Elastic precursor effects during the Ba${}_{1-x}$Sr${}_x$TiO${}_3$ ferroelastic phase transitions

Abstract

Elastic softening in the paraelastic phases of Ba${}_{1-x}$Sr${}_x$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 SrTiO${}_3$\ and 0.2 in 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 SrTiO${}_3$% \ to BaTiO${}_3$\ indicating a change from a displacive to a weakly order/disorder character of the elastic precursor. Mixed crystals of Ba${}_x$% Sr${}_{1-x}$TiO${}_3$ possess intermediate behaviour. The amplitudes of the precursor elastic softening increases continuously from SrTiO${}_3$ to BaTiO. Using power law fittings reveals that the elastic softening is still of the unsoftened Young's modulus at temperatures as high as 750 K in BaTiO${}_3$ with $\kappa \simeq$ 0.2. This proves that the high temperature elastic properties of these materials are drastically affected by elastic precursor softening.

9 pages, 7 figures