high resolution x-ray experiments

52 Nuclear Instruments and Methods in Physics Research B24/25 (1987) 52-58 North-Holland, Amsterdam

HIGH RESOLUTION X-RAY EXPERIMENTS

Richard D. DESLATTES

National Bureau of Standards, Gaithersburg, Mayland 20899, USA

Accelerators provide highly charged ions for a variety of experimental studies. In applications involving accurate high resolution

spectroscopy, care is needed to keep Doppler corrections manageable and reduce satellite problems. Some aspects of the motivation

for such work are reviewed along with techniques for trying to produce clean spectra.

1. Theoretical background and motivation

While the central questions of atomic structure and spectra are often regarded as having been generally understood for a long time now, there remain open questions of a fundamental character even regarding hydrogenlike systems and a proper Hamiltonian formulation is not yet at hand for heliumlike systems. The problems of relativistic and quantum electrodynamic effects in heavy atoms were the subjects of a recent workshop which focussed on one- and few-electron ions

[l] where it was evident that even among persons who tend to be critical of the status of relativistic quantum

theory, there is relative satisfaction regarding the one- electron problem [2]. Starting from the external field

Dirac equation, the well-appreciated difficulties associated with divergences, limitations of perturbative meth-

ods and the needed re-normalization have come to be matters resolvable by teachable methods. However, when one considers accurate measurements and refined calculations, there is still considerable room for di- alogue. State-of-the-art theoretical calculations are conveniently available in the all-Z results recently published by Johnson and Soff [3] who used procedures developed by Mohr [4] for radiative corrections and a uniform nuclear model with a diffuse boundary and a mean radius varying as A113.

Although interpretation of experimental results will

ultimately derive from comparison with such careful and accurate calculations, it is convenient to develop

0 I I I I 1 1 I I 1 I 0 20 40 60 80 100

Atomic Number 2

Fig. 1. Estimated values of Q = S/y (see text) for Lyman transitions (An = 1, 2. 3) in H-like ions versus Z. In general, higher Q

values permit more accurate measurement of the radiative correction.

R. D. Desluites / High resolution X-rqv experiments 53

scaling rules from simpler analytic approximations for the design of experiments and to indicate which questions become important in various ranges of Z. A simple argument follows from the review article of Kugel and Mumick [5] who summarize pertinent sec- tions of Bethe and Salpeter. Consider the Lamb shift as a test of the self-energy (dominant) and vacuum polari- zation. Any experiment probing this area can be characterized by a dimensionless Q-value taken to be the ratio of the radiative correction S to the line width y. Q-values obtainable for Lyman series lines are indicated in fig. 1 for the first three members; in contrast, measurements based on high Z analogs of the classical Lamb-Retherford experiment are limited to Q values 8 times smaller than those shown for the 2p level [6]. There is, however, a serious technical disadvantage in the Lyman series approach since considerable measurement accuracy is required to discern a precise value for the Lamb shift because it amounts to only a very small part of the measured interval.

Moreover, to reach informative comparisons between theory and experiment, one must confront the question of the nuclear finite size correction. For- tunately there is a straightforward approach to this question. Firstly, one notes that a considerable body of systematic theory and experiment is available both con- cerning rms nuclear radii and their internal charge distribution. Secondly, for many values of Z there are two or more stable isotopes often having quite different nuclear sizes. What is needed then is to carry out systematic studies as functions of Z and A with the aim

of extracting Z dependent, A independent trends, if any. We are at present at an earlier and simpler stage in this quest. Fig. 2 summarizes all available data for Z B 1 and includes also representative recent results for hydrogen and helium. The reference level (origin) is that due to Johnson and Soff; only in the most accurate data at Z = 18 does one reach even the level of the (technical) difference between the calculation due to Mohr [4] and that due to Erickson [7]. Perhaps one more order of magnitude is needed before experiments can seriously challenge present theory. From fig. 2 it is also clear that significant data above Z = 20 are rare.

