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7. Comparison of agreement of earlier tabulations with the New Result

The worst cases of this convergence error in C95 are represented by Zn Z = 30 and Pm Z = 61 fig. 7. The imprecision of theory increases towards lower energies below the edge. A rough estimate of the imprecision for experiment, Henke [33], C95 and this work is given by the difference between Henke and C95. Hence this uncertainty reaches 50 % below 300 eV, and lies around 10 % at about 600 eV. The uncertainty in theory within 10 % of the edge is estimated to be about 10 %, so between 600 eV and 900 eV we might expect agreement of theoretical approaches at the 3 % level. The discrepancy with Henke lies at the 10 % level which in this region we attribute to solid state structure or to the synthesis used by Henke.

In the high-energy region convergence of theory would expect a 1 % accuracy, but discrepancies of 6 % are observed. These areas must be the subject of future experiments in this field.

The convergence errors of C95 near the edge represent 1.5 σ errors, where σ is estimated as 50 %, as stated earlier. In these and similar cases the Scofield result yields 80 % and 220 % errors near the edge (or 4 σ to 5 σ errors); conversely, C95 yielded maximum 68 % and 87 % errors, respectively, at the same locations. We assume that the cause of the Scofield discrepancies lie in the same problem regarding the electron distribution. This will be affected by the formalism used to derive the wavefunctions. The C95 convergence errors tended to be extended over slightly larger energy ranges (i.e., 40-50 % versus 20-30 % above the edge). For Z = 61, fig. 7.

Henke [33] displays 30 % discrepancies in the near edge region. Henke includes a weighting for a theoretical prediction, but may be affected more by the scatter of available experimental results, or by the Z-interpolation scheme used. Figure 8 and fig. 9 illustrates these percentage deviations explicitly compared to this work (which also has an uncertainty, but yields a correct IPA structure).

As stated, usually the experimental data is inadequate to make a critical comparison of C95 or of the current work with respect to other databases. A nice comparison is however given by the noble gas Kr, Z = 36 (fig. 10) [57]. Here the structure suggested by C95 is clearly incorrect, although the theoretical uncertainty was almost equal to the difference between C95 and experiment. The structures of Henke [33] and Scofield [28] are also seen to be in error, particularly for the L III edge region, although for krypton this maximum error is 50 % rather than the larger error of C95. The experimental data set plotted here is quite dated, and we estimate experimental uncertainties to be 4 %. The precision appears to be better than this, and possibly approaching the 1 % level. This is therefore a good data set, and there is also the advantage of this refering to a monatomic gas, so that the independent particle approximation should be valid.

Other experimental data are plotted in the tabulation. Results for Yb (Z = 70), Lu (Z = 71) and Ta (Z = 73) suggest a smoothed M V edge structure, although this may be partly due to detector and monochromator resolution.

Current experimental data for rhenium (Z = 75), gold (Z = 79), lead (Z = 82) and bismuth (Z = 83) also give strong evidence against the oscillation of C95. In particular, data for gold (Z = 79) and lead (Z = 82) appear to favor the current work rather than Henke [33] and Scofield [28], certainly in the near-edge region for the M IV/M V edges. The predicted structure matches up very well with the current result, as opposed to alternatives. There is some indication of M V smoothing, which may also be due to detector and monochromator resolution. Scattering contributes to the experimental data at the 0.1 % to 0.25 % level. These plots also show some absolute offsets at the 1 σ to 2 σ level, where σ is given by experiment. There is strong motivation for high accuracy experiments to address these sorts of discrepancies and to reduce the experimental uncertainties by a factor of three or so.

A confirmation that the region of interest has been fully addressed is given by the result for Z = 37 (Rb) in fig. 11 and fig. 12, and by the result for Z = 59 (Pr) in fig. 13 and fig. 14. Here the revised approach is indistinguishable from the earlier result, and the signature of the previous lack of convergence is absent. Figures 13 and 14 show all L and M edge regions for completeness. Hence the earlier tabulation is not reproduced for the elements lying between these in the periodic table.

Neodymium Z = 60 and actinium Z = 89 are included in this tabulation. Although the results for Z = 89 was not obviously affected by the earlier lack of convergence, the new results show a very minor variation which is therefore also presented.

A recent experimental program by Chantler et al. [47] is proceeding to address the experimental variation in the literature, by measuring attenuation coefficients to much better than 1 % over central energy ranges for important elements. Other work is also in progress by several experimental groups. A number of detailed XAFS studies have been made, which often show high resolution relative structure but without an absolute calibration to compare directly to theory. Of course, the near-edge region of direct relevance here is also strongly affected by XAFS, which are intrinsically solid state interactions not represented in the current series of tabulations. The main exception to this rule are the noble gases He, Ar and Kr as discussed above.

All these experimental programs hold the prospect of reducing the experimental uncertainty to much less than the theoretical variation, which will allow much more critical investigation of atomic and solid form factors. A number of detailed theoretical issues including the near-edge "offset" from IPA theory will then become accessible to investigation.

Even where experimental accuracy is inadequate to discriminate between theoretical alternatives, the anticipated atomic edge structure is reasonably well represented by the new results tabulated below, and differs from that of all of the earlier tabulations represented in the plots. The tables and plots show that the experimental structure represented by the experimental data for krypton, Z = 36, and other elements support the edge structure of this work rather than that of earlier references.


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