The distribution function of weak interaction Fermi gas in weak magnetic field is derived by the way of pseudo-potential method and larger number approximate method.At the same time,the magnetizable rate of Pauli-paramagnetism is analytic calculated.We also analyse the effect of interaction of particles on the rnagnetizable rate.
By considering strict relativistic energy of single particle and using the methods of quantum statistics, the relativistic paramagnetism of a weakly interacting Fermi gas in a weak magnetic field is studied, and the relativistic most probable paramagnetic susceptibility as well as the average magnetic susceptibility of the system are solved. On the basis, the influences of relativistic effect on the most probable paramagnetic susceptibility of the system are discussed, and the relativistic critical value of particle number is given. It is shown that, comparing with nonrelativistic situation, when T≫TF, the relativistic most probable magnetic susceptibility and the relativistic critical value of particle number have not changed. When T≪TF, the relativistic effects make the system display paramagnetism easily and susceptibility increase, but the relativistic effects also amplified the impact of the interaction on the magnetic susceptibility.
On the basis of the probability distribution function of particle number,the Pauli paramagnetism of an ideal Fermi gas with finite number of particles is investigated by using a combination of theoretical analysis and numerical simulation.The analytical expressions of critical value of particle number and upper magnetic field are given and the effects of temperature and particle number on the paramagnetism are analyzed.It is shown that,when the temperature is close towards Fermi temperature,the difference between limit magnetic susceptibility and average magnetic susceptibility decreases.When the temperature is higher than the Fermi temperature,the resulting difference between limit magnetic susceptibility and average magnetic susceptibility increases first and then decreases with the increase of temperature.
By considering strict relativistic energy of single particle and using the methods of quantum statistics,the relativistic paramagnetism of a weakly interacting Fermi gas in a weak magnetic filed is studied,and the relativistic most probable paramagnetic susceptibility as well as the average magnetic susceptibility of the system are solved.On the basis,the influences of relativistic effect on the most probable paramagnetic susceptibility of the system are discussed,and the relativistic critical value of particle number is given.It is shown that,comparing with nonrelativistic situation,when TT_F,the relativistic most probable magnetic susceptibility and the relativistic critical value of particle number have not changed.When TT_F,the relativistic effects make the system display paramagnetism easily and susceptibility increase,but the relativistic effects also amplified the impact of the interaction on the magnetic susceptibility.
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.
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).
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.