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Physical Review

Magnetic Moment of a Magnetized Fermi Gas

Vittorio M. Canuto, Hong‐Yee Chiu

Published Sep 25, 1968• 89 citations
Publisher

Abstract

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 $\mathfrak{M}$ was found to be proportional to the square of the electron density, but at higher densities $\mathfrak{M}$ 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.

DOI
10.1103/physrev.173.1229

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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.
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 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.
The calculation of the induced spin density of a normal fermion system, such as the electron gas, in the limit of small wave numbers, is carried beyond the random phase approximation (R.P.A.) by the method of canonical transformation so as to include the first non-R.P.A. corrections. An expression for the induced spin density in the same limit, exact to all orders of particle-particle coupling, is then derived by more general methods of many-body perturbation theory. It depends only on the knowledge, to all orders of interparticle coupling, of the effective mass and the forward scattering of quasi-particles at the Fermi surface, and also yields the results of the extended R.P.A. on appropriate expansion. In the limit of zero wave number, the resulting expression for the magnetic susceptibility is found identical with that deduced by Landau from a phenomenological basis and may be regarded as an additional confirmation of the microscopic validity of the theory of the Fermi liquid.
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.
The spin-polarized ferromagnetic state of a cold Fermi gas is investigated for interacting and noninteracting charge-neutral and $\ensuremath{\beta}$-equilibrated gases. The standard minimal couplings between the magnetic field and the fermions' charges and magnetic dipole moments define the fermions' interaction with the magnetic field. Assuming a variable coupling strength between the magnetic field and the fermion (baryon) dipole moments, it is shown that a ferromagnetized state can be achieved that corresponds to a lower energy spin-polarized state with a magnetic field entirely due to the gas's magnetic response. We find that, depending on the density, a very large increase in the baryon dipole moments is needed to achieve this ferromagnetized state. While the required increase seems unlikely, the induced magnetic field is of the order $\ensuremath{\approx}{10}^{17}$ gauss. Furthermore, while externally magnetized Fermi gases have an anisotropic pressure, the pressure of the ferromagnetized gas is completely isotropic and in the thermodynamically preferred magnetized state.
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.