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The metal atoms in the pyrochlore system of compounds (A₂B₂O₇, where A and B are metals) form an infinite three-dimensional network of corner-sharing tetrahedra with cubic symmetry. For antiferromagnetic nearest-neighbor interactions and only B atoms magnetic, there is a very high degree of frustration, and no long-range order is predicted in the absence of further neighbor interactions. A general form of the mean-field theory is developed for dealing with n-component classical vector spins on any lattice. Calculations for the pyrochlore problem show that the Fourier modes of the system are completely degenerate for all wave vectors in the first Brillouin zone. In some cases further neighbor interactions will select the q=0 or incommensurate modes. A comparison is made with long-range order known to exist in the pyrochlore form of FeF₃. The highly degenerate ordered phases of more complicated systems, where both A and B atoms are magnetic, will also be discussed. A comparison is made of the corner-sharing tetrahedral lattice and the more familiar stacked triangular antiferromagnets, with regard to the degree of frustration in both systems. Results for the Kagom\'e lattice and the square lattice with crossings, which are the two-dimensional analogs of the corner-sharing tetrahedral lattice, are also briefly discussed.
Reimers et al. (Tue,) studied this question.