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Communications in Theoretical Physics

Third-Order Magnetic Susceptibility of an Ideal Fermi Gas*

L. Q. Lai, Zhao 照 Li 李, Yabin Yu, Quanhui Liu

Published Nov 1, 2018• 2 citations
Publisher

Abstract

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.

DOI
10.1088/0253-6102/70/5/619

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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.
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.
It has been suggested that either the diamagnetism or paramagnetism of Bose gases, due to the charge or spin degrees of freedom respectively, appears solely to be extraordinarily strong. We investigate the magnetic properties of charged spin-1 Bose gases in an external magnetic field, focusing on the competition between the diamagnetism and paramagnetism, using the Lande-factor g of particles to evaluate the strength of the paramagnetic effect. We propose that a gas with g < 1/√8 exhibits diamagnetism at all temperatures, while a gas with g > 1/2 always exhibits paramagnetism. Moreover, a gas with the Lande-factor in between shows a shift from paramagnetism to diamagnetism as the temperature decreases. The paramagnetic and diamagnetic contributions to the total magnetization density are also calculated in order to demonstrate some details of the competition.
In this paper we have calculated the exact expression for the magnetic moment of a Fermi gas in a magnetic field (magnetized Fermi gas). The susceptibility obtained from our result becomes the classical Curie-Langevin law in the weak-field limit for a nondegenerate and nonrelativistic gas. The induced magnetic moment M\mathfrak{M}M was found to be proportional to the square of the electron density, but at higher densities M\mathfrak{M}M passes through a maximum and eventually becomes negative; hence a magnetized gas is first paramagnetic but later becomes diamagnetic. We have also shown that spontaneous magnetization cannot take place in a noninteracting electron gas.
The magnetic susceptibilities of the degenerate (below the Bose–Einstein condensation temperature) and nondegenerate ideal gases of nonrelativistic charged spinless bosons are presented. In both cases, the boson gas is diamagnetic. The magnetic susceptibility of the degenerate boson gas below the Bose–Einstein condensation temperature increases in modulus as the temperature increases. As expected, the magnetic susceptibility of the nondegenerate boson gas decreases in modulus with increasing temperature according to the Curie law in low magnetic fields.
Asymptotic expansions for the thermodynamic functions of an ideal spin-\textonehalf{} Fermi gas in a weak magnetic field are obtained about the point z=1z=1z=1, where zzz is the fugacity. We consider paramagnetism, diamagnetism, and paramagnetism plus diamagnetism together. To relate to liquid 3He^{3}\mathrm{He}3He films and bulk fluid, we study the spin paramagnetism in two and three dimensions.
The magnetic properties of a charged spin-1 Bose gas with ferromagnetic interactions are investigated within mean-field theory. It is shown that a competition between paramagnetism, diamagnetism and ferromagnetism exists in this system. It is shown that diamagnetism, being concerned with spontaneous magnetization, cannot exceed ferromagnetism in a very weak magnetic field. The critical value of reduced ferromagnetic coupling of the paramagnetic phase to ferromagnetic phase transition I[combining overline](c) increases with increasing temperature. The Landé-factor g is introduced to describe the strength of the paramagnetic effect which comes from the spin degree of freedom. The magnetization density M[combining overline] increases monotonically with g for fixed reduced ferromagnetic coupling I[combining overline] as I[combining overline] > I[combining overline](c). In a weak magnetic field, ferromagnetism makes an immense contribution to the magnetization density. On the other hand, at a high magnetic field, the diamagnetism tends to saturate. Evidence for condensation can be seen in the magnetization density at a weak magnetic field.