Exchange striction, the origin of polycrystalline magnetoelastic anisotropy

It is analytically shown that the origin of the magnetoelastic anisotropy term $\Delta K$, previously established as specific of the polycrystalline state, is related to the product of the strong isotropic virtual "exchange stresses" due to the exchange striction by the anisotropic magnetostrictive deformation constrained by the neighboring grains. A linear law of the variation of $\Delta K$ versus the exchange-striction volume anomaly $\frac{\Delta V}{V}$ is obtained for Ni${\mathrm{Fe}}_2$ ${\mathrm{O}}_4$ and yttrium iron garnet. An analytical expression of $\Delta K\left(T\right)\mathrm{}$ as a function of ionic structure parameters is established.