Mean-square displacements of atoms in thin crystal films

On the basis of a simple model of a simple cubic crystal, calculations are carried out of the mean-square displacement of an atom in a thin film of $L$ layers bounded by two free (001) surfaces. The results are compared with those for a semi-infinite crystal bounded by a single free (001) surface for atoms at the same distance from the surface. The conditions under which the two sets of results are in essential agreement are established.