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Journal of Mathematical Physics

Chemical potential of a D-dimensional free Fermi gas at finite temperatures

M. Howard Lee

Published Aug 1, 1989• 14 citations
Publisher

Abstract

The chemical potential of a D-dimensional free Fermi gas at low temperatures has been obtained using some ideas due to Barker and Blankenbechler [J. Math. Phys. 97, 302 (1986); Am. J. Phys. 25, 279 (1957)]. There is an interesting even–odd-dimensional effect in the behavior of the chemical potential, also observed in other properties of a free Fermi gas, such as the susceptibility. For D=2, it is possible to give a closed form expression, thus valid both in the high- and low-temperature regions, probably not possible in any other even or odd dimensions.

DOI
10.1063/1.528269

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The chemical potential of a D-dimensional free Fermi gas at low temperatures has been obtained using some ideas due to Barker and Blankenbechler [J. Math. Phys. 97, 302 (1986); Am. J. Phys. 25, 279 (1957)]. There is an interesting even–odd-dimensional effect in the behavior of the chemical potential, also observed in other properties of a free Fermi gas, such as the susceptibility. For D=2, it is possible to give a closed form expression, thus valid both in the high- and low-temperature regions, probably not possible in any other even or odd dimensions.
The chemical potential and fluctuations in number of particles in a D-dimensional free Fermi gas at low temperatures are obtained by means of polylogarithms. This idea is extended to show that the density of any ideal gas, whether Fermi, Bose, or classical, can be expressed in polylogarithms. The densities of different statistics correspond to different domains of polylogarithms in such a way that there emerges a unifying picture. The density of the classical ideal gas represents a fixed point of polylogarithms. Inequalities for polylogarithms are used to provide a precise bound on errors in the fermion chemical potential at low temperatures.
In this paper, the chemical potential of two-dimensional (2D) and quasi-one-dimensional (Q1D) multisubband charged Fermi gases is evaluated. We start with a rather general formula for the thermodynamic potential of an ideal quantum statistical system with arbitrary occupation-number to calculate, as a particular case, the chemical potential of the multisubband 2D Fermi gas described by a quadratic energy spectrum. The chemical potential is also studied in the case of a low-dimensional Fermi gas in the presence of a quantizing magnetic field. The same approach is used to study the chemical potential of a multisubband Q1D Fermi gas. The influence of temperature is considered. We found interesting analytical results for special limits as well as numerical results that may be important for the description of both the thermodynamic and kinetic properties of Fermi systems in low dimensionality.
The properties of an ideal Fermi gas in an external potential in any dimensional space are studied, based on the semiclassical (Thomas-Fermi) approximation. The general analytical expressions of the total particle number N,N,N, density of states $D(\ensuremath{\varepsilon}),$ Fermi energy EF,{E}_{F},EF​, total energy E,E,E, and heat capacity CCC have been derived, where N,N,N, E,E,E, and CCC are expressed by the Fermi integration. Moreover, the analytical expressions of the total energy E,E,E, chemical potential \ensuremath{\mu}, and heat capacity CCC in the high- and low-temperature approximations are given. From these results, how the characteristics of the Fermi gas depend on an external potential and the dimension of space is discussed.
The effects of low dimensionality on the thermodynamics of a Fermi gas trapped by isotropic power-law potentials are analyzed. Particular attention is given to different characteristic temperatures that emerge, at low dimensionality, in the thermodynamic functions of state and in the thermodynamic susceptibilities (isothermal compressibility and specific heat). An energy-entropy argument that physically favors the relevance of one of these characteristic temperatures, namely, the nonvanishing temperature at which the chemical potential reaches the Fermi energy value, is presented. Such an argument allows interpreting the nonmonotonic dependence of the chemical potential on temperature, as an indicator of the appearance of a thermodynamic regime, where the equilibrium states of a trapped Fermi gas are characterized by larger fluctuations in energy and particle density as is revealed in the corresponding thermodynamics susceptibilities.
We study normal state properties of an interacting Fermi gas in an isotropic harmonic trap of arbitrary dimensions. We exactly calculate the first-order perturbation terms in the ground-state energy and chemical potential, and obtain simple analytic expressions of the total energy and chemical potential. At zero temperature, we find that the Thomas–Fermi approximation agrees well with the exact results for any dimension even though the system is small. In the high-temperature (classical) region, we find the interaction energy decreases in proportion to T − d /2 , where T is the temperature and d is the dimension of the system.
A system of a finite number of harmonically trapped fermions in one dimension, in the presence of a static magnetic field, is studied within the framework of the static fluctuation approximation, for different repulsive and attractive potential strengths. Specifically, the thermodynamic properties of the system (the chemical potential, total energy, heat capacity, and entropy), as well as its magnetic properties (the magnetization and susceptibility), are calculated. It is observed that the system remains in an ordered phase for a small number of particles N, even at high temperatures T. Disorder sets in for large N, even at low T. The effect of the potential strength on the heat capacity is particularly tangible in the region bordering the quantum and classical regimes. The effect of the temperature (representing disorder) is directly opposite to that of the magnetic field (representing order), as expected on basic physical grounds. These features are consistent with experimental results.