New method for the calculation of atomic phase shifts: Application to extended x-ray absorption fine structure (EXAFS) in molecules and crystals

Abstract

The scattering of electrons of kinetic energy up to 1000 eV by an atom is of special interest in the understanding of extended x-ray absorption fine-structure (EXAFS) spectra. An important physical feature is the reduction of the exchange and correlation potential as the kinetic energy of the electron increases. This is taken into account by replacing the atom by an electron gas with spatially varying density and calculating the self-energy using the plasmon pole approximation. This results in a set of complex phase shifts which is then applied to the EXAFS problem. Comparison is made with phase shifts extracted from experimental EXAFS spectra and excellent agreement is obtained. Direct comparison of the theoretical and experimental spectra again shows excellent agreement in both the amplitude and the phase. We also analyze the EXAFS spectra by a Fourier-transform technique which first removes the amplitude and phase shift using the calculated result. The importance of a proper choice of zero of energy ${E}_{0}$ is emphasized. We choose ${E}_{0}$ by the requirement that the imaginary part and the absolute value of the Fourier transform should peak at the same distance, thus assuring that the absolute phase is given correctly. Using this procedure the nearest-neighbor distances in ${\mathrm{Br}}_{2}$, Ge${\mathrm{Cl}}_{4}$, and crystalline germanium are determined. In all cases the results are within 0.01 \AA{} of the known distances. Several shells in germanium are also determined, with accuracy of better than 1%. Application of our method to crystalline copper indicates that the outer shells are more seriously affected by multiple-scattering problems and our procedure permits us to discard peaks that are spurious or unreliable. The present determination of the nearest-neighbor distance in copper is found to be in error by 0.014 \AA{}. Results of the application of this method to the determination of the bond lengths of a variety of compounds are summarized.