For ideal particles in a finite multilayer system with free boundary conditions and a hopping amplitude, t, between neighboufing layers, we calculate the temperature dependence of the specific heat and the magnetic susceptibility. In spite of the fact that the excitation gap in the discrete direction is wiped out by the continuous 2-D spectrum, Schottky anomaly persists for both classical and quantum particles. Although finite-temperature Bose condensation only occurs when the layer number, n, is strictly infinite, we find the tendency towards condensation to happen much earlier and to be responsible for a second peak in the specific heat of bosons when n is large. The temperature dependence of the susceptibility for bosons also exhibits a similar dimensional transition as n increases, but the critical n is different from that for the specific heat.
Recent subjects, like the high-temperature superconducting cuprates and magnetic multilayers, renewed a lot of interest in the effect of layer structures. In thallium or bismuth oxide [ 1], it was observed that the superconducting transition temperature increases with the number of Cu-O layers up to three layers. It is then natural to ask, knowing that finite-temperature Bose condensation occurs in 3-D but not in 2-D, whether there is a dimensional crossover or a similar enhancement in the condensation temperature when we increase the layer number, n, or the hopping amplitude, t, between neighbouring layers of a multilayer system. The answer turns out to be No, and Bose condensation is only possible when n is strictly infinite. The reason is that, as long as n is finite, the density of states most bosons experience remains that of two dimensions at temperatures lower than the level difference i in the discrete direction. Only when n is I If we assume a constant hopping amplitude, t, between neighbouring layers, the eigenenergies in the discrete direction are nothing but the eigen-modes of a series of springs with fixed boundary conditions: E~ = 2t cos (,~'/(n + 1