AtomicL-shell Compton profiles and incoherent scattering factors: Theory

Compton profiles are calculated for $L$-shell electrons within the Born approximation by use of nonrelativistic "exact" hydrogenic bound- and continuum-state wave functions. The wave functions are expressed in parabolic coordinates and the resulting matrix elements are evaluated following the method of F. Bloch (1934). Impulse-approximation profiles are calculated, and comparisons with the "exact" profiles show that, as expected, the two profiles lie very close to one another for weak binding and high incident photon energies. However, even when these conditions are not fulfilled, the two curves have a tendency to cross one another in the neighborhood of the profile center. This tendency has been observed in previous studies of $K$-shell profiles. However, unlike $K$-shell profiles, the $2S$ Compton profiles exhibit a secondary maximum. The secondary maxima occur at approximately the same region in $q$, where the impulse-approximation profiles exhibit a plateau. The location of this plateau is shown to be related to a node in the bound-state wave function and is around $q=\frac{Z}{2}$ for the $2S$ profiles. The impulse approximation, being a monotonic decreasing function in $\mathrm{}\left|q\right|\mathrm{}$, cannot exhibit the secondary-maximum structure appearing in the "exact" hydrogenic profiles. The intensity of the secondary maximum in the $2S$ profiles is reduced by over an order of magnitude from the central peak. The $2P^{\mathrm{}\left(0\right)\mathrm{}}$ Compton profiles also exhibit structure, however, unlike the relatively small secondary maximum in the $2S$, the two maxima in the $2P^{\mathrm{}\left(0\right)\mathrm{}}$ profiles are of the same order of magnitude. Integrated profiles (incoherent-scattering factors) are calculated and the impulse-approximation results agree with the "exact" results over a wide range of binding energies owing to profile crossover near the center. Waller-Hartree incoherent-scattering factors give a closer agreement with the "exact" results than observed for $K$-shell electrons; however, for low-momentum transfer the Waller-Hartree results can differ from "exact" results by more than 50%. In such regions, impulse-scattering factors represent a considerable improvement over the Waller-Hartree factors.