Generalization of the Luttinger-Tisza method: Application to the case of highly anisotropic dipole-dipole interactions

A previous generation of the Luttinger-Tisza method is further extended to cover the case of highly anisotropic dipole-dipole interactions ($g_{\mathrm{zz}}\ne 0$, $g_{\mathrm{xx}}=g_{\mathrm{yy}}\approx 0$) for $2^m(m\ge 2)$ ions per unit cell. It is found that the Luttinger-Tisza method using permutation groups is applicable provided the ions occupy special positions; the resulting configurations are such that all ionic spins related to each other by ${\overrightarrow{\Gamma }}^2(\overrightarrow{\Gamma }=l^{\prime }\overrightarrow{\mathrm{a}}+m^{\prime }\overrightarrow{\mathrm{b}}+n^{\prime }\overrightarrow{\mathrm{c}})$ translations point in the same direction. In addition, attention is drawn to certain other cases of magnetic configurations which can be treated by the Luttinger-Tisza method; in these cases all ionic spins which are related to each other by $\overrightarrow{\Gamma }$ translations, point in the same direction.