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Jiangxi kexue

Thermodynamic Properties of A Relativistic Fermi Gas Trapped in a General External Potential

Fan Zhao-Lan

Published Jan 1, 2009• 0 citations
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Abstract

Based on the semiclassical(Thomas-Fermi) approximation,the macrocanonical partition function and macrocanonical potential of a relativistic Fermi gas in a general external potential in any dimensional space are studied.The general analytical expressions of the total particle number N,total energy U and heat capacity C have been derived with thermodynamic fundamental equations,where N,U and C are expressed by the modified Bessel function.Moreover,the analytical expressions of the total energy,chemical potential and heat capacity in the high-and low-temperature approximations are given.From these results,how the characteristics of the Fermi gas depend on various factors is discussed.

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Based on the semiclassical(Thomas-Fermi) approximation,the macrocanonical partition function and macrocanonical potential of a relativistic Fermi gas in a general external potential in any dimensional space are studied.The general analytical expressions of the total particle number N,total energy U and heat capacity C have been derived with thermodynamic fundamental equations,where N,U and C are expressed by the modified Bessel function.Moreover,the analytical expressions of the total energy,chemical potential and heat capacity in the high-and low-temperature approximations are given.From these results,how the characteristics of the Fermi gas depend on various factors is discussed.
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 properties of an ideal Fermi gas in finite gravitational field are studied,based on the semiclassical(Thomas-Fermi) approximation.The general analytical expressions of the total energy,Fermi energy,chemical potential and heat capacity are derived.Moreover,the analytical expressions of the chemical potential and heat capacity in the high-and low-temperature approximations are given,and then,the example of electron gas is used to calculate exactly.All these results are compared with those of free Fermi gas and Fermi gas in infinite gravitational field.The effects of finite gravitational field on properties of Fermi gas are discussed.
Using the methods of numerical simulation and quantum statistics, based on the thermodynamic quantities of Fermi gas trapped in a general external potential, we studied the thermodynamic properties of a relativistic Fermi gas trapped in hard-sphere potential, gave the analytical expressions of the energy and heat capacity of the relativistic Fermi system, and analyzed the influence of the relativistic effect on the energy and heat capacity. The results showed the energy and heat capacity of the relativistic Fermi system is larger than those of the nonrelativistic case. The larger the relativistic eigenvalue, the lower the change temperature of heat capacity. As the temperature rises, the larger the relativistic eigenvalue, the greater the energy.
In this paper, we provide comprehensive study of the thermodynamic quantities of the ideal Fermi gas confined in a three-dimensional harmonic trap by using the properties of Fermi – Dirac integral function both analytically and numerically. The dependences of the chemical potential, total energy and heat capacity on the temperature are obtained via the appropriately approximated analytic formulae. Afterwards, the results are compared with the exact numerical ones in order to evaluate the applicability of these formulae.