Density profile of a harmonically trapped ideal Fermi gas in arbitrary dimension

TL;DR

This paper derives exact formulas for the density distribution of a noninteracting Fermi gas in harmonic traps across any dimension, including shell effects and finite particle corrections.

Contribution

It provides the first closed-form analytic expressions for the density profile in arbitrary dimensions, incorporating shell structure effects beyond the local density approximation.

Findings

Analytic density profiles for all dimensions

Shell effects scale as N^{-(1+1/d)/2}

Finite particle corrections are characterized

Abstract

Closed form analytic expressions are derived for the density profile of a harmonically trapped noninteracting Fermi gas in $d$ dimensions. Shell structure effects are included to leading order in 1/N, where $N$ is the number of particles. These corrections to the local density approximation scale as $\delta n/n\sim N^{-\alpha}$, where $\alpha=(1+1/d)/2$.