Interest attaches also to He-like spectra since they represent the simplest case in which electron-electron interactions must be considered along with the (nuclear) Coulomb and radiative interactions present in H-like systems. Heliumlike ions are also attractive for study in that in the nonrelativistic limit, extremely accurate wavefunctions are available from variational calculations. At the same time, there are increasing contribu- tions from relativistic effects with increasing Z as well as from specifically two-electron QED effects. On the other hand, as recently emphasized by Desclaux and collaborators [8], a rigorous formulation of this problem is elusive. For example, if one begins with the relativistic counterpart of the Schriidinger equation, i.e. the Bethe-Salpeter equation, generalized to include an external field, no evident path leads to a solution since the equation involves an integral kernel that cannot be obtained in closed form and depends on two time variables. Instead all authors to date have had recourse

160

120

“0

X 100 b

w” I o1 60 Y

W’

-

l-

b-

I-

I-

0 1s

0 2s

4 6 6 10 20 40 z

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Fig. 2. An overview of the current state of the one-electron problem as revealed by Lamb shifts of the 2s and 1s levels. The reference level from the calculational results of Johnson and Soff for the inner s-levels. Experimental data presently available are indicated with

error estimates.

I. ATOMIC PHYSICS / RELATED PHENOMENA

54 R. D. Des1utre.s / High resolution X-ruq’ experiments

to some form of effective interaction between the electrons leading to a corresponding effective Hamiltonian. Overall understanding of the relationship among available calculations is only recently becoming available. In addition, a possibly fundamental reformulation is un- derway by Mohr [9] wherein radiative and correlation effects (which give comparable corrections Z B 1) are being formally treated in a similar fashion for the first time. The overall experimental and theoretical situation for He-like systems is summarized in fig. 3. The reference theory is the 2 expansion results of Vainstein and Safranova [lo] even though there are problems due to the limited numerical accuracy of this work and some difficulties below Z = 10 are suggested by comparisons with experiment and more recent theoretical estimates. Theoretical estimates from the work of Drake [ll], Hata and Grant [12] and Indelicate et al. [13] are in substantial agreement with one another and with the accurate experimental data summarized by Berry, DeSerio and Brooks [14] for 23P-23S transitions. This case is indicated in part (a) of fig. 3, Drake’s results are indicated by the solid curves above 2 = 10. These show significant systematic departures from the reference calculation which are most clear in the region of higher Z where no experimental data are available. For 11S-23P transitions, as shown in (b), the Drake results are also indicated by solid curves for the more prominent reso- nance transitions. Where there are overlapping results

g +lOr I 23s,-23~3

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s ’ 2 5 -5 - 30 20 30 40 4 -IO-

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from Hata and Grant at lower Z values, they agree well with Drake’s calculations and with experiments. As Z increases Drake’s results depart from the Z-expansion values used as reference and experimental data are few: those which are available are indicated by symbols with error flags. It seems clear that the few experiments carried out thus far are almost able to make useful statements about the available theoretical estimates. This is quite good news since the path toward appreciable improvement is clear. One notes that data are not broadly available even at lower Z and are absent or very rare above Z= 20; thus trends which may be present await discovery by systematic extension of the domain of available data.

2. Strategies for producing clean one- and two-electron spectra

It is important that spectra be obtained from ions which are effectively isolated from environmental per- turbations if they are to be suitable for comparison with basic theoretical calculations. One is thereby restricted to certain accelerator technologies and to extracted beams from ion sources. Spectra having the needed cleanliness are still not generally available since there are additional requirements that the emitting ions should not only be isolated from other ions but also that they

Fig. 3. An overview of current knowledge of some important transitions in He-like ions. Results are presented as deviations from a reference theory, in this case, the Z-expansion work of Vainshtein and Safranova [IO]. Part (a) concerns certain An = 0 transitions in

the n = 2 manifold while part (b) deals with the prominent An = 1 transitions from this manifold which terminate in the ground

level. The solid curves and dots are relativistic calculations by Drake [ll] while recent MCDF calculations by Indelicate et al. [13] are

shown by open diamonds. In the range 2 s 10 the reference theory has insufficient numerical precision for current purposes. In this

region however, the three main recent calculations are in substantial agreement with one another and with experimental data as

summarized by Berry, DeSerio and Brooks [14] for the An = 0 case.

R. D. Desiutfes / High resofutiort X-my experiments 55

be free of “spectator” electrons and that their state of

motion be sufficiently well-defined and slow that ap-

propriate Doppler corrections are robust at the targeted accuracy levels. Since this story is not yet widely known, a brief overview of the technologies currently available may be in order here; the overall situation has been reviewed earlier [15].

