2. Calculating Rayleigh amplitudes and cross sections

Considerable progress has been made in the last half of the twentieth century in the practical evaluation of elastic photon scattering by atoms. Three main avenues of increasing computational complexity have emerged to yield practical predictions: the form factor approximation, anomalous scattering factors for forward-angle scattering, and predictions based on the second-order S-matrix element. For an assessment of the validity of simpler approaches to scattering see Kissel et al. (1995); for a review of elastic scattering focused on the low-energy gamma-ray region see Kane et al. (1986).

The form factor approximation, valid for photon energies much greater than electron binding energies and for non-relativistic momentum transfers, has been extensively tabulated. Tables of nonrelativistic form factors (Hubbell et al., 1975), relativistic form factors (Hubbell and Ųverbų, 1979) and modified relativistic form factors (Schaupp et al., 1983) for all neutral atoms have been published.

Anomalous scattering factors are a forward-angle energy-dependent result computed via the dispersion relation and optical theorem. Predictions are now readily available for all atoms for keV energies due to a work pioneered by Cromer and Liberman (Cromer and Liberman, 1970a,b, 1976, 1981; Cromer, 1974, 1983; Creagh and McAuley, 1992; Henke et al., 1981, 1993). Anomalous scattering factors have been used in conjunction with form factors to yield corrected differential scattering predictions.

Numerical evaluation of the S-matrix element is a more sophisticated approach to the evaluation of elastic photon scattering by atoms. Contained within this approximation is the high-energy form factor result, as well as the forward-angle energy-dependent anomalous-scattering-factor result. It goes beyond both these approximations to yield predictions valid at all scattering angles and energies. Computationally, it is the most complex of the schemes discussed here, requiring considerable computer time. Due to the dramatic increase in the availability of computers, systematic evaluation of the S-matrix element has now become practical.

Numerical evaluation of the second-order S matrix for single-electron transitions in a potential was first attempted in the 1950's by Brown and co-workers (see, for example, Brown et al., 1955). Further refinements in the technique were introduced by Johnson and co-workers in the 1960's (see, for example, Johnson and Feiock, 1968; Johnson and Cheng, 1976). Kissel, Pratt and co-workers have made extensive studies of the scattering process, focused on predictions utilizing the S-matrix approach (see, for example, Kissel and Pratt, 1985; Kane et al., 1986; Pratt et al., 1994; Kissel et al., 1995). Kissel and Pratt have developed a prescription for practical and accurate predictions of elastic scattering that utilizes S-matrix amplitudes for the contribution from inner-shell electrons (h-bar*omega <= 300e, e is electron binding energy), and modified relativistic form factors to estimate the contribution from outer-shell electrons. This effort has resulted in availability of systematic S-matrix scattering predictions for all atoms, all angles and photon energies from 0.0543-2754 keV.

For x-ray and low-energy gamma-ray energies, it is customary to compute the total elastic photon-atom scattering amplitude as the sum of separate amplitudes:

R - the Rayleigh amplitude, for scattering from the atomic electrons;
NT - the nuclear Thomson amplitude, for scattering from the nucleus modeled as a point charge;
D - the Delbruck amplitude, for scattering from the field of the nucleus;
NR - and the nuclear resonance amplitude, for scattering from the internal structure of the nucleus modeled by the giant dipole resonance.

For the x-ray and low-energy gamma-ray energies considered here, the Rayleigh amplitude dominates the total scattering amplitude for most energies and angles. The nuclear Thomson amplitude is important in the 100-keV range and higher for heavy atoms at intermediate and back angles. Except for heavy atoms and photon energies above about 1 MeV, the Delbruck and nuclear-resonance amplitudes are not expected to contribute significantly to the total scattering amplitude.

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Information date: Sep. 2, 2000 lk