Theoretical X-Ray Scattering Factors Based on Energy-Band Structure

The energy bands for the $3d$ and $4s$ states of copper and the $2p$ states of aluminum are calculated by the augmented-plane-wave method. The crystal charge density is calculated for copper and aluminum and is used to find the scattering factors. For copper these factors are in better agreement with experiment than are those determined from Hartree-Fock atomic calculations. This improvement is shown to be due to the fact that the copper valence-electron charge density is more spread out in the solid than in the atom. The calculated scattering factors are sensitive to the starting crystal potential because of the difficulty of handling exchange in the solid. For aluminum, no improvement over Hartree-Fock scattering factors is realized by performing an energy-band calculation.