On the basis of the results derived from pseudopotential method and ensemble theory, thermal stability of a weakly interacting Fermi gas in a weak magnetic field is studied by using analytical method of thermodynamics. The exact analytical expressions of stability conditions at different temperatures are given, and the effects of interactions as well as magnetic field on the stability of the system are discussed. It is shown that there is an upper-limit magnetic field for the stability of the system at low temperatures, and there is an attractive dividing value at high temperatures. If attractive interaction is lower than the critical value, the stability of the system has no request for magnetic field, but if attractive interaction is higher than the dividing value, a lower-limit magnetic field exists for the stability of the system.
On the basis of the results derived from pseudopotential method and ensemble theory, thermal stability of a weakly interacting Fermi gas in a weak magnetic field is studied by using analytical method of thermodynamics. The exact analytical expressions of stability conditions at different temperatures are given, and the effects of interactions as well as magnetic field on the stability of the system are discussed. It is shown that there is an upper-limit magnetic field for the stability of the system at low temperatures, and there is an attractive dividing value at high temperatures. If attractive interaction is lower than the critical value, the stability of the system has no request for magnetic field, but if attractive interaction is higher than the dividing value, a lower-limit magnetic field exists for the stability of the system.
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.
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.
Based on the analytical expression of relativistic free energy for a weakly interacting Fermi gas in a weak magnetic field, by using the method of quantum statistics, the stability conditions of the system at both high and low temperatures are given, and the effects of magnetic field and interparticle interactions on the stability of the system are analysed. It is shown that at high temperatures, the stability conditions of the system are completely the same, no matter whether it is the ultrarelativistic case or nonrelativistic case. At extremely low temperatures, the mechanical stability conditions of the system show a similar rule through a comparison between the ultrarelativistic case and nonrelativistic case. At the same time, thermal stability of a relativistic Bose gas in a weak magnetic field is discussed, and the influence of the effect of relativity on the thermal stability of the system is investigated.
Based on the pseudopotential method and the local-density approximation, the thermodynamic properties of a weakly interacting Fermi gas in a strong magnetic filed are studied, the integrated analytical expressions of thermodynamic quantities of the system are derived, and the effects of magnetic field as well as interparticle interactions on the thermodynamic properties of the system are analyzed. It is shown that at both high and low temperatures, magnetic field may adjust the effects of interacting. At low temperatures, magnetic field can lower the chemical potential, total energy and heat capacity of the system compared with the situation of Fermi gas in the absence of the magnetic field. The repulsive interactions may increase the chemical potential, but reduce the total energy and heat capacity of the system compared with the situation of non-interacting Fermi gas. At high temperatures, magnetic field as well as repulsive interactions can reduce the total energy and increase heat capacity of the system, moreover, strong magnetic field may change the effects of interaction on the total energy and the heat capacity of the system.
Taking the anomalous magnetic moment into consideration, the analytical expression of the free energy for a weakly interacting Fermi gas in a weak magnetic field is derived by using the pseudopotential method and thermodynamic theory, therefore the thermodynamic properties can be studied. Based on the derived expression, the thermodynamic properties of this system at both high and low temperatures are given, and the effect of anomalous magnetic moment on thermodynamic properties can be analyzed. The effect of anomalous magnetic moment on the thermodynamic properties is related to temperature, and with the rise of temperature this effect increases in the low temperature zone, but decreases in the high temperature zone. For the chemical potential and internal energy of the system, the anomalous magnetic moment strengthens the influence of the magnetic field, but weakens the influence of the interaction. Under the influence of anomalous magnetic moment, the heat capacity of the system decreases in the low temperature zone, but increases in the high temperature zone.
Based on the fact that most of low temperature experiments of quantum systems are explored in an external field on condition that the particle numbers, volumes and energies of systems may be changed, the thermodynamic properties of weakly interacting Fermi systems in weak magnetic field are studied by using the statistical distribution of the completely open system with variable particle number, volume, and energy (N-E-V distribution). Firstly, the analytical expressions of internal energy and heat capacity, which are in the Fermi integral form, are obtained in the general case, and the analytical expressions and numerical results of energy and heat capacity are given under the extreme condition of supper-low temperature. The calculation results by the N-E-V distribution (with particle number density being variable) are compared with those by a pseudopotential method (with particle number density being unchanged). It can be found that the deviations of the internal energy and heat capacity calculated by the two different methods are very small, and the N-E-V distribution method can partially compensate for the error caused by the pseudo potential method. The most interesting point of the results obtained by the N-E-V distribution method is that there is a phase transition temperature in the weakly interacting Fermi system in weak magnetic field under the low temperature condition. The phase transition temperature is just in the range where occur the Fermi systems, Bose-Einstein condensation (BEC), Bardeen-Cooper-Schrieffer (BCS) phase transition, and BEC-BCS crossover, and does not vary with strength nor characteristic (attraction or repulsion) of the weak interaction, but it decreases with the strengthening of the external magnetic. When there is no external magnetic, the phase transition temperature is highest (more than 0.184 times Fermi temperature).