Inelastic scattering from core electrons: A multiple scattering approach

The real-space multiple-scattering approach is applied to model nonresonant inelastic scattering from deep core electron levels over a broad energy spectrum. This approach is applicable to aperiodic or periodic systems alike and incorporates ab initio, self-consistent electronic structure and final state effects. The approach generalizes to finite momentum transfer a method used extensively to model x-ray absorption spectra (XAS), and includes both near-edge spectra and extended fine structure. The calculations can be used to analyze experimental results of inelastic scattering from core electrons using either x-ray photons or electrons. In the low momentum transfer region (the dipole limit), these inelastic loss spectra are proportional to those from XAS. Thus, their analysis can provide similar information about the electronic and structural properties of a system. Results for finite momentum transfer yield additional information concerning monopole, quadrupole, and higher couplings. Our results are compared both with experiment and with other theoretical calculations.

Figure 1

LDOS ${\rho }_l^0\left(E\right)$ for the central site of fcc Al, in the absence of other scatterers: $l=0$ (solid); $l=1$ (long dashes); $l=2$ (dashes); and $l=3$ (short dashes). Note how the successive terms ${\rho }_l^0\left(E\right)$ with increasing angular momentum $l$ peak at successively higher energies and overlap each other.

Figure 2

Comparison between experiment (Ref. 18) (top panels) and inelastic loss spectra calculated using the BSE (middle panels) and the RSMS method of this work (lower panels) for the Be $K$ edge. The spectra are shown for $\mathbf{q}$ perpendicular to the $c$ axis (left panels), and $\mathbf{q}$ along the $c$ axis (right panels). Spectra are shown for two values of momentum transfer $q$ as indicated in the figure labels. The theoretical spectra was shifted by 4.5 eV to align with the experimental edge.

Figure 3

Local projected DOS for Be together with inelastic scattering spectra for two values of momentum transfer $q$ along the $x$ axis. The Fermi level is at $-9.4\mathrm{eV}$.

Figure 4

Comparison of contributions to the inelastic loss spectra from monopole and dipole transitions, for momentum transfer $q=4.78\mathrm{a}.\mathrm{u}.$ along the $c$ axis.

Figure 5

Changes in the shape of different contributions to the Be $K$-edge spectra, as a function of the magnitude of the momentum transfer along the $c$ axis for two values of $q$ listed in the figure labels. The upper panel shows the changes for the monopole contribution and the lower panel, those for the dipole allowed transitions. The spectra were normalized so that the height of the first peak is the same for both values of $q$.

Figure 6

Comparison of inelastic scattering and x-ray absorption spectra at the Al $L_1$ edge using the path expansion for several values of momentum transfer as indicated in the figure labels; the curve $\sigma \left(\omega \right)∕\omega$ is the result for $q=0$. For clarity, the inelastic scattering spectra were slightly shifted vertically. All the spectra were scaled to be shown on the same figure.

Figure 7

Fine structure ${\chi }_{\mathbf{q}}\left(k\right)$ for Al at the $L_1$ edge for different values of momentum transfer as listed in the figure labels, together with the x-ray absorption fine structure $\chi \left(k\right)$, i.e., $q=0$. The curves are shifted for clarity.

Figure 8

Fourier transform of Al fine structure for several values of momentum transfer as listed in the figure labels. The differences reflect the interference between different channels contributing to the spectra.