Electron Energy Levels in LaSn3. II. Relativistic Connections Using Perturbation Theory

In a separate paper, electron energy levels in perfectly ordered La${\mathrm{Sn}}_3$ were calculated nonrelativistically for the equivalent of 64 points in the reciprocal-space lattice using a modified orthogonalized-plane-wave method. We now use perturbation theory to determine the relativistic corrections. Starting with Dirac's theory of the relativistic spinning electron and following Slater we write the final $E\left(\overrightarrow{\mathrm{k}}\right)$ as $E^0\left(\overrightarrow{\mathrm{k}}\right)$ plus mass-velocity, Darwin, and spin-orbit corrections where the $E^0\left(\overrightarrow{\mathrm{k}}\right)$ are the nonrelativistic energy levels previously calculated. The mass-velocity and Darwin operators do not affect symmetry and thus involve no mixing between different irreducible single-group representations. These two corrections thus cause only a shift in energy. This shift is calculated by nondegenerate first-order perturbation theory. The spin-orbit operator does affect symmetry so that those nonrelativistic levels associated with two- and three-dimensional single-group representations can split. This splitting is calculated by degenerate first-order perturbation theory. The spin-orbit operator can also mix levels belonging to different irreducible single-group representations providing the nonrelativistic levels are not too widely separated in energy. Perturbation theory has been used to determine this mixing in most of the appropriate cases with particular emphasis given to those levels near the Fermi level. Relativistic $E\left(\overrightarrow{\mathrm{k}}\right)$ curves are shown for six directions in $\overrightarrow{\mathrm{k}}$ space. Using the final $E\left(\overrightarrow{\mathrm{k}}\right)$, a Fermi level of - 0.50 Ry has been computed. This is about 0.03 Ry lower than the nonrelativistic Fermi level. Comparison with the limited amount of existent La${\mathrm{Sn}}_3$ experimental data is discussed briefly.