Configurations magnétiques. Méthode de fourier

Abstract

Spin configurations are directly related to the eigenvectors of a matrix. Each Bravais lattice j, occupied by magnetic ions, contributes to the exchange energy a value − 2λ j . The scalar parameters λ j are invariant in the symmetry operations of the crystallographic group. If the magnetic ions only occupy one Bravais lattice ( j = 1), λ 1 is the fourier transform of the exchange integral and the matrix method is reduced to the Villain method. In the case of a set { A} of n equivalent atoms, the λ j are the eigenvalues of a matrix. When the chemical unit cell is conserved, the eigenvectors which describe the spin configurations form the basis of irreducible representations which fact relates the matrix method to the group theoretical method of D zyaloshinsky and T urov. A trivial mode is the ferromagnetic one which corresponds to the identity representation. In coupled systems { AB} the eigenvectors are linear combinations of the eigenvectors of { A} and { B}. Néel's ferrimagnetism appears as the result of the coupling of the “trivial” ferromagnetic modes of { A} and { B}. Stability conditions of the solutions are expressed by inequalities between exchange integrals. The study is limited to the determination of “fundamental modes”, i.e. complete modes with a single propagation vector k. The following applications of the matrix method are studied : (a) the body-centered cube (example of one Bravais-lattice) (b) structures related to the corundum type as Fe 2O 3, Cr 2O 3, ilménites (example of n equivalent Bravais-lattices). (c) cubic spinels and (d) garnets (examples of coupled systems).