Statistical Mechanics of Monolayers of Bose and Fermi Atoms Adsorbed on Crystalline Surfaces

Abstract

We have calculated the heat capacity and entropy of Bose and Fermi atoms adsorbed on crystalline surfaces at low temperatures. The treatment is quantum mechanical at the outset, the influence of tunneling of atoms between adsorption sites being discussed in detail. It is shown by general qualitative arguments that the energy spectrum of a low-coverage monolayer contains a low-lying tunneling band of translational surface states. The finite energy width of the tunneling band ensures that the monolayer entropy due to the distribution of particles among the translational states will vanish at 0\ifmmode^\circ\else\textdegree\fi{}K. This entropy is shown to be equivalent to the configuration entropy at relatively high temperatures and therefore demonstrates the manner in which the configuration entropy is removed at 0\ifmmode^\circ\else\textdegree\fi{}K. At higher temperatures, particles become excited to higher bands which may be separated from the tunneling band by an energy gap. Quantum-statistical properties of non-interacting bosons and fermions in a two-band model are analyzed in terms of the number of adsorption sites, the tunneling band width, and the interband energy gap. When the interband gap is larger than the tunneling band width, the monolayer heat capacity displays two distinct peaks, at temperatures $\mathrm{kT}$ characteristic of the interband energy gap and of the tunneling band width. The heat capacity approaches ideal two-dimensional gas behavior at zero interband gap. The systems are studied over a range of other parameters, including the ratios of densities of states in the two bands and the number of atoms adsorbed on the surface. A qualitative discussion is given of the influence of interactions among the adsorbed atoms, and of their modifications by the substrate. The relevance of the theory to current experiments on helium monolayers is discussed.