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Journal of Yantai University

Effects of Temperatures and Particle Number on Pauli Paramagnetism of Fermi Gas

Tian Chen-xi

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

On the basis of the probability distribution function of particle number,the Pauli paramagnetism of an ideal Fermi gas with finite number of particles is investigated by using a combination of theoretical analysis and numerical simulation.The analytical expressions of critical value of particle number and upper magnetic field are given and the effects of temperature and particle number on the paramagnetism are analyzed.It is shown that,when the temperature is close towards Fermi temperature,the difference between limit magnetic susceptibility and average magnetic susceptibility decreases.When the temperature is higher than the Fermi temperature,the resulting difference between limit magnetic susceptibility and average magnetic susceptibility increases first and then decreases with the increase of temperature.

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On the basis of the probability distribution function of particle number,the Pauli paramagnetism of an ideal Fermi gas with finite number of particles is investigated by using a combination of theoretical analysis and numerical simulation.The analytical expressions of critical value of particle number and upper magnetic field are given and the effects of temperature and particle number on the paramagnetism are analyzed.It is shown that,when the temperature is close towards Fermi temperature,the difference between limit magnetic susceptibility and average magnetic susceptibility decreases.When the temperature is higher than the Fermi temperature,the resulting difference between limit magnetic susceptibility and average magnetic susceptibility increases first and then decreases with the increase of temperature.
By using theoretical analysis and numerical simulation, the stability and the paramagnetism of Fermi gas in a strong magnetic field at low temperatures are studied, and the influences of temperature and magnetic field on the stability and susceptibility of the system are analyzed. The results show that the strong magnetic field causes variation of a system stability, and such a change relates to temperature. While the effects of temperature on stability and susceptibility are relevant to the chemical potential (i.e. the particle number density) of free gas. When the chemical potential is even, the magnetic field weakens the stability of the system, and the increasing of temperature leads to the reduction in the susceptibility of the system. When the chemical potential is odd, the magnetic field strengthens the stability of the system, while the increasing of temperature can increase the susceptibility of the system. With temperature increasing, the effect of the magnetic field on the stability is weakened. With magnetic field increasing, the change of the stability of the system becomes oscillatory, and the oscillation center of the susceptibility shifts down.
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.
Abstract At low temperature and under weak magnetic field, non-interacting Fermi gases reveal both Pauli paramagnetism and Landau diamagnetism, and the magnitude of the diamagnetic susceptibility is 1/3 of that of the paramagnetic one. When the temperature is finite and the magnetic field is also finite, we demonstrate that the paramagnetism and diamagnetism start to deviate from the ratio 1/3. For understanding the magnetic properties of an ideal Fermi gas at quite low temperature and under quite weak magnetic field, we work out explicitly the third-order magnetic susceptibility in three cases, from intrinsic spin, orbital motion and in total. An interesting property is in third-order magnetic susceptibilities: when viewing individually, they are both diamagnetic, but in total it is paramagnetic.
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.