Elastic properties in quantum ferroelectricKTa1−xNbxO3

Elastic compliance measurements are reported in $\mathrm{K}{\mathrm{Ta}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ mixed crystals exhibiting a concentration-dependent ferroelectric phase-transition temperature $T_c$. For concentrations $x\ge 0.008$, an anomaly is observed at $T_c$ in the temperature dependence of the elastic compliance $s_{11}$, consisting of a steplike discontinuity with a superimposed Curie-Weiss-type singularity. The critical exponent $\mu$, defined by $s_{11}\sim {(T-T_c)}^{-\mu }$, varies from $\mu =1.3\mathrm{}$ for $x=0.05\mathrm{}$ to $\mu =1.9\mathrm{}$ for $x=0.008\mathrm{}$. As $x$ is lowered towards 0.008, the amplitude of the step discontinuity is largely unaffected. For $x<\mathrm{}0.008\mathrm{}$, no anomaly occurs. The experimental results are compared to dielectric data sets previously obtained, and interpreted in the light of renormalization-group predictions. A specific quantum critical behavior is obtained for transition temperatures $T_c<30\mathrm{}$ K.