Compton Scattering of X-Rays from Nonspherical Charge Distributions

The Waller-Hartree theory for the incoherent Compton scattering of x-rays is extended to include the effects fo the inherent nonsphericity of atomic charge distributions. The dependence of the scattering on atomic orientation is treated by a general formalism which depends on the use of the matrix elements of a unitary representation of the group of three-dimensional rotations. Application is made to derive the one-electron scattering matrix elements from atoms with $s$, $p$, and $d$ electrons. It is shown that by a proper averaging over all orientations of the scattering vector, "mean" scattering formulas result which may be used directly for predicting the scattering from monatomic gases.