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Quantum Physics Letters

A Rigorous Approach for Calculating the Chemical Potential of an Ultrarelativistic Spherical Degenerate Fermi Gas Interacting with Nuclear Matter

M. A. Grado-Caffar, M. Grado-Caffaro

Published Oct 20, 2013• 2 citations
Publisher

Abstract

Abstract: We present an accurate analytical formulation to determine the chemical potential of an ultrarelativistic spherical degenerate Fermi gas interacting with nuclear matter by examining rigorously the three terms of the general mathematical expression for the chemical-potential energy namely, the Fermi energy, the optical-potential energy, and a third term which is negligible if the Fermi gas is dilute. In fact, we derive the above expression as a function of distance and absolute temperature. The Fermi velocity is also determined. In addition, essential aspects of the optical potential are discussed.

DOI
10.12785/qpl/020301

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Abstract: We present an accurate analytical formulation to determine the chemical potential of an ultrarelativistic spherical degenerate Fermi gas interacting with nuclear matter by examining rigorously the three terms of the general mathematical expression for the chemical-potential energy namely, the Fermi energy, the optical-potential energy, and a third term which is negligible if the Fermi gas is dilute. In fact, we derive the above expression as a function of distance and absolute temperature. The Fermi velocity is also determined. In addition, essential aspects of the optical potential are discussed.
Abstract Knowledge of the chemical potential is essential in application of the Fermi–Dirac and the Bose–Einstein distribution functions for the calculation of properties of quantum gases. We give expressions for the chemical potential of ideal Fermi and Bose gases in 1, 2 and 3 dimensions in terms of inverse polylogarithm functions. We provide Mathematica functions for these chemical potentials together with low- and high-temperature series expansions. In the 3d Bose case we give also expansions about TBT_{{{{\mathrm {B}}}}}TB​ T B . The Mathematica routines for the series allow calculation to arbitrary order.
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.
Thermodynamic properties of matter generally depend on the details of interactions between its constituent parts. However, in a unitary Fermi gas where the scattering length diverges, thermodynamics is determined through universal functions that depend only on the particle density and temperature. By using only the general form of the equation of state and the equation of force balance, we measured the local internal energy of the trapped gas as a function of these parameters. Other universal functions, such as those corresponding to the Helmholtz free energy, chemical potential, and entropy, were calculated through general thermodynamic relations. The critical parameters were also determined at the superfluid transition temperature. These results apply to all strongly interacting fermionic systems, including neutron stars and nuclear matter.
Thermodynamic properties of matter generally depend on the details of interactions between its constituent parts. However, in a unitary Fermi gas where the scattering length diverges, thermodynamics is determined through universal functions that depend only on the particle density and temperature. By using only the general form of the equation of state and the equation of force balance, we measured the local internal energy of the trapped gas as a function of these parameters. Other universal functions, such as those corresponding to the Helmholtz free energy, chemical potential, and entropy, were calculated through general thermodynamic relations. The critical parameters were also determined at the superfluid transition temperature. These results apply to all strongly interacting fermionic systems, including neutron stars and nuclear matter.