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12. Other Current Concerns

Most of this description has been concerned with problems of f′ or f1 and f″ or f2. Difficulties inherent in the form factor formalism as opposed to the Modified Form Factor formalism or general S-matrix formalisms are discussed elsewhere [46,47]. Use of Modified Form Factors is of value at or above MeV energies for various angles, where the separability of f0 and f1 becomes questionable [10]. Sum rules have been discussed and investigated, particularly at relatively low energies [48,49], and may be more generally applied to theoretical or experimental data. Comparisons of form factor predictions to others or to experiment without allowance for the issues raised in this paper will endanger the validity of conclusions drawn.

One of the major differences between these classes of theory, and in particular between RDP, RMP [38,39] and S-matrix methods [12] relate not to f1 or f2 per se, but relate to the relativistic correction factor frel. This contribution indicates the correction for the high-frequency limit. For uranium at low energies frel is of order -2.5 e/atom [17-20], -1.5 e/atom [38,39], or -1.2 e/atom [12,39]. The differences between these theories are large, with a range at the 1 e/atom level and an uncertainty quoted as ± 0.36 e/atom [39]. Equally, for calcium (Z = 20), quoted values vary from -0.06 eu to -0.036 eu to -0.034 eu with, again, a quoted uncertainty of ± 0.009 eu [39]. These are important and significant differences (particularly for medium and heavy elements), although it should be clear that any comparison of theories or experiments is difficult without correcting for the effects discussed herein with their associated many-electron errors. Issues regarding the forms for frel demand that input values for f1 be calculated to better precision than the uncertainty in the relativistic component. Errors of the preceding sections or of integration or interpolation precision at this level are unnecessary and obscure comparison of different models with experiment.

The integration method adopted and discussed herein makes direct use of the separation of orbitals and the assumption that the orbital absorption coefficient is exactly zero below the edge. Correlation between orbitals, excited bound states, level widths and resonance phenomena may require alternate transforms based on hbarω = (0,∞) to (0,1) intervals. This general issue was addressed by Cromer and Liberman [17] and has been addressed more recently by Wang and Pratt [50]. The presence of multiple edges and near-singularities creates difficulty for most standard integration methods. For comparison of theory to experiment, direct measurement of Re(f) is required, but the current revised theoretical calculation has an advantage over methods based on limited ranges of experimental or synthesized attenuation coefficients, in avoiding smoothing and other problems near edges.

A secondary difficulty, so far as published results are concerned, relates to tabulations often being restricted to a small set of Kα lines, or to other discrete sets of energies which fail to reveal major variation of structure with energy.

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