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5. General Discussion of Recent Issues, and a Summary of Earlier Issues

Hydrogen: C95 uses a simplified approach to give the form factor for hydrogen itself. This is extrapolated to high energies, and it may be noted that at very high energies there is an approximation error for the result for a single isolated hydrogen atom. The primary purpose of that tabulation (and the current work) is to address the need in crystallographic and synchrotron communities for accurate form factors for structural and other investigations. Hence the primary target lies over the range of x-ray energies. We are grateful to P. Mohr for raising this issue. Of course, for many investigations the form factor of bonded hydrogen is non-spherical and completely different from that for atomic hydrogen. In these cases a form factor for atomic hydrogen may be used to directly investigate the bonding patterns, and so the tabulated values remain useful. Results may alternatively be obtained for the assumptions of bonded floating sphere hydrogen [37], and/or hydrogen in the H2 molecule [12] and [38]. However, it is worthwhile investigating the actual limitation of C95 across the range of tabulated energies. This is presented in fig. 1, where a variety of models are given for the hydrogen atomic form factor. The Sauter relativistic Born approximation is actually very poor for x-ray energies, but indicates the asymptotic limit at high energies [39]. This functional dependence is not observed in the earlier tabulation, and reference should be made to other sources listed here for energies above 433 keV.

Henke et al. [32] covers a very restricted energy range, and the Sauter formula (e.g., ref. [39]) only becomes a useful approximation at energies above 80 keV. With these two exceptions, all approaches appear very similar across several decades of energy and form factor. C95 is accurate to within approximately 2 % up to 330 keV for an isolated hydrogen atom. The original tabulation presented results by extrapolation to 433 keV, where the relativistic high-energy correction to the simple result has a magnitude of 13 % to 15 %. Although this correction is beginning to be significant at this level, the magnitude of coherent and incoherent scattering dominates by seven orders of magnitude. Other comments regarding the utility of the earlier presentation were given in C95.

Singularities, Integration Precision, Interpolation: C95 detailed the correct approach to these issues, and discussed particular tabulations where problems of these types have been noted earlier. The main problems are related to the use of a relatively sparse set of values of f2 as a function of energy, and the use of inappropriate formulae for the determination of the imaginary and real components of f from the atomic orbital wavefunctions. Both C95 and the current work are free from such problems.

Several approaches have major problems with extrapolation, interpolation and integration approaches to the determination of Re(f) and of Im(f). The work of Creagh and Hubbell [4] suffers from some generally minor limitations in this regard, and theory reported in Saloman, Hubbell and Scofield [34] is relatively free from these effects. This paper does not relate directly to regions of failure of extrapolation, integration or interpolation. However, the specific near-edge problems discussed below reveal new limitations which in some cases may be related to problems of extrapolation, depending on the computational approach used.

Comparison of recent tabulations for helium, Z = 2: Helium is a near-perfect system for study. The gas is monatomic so the isolated atom approximation is valid. There are only two electrons, but correlations of the two electron wavefunctions are large. The independent particle approximation (IPA) can be very good, except for direct correlations of the two motions of the electrons during transitions. Figure 2 indicates that Scofield (unrenormalized) [34] deviates from experiment by generally 3 σ to 4 σ in the soft-to-medium x-ray regime, as opposed to C95, who lies within a fraction of σ deviation from experiment. C95 provided a simple computation of scattering coefficients to complement the more detailed computation of form factors contained therein. The differences between the simple coherent cross-section of C95 and that given in Saloman are significant at the 1.5 σ level in this region.

If coherent scattering follows Bragg-Laue processes (such as for crystals and diffraction peaks) or Thermal Diffuse Scattering approximations (usually for crystals, but with explict alignment away from Bragg peaks) then the estimates of Chantler or Saloman et al. may be inappropriate and the actual scattering cross-section may be larger or smaller than that predicted, by an order of magnitude or more. However for isolated atoms such as helium, or for systems where the Rayleigh scattering approximation is good, the estimates of ref. [13] (and herein) are expected to be good approximations to the experiments.

