Static Magnetoelastic Coupling in Cubic Crystals

The static magnetoelastic coupling in ferromagnetic or antiferromagnetic cubic crystals is analyzed in terms of a general formalism dictated by symmetry considerations. Besides the coupling of the spins to the external strains, resulting in external magnetostriction, the spins can also couple to internal strain modes. Only particular types of ionic displacements can couple to the spins, and these are classified. The spin operators which enter the theory are analyzed in terms of Tensor Kubic Operators, which are operator analogs of the Kubic harmonics, and which generate the irreducible representations of the cubic group. All equilibrium ionic displacements are found explicitly, and their temperature dependence is obtained. These equilibrium strains then lead to a general expression for the magnetoelastic contribution to the anisotropy energy and to the specific heat. On the basis of the usual $\frac{l(l+1\mathrm{})\mathrm{}}{2}$ power law we derive the temperature dependence of the magnetoelastic coupling coefficients and of their contributions to the anisotropy energy and specific heat. The available experimental data on magnetostriction, magnetization, and elastic constants for nickel are specifically analyzed. In general, magnetically induced strains lower the symmetry from cubic, depending on the direction of the magnetization and on the particular strain modes supported by the crystal. We analyze these deviations from cubic symmetry and show which symmetry groups remain below the magnetic transition.