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arXiv:1212.2830

Relativistic Thermodynamics of Magnetized Fermi Electron Gas

Nodar L. Tsintsadze, Levan N. Tsintsadze

Published Dec 12, 2012• 6 citations
arXivHTMLPDF

Abstract

To study the relativistic thermodynamic properties of a Fermi gas in a strong magnetic field, we construct the relativistic thermodynamic potential by the relativistic Fermi distribution function taking into account that the motion of particles in a plane perpendicular to the magnetic field is quantized. With this general potential at hand, we investigate all the thermodynamic quantities as a function of densities, temperatures and the magnetic field. We obtain a novel set of adiabatic equations. Having the expression of the pressure and adiabatic state equations, we determine the sound velocity for several cases revealing a new type of sound velocity. Finally, we disclose the magnetic cooling in the quantized electron Fermi gas, which is based on an adiabatic magnetization in contrast to the known adiabatic demagnetization.

DOI
10.48550/arxiv.1212.2830

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To study the relativistic thermodynamic properties of a Fermi gas in a strong magnetic field, we construct the relativistic thermodynamic potential by the relativistic Fermi distribution function taking into account that the motion of particles in a plane perpendicular to the magnetic field is quantized. With this general potential at hand, we investigate all the thermodynamic quantities as a function of densities, temperatures and the magnetic field. We obtain a novel set of adiabatic equations. Having the expression of the pressure and adiabatic state equations, we determine the sound velocity for several cases revealing a new type of sound velocity. Finally, we disclose the magnetic cooling in the quantized electron Fermi gas, which is based on an adiabatic magnetization in contrast to the known adiabatic demagnetization.
Based on the single particle energy spectrum of weak relativity and Poissons formula,the thermodynamic potential function of Fermi gas in a strong magnetic field is derived. Furthermore, by using the thermodynamic relationships, the analytical expressions of statistic characteristic quantities of the system at low temperatures are obtained, and the influence mechanism of relativistic effect on the statistic properties of the system is analysed. It is shown that the relativistic effect becomes more significant with the magnetic field increasing. Compared with the corresponding oscillating amplitude, the monotonic term, which is caused by the relativistic effect, is much larger than the amplitude for the total energy, however, for the chemical potential and magnetic moment, they are almost of the same order.
Based on the Mean Field Theory, the relativistic thermodynamic potential function (RTPF) of interacting Fermi gases in a strong magnetic field is derived. On this basis, by taking the interacting term from the RTPF and using the thermodynamic relations, analytical expressions of the energy, the heat capacity and the chemical potential are calculated at low temperatures. And combined with numerical simulation analysis, the effect of interaction on the thermodynamic properties of the system is analyzed and the regulation mechanisms on the interaction impacts of both the magnetic field and the relativistic effect are discussed.
We propose an adiabatic magnetization process for cooling the Fermi electron gas to ultra-low temperatures as an alternative to the known adiabatic demagnetization mechanism. We show via a new adiabatic equation that at the constant density the increase of the magnetic field leads to the temperature decrease as T∼1/H2T\sim 1/H^2T∼1/H2.
By using the methods of theoretical analysis and numerical simulation, the relativistic effects of the thermodynamic properties of Fermi gas in a strong magnetic field are studied, and the influential mechanisms of temperatures and the magnetic field on the relativistic effects of the system are analyzed.It is shown that, at low temperatures, the effects of temperatures on the relativistic effect relate to the chemical potential of the free system.The magnetic field's effects on the relativistic effects of heat capacity and chemical potential display oscillating characteristics.At high temperatures, the relativistic effects of the thermodynamic quantities trigger oscillations with increasing of temperatures and magnetic field, and its amplitude decreases obviously to disappearing with the increasing of temperatures.However, its amplitude increases evidently due to the amplified magnetic field.
This paper derives the analytical expression of free energy for a weakly interacting Fermi gas in a weak magnetic field, by using the methods of quantum statistics as well as considering the relativistic effect. Based on the derived expression, the thermodynamic properties of the system at both high and low temperatures are given and the relativistic effect on the properties of the system is discussed. It shows that, in comparison with a nonrelativistic situation, the relativistic effect changes the influence of temperature on the thermodynamic properties of the system at high temperatures, and changes the influence of particle-number density on them at extremely low temperature. But the relativistic effect does not change the influence of the magnetic field and inter-particle interactions on the thermodynamic properties of the system at both high and extremely low temperatures.
In a previous paper, we calculated the equation of state of a Fermi gas at arbitrary temperature in a uniform and constant magnetic field of arbitrary strength. (Such a gas is now referred to as a magnetized Fermi gas.) In the present paper, we have studied the properties of the equation of state and have obtained simplified expressions for the normal stresses and the energy and particle densities. These expressions can be used in astrophysical applications. We have found that under suitable conditions of degeneracy (the temperature approaching 0), a magnetized Fermi gas behaves exactly as a one-dimensional gas, which has been studied extensively as a theoretical problem. We have also obtained expressions for the grand partition function.