Physica B 165&166 (1990) 595-596 North-Holland BOSE-EINSTEIN CONDENSATION IN QUASI-TWO-DIMENSIONAL SYSTEMS Humam B. GHASSIB and Yahya F. WAQQAD Department of Physics, University of Jordan, Amman, Jordan Bose-Einstein condensation in an ideal quasi-two-dimensional Bose gas is recon- sidered with a view of establishing a simple criterion for crossover effects from two- to three-dimensional systems. The main attention is focused on the density of states involved. It is deduced that the critical value at which these effects occur is universal, apart from a geometrical factor of the order of unity. The implications for strongly interacting Bose systems are discussed very briefly. 1 . INTRODUCTION It is well known that Bose-Einstein condensation (BEE) does not occur in strictly two-dimensional ideal Bose systems (l-3), unlike the three-dimension- al case. However, it remains an interest- ing problem to see how, in an (mxmxd) system, EEC reappears and: furthermore, how the critical value d can te deter- mined at which crossoverzfferts from two- to three-dimensional behaviour manifest themselves. This is attempted in Section 2, using the simplest possible techniques. The results obtained are then discussed in Section 3. 2. DEVELOPMENT 2.1 BEC in Strictly Zd-Systems To establish the notation we first revisit the strictly two-dimensional ideal Bose system. The number of excited states in this system is gi;en by dp , (I) r-lexp{6E(p)1-1 where S is the area of the system, h is Planck's constant, 8 is the temperature parameter, l/kBT, kB being Boltzmann's constant and T the temperature, E(p) is the energy of a Bose particle of momentum p, and z is the fugacity 5 exp(-BP), p being the chemical potential. Performing the angular integration in eq.(l) yields N_ 2nm In\ “=-_ g1(2), iLi S hZ6 where m is the mass of the boson and gn(z) is the familiar Bose function of order n and argument z (4,5). Clearly, N, is a maximum only when gl(z) is a maximum - i.e., when 2~1. accordingly, However, gl(l) diverges; all particles In the two- dimensional ideal Bose gas are excited. This means that no BEC occurs for systems whose density of states is constant, which is the case for infinite two-dimen- sional systems. 2.2 BEC in Quasi-Zd-Systems Suppose the ideal Bose gas is now con- fined to an infinite plane sheet whose thickness is d. BEC can now be probed in a simple manner by specifying those terms which depend on dimensionality and those which domnot. While N, is still given by Ne = g(E;d);n(E)>dE, [ (3) j0 its spatial dependence now enters through the density of states g (E;d). The problem is then reduced to evaluating this last function. To calculate g(E;d) the total number of states u up to an energy E should be determined first; the derivative ao/aE will then ztve g(E;d). Specifically, where f(n;d) is 1 if n=O,?l, i n0 is the maximum value of n with the total energy E: i E(n) ?? hZ kZ - (nn/d)' 7iii[ 1 so that i nOzdmax=[k.]int; k being the wavenumber. It follows that n0 (4) 2 * . f and cA;patible (5) a(E(n);d)= I 4nS mE - h* 0921-4526/90/$03.50 @ 1990 - Elsevier Science Publishers B.V. (North-Holland)