AtomicL-shell Compton profiles: Calculations for atoms and comparison with experiment

The techniques we developed previously for calculating $L$-shell Compton profiles within the Born approximation are applied to the calculation of improved core Compton profiles for real atoms. The method is referred to as the exact hydrogenic (EH) method. We introduce a new method for determining effective nuclear charges $Z^{*}$ to use in hydrogeniclike wave functions and our EH method. These nuclear charges $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ are determined by imposing the requirement for each atomic orbital that the impulse hydrogenic (IH) Compton profile at its center $q=0.0\mathrm{}$ match the impulse Hartree-Fock (IHF) Compton-profile value at $q=0.0\mathrm{}$. It is then demonstrated that the two $L$-shell impuse profiles are found to lie very close to one another as we go away from the profile center, the agreement improving as we go to atoms of higher atomic number. The wave functions for the $2s$ and $2p$ orbitals are studied and it is observed that hydrogenic wave functions with our values of $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ fit the Hartree-Fock wave functions much more accurately than if a $Z^{*}$ associated with the binding energy is used. We demonstrate in the case of Mo $K\alpha$ radiation scattering from neon and aluminum that our EH method using $Z^{*}\mathrm{}\left(0\right)\mathrm{}$ gives results in better agreement with experiment than any impulse calculation. In the neon case, the EH correction to the IHF result is ∼2.4% at the profile center. Dependence of the Compton profile on incident photon energy is demonstrated theoretically and observed to be in agreement with experiment.