The nonrelativistic limit of the second order S-matrix element for the   elastic scattering of photons by K-shell bound electrons
  In this paper, the right expressions of the fully nonrelativistic K-shell Rayleigh scattering amplitudes and cross-sections are obtained by using the Coulomb Green function method. Our analytical result has no spurious poles as the old nonrelativistic result with retardation [M. Gavrila and A.Costescu Phys. Rev. A 2, 1752 (1970)] presents. Taking into account the exact expression of the second order S-matrix element in the case of the elastic scattering of photons by K-shell bound electrons we obtain its nonrelativistic limit (valid for photon energies omega up to alpha*Z*m, for intermediate Z), by removing in a consistent and appropriate way the higher order terms in alpha*Z and omega/m. We obtain the imaginary part of the Rayleigh amplitudes in terms of elementary functions. Thereby the exact nonrelativistic total photoeffect cross-section is obtained via the optical theorem. Comparing the predictions given by our formulae with the full relativistic numerical calculations of Kissel et al [Phys. Rev. A 22, 1970 (1980)], and with experimental results, a fairly good agreement within 10% is found for the angular distribution of Rayleigh scattering for photon energies up to alpha*Z*m, for intermediate Z, and up to twice the photoeffect threshold energy for high Z elements. Considering the right relativistic location for the photoeffect threshold very good predictions are found below the first resonance, even in its vicinity. We present our numerical results for Rayleigh angular distribution and the photoeffect total cross-section, for various photon energies, scattering angles and Z, compared with other theoretical and experimental results. 
A. Costescu, S. Spanulescu, C. Stoica