More detailed evaluation of scattering coefficients is given by Hubbell and Øverbø [13] (σcoh), and Hubbell et al. [12] (σincoh), tabulated in ref. [34]. Use of these (generally more accurate) scattering coefficients with the attenuation coefficients of C95 yields very good agreement with the precision experiment of Azuma et al. [40]

The discrepancy shown in fig. 2 is primarily due to the use by Scofield of Hartree-Slater orbitals, hence omitting certain relativistic corrections. At some level, this limitation would be expected to yield lower accuracy than the self-consistent Dirac-Hartree-Fock approach (ref. [16] and this work). The general approach for new theoretical work is certainly to use a Multi-Configurational Dirac-Hartree-Fock approach whenever possible, and this argues for the approach of this work rather than that of ref. [34]. The DHF approach more accurately incorporates relativistic effects which become more significant for higher Z elements.

For Z = 2 to 54, Scofield provided estimated renormalization factors to convert to values which might be expected from a relativistic Hartree-Fock model. The difference between renormalized and unrenormalized results vary from about 5 % to 15 % or more for lower energies or outer orbitals, so is very significant in the current discussion. There are other differences between Scofield and Chantler beyond simply the Dirac-Hartree-Fock versus Hartree-Slater approach. The exchange potential of the Chantler approach follows that of Cromer and Liberman [25] and Brennan and Cowan [30] and is quite distinct from the approximation used by Scofield. On this issue the preferred approach is not clear a priori.

In the context of Helium, application of renormalization would improve agreement with experiment, but by only a fraction of a standard deviation, and hence would not resolve the discrepancy. This large and significant discrepancy is several σ, but only about 8 % to 10 % in magnitude. Other discrepancies for higher Z elements show discrepancies many times this value.

Much recent theoretical and experimental work has investigated helium, particularly in the VUV region. These extensive calculations offer improvements in precision, particularly in the energy ranges below 300 eV and above 300 keV, while having similar quoted precision in the central x-ray range. A review has shown consistency of recent detailed calculations by Hino [41] and Anderson and Burgdörfer [42] with C95 in the region plotted in fig. 2 [43]. This review also showed the consistency of experimental results of Samson et al. [44] with Azuma et al. [40] and the inconsistency of these results with Henke et al. [33] and Viegele et al. [45]. Detailed investigations of sum rules by Berkowitz [46] has supported the approach of C95. Undoubtedly further theoretical and experimental work is needed, particularly for the high energy regions.

Causes of uncertainty near absorption edges: The above examples concentrated on regions where alternate theories claim convergence to 0.1 % and hence can claim accuracies of 1 %. However, the greatest discrepancies between these theories occur near edges, with deviations by factors of 5 or more between alternate results.

The cause of near-edge error in theoretical computations is often due to inadequate interpolation, extrapolation or integration methods, which introduce apparent oscillations or discontinuities into the data [31]. The cause of near-edge error in experimental compilations is often due to neglect of the edge region or smoothing through edge structure [33]. The cause of near-edge error in specific experiments is often due to the dramatic variation of form factor with energy, requiring both accurate absolute intensity measurement and also precision energy calibration [47].

Assuming that these issues have been correctly addressed, theory will disagree with experiment near edges by large factors due to XAFS and related structure. This can reach a 200 % discrepancy between IPA theory and a solid-state experiment [17]. Even if the experiment is performed on a monatomic gas, there may be pressure-dependent structure and other strong oscillatory behavior near edges. Some of this structure (shape resonances and Cooper minima) may be qualitatively predicted by some theoretical approaches, but often the detailed experimental result will show significant quantitative discrepancy [48].

The largest discrepancies between C95 and the Scofield theory are not due to any of these causes. C95 claims uncertainties of up to a factor of two (50 %) in soft x-ray near-edge regions. Reference [34] refers to 10 % to 20 % discrepancies from experimental data in the medium-Z regime, which may be taken as an uncertainty estimate.

In most elements and regions, the near-edge variation falls within these error bars. This is illustrated for copper in fig. 3. Such experimental data is not sufficiently precise to distinguish between these two theories, or even to observe edge structure which would diverge from refs. [32,33]. Figure 3 indicates the 1 keV lower limit of the range of ref. [31]. The current result is slightly modified in the near edge region, but the accuracy here is not improved, the difference between the earlier result of Chantler and the current are within one standard deviation, and we do not present this in the following tabulation. A detailed experimental work claiming 4 % accuracy has recently demonstrated good agreement of C95 with experiment and with Creagh and McAuley [17] for copper [49].

