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Relative to appropriate high-energy theory, which would yield well-defined edges and smooth behavior for each orbital on a log-log plot, the results of C95 and refs. [32,33] and [34] are all in error. The structure from C95 could be interpreted as a sharp shape resonance, but it is a fictitious one.
This error arises from an accumulation of minor errors in inner shells and the electronic wavefunction distributions. Particularly for near-edge energies, these errors accumulate, which is a strong reason for the low accuracy claimed by theory in this region. The K shell (1s orbital) and LI shell (2s orbital) are usually accurately computed, and the form factors for these sub-shells are accurate; but the errors for LII and LIII (2p) are amplified, and also fall in increasingly difficult soft x-ray energies. Hence the wavefunction solution for the orbital radial electron density, which leads to the computation of the near-edge form factor, becomes unreliable and increasingly inaccurate. Similarly, the MI-MIII (3s, 3p) edges are well defined, but the MIV and MV (3d) structure is poor near the absorption edge.
For C95, this yields a sharp slope for the LII/III-edges for Z = 30 to 36, and for the MIV/V-edges for Z = 60 to 88. For ref. [31], this effect appears periodically in a less well defined manner.
Within the convergence criteria for the DHF wavefunctions, this may be more or less difficult to address, depending upon the exchange potential and method used. In the case of C95, we have been able to retain the original formalism but simply to require a better and more uniform convergence in these regions.
When the wavefunctions of C95 are improved and this issue is addressed, we obtain the "New" or "Current Result" (figs. 3 et seq.). These new results are tabulated for the regions of atomic number and energy where any significant imprecision was observed. The results of this work also appear to reliably obtain the theoretically expected IPA edge structure. The precision of these results is clearly dramatically improved; but the accuracy is still limited for the reasons discussed above. Hence we would claim no better than 20 % to 30 % accuracy in this region, even though in some cases experiment may agree to better than 10 %.
This paper emphasizes the results of this investigation for the imaginary component of the form factor. The same structures are seen on an expanded log plot of [µ/ρ] as illustrated by fig. 5. Due to space constraints, we present plots of the real and imaginary components of the form factor for all energies and all atomic numbers affected, but we present [µ/ρ] only in the tables. As indicated in (eq 7), there is a simple relation between the two.
The same qualitative errors in structure for f2 or [µ/ρ] are transformed following (eq 4) and (eq 6) into qualitative errors in the structure of Re(f) as a function of energy, as indicated in fig. 6. The result reported here is in better agreement with Henke et al. [32,33] than C95, and includes features for all edges. The most common spurious structural effects in Re(f) are seen just above the edge, where a spurious peak may appear, and in subsequent waves of dips and peaks extending up in energy for some keV or so. This same structure also leads to an accentuated minimum in Re(f) at the edge location, and also to an apparent decrease in Re(f) or f1 below the edge, by perhaps 1 e/atom. These secondary effects are quite variable, depending on the nature of the approach to convergence of the wavefunctions. However, this seems to represent the most common signature in problem cases in C95.
The transform of the erroneous structure shows significant deviations from the new result, in some cases down to 100 eV. Hence the plots and tabulations cover regions down to 100 eV even though the error in convergence of f2 only exists in the 1 keV to 4 keV region. By providing this full region, we allow the new tables to be continuous with the older tabulation of C95, so that a simple replacement of the new material for the old yields a smooth and continuous result. We have taken the opportunity suggested by colleagues to implement a finer grid spacing in this near-edge regime, simplifying any interpolation which may be applied to the data.
C95 stated low energy, high energy and near-edge limitations of this tabulation, which also apply to this current work. The main difference is the new precision in the computation of soft x-ray near-edge regions. However, we tabulate these estimates of precision in Table 2, and give an indication of the effects which limit the accuracy in these regions. Two types of inaccuracy may be identified. The first listed is the estimate of convergence precision (intrinsic to the computation), while the second is an estimate of additional structure (such as XAFS or solid state effects) in particular applications. Correlation between electrons contributes to both of these error estimates.
| Regions of energy within tabulated region 0.001 eV to 1 MeV |
Estimated typical uncertainty | |||
| f2, [µ/ρ]PE
and µPE Monatomic Gases |
f2, [µ/ρ]PE
and µPE Solids, Liquid |
f′ f1 - Z * |
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|---|---|---|---|---|
| Below 200 eV [correlations, phonons] | 50 % to 100 % [33] | 100 % to 200 % [33] | 50 % to 100 % [33] | |
| 200 eV to 500 eV | 20 % to 30 % [33] | 50 % to 100 % [33] | 20 % to 50 % [33] | |
| 500 eV to 1 keV | 3 % to 10 % [33] | 5 % to 20 % [33] | 5 % to 15 % | |
| Near edges (within 0.1 %) | 20 % to 30 % | 50 % | 50 % to 100 % | |
| Near K edges (within 10 %) | 10 % | 10 % to 20 % | 30 % | |
| Near K edges (1.1 < E/EK < 1.2) | 3 % | 3 % | 10 % | |
| Well above K edges (E/EK > 1.2) | 1 % | 1 % | 1 % to 2 % | |
| Near LI, MI -
MIII edges (within 15 %) |
15 % | 15 % to 30 % | 30 % | |
| Near LI, MI -
MIII edges (1.15 < E/Eedge < 1.4) |
4 % | 4 % | 10 % | |
| Well above LI, MI -
MIII edges (E/Eedge > 1.4) |
1 % | 1 % | 1 % to 2 % | |
| Near LII/III, MIV/V edges (within 15 %) |
20 % | 20 % to 40 % | 30 % | |
| Near LII/III, MIV -
MV edges (1.15 < E/Eedge < 1.4) |
4 % | 4 % | 10 % | |
| Well above LII/III,
MIV - MV edges (E/Eedge > 1.4) |
1 % | 1 % | 1 % to 2 % | |
| Above 200 keV (see also section 5 and section 8.9) |
2 % to 3 % | 2 % to 3 % | 1 % to 2 % | |
| * See section 9 | ||||
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