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 ${T}_{c}(n)$ obtained by ratio and Pad\'e approximant techniques are presented. These results are consistent with the conjectures that ${T}_{c}(\ensuremath{\infty})\ensuremath{-}{T}_{c}(n)$ varies with thickness $n$, as ${n}^{\ensuremath{-}\ensuremath{\lambda}}$ with $\ensuremath{\lambda}=1$ in case (a) and $\ensuremath{\lambda}=\frac{1}{{\ensuremath{\nu}}_{c}}\ensuremath{\simeq}1.56$ in case (b); but other, somewhat larger, values of $\ensuremath{\lambda}$ are not excluded.