Effect of Spin-Orbit Coupling on the Energy Levels in the6dBand for Actinide Metals

The importance of spin-orbit coupling in modifying the energy levels in the $6d$ band for the actinide metals is discussed. The energy matrix, $H_0$, used in this paper, neglecting spin-orbit effects, is derived from Kohn's variational principle by expanding the wave function in terms of the five atomic $d$ orbitals having cubic symmetry. A single disposable parameter, $E_d$, occurs in this model such that $2E_d$ is the width of the $d$ band. The spin-orbit energy matrix, $H_{\mathrm{so}}$, contains the spin-orbit splitting parameter, $E_{\mathrm{so}}$, which is equal to the $d_{\frac{3}{2}}-d_{\frac{5}{2}}$ level separation. The complete 10×10 energy matrix has been diagonalized analytically along a fourfold symmetry axis and numerically at 150 points in the Brillouin zone for the case $\frac{E_{\mathrm{so}}}{E_d}=0.2\mathrm{}$ appropriate to face-centered cubic thorium with $E_{\mathrm{so}}=0.4\mathrm{}$ ev and $E_d=2\mathrm{}$ ev. Plots of the energy variation along the two, three, and fourfold axes in the Brillouin zone are presented. Calculations based upon this model are also presented which show that the usual Pauli spin paramagnetism of the conduction electrons can be modified quite significantly by spin-orbit coupling.