Brillouin spectroscopy of incommensurate ${\mathrm{Ba}}_2$${\mathrm{NaNb}}_5$${\mathrm{O}}_{15}$ at its 105-K phase transition

We have performed Brillouin studies of acoustic-phonon behavior in barium sodium niobate above and below its phase transition at $T_0$=105 K. Anomalies in sound velocity and attenuation are observed for the sound waves corresponding to the $C_{11}$ and $C_{22}$ elastic coefficients. Above $T_0$ the sound-velocity anomalies display lambda-shaped dips of algebraic form $V_{\mathrm{jj}}$(T) =$V_{\mathrm{jj}}$(∞)-β/(T-$T_C$), with $T_C$ several degrees lower than the actual transition temperature $T_0$; this is the form expected for free energies dominated by linear coupling between strain and the order parameter. Below $T_0$, $V_{\mathrm{jj}}$(T)=$V_{\mathrm{jj}}$(∞)-β’/($T_c^{\mathcal{’}}$-T The attenuation displays an increase of approximately 100% as the transition temperature is approached from above or below. By combining sound-velocity and attenuation data through a Landau-Khalatnikov approach, we are able to extract a single relaxation time and find that this time τ satisfies the expected mean-field dependence characteristic of critical slowing down: ${\tau }^{\mathrm{-}1}$(T)=${\tau }_1^{\mathrm{-}1}$(T-$T_C$)/$T_C$+${\tau }_2^{\mathrm{-}1}$.