Theory of the Diamagnetism of Bloch Electrons

This paper deals with the diamagnetic susceptibility of a degenerate gas of Bloch electrons in a cubic lattice. It is shown that the effective Hamiltonian of such electrons in a magnetic field is rigorously a particular power series in P of the form $\tilde{H}=E\left(\mathrm{P}\right)$, where $\mathrm{P}=\left(\frac{\hslash }{i}\right)\nabla -\left(\frac{e}{c}\right)\mathrm{A}$ and $E\left(\hslash \mathrm{k}\right)$ is the energy of the band in question. (Because of the noncommutativity of the components of P, $E\left(\hslash \mathrm{k}\right)$ does not determine $E\left(\mathrm{P}\right)$ uniquely. $E\left(\mathrm{P}\right)$ is not a symmetrized power series in the components of P.) By expanding $E\left(\mathrm{P}\right)$ in powers of P, one is led to a series expansion for the diamagnetic susceptibility of the form $\chi =-\frac{e^2{k}_0}{12{\pi }^{2}m{c}^2}\left(\frac{m}{m^{*}}+c_2{k_0}^2+c_4{k_0}^4+\cdots \right),$ where $k_0={\left(3\mathrm{}{\pi }^{2}n\right)}^{\frac{1}{3}}$ and $n$ is the number of electrons per unit volume. (For a spherical band, $k_0$ is the wave number on the surface of the Fermi sea.) The first term of this series is the well-known Landau-Peierls expression, the higher terms are corrections to it. In the tight binding approximation the second term becomes dominant and reduces correctly to the atomic diamagnetism. We have calculated the first two terms of this series for Li and Na. For Na the second term was found to be very small; for Li it is more than half as large as the first and of opposite sign. Our numerical results are ${\chi }_{\mathrm{Li}}=-0.074\mathrm{}\times {10}^{-6\mathrm{}}$ and ${\chi }_{\mathrm{Na}}=-0.26\mathrm{}\times {10}^{-6\mathrm{}}$ cgs volume units.