Capture The Finite Particle Number Effect In Atoms

Based on the Thomas-Fermi approximation, the finite number effect, along with dimensionality, has been discussed for a Bose system and Fermi system trapped in 3D, 2D, 1D anisotropic harmonic oscillator potential, without considering the inter-atom interaction. We indeed found the remarkable differences between the finite number case and the thermodynamical case, including dimensionality.For Bose system, the critical temperature Tc increase with atom number N. And we also found that the dimensionality had a great effect on such system. The condensate fraction is proportional to T3 in 3D, T2 in 2D, whereas T in 1D. This relation is hold for specific heat c. And the critical temperature T0c under thermodynamical limit is also proportional to in 3D, in 2D, whereas a little difference in 1D (see Eq.(2.33)). This law can be generalized to a system in any dimension d, i.e . It is also right for and c, i.e,c Td. However, the finite number effect decrease the critical temperature and make an obvious changes in and c. The difference between finite number effect and thermodynamical limit depends on the shape of anisotropic harmonic oscillator potential and the total number N. It is interesting to note that the finite number effect is decreased with the reduction of dimension d.For Fermi system, the finite number effect adds a negative correction to the Fermi temperature in 3D, 2D cases, whereas has no effect on 1Dcase. Fermi temperature under thermodynamical limit is proportional to in 3D, in 2D, whereas Nω in ID. In addition, thespecific heat c have been manipulated under control of dimensionality, i.e The chemical potential μ have been discussed in lower-dimension system. The results is interesting that μ, dose not intend to Fermi energy F when temperature is very low in 2D, whereas is a constant in 1D. This means that it is impossible to observe phase transition in ID Fermi system.