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Journal of Sichuan University

The effects of relativistic correction on Fermi gas in a weak magnetic field

Yi Zhao

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

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.

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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.
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 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 pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.
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.
The partition function of system with finite number of particles is derived is derived by using partition function of a weakly interacting Fermi gas trapped in weak magnetic field.Based on the derived expressions,the analytical expressions of thermodynamic quantities of a weakly interacting Fermi gas for finite particle number are solved by using statistical average method and the thermodynamic properties under different temperature conditions are given.It is shown that each thermodynamic quantity has a correctional term due to effects of a finite number of particles.The particle number has a direct effect on correction of chemical potential but has no direct effects on corrections of total energy and heat capacity except T=0.Furthermore,effects of a finite number of particles always reduce chemical potential,thereby zero point of chemical potential shifts towards low temperatures.The increase of particle number weakens the effect,but the repulsive interactions between particles strengthen the effect.
Taking the anomalous magnetic moment into consideration, the analytical expression of the free energy for a weakly interacting Fermi gas in a weak magnetic field is derived by using the pseudopotential method and thermodynamic theory, therefore the thermodynamic properties can be studied. Based on the derived expression, the thermodynamic properties of this system at both high and low temperatures are given, and the effect of anomalous magnetic moment on thermodynamic properties can be analyzed. The effect of anomalous magnetic moment on the thermodynamic properties is related to temperature, and with the rise of temperature this effect increases in the low temperature zone, but decreases in the high temperature zone. For the chemical potential and internal energy of the system, the anomalous magnetic moment strengthens the influence of the magnetic field, but weakens the influence of the interaction. Under the influence of anomalous magnetic moment, the heat capacity of the system decreases in the low temperature zone, but increases in the high temperature zone.