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(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, δc_αβ=c_αβ(T)-c_αβ(∞), 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 →k₀=(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 ${{ω}}²(k)={{ω}}²({k}₀)+{{α}}ₓ{({k}ₓ{-}{{π}}{a})}²+{{α}}y×{}{({k}y{-}{{π}}{b})}²+{{α}}z{q}z²$, with ${{α}}ₓ={{α}}y{}{{α}}z$. A comparison is made with the elastic tensor in quartz.