On the Bose-Einstein condensation

Abstract

The Bose-Einstein condensation of a gas is investigated. Starting from the well-known formulae for Bose statistics, the problem has been generalized to include a variety of potential fields in which the particles of the gas move, and the number w of dimensions has not been restricted to three. The energy levels are taken to be $\epsilon _{i}\equiv \epsilon _{s_{1},\ldots,s_{w}}$ = constant $\frac{h^{2}}{m}\left\{\frac{s_{1}^{\alpha}-1}{a_{1}^{2}}+\cdots +\frac{s_{w}^{\alpha}-1}{a_{w}^{2}}\right\}$ (1 $\leq \alpha \leq $ 2), the quantum numbers being s$_{1,\ldots,w}$ = 1, 2,..., and a$_{1}$, $\cdots $,a$_{w}$ being certain characteristic lengths. (For $\alpha $ = 2, the potential field is that of the w-dimensional rectangular box; for $\alpha $ = 1, we obtain the w-dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones & Jones (1938). It is shown that the number $q$ = w/$\alpha $ determines the appearance of an Einstein transition temperature T$_{0}$. For q $\leq $ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q $\leq $ 2, the mean energy $\overline{\epsilon}$ per particle and the specific heat d$\overline{\epsilon}$/dT are continuous at T = T$_{0}$. For q > 2, the specific heat is discontinuous at T = T$_{0}$, giving rise to a $\lambda $-point. A well-defined transition point only appears for a very large (theoretically infinite) number N of particles. T$_{0}$ is finite only if the quantity $\nu $ = N/(a$_{1}\cdots $a$_{w}$)$^{2/_{\alpha}^{-}}$ is finite. For a rectangular box, $\nu $ is equal to the mean density of the gas. If $\nu $ tends to zero or infinity as N $\rightarrow \infty $, then T$_{0}$ likewise tends to zero or infinity. In the case q > 1, and at temperatures T < T$_{0}$, there is a finite fraction N$_{0}$ /N of the particles, given by N$_{0}$/N = 1 - (T/T$_{0}$)$^{q}$, in the lowest state. London's formula (1938b) for the three-dimensional box is an example of this equation. Some further results are also compared with those given by London's continuous spectrum approximation.