关键词: Fermi gas, interaction, instability conditions

Abstract: 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 $\alpha n+4\varepsilon_{\rm F}$ (n and $\varepsilon_{\rm F}$ denote the particle-number density and the Fermi energy respectively, $\alpha=4\pi a\hbar^2/m$, 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 $(\alpha n+4\varepsilon_{\rm F}/3)$ 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.

Key words: Fermi gas, interaction, instability conditions