Low-temperature specific heat of a thin film

On the basis of a simple model of a simple cubic crystal, calculations are carried out of the low-temperature specific heat of a thin film of $L$ layers bounded by two free (001) surfaces. The specific heat is given as the sum of four terms, namely: $C^{\mathrm{}\left(B\right)\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$, $C^{\mathrm{}\left(s\right)\mathrm{}}\mathrm{}\left(T\right)\mathrm{}$, $C^{\left(s^{\prime }\right)}\mathrm{}\left(T\right)\mathrm{}$, and $C^{\left(s{s}^{\prime }\right)}\mathrm{}\left(T\right)\mathrm{}$, corresponding, respectively, to the bulk specific heat, the two surface specific heats, and a term which depends on the proximity of the two surfaces.