Pressure-induced ferroelastic transition and internal displacement in TeO2

The soft effective elastic constant $C_s=\frac{1}{2}(C_{11}-C_{12})$ of Te${\mathrm{O}}_2$ has been measured at 77 K as functions of the uniaxial-stress loads in the [110], [$\overline{1}10$], and [001] directions up to 4 kbar. By using the method of Thurston and Brugger, we have determined the values of the third-order elastic constants as $C_{111}-C_{122}=3.85\mathrm{}$ and $C_{113}-C_{123}=-0.39\mathrm{}$ in ${10}^{12}$ dyn/${\mathrm{cm}}^2$. These values are not big in comparison with those of other materials. The origin of the pressure-induced ferroelastic transition found by Peercy and Fritz can be attributed to the anomalously small elastic constant $C_s$ at 1 bar as well as the negative pressure coefficient of $C_s$. For the purpose of evaluation of the elastic constant, by making use of a model lattice with two-dimensional covalent bondings, we have calculated Keating's strain energy which includes the homogeneous strain and the internal displacement as variables. An internal displacement of oxygen atoms, which corresponds to the spontaneous internal diaplacement previously found by Worlton and Beyerlein in the high-pressure ferroelastic phase, is found concurrently to follow the homogeneous strain $e_{\mathrm{xx}}-e_{\mathrm{yy}}$ so as to suppress the increase of an O-Te-O bond-bending energy: This leads to the result $C_{11}-C_{12}\simeq 0$. The pressure dependence of the spontaneous internal displacements and the supralinear changes of the $A_1$- and $B_1$-optical-phonon frequencies under the [100] uniaxial pressure are interpreted by using Landau's free energy which includes the interaction terms between the homogeneous strain and the internal displacements, and the parameters in the free-energy are determined.