The Luttinger-Tisza method

The Luttinger-Tisza method and generalizations of this method for determining the minimum energy spin arrangement in a crystal subject to certain strong conditions are reviewed. It is shown that one can always replace the strong conditions by a set of additional conditions which in the simplest cases is identical with the single weak condition introduced in the Luttinger-Tisza method. A general method of calculating the minimum energy spin arrangement, based on the space group symmetry of the magnetic atom arrangements, is given, and the method of Niemeyer, based on permutation groups, is shown to be equivalent to this method. It is also known that the Luttinger-Tisza conjecture concerning the periodicity of the minimum energy spin arrangement is equivalent to a sufficient, but not necessary, condition for a minimum of the energy.