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10. How to Use These Tables

These tabulations provide f1 and f2 form factors for all elements up to Z = 92 from 0.001-0.01 keV and 400-1000 keV. This exceeds the range for normal x-ray diffraction and crystallography studies but allows limitations and specialized experiments to be investigated with reference to updated and corrected theory. It also allows a sufficiently fine grid with accurate atomic edge structure to allow such experiments as DAFS (diffraction anomalous fine structure) to investigate fine structure and spatial distribution of atoms and electrons within materials [36]. Multilayer diffraction experiments may be pursued at lower energies in an analogous manner.

In these cases values for f1 and f2 should be extracted from the tables for the given element(s) and energies required. Linear interpolation of f1 should be adequate, while linear log-log interpolation of f2 should be adequate on this scale, if required. Equation (3) should then be used to obtain f′ using the negative value of frel as included at the top of each table for each element. For comparison to old data or computations, the value corresponding to Ref. [6], denoted H82, should be used (following Cromer and Liberman but omitting the Jensen energy-dependent correction) [17-20,37]. More recent work has suggested not only that the Jensen term should be omitted but also that the appropriate relativistic correction is 3/5ths of the Cromer-Liberman value [38,39]. This latter value is denoted 3/5CL at the top of each table for each element. Likewise, the nuclear Thomson term (also negative with respect to the atomic phase) is provided at the top of each table for each element.

For comparisons to other results in the forward scattering limit where the momentum transfer q = 0, the value of f0 = Z may be used and the real and imaginary components of f are then fully defined. For large scattering angles it is necessary to use a more appropriate value of f0, as may be gained from Refs. [40,41]. Then the composition and arrangement of the material may be used as indicated in the introduction to provide structure factors, refractive indices, and Fresnel coefficients, together with scattered, diffracted or transmitted intensities.

For filters or filter materials, the photoelectric attenuation coefficient is provided in order to compare to appropriate experiments or to allow for objects in a beam-line. The conversion to this from f2 in appropriate units is provided at the top of the table. Use of barns/atom is also common, and the conversion factor for this is also provided. Often this column is not measured, and only the total observed attenuation coefficient µTOT = μPE + σcoh + σinc is observed. These latter two coefficients are angle-dependent and may in part be determined from appropriate structure factors for a given crystal orientation. However, a column is provided for the sum of these two latter coefficients in an average-over-angles for an atomic scatterer [34,35]. These references should be consulted for details concerning the approximation involved, although the column in the current tabulation is a new computation of the sum. The main assumption is that Bragg-Laue peaks and troughs are avoided, or that the material is randomly oriented and preferably mosaic. If this is not true, it may be necessary to compute the dynamically diffracted intensities from the structure factor rather than rely on the approximation. Further comments are provided in the next section. However, simply summing these two columns allows the comparison of theory to experimental attenuation data.

The column providing the photoabsorption coefficient for the K-shell only is included for two purposes. The first is that at high energies this is the dominant contribution to the total photoabsorption, and provides a guide for the local energy dependence of the cross-section. It serves as an illustration of the isolation of individual orbital cross-sections, particularly for higher energies.

The isolated K-shell cross-section is also important for experimental diagnostics and corrections. In particular, fluorescence yields from atoms are negligible for almost all orbitals except the K-shell, when compared to Auger and Coster-Kronig transitions. However, the fluorescence yield fraction for the K-shell is large, so the dependence of the cross-section upon energy is equally important. The qualitative result in an experimental ion chamber is significant - in these processes, a likely result is that the fluorescent x-ray shall escape from the ion chamber without conversion to (detectable) ion pairs. A more detailed discussion of this is provided elsewhere [42].

The plots provide additional data not contained in the tables. This includes a plot of the K-shell contribution to the imaginary component of the form factor f2. This is related, as discussed, to the K-shell contribution to the photoabsorption cross-section. The enhanced energy range provides an indication of the appropriate extrapolation to higher energies. It also provides the atomic orbital computation of the lower energy behavior, despite increasing imprecision as the energy falls below 0.010 keV.

The plots also provide comparison to the results of Henke et al. [15,16]. This is considered by the author to be the most useful and convenient publication available for comparison. It indicates limitations regarding restricted ranges and tabulation steps, shows good agreement over much of the energy range for many elements, and indicates regions of divergence, difficulty or concern. Some of these concerns have been addressed directly in this paper, while others are left to the reader. In many cases the best naive statement of uncertainty in either computation arises from the divergence between the two. This may relate to local structure, absolute values, or global structure. An alternate error estimate is provided in the following section.

For less common use, the edge energies used are provided at the top of each table so that criticism (or experimental investigation) may indicate a shift of the local energy scale which may be appropriate in a specific material or experiment. This is not encouraged or recommended; nonetheless, it is provided as a statement of the assumptions and basis of the computation.

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