Experimental data have large scatter and large uncertainties compared to the theoretical discrepancies discussed here, and hence cannot distinguish between the alternatives. This is generally true for this near-edge soft x-ray region, and has made comparisons of theory difficult. Figure 3 also plots experimental data plotted for f2 rather than µ/ρ. This involves a straightforward scaling of attenuation data and subtraction of scattering contributions to attenuation cross-sections. The coherent and incoherent scattering functions contribute a maximum of 1.5 % in the region tabulated, and a maximum of 0.2 % in the regions near edges. The uncertainty in this subtraction should generally be less than 0.2 % and hence will not add to the experimental uncertainty. The experimental references in the figures are taken from the comprehensive database of ref. [31]. References [50] indicate the range of references used in compiling fig. 3, as a typical example.

Isolated Atoms, Independent Particles and the formalism: In this work we use the same formalism as described in C95. This also follows the DHF SCF approach of Cromer and Liberman [22-25] and uses the Kohn-Sham potential [51] and experimental energy levels [76] to compute partial photoelectric absorption coefficients using the Brysk-Zerby program [52] (modified). The modifications introduced are to improve computational precision rather than a change of the formalism. We then use f2 to compute f1 using the standard Kramers-Kronig dispersion formula (eq 6).

Hence we treat each atom as an isolated system, not influenced by any other atoms or particles (this is the isolated atom approximation). Additionally, we determine each wavefunction including correlation according to the DHF procedure, and allow for the electron-electron interactions via the use of the central field and Kohn-Sham potential. In other words, we use Dirac relativistic wavefunctions with full antisymmetrization of product wavefunctions within the DHF method.

We make the assumption of the independent particle approximation (IPA) so that each electron is considered to move in an effective potential of the nucleus with the average repulsive force of the electrons. This effective screening neglects some correlation and also neglects the fact that the potential for one electron is really not identical to that of a different electron. This assumption is quite general - the only choice is the selection of the form of the central potential.

This may be contrasted with other procedures including, e.g., the use of Hylleraas, MCDF or Configuration Interaction wavefunctions. The most important question is to ask what the consequences of a refined treatment of the wavefunction might yield, and this is a worthy and valuable issue for the future. Our understanding is that, within the context of the current discussion, such issues affect edge energies dramatically, but do not have a great effect on the issues and results presented in this paper. The computational cost of such approaches generally allows them for investigation of specific energies, or perhaps a specific edge, but otherwise gives a major limitation in developing general solutions for all Z and all energies.

Convergence: The estimation of the "accuracy" or "precision" of a theoretical work is always difficult. We have investigated several types of consistency which lead us to give the specifications (i.e., estimates) listed in section 9. We have investigated plots similar to fig. 3 in C95 which lead us to estimate convergence of that kind for f1 at the 0.2 electron/atom level (away from edges) for uranium, or in general at the 0.2 % level. Near edges this increases, and for difficult regions of f2, this may lead to an additional offset error in f1 over a wide energy range (as discussed in detail below).

We have investigated convergence in computational detail of the sort represented by fig. 4 in C95 which lead us to estimate convergence of that kind for f2 approaching the 0.4 electron/atom level (away from edges) for uranium, or in general approaching the 1 % level for "good" regions "away from edges." This imprecision has a secondary effect upon the determination of offset errors for f1, but in general at a lower level. We have estimated convergence of the intrinsic wavefunction itself (related to this), which suggests the values we have indicated. We have noted that convergence of theoretical edge energies is not required nor appropriate using this approach, and we do not do this. We have compared where possible our experimental results to a selected set of the best experimental data, and this has so far supported our uncertainty estimates. We have noted further details in C95, including specific caveats for particular regions, which we do not reproduce here. For example, we note in C95 that the result for Rb (Z = 37) below 0.112 keV is invalid, and that significant imprecision remains (even within the formalism) below 0.2 keV. We also note that solid state effects and correlated electron excitations (phonons, double-electron resonances, etc.) do occur at low energies and are not accounted for in this formalism.

Tseng et al. [53] point out that f is rather insensitive to electron correlation effects, at least for the elements Z = 2 to Z = 6 for which such effects have been studied by Kim and Inokuti [54] and by Brown [55]. Although the incoherent scattering function may have electron correlation effects of 20 % to 30 % due to inadequacies of the independent particle model or the implementation used for wavefunctions, such effects on the atomic form factor f were found to be 1 % or less. Weiss also confirms this [56].

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