced charge wCox(% - V, - V,) for unit length is mobile, so that g(Vo) is smaller, and is given by [compare, e.g., van der Ziel (Ma)]

g(V0) = pwCOx(& - VT - V,) - + Vo - K)l1/’ (129)

where E is the relative dielectric constant of the semiconductor, E, = 8.85 x F/m, and N , is the acceptor concentration in the substrate. The

device again saturates if g ( & ) = 0. When S , ( f ) and gmax are evaluated, it is found that a can be somewhat larger than in the previous case (55).

d. JFET or MOSFET with a very short channel. In JFETs or MOSFETs with a very short channel the field in the conducting channel becomes so large that the mobility ,u becomes field dependent and the electron temperature T , rises above the lattice temperature T o . It is difficult to find

c. MOSFET with substrate of higher Conductivity.


accurate closed-form expressions for p ( E ) and T , ( E ) as functions of the field E. The literature abounds with approximations; the one most commonly used is (56,57)

p = po/(1 + E/E,), T , = To(1 + f iE/E,) (130)

where p, is the low-field value of p, E , is a critical field strength, and f i a constant. It is now found that CI increases with decreasing channel length and can rise above unity. This agrees with the evidence presently available, but does not indicate how accurate the approximation is.

2. Noise Due to Traps in the Channel of J F E T s (58)

the type electron + empty trap neutral donor.

to Section II1,C the current fluctuation 61 is

In an n-channel JFET carrier fluctuations occur due to processes of filled trap and electron + ionized donor +

Let a section Ax have AN carriers and let 6 A N 2 = fi AN. Then, according

61 = 106 AN/AN, S A X , f) = ( ~ o / A N ) ~ ~ A N ( ~ ) (131)

where

SA,(f) = 4PANz/(l + 02z2) Now substituting 1, = g( V,) dVo/dx = qp(AN/Ax) dVo/dx

(dVO/dXb 1 + 0 2 z 2 F(x, f) = S A X , f ) A x = 4qpP1,

and consequently, if f l is the average value of f i ,

(131a)

yields

SJf) = (4qpfl10&/L2)z/(1 + 02z2) = 4kTR,, ,gi (133)

This defines the g-r noise resistance Rngr [see also van Vliet and Hiatt (59)l. Usually

z = zo e x p ( q E , / k T ) (1 33a)

where E, is the activation energy of the center involved. Typical values are E, = 0.36 eV and E , = 0.17 eV, indicating trap levels (60). For a similar effect at room temperature, and with E , = 0.55 eV, see Section V,B,3.

3. Generation Noise Due to Centers in the Space Charge Region of J F E T s

In the JFET the junctions are back biased, and a recombination center in the space charge region will alternately generate an electron and a hole that are rapidly collected. This gives rise to a fluctuating charge on the


center, which causes a locally fluctuating width of the channel, which in turn produces a fluctuating current in the external circuit.

The one-dimensional problem can be solved with the help of the Klaassen-Prins schematic; Sah’s (61) and van der Ziel’s (9) one-dimensional treatments can easily be cast into this framework. Here S,( f ) shows a logarithmic divergence at saturation for the one-dimensional problem; according to Lauritzen (62) this is removed by the two-dimensional model. A noise resistance R, can thus be defined, and it can be expressed in the form

R, = A( V,)Z /~ + 0’7’) (134)

where t is the time constant of the centers. Lauritzen’s computer solution could evaluate the function A( V,) numerically. Experimentally A( V,) increases when V , is made more negative (63). The time constant obeys Eq. (133a) with E, = 0.55 eV.

The effect is also present on the space charge region of a MOSFET, but here the noise is usually masked by flicker noise (64). The theory is essentially the same.

4. High-Frequency Behavior of FETs

At high frequencies the ac current AI(x, t ) in the channel depends on position, because of capacitive leakage to the gate. That is, Eq. (115) must be written

Al(x , t ) = -d [g (T / , )AV] /dx (135)

We have here ignored the distributed noise source h(x, t ) and changed the direction of current flow, by considering the current positive when flowing from source to drain. We further change to the voltage Vb between gate and channel; then AV = - A V O , so that

A ~ ( x , t ) = d [ g ( V b ) A V b ] / d ~ (135a)

Now, as is seen from Fig. 20a

d AI/dx = d Al,/dx = j o C ( x ) AVb(x) (136)

where C ( x ) is the small-signal capacitance per unit area between gate and channel at x, and w is the width of the channel. Hence

(137) d*[q( Vo) AF/b(x)]/dx2 = ~ w C ( X ) W A V ~ ( X )

This is the wave equation of the nonlinear RC transmission line describing the device; it holds for both JFETs and MOSFETs.

The frequency dependence of the device parameters and of the noise is governed by the normalized frequency 8:

(1 37a) a = oL’/[p(< - V,)]

362 A. VAN DER ZIEL AND E. R. CI-ENETTE

L I q1 tZ$)

(b ) FIG. 20. Schematic diagram for deriving the wave equation of the active distributed RC

transmission line representing the FET. (b) Schematic diagram of the transmission noise model used for calculating the noise response of the FET.

The device ceases to operate as a low-noise device for a > 3. It is thus important to make L small, p large, and V , - V, relatively large. These conditions are best met in GaAs FETs.

It is possible to express a in terms of device parameters. To that end we observe that the gain-bandwidth product of the device gm/(2nCgS) in saturation may be written

(137b)

since gm = pCOxw(V, - VT)/L, C,, = @2,,,wL, where C,, is the gate-source capacitance in saturation (19).

(137c)

so that the condition < 3 corresponds to f < 2Bmax. This makes it possible to read the maximum frequency for low-noise operation directly from the the specification sheet of the device.

This derivation holds for MOSFETs, but should be approximately true for JFETs. For short channels hot-electron effects occur.

We now solve the noise problem as follows (65): We short-circuit source and drain; this means that our transmission

line is short-circuited at both ends. Next we assume a noise emf with Fourier component Au,(x) between x and x + Ax. Let the short-circuited transmission lines, when seen from x, have impedances Z,(x) and Z,(x), respectively, and let noise components Au,(x) and Aul(x) developed across


them (Fig. 20b), then

AUI(X) = Avn(X)z , / (z , + ZZ), AvZ(x) = -Au,(x)ZZ/(Z, + Zz) (138)

where Zl(x) and Z,(x) can be evaluated from the transmission line model. Next the noise currents with Fourier components Ai,(O) and Ai,(L) in

the leads short-circuiting input (x = 0) and output (at x = L ) are evaluated and expressed in terms of Au,(x), i.e.,

Ai,(O) = -u,(x)AzJ,(x), Ai,(L) = ~, , (x)AK(x) ( 139)

The gate noise thus has a Fourier coefficient

Aign = [bn(x) - an(x)]Aun(x) (1 39a)

We now make up Ai,(L)A$(L), AignAi;,,, and AignAi:(L), sum over all sections Ax, and so obtain the corresponding gate noise spectrum S , ( f ) , drain noise spectrum s d ( f ) , and cross-correlation spectrum sgd(f ) , bearing in mind that

Sv,( f ) = lim 2TAu,(x)Au,*(x) = 4kTAx/g(VO) (1 39b)

The problem was solved by Shoji (65). Klaasen and Prins (66,67) used a somewhat different approach, whereas van der Ziel gave an approximate solution (68,69). Usually it is most convenient to make a Taylor expansion of the gate and the drain noise, and terminate the expansions at the lowest nonzero term. This yields a gate noise spectrum S , ( f ) , varying as w2, a drain noise spectrum & ( f ) independent of frequency, and a cross-correlation spectrum s g d ( f ) varying asjo, in full agreement with van der Ziel’s approximate solution. The results are valid for up to moderate frequencies. In that approximation the correlation coefficient

T- m

is imaginary and has an absolute value of about 0.40. For the reason why IcJ is so much smaller than unity, see van der Ziel(9).

When one evaluates the high-frequency gate conductance gg for moderately high frequencies (70), gg varies as o2 over a wide frequency range. One may thus express S , ( f ) in terms of gg. At saturation

where y = 3 for a MOSFET and varies between 1 .O and 4 for a JFET. When one evaluates the complex transconductance Y, of the device

at very high frequencies (19,70) one sees that Y, decreases slowly with increasing frequency. By extending Shoji’s analysis to higher frequencies, Choe et a/. (71) were able to show that & ( f ) increases slowly with increasing frequency, in agreement with Baril’s experiments (72). As a consequence the noise resistance R, increases slowly with increasing frequency.


C . Other Noise Sources in JFETs and MOSFETs

1. Shot Noise

The gate-channel junction of a JFET behaves as a p-n junction and hence the gate current I , = I g l ( V ) - I g 2 . According to Section I11 the two currents should show full shot noise, that is,

S,,(f 1 = W g l ( V ) + W g 2 (142)

For a floating gate 161( V ) = IB2 = I , , , and

Srs( f) = 4kTI,o = 4kTjR,o ( 142a)

where In principle it would be better if the gate were back-biased, but in practice

some forward bias may be tolerated as long as this does not seriously affect the noise figure of the FET stage.

The substrate-channel junction of a MOSFET also behaves as a p-n junction and hence the above discussion also applies to it. For a discussion of shot noise at elevated temperatures see van der Ziel(73).

= qI,,/kT is the ac input resistance for floating gate.

2. Avalanche Noise

When a JFET or MOSFET is biased far beyond saturation, the field in front of the drain becomes so large that avalanche multiplication occurs. The minority carriers are immediately taken out of circulation, by the gate in the JFET case and by the substrate in the MOSFET case, so that only the majority carriers are retained (74).

Let n, be the number of carriers arriving in the avalanche region and pz, the number arriving at the drain; then the multiplication factor is M = iid/ii,. Burgess' variance theorem applies and hence

var nd = ( M ) 2 var n, + ii, var M (143)

If we now change to currents and introduce the spectra SId(f) and S I s ( f ) of the drain and the source current, respectively, we have

(143a) ' , d ( f ) = ( M ) 2 s I s ( f ) + 2q'S var

where S , , ( f ) = a4kTg,,, in saturation. We shall see that

var M = M(M - 1) ( 144)

Since the ionization processes are independent random events, the added current AI in section Ax has shot noise. The current equation for the avalanche region is now

d l = a(x)I d x (145)


with I = I, at x = 0. Solving for I yields

I / l , = exp [ st a(u) du] or R = exp [ a(u) du] (146)

M(0, x) = exp[st a(u)du], M(x, d ) = e n p [ l a(u)du] (146a)

The added noise is any section dx is therefore

dSI = 2q d l = 2qla dx = &I, exp [st a(u) du] a(x) dx (147)

This is multiplied by [M(x, 41’. Therefore, the section Ax gives a contribution dSId to SId, and the total effect 2q1,varM is found by integration. Hence

2qI, var M = 2 q 1 , ( ~ ) 2 sod exp [ - J; a(u) du] a(x) dx = ~ ~ I , M ( M - 1)

from which (144) follows. We have here substituted

J. CI(u) du = y (x ) , a(x) dx = dy

D. Comparison with Experiment

The problem of comparison with experiment is challenging because, as the above discussions have indicated, the relative importance of the various noise sources depends strongly on the device materials and geometries.

Fortunately the early work by Bruncke and Bruncke and van der Ziel demonstrated the validity of the thermal noise model for germanium alloy junction devices ( 4 9 9 ) . Here the main noise source was the white thermal noise of the conducting channel. The equivalent noise resistance was found to be in good agreement with the classical expression

Rn = CIIglno (148)

where CI z 5 for the ideal FET. This work was done on relatively large, low-frequency devices of cylindrical geometry. The effects of extrinsic source resistance on the small-signal and noise performance were calculated and found to be in good agreement with the experimental results.

As device geometries were reduced, the effects of high fields became of increasing significance. Klaassen (56) studied the influence of hot-carrier effects on the thermal noise of FETs. He took account of both the mobility saturation and the increased free-carrier temperature as described in Eq.


3.0

0.2 0.3 0.4 0.5 0.6 0.7

(Vgs - Vth)

FIG. 21. The normalized saturation noise resistance a = R,g,, of a MOSFET tetrode as a function of the applied input gate-source voltage V, - VT. Solid curve: theory; data points: experimental. Channel length 1.5 jim (F. M. Klaassen I.c.).

(130). Figure 21 shows the results as a plot of the normalized saturation- noise resistance of a short-channel MOSFET as a function of the applied gate-source voltage at room temperature. The effect is much stronger at lower temperatures, because in Eq. (130) E , decreases with increasing temperature, whereas increases. Near 77"K, however, the effect could be completely masked by g-r noise effects for JFETs. A careful analysis is then needed to determine what part of the effect is due to hot electron noise and what part is caused by g-r noise involving donors.

G-r noise effects were extensively studied by Hiatt (63). Figure 22 shows noise at room temperature due to g-r processes in the space charge region, as a function of gate bias; apparently A(<) decreases if 5 is made less

I lo 102 lo3 0' 10' 102 . ..,...I " . . . . . ( 1 . ' . . - d

Fnq. (k)

FIG. 22. Noise resistance spectra at room temperature due to g-r processes in the space charge region with the gate bias as a parameter.


I W D2 10' 10' lo5 FREO. (W

FIG. 23. Noise at three temperatures near room temperature to deduce tht activation energy of the centers involved in the g-r process.

negative. Figure 23 shows noise at three temperatures near room temperature, from which it was deduced that the activation energy of the centers involved was 0.55 eV. Figure 24 shows noise at three different temperatures around 180°K due to traps in the channel, from which an activation energy of 0.36 eV was deduced. A similar set of spectra around 120°K indicated an activation

W I K) W* d 10' lo5

FREQ. (Hz)

FIG. 24. Noise at three temperatures around 180°K due to g-r noise from traps in the channel. An activation energy of 0.36 eV was deduced from these data.

