Some Results Concerning the Crossover Behavior of Quasi—Two-Dimensional and Quasi—One-Dimensional Systems

Abstract

A magnetic system with intraplanar and interplanar interaction strengths $J$ and $\mathrm{RJ}$ is is treated. Rigorous relations are established concerning the first few derivatives with respect to $R$ of the susceptibility $\ensuremath{\chi}(R)$. Considering $\ensuremath{\chi}(R)={b}_{0}+{b}_{1}R+{b}_{2}{R}^{2}+\ensuremath{\cdots}$, we find ${b}_{1}$ and the order of magnitude of ${b}_{2}$. Hence we can predict when the system crosses over from $d$-dimensional to $d$-dimensional behavior (e.g., for quasi---two-dimensional systems, $d=2$, $\overline{d}=3$, while for quasi---one-dimensional systems, $d=1$, $\overline{d}=3$). These results also support scaling with respect to the anisotropy parameter $R$.