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3. Available Tabulations and Basis of Current Formalism

This section discusses other available theoretical, experimental and synthesized tabulations for form factors and attenuation. The main sources for comparison are the theoretical development of Cromer, Mann and Liberman [17-20], the experimental-theoretical synthesis by Henke et al. [6] and recent developments along both lines. [21,15]

These works have employed numerous simplifications compared to detailed relativistic S-matrix calculations [12], but the latter do not lend themselves to convenient tabular application for the range of Z and energy of general interest. Conversely, the earlier tables appear to have large regions of limited validity throughout the range of Z and energies, and in particular have limitations with regard to extrapolation to energies outside tabulated ranges. This is well known, and other tabulations of attenuation coefficients or form factors based on experiment or on alternate theoretical bases [22,23] have met with partial success in some regimes, generally with corresponding areas of limitation elsewhere.

With the exception of high-order S-matrix calculations, the method of Refs. [17-20] may be expected to produce greatest theoretical precision and is perhaps predominant in the literature for the last decade. A number of intermediate theoretical and procedural assumptions limit the precision and applicability of this method. The current method, serving as a basis for the present tabulation, allows significant improvement both in the earlier tabulated range and in regions extrapolated with the formulae. A particular concern of the current author lies in suitable extrapolation to high and low energies, and interpolation to energies not covered by the Kα values presented in the literature for these tables.

For this purpose a Fortran version of the code was gratefully received from S. Brennan [24] which contains the updates of Cromer and Liberman [20] in their source code presented in Ref. [18]. In this method of Cromer and Liberman, relativistic SCF (self-consistent field) wavefunctions using the Kohn-Sham potential [25] and experimental energy levels [26] were used to compute partial photoelectric absorption coefficients (σ(E)) at ten or eleven selected energies, with the Brysk and Zerby program [27]. This introduces approximations regarding the potential and edge level derivation, but more significantly uses a model with no molecular or solid-state fine structure. Application of this code [20,24] to near-edge regions of bound atoms is therefore limited by this procedure.

The development of Cromer and Liberman will be of prime consideration in the current discussion, together with the formulation for avoiding difficulties therein [28]. In the code of Ref. [20], separate partial cross-sections were calculated for most orbitals of each element in the range of Z = 3 to 92. However, some outer orbitals of higher Z elements were omitted or displaced (with respect to their free-atom ordering or typical bonding patterns), which leads to inadequacies in these cases at lower energies. In the extreme case of Rb (Z = 37), 9 electrons and hence four sub-shells were omitted (4s, 4p1/2, 4p3/2, 5s) leading to unnecessary termination of the tables below 0.112 keV, and significant imprecision below 0.160-0.200 keV, whereas formal results could have been given down to 0.004 keV. Most elements include all edges down to about 0.050 keV, and light elements (to Ca) generally include all orbitals.

For each orbital, five cross-sections σ(E) covered energies of 5-23 keV [18] or 1-80 keV [24], for interpolation and determination of f″(E) (the imaginary, absorptive component of the scattering factor). If the binding energy lay in the range 1 keV < -ε1 < 70 keV, an additional point applied to E = -1.001 ε1. An additional five values covered the above-edge range (with energies set for 5-point Gauss-Legendre integration). This second group of 5 points applied to the integral of f″(E) to give f′(E): the locations of E and the form of the integral chosen depends on whether -ε1 < 1 keV, -ε1 > 70 keV, or -ε1 is of intermediate energy.

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