Critical Temperatures of Ising Lattice Films

Abstract

Ising lattices consisting of $n=2, 3, 4, \mathrm{and} 5$ interacting plane square lattice layers are studied by hightemperature series expansions. Specifically, seven to nine terms of the zero-field susceptibility expansion have been obtained for (a) free-surface boundary conditions (in which each surface spin interacts with only five nearest neighbors); and (b) periodic boundary conditions (in which all spins interact equivalently with six nearest neighbors). Estimates of the critical temperatures $T_c\mathrm{}\left(n\right)\mathrm{}$ obtained by ratio and Padé approximant techniques are presented. These results are consistent with the conjectures that $T_c\left(\infty \right)-T_c\mathrm{}\left(n\right)\mathrm{}$ varies with thickness $n$, as $n^{-\lambda }$ with $\lambda =1\mathrm{}$ in case (a) and $\lambda =\frac{1}{{\nu }_c}\simeq 1.56$ in case (b); but other, somewhat larger, values of $\lambda$ are not excluded.