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Physical Review

Electrodynamic Properties of a Gas of Spin-½ Particles in a Uniform Magnetic Field

V. Arunasalam

Published Jun 10, 1967• 4 citations
Publisher

Abstract

A rather complete discussion of the electrodynamic behavior of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is presented. The method presented here is based on the general principles of elementary nonrelativistic quantum theory and takes account of the Doppler-broadening effects arising from the random thermal motion of the particles and the Compton recoil effects arising from the emission (spontaneous plus the induced or the stimulated emission) and absorption of photons by the spin-\textonehalf{} particles. In the first half of the paper a method is presented for calculating the spin permeability tensor which determines the propagation of transverse electromagnetic radiation through a gas of spin-\textonehalf{} particles in a uniform external magnetic field. It is shown that the spin permeability tensor thus obtained yields the familiar results of solid-state physics, such as the nuclear and electronic-spin radiofrequency susceptibility and the spin paramagnetism. In the second half of this paper the subject of spin-radiative equilibrium of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is discussed in some detail. Simple closed-form expressions for the steady-state photon number density and the spin-radiative relaxation time of the system under study are thus derived. It is shown that the steady-state photon number density is the familiar distribution function corresponding to Bose---Einstein statistics for "complete thermodynamic equilibrium." The conditions under which the system of spin-\textonehalf{} particles in a uniform magnetic field can become unstable for transverse electromagnetic radiations are also discussed.

DOI
10.1103/physrev.158.234

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Soviet Journal of Quantum Electronics•Nov 30, 1982•1 citation
A rather complete discussion of the electrodynamic behavior of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is presented. The method presented here is based on the general principles of elementary nonrelativistic quantum theory and takes account of the Doppler-broadening effects arising from the random thermal motion of the particles and the Compton recoil effects arising from the emission (spontaneous plus the induced or the stimulated emission) and absorption of photons by the spin-\textonehalf{} particles. In the first half of the paper a method is presented for calculating the spin permeability tensor which determines the propagation of transverse electromagnetic radiation through a gas of spin-\textonehalf{} particles in a uniform external magnetic field. It is shown that the spin permeability tensor thus obtained yields the familiar results of solid-state physics, such as the nuclear and electronic-spin radiofrequency susceptibility and the spin paramagnetism. In the second half of this paper the subject of spin-radiative equilibrium of a gas of spin-\textonehalf{} particles in a uniform external magnetic field is discussed in some detail. Simple closed-form expressions for the steady-state photon number density and the spin-radiative relaxation time of the system under study are thus derived. It is shown that the steady-state photon number density is the familiar distribution function corresponding to Bose---Einstein statistics for "complete thermodynamic equilibrium." The conditions under which the system of spin-\textonehalf{} particles in a uniform magnetic field can become unstable for transverse electromagnetic radiations are also discussed.
The thermodynamic properties of a Bose gas of particles with nonzero spin and magnetic moment in states with and without a Bose-Einstein condensate are studied as a function of the external magnetic field. It is shown that there exists a temperature interval where a transition into a state with a Bose-Einstein condensate occurs in a Bose gas with constant temperature and particle number density when the magnetic field is increasing and reaches a critical magnitude. The temperature and magnetic field derivatives of the specific heat at constant volume and the magnetic susceptibility undergo a jump at such a transition.
A two-body-interaction model characterized by the scattering length a is applied to study the thermodynamic and magnetic properties of nonrelativistic interacting systems of spin-(1/2) particles in the presence of an external homogeneous magnetic field under the assumption that the particles obey generalized statistics of order l. Here one assumes that the occupation number of an energy state can take values 0,1,2,3,...,l and no more, where l is an integer greater than unity. Exact expressions for thermodynamic and magnetic properties due to the generalized nature of the statistics are obtained and the characteristic departure of the present results from those of the Fermi-Dirac case are pointed out. Variation of thermodynamic properties with the interaction parameter ${\mathrm{av}}^{\mathrm{\ensuremath{-}}1/3}$ is also obtained.
The possibility for the photon component to affect the magnetic properties of a system of quantum gases of two-level atoms staying in thermodynamic equilibrium with radiation (photons) has been studied. A corresponding simple model has been proposed, which enabled the general equations describing the thermodynamic equilibrium in this system to be derived. The resulting equations are solved in the temperature interval far from the degeneracy temperatures of all three system components. The analysis of the solutions testified to a non-trivial behavior of the system’s magnetic state as a response to changes in the photon density and the intensity of the external magnetic field. It is shown that the growth of the photon density induced in the system by external sources can increase both system’s magnetization and the density of excited atoms. Such a conclusion is not trivial a priori given the fact that photons in the vacuum have no magnetic moment.
In this chapter, within the general method of Gel'fand–Yaglom, starting with an extended set of representations of the Lorentz group, we have constructed a relativistic generalized system of the first-order equations for a spin 1 particle with two additional characteristics, anomalous magnetic moment and polarizability. We have taken into account the presence of external electromagnetic fields. After eliminating the accessory variables of the complete wave function, we derive the minimal 10-component form of equations, the last includes two additional interaction terms which are interpreted as related to anomalous magnetic and moment and polarizability of the particle. This approach is extended to space-time models with pseudo-Riemannian structure, within the tetrad method. We specify the basic equations to the cylindrical coordinates, and also in presence of external uniform magnetic field. After separating the variables, we get the system of ten first-order-differential equations in polar coordinate. To resolve this system, we apply the method of Fedorov–Gronskiy; and the complete 10-component wave function is decomposed into the sum of three projective constituents, where dependence of each component on the polar coordinate is determined by only one function, https://www.w3.org/1998/Math/MathML" display="inline"> F i ( r ) , i = 1 , 2 , 3 . We construct them in terms of confluent hypergeometric equations; at this a quantization rule for some spectral parameter exists. Additionally arises an algebraic homogeneous system of ten equations, which completely determines the structure of 10-component solutions. By vanishing the determinant of the linear system, we derive a cubic algebraic equation with respect to energy parameter ϵ 2 . Its solutions are found in analytical form, and studied numerically. For different values of parameters, anomalous magnetic moment and polarizability, we have found three series of physically interpretable energy levels. For more details see [ 111 ]–[ 119 ].
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.
An investigation is made of the change in the magnetic properties of a diamagnetic molecule (atom) induced by a resonance optical radiation field. Calculations are made for the 0→1 molecular transition in the presence of an external magnetostatic field. Expressions are derived for the magnetization and the magnetic susceptibility of the molecule, which depend nonlinearly on the light wave intensity, in the case of longitudinal and transverse directions of the external magnetic field with respect to the direction of propagation of the radiation. It is shown that the field-induced nonlinear magnetic susceptibility of a diamagnetic substance can change sign and increase considerably (by several orders of magnitude compared with the linear diamagnetic susceptibility). The field-induced nonlinear magnetization of the substance does not depend on its magnetic properties and is the same for diamagnetic and paramagnetic substances.