368

e I,,(MERMAL)

A. VAN DER ZIEL AND E. R. CHENETTE

energy of 0.17 eV. Figure 25 shows a spectrum at 80"K, indicating practically white noise due to g-r effects involving donors.

When transit time effects become important, the noise increases monotonically with frequency. This problem was investigated experimentally by studying the noise performance of specially fabricated long-channel (1 -3 mil

I0 IM 3M K)M x)M 50M

FREO. (Hd

FIG. 26. Spectra of long-channel SOS MOSFETs, indicating an increase in noise at high frequencies due to transit time effects in the channel. At low frequencies this thermal noise is masked by excess noise.


channel length, 30-90 mil channel width) silicon-on-sapphire devices (71,72) so that the frequency effects would be in a convenient frequency range (Fig. 26). Precise evaluation of the data is difficult because of the presence of excess low-frequency noise, but the high-frequency results were in excellent agreement with the results of the explanation based on transit time effects, as Fig. 27 indicates. The theoretical expression for frequency dependence of the output noise current generator at saturation was shown to be expressible in terms of Bessel functions of fractional order. At lower frequencies, &( f ) is expressible in a Taylor expansion in W involving the low-frequency transconductance gmo :

s d ( f) = 4kTg,,[# f 1.29 X lo-2(W)2 -k 9.1 X 10-4(w)4 f.. .] (149)

Unfortunately this Taylor series converges very poorly for W > 2, so that one has to use the rigorous expressions above that frequency. Note that Eq. 149 reduces to the well-known result o! = $ for 5 = 0.

0'5 t

E. Noise Figure of F E T Amplifiers (9)

The noise figure of an FET amplifier can be calculated with the help of the equivalent circuit shown in Fig. 28. Here the circuit shown is for a


FIG. 28. Equivalent noise circuit of a JFET or MOSFET amplifier.

common-source amplifier and it is assumed that the gate to drain capacitance has been neutralized. It is a well-known theorem that the noise figure of an amplifier is independent of the connection if feedback effects can be neglected. Hence the result that follows is valid for neutralized common-gate and common-drain amplifieres as well as for the common-source connection. It is also valid for both JFETs and MOSFETs.

1. Theory

The drain noise is described by the noise current generator id, and the gate noise is described by the generator i, = ib + ii. Here il, is assumed to be fully correlated with id, ii is uncorrelated with id, is represents full thermal noise of the source conductance g, , y, is the complex source admittance, y , represents the complex gate admittance, and y , the complex transadmittance of the FET. The model is general and valid at the highest frequencies. The mean-squared value of the output current can be written as

We now equate this to the output noise current caused by the apparent input noise current (z)f'2; that is,

2 - it = F * 4kTg, Af

The result is

It is convenient to define the noise conductance g,, the noise resistance R,, and the correlation admittance y,,, as follows:

9, = p/4kT df; Rn = z/[ly,l24kT df]; ye,, = i;Ym/id

Thus F can be written

F = 1 + (gn/gs + (Rnlys + Y g + Yeorl2/gs)


which has the tuned value

~t = 1 + (gn/gs) + [Rn(gs + gg + gcorY/gsl (153)

when

b, + b, + bco, = 0 (153a)

It is a straightforward calculation to show that this tuned-noise figure has the minimum value

(154) 2 112 Fmin = 1 + 2 R n ( g g + g o , ) + 2[Rngn + R,2(gg + gcor) I

when

gs = [(gg + gcorl2 + ~ n / R n l ’ ‘ ~ (1 54a)

Halladay and van der Ziel demonstrated the use of measurements of the tuned-noise figure as a tool in characterizing MOSFETs (54).

For a junction FET the theoretical values of the noise parameters are

Rn = a/grnO > g n gg 9 Scor << gg

These expressions are valid up to high frequencies, provided that it is taken into account that the factor CI increases slowly with increasing frequency. Hence to a good approximation

Fmin = 1 + 2agg/gmo + 2[(agg/gm,)(l + agg/gmJl”2 (154’)

This gives relatively low noise up to about a = 3.

2. Experiments

Recent advances in materials and in device fabrication have yielded GaAs FETs with superior high-frequency, low-noise performance. Devices with channel lengths of less than 1 pn show maximum frequencies of oscillation greater than 100 GHz. Tuned-noise figures in 50 0 systems of the order of 1 . 2 dB at 4 GHz, 1.5 dB at 6 GHz, 2dB at 10 GHz, and 4 dB at 18 GHz have been reported (75). The device measured at 18 GHz yielded a gain G,,, = 12 dB.

It is interesting to compare these data with the best reported results for silicon BJTs. Devices with fT N 9 GHz showed tuned-noise figures of about 3 dB at 6 GHz with G,,, N 8 dB (75).

Figure 29 shows the improvement that may be available with the advent of practical fabrication of devices using terniary compounds such as InGaAs (76).

372

F=8 OH2

A. VAN DER ZIEL AND E. R . CHENETTE

-2

I

3. Noise Figure of the Tetrode F E T (77)

The tetrode FET can be thought of as an “integrated cascode circuit” (a common-source stage followed immediately by a common-gate stage). It has great appeal for high frequency applications because, in many cases, it is not necessary to neutralize the gate-drain capacitance to obtain stable operation. However, the effect of this capacitance and the contribution of the common-gate part of the circuit to the noise figure can be seen in the exact expression for the noise figure of the cascade FET. It can be written as

Here RH, accounts for the contribution of the common-gate portion to the total noise of the cascade circuit. It has the form

(155a)

FIG. 29. Predicted behavior of GaAs and InCaAs FETs (15% InAs) with uniform velocity saturation profiles. Noise plotted vs. lD/lDs. Note the improvement possible with InGaAs (76).


Here the terms are as defined in the discussion of noise figure above. The subscripts 1 and 2 refer to the first (common-source) and second (common- gate) parts of the circuits, respectively. For example, yml is the transadmittance and xu, the output admittance of the common-source transistor.

At relatively low frequencies, WCdg << J y m l J , Ix,, + y,,l << lym1/, and gn2 << y,, . In that case Rh, << R,, and the noise figure is simply that of the input stage.

Some improvement in the high-frequency noise performance of the cascade circuit can be obtained by neutralizing C,, and also by tuning at the interstage. This latter tuning has the effect of modifying the (Gut + ygsz + ycorz) term in Rh, . An example of the results that can be obtained are shown in Fig. 3 4 which shows a comparison of the noise figure for a single device operating in a common-source neutralized tuned amplified and with the same devices operating in cascade with and without c d g neutralized. Also shown is the additional improvement that can be obtained by tuning at the interstage. (77).

10

a

.E 6

B w

E U

4

2 I

TUNING

fwa 4

FIG. 30. Noise figure vs. frequency for the cascade circuit, the cascade circuit with neutralization of the first half, the cascade circuit with neutralization and with tuned interstage between first and second half, and the FET triode with neutralization (77).

4. Integrated Low-Frequency Amplijiers with JFET Direrential Input

Recent developments in integrated amplifiers have led to a differential JFET input stage in front of a normal BJT amplifier. The advantages are an extremely high input impedance and a very low offset current when compared to a BJT differential stage.


Figure 31 shows data for an amplifier with a well-designed low-noise JFET differential input stage (77a). This amplifier also shows a very low corner frequency for excess low-frequency noise. For low-source-resistance applications the BJT amplifiers show somewhat lower equivalent input noise emf. However, the equivalent input noise current of the JFET amplifier is less than 0.01 pA/Hz1I2.

EWIWENT NFUTNOE WLTAGE (EXFANDED SCALE)

FREQUENCY (Hz)

FIG. 31. Equivalent input noise voltage (in nV/Hz''') for an integrated circuit with JFET differential input. (Courtesy National Semiconductor Corp.) (774.

VI. MISCELLANEOUS SOLID-STATE DEVICE NOISE PROBLEMS

A. Shot Noise in Josephson Junctions

A Josephson junction in its simplest form consists of a superconducting point pressed against a superconducting plate with a very thin (10-20 A) oxide layer between. The current flow is by tunneling, but there are some unexpected features due to the superconducting properties of the materials.

In a superconductor there are two types of electrons: normal, single electrons, denoted by n and carrying a charge q, and Cooper pairs, denoted by p and carrying a charge 2q. The latter are responsible for the superconducting properties. Both the normal electrons and the Cooper pairs take part in the tunneling process. The normal electrons carry a current I , = I,, - In2, where Znl and In2 flow in opposite directions, and the paired electrons give rise to a pair current I , = IP1 - ZPz , where ZP1 and IPz flow in opposite directions. Hence

(156) I = I , + I , = I , , - I n 2 + I , , - I,2


Each current fluctuates independently and all show full shot noise. However, since the pairs carry a charge 2q,

S,n(f) = 2q(In1 + I n 2 ) ~ ( f ) = 2qInp(f)(1n1 + I n 2 ) / ( I n 1 - I n 2 1 (157)

S,,(f) = 4qUP1 + Ip2)p(f) = 4ql,p(f)(lPl + ~p2)/(Ipl - Ip2) (158)

where p ( f ) is the quantum correction factor. A calculation shows that

(159) ( I n 1 + 1n2)/(1n1 - I n J = coth(qV/2kT)

V p 1 + 1,2)&1 - l p 2 ) = coth(2q’WT)

Consequently the sum of the two noises is

S J f ) = 2qlnp(f)coth(qV/2kT) + 4qI,p(f) coth(2qV/2kT) (160)

For small values of qV/2kT this may be written

S, ( f ) = 4kTp(f)(ln + Ip)/’V = 4kT(I/V)p(f) ( 161)

The characteristic shows a kink at I/ = 0, since a maximum pair current I , can flow across the junction at zero bias in either direction. Consequently I /V # dI/dV, even in the limit V -+ 0.

We have hereby added the quantum correction factor to an otherwise well-known formula (78); the factor is important for Josephson junction mixers in the high kMHz range and beyond. The low-frequency formula has been well verified by experiment (79).

B. Space-Charge-Limited Solid-state Diodes

In single-injection diodes one electrode injects carriers into the semiconductor material, and the other electrode collects them. The current flow is limited by the space charge of the injected carriers (hence the name) and the characteristic is

I , = &,poAV;/d3 ( 162)

where V , is the applied voltage, d the electrode distance, E the relative dielectric constant of the semiconductor, A the cross-sectional area of the device, and p, the carrier mobility.

The noise in each section Ax is thermal noise, but due to the nonlinear characteristic of the device, the noise is twice the thermal noise of the ac conductance y (80) :

S A f ) = 8kTg ( 163)


at all frequencies. At low frequencies g = go = dl,/dV, = 21,/V,, or

SAf) = 16kTI,/V,, S d f ) = S I ( f ) / g g = 4kTV,/I, (163a)

These equations were verified by Liu (81) and by Nicolet and Golder (82). In double-injection diodes electrons are emitted by the one electrode and

collected by the other, whereas holes are emitted by the other electrode and collected by the one. There is now approximate space charge neutrality and the carriers have a lifetime z. The characteristic can have different forms, but at high frequencies (07 >> 1) the ac conductance is g = la/V, and the noise is (83)

Sr(f) = 4kTg ( 1 64)

This has been verified by experiment (84). For further details we refer to two recent review papers (84a,b).

C. Negative-Conductance and Negative-Resistance Amplijiers

Figure 32a shows a negative-conductance amplifier of source conductance gs and negative conductance -&, in parallel; thermal noise is associated with gS, whereas - g d has a short circuit noise current of spectral intensity S r ( f ) . The noise figure of the circuit is seen by inspection to be

F = 1 + Sr( f ) /4kTgs (165)

The value of F for g8 = gd is denoted as F , , because the available power gain is then infinite, and F , - 1 is called the noise measure M of the device. Obviously

M = S I ( f )/4kTg* (165a)

Figure 32b shows a negative-resistance amplifier of source resistance R, and negative resistance R in series; thermal noise is associated with R,, whereas - Rd has an open-circuit noise voltage of spectral intensity S,(f). The noise

FIG. 32. (a) Negative conductance amplifier in which device has a negative conductance -gd and its noise is represented by a parallel noise current generator [S,(f) Af]’’’. (b) Negative

.resistance amplifier in which device has negative resistance - R, and its noise is represented by a series noise emf [S,(j)Af]1’2.


figure of the circuit is seen by inspection to be

F = 1 + S,( f ) / 4 k T R , (166)

The value of F for R, = Rd is denoted as F , , because the available power gain is then infinite, and F , - 1 is called the noise measure M of the device. Obviously

M = S,( f ) / 4 k T R d (1 66a)

We now apply this to several cases.

1. Tunnel Diode Arnplijier

negative conductance is - g d . Consequently, The limiting noise of the diode is shot noise of the current Id and the

S,(f 1 = w, (167)

= (ql2 )Id/gd (167a)

Typically M is of the order of 1, and so the device is a rather low-noise device.

2. Transferred Electron Negative Conductance ArnpliJer (85,86)

In these devices the part n, of the carrier density n is in the lower valley with high mobility and the part n2 is in the upper valley with low mobility. Let these carriers have mobilities p, and p2 and diffusion constants D , and D , , respectively. Then the effective diffusion constant is defined as

D = ( n l D , + n2D2) /n (168)

and the effective mobility is defined as

P = ( W l + n,CLz)/n (1 68a)

so that the average drift velocity

ud = PE (168b)

where E is the field strength. D , , D 2 , pl, and p,, as well as n , and n 2 , are functions of the field E. The differential mobility p' is defined as

p'(E) = dud/dE = p + E dp/dE (169)

and for appropriately chosen fields it is negative. Hence the small-signal negative conductance is

- gd = qnp'(E) A / L ( 169a)

where A is the cross-sectional area of the device and L its length.


