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arXiv:quant-ph/0003030 · Physica Scripta

Fermi Gas in Harmonic Oscillator Potentials

X. X. Yi, J C Su

Published Aug 1, 1999• 6 citations
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Abstract

Assuming the validity of grand canonical statistics, we study the properties of a spin polarized Fermi gas in harmonic traps. Universal forms of Fermi temperature T F , internal energy U and the specific heat per particle of the trapped Fermi gas are calculated as a function of particle number, and the results are compared with those of infinite number particles.

DOI
10.1238/physica.regular.060a00117

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•Dec 31, 2005•0 citations
Assuming the validity of grand canonical statistics, we study the properties of a spin polarized Fermi gas in harmonic traps. Universal forms of Fermi temperature T F , internal energy U and the specific heat per particle of the trapped Fermi gas are calculated as a function of particle number, and the results are compared with those of infinite number particles.
In view of ongoing experiments to trap ultracold spin-polarized 6Li,{}^{6}\mathrm{Li},6Li, we study various properties of an interacting Fermi gas in a harmonic trap, taking the discrete nature of the unperturbed harmonic trap levels into account exactly. As 6Li{}^{6}\mathrm{Li}6Li has a rather large and negative scattering length, we focus on the effects of the attractive atom-atom interaction on several thermodynamic properties, and on the momentum and density distributions. The dependence of the chemical potential, the specific heat, and the density and momentum distributions on the number of particles in the trap is obtained. We also calculate the energy of the gas. Comparison is made with results of a semiclassical calculation and with the properties of a noninteracting gas. We find that the effect of the interactions is rather large for realistic trap frequencies. Hence it is important to include these interactions in any quantitative predictions relevant for experiments.
At zero-temperature and finite-temperature, the thermodynamic properties of finite unitary Fermi gas in a three-dimensional harmonic trap are investigated by using fractional exclusion statistics, and the results are compared with those of the system which satisfies the thermodynamic limit. At zero-temperature, Fermi energy and average energy of per particle increase with the increase of the number of particles for finite unitary Fermi gas, and their limits are the corresponding parameters of the system which satisfy thermodynamic limits. Fermi energy and average energy of per particle each have a maximum value changing with the boundary of the potential well. For the finite-temperature trapped unitary Fermi system, when the number of particles is certain the average energy of per particle, average entropy of per particle, average heat capacity of per particle each have a characteristic temperature, respectively, when the temperature is equal to the characteristic temperature of the physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the temperature is lower (or higher) than the characteristic temperature of parameter, the physical parameter of the finite system will be greater (or less) than the corresponding parameter of the thermodynamic limit system. The characteristic temperature has particle number effect and boundary effect. When the temperature is determined, the average energy of per particle, average entropy of per particle and average heat capacity of per particle each have a characteristic number of particles, respectively, when the number of particles is equal to the characteristic number of particles for physical parameter, the corresponding parameters for the finite system and the thermodynamic limit system are equal, when the number of particles is less (or more) than the characteristic number of particles for corresponding parameter, the corresponding parameter of the finite system will be less (or larger) than the thermodynamic limit of system.
We utilize a fractional exclusion statistics of Haldane and Wu hypothesis to study the thermodynamics of a unitary Fermi gas trapped in a harmonic oscillator potential at ultra-low finite temperature. The entropy per particle as a function of the energy per particle and energy per particle versus rescaled temperature are numerically compared with the experimental data. The study shows that, except the chemical potential behavior, there exists a reasonable consistency between the experimental measurement and theoretical attempt for the entropy and energy per particle. In the fractional exclusion statistics formalism, the behavior of the isochore heat capacity for a trapped unitary Fermi gas is also analyzed.
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 is 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. © 2005 Elsevier Inc. All rights reserved.