7. Anomalous scattering factors (tables_ASF, data_ASF)

Sample anomalous scattering factors extracted from the RTAB database are shown in Table 5.

An interesting feature of our ASF values that differs from other authors is the explicit inclusion of bound-bound resonant transitions. In our underlying model of single-electron transitions in a potential, a bound-bound resonant transition occurs at a single energy (our levels have no widths), the difference of the energies of the two orbitals involved in the transition. This infinitely narrow transition is manifested as a delta-function spike in the imaginary scattering amplitude and a resonance approaching infinity in the real scattering amplitude. Although these explicit spikes and infinities are unphysical, the underlying strength of the transition is important and contributes significantly to the scattering at low energies. The inclusion of bound-bound transitions in our anomalous scattering factors is important for satisfying the Thomas-Reiche-Kuhn sum rule (see, Kissel et al., 1995), wherein an appropriate integral over all energies of the imaginary scattering amplitude should equal the number of electrons in the atom. In many cases, bound-bound transitions contribute 30% or more of the contribution to the TRK sum rule.

A challenge for evaluating anomalous scattering factors via the relativistic dispersion relation and the optical theorem involves energies above 2mc^2. As discussed by Pratt et al. (1994), partitioning of the many particle scattering amplitude to yield R+NT+… introduces an additional contribution to the imaginary amplitude for Rayleigh scattering as computed by the optical theorem. Above 2mc^2, to lowest order, one needs to subtract the contribution of bound-electron pair production from that of photoeffect to correctly compute the total cross section. [footnote - By bound-electron pair production, we mean pair production in which the electron of the pair is created is a bound state of the atom. Ordinarily, discussions of pair production involve the situation where both the electron and positron are created in the continuum.] This insight settles a long standing quandary as it is known that the relativistic photoeffect cross section goes as 1/E at high energies, and as a consequence, the dispersion integral is not convergent if photoeffect were the only contribution to lowest order at high energy. As with photoeffect, bound-electron pair production goes as 1/E at high energies so the integration is naturally cut off at high energies. We utilize an estimate of the bound-electron pair production cross section for the non-relativistic K shell due to Costescu (see, for example, Bergstrom et al., 1997). In our evaluation of the anomalous scattering factors on 0-10 MeV, we carry out the integration for energies of 0-100 MeV.

The validity of this estimate for the total photon-atom cross section above 2mc^2 has not been examined in detail. It is possible that this approximation in our evaluation of the anomalous scattering factors is responsible for the decreasing validity of our MFASF predictions at non-forward angles for energies above 100 keV.

Previous evaluations of anomalous scattering factors have ignored the issue of energies above 2mc^2 by simply cutting off the integration at some suitably high energy. Besides being important for the evaluation of anomalous scattering factors in the 100-10,000 keV range, the behavior of the total cross section above 2mc^2 effects values at lower energies by an overall constant. This is related to the high-energy limit corrections to the anomalous scattering factors of Cromer and Liberman prepared by Kissel and Pratt (1990).

All our anomalous scattering factors satisfy the TRK sum rule to a fraction of 1% (typically at the 0.01% level). The sum rule check is included as part of the output in the ASF files in 'data_ASF'. As an example, the TRK sum rule check in 'data_ASF/006_asf0sl' is

GPRIME: SUM-RULE CHECK ON CROSS SECTIONS;
     COMPUTED=   1.25405E+00 FROM BOUND-BOUND TRANSITIONS
                 4.74291E+00 FROM BOUND-FREE  TRANSITIONS
                 2.07488E-03 FROM HIGH-ENERGY LIMIT
                 5.99903E+00 TOTAL
    PREDICTED=   6.00000E+00
   DIFFERENCE=  -9.69774E-04  (-1.62E-02%)

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Information date: Sep. 2, 2000 lk