Harmonically trapped ideal quantum gases

M. Grether; M. Fortes; M. de Llano; F. J. Sevilla; M. A. Solís; A. A. Valladares

doi:10.1140/epjd/e2003-00005-1

EPJ

Harmonically trapped ideal quantum gases

M. Grether¹^a, M. Fortes², M. de Llano³, J.L. del Río⁴, F. J. Sevilla², M. A. Solís² and A. A. Valladares³

¹ Facultad de Ciencias, UNAM, Apdo. Postal 70-542, 04510 México DF, Mexico
² Instituto de Física, UNAM, Apdo. Postal 20-364, 01000 México DF, Mexico
³ Instituto de Investigaciones en Materiales, UNAM, Apdo. Postal 70-360, 04510 México DF, Mexico
⁴ Universidad Autónoma Metropolitana, Unidad Iztapalapa, 09340 México DF, Mexico

Received: 17 July 2002
Revised: 14 October 2002
Published online: 21 January 2003

Abstract

We solve the problem of a Bose or Fermi gas in d-dimensions trapped by $\delta \leq d$ mutually perpendicular harmonic oscillator potentials. From the grand potential we derive their thermodynamic functions (internal energy, specific heat, etc.) as well as a generalized density of states. The Bose gas exhibits Bose-Einstein condensation at a nonzero critical temperature T_c if and only if $d+\delta >2$ , along with a jump in the specific heat at T_c if and only if $d+\delta >4$ . Specific heats for both gas types precisely coincide as functions of temperature when $d+\delta =2$ . The trapped system behaves like an ideal free quantum gas in $d+\delta$ dimensions. For $\delta =0$ we recover all known thermodynamic properties of ideal quantum gases in d dimensions, while in 3D for $\delta =$ 1, 2 and 3 one simulates behavior reminiscent of quantum wells, wires and dots, respectively. Good agreement is found between experimental critical temperatures for the trapped boson gases $_{37}^{87}$ Rb, $_{3}^{7}$ Li, $_{37}^{85}$ Rb, $_{2}^{4}$ He, $_{19}^{41}$ K and the known theoretical expression which is a special case for $d=\delta =3$ , but only moderate agreement for $_{11}^{27}$ Na and $_{1}^{1}$ H.

PACS: 05.30.Fk – Fermion systems and electron gas / 05.30.Jp – Boson systems / 03.75.Fi – Phase coherent atomic ensembles; quantum condensation phenomena / 05.70.Ce – Thermodynamic functions and equations of state

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EPJ

Harmonically trapped ideal quantum gases

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