If we neglect noise due to intervalley transitions, the limiting noise in this amplifier is diffusion noise of the carriers. According to Section 11,

S,( f ) = 4q2(Dlnl + D2n2)A/L = 4q2DnA/L (170)

M = qD(E)/kTI”E)I (171)

Substituting (169a) and (170) into (165a) yields

According to Sitch and Robson (87) the lowest theoretical noise figures for GaAs and InP are 7 and 4 dB, respectively; the best.experimenta1 results reported so far are 10.5 dB for GaAs and c8 dB for InP, as quoted by Sitch and Robson. According to Littlejohn et al. (88) even better results might be expected for Ga, -Jn,P1 - y A ~ y alloys.

3. I M P A T T Diode Amplijiers

IMPATT diodes have a dynamic negative resistance in a certain frequency range and hence they can be used as microwave amplifiers.

Gummel and Blue (89) calculated the noise measure M for germanium diodes and found a minimum measure of 20-30 dB, depending on the parasitic (series) resistance rp and on the current density; the noise measure decreased with decreasing series resistance and with increasing current density. The calculated frequency response of M showed a sharp dip in M at an optimum frequency Apt.

According to Rulison et al. (90) germanium structures with a noise figure of 30 dB have been reported. According to Gummel and Blue, similar silicon structures of the same doping profile have typically 6-8 dB higher noise figures. These noise figures are so high that these devices are not very useful as amplifiers.

The dependence of F on the parasitic series resistance rp is easily understood from Fig. 33. According to this figure,

F - 1 = rp/Rs + S,( f ) / 4 k T R s (172)

FIG. 33. Negative-resistance amplifier with parasitic series resistance rp having thermal noise.


and the maximum power transfer occurs if R, = Rd - rp. Substituting this into (172) yields

(1 72a)

where M o = SV(f ) /4kTRd is the noise measure without series resistance. We see that M goes to infinity if Rd approaches r p .

= rp/(Rd - r p ) + MORd/(Rd - r p )

D. Switching Devices

We discuss here the unijunction transistors and the p-n-p-n diodes.

1. The Unijunction Transistor

A unijunction transistor consists of a resistive element between two electrodes B , and B2 and a diode E that is alloyed in at one side and that injects minority carriers into the resistive element when forward-biased. These injected carriers modulate the conductance of the resistive element and are the reason why the characteristic (IE, V,,,) shows a negative conductance region for forward bias.

Baertsch and van der Ziel (91) studied the noise to elucidate the device behavior. They found two sources of noise in the current I , .

(1) Thermal noise of the real part of the impedance ZEB. This comes about because the diode can be considered a diode with a large series resistance so that the noise is mostly thermal noise of this resistance.

(2) Noise due to the recombination of the injected minority carriers with majority carriers. This has a spectrum of the form

KIy(1 + 0 2 2 2 ) (173)

characteristic of recombination noise. At higher currents the high-frequency part of that spectrum varies as which is typical for diffusion-dominated recombination noise, caused by the fact that the recombination mostly takes place at the surface (the carriers have to diffuse to the surface in order to recombine).

These results are quite compatible with what is known about the operation of the device.

2. The p-n-p-n Diode

The p-n-p-n diode is a bistable device, i.e., it has the two stable states “on” and “off.”

The noise is 1,” noise, which is indicative of a surface mechanism. The noise is orders of magnitude larger than in a normal p-n diode, because we


P n P T 1 I

n P

deal here with a device that is governed by an internal feedback mechanism; the device can be considered as the parallel connection of a p-n-p and an n-p-n transistor, in which the collector of the first transistor is connected to the base of the second transistor and the collector of the second transistor (Fig. 34). This gives rise to a considerable amount of feedback, which increases the noise by several orders of magnitude (92).

I * n

FIG. 34. p-n-p-n structure represented by a p-n-p and an n-p-n transistor partly

In addition they found that there was a sharp peak in the equivalent saturated-diode current* I,, of the short-circuited device noise current in at the particular current I for which dV/dZ was zero and V as a function of f had a minimum. Apparently the open-circuit emf en behaves more or less normally, and hence, if 2 is the ac device impedance we have

connected in parallel, to illustrate the feedback effect in the device.

= ~ / p l z (174)

which shows a sharp peak where 121 has a minimum. The noise behavior is thus quite compatible with what is known about the device otherwise.

REFERENCES

I. General Review Papers

E. R. Chenette, Noise in semiconductor devices. Ado. Electron. 23, 303 (1967). A. van der Ziel, Noise in solid state devices and lasers. Proc. IEEE 58, 1178 (1970). A. van der Ziel, The state of solid state device noise research. .Fhysicu (Urrechr) 83B, C , 41

(1976).

11. Detailed References

1 . H. Nyquist, Phys. Reo. 32, 110 (1928). 2. H. B. Callen and T. E. Welton, Phys. Reu. 83, 34 (1951). 3. B. M. Oliver, Proc. IEEE 53,436 (1965).

* is defined by the relationship Zqf,, = p/Af, when Af is the frequency interval for which 7 is defined and q is the electron charge.


4. See K. M. van Vliet, Solid Stare Electron. 13,649 (1970); J. Math. Phys. 12, 1998 (1971). An early presentation of the ideas is found in K. M. van Vliet and J. Blok, Physica (Utrecht) 22, 231 (1956).

5. A. van der Ziel, “Noise, Sources, Characterization, Measurement.” Prentice-Hall, Englewood Cliffs, New Jersey, 1970. Chapter 5 gives a review of unpublished work by the late Dr. A. G. Th. Becking on this subject. K. M. van Vliet and A. van der Ziel, Solid State Electron. 20, 931 (1977). The basic ideas presented here were first presented in lectures on linear response theory by van Vliet, and in lectures on noise by van der Ziel.

7. A. van der Ziel, Physica (Utrecht) 81B, C , 107 and 230 (1976). Unfortunately the zero point energy term in p(f) is omitted there.

8. A. van der Ziel, Physica (Utrecht) WB, C , 262 (1977). 9. A. van der Ziel (“Noise, Sources, Characterization, Measurement.” Prentice-Hall,

Englewood Cliffs, New Jersey, 1970) discussed this problem in detail. 10. K. M. van Vliet and J. R. Fassett, in “Fluctuation Phenomena in Solids” (R. E. Burgess,

ed.), Chapter 7. Academic Press, New York, 1965.

6.

11. A. van der Ziel, Physica (Utrecht) 81B, 1 1 1 (1976); J . Appl. Phys. 47, 2059 and 5487 (1976).

12. A. van der Ziel, Proc. IRE 43, 1639 (1955). 12a. A. van der Ziel and A. G. T. Becking, Proc. IRE 46, 589 (1958). 13. M. J. Buckingham and E. A. Faulkner, Radio Electron. Eng. 44, 125 (1974). 14. A. van der Ziel and K. M. van Vliet, Solid State Electron. (1978) (in press). 15. M. L. Tarng and K. M. van Vliet, Solid State Electron. 15, 1055 (1972). 16. H . S. Min, K. M. van Vliet, and A. van der Ziel, Phys. Status Solidi A 10, 605 (1972);

13, 701 (1973). 17. H. S. Min and K. M. van Vliet, Phys. Status Solidi A 11, 653 (1972). 18. K. M. van VIiet and H. S . Min, Solid State Electron. 17, 267 (1974). 19. A. van der Ziel, “Solid State Physical Electronics,” 3rd ed., Chapter 16. Prentice-Hall,

Englewood Cliffs, New Jersey, 1976. 19a. A. van der Ziel and K. M. van Vliet, Solid State Electron. 20, 721 (1977). 20. K. M. van Vliet, Phys. Status Solidi A 16, K13 (1973). 21. K. M. van Vliet, Solid State Electron. 15, 1003 (1972). 22. P. 0. Lauritzen. IEEE Trans. Electron Devices 4-15, 770 (1968). 23. See A. van der Ziel, “Noise,” Chapter 6. Prentice-Hall, Englewood Cliffs, New Jeisey,

1954. 24. A. van der Ziel, Solid State Electron. 18, 969 (1975). 25. K. M. van Vliet, IEEE Trans. Electron Devices ed-23, 1236 (1976). 26. K. M. van Vliet and A. van der Ziel, IEEE Trans. Electron Devices 24, 1127 (1977). 27. L. Scott and M. J. 0. Strutt, Solid State Electron. 9, 1067 (1966). 28. T. E. Wade and A. van der Ziel, Solid State Electron. 19,909 (1976). 29. R. J. McIntyre, IEEE Trans. Electron Devices ed-13, 164 (1966). 30. W. Lukaszek, A. van der Ziel, and E. R. Chenette, Solid State Electron. 19, 57 (1976). 31. A. van der Ziel, “Noise in Measurements.” Wiley (Interscience), New York, 1976. 32. A. van der Ziel, J. Appl. Phys. 47,2059 (1976). 33. M. McColl. D. T. Hodges. and W. A. Garber, 2nd In?. Conf Ginter Sch. Sub-mm Gaves

and Their Appl.. 1976 Conf. Rep. No. 62 (1976). 34. J . M. Small, G. M. Elchinger, A. Javan, A. Sanchez, F. J. Bachner, and D. L. Smythe,

Appl. Phys. Lett. 24, 275 (1974). 35. H. C. Montgomery and M. A. Clark, J. Appl. Phys. 24, 1337 (1952). 36. A. van der Ziel, J. Appl. Phys. 25,815 (1954).


37. L. J. Giacoletto, in “Transistors I,” p. 296. R. C. A. Laboratories, Princeton, New Jersey, 1956.

38. A. van der Ziel, “Fluctuation Phenomena in Semiconductors.” Butterworth, London, 1959.

39. A. van der Ziel, Solid State Electron. 20, 715 (1977). 39a. H. S . Min and K. M. van Vliet, Solid State Electron. 17, 285 (1974). 40. A. H. Tong and A. van der Ziel, IEEE Trans. Electron Devices ed-15,307 (1968). 41. T. E. Wade and A. van der Ziel, Solid State Electron. 19, 381 (1976). 42. T. E. Wade, K. M. van Vliet, A. van der Ziel, and E. R. Chenette, ZEEE Trans. Electron

Devices 4-23 , 1007 (1976). 43. T. E. Wade, A. van der Ziel, E. R. Chenette, and G. Roig, ZEEE Trans. Electron Devices

ed-23, 1007 (1976). 44. E. G. Nielsen, Proc. IRE 45, 957 (1957). 45. A. van der Ziel, J. A. Cruz-Emeric, R. D. Livingstone, J. C. Malpass, and D. A. McNa-

mara, Solid State Electron. 19, 149 (1976). 46. Device specification sheet for Motorola 2N4957, 2N4958, 2N4959,2N5829. 47. F. M. Klaassen and J. Prins, Philips Res. Rep. 22, 505 (1967). 48. A. van der Ziel, Proc. I R E 56, 1808 (1962). 49. W. Bruncke, Proc. IRE 57,378 (1963). 50. W. Bruncke and A. van der Ziel, ZEEE Trans. Electron Devices ed-13, 323 (1966). 51. R. A. Pucel, H. A. Haus, and H. Statz, Adv. Electron. 38, 148 (1965). 52. A. G. Jordan and N. A. Jordan, ZEEE Trans. Electron Devices 4-12, 148 (1965). 53. L. D. Yau and C. T. Sah, Solid State Electron. 12,927 (1969). 54. H. E. Halladay and A. van der Ziel, Electron. Lett. 4, 366 (1968); Solid State Electron.

12, 161 (1969). 54a. A. van der Ziel, “Solid State Physical Electronics,” 3rd ed., Chapter 18. Prentice-Hall,

Englewood Cliffs, New Jersey, 1976. 55. C. T. Sah, S. Y. Wu, and F. H. Hielscher, ZEEE Trans. Electron Devices ed-13, 410

(1966). 56. F. M. Klaassen, IEEE Trans. Electron Devices ed-17,858 (1970). 57. K. Takagi, Y. Sumino, and K. Tabata, Solid State Electron. 19, 1043 (1976). 58. A. van der Ziel, Proc. IRE 51, 1570 (1963). 59. K. M. van Vliet and C. F. Hiatt, IEEE Trans. Electron Devices ed-22,616 (1975). 60. C. F. Hiatt, A. van der Ziel, and K. M. van Vliet, ZEEE Trans. Electron Devices 4-22 ,

614 (1975). 61. C. T. Sah, Proc. ZEEE 52,795 (1964). 62. P. 0. Lauritzen, Solid State Electron. 8,41 (1965). 63. C. F. Hiatt, Ph.D. Thesis, University of Florida, Gainesville (1974). 64. L. D. Yau and C. T. Sah, ZEEE Trans. Electron Devices ed-16, 170 (1969). 65. M. Shoji, IEEE Trans. Electron Devices 4-19 , 520 (1966). 66. F. M. Klaassen and J. Prins, IEEE Trans. Electron Devices ed-16,952 (1969). 67. F. M. Klaassen and J. Prins, Philips Res. Rep. 23,478 (1968). 68. A. van der Ziel, Proc. ZEEE 51,461 (1963). 69. A. van der Ziel and J. W. Ero, ZEEE Trans. Electron Devices 4-11 , 128 (1964). 70. J. Geurst, Solid State Electron. 8, 88 and 563 (1965). 71. H. M. Choe, W. B. Baril, and A. van der Ziel, Electronics 21, 589 (1978). 72. W. B. Baril, M.Sc. Thesis, University of Minnesota, Minneapolis (1975). 73. A. van der Ziel, Solid State Electron. 12, 861 (1969). 74. C. S. Kim, Ph.D. Thesis, University of Florida, Gainesville (1971). 75. H. F. Cooke, Avantek, Santa Clara, California. private communication. 76. R. L. Bell and S. G. Bandy, Varian Associates, Palo Alto, California. 1977. Final Tech-

nical Report N00014-76-C-1056.


