Rotationally Invariant Theory for the Effect of Magnetoelastic Interactions on the Elastic Constants of the Heavy-Rare-Earth Metals

A theory developed by Toupin, Tiersten, Brown, and Melcher employing finite strains and angular-momentum invariants is applied to the rare-earth metals of hcp structure. A Hamiltonian is written down which includes Heisenberg-exchange, crystal field, and magnetoelastic terms and is invariant under combined rotations of the magnetic and elastic systems. When the approximations of small-strain theory are subsequently carried out, there appear new terms originating in the crystal field that are linear in the antisymmetric strains ${\omega }_{\mu \nu }$ and correspond to rotations of the elastic medium. The coupling of transverse acoustic waves to the magnetic system is studied and expressions are derived for the dependence of the elastic constants $c_{44}$ and $c_{66}$ on an applied magnetic field in the ferromagnetic phase. The terms involving the antisymmetric strains result in new effects similar to those found by Melcher in Mn${\mathrm{F}}_2$, from which it should be possible to obtain in a direct manner the values of certain magnetoelastic constants and anisotropy constants. Using available data on magnetic anisotropy and magnetostriction, we have estimated the size of the effects that may be expected to be found in Gd, Tb, Dy, Ho, and Er. Fractional changes in $c_{44}$ and $c_{66}$ as large as ${10}^{-2\mathrm{}}$ are predicted for Tb, Dy, Ho, and Er in a field of about 50 kOe, while the maximum change for Gd is predicted to be about ${10}^{-4\mathrm{}}$. Calculations have also been performed for the field-dependent changes in $c_{11}$ and $c_{33}$ for longitudinal waves in the paramagnetic region. These changes result from the fact that the finite strains $E_{\mu \mu }$ include terms of the form ${\epsilon }_{\mu \mu }^2$. The resulting changes in $c_{11}$ and $c_{33}$ depend linearly on the magnetoelastic constants and vary as $H^2$ in the paramagnetic region. Estimates of certain combinations of these constants are made from the experimental measurements of Moran and Lüthi on Dy and Ho.