Sever Spanulescu

Rayleigh scattering of x-ray and γ-ray by 1s and 2s electrons in ions and neutral atoms

Using the Coulomb–Green function method and considering the nonrelativistic limit for the two-photon S-matrix element, the right nonrelativistic 2s Rayleigh scattering amplitudes are obtained. Our result takes into account all multipoles, retardation and relativistic kinematics contributions, and the old dipole approximation result of Costescu [1] is retrieved as a limit case. The total photoeffect cross-section which is related to the imaginary part of the Rayleigh forward scattering amplitude through the optical theorem is also obtained. Our Coulombian formulae are used in the more realistic case of elastic scattering of photons by bound 1s and 2s electrons in ions and neutral atoms. Screening effects are considered in the independent particle approximation through the Hartree–Fock method. The effective charge Zeff is obtained by fitting the Hartree–Fock charge distribution by a Coulombian one. Good agreement (within 10%) is found when comparing the numerical predictions given by our nonrelativistic formulae with the full relativistic numerical results of Kissel [2] in the case of elastic scattering of photons by 1s and 2s electrons and Scofield [3] in the case of K-shell and 2s subshell photoionization for neutral atoms with 18 ≤ Z ≤ 92 and photon energies ω ≤ αZm.

The second-order S-matrix element for the elastic scattering of photons by K-shell bound electrons: the nonrelativistic limit

The right expressions of the nonrelativistic K-shell Rayleigh scattering amplitudes and cross-sections are obtained by using the Coulomb Greens function method. Our analytical result does not have the spurious poles that occur in the old nonrelativistic result with retardation (Gavrila and Costescu 1970 Phys. Rev. A 2 1752). Starting from the expression of the second-order S-matrix element for the case of the elastic scattering of photons by K-shell bound electrons, we obtain the correct nonrelativistic Rayleigh angular distribution (valid for photon energies ω up to αZm) by removing the relativistic higher order terms in αZ and ω/m. The imaginary part of the Rayleigh amplitudes is obtained for any scattering angles in a closed form in terms of elementary functions. Thereby a simple formula for the exact nonrelativistic photoeffect total cross-section is obtained via the optical theorem, giving significantly better predictions than Fischers nonrelativistic photoeffect formula. Comparing the predictions given by our formulae with the full relativistic numerical calculations of Kissel et al (Phys. Rev. 1980 A 22 1970), and with experimental results, a fairly good agreement within 10% is found for the angular distribution of Rayleigh scattering for photon energies up to 200 keV and both below and above the first resonance.