# Third- and fourth-order virial coefficients of harmonically trapped fermions in a semiclassical approximation

## TL;DR

This paper derives analytical expressions for third- and fourth-order virial coefficients of trapped fermions using a semiclassical approximation, providing insights into their temperature, trapping, and interaction dependence.

## Contribution

It introduces a simple semiclassical method to calculate higher-order virial coefficients for trapped fermions across various dimensions and interaction strengths.

## Key findings

- Analytic formulas for virial coefficients agree with numerical results in certain regimes.
- Interaction effects on virial coefficients are characterized as functions of temperature and coupling.
- Results are applicable to understanding thermodynamics of trapped Fermi gases.

## Abstract

Using a leading-order semiclassical approximation, we calculate the third- and fourth-order virial coefficients of nonrelativistic spin-1/2 fermions in a harmonic trapping potential in arbitrary spatial dimensions, and as functions of temperature, trapping frequency and coupling strength. Our simple, analytic results for the interaction-induced changes $\Delta b_3$ and $\Delta b_4$ agree qualitatively, and in some regimes quantitatively, with previous numerical calculations for the unitary limit of three-dimensional Fermi gases.