Theory of the Magnetic Susceptibility of Bloch Electrons

It has heretofore always been assumed that the magnetic susceptibility of a crystal could be written $\chi ={\chi }_{\mathrm{dia}}^{\mathrm{core}}+{\chi }_{\mathrm{dia}}^{\mathrm{val}}+{\chi }_{\mathrm{spin}}^{\mathrm{val}}$, where ${\chi }_{\mathrm{dia}}^{\mathrm{core}}$ is the contribution of the core electrons, ${\chi }_{\mathrm{dia}}^{\mathrm{val}}$ is the contribution of the orbital motion of Bloch valence or conduction electrons completely neglecting spin, and ${\chi }_{\mathrm{spin}}^{\mathrm{val}}$ is the Pauli spin paramagnetism but with the free-electron $g$ factor replaced by the effective $g$ factor. The entire effect of spin-orbit coupling is assumed to be included in the effective $g$ factor. We show that this is not the case and that there is a large fourth contribution to $\chi$, the effect of the spin-orbit coupling on the orbital motion of the Bloch electrons ${\chi }_{\mathrm{so}}$. We construct a many-band Hamiltonian using the Bloch representation and derive the susceptibility directly from this Hamiltonian avoiding the ambiguity of the usual decoupling transformations. Our result agrees with the expression derived by Roth but is in a much more transparent form.