77. 77u.

78. 79.

80. 6ri. 82. 83. 84.

84a. 84h.

85. 86. 87. 88. 89. 90. 91. 92.

A. van der Ziel and K. Takagi, IEEE J. Solid-State Circuits sc-4, 170 (1969). Specification Sheet of LF155/LF156/LF157 monolithic JFET input operational amplifiers. National Semiconductor Corp.. ,Santa Clara, California. S. Stephen, Phys. Reo. 182, 531 (1969). H. Kanter and F. L. Vernon, Jr., Phys. Let!. A 32, 155 (1970) Phys. Rev. Lett. 25, 588 (1970). A. van der Ziel. Solid State Electron. 9, 899 (1966). S. T. Liu. Solid Stute Electron. 10, 253 (1967). M. A. Nicolet and J. Colder. Pliys. Status Solidi A 17, K49 (1973). C. H. Huang and A. van der Ziel, Physica (Urrechr) 78,220 (1974). C . H. Huang and A. van der Ziel, Physica (Utrecht) 76, 172 (1974). M. A. Nicolet, H. R. Bilger, and R. J. J . Zijlstra, Phys. Status Solidi B 70, 9 (1975). A. van der Ziel. in “Space-Charge-Limited Solid State Diodes, Semiconductors and Semimetals” (A. K . Willardson and A. C. Beer, eds.). Academic Press, New York. H. Thim, Electron. Lett. 7 , 106 (1971). H. A. Haus, IEEE Trans. Electron Devices ed-20, 264 (1973). J. E. Sitch and P. N. Robson, IEEE Trans. Electron Devices 4-23, 1086 (1976). M. A. Littlejohn, J. R. Hauser, and T. H. Glissen, Appl. Phys. Lett. 30,242 (1977). H. K. Gummel and J. L. Blue, IEEE Trans. Electron Devices ed-14,569 (1967). R. L. Rulison, G. Gibbons, and J. G. Josenhans, Proc. IEEESS, 223 (1967). R . D. Baertsch and A. van der Ziel, IEEE Trans. Electron Devices ed-13, 683 (1966). D. L. Presthold and A. van der Ziel, IEEE Trans. Eleciron Devices ed-14, 336 (1967).

AUTHOR INDEX

Numbers in parentheses are reference numbers and indicate that an author’s work is referred to although his name is not cited in the text. Numbers in italics show the page on which the complete reference is listed.

A

Aamodt, L. C., 219(44), 235. 252, 253(44),

Aberth, W., 118, 124, 125 Abgrall, M. H . , 95, 126 Abrams, R. L., 137, 175, 199 Ackerman, I . , 16.50 Adams, A., 141, 142,200, 203. 205 Adams, M. J . , 260(70), 288, 209 Afromowitz, M. A . , 234(48). 235(48), 309 Ahnell, J. E., 123, 124, 125 Albach, G . G . , 154, 199 Albain, J . L., 27, 28,51 Alcock. A . J. . 150,204 Alexander, M. H . , 186, 195, 199 Allen, W. D., 77, 128 Allman, S. L.. 202 Altamura, O., 161,202 Anbar, M.. 115, 118, 122, 124,125, 128, 129 Anderson, L. W.. 148, 173,203 Anderson, R . W. , 182, 183,199 Anderson, V . E. , 126 Anstis, G . R. , 7, 13.50, 52 Antonini, E., 283(86), 310 Arecchi, F. T., 163, 199 Armstrong, L., Jr., 133, 199 Arnold, H. D., 223(46), 309 Arrathoon, R., 177, 199 Asundi, R. K., 63, 78,129 Atkinson, J. B., 179, 180, 182,204 Auchet, J. C., 160,205 Azira, R., 78, 125

309

B

Bachner, F. J., 335(34), 381 Bader, H. , 157, 199 Baertsch, R. D., 379,383 Bafus, D. A., 109, 127

Bakale, G., 115, 125 Baker, R. F., 4,52 Bakos, J . , 136, 199 Balling, L. C., 184, 185,201, 202, 206 Bandy, S. G., 371(76), 382 Bansal. K. M., 95, 125 Bansal, M., 305(125), 311 Bard, A. J . , 242(56), 309 Bardhan, P., 49,52 Bardsley, J. N., 56,125 Baril, W . B., 363(71, 72), 382 Barnes, D., 290(94), 310 Bashkin, S., 139, 199 Bass, M., 144, 145, 146, 199 Basting, D., 146, 147, 199 Bates, D. R., 179 Batley, M., 95, 116. 125 Baumann, H., 123, 124, 125 Baumgartner, G., 134, 199 Beadle, B. C., 260(70), 288(70), 309 Beauchamp, J. L., 71, 73, 95, 125, 127 Beaven, G. H., 302(121), 311 Beck, G., 151, 199 Becker, R . S . , 95.126, 129 Becking. A . G. T., 322(12a), 337(12a),

341(12a). 374(12a), 381 Beenakker, J. J. M., 192,204 Begum, G . M., 95, 126 Bell, A. G., 210, 215, 242,308 Bell, R. L., 371(76), 382 Bennett, H. S., 219(45), 235,309 Bennett, R. A., 126 Bennett, R. G., 158, 199 Bennett, W. R. Jr., 140, 156, 199 Bentley, J . , 36, 51 Berg, R. A., 3 0 3 124). 311 Berger, A. S., 157, 199 Bergmann, K., 137.200 Bergstedt, K . , 185, 200 Berlande, J., 136, 182,200, 202 Bernstein, R. B., 191, 203

385

AUTHOR INDEX

Bersohn, R., 182,200 Bertoncini. P. J . , 187,205 Bethe, H. A , , 12,50 Bethe, K., 123, 124,125. 126 Bevington. P. R., 166, 169,200 Beyer, R. A , , 188,200 Bilger, H. R., 376(84a), 383 Biondi, M. A,, 126 Bischel, W. K., 199,203 Blau, L. M., 77, 126 Blickensdelfer, R. P . , 184,200 Blue, J. L., 378(89), 383 Boersch, H. , 4,51 Boesl, O., 188,200 Bolden, R. C., 157, 175,203 Bosquet, P., 208(1), 308 Bott, 3 . F., 191,200 Bouby, L., 95, 126, 127 Bouchoule, A., 175,200 Bouldin, D. W., 126 Boulmer, J . , 143, 157, 173, 176, 177, 178,

Bourhe, M., 178 Bowen, D. K . , 23,51 Bowers, M. T., 64, 129 Bowie, J. H., 126 Bowman, H. F., 300(112),310 Bradley, D. J. , 150,200 Brau, C. A., 164,200 Breckenridge, W. H . , 184, 200 Brehm, B., 77, 126 Brewer, L., 133,200 Bridgett, K. A., 140, 200 Briegleb, G . , 95, 126 Broida, H. P., 188,200, 205 Brown, L. M., 17, 35, 36, 49,51 Brown, 0. B., 157, 199 Broyer, M., 140, 193,200 Bruncke, W., 359(49, 50), 365,382 Brunori, M. , 283(86), 310 Buckingham, M. J. , 324(13), 381 Budick, B., 135,200 Burke, P. G., 76, 126 Burrel, C. F., 177,200 BUKOW, P. D., 76, 126 Buxton, B. F., 44,51

200, 205

C

Callen, H. B., 314(2), 315(2), 380 Callender, R. H., 138,200

Calloway, A. R., 153,203 Camhy-Val, C., 142,200 Capelle, G., 187, 188,200, 205 Capelle, G. A., 149,205 Carlson, L. R., 198,205 Carlsten, J . L., 161, 201 Carpenter, R. W., 36,51 Carroll, M. M., 242(60), 243(60), 309 Carter, J. G., 67,68,69,70,89,95, 101, 115,

116, 117, 126, 127. 128 Catherinot, A., 175, 200 Cehelnik, E. D., 164,200 Celotta, R. J., 126 Chaney, E. L., 67, 101. 116, 117, 126, 127 Chanin, L. M., 126 Chase, L. H. , 276(81), 309 Chafelain, C. L., 214(36), 308 Chaudhuri, J . , 126 Chen, C. H., 173, 177,204 Chen, E., 95, 104, 126. 129 Chen, E. C. M., 126 Chenette, E. R., 332(30), 335(30), 338(30),

Chevalier, J.-P., A. A., 36, 48, 51 Chw, H. M., 363(71),382 Christcdoulides, A. A., 126 Christophorou, L. G., 56, 57, 58, 59, 60, 63,

64, 65, 67, 68, 69, 70, 80, 81, 82, 85, 86, 87, 88, 89, 95, 96, 99, 101, 102, 115, 126, 127, 128, 129

Chupka, W. A,, 126 Church, D. A.. 140,200 Chutjian. A., 133,200 Clapp, P. C., 49.51 Clark, J. H., 187,202 Clark, M. A. , 335(35), 381 Clark, R. , 185, 195,203 Coates, P. B., 153, 157,200 Cochran, W. G., 169,205 Cockayne, D. J . H., 5 , 16, 21,51 Cohen, J. B., 49.52 Cohen, J. S., 175.200 Collins, C. B., 157, 173, 192,200 Collins. P. M., 67,68,95, 101, 116, 117,125,

Comer, J. , 78, 127. 128 Compton, R. N., 63, 95, 96, 100, 102, 104,

106, 111, 116, 118, 119, 120, 121, 122, 126, 127, 128

38 I

126, 127

Cook, T. B., 178,201, 205 Cooke, H. F., 371(75), 382

AUTHOR INDEX 387

Cooper, C. D., 95. %, 100, 104. I l l , 116, 118, 119, 120, 121, 122, 127, 128

Corey, R. B., 300(16), 301(116),31/ Cotton, D. W. K . , 293(101), 310 Cottrell, T. L., 213(22), 308 Courtens, E., 163, 199 Cowley, J. M. , 4, 5 , 8, 9, I I , 12, 14, 18, 19,

22, 23, 26, 27, 28, 29, 31, 36, 39, 46, 47, 48, 5 0 , 5 / . 53

Coxon, J. A, , 178,205 Craig, N . C., 191,200 Crandall, I . B., 223(46), 309 Cravalho, E. G., 300(112), 310 Craven, A. J. . 17, 35, 36, 48, 49.51 Crewe, A . V., 2, 22.51 Crick, F. H. C., 300( 117), 311 Cruz-Emeric, J. A., 354(45), 382 Cubeddu, R., 149,200 Cuellar, E., 264(73), 309 Curry, S. M . , 149, 200 Cuvellier, J., 136, 182,200, 202 Cvejanovir, D., 141,200. 203

D

Dagdigian, P. J . , 186, 189, 190, 195, 199,

Dalby, F. W., 188,200 Damgaard, A . , 179, 199 Davidson, J . A , , 185,200 Davies, C. C., 140, 157, 199, 200, 203 Davies, D. D., 191, 202 Davis, F. J., 95, 127 Davy, P., 175,200 Dawson. J. F., 184,206 DeCorpo, J . J . , 109, 127 de Diego, N . , 36.51 Deech, J . S. , 163, 179, 185,200 De Gennes, P. G . , 264(74), 309 Delaney, M. E., 213,308

200,206

Delwch, J.-F., 143, 154, 157, 168, 171, 173, 174, 176, 177, 178, 180, 188,200,201, 205

Demtrader, W., 134, 137, 187, 199, 200, 201 Depristo, A. E., 186, 199 Deutsch, T. F., 144, 145, 146, 199 Devoret, M., 142, 201 Devos, F., 157, 171, 174, 176, 177, 180, 188,

de Vries, A. E., 191, 192, 203 200, 201

Dewey, C. F., Jr., 213(29), 214(32), 308 de Witte, O., 149, 202 Dexter, D. L., 276(83), 310 Dickson, H. W., 63, 126 Dispert, H . , 95, 127 Docken, K. K., 187, 190,201 Doemeny, L. J . , 158, 160,203, 205 Dolder, K . T., 76, 122, 123, 127, 129 Donlop, G. L., 36,5/ Donnelly, V. M., 187, 188, 201 Donohue, D. E., 157,201 Donovan, J . W., 302,311 Donzel, B., 160,205 Dorelon, A., 136,201 Dorset, D. L., 46, 51 Dougheny, R. C., 124, 127 Dowell, W. C. T., 20,5/ Dreux, M., 142,200 Drexhage, K. H., 146,201 Dube, L., 78, 127 Dubochet, J. , 35,5/ Dubrevil, B., 175, 200 Ducas, T. W., 187,20/ Dufay, M . , 139,201 Duff, 1. S., 165,201 Dumont, A. M., 142,200 Dunning, F. B., 150, 178,201, 205 Dupouy, G., 24,5/ Durup, J. , 124, 127 Duschinsky, F., 133,201 Dyson, R. D., 166,203

E

Eades, J . A,, 44, 51 East, E. J. , 292(97), 310 Eck, T. G. , 135,201 Edelson, D., 65, 127 Edelstein, S . A., 181, 182, 201 Edwards, 1. G., 151,201 Ehrhardt, H., 78, 127, 129 Eland. J . H. D.. 142,201 Elchinger, G. M., 33334). 381 Eng, S. T., 214(30), 308 Engel, A., 35,5/ Epstein, W. L., 293(102), 310 Erdmann, T. A., 180,201 Eman, P., 141,201 Ero, J . W., 363(69), 382

