Elastic precursor effects during <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>Ba</mml:mi><mml:mrow><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>x</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi>Sr</mml:mi><mml:mi>x</mml:mi></mml:msub><mml:msub><mml:mi>TiO</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:math> ferroelastic phase transitions
F. Cordero, F. Trequattrini, Paulo Sergio Silva, Michel Venet, Oktay Aktas, Ekhard K. H. Salje
Abstract
Elastic softening in the paraelastic phases of ${\mathrm{Ba}}_{1\ensuremath{-}x}{\mathrm{Sr}}_{x}{\mathrm{TiO}}_{3}$ 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 $\ensuremath{\delta}S\ensuremath{\propto}|T\ensuremath{-}{T}_{\mathrm{C}}{|}^{\ensuremath{-}\ensuremath{\kappa}}$ with a characteristic exponent $\ensuremath{\kappa}$ 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 ${\mathrm{Ba}}_{x}{\mathrm{Sr}}_{1\ensuremath{-}x}{\mathrm{TiO}}_{3}$ 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 $\ensuremath{\kappa}\ensuremath{\simeq}$ 0.2. This proves that the high-temperature elastic properties of these materials are drastically affected by precursor softening.