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arXiv:math-ph/0605013 · Journal of Mathematical Physics

Diamagnetic expansions for perfect quantum gases

Philippe Briet, Horia D. Cornean, Delphine Louis

Published Aug 1, 2006• 8 citations
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Abstract

In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated with the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω≔eB∕c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).

DOI
10.1063/1.2259582

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In this work we study the diamagnetic properties of a perfect quantum gas in the presence of a constant magnetic field of intensity B. We investigate the Gibbs semigroup associated with the one particle operator at finite volume, and study its Taylor series with respect to the field parameter ω≔eB∕c in different topologies. This allows us to prove the existence of the thermodynamic limit for the pressure and for all its derivatives with respect to ω (the so-called generalized susceptibilities).
The system we consider here is a charged fermions gas in the effective mass approximation, and in grand-canonical conditions. We assume that the particles are confined in a three dimensional cubic box ΛΛΛ with side L≥1L\geq 1L≥1, and subjected to a constant magnetic field of intensity $ B \geq 0 $. Define the grand canonical generalized susceptibilities χLNχ_L^NχLN​, N≥1N\geq 1N≥1, as successive partial derivatives with respect to BBB of the grand canonical pressure PLP_LPL​. Denote by P∞P_{\infty}P∞​ the thermodynamic limit of PLP_LPL​. Our main result is that χLNχ_L^NχLN​ admit as thermodynamic limit the corresponding partial derivatives with respect to BBB of P∞P_{\infty}P∞​. In this paper we only give the main steps of the proofs, technical details will be given elsewhere.
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.
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.
Diamagnetism of 2D-Fermions in the Strong Nonhomogeneous Static Magnetic Field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) : gas magnetization, ... and gas compressibility
We study diamagnetism of a gas of fermions moving in a nonhomogeneous magnetic field (B=B(0,0,1/cosh2(x−x0δ)))( {\bf B} =B( 0, 0, 1/cosh^{2}( \frac{x-x_{0} }{ δ})))(B=B(0,0,1/cosh2(δx−x0​​))) The gas magnetization, the static magnetic susceptibility, the chemical potential and the gas compressibility are discussed and compared with the uniform field case. General need to study dynamics of electrons in different types of magnetic fields follows from a large number of experimental situations in which its understanding enables physicists to obtain new information.