Elastic constants and soft optical modes in cadolinium 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(T)=c()+A|T0-T|-K, where K ranges from 0.76 to 0.90. Some components of the critical part of the elastic tensor, δcαβ=cαβ(T)-cαβ(), are interrelated. The relations are δc122=δc11δc22 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 k→0=(πa, πa, 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 k0 of the form ω2(k)=ω2(k0)+αx(kx-πa)2+αy×(ky-πb)2+α zqz2, with αx=αyαz. A comparison is made with the elastic tensor in quartz. © 1972 The American Physical Society.