The magnetic susceptibility of an electron gas

The cluster expansion of the quantum-mechanical grand partition function is obtained for a gas of interacting charged particles in a uniform magnetic field. The magnetic field is conveniently included by using a Green function for a charged particle in a uniform field. The theory is applied to calculate the magnetic susceptibility of an electron gas for small magnetic fields. For Boltzmann statistics the equation of state is unaltered to the DebyeHuckel approximation and the field only enters into the quantum-mechanical corrections to this equation. The first terms in a low-density expansion of the susceptibility are obtained. For Fermi-Dirac statistics an exact high-density expansion of the susceptibility to first order in the coupling constant is obtained at zero temperature. The first-order exchange energy is divergent but this divergence is removed by including the ring diagrams.