Bean-foil studies of projectile spectra are simple to carry out and result in stripping and excitation down to the shell whose effective orbital electron speeds are comparable to that of the ion in the laboratory. For a specified velocity or specific energy, ions emerge in a distribution of charge states characterized by a mean

value and a spread. The distribution of excitation over

these charge states is complex and contingent on many factors but generally any charge state which can be produced is also found “optically” excited as well. Results and limitations characteristic of this class of

experiment may be appreciated from the example of a fairly careful experiment carried out on hydrogenlike

chlorine using the Emperor tandem accelerator at the Brookhaven National Laboratory [6]. The line shapes were not simple and significant changes accompanied changes in the beam energy due to the presence of spectator electrons whose population distributions de- pend on beam velocity; the line shapes become more distorted with decreasing velocity. Low velocity data are, however, especially important in facilitating ex- trapolation to zero velocity as is needed to correct for Doppler shifts. The final result obtained in this work had an inaccuracy estimated at the level of approximately 10% of the Lamb shift and, for the reasons indicated above, appears unimproveable.

Nevertheless, beam-foil methods have been useful in the past and remain valuable even at the present time particularly for exploratory work. In some of the earliest work, Briand and collaborators studied the Lyman (Y of Fe”+ using sufficiently high velocities that the lines appeared simple in shape [7]. Unfortunately this in- creased the size and uncertainty of the Doppler correction to the extent that it was the dominant component of the reported error of 15% of the Lamb shift. More recently, Briand and other collaborators have extended the explored range of crystal diffraction spectroscopy to include measurement of the Lyman alpha spectrum of Kr35+ with a reported level of accuracy amounting to 4% of the Lamb shift again dominated by uncertainty in the Doppler correction needed by the data [8].

An important method for avoiding significant Dop- pler correction altogether is offered by the so-called recoil mechanism for producing ionization and excitation. In this procedure, highly charged, heavy ions are delivered to a neutral light gas target at high velocity, typically 0.1~ (= 5 Mev/amu). The soft collisions which predominate are very effective in stripping the light target ions to rather high charge states without, how-

ever, imparting appreciable momentum to these in the process. As was the case with beam-foil excitation,

these collisions produce appreciable populations of excited ions in all charge states achieved. Recoil energies are typically of the order of 10 eV producing a small Doppler broadening, noticeable in a high resolution measurement, but guaranteed by symmetry to yield no net shift. In spite of the considerable advantages of this

procedure, it too is seriously flawed by the presence of abundant spectator electrons leading to important limitations in the accuracy which may be achieved in this.

In two recent excercises, data were obtained from H-like [19] and He-like [20] argon produced by impact of 66-times ionized uranium at a specific energy of

approximately 5.6 Mev/amu or a total laboratory energy of about 1.3 GeV. The target was a low pressure cell provided with very thin Ni windows for entrance and exit of the uranium beam which was stripped to equilibrium in the entrance window. The beam was produced by the heavy ion linear accelerator, UNILAC, at the Gesellschaft fur Schwerionenforschung (GSI) in Darmstadt, FRG. The data exhibit considerable statistical precision, reflecting the fact that the recoil production is relatively efficient with each uranium ion excit- ing several argon ions although clearly not all of these yield the specific 2p excitations of interest. Limitations inherent in this procedure are evident; spectator elec-

trons abound leading to complex structures partly re- solved and partly underlying the features of interst. In the case of H-like spectra, serious efforts were under- taken to model and analyze these satellite features and determine in this way corrected locations for the Lyman lines [19]. In one approach, model calculations led to an estimated population distribution among accessible con- figurations for each of which Hartree-Fock calculations were used to produce “stick” spectra which were finally smeared by an effective resolution function and sub- tracted from the experimental data. An alternative approach involving entirely empirical fitting gave results for the main line locations in good agreement with the

model approach. The final accuracy was sufficient to permit a 1.5% estimate of the Lamb shift. Although this result represented a definite improvement over previous experiments, it nonetheless escapes significance in that it is capable of resolving merely technical issues on which there were other sources of information having comparable force. While one can think of several im- provements in experimental tactics for such a recoil measurement there seems to be no obvious path toward say a factor of 10 which even at this relatively low value of Z would already be of interest.

