Equation of state, Debye-Waller factor, and electrical resistivity of ferroelectrics near their critical point

The Larkin-Khmelnitskii theory of uniaxial ferroelectrics is used to derive further predictions for the critical behavior of ferroelectrics, including "local properties" and transport properties. Special attention is paid to the experimental observability of the predicted logarithmic correction terms. In particular, in the expansion of the electric field E in powers of the dielectric polarization $P$, i.e., $E=P\Sigma i^{}{f}_{2i}\mathrm{}\left(T\right)\mathrm{}{P}^{2i}$, the temperature dependence of the coefficients $f_2\propto {\left|\ln \right(\frac{T}{T_c-1\mathrm{}}\left)\right|}^{-1\mathrm{}}$ and $f_4\propto {\left(\frac{T}{T_c-1\mathrm{}}\right)}^{-1\mathrm{}}{\left|\ln \right(\frac{T}{T_c-1\mathrm{}}\left)\right|}^{-\frac{4}{3}}$ is obtained, deviating significantly from $f_2=\mathrm{const}$ and $f_4=\mathrm{const}$ of the simple Landau theory. We argue that the nonanalytic behavior of $f_2$ could be measured more easily than either the logarithmic correction in $f_0\propto \