Statistical Mechanics of Ideal Fermions in a Thin Film

Starting from the partition function for ideal fermions in a thin film with box boundary conditions, the influence of size effects on the specific heat is examined both numerically and analytically. For fairly thin films the specific heat is depressed at low temperatures and raised at high temperatures. In the very thin case, the behavior is that of a two-dimensional gas at low temperatures and that of a three-dimensional Boltzmann gas at high temperatures. The relevance to ${}_{}{}^3{}_{}{}^{}\mathrm{He}$ thin-film experiments is briefly discussed.