Ideal Fermi gases in harmonic oscillator potential traps

TL;DR

This paper analytically investigates the thermodynamic properties of ideal Fermi gases in harmonic traps, revealing oscillatory behaviors in chemical potential and specific heat linked to de Haas-van Alphen effects across different dimensions and trap anisotropies.

Contribution

It provides analytical results for chemical potential and specific heat in harmonic traps, highlighting oscillations and deviations from classical behavior due to quantum effects.

Findings

Chemical potential exhibits step-like behavior.

Specific heat shows oscillations with particle number.

Anisotropic traps cause jumps in specific heat.

Abstract

We study the thermodynamic properties of an ideal gas of fermions in a harmonic oscillator confining potential. The analogy between this problem and the de Haas-van Alphen effect is discussed and used to obtain analytical results for the chemical potential and specific heat in the case of both isotropic and anisotropic potentials. Step-like behaviour in the chemical potential, first noted in numerical studies, is obtained analytically and shown to result in an oscillatory behaviour of the specific heat when the particle number is varied. The origin of these oscillations is that part of the thermodynamic potential responsible for the de Haas-van Alphen-type effect. At low temperatures we show analytically that there are significant deviations in the specific heat from the expected linear temperature dependence, again as a consequence of the de Haas-van Alphen part of the thermodynamic potential. Results are given for one, two, and three spatial dimensions. In the anisotropic case we show how the specific heat jumps as the ratio of oscillator frequencies varies.