388 AUTHOR INDEX

Estberg, G. N., 77, 124, 127 Everett, M. A., 295( 109), 310

F

Fabelinskii, I . L., 150,201 Fabre, C., 163, 165,202 Fairbank, W. M., 195.201, 241(55), 309 Farragher, A. L., 89, 116, 127 Fassett, J. R., 320, 374(10), 381 Faulkner, E. A, , 324(13), 381 Fehsenfeld, F. C . , 100, 127 Feinberg, R., 186, 194,201 Feldrnan, D. W., 276(82), 309 Feldrnan, P., 60, 100, 127 Feneuille, S., 133, 199 Fenster, A., 153,201 Fessenden, R. W. , 95,125 Fichtner, W., 151,201 Fields, P. M., 12, 51 Figger, H., 180, 201 Fink, D., 139,199 Fiquet-Fayard, F., 83, 126, 129 Fishrnan, V. A., 242(56), 309 Fitzsimmons, W. A., 148, 173,203 Flach, R., 148,201 Flynn, G., 191,205 Flynn, G . , 191, 205 Flynn, G. W., 187, 190,202, 203 Foldy, L. L., 135,201 Foman, R. A., 219(45), 235(45,49), 309 Forst, W., 103, 127 Foster, M. S., 71, 73, 95, 127 Fournier, P. R., 136. 182,200, 202 Fowler, R. G., 140,203, 205 Fowler, W. B., 276(83), 310 Frank, J . , 7, 13.53 Franken, P. A., 135,201 Franklin, J . L., 109, 127, 128, 164,204 Fraser, R. D. B., 293(100), 300(100),

Frazia, J. R., 96, 127 Frernlin, J. H., 124, 127 Freund, R. S . , 136,201 Frey, A. R., 253(64), 309 Friesem, A. A., 149,201 Fues, E., 16,51 Fujirnoto, F.. 22, 39,51 Fukayama, K., 293(102), 310 Furcinitti, P. S., 184, 201

301(100), 304(100), 305(100),310

Fusaro, R. M., 293(104),310 Futrell, J . H . , 71, 72, 73, 128

G

Gallagher, T. F., 181, 182,201 Ganiel, U . , 146, 149,201 Gant, K. S., 95, 99, 126, 127 Garber, W. A,, 335(33), 381 Gardner, D. G., 167,201 Gardner, J . C . , 167,201 Gaussorges, C., 179,201 Gauthier, J.-C., 154, 168, 171, 172, 174, 180,

Gautschi, M. , 156,202 Gear, C. W., 165, 171,201 Geindre, J.-P., 154, 168, 174,201, 204 Geiss, R. , 27, 36,49,51 Gemmill, C. L., 304(122), 311 German, K. R., 188,201 Gershenzon, M., 260(68), 309 Gersho, A., 216(42, 43), 223(43), 308 Gersten, J . I., 137, 138,200, 201 Geurst, J . , 363(70), 382 Giacoletto, L. J. , 336, 382 Giannelli, G., 161, 202 Gibb, 0. L., 149,202, 205 Gibbons, G., 378(90), 383 Gibson, J . R., 76, 127 Giniger, R . , 187,203 Gjannes, J. K., 39, 43.44.51 Glaeser, R. M., 32, 33,52 Glennon, B. M., 141,206 Glissen, T. H., 378,383 Goans, R. E., 80, 81, 82, 87, 88, 126, 127 Gcdard, B., 149,202 Goddard, T. P., 182, 183, 199 Golay, M. J . E., 160,205 Gold, B., 161, 202 Golden, D. E., 76, 127 Golder, J., 376,383 Goldwasser, A,, 161,205 Golebiewski, A., 56. 129 Goode, G. C., 128 Goodman, P., 5, 12, 13, 16, 21, 39, 40, 42,

43, 44, 46, 47,51, 52 Gordon, H. R. , 157, 199 Gordon, J . G., 262(72), 309 Gorelik, G. , 212(20), 213(20), 308 Gornik, W. , 180, 181, 184,202

188,200, 204

AUTHOR INDEX 389

Gounand, F., 136, 182,200, 202 Gowrnay, L. S., 242(58). 309 Grabiner, F. R., 191.202 Gragg, J. E., Jr., 49, 52 Grant, M. W., 56, 57, 58. 126 Gray, H. B., 262(72), 309 Gray, R. C., 242(56), 309 Green, S . , 137,202 Greenspan, H., 165, 202 Gregg, E. C., 115, 125 Griffith, A., 17, 36, 49, 51 Grifiths, J . E., 65, 127 Grigson, C. W. B. , 22, 52 Grinvald, A., 160, 202 Grischokowsky, D. R., 199,202 Gronsky, R.. 29,53 Gross, M.. 163, 165,202 Grossman, L. W., 196, 197,202 Guillard, R. R. L., 286(89), 310 Gummel, H. K. , 378(89), 383 Gunther, F. , 4, 53 Gupta, A., 257(66), 270(77), 309 Gupta. J. P., 257(66), 309 Gusinow, M. A,, 77, 126 Guthohrlein, G., 140,202 Giittinger, H., 156,202

H

Haas, Y., 187, 202 Hacker, W., 151,201 Hackett, C. E., 213(29), 308 Hadek, V . , 264(73), 309 Hadjiantoniou, A., 67,68,69,89,95, 115,126,

Hafner, W., 165,202 Hall, C. R. , 23, 5 / Hall. J . L.. 77, 126 Hall, S . S . , 209(10), 271(10), 272(10),308 Halladay, H. E., 359(54), 382 Halpern, J . B.. 188,202 Hamming. R. W . , 161,202 Hancock. G . , 188.202 Hansch. T. W., 148, 149, 186, 190, 194,200,

201, 202, 203, 205, 241(55), 309 Happer, W., 135, 185,202. 203, 205 Harde, H., 140, 202 Hardy, A., 146,201 Harland, P. W., 95, 109, 128, 129 Haroche, S . , 163. 165, 185,202

127, I28

Harshbarger, W. R., 214,308 Hart, E. J . , 115, 128 Hanig, P. R., 151, 152, 157,202. 203 Hashimoto, H., 29,52 Haus, H. A. , 359:51), 377(86), 382, 383 Hauser, J. R. , 378(88), 383 Havey, M. D., 185,202 Haxo, F. T., 286(90), 310 Hazan, G . , 160, 202 Heavens, 0. S., 179,202 Hecht, H. G . , 208(2), 308 Heernskerk, J. P. J . , 192,204 Heidner, R. F., 191, 200 Heinicke, E., 123, 124, 125 Helman, W. P., 158,202 Hernbree, G. G . , 27, 28.51 Henglein, A., 95, 104. 128 Henis, J . M. S., 71, 74, 128 Hermann, Y., 78, 129 Herrick, D. R. , 123, 128 Herzberg, G . , 61, 128 Herzenberg, A., 78, 127 Hessel, M. M., 187. 205 Hewes, R. A., 276(80), 309 Hiatt, C. F., 360(59, 60), 361(63), 366,382 Higgs, L. A., 160,204 Hill, N . R., 160,202 Hill, R. M., 181, 182. 199,201. 203 Hillier, J . , 4, 52 Hirnmel, G . , 185,200 Hinchen, J. J . , 193,202 Hinze, J . , 187, 190,201 Hiraoka, H . , 76, I28 Hirsch, P. B., 12, 23,52 Hirschfeld, T., 208(3), 308 Hobbs, R. H., 193,202 Hocker, L. O., 190,202 Hodges, D. T., 335(33), 381 Hoerni, J . , 16, 47,52 Hoffman. M. V. , 276(80), 309 Hogan, P. , 191.202 Holbrook. K. A., 103. 129 Holiday, E. R. , 302(1211,3/1 Holstein, T., 164, 202 Holt, H. K. , 164, 202 Hordvik, A., 237(52), 309 Horwitz. H., 182,200 Horwitz. J.. 292(98). 310 Horz, G., 36.51 Hotop, H.. 77. 128 Houk, K . N . , 95, 127, I28

390 AUTHOR INDEX

Houston, P. L.. 187,202 Hove, C. A. G., 300(113), 310 Howard, C. J., 185, 200 Howie, A., 12, 23, 49,52 Hsu, G. H., 264(73), 309 Huang, C. H., 376(83, 84), 383 Huebner, R. H., 96, 127 Huetz-Aubert, M., 190,202 Hughes, B. M., 95, 95, 104. 106, 128 Hurst, G. S., 95, 102, 106, 173, 174, 195, 196,

197,127, 202, 204

I

Iijima, S., 2, 31.52 Imhof, R. E., 141, 142,203, 205 Inoue, N., 293(102), 310 Inouye, K., 303124). 311 loup, G. E. , 160,202. 203 Ippen, E. P., 150, 205 Isaacson, M., 33, 35,52 Isenberg, I . , 166,203 Itzkan, l . , 197,205 Ivanov,A. J., I l l , 128

J

Jaeger, T., 144, 150,204 Jagur-Grodzinski, J . , 126 James, D. R. , 126 James, G., 104, 128 Jameson, D. G., 154,203 Jannes, G., 128 Janney, R., 156,203 Jap, B. K., 14, 19, 46, 50.51 Jauan, A., 190,202, 333341,381 Jefferies, R. , 151,201 Jeffrey, S . W., 286(90), 310 Jenkins, F. P., 290(94), 310 Jennings, D. A,, 185,200 Jensen, A., 286(89), 310 Jeyes, S. R., 185, 194, 195,203 Johns, H. E., 153,201 Johnson, A. W. , 140,203 Johnson, A. W. S., 44,52 Johnson, B. W., 157, 173, 192,200 Johnson, J. P., 67,68,70,95,96,101, 106, 1

Johnson, L. F., 260(68), 309 Johnson, M. L., 166,203

114, 128 112,

Johnson, S. A,, 198, 205 Jones, L. G. P., 17, 35, 36, 49.51 Jones, P. F., 153,203 Jordan, A. G., 359(52), 382 Jordan, N. A., 359(52), 382 Jore, A., 27,53 JQrgensen, S. W., 188.203 Josenhans, J . G., 378(90), 383 Jost, R., 136,201 Joy, D. C., 50.52 Joyez, G., 78, 128 Judish, J . P., 173, 174,202

K

Kaiser, D., 180, 181,202 Kaiser, H . J., 123, 124, 125 Kakivaya, S . R., 300(113), 310 Kamm, R. D., 213(29), 308 Kanter, H., 375(79), 383 Kao, L. W., 95, 129 Kasatochkin, V . I . , 24.52 KatB, H., 185, 194, 195,203 Kaufman, F., 187, 188,201 Kaya, K., 214(35, 36), 308 Keck, J., 176,204 Kellenberger, E., 35, 51 Kenik, E. A., 36.51 Kenyon, N. D., 213(27), 308 Keto, J . W., 149,204 Keubler, N. A., 214(36, 37), 308 Khadjavi, A., 135,203, 205 Kielkopf, J. F., 180,203 Kim, C. S., 364(74), 382 Kindlmann, P. J., 140, 156, 199 King, A . A., 260(70), 288(70), 309 King, G. C., 141, 142,200, 203, 204 King, T. A., 140, 157,200,203 Kinder, L. E., 253,309 Kirkhright, G. F., 260(70), 288(70), 309 Kiser, R. W., 129 Kishida, Y., 305(124), 311 Kistemaker, P. G., 191,203 Kittel, F. K., 150, 201 Klaassen, F. M., 357(47), 360(56), 363, 365,

Klar, H., 137,203 Kleppner, D., 187,201 Klots, C. E., 106, 128, 173, 174,204 Knaap, H. F. P., 192,204

382

AUTHOR INDEX 39 1

Kniseley, R. N., 237(50), 242(50), 309 Knox, W. A,, 33, 34,52 Kobayaski, Y.. 303124). 311 Koch, F. A., 27, 28.51 Koehler, W. R . , 294(108),310 Koepke, J . W . , 261(71),309 Kohanzadeh, Y., 242(60), 243(60), 309 Koike, F., 83, 84, I28 Koski. W. S., 123. 124, 125 Kossel, W., 4, 14, 52 Kovacs, M. A , , 190,202 Kramer, S . D., 197.202 Krause, L., 136, 180. 182, 184, 185,203. 205 Krauss, M., 78. 100. 120, 128 Kreuzer, L. B.. 213. 251(25),308 Kubota, S . , 140, 203 Kuck, J . F. R . , Jr., 292(98), 310 Kuhl, J.. 144, 150,203 Kunii, T. L., 104, 116, 128 Kunze, H. J.. 104, 116, 128 Kunze. H. J.. 177,200 Kuroda, H., 104. 116, I28 Kurzel , R . B . , 29 1196). 3 10 Kwong, H . S . . 136, 180,205

LaBahn, R. W., 77, 124, 127 La Budde, R. A,, 191,203 Lacmann, K., 95, 127 Lahmann, N. , 244, 246,309 Laiken, S. L., 166,203 Landry, J. D., 27, 28,51 Lange, W., 180, 181, 184,202 Langhans, L., 78, 79, 127 Langmore, J . , 33, 35,52 Lankhard, J. R . , 199, 202 Larson, R. S., 242(5Y), 309 Latimer, C. J., 178, 205 Lau, A. M. F., 199,203 Lauritzen, P. O . , 328(22), 329, 330, 361,381

Laush, G., 167,201 Lawler, J . E., 148, 173, 203 Lawton, M., 157, 175,203 Leach, S . , 142,201 Leblane, J. C., 153,201 Le Calve, J., 178,203 Lee, J. H., 140,203 Leenhouts, A , , 27,53 Lehman, M. D., 294( 108). 310

382

Lehmann, J. C., 193, 200, 203 Lehmann, W . , 276(82), 309 Lehmpfuhl, G., 5,13,16,21,22,36,40,42,43,

