Phase Transition of the Uniaxial Disordered Ferroelectric Sr$_{0.61}$Ba$_{0.39}$Nb$_2$O$_6$

We report a neutron scattering study of a ferroelectric phase transition in Sr$_{0.61}$Ba$_{0.39}$Nb$_2$O$_6$ (SBN-61). The ferroelectric polarization is along the crystallographic $c$-axis but the transverse acoustic branch propagating along the $<$1, 1, 0$>$ direction does not show any anomaly associated with the this transition. We find no evidence for a soft transverse optic phonon. We do, however, observe elastic diffuse scattering. The intensity of this scattering increases as the sample is cooled from a temperature well above the phase transition. The susceptibility associated with this diffuse scattering follows well the anomaly of the dielectric permittivity of SBN-61. Below T$_\mathrm{c}$ the shape of this scattering is consistent with the scattering expected from ferroelectric domain walls. Our results suggest that despite apparent chemical disorder SBN-61 behaves as a classic order-disorder uniaxial ferroelectric with critical fluctuations in the range $<10^{-11}$ s.