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arXiv:cond-mat/9608032 · Physics Letters A

Density of states for Bose-Einstein condensation in harmonic oscillator potentials

Klaus Kirsten, David J. Toms

Published Oct 1, 1996• 51 citations
arXivHTMLPDF
DOI
10.1016/0375-9601(96)00630-5

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The thermostatistic properties of a q-generalized boson system trapped in an n-dimensional harmonic oscillator potential are studied, based on the generalized statistic distribution derived from Tsallis' entropy. The density of states, total number of particles, critical temperature at which Bose-Einstein condensation occurs, internal energy, and heat capacity at constant volume are derived. The characteristics of Bose-Einstein condensation of the system are discussed in detail. It is found from the results obtained here that the thermostatistic properties of such a system depend closely on parameter q, dimensional number of the space, and frequency of the harmonic oscillator; and the external potential has a great influence on the thermostatistic properties of the system.
Low-dimensional Bose-Einstein condensations (BEC) for g(e) (the density of states) = Cde(superscript d-1) with d ≤ 1, corresponding to a dimensionality of not larger than 2, are theoretically analyzed. When the energy quantization of the first excited state is taken into account, the boson number in the excited states is finite even for d ≤ 1 with the density of states remaining proportional toe(superscript d-1). The condensation temperature increases with the particle number faster for d ≤ 1 than for d ＞ 1. The condensation depends highly on the quantized energye1 of the first excited state. Whene1 just deviates slightly from zero, the condensation temperature quickly increases withe1. Theoretical estimation shows that a low-dimensional BEC can occur at a temperature above 1 K for excitons in semiconductor quantum wells even without the harmonic-oscillator potential.
Bose–Einstein condensation is the anomalous accumulation of particles in the ground state of a system of bosons, when compared with the population in a system of particles obeying classical statistics. I use undergraduate-level statistical mechanics and a symbolic algebra computer program to study the occupation numbers of the energy levels of a finite number of noninteracting particles confined in a three-dimensional harmonic oscillator potential. I also calculate the heat capacity of the gas. The harmonic oscillator potential simplifies the calculations and approximates the conditions of the recent experiments achieving Bose–Einstein condensation in laser-cooled alkali vapors.