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Acta Physica Sinica

The criterion of instability for a weakly interacting Fermi gas

Yuan Du-Qi, 宝鸡文理学院物理系，宝鸡 721007

Published Jan 1, 2006• 9 citations
Publisher

Abstract

Using the equation of free energy for a weakly interacting Fermi gas, the stability of a weakly interacting Fermi gas in the absence of external potential is studied by means of thermodynamics. The instability arising from the weakly attractive interacting of a Fermi gas is pointed out. The particle density criterion and the temperature criterion of instability for the imperfect Fermi gas with weakly attractive interaction are given, and the dependance of the critical density of particles on the temperature and the mass of particle and the attractive interaction is discussed.

DOI
10.7498/aps.55.3912

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Using the equation of free energy for a weakly interacting Fermi gas, the stability of a weakly interacting Fermi gas in the absence of external potential is studied by means of thermodynamics. The instability arising from the weakly attractive interacting of a Fermi gas is pointed out. The particle density criterion and the temperature criterion of instability for the imperfect Fermi gas with weakly attractive interaction are given, and the dependance of the critical density of particles on the temperature and the mass of particle and the attractive interaction is discussed.
In this paper the analytical expression of free energy expressed by small parameter r of a weakly interacting Fermi gas trapped in weak magnetic field is derived by using `the maximum approximation' method and the ensemble theory. Based on the derived expression, the exact instability conditions of a weakly interacting Fermi gas trapped in weak magnetic field at both high and low temperatures are given. From the instability conditions we get the following two results. (1) At the whole low-temperature extent, whether the interactions are repulsive or attractive with (α n +4ε F /3) ( n and ε F denote the particle-number density and the Fermi energy respectively, α = 4π a ℏ 2 / m , and a is s-wave scattering length) positive, there is a lower-limit magnetic field of instability; in addition, there is an upper-limit magnetic field for the system of attractive interactions with (α n +4ε F /3) negative. (2) At the whole high-temperature extent, the system with repulsive interactions is always stable, but for the system with attractive interactions, the greater the scattering length of attractive interactions | a | is, the stronger the magnetic field is and the larger the particle-number density is, the bigger the possibility of instability in the system will be.
The energy density of an interacting Fermi gas in a harmonic potential is derived within localdensity approximation framework,in which the gas is assumed at each point to be locally homogenous. Based on the derived expressions,the thermodynamic stability conditions of trapped gas at zero temperature are studied,and the analytical expressions of stability conditions for d=l,2,3 dimensions are given. Furthermore,the stability phase diagrams are plotted for different dimension.It is shown that it is always stable when the system has repulsive interactions or has no interactions in condition of certain density of particle number,no mater what the dimension is.However,the system with attractive interactions is stable only under the condition of weak interaction strength.For d=1 and d = 3 cases,the critical values of interaction strength have close relationship with particle number density,but as for d= 2 case,the corresponding critical value does not depend on the particle number density explicitly.
By means of the single particle energy spectrum of weak interaction between fermions and Poisson formula,the thermodynamic potential function of relativistic Fermi gas in a strong magnetic field is derived.Based on this,we obtained the criterion of stability for the system.The results show that the mechanics stability of a Fermi gas with weak interacting is influenced by the interacting.While the magnetic field is able to regulate the influence and the relativistic effect has almost no effect on it.
Using the free energy and the equation of state for an imperfect Bose gas, the analytical expression between the critical temperature and the length of dispersion wave and the pressure for the imperfect Bose gas is derived. The unusual properties of the specific heat at constant pressure and the expansion coefficient at constant pressure and isothermal compressibility by the attractive force for the imperfect Bose gas are revealed. The instability arising from the attractive force is pointed out, and the temperature criterion and particle density criterion of instability is given for the imperfect Bose gas under the action of attraction.
Based on the theoretical results derived from pseudopotential method and local approximation, this paper studies the thermodynamic stability of a weakly interacting Fermi gas trapped in a harmonic potential by using analytical method of thermodynamics. The effects of the interparticle interactions as well as external potential on the thermodynamic stability of the system are discussed. It is shown that the system is stable as for the complete average, but as for local parts, the system is unstable anywhere. This instability shows that the stability conditions of mechanics cannot be satisfied anywhere, and the stability conditions of thermostatics cannot be satisfied somewhere. In addition, the interactions and external potential have direct effects on the local stability of the system.