In the case of He-like argon, the experimental arrangements were quite similar although a different analyzing crystal and a different (and inferior) X-ray line was used as a transfer standard [20]. The main difference is that the threshold for theoretical interest in

I. ATOMIC PHYSICS / RELATED PHENOMENA

56 R. D. Deslattes / High resolution X-t-q experiments

He-like spectra is reachable within the limited accuracy to which one has ready access. In one sense this can be attributed to unresolved questions regarding the proper formulation of the two-electron problem. The experimental data appear to support the two most recent and, in fact, most refined calculations. The He-like data, no less than the H-like data, are notably troubled by the presence of spectral distortions arising from spectator electrons. One other effect was noted: lifetimes are sufficiently long that appreciable quenching changes are seen between the two cell pressures used. This means that it would be desirable to repeat the measurement at still lower pressures in order to finally come to a stable picture. Unfortunately such reduction in the target density leads to a corresponding reduction of counting rate with longer measuring times then required to retain statistical precision.

In response to the problems and limitations just described, more recent work has evolved in the direction of a more complex (and less efficient) approach to the production of clean spectra in one- and two-electron ions. The idea is to accelerate ions to sufficiently high speeds that foil stripping or gas stripping yields good abundances of totally naked ions. These naked ions can then be decelerated to a range where electron capture into orbits with principal quantum number 2, 3, 4, . . . is efficient. At these speeds Doppler corrections can be managed if a sufficient range of speeds can be sys- tematically explored by the accelerator system. By mag- netic analysis of the beam just prior to the capture stage it is possible to assure that only a single charge state enters the region of observation. By use of a beam of atomic hydrogen as the source of electrons for the capture process it would be possible to guarantee that no extra electrons are available that might act as spec- tators during the emission transition of interest. Actu- ally, a low density atomic helium target proved to be rather effective in this regard and was appreciably easier to manage in the demonstration experiment described next.

This work was carried out at the Max Planck In- stitute for Nuclear Physics (Heidelberg) using their tandem-linac with the linac post-accelerator operated as decelerator [21]. Negative chlorine ions were accelerated to the terminal voltage and stripped (in the terminal) to 9 + or 11 + ; these then emerged at energies near 135 MeV where they were again foil-stripped to obtain an appreciable yield of naked ions. They were decelerated to final energies between 22 and 80 MeV and magneti- cally analyzed to assure a clean beam of naked ions [22]. These were collimated by defining apertures and en- tered an open cell containing low pressure helium through several stages of strong differential pumping. The total beam entering the cell was in the range of one to several hundred picoamperes (electrical). The range of energies available meant that capture was primarily

into n = 2 with the yield falling rapidly with increasing energy roughly as Ee9. This procedure was effective in removing satellite features from the observed spectra; unfortunately, our ability to carry through a full-scale Doppler analysis was limited by available intensities and beam time and ultimately by the low energy deceleration limit constrained by limitations of the Heidel- berg tandem-linac.

3. Doppler corrections

Systematic application of the accel-strip-decel- capture approach described above requires serious at- tention to management of the Doppler correction. Sim- ply operating at a laboratory angle of 90’ and reducing projectile speed to give a “small” correction is a de- cidedly poor approach for several reasons: Firstly, it represents a serious restriction on the Z-range accessible to a given number of linac stages i.e., to a specific accelerator. Secondly, deceleration is costly in terms of beam losses both by general attrition and because of Liouville’s theorem. Lastly, final energies often need to be chosen to enhance the cross-section for population of particular states hence are no longer free parameters.

The starting point is, of course, the standard Dop- pler formula for the laboratory wavelength, h, corresponding to a center of mass (CM) value, X0, when the projectile speed is /3 = v/c and the direction of observation is at a (laboratory) angle OL, namely:

h=X 1-P-a O (1-p2) ’

(1)

whence

AX/A= F(a, ,8)Aa+ G(a, /3)A/3. (2)

There are two “good” regions for experiments according to eq. (2), specifically in the region F = 0 and around G = 0. The neighborhood of F = 0 corresponds to forward or backward emission (cx = 0 or (Y = g) where relaxed restrictions on directional stability and beam emittance are relaxed but at the cost of large Doppler corrections and of maximum sensitivity to beam velocity. The second “good” region is where G = 0, (i.e., cos a = /3), which gives very small Doppler corrections but at the cost of requiring great stability in beam direction and permitting only a small range of directions, i.e., requiring low emittance beams. Finally, fig. 4 shows E,,,/&, for a range of /3 values for each of a number of (laboratory) angles of observation near 90“. For angles greater than 90°, the curves cross E,,,/Ec, = 1 in two places, the second corresponding to cos a = /3; there is also a region between these crossings where the correction depends weakly on /3, i.e. where a perhaps sizeable Doppler correction is substan- tially independent of /?. Finally there is a high p region

