ASF Angle Dependence

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Angle Dependence Calculation

Atomic number or symbol
Energy (keV)
Minimum angle (degrees)
Maximum angle (degrees)
Number of Angles

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A commonly used approximation to Rayleigh or elastic scattering of photons from bound electrons is the form factor with angle-independent anomalous scattering factors (FFASF) approach (see Kissel et al., 1995). In the case of forward scattering, only one of the two complex amplitudes, necessary for a complete description of the elastic photon scattering process at arbitrary angle, contributes and this remaining amplitude may be written in terms of the form factor and two anomalous scattering factors. The term "anomalous scattering" has historically been used to denote departures from form factor predictions. The form factor is simply the Fourier transform of the scattering charge density. The imaginary anomalous scattering factor can be obtained from the total cross section for inelastic photon-atom interactions using the optical theorem. The real anomalous scattering factor is related to the imaginary anomalous scattering factor by a dispersion integral.

While the FFASF approach can give the exact nonrelativistic result for forward scattering, it neglects the angle dependence of the anomalous scattering factors and it completely neglects the second complex amplitude necessary for a complete angle-dependent description of the elastic scattering process. Recently, Costescu et al, (1994) gave simple expressions for the angle dependence of both amplitudes for photons scattering from the ground state of hydrogenic ions. This web page runs a computer code that utilizes the Costescu et al. expressions for the angle dependence for non-relativistic scattering from Hydrogen to estimate the angle dependence of the real and imaginary anomalous scattering factors.

These results are expected to have limited usefulness. Only the angle dependence of the K shell is considered. In many cases the K shell dominates the total anomalous scattering factors, but often the contributions from other shells are also important, and their angle dependence is different from that of the K shell. Also, relativistic effects are known to be important for the inner shells of heavy atoms, but the angle dependence computed here is a nonrelativistic result. Finally, screening effects are ignored by these hydrogenic predictions.

Nevertheless, we present this estimate for the angle dependence of anomalous scattering factors as so few predictions are available from any source. More rigorous estimates can be obtained from the S-matrix predictions, which are available as data files at this site.

For more information on anomalous scattering factors, see

"Validity of Form-Factor, Modified-Form-Factor and Anomalous-Scattering-Factor Approximations in Elastic Scattering Calculations," by Lynn Kissel, B. Zhou, S. C. Roy, S. K. Sen Gupta and R. H. Pratt, Acta Crystallographica A51, 271-288 (1995).

For more information on the region of validity of the present results, see

"Retardation and Multipole Effects in Rayleigh Scattering by Hydrogenlike Ions at Low Photon Energies," by A. Costescu, P. M. Bergstrom, Jr., C. Dinu and R. H. Pratt, Physical Review A 50, 1390-1398 (1994).

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Information date: August 8, 2000 lk