Influence of Collisions on the Magnetic Susceptibility of an Electron Gas. II. The Constant Susceptibility

An expression is derived for the collision correction to the constant magnetic susceptibility of an electron gas. Using a perturbation technique, a general expression is obtained for the free energy of a collection of electrons moving in a uniform magnetic field and scattered by a set of randomly distributed impurity centers. The expansion of the free energy is carried out in ascending powers of the scattering potential to terms of fourth order; this is necessary in order to determine the effect of collisions on the nonperiodic part of the susceptibility. The free energy is evaluated for impurities represented by a short-range screened Coulomb potential. The second-order correction to the free energy, needed in the evaluation of the constant susceptibility, is in agreement with earlier results; the third-order contribution to the free energy gives a further correction to the periodic susceptibility. The fourth-order correction to the free energy contains the first correction to the constant susceptibility along with additional corrections to the periodic part. In agreement with the prediction of Peierls, it is found that the influence of collisions on the constant susceptibility is small, provided $\eta \gg \frac{\hslash }{\tau }$, where $\eta$ is the Fermi energy and $\tau$ is the effective collision relaxation time, for which an explicit expression is obtained. The steady diamagnetism is shown to be increased in magnitude by the collision of electrons with impurities.