Lattice Theory of Spin Configuration

Abstract

In the frame of a given crystallographic symmetry a matrix ζ(k) is constructed, the eigenvectors and eigenvalues of which are directly related to the spin configuration and magnetic exchange energy, respectively. For Bravais lattices the matrix equation reduces to Villain's equation. When chemical and magnetic unit cells coincide, the eigenvectors of ζ(0) are shown to be identical to the ``basis of irreducible representations'' used by Turov and Dzialoshinski in the construction of an invariant Hamiltonian. The usefulness of both methods is discussed. The matrix theory is more general in the sense that it remains valid for magnetic cells different from the chemical one. Although the theory presented here starts from conventional crystallographic symmetry, it contains all the configurations possible in the so-called magnetic groups and even those not contained there (helical configurations). The theory includes isotropic as well as anisotropic (crystalline field, dipolar, pseudodipolar, antisymmetric) coupling in the order 2 approximation. Various examples of magnetic structures in the fields of perovskites, ilmenites, spinels, garnets, etc., are presented.