Generalization of the Luttinger-Tisza method. II. Application to the case of extended spin components

It is shown, with the use of permutation groups, that the Luttinger-Tisza (LT) method can be used to treat the low-temperature ordering due to dipole-dipole interaction for the case when there are four and eight magnetic ions present per unit cell, provided that these ions occupy special positions. Further, it is shown that, by using the idea of splitting the $x$, $y$, $z$ components of the ionic spin and placing them at special positions, one can also apply the LT method using permutation groups to calculate the ground states corresponding to the cases of 12 and 24 magnetic ions per unit cell. It is found that these considerations are applicable to the low-temperature ordering of dysprosium, terbium, and holmium aluminum garnets.