Tensor order parameters for magnetic-structural phase transitions in crystals with strong spin-lattice coupling

We formulate the thermodynamic theory of phase transitions in magnetically ordered systems in terms of a tensor, or coupled, order parameter. This basis is constructed by coupling atomic spin and lattice displacement. Symmetry lowering is predicted at the second-order phase transition point (tricritical points are not considered here). Lower-symmetry phases should in general be classified according to the Shubnikov symmetry space group Sh, which will reveal the total broken symmetry due to the coupled order parameter. In case the apparatus is "blind" to one portion of the order parameter: either spin or displacement, the apparent symmetry group will not be Sh, but a related space group, which will reveal "partial information." Comparing this formulation and the usual (uncoupled) theory, new results are obtained here: for example "pseudoscalar order parameters" can arise and different "symmetry-broken" groups. An illustration is given by applying the formulation to the spinel-structure space group: $O_h^7-Fd3m$. It is conjectured that for Tb${\mathrm{Ni}}_2$ the tensor order parameter $\Gamma 1$— may be relevant, so that the phase transition which has been identified as $O_h^7\to {\mathrm{Sh}}_{166}^{101}$ may actually be $O_h^7\to {\mathrm{Sh}}_{227}^{132}$, caused by a pseudoscalar.