Elastic electron-atom scattering in amplitude-phase representation with application to electron diffraction and spectroscopy

Abstract

The amplitude-phase representation (APR) [W. E. Milne, Phys. Rev. 35, 863 (1930)] is applied to the relativistic radial Schr\"odinger equation corresponding to the Dirac equations with central potential. The initial conditions, hitherto unspecified, for the nonlinear second-order amplitude equation at the finite radius of a muffin-tin (MT) sphere are established by a semiconvergent method. This opens the possibility of using APR for the calculation of electron scattering phase shifts with a finite MT radius as well as with a large atomic radius, whereby the first wave node is phase of reference equal to a multiple of $\ensuremath{\pi}$, adding nothing to the phase shifts. Furthermore, APR is used for benchmarking wave functions obtained by ordinary differential equation integration of the Dirac equations up to high energy and high orbital quantum number. The present APR procedure is discussed with reference to earlier numerical methods. To complete the physical picture, the paper ends with a discussion on exchange-correlation dependent scattering potential in MT spheres of optimized radii, a crystal potential model whose dependence on radii and energy dependent inner potential has recently been corroborated by low energy electron diffraction in oxides with many atoms per unit cell.