Elastic softening in the paraelastic phases of ${\mathrm{Ba}}_{1-x}{\mathrm{Sr}}_x{\mathrm{TiO}}_3$ is largest near the transition temperatures and decreases on heating smoothly over extended temperature ranges. Softening extends to the highest measured temperature (850 K) for Ba-rich compounds. The temperature evolution of the excess compliance of the precursor softening follows a power law $\delta S\propto |T-T_{\mathrm{C}}{|}^{-\kappa }$ with a characteristic exponent $\kappa$ ranging between 1.5 in ${\mathrm{SrTiO}}_3$ and 0.2 in ${\mathrm{BaTiO}}_3$. The latter value is below the estimated lower bounds of displacive systems with three orthogonal soft phonon branches (0.5). An alternative Vogel-Fulcher analysis shows that the softening is described by extremely low Vogel-Fulcher energies $E_a$, which increase from ${\mathrm{SrTiO}}_3$ to ${\mathrm{BaTiO}}_3$ indicating a change from a displacive to a weakly order-disorder character of the elastic precursor. Mixed crystals of ${\mathrm{Ba}}_x{\mathrm{Sr}}_{1-x}{\mathrm{TiO}}_3$ possess intermediate behavior. The amplitude of the precursor elastic softening increases continuously from ${\mathrm{SrTiO}}_3$ to ${\mathrm{BaTiO}}_3$. Using power-law fittings reveals that the elastic softening is still $33\%$ of the unsoftened Young's modulus at temperatures as high as 750 K in ${\mathrm{BaTiO}}_3$ with $\kappa \simeq$ 0.2. This proves that the high-temperature elastic properties of these materials are drastically affected by precursor softening.

• Received 10 December 2022
• Accepted 24 January 2023

DOI:https://doi.org/10.1103/PhysRevResearch.5.013121

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Published by the American Physical Society