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EFFECT OF A PERIODIC POTENTIAL ON A DEGENERATE FERMI GAS

S. A. Moszkowski, L. Wilets

Published Jan 18, 1962• 1 citation
OpenAlex

Abstract

The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It is found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed.

Related papers

EFFECT OF A PERIODIC POTENTIAL ON A DEGENERATE FERMI GAS. Technical Note No. 14

S. A. Moszkowski, L. Wilets

The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It was found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed. (auth)

•Jan 18, 1962•0 citations

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Gradient corrections to the kinetic energy density functional of a two-dimensional Fermi gas at finite temperature

Brandon P. van Zyl, K. Berkane, K. Bencheikh +1 more

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arXiv:1012.0347•Physical Review B•May 31, 2011•6 citations
The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It is found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed.
The effect of a periodic potential on the particle and energy density of a degenerate Fermi gas is considered. It was found that for a weak and long wavelength potential the inhomogeneity correction (deviation from the Thomas Fermi statistical model) to the kinetic energy density is only 1/9 as large as the Weizsacker estimate. The potential energy density due to Majorana exchange two-body interactions of short range is also given to a high accuracy by the statistical model; the corrections are only 1/9 as large as for a short range Wigner (ordinary) interaction. For the analogous two dimensional Fermi gas, the inhomogeneity corrections vanish at least to high order unless the periodic potential contains short wavelength components. Relevance to the problem of the nuclear surface is discussed. (auth)
A degenerate Fermi gas having only a long-wavelength periodic density modulation is considered. We show that the exchange and correlation potentials which act on particles near the Fermi surface are not those derived from the exchange and correlation contributions to the local chemical potential. For each potential there is an additional term related to the change in effective mass caused by exchange and correlation. This result calls attention to an unresolved theoretical question: whether m∗{m}^{*}m∗, the density-of-states effective mass at the Fermi surface, is greater than or less than mmm (for metallic densities).
In a recent paper [B. P. van Zyl et al., Phys. Rev. A 89, 022503 (2014)], the average density approximation (ADA) was implemented to develop a parameter-free, nonlocal kinetic energy functional to be used in the orbital-free density functional theory of an inhomogeneous, two-dimensional (2D) Fermi gas. In this work, we provide a detailed comparison of self-consistent calculations within the ADA with the exact results of the Kohn-Sham density functional theory and the elementary Thomas-Fermi (TF) approximation. We demonstrate that the ADA for the 2D kinetic energy functional works very well under a wide variety of confinement potentials, even for relatively small particle numbers. Remarkably, the TF approximation for the kinetic energy functional, without any gradient corrections, also yields good agreement with the exact kinetic energy for all confining potentials considered, although at the expense of the spatial and kinetic energy densities exhibiting poor pointwise agreement, particularly near the TF radius. Our findings illustrate that the ADA kinetic energy functional yields accurate results for both the local and global equilibrium properties of an inhomogeneous 2D Fermi gas, without the need for any fitting parameters.
We present a simple implementation of a density-dependent, zero-range interaction in a degenerate Fermi gas using hyperspherical coordinates in the large particle number limit. The method produces a 1D effective potential that accurately describes the ground-state energy as a function of the hyperradius of the two spin component gas throughout the vicinity of the unitarity regime. In the unitarity regime the breathing mode frequency is found to limit to the non-interacting value. A dynamical instability, similar to the Bosenova, is predicted to be possible for a gas containing more than three spin components, for large, negative, two-body scattering lengths. While the situation is less clear for the gas with three components, collapse might also be possible.
The structure and stability of dilute degenerate Fermi gases trapped in an external potential is discussed with special emphasis on the influence of s- and p-wave interactions. In a first step an effective contact interaction for all partial waves is derived, which reproduces the energy spectrum of the full potential within a mean-field model space. Using the s- and p-wave part the energy density of the multicomponent Fermi gas is calculated in Thomas-Fermi approximation. On this basis the stability of the one- and two-component Fermi gas against mean-field induced collapse is investigated. Explicit stability conditions in terms of density and total particle number are given. For the single-component system attractive p-wave interactions limit the density of the gas. In the two-component case a subtle competition of s- and p-wave interactions occurs and gives rise to a rich variety of phenomena. A repulsive p-wave part, for example, can stabilize a two-component system that would otherwise collapse due to an attractive s-wave interaction. It is concluded that the p-wave interaction may have important influence on the structure of degenerate Fermi gases and should not be discarded from the outset.
We examine the leading-order semiclassical gradient corrections to the noninteracting kinetic-energy density functional of a two-dimensional Fermi gas by applying the extended Thomas-Fermi theory at finite temperature. We find a nonzero von Weizs\"acker-like gradient correction, which in the high-temperature limit goes over to the functional form $(\ensuremath{\hbar}{}^{2}/24m)(\ensuremath{\nabla}\ensuremath{\rho}){}^{2}/\ensuremath{\rho}$. Our work provides a theoretical justification for the inclusion of gradient corrections in applications of density-functional theory to inhomogeneous two-dimensional Fermi systems at any finite temperature.