Density profile of the weakly interacting Fermi gas confined in a potential well: Nonzero temperature

Abstract

A theory is developed for the low-temperature density profile of a weakly interacting normal Fermi gas confined within an arbitrary potential well. This problem is of interest in connection with magnetically confined spin-aligned atomic deuterium (${\mathrm{D}}_{\mathrm{\downarrow }}$). The approach used is based on the statement of constancy of the chemical potential in equilibrium. This involves the internal chemical potential, a functional of the density, which we treat within the local-density approximation. We develop an accurate approximate form for the local internal chemical potential: For a given low temperature, it is taken equal to the known forms at low density (classical) and higher density (quantum); for intermediate density we use an appropriate interpolant which smoothly connects the low- and high-density forms. Finite-temperature density profiles are evaluated for the ideal Fermi gas for parameters appropriate to magnetically confined ${\mathrm{D}}_{\mathrm{\downarrow }}$ for both symmetric linear and quadratic one-dimensionally varying wells. We also evaluate density profiles including the leading-order interparticle interaction effect. For conditions relevant to hypothetical ${\mathrm{D}}_{\mathrm{\downarrow }}$ at moderate densities this effect is found to be small but measurable.