Dimensional Crossover for the Bose -- Einstein Condensation of an Ideal Bose Gas

TL;DR

This paper investigates the behavior of an ideal Bose gas confined between two parallel surfaces, revealing how dimensional crossover influences Bose-Einstein condensation, especially in nanoscopic systems, with findings on condensation temperature and state occupation.

Contribution

It provides a detailed analysis of Bose-Einstein condensation in a confined geometry, highlighting the effects of dimensional crossover and surface shape on condensation properties.

Findings

Condensation occurs at T ~ 1/log(N) in the 2D limit.

Macroscopic occupation of low-lying excited states when condensed.

Condensation temperature and behavior are sensitive to surface shape and dimensional parameters.

Abstract

We study an ideal Bose gas of N atoms contained in a box formed by two identical planar and parallel surfaces S, enclosed by a mantle of height a perpendicular to them. Calling r0 the mean atomic distance, we assume S >> r0^2 while a may be comparable to r0. In the bidimensional limit (a/r0 << 1) we find a macroscopic number of atoms in the condensate at temperatures T ~1/log(N); therefore, condensation cannot be described in terms of intensive quantities; in addition, it occurs at temperatures not too low in comparison to the tridimensional case. When condensation is present we also find a macroscopic occupation of the low--lying excited states. In addition, the condensation phenomenon is sensitive to the shape of S. The former two effects are significant for a nanoscopic system. The tridimensional limit is slowly attained for increasing (a/r0), roughly at (a/r0) ~ 10^{2}-10^{3}.