On the Bose-Einstein condensation

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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 ε i ≡ ε s 1 , . . . . , s 10 = constant h 2 m s 1 α − 1 a 1 2 + . . . + s w α a w 2 ( 1 ≤ α ≤ 2 ) the quantum numbers being s 1 , w = 1, 2, ..., and a 1 , ..., a w being certain characteristic lengths. (For α = 2, the potential field is that of the w -dimensional rectangular box; for α = 1, we obtain the w -dimensional harmonic oscillator field.) A direct rigorous method is used similar to that proposed by Fowler & Jones (1938). It is shown that the number q = w /α determines the appearance of an Einstein transition temperature T 0 ·For q≤ 1 there is no such point, while for q > 1 a transition point exists. For 1 < q≤ 2, the mean energy ϵ - per particle and the specific heat dϵ - /dT are continuous at T = T 0 · For q > 2, the specific heat is discontinuous at T = T 0 , giving rise to a A λ-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 v = N/(a 1 .... a w )2/ α ¯ is finite. For a rectangular box, v is equal to the mean density of the gas. If v tends to zero or infinity as N→ ∞, 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 (1938 b ) 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.