Lehning, H., 121, 128 Leigh, R . W., 138,200 Lemont, S., 187,203 Lempicki, A, , 144, 146,203, 204 Lenzi, M.. 188,202 Lepoutre. F., 190,202 Lesech, C., 179,201 Leskovar, B., 151, 152, 157,202. 203 Letokhov, V. S., 196,203 Levin, L.. 197.205 Levy. R. H. , 197,205 Levy-Leblond, J . M., 123, 128 Lewis, C.. 158, 160,203 Lewis, N. S . , 262(72), 309 Liaaen-Jensen, S., 286(89), 310 Lifshitz, C., 95. 96. 104, 106, 128 Linder, F., 78, 79, 80, 83, 84, 127. 128. 129 Lindquist, L., 189, 203 Lineberger, W. C., 77, 128, 188,200. 206 Link, J . K., 133,200 Lipsch, J . M. J. G. , 266(75), 309 Littlejohn. M. A , , 378(88), 383 Littman, M. G., 187,201 Liu, C. H., 140,200 Liu, S. T., 376,383 Livingstone, R. D.. 354(45), 382 Lo, C. C., 151, 152, 157,203 Lombardi, M., 136,201 Lopez-Delgado, R., 189,203 Lotem, H., 149,203 Lovelock, J . E. , 95, 104, 129 Lucas, N. S . , 293(103), 310 Ludewig, H. J., 244(62), 246(62), 309 Luft, K. F. , 212( 18). 308 Lukaszek, W., 332(30), 335(30), 381 Lurianovich, V. M., 24.52 Lurio, A., 135, 203. 205 Luther, J., 180, 181, 184,202 Luypaert. R. , 163, 179, 185,200 Lynch, D. F., 43.52 Lynch, R. T., 149,203 Lyons. L. E., 95, 116, 125, 128

47 ,5 / , 52

M

Mabie, C. A,, 71, 74, 128 McAfee, K. B., Jr., 65, 127

392 AUTHOR INDEX

McCaffey, A. J . , 185, 194, 195,203 MacCarrol, R., 179,201 McClelland, J. F., 231, 242(50), 309 McClure, D. S., 257,309 McColl, M., 335(33), 381 McCorkle, D. L., 56, 57, 58, 68, 70, 95, 96,

101, 106, 112, 114, 126, 128 McCoubrey, J . C., 213(22). 308 McCreary, R. D., 115, 125 McDonald, F. A,, 239(54), 242(54), 252,309 MacDonald, R. G. , 190,203 MacFarlane, S., 49,52 MacGillavry, C. H., 4, 15.52 McGowan, J. W., 76, 128 Mcllrath, T. J., 161, 162, 204 Mclntyre, R. J . , 332(29), 381 McMahon, A. G., 47.52 McNamara, D. A , , 354(45), 382 MacRae, T. P., 293(100), 300(100), 301(100).

Magyar, G., 146,204 Mahan, B. H., 128 Maher, D. W., 50,52 Malmberg, P. R.. 139, 199 Malpass, J . C., 354(45), 382 Mandl, F., 56, 125 Mann, K. R., 262(72), 309 Mannami, M., 29,52 Mansbach, P., 176,204 Marek, J., 179, 180,204 Margoliash, E., 281(85), 310 Margulies, S., 169,204 Marowsky, G., 149,204 Martin, J. J . , 154,203 Martin, M. M., 189,203 Martinson. I . , 140,204 Masnou-Seeuws, F., 179, 201 Mason, H. S., 304( 123), 311 Masterson, K. D., 140,204 Masurant, T., 294(108), 310 Mathis, R. A, , 126 Matolsky, A. J., 293(99), 301(99), 310 Matsen, F. A., 112, 128, 164,204 Maugh, T. H. , 11, 209(6), 308 Max, E., 214(30), 308 May, C. A., 198,205 Maytal, M., 160,202 Measures, R. M., 177, 183,204 Meier, K. , 180, 181, 184 Meinel, A. B., 139, 199 Meinke, W. W., 167,201

304(100), 305(100), 310

Melton, L. A., 186, 190,206 Mercadier, M. E., 215,308 Mercer, G. N., 140, 156 Merer, K.. 180, 181, 184,202 Merkle, L. D., 209(11), 275, 276(11, 84),

278(11), 308, 310 Meyer, J . , 154, 199 Michl, J . , 95, 128 Mielenz, K. D., 164, 200 Mier, P. D., 293(101). 310 Mies, F. H., 78, 179, 182, 128. 204 Miller, T. A., 136, 201 Mills, J. C., 5, 16, 21.51 Min, H. S., 326(16, 17, 18). 327(16, 17, 18).

337(39a), 344(39a), 345(39a), 347(39a), 374(16, 17, 18), 381, 382

Misell, D. L., 32, 33, 52 Miyake, S., 39, 51 Mohamed, K . A., 142,204 Mollenstedt, G., 4, 14.52 Monahan, E. M., Jr., 237,309 Montgomery, H. C., 335.381 Mocdie,A. F . , 5 , 12, 16,20,21,39,43,44,5/ Mooradian, A., 144, 150,204 Moore, C. B. , 187, 190, 191,200, 202 Moore, D. S., 184,200 Morrison, D., 165,204 Morse, D. L., 268(76), 309 Morton, R. A., 118, 128 Moss, S. C., 49, 52

Muccini, G. A., 95, 104, 128 Munchausen, L. L., 95, 128 Munro, 1. H., 189,203 Murphy. J . , 276(82), 309 Murphy, J . C., 219(44), 235(44), 252(44),

MOY, J.-P., 154, 174,204

253(44), 309

N

Naff, W. T., 95, 96, 100, 111, 127. 128 Naiki. T., 29, 52 Nathan, R. , 19, 46,52 Nayfeh, M. H., 173, 174, 177, 195,202. 204 Nazaroff, G . V . , 56, 129 Nelson, D. F., 260(68), 309 Nelson, D. R.. 95, 127 Nemzek, T. L., 158, 160,205 Nesbet, R. K., 76, 128 Neumann, D., 100, 120, 128 Neumann, G., 146, 149,201

AUTHOR INDEX 393

Neusser, H. J. , 188,200 Npuyen, T. D.. 178,204 Nicholas, D. J.. 77, 128 Nicholas, J . V., 150,200 Nicholson, R. B., 12, 23,52 Nicolet, M. A. , 376(84a), 383 Nielsen, E. G., 352,382 Nielsen, P. E. Hvjlund, 14, 27, 28,51 Niemax, K., 180,204 Nolle, A. W., 237,309 Nordieck, A . . 171.204 Norgard, S . , 286(89), 310 Novick, R., 60, 77. 124, 126, 127 Nowacki , W., 243,309 Nyquist. H.. 314, 321(1),380

0

O'Connor, C. M., 118, 129 Odom, R. W., 71, 72, 73, 128 Oka, T., 192,204 O'Keefe, M. A. , 7, 12, 13, 52 Oliver, B. M., 314, 3133). 380 Olsen, R. L., 295(109), 310 Olson, R. E., 182, 204 Ortner, P. B . , 285(88), 286(88),3/0 Ottesen , K . K . , 268(76), 309 Ottinger, C., 137, 138,204 Ovenall, D. W., 121, 128

P

Pace, M. 0.. 126 Pace, P. W., 179, 180, 182,204 Paech, F., 187,204 Page, F. M., 89, 95, 116, 127. 128 Pai, R . Y.. 126 Paisner, I . A . . 185, 198,202. 205 Palmer, L. D., 95, 128 Pankove, J . I . , 258(67), 309 Papir. Y . , 299(110), 300(110),310 Pappalardo, R., 146, 204 Park, R., 287(92), 310 Parker, J . G . , 216, 219(44). 23344). 252(44)

253(44), 308. 309 Parker, J . W., 173, 203 Parlant, G., 83, 129 'Parravano. C., 182, 183, 199 Pascale, J . , 136.200 Pashley, D. W., 12, 2 3 , Z

Patel, C. K. N., 214(31), 308 Pauline, L., 300, 301(116),3ll Payne, M. G., 173. 174, 177, 197,202, 204 Peart, B., 122, 123, 129 Pebay-Peyroula, J. C., 178,204 Pendrill, L. R., 179, 200 Perkampus, H. H . , 272(79), 309 Peterson, E. W., 304( 123). 3f 1 Petty-Sil, G., 188,200 PFund, A. H. , 211,308 Phelps, A. V., 126 Pietenpol, 1. L., 77, 124, 129 Pike, C. T.. 197,205 Pillet, P., 163, 165, 202 Pines, E., 294(106, 107), 293107). 296(106,

Pinnekamp, F., 185,200 Polanyi, J . C., 193,204

107). 297(107), 298(107), 303(107), 310

Ponomarev.0. A., 1 1 1 , 128 Popov, N. M., 24.52 Porter, D. A., 36,5 / Potternas, J . P., 104, 128 Powell, R. C., 209(11). 27311). 278(11), Prangsrna, G. J. , 192,204 Prasad, A . N., 82, 129 Preece. W. H., 216,308 Presthold, D. L., 380(92), 383 Prins, J. . 357(47), 363(66, 67). 382 Printz, M. P., 166, 203 Prockop, D. J., 305(124), 3 / l Provencher, S. W., 167,204 Pruett, J. G. , 189, 190,204 Pryce, M . H. L., 188,200 Pucel, R . A. , 359(51),382

308

R

Rabinovitch, B. S., 103, 129 Rackham, G. M., 44.51 Rademacher, J.. 85, 86, 129 Rader, C. M., 161,202 Radloff, H. H. , 180, 181, 184, 202 Ramachandran, G. N., 305( 125), 311 Ramakrishnan, C., 305(,125), 311 Randerath, K., 271(78), 309 Rayleigh , Lord, 2 15,308 Read, A. W., 213(24), 308 Read, F. H. , 59, 78, 129, 141, 142,203, 204.

Rees, A. L. G., 4 ,51 205

394 AUTHOR INDEX

Reinhardt, P. W., 95, 102, 104, 106, 127 Rembaum, A., 270(77), 309 Rhodes, C. K., 190,202, 203 RibariE, M . , 165,202 Ricard, D., 146,204 Richardson, M. C., 150,204 Riecke, W. D., 5 , 24, 52 Riera, A., 179, 201 Ristau, W., 95. 100, 129 Robben, F., 153,204 Roberts, G., 95, 104, 129 Robertson, W. W., 136, 165, 174,205 Robin, M. B., 214(33, 34, 35, 36, 37). 308 Robinson, P. J., 103, 129 Robrish, P., 187,202 Robson, P. N . , 378(87), 383 Rodrigo, A. B., 183,204 Roentgen, W. C., 211,308 Rogers, G. E., 293(100), 300(100), 301(100),

304(100), 305(100),310 Rohr, K., 78, 129 Rollett, J. S., 160, 204 Rosen, H., 187,202 Rosencwaig, A., 208(5), 209(5, 7, 8, 9, lo),

216(42, 43), 228(47), 229(47), 230(47), 231(47), 232(47), 236(7), 238(47, 53), 239(53), 241, 247(53), 250(53), 251(53), 254(53), 259(5, 9). 261(71), 265(5, 9), 266(9), 267(9), 271(10), 272(10), 274(5, 9), 275(9), 279(9), 280(9), 281(8), 282(8), 283(5, 8). 2850, 88), 286(88), 287(5), 289(5), 290(5), 294(106, 107), 295(5, 8, 9, 107), 296(106), 297(107), 298(107), 303( 107). 308, 309

Rosengren. L. G . . 214(30), 308 Rothman, S., 300(115), 310 Rowe, M. D., 185, 194, 195,203 Rubbmark, J . , 194,201 Rubinov, A. N., 150,204 Rudee, M. L., 49,52 Ruhle, M., 36,51 Rulison, R. L., 378(90), 383 Runge, W. J., 293(104), 310 Rupley, J. A., 293(105), 310 Ryan, F. M., 276(82), 309

S

Sackett, P. B., 187,205 Sadeghi, N., 178,204

Sadowski, C. M., 185,200 Sah, C. T.. 359(53), 361(61), 382 Sakakibara, S., 305(124),3ll Sakurai, K., 187, 188,200, 205 Sala, K., 150,204 Samelson, H., 144, 146,203, 204 Samson, S., 264(73), 309 Sanche, L., 76, 78, 126, 129 Sanchez, A., 335(34), 381 Sandeman, I . , 272(79), 309 Sanders, J. V . , 7, 13.52 Sargent Janes, G., 197, 205 Sauer, K., 151, 152, 157,202, 203, 287,310 Savitzky, A, , 160,205 Sayre, R. M., 295(109),310 Schafer, F. P., 144, 146, 149, 199, 204, 205 Schawlow, A. L., 148, 185, 186, 194, 201,

202, 205, 241(55), 309 Scheuer, P. J., 285(87), 310 Scheuplein, R. J., 300( 114), 310 Schiebold, I. , 4.53 Schiff, H. I . , 185, 200 Schijter, A., 281(85), 310 Schlag, E. W., 188,200 Schlossberg, H., 237(52), 309 Schmeltekopf, A. L., 100, 127, 185,200 Schmidt, W., 144,203 Schmieder, R. W., 135,205 Schmiedl, R., 187,204 Schmit, G. C. A., 266(75), 309 Schnitzer, R., 118, 122, 124, 125, 129 Schodt, K. P., 301(118),311 Schulz, G. J., 56, 62, 63, 76, 78, 110, 124,

