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The Journal of Atomic and Molecular Sciences

Relativistic Paramagnetism of a Weakly Interacting Fermi Gas

Men Fu-Dian, Xiao-Gang He Xiao-Gang He, Hui Liu Hui Liu, Bin Dong

Published Feb 1, 2011• 2 citations
Publisher

Abstract

By considering strict relativistic energy of single particle and using the methods of quantum statistics, the relativistic paramagnetism of a weakly interacting Fermi gas in a weak magnetic field is studied, and the relativistic most probable paramagnetic susceptibility as well as the average magnetic susceptibility of the system are solved. On the basis, the influences of relativistic effect on the most probable paramagnetic susceptibility of the system are discussed, and the relativistic critical value of particle number is given. It is shown that, comparing with nonrelativistic situation, when $T\gg T_F,$ the relativistic most probable magnetic susceptibility and the relativistic critical value of particle number have not changed. When $T\ll T_F$, the relativistic effects make the system display paramagnetism easily and susceptibility increase, but the relativistic effects also amplified the impact of the interaction on the magnetic susceptibility.

DOI
10.4208/jams.121610.012211a

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By considering strict relativistic energy of single particle and using the methods of quantum statistics, the relativistic paramagnetism of a weakly interacting Fermi gas in a weak magnetic field is studied, and the relativistic most probable paramagnetic susceptibility as well as the average magnetic susceptibility of the system are solved. On the basis, the influences of relativistic effect on the most probable paramagnetic susceptibility of the system are discussed, and the relativistic critical value of particle number is given. It is shown that, comparing with nonrelativistic situation, when T≫TF,T\gg T_F,T≫TF​, the relativistic most probable magnetic susceptibility and the relativistic critical value of particle number have not changed. When T≪TFT\ll T_FT≪TF​, the relativistic effects make the system display paramagnetism easily and susceptibility increase, but the relativistic effects also amplified the impact of the interaction on the magnetic susceptibility.
By considering strict relativistic energy of single particle and using the methods of quantum statistics,the relativistic paramagnetism of a weakly interacting Fermi gas in a weak magnetic filed is studied,and the relativistic most probable paramagnetic susceptibility as well as the average magnetic susceptibility of the system are solved.On the basis,the influences of relativistic effect on the most probable paramagnetic susceptibility of the system are discussed,and the relativistic critical value of particle number is given.It is shown that,comparing with nonrelativistic situation,when TT_F,the relativistic most probable magnetic susceptibility and the relativistic critical value of particle number have not changed.When TT_F,the relativistic effects make the system display paramagnetism easily and susceptibility increase,but the relativistic effects also amplified the impact of the interaction on the magnetic susceptibility.
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.
Based on the methods of quantum statistics, by considering relativistic correction of single particle and introducing the nonrelativistic free energy of the system of spinless and free fermions,the free energy for Fermi gas in a weak magnetic field is derived, thereout the analytical expressions of thermodynamic quantities of the system under various temperature condtions are given, the effects of relativistic correction on the properties of the nonrelativistic system are exhibited particular and concrete, and the mechanism of the effects is analysed. Compared with those of nonrelativistic situation, it is shown that the effects of relativistic correction always reduce chemical potential. The effects of relativistic correction decrease the total energy and heat capacity at low temperatures, while increase them at high temperatures. The changes of the total energy and heat capacity (not include the low-temperature heat capacity ) are proportional to the square of magnetic field.
By means of the single particle energy spectrum of weak interaction between fermions and Poisson formula,the thermodynamic potential function of relativistic Fermi gas in a strong magnetic field is derived.Based on this,we obtained the criterion of stability for the system.The results show that the mechanics stability of a Fermi gas with weak interacting is influenced by the interacting.While the magnetic field is able to regulate the influence and the relativistic effect has almost no effect on it.
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 analytical expression of relativistic free energy for a weakly interacting Fermi gas in a weak magnetic field, by using the method of quantum statistics, the stability conditions of the system at both high and low temperatures are given, and the effects of magnetic field and interparticle interactions on the stability of the system are analysed. It is shown that at high temperatures, the stability conditions of the system are completely the same, no matter whether it is the ultrarelativistic case or nonrelativistic case. At extremely low temperatures, the mechanical stability conditions of the system show a similar rule through a comparison between the ultrarelativistic case and nonrelativistic case. At the same time, thermal stability of a relativistic Bose gas in a weak magnetic field is discussed, and the influence of the effect of relativity on the thermal stability of the system is investigated.