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arXiv:cond-mat/0405425 · Physical Review Letters

Density Profile of a Harmonically Trapped Ideal Fermi Gas in Arbitrary Dimension

Erich J. Mueller

Published Nov 4, 2004• 30 citations
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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 $\ensuremath{\delta}n/n\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\alpha}}$, where $\ensuremath{\alpha}=(1+1/d)/2$.

DOI
10.1103/physrevlett.93.190404

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Closed-form analytic expressions are derived for the density profile of a harmonically trapped noninteracting Fermi gas in ddd dimensions. Shell structure effects are included to leading order in 1/N1/N1/N, where NNN is the number of particles. These corrections to the local density approximation scale as \ensuremathδn/n\ensuremath∼N\ensuremath−\ensuremathα\ensuremath{\delta}n/n\ensuremath{\sim}{N}^{\ensuremath{-}\ensuremath{\alpha}}\ensuremathδn/n\ensuremath∼N\ensuremath−\ensuremathα, where \ensuremathα=(1+1/d)/2\ensuremath{\alpha}=(1+1/d)/2\ensuremathα=(1+1/d)/2.
We develop a Green's function method to evaluate the exact equilibrium particle-density profiles of noninteracting Fermi gases in external harmonic confinement in any spatial dimension and for arbitrary trap anisotropy. While in a spherically symmetric configuration the shell effects are negligible in the case of a large number of particles, we find that for very anisotropic traps the quantum effects due to single-level occupancy and the deviations from the Thomas-Fermi approximation are also visible for mesoscopic clouds.
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We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas-Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation tau(TF)[rho] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities tau(r) and rho(r) including their leading-order oscillating terms.
We investigate the particle and kinetic energy densities of harmonically trapped fermion gases at zero temperature in arbitrary dimensions. We derive differential equations connecting these densities, which so far have been proved only in one or two dimensions, and give other interesting relations involving several densities or the particle density alone. We show that in the asymptotic limit of large particle numbers, the densities go over into the semi-classical Thomas–Fermi (TF) densities. Hereby the Fermi energy to be used in the TF densities is uniquely identified. We derive an analytical expansion for the remaining oscillating parts and obtain very simple closed forms for the leading-order oscillating densities. Finally, we show that the simple TF functional relation τ TF [ρ] between kinetic and particle densities is also fulfilled for the asymptotic quantum densities τ( r ) and ρ( r ) including their leading-order oscillating terms.