X-ray incoherent scattering total cross sections and energy-absorption cross sections by means of simple calculation routines

Simple but accurate expressions for the total cross sections and energy-absorption cross sections for inelastic scattering of x rays from atomic, molecular, and condensed matter are developed. The starting point is the relativistic impulse approximation for Compton scattering of x rays. This gives the inelastic-scattering cross-section differential with respect to energy and solid-angle interval. Integration over energy of this expression up to all the shells $K$, $L$, $M$,... for the orbitals gives a simple result in the "linear" approximation [R. Ribberfors and K.-F. Berggren, Phys. Rev. A 26, 3325 (1982)]. The result is a cross-section differential with respect to solid angle. Numerical results for Al, Ag, and Pb are in good agreement with available experimental values and with the Waller-Hartree theory. It is shown that a further integration of the differential cross section with respect to the solid angle results in a simple formula for calculating the total cross section for inelastic scattering of x rays. A comparison with integrated values of the incoherent scattering function (Waller-Hartree theory) is carried out. Calculations of the (electron) energy-absorption cross-section result in a compact formula. In this case also, we make comparisons with the double-differential cross section integrated twice and with the Walter-Hartree theory.