Critical attenuation and dispersion of longitudinal ultrasound near a nematic—smectic-Aphase transition

We report a detailed study of the anomalous attenuation and velocity dispersion of longitudinal ultrasound near a nematic—smectic-$A$ phase transition in the liquid-crystal $\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{p}\mathrm{h}\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{l}-\mathrm{b}\mathrm{i}\mathrm{s}-p-p^{\prime }-\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{y}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{i}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}$. The measurements were performed at various frequencies and at various angles between the sound propagation direction and the symmetry direction. The attenuation anomaly is found to be dominated by an isotropic contribution in contradiction with older theories but in qualitative agreement with the recent ones. The results are consistent with the picture where the critical slowdown of fluctuations contributes to the anomaly both above and below the transition while a Landau-Khalatnikov-type relaxation of the smectic order parameter contributes below the transition in a way analogous to the $\lambda$ transition in helium. From the anisotropy of the longitudinal sound we extracted the velocity of the "second-sound" mode which softens as the transition is approached from below and collapses to zero at the transition. Moreover, we attribute the pronounced dip in sound velocity to the presence of a Pippard-Buckingham-Fairbank anomaly. The results are discussed in the light of the current understanding of the dynamics of the nematic—smectic-$A$ phase transition.