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5. Avoidance of Singularities in f′(E) Above Edges

The functional forms of the integral of f″ as indicated in Refs. [17-20] are formally continuous above each edge but include singularities in both numerator and denominator (which cancel). Consequently, the code of Cromer and Liberman [17-20] yields divide-by-zero errors for Z = 18 (argon, 1.28 keV) and elsewhere at energies for Z > 2 corresponding to cancellations (5 per orbital).

The functional forms indicated [17-20] were otherwise appropriate and valid in the 5 keV-50 keV range. This was the main concern of the authors, but extension beyond this range in either direction requires reevaluation of the construction. In particular, the forms pertain to E >>  -ε1, E simeq  -ε1 and E << -ε1 and not to -ε1 < 1 keV, -ε1 > 70 keV. Further, that for -ε1 > 70 keV omitted a correction which is increasingly important as E approaches -ε1. The nature and magnitude of this latter problem is depicted in Fig. 1 for uranium. The apparent "edge" is displaced and offset by several keV with five discontinuities, and at 500 keV (four times the edge energy) the error is of the same magnitude as Z. This energy range is much higher than that of Refs. [15,17-21]. In these figures and in the text, the current formulation avoids or overcomes all problems discussed, as far as is possible within the prescriptions provided. The use of f1 as the plotting variable follows the standard of Refs. [7,16] but more importantly places the effects on an approximate absolute footing for the real component of the atomic scattering factor in the forward direction.

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