Rotational invariance, finite strain theory, and spin-lattice interactions in paramagnets; application to the rare-earth vanadates

The propagation of transverse elastic waves in paramagnetic crystals in the presence of an applied magnetic field is investigated both theoretically and experimentally. A theory is developed which is based upon finite-deformation elasticity theory and a description of the paramagnetic ions as pseudospins. Although the spatial rotational properties of the pseudospin operators are not known, the theory correctly includes in the thermodynamic (i.e., quasistatic) limit the spin-lattice coupling arising from the rotational motion associated with a transverse elastic wave. Explicit calculations of the changes in the elastic constant $c_{44}$ resulting from the spin-lattice coupling are presented for several systems in the thermodynamic limit. The lattice distortions involved correspond to transverse elastic waves propagating in the $z$ and polarized in the $x$ directions ($u_{\mathrm{xz}}$ distortion) and propagating in the $x$ and polarized in the $z$ directions ($u_{\mathrm{zx}}$ distortion). The infinitesimal strains $e_{\mathrm{xz}}$ associated with these waves are identical; the rotation ${\omega }_{\mathrm{xz}}$ is of opposite sign for the two waves; the second-order lattice compression is along the respective propagation directions. The latter two contributions are not included in the usual small-strain theory but do make significant contributions to the elastic properties of the paramagnetic system. Because of these contributions, the shift in $c_{44}$ due to the spin-lattice coupling is different for the two types of waves. In addition the shift can be either positive or negative depending on the relative magnitude and sign of the several coupling coefficients involved. Measurements of the shift in $c_{44}$ in the presence of a magnetic field have been performed in the tetragonal phases of the four rare-earth vanadates: TmV${\mathrm{O}}_4$, NdV${\mathrm{O}}_4$, TbV${\mathrm{O}}_4$, and DyV${\mathrm{O}}_4$. These measurements confirm the predictions of the theory presented here and are in clear contradiction with the predictions of an infinitesimal-strain magnetoelastic theory. In all four cases the measured shifts in the elastic constant $c_{44}$ are different for the $u_{\mathrm{xz}}$ and $u_{\mathrm{zx}}$ distortions. In the case of TmV${\mathrm{O}}_4$ the one is positive and the other negative.