R. D. Desluttes / High resolution X-ray experiments 51

0.996 1 ’ 1 0.000 0.20 0.40 0.60 0.80 0.100

BETA = V/C

Fig. 4. Doppler effect modifications of observed transition energies as functions of projectile speed, /3 = u/c for various laboratory angles as marked on the curves of E,,,/Ec,. These curves illustrate several “Doppler engineering” concepts including the special speed-angle regime for “zero” correction and the use of high velocity data to facilitate “internal”

determination of laboratory angle of observation.

where two or more measurements of the Doppler shift with different beam speed can lead to a unique and precise determination of the laboratory angle of observation assuming that beam direction is independent of beam energy.

Considering the factors discussed in the preceding paragraphs it appears that a specific experiment at a definite accelerator can be optimized by Doppler design. This has mainly to do with minimizing the experiment’s sensitivity to the less well-controlled parameters and evaluating the rest by robust and redundant procedures. In particular, working with fastest beam usable consistent with Doppler correction accuracy and charge transfer cross-section provides a practical optimization at least with regard to beam intensity and range of charge-states which can be made available at a specific accelerator facility.

4. Efficient and accurate high resolution spectroscopy

The picture which emerges from the preceding dis- cussion suggests that there is considerable scientific potential associated with adequate systematic study of the main spectral features obtainable from one- and

two-electron ions. With judicious choices as to sources of excited systems and their well-considered operation one can hope to obtain meaningful spectra. But these must be well-measured to discover this meaning. Clearly, high resolution spectroscopy is called for; lines of interest are generally much narrower than normal X-ray lines and fully justify efforts to obtain resolving power well beyond what is called for in conventional X-ray spectroscopy. At the same time we must deal with weak sources in an environment characterized by high background and little resembling the well-controlled sur- roundings in which one would prefer to carry out high resolution spectroscopy. At the same time numerical results from these measurements need to be connected in a robust fashion back to the spectrum of atomic hydrogen itself which is our source of knowledge con- cerning the Rydberg constant which provides the quantitative link between theory and experiment.

While all of the above considerations must, in some sense, be satisfied it is evident that they conflict with one another. For instance, it is satisfying to do one-step measurements deriving accurate results directly from the experiment at hand. However, accurate spectrome- ters are seldom efficient and tend to be environmentally sensitive as well. On the other hand, focussing spec- trometers tend to be relatively efficient but cannot be operated as direct-reading devices. Furthermore, when adjusted for maximum efficiency, they tend to acquire data in a point-by-point manner placing the results at the mercy of accelerator operation which is notoriously unstable in general. Finally, we note that, while X-ray lines are convenient transfer standards (readily available, conveniently located and intense), they are much broader than are the most interesting ion lines; this requires either discernment of structural detail internal to the X-ray lines or the introduction of synthetic crystal filters capable of maintaining stability between calibrations and able to extract a small slice from an X-ray line to serve as a transfer standard.

One approach which has several attractive features uses high resolution curved-crystal optics operated with the source internal to the focal circle and coupled to an imaging detector which is often a high performance position-sensitive proportional counter. Profiles from the ionic spectra are accumulated alternating with profiles from the transfer spectrum used for calibration. Where possible this calibration is effectively continuous, for example with calibration being done between pulses of the accelerator. Normalization of the transfer standard is accomplished entirely off-line by accurate angle measurements (goniometry) using flat crystals whose spacing is established in a measurement chain going back to an iodine stabilized HeNe laser. This particular laser has been used to anchor all recent measurements of the hyperfine components of the Balmer (Y line of atomic hydrogen, i.e. to determine the Rydberg con-

1. ATOMIC PHYSICS / RELATED PHENOMENA

58 R. D. Desluttes / High resolutton X-r-q experiments

stant; it thus serves as a convenient surrogate for an

atomic hydrogen beam.

References

[l] Atomic Theory Workshop on Relativistic and QED Ef-

fects in Heavy Atoms, eds., H.P. Kelly and Y.-K. Kim

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[ll] G.W. Drake, private communication (1986); Phys. Rev.

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[12] J. Hata and I.P. Grant, J. Phys. B17 (1984) 931.

[13] P. Indelicate. 0. Gorceix and J.P. Desclaux. J. Phys. B, to

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K.-H. Schartner, Phys. Rev. A29 (1984) 2939.

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Gould, Phys. Rev. Lett. 50 (1983) 832.

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