Schulz, H. H., 180, 181, 184,202 Schuster, T. M., 166,203 Schwartz, S. E., 188,205 Schweinler, H. C., 95, 100, 127, 129 Scott, L.. 330(27), 381 Sealer, D. B., 177, 199 Secomb, T. W., 43, 44.52 Senum, G. I., 188,205 Series, G. W., 163, 179, 185,200 Serrallach, E., 156,202 Sessler, G. M . , 251, 309 Setser, D. W., 178,205 Shahin, I. S., 148,201 Shanin, I. S., 148,201, 202 Shank, C. V., 150,205 Sharp, B. L., 161,205 Shaw, D. A., 141,205

125, 129

AUTHOR INDEX 395

Shaw, J. R. D., 150,200 Shaw, M. J . , 157, 171, 175,200, 203, 205 Shishido, T., 47,52 Shoji, M., 362(65), 363,382 Shuman, H., 27, 28.51 Siara, I. N. , 136, 180, 185,205 Siebert, D., 191,205 Sielicki, M., 286(90), 310 Silcox, J. , 50, 52 Sinclair, R., 29, 53 Singer, E. J. , 294( 108). 310 Sitch, J. E., 378(87), 383 Skinner. H. W. B., 140,203 Slobcdskaya, P. V., 213,308 Small, J. M., 335(34),38/ Smith, D. J . , 22,53 Smith, D. L., 71, 72, 73, 128 Smith, I. W. M., 190,203 Smith, M. W., 141,206 Smith, W. H., 133, 139, 140,205 Smith-Saville, R. I.. 140, 200 Smythe, D. L., 335(34), 381 Snedecor, G . W., 169,205 Snider, N. S., 164, 205 Solarz, R. W. , 198,205 Somoano, R. B., 260(69), 263(69), 264(73),

268(69), 269(69), 270(69, 70), 289(69), 309 Sgrensen, S., 188,203 Sorokin, P. P., 199,202 Spence, D., 59, 78, 126, 129 Spence, J. C. H., 12, 13, 14,53 Spitschan, H., 150,203 Stapleton, B. J. , 126 Statz, H. , 359(51),382 Stebbings, R. F., 150, 172, 178,20/, 205 Stedman, D. H., 178,205 Steeds, J. W., 44,51 Steele, D., 95, 129 Steinberg, I., 160,202 Stephen, S., 375(78), 383 Stem, R. C., 157,201 Stetzenbach, W., 187,201 Stevefelt, J. , 143, 157, 173, 176, 178, 200,

Stevens, C. M.. 126 Stevens; W. J . , 187,205 Stewart, W. D. P., 286(91), 310 Steyer, B., 146, 147, 149, 199, 205 Stillinger, F. H., 123, 128 Stobbs, W. M., 48,51 Stoble, W. M., 17, 35, 36, 49.51

205

Stock, M., 134, 201 Stockdale, J. A. D., 95, 129 Stockmann, F., 151,205 Stokseth, P., 144, 150,204 Stoner, J. O., 140,204 Streit, G . E., 185, 200 Strutt, M. J. O., 330(27), 381 Stuckey, W. K., 129 Su, T., 64, 129 Suchard, S. N., 149,205 Sumino, Y., 360(57), 382 Sutton, D. G., 149,205 Svec, W. A., 286(89), 310 Sweetman, D. R., 77, 129 Szent-Gyorgyi, A, , 115, 129 Szigeti, J . , 136, 199 Szwarc, M., 126

T

Tabata, K. , 360(57), 382 Tai, C., 188,200 Takagi, K., 360(57), 372(77), 373(77), 383 Takagi, S., 39.51 Tanaka, M., 47,52 Tandon, S. P., 257,309 Tango, W. J., 187,205 Tarng, M. L., 326(15), 327(15),38/ Taylor, H. S., 56, 78, 79, 122, 124, 127,

Taylor, W. B., 153,201 Teets, R. E., 186, 194,201. 205 Thim, H., 377(85), 383 Thomas, B. S . , 160,203 Thomas, G., 29,53 Thomas, L. D., 122, 124,129 Thompson, R. T., 140,205 Thynne, J . C. J . , 95, 109, 128, 129 Tieman, T. O., 95, 96, 104, 106,128 Tilford, S. G . , 139, 199 Timons, C. J., 272(79), 309 Tittel, F. K., 149,204 Tom, A., 192,203 Tonelli, A . G., 305(124),3ll Tong, A. H., 345(40), 347,382 Tramer, A., 189.203 Treves, D., 146,201 Trowbridge, C. W., 77, 128 Tyndall, J . , 210,308

129

396 AUTHOR INDEX

V

van der Ziel, A, , 315(5), 317, 318(8), 319(9), 321(11), 322(12, 12a). 323(12), 324(14), 325(19), 326(14, 16, 17, 18, 19), 327(16, 17, 18, 19, 20), 329(23, 24), 330(26, 28), 332(30), 335(9, 32), 336(36), 337(12, 12a, 39), 339(19, 39), 341(12a, 38, 39), 343(19, 38, 39), 345(12, 40, 41), 346, 347(40, 43), 352(9), 354(45), 358(48, 50), 359(54), 360(58, a), 361, 362(19), 363(19), 364, 365(50), 369(9, 73), 372(77), 373(77), 375(79), 376(83, 84, 84b), 379(91), 380(92), 381, 382, 383

van Heyningen, R., 291(95), 310 Vannier, M., 149,202 van &strum, K. J., 27,53 van Vliet, K. M.. 315(4), 317(10), 320, 324(4,

14). 325(19), 326(14, 15, 16, 17, 18), 327(15, 16, 17, 18, 19, 20, 21), 329(25), 330(26), 335(30), 338(30), 346(20), 360(59, 60), 381, 382

Velapoldi, R. A., 164, 200 Velasco, R., 137,204 Vernon, F. L., Jr., 375(79), 383 Vestal, M. L., 129 Vestervelt, P. J., 242(59), 309 Viengerov, M. L., 211, 212(19),308 Vigue. J., 193,200 Vinson, L. J . , 294(108),310 Vitry, R., 142 Volksen, W., 270(77), 309 von Ardenne, M., 4,53

W

Wade, R. H., 7, 13.53 Wade, T. E., 330(28), 345(41), 347(42, 43),

Wagner, E. B., 173, 174,202 Wagner, E. H., 16.51 Wahl, A. C., 187,205 Wahl, P., 160,205 Walkley, K., 299(111), 300(111),310 Wall, I. , 2,22, 33, 35,51, 52 Wallenstein, R., 148, 149, 163,205 Walther, H., 146, 147, 180,201, 205 Walton, A. G., 301(118),3/1

381,382

Walton, D. S., 122, 123,129 Wang, C. P., 149,205

Ware, W. R., 158, 160,205 Warner, J., 182, 183,199 Warren, K. D., 95, 104, 129 Weber, M. J. , 144, 145, 146, 199 Webster, M. J . , 171, 175,205 Weigle, J., 47, 52 Weinflash, D., 77, 126 Weingartshofcr, A., 78, 79, 127, 129 Welge, K. H., 188,202 Wellenstein, H. F., 136, 165,205 Welling, H., 244(62), 246(62), 309 Welsh, L. W., Jr., 76, 128 Welta, D., 290(94), 310 Welton, T. E., 314(2), 315(2), 380 Wendlandt, W. W., 208(2), 308 Wentworth, W. E., 95, 100, 104, 126, 129 Wertheim, G. K., 161,205 West, 1. E., 251(63), 309 West, W. P., 178,201, 205 Wetlaufer, D. B., 301(119),311 Wetsel, G. C., 239, 242(54), 252,309 Whelan, M. J., 12, 23.52 Whiffen, D. H., 95, 129 Whinnery, J. R., 242(60), 243(60), 309 White, R. M., 242,309 Whitten, G. Z., 103, 129 Wieder, H. . 135,201 Wieman, C., 186, 194,205 Wiener, N., 159,205 Wiese, W. L., 140, 141,205, 206 Wildnauer, R., 299( 1 lo), 300( 1 lo), 310 Wilkins, R. L., 191,206 Wilks, P. A., Jr. , 208(3), 308 Williams, R., 264(73). 309 Wilson, C. J.. 17, 35, 36, 49,51 Wine, P. H., 186, 190,206 Wodarezyk, F. J. , 190,203 Wolbarsbt. M. L., 291(96), 309 Wolga, G. J., 137, 175, 199 Wolstenholme, G. E. W., 118, 129 Wong, S. F., 78, 110, 125, 129 Wood, M., 300(112), 310 Woodall, K. B., 193,204 Worden, E. F., 198,205 Wright, G. B.. 208(4), 308 Wright, J. J., 184, 185,201, 202. 206 Wrighton, M. W., 268(76), 309

Wang, H.-T., 59, 129

AUTHOR INDEX 397

Y

Yamanashi, B. S . , 291(96), 310 Yamashita, M . , 154,206 Yang, J . L . , 138,200 Yardley, J . T. , 144, 187,206 Yasunobu, K . T., 304(123), 311 Yau, L. D., 359(53), 361(64),382 Ycargers, E., 295(109), 310 Y e , S., 234(48), 235(48), 309 Yeh, P. S., 234(48), 235(48), 309 Yen, W. M., 148,201 Young, C. E. , 128

Young, J . P . , 195, 197,202 Yu, N. T., 292(97), 310

Z

Zare, R. N . , 135, 137, 186, 187, 189, 190, 202, 204, 205, 206

Zecca, A., 76.127 Zegarski, B . R . , 136,201 Zijlstra, R. J . J . , 376(84a), 383 Zimmermann, M. L., 187,201 Zittel, P. F., 188,200, 206

SUBJECT INDEX

A

ADP crystal, 150 Algae, photoacoustic spectra of, 284-286 Amplifier

IMPATT diode, 378 negative-resistance, 376-379 transferred electron negative conductance,

tunnel diode, 377 Analog sampling gate, 155 Argon, metastable, 178 Atomic ground state, electron affinity of, 59 Atomic lifetimes, in pulsed-laser fluorescence

377-378

spectroscopy, 172-186 see also Lifetimes

Atomic negative ions, metastable, 75-79 Autodetachment lifetimes

of doubly charged negative ions, 122-124 TOF vs. ICR measurements of, 65-7 1 . 7 4 7 5 variation of with incident electron energy,

Autodetachment process, theoretical treatment

Avalanche multiplication, 33 1-332

Avalanche noise, in JFETs and MOSFETs, 364 Avalanching, Zener breakdown and, 332-335

100-102

of, 102-111

in p-n junction diodes, 33 1

B

Bacillus subtilis, 289 Beam-foil spectroscopy, 139-140 Benzene ion, lifetimes of, 8 5 8 8 Biology, photoacoustic spectroscopy in,

BJT amplifiers, noise figure of, 352-356 Boxcar averager, for fluorescent signals, 156

280-289

C

Catalysis, photoacoustic spectroscopy in, 266-273

CBED, see Convergent beam electron diffraction Charge integration technique, 154 Chemical reaction studies, photoacoustic spec-

Coccolitus huxleyii, 287 Collision-induced excitation transfer, 13 1 Collision-induced fluorescence method, 136 Contamination mounds, in electron microdiffrac-

Continuous-energy density expression, 103 Convergent beam electron diffraction patterns, 4,

troscopy in, 266-273

tion, 34-35

13-19, 25-27 crystal symmetry in, 44-45 defocused, 19-20 Grigson scanning, 22-23 rocking curves for, 47 symmetry in, 42 variants in, 19-23 wide-angle, 20-22

Convergent transmission electron microscopy,

identification of crystalline phases with, 36-38 SAED in, 23-24

7-9, 13. 24, 34, 49-50

Core-excited resonances, 58-59 Crystal analysis, in electron microdiffraction,

CTEM, see Convergent transmission electron

CW (continuous wave) dye laser, 195

46-43

microscopy

see also Dye laser

D

Decay function, 157-158 Deconvolution, 157-160 Deexcitation studies, photoacoustic spectroscopy

in, 273-280 Diffraction, of plane waves, 10-12 Diffraction techniques

convergent beam electron diffraction and,

defocused and wide-angle CBED in, 19-22 microbeam selected-area pattern in, 24-27

14-19

398

SUBJECT INDEX 399

Dinegative ions, 124 Diodes

noise in, 320-335 single-injection, 375-376 space-charge-limited solid-state, 375-376

Disordered systems, in electron microdiffraction, 48-49

Doubly charged negative ions, autodetachment lifetimes of, 122-124

Dunaliella tertiolecta. 287 Dye lasers, ]&I50

see also Lasers; Nd-YAG lasers; Pulsed dye lasers; Pulsed laser fluorescent spectroscopy; Ruby laser CW, 195

frequency doubling in, 150 laser-pumped, 149 optical properties of dyes used in, 144-146 in photoacoustic spectroscopy, 249 pulsed, see Pulsed dye lasers pumping techniques in, 147-149 special use of, 149-150 tunable, 196- I97

Dynamical scattering, 12-14

E

EA (electron affinity) of atomic or molecular ground state, 59 energy difference and, 118 estimates of, 104 for nitrobenzenes, 112 perfluorination and, 89, 96 positive, 97 reduction in, 107

Electron attachment data, dissociative, 62 Electron attachment experiments, high-pressure,

Electron capture, in ground electronic state, 89-1 I6

Electron diffraction, low- and high-energy , 3 Electron energy

63-65

electron affinity and, 118 negative-ion current and, 101

Electronically excited states, lifetimes and quenching cross sections of, 186-189

Electron microdiffraction, 1-50 contarnination in, 33-35 crystal symmetry in, 4 4 4 6 disordered systems in, 48-49 forbidden reflections in, 39

for germanium films, 49 identification of crystalline phases in, 3638 instrumental stability in, 35-36 intensities in, 46-48 interpretation and application of, 36-50 operational factors in, 31-36 radiation danger in, 31-33 related techniques and, 49-50 structural analysis of crystals in, 46-48 symmetry in, 38-46 three-dimensional symmetry in, 43-44 two-dimensional symmetry in, 3%43

Electron microscopy, see Convergent transmission electron microscopy; Scanning transmission electron microscopy

Electron photon delayed coincidence method, 141

Electron scattering, negative-ion resonance and, 56, 62-63

Emitter space charge region, recombination in, 347

Excitation transfer, collisionally induced, 13 1 Excited-state kinetic studies

direct and indirect methods for, 132-142 stationary methods for, 133-138 time-resolved methods for, 138-142

Eye lenses, photoacoustic spectroscopy of, 29 1-293

F

Fabry-Perot etalon, 148 Feshbach resonances

electron-excited. 116-1 18 nuclear-excited, 57, 59, 89-1 16

FET (field effect transistor), high-frequency be-

FET ampIifiers, noise figure in, 369-372 Fluorescence decay curves, 159 Fluorescence method, collision-induced,

Fluorescence photons, pumping source of,

Fluorescence signals

havior of, 361-363

136-138

161-162

analog techniques for, 156-157 digital techniques for, 15&156 time-resolved measurement of, 154-156

Fluorescence spectroscopy, pulsed-laser, see Pulsed-laser fluorescence spectroscopy

400 SUBJECT INDEX

Fluorescent studies, photoacoustic spectroscopy K in, 274-279

Focusing, imaging and, 5-10 Franck-Condon factors, 97

KDP crystal, 150 Kikuchi line patterns, 14 Klassen-Prins schematic, applications of,

357-358

G

Generation-recombination noise, 3 19-320 Grandparent-parent-daughter states, 59 Grigson scanning CBED, 22-23

H

L

Langevin equation, 319 Laser(s)

dye, see Dye lasers nitrogen, see Nitrogen lasers

Laser fluorescence spectroscopy, time-resolved, 131-199

Laser time profile, deconvolution in, 160 LEED (low-energy electron diffraction), 3,

Level-crossing technique, 135 Level populations, master equation in, 164-166 Lifetimes

Hanle effect, 135 HEED (high-energy electron diffraction), 22 Helium, in pulsed-laser fluorescence spectros-

Hemoglobin, heme absorption spectrum in, 283 Hemoproteins, photoacoustic spectroscopy of,

Hexafluorobenzene, long-lived parent in, 88,91,

27-28 COPY, 173-177

atomic, 172-186 autodetachment, 65-71, 74-75, 100-102,

28 1-282

99, 108-110 122-124

1

ICR. see Ion cyclotron resonance techniques Imaging

focusing and, 5-10 theory of, 5-14

IMPAl'T (impact avalanche and transit time) diode amplifiers, 378-379

Incident electron energy, variation of autodetachment lifetime with, 100-102

Ion, dinegative, 124-125 Ion cyclotron resonance techniques, 61, 71-75

of benzene ion, 85-88 of metastable molecular negative ions, 79-125 of metastable negative ions, 55-125

Long autodetachment lifetimes, vs. large

Long-lived parent negative ions molecular attachment cross sections, 102

excited electronic state and, I 1 6 1 18 formation of, 89-100

tial input, 373-374 Low-frequency amplifiers, with JFET differen-

M

Ionization potential, of neutral state, 59 Iridium carbony1 compounds, in photoacoustic Marine phytoplankton, photoacoustic spectra of,

Master equation, least-squares fitting of 285-286

spectroscopy, 261-262

coefficients of, 168-371 -

Medicine, photoacoustic spectroscopy in,

Metal-oxide metal detectors and mixers, 335 Metastable atomic negative ions, 75-79

Metastable fragment negative ions, long-lived,

J 289-306

JFET (junction field effect transistor) noise in, 356374 noise due to traps in channel of, 360

lifetimes of, 123-124

Josephson junctions, shot noise in, 374-375 118-122

SUBJECT INDEX 40 1

Metastable molecular negative ions extremely short-lived, 77-79 lifetimes of, 79-125 vs. metastable atomic negative ions, 75-76 moderately short-lived, 79-88

biological significance of, 115-1 16 experimental methods with, 61-75 formation modes for, 56-60 lifetimes of, 55-125 range of variation in lifetimes of, 60-61 resonances in, 56-60

Metastable rare gases, 178 Microdiffraction

defined, 1-2 electron, see Electron microdiffraction optical, 28-3 I reflection, 27-29

Metastable negative ions

Molecular attachment cross section, vs. long

Molecular negative ions, metastable, see Metast-

Monosubstitutedbenzenes, parent negative ions

MOSFET (metal-oxide-silicon field effect tran-

autodetachment lifetimes, 102

able molecular negative ions

of, 96

sistor) with low-conductivity substrate, 359 noise in, 356-374

Multichannel photon counter, 157 Multicomponent exponential decay analysis,

166-168

N

Nd:YAG laser, 188 doubled, 182 pumping and, 146-147

Negative-conductance amplifiers, 376379 Negative-ion current, electron energy and, 101 Negative-ion lifetimes, incident-electron energy

Negative-ion resonance, 56-60

Negative ions

and. 107

electron scattering and, 62-64

core-excited resonance for, 58-59 metartable, see Metastable negative ions shape resonances for, 57

Negative ion stares, vibrationally excited, 80 Negative-resistance amplifiers, 376-379 NIR, see Negative-ion resistance

Nitric oxide states, lifetime problem of, 187 Nitrobenzene, parent negative ions of, 1 1 1-1 15 Nitrogen lasers, 146-147 Noise

collective and corpuscular approaches to,

in diodes, 320-335 generation-recombination, 3 19-320 in miscellaneous solid-state devices, 374-380 shot, 317-318 sources of, 314-320 thermal, 314-317 in transistors, 335-356

Noise figure, of BJT amplifiers, 352-356 Noise resistance, 350-351

327-328

0

Optical diffraction pattern, 30 Optical microdiffraction, 28-3 1 Optical spectroscopy, two categories of, 208

see also Photoacoustical spectroscopy

P

Parent molecular ' negative ions, long-lived,

Parent negative ions 8%l16

formation of by electron capture in field of

long-lived, 89-100 of nitrobenzenes, 1 11-1 15 parent neutral molecule ionization potential

excited electronic state, 116-1 18

and, 113 PAS, see Photoacoustic spectroscopy PBD

[2-phenyl-5(4-biphenyl)- 1,3,4-oxadiazole], 145

Pendellosung solution, 15 Pertluorination

effect of on lifetime of parent negative ions of monosubstituted benzenes, 96

electron affinity and, 89, 96 Phase shift lifetime measure-

ments, 133-134 Photoacoustic effect

absorption coefficient and, 237 experimental methodology in, 247-256

402 SUBJECT INDEX

experimental verification of, 235-240 in gases, 21 1-214 gas-microphone coupling and, 241-242 general case for, 227-233 heat flow equations for, 217-220 liquid-coupling piezoelectric transducer and,

in liquids, 241-247 modulation frequency and, 228-23 I in nonhomogeneous solids, 233-235 normalized length in, 231-232 RG theory of, 216-224 in solids, 210 special cases for, 225227 and temperature distribution in cell, 220-224

Photoacoustic signal, log-log plot of, 239 Photoacoustic spectrometer, commercial types

Photoacoustic spectroscopy, 207-308

244-247

of, 256

see also Photoacoustic effect in bacterial studies, 289 in biology, 28@289 in catalysis and chemical reaction studies,

data acquisition in, 255-256 in deexcitation studies, 273-280 dye lasers in, 249 experimental chamber in, 249-255 experimental methodology in, 247-256 future trends in, 306307 of hemoproteins, 28 1-282 human eye lenses and, 291-293 inorganic and organic semiconductors in,

inorganic insulators and, 257-258 in in vivo studies, 306 of liquid crystals, 264-266 in medicine, 289-306 of metals, 264 microphone configurations in, 250-251 in photochemical studies, 279-280 in physics and chemistry, 25&266 of phytoplankton, 285-286 of plant matter, 283-289 surface studies of, 270-273 TCSA and, 289-290 in tissue studies, 293-305

266273

258-264

Photoacoustic spectrum, power spectrum and,

Photochemical studies, photoacoustic spectros- 236

copy and, 279-280

Photoelectric effect, liquid-microphone coupling in, 242-244

Photomultipliers applications of, 151-154 signal-to-noise ratio for, 153

Photon counter, multichannel, 157 Photon detectors, 150-154 Photon-photon delayed coincidence method,

Plane waves, diffraction of, I(!-I2 Plant matter, photoacoustic spectroscopy of,

Plasma diagnostics, pulsed selective-excitation spectroscopy in, 177

Platymonas sp., 287 p-n devices, noise in, 337-356 p-n junction, generation-recombination in space

charge region of, 328-330 p-n junction diodes

141-142

283-289

breakdown in, 331 shot noise in, 321-328

p-n-p-n diode, 379-380 p-n-p transistors, injected electron emitter current

in, 341-342 Princeton Applied Research photoacoustic spec-

trometer, 256 Pulsed dye lasers

see also Dye lasers linear polarized light from, 163-164 spectral fluctuations in, 149

Pulsed electron excitation, 140 Pulsed laser fluorescence spectroscopy

see also Dye lasers; Pulsed dye lasers alkali atoms in, 179-184 atomic and molecular physics applications of,

atomic lifetimes in, 172-186 chemical reaction rates in, 189-190 data analysis and reduction in, 164-172

Pulsed-laser fluorescence spectroscopy experimental techniques for, 143-164 helium and other rare gases in, 173-179 least-squares method in, 168-171 master equation of level populations and,

miscellaneous atoms in, 184-186 molecular energy relaxation in, 186-195 multicomponent experimental decay analysis

parasitic phenomena in, 163 pressure fits in, 171-172

172-1 95

164-166

in, 166-168

SUBJECT INDEX 403

pumping source vs. chamber of fluorescence

recent developments in, 195- 199 rotational energy transfer and, 191-195 systematic errors in, 163-164 vibrational excitation transfer and, 19C-191

Pulsed selective-excitation spectroscopy, 177 Pumping source, fluorescence photons and,

Pyraminmonas sp., 287

photons in, 161-162

I6 1-162

R

Radiative deexcitation, 13 I Rare-gas-halide lasers, 149 Reciprocity principle, 26 Reflection high-energy electron diffraction, 3.

Reflection microdiffraction, 27-28 Resonance ionization techniques. 197 Resonances

27-28

molecular orbits and, 59 types of, 57-59

69, 116-118

2 16-224

tion), 3, 27-28

Retarding potential difference grid assembly, 65,

RG theory, of photoacoustic effect in solids,

RHEED (reflection high-energy electron diffrac-

Rotational energy transfer, 191-195 RPD, see Retarding potential difference grid

Ruby lasers, frequency-doubled, 147 Rydberg states, studies of, 198-199

assembly

S

SAED model, see Selected area electron diffrac-

Scanning transmission electron microscopy, 5.9 ,

identification of crystalline phases with, 36-38 instrument for, 16-17 mechanical and electrical stability in, 35

dynamical, 12-14 electron, see Electron scattering

tion model

13, 34.49-50

Scattering

Scherzer optimum defocus. 29 Schottky barrier diode, 318

characteristics of, 320 detectors and mixers using, 335

Schottky barrier gate, 359 Schottky’s theorem, 3 17 Selected-area electron diffraction, 2, 4 , 23-25

Semiconductors spherical aberration in, 24

see also Transistor direct and indirect bandgap, 260 inorganic and organic, 258-264

Shape resonances, for negative ions, 57 Shockley-Read.Hal1 centers, 319, 323, 328 Shot noise, 317-318

in JFETs and MOSFETs, 364 in Josephson junctions, 374-375 in p-n junction diodes, 321-328

Sodium atoms, quenching of by iodine

Sodium dioxide ion, lifetime of at thermal ener-

Solids, photoacoustic effect in, 214-241 Solid state devices, noise in. 31S380

Spectroscopy, pulsed-laser fluorescence, see Pulsed-laser flourescence; see also Photoacoustic spectroscopy

STEM, see Scanning transmission electron microscopy

Symmetry, in electron microdiffraction, 38-46

molecules, 182

gies, 85-88

see also Transistor

T

TCSA (tetrachlorosalicylanilide), 289-290 Tetrode FET, noise figure of, 372-373 Thalassiosira pseudonana, 287 Thermal noise, 314-317

in conducting channel, 358-365 as velocity-fluctuation noise, 315-3 17

Thin-layer chromatography, 270-272 Time-of-flight mass spectroscopy, in metastable

negative ion studies, 61. 65-75, 89,

Time-resolved laser fluorescence techniques,

Time-resolved methods, in excited-state kinetic

Tissue studies, photoacoustic spectroscopy of,

TOF, see Time-of-flight mass spectroscopy

107-108, 118

183-184

studies, 138-142

293-305

404 SUBJECT INDEX

Transferred electron negative conductance

Transistor, unijunction. 379-380 Transistor noise, 335-336

Tunable dye laser, 196-197 Tunnel diode amplifier, 377

amplifier, 377-378

equivalent circuits and, 343-347

U

Unijunction transistor, 379-380

W

Wave function symmetry, in electron microdiffraction, 42-43

X

X-ray diffraction crystal structure analysis in, 46 microdomains and, 48-49

X-ray diffraction patterns. symmetry in, 38-39. 44

Van de Graaf accelerator, in beam-foil spectros-

Vibrational excitation transfer, 190-191 copy, 139 Zener breakdown. avalanching and, 332-335

Zero-point energy, 103

A B C B D B E D F 1 6 2 H 3 1 4 J 5

advances in electronics and electron physics, volume 46

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