A general formulation is developed for determining the free energy of a Fermi gas contained in an arbitrary smooth external potential barrier and in a weak magnetic field, in the low-temperature limit. The Wigner phase-space formalism of quantum mechanics is used as a calculational tool. Explicit formulas are given, which enable one to compute surface and temperature effects on various physical properties (susceptibility, specific heat, etc.) of the system. Some simple examples are constructed for the diamagnetic-susceptibility calculation, which show that the corrections depend on the form of the surface potential barrier and the size of the material, but, in general, they are small in comparison with the dominant Landau result when the size is much larger than ∼100 Å. The general formalism that we present can also be applied to other kinds of Fermi gas (for example, nucleons) contained in an external potential. For example, we show how the modified Thomas-Fermi theory may be extended to include temperature effects.

• Received 18 February 1986

DOI:https://doi.org/10.1103/PhysRevB.34.5160