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Journal of Physics A Mathematical and General

Trapped mesoscopic quantum gas in a magnetic field

P. N. Vorontsov‐Velyaminov, R. I. Gorbunov, Sergei D. Ivanov

Published Feb 28, 2000• 1 citation
Publisher

Abstract

Feynman-type series for grand potential , number of particles, internal energy, magnetization and susceptibility are derived for systems of noninteracting quantum particles in an external magnetic field. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in the presence of a homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single-particle states. N -convergence to universal curves for chemical potential and energy is presented. At low temperatures magnetization of the Fermi gas exhibits a pronounced quantum behaviour.

DOI
10.1088/0305-4470/33/9/310

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Feynman-type series for grand potential , number of particles, internal energy, magnetization and susceptibility are derived for systems of noninteracting quantum particles in an external magnetic field. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in the presence of a homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single-particle states. N -convergence to universal curves for chemical potential and energy is presented. At low temperatures magnetization of the Fermi gas exhibits a pronounced quantum behaviour.
Previously developed approach [1 is extended to systems of noninteracting quantum particles in an external magnetic field. Feynman type series for grand potential, number of particles, internal energy and magnetization are derived. Numerical calculations at constant N for harmonically trapped systems of fermions and bosons in presence of the homogeneous magnetic field are carried out with the aid of both Feynman-type series and series over single particle states. Particular attention is paid to low temperature behavior of fermionic systems.
A system of a finite number of harmonically trapped fermions in one dimension, in the presence of a static magnetic field, is studied within the framework of the static fluctuation approximation, for different repulsive and attractive potential strengths. Specifically, the thermodynamic properties of the system (the chemical potential, total energy, heat capacity, and entropy), as well as its magnetic properties (the magnetization and susceptibility), are calculated. It is observed that the system remains in an ordered phase for a small number of particles N, even at high temperatures T. Disorder sets in for large N, even at low T. The effect of the potential strength on the heat capacity is particularly tangible in the region bordering the quantum and classical regimes. The effect of the temperature (representing disorder) is directly opposite to that of the magnetic field (representing order), as expected on basic physical grounds. These features are consistent with experimental results.
An exact Feynman-type presentation of the grand canonical partition function and averages as series over cycles for a system of noninteracting identical particles with a spin in an arbitrary external field is derived, and a numerical procedure for obtaining $\ensuremath{\mu}(\ensuremath{\beta})$ and other dependencies at constant N is developed. It is shown that the same series can be obtained also from the conventional form of the grand potential $\ensuremath{\Omega}$ (i.e., a sum over single-particle energy states). Numerical calculations at constant N are carried out for quantum gas of bosons and fermions in three-dimensional harmonic field and in the P\"oschl-Teller potential.
We calculate the quantum corrections of the thermodynamic quantities of a system of confined bosons at finite temperature. Systematically quantum corrections are written in a series of $\ensuremath{\Elzxh},$ which is convergent when kT\mathrm{kT}kT is much larger than the spacing between energy levels of the system. We apply this method to the different physical systems. First we calculate the thermodynamic potential of spin-1 bosons confined in a general power-law potential, and calculate the paramagnetic susceptibility of the gas including the quantum corrections. Local-density approximation for charged bosons in the weak magnetic field is improved by the semiclassical expansion. Diamagnetic susceptibility of a system of confined charged bosons is calculated using this method. Finally we apply this method to calculate analytically the thermodynamic potential of a weakly interacting Bose-gas confined in at three-dimensional harmonic-oscillator potential. For a large number of particles, quantum corrections become small, and contribute to the finite-size corrections to scaling.
The analytical expressions of internal energy, chemical potential and heat capacity of the little parameter r for a weakly interacting Fermi gas in weak magnetic field are derived by using “pseudopotential" method and ensemble theory. Based on the derived expressions, the thermodynamic properties of a weakly interacting Fermi gas in weak magnetic field at both high and low temperatures are given. The effects of magnetic field and interparticle interactions on the thermodynamic properties are discussed. The difference in the effects of magnetic trap and three-dimensional harmonic trap on the properties of the system and their reasons are analyzed.