Elastic Constants and Soft Optical Modes in Gadolinium Molybdate

The complete set of elastic constants in single-crystal gadolinium molybdate (GMO) is presented in a temperature range from 20 to 200 °C. This covers the structural phase transition at 159 °C, below which GMO is both ferroelastic and ferroelectric. Elastic constants in both phases behave as $c\left(T\right)=c\left(\infty \right)+A{|T_0-T|}^{-K}$, where $K$ ranges from 0.76 to 0.90. Some components of the critical part of the elastic tensor, $\delta c_{\alpha \beta }=c_{\alpha \beta }\mathrm{}\left(T\right)-c_{\alpha \beta }\left(\infty \right)$, are interrelated. The relations are $\delta c_{12}^2=\delta c_{11}\delta c_{22}$ for both phases. From this relation and from the Curie-Weiss-like behavior of the elastic constants we conclude that (i) the soft mode near the $M$ point of the tetragonal cell ${\overrightarrow{\mathrm{k}}}_0=(\frac{\pi }{a}, \frac{\pi }{a}, 0)$ interacts anharmonically with acoustic phonons and gives rise to critical softening of specific elastic constants; and (ii) the findings are compatible with a dispersion relation near $k_0$ of the form ${\omega }^2\mathrm{}\left(k\right)={\omega }^2\left(k_0\right)+{\alpha }_x{(k_x-\frac{\pi }{a})}^2+{\alpha }_y\times {(k_y-\frac{\pi }{b})}^2+{\alpha }_z{q}_z^2$, with ${\alpha }_x={\alpha }_y\gg {\alpha }_z$. A comparison is made with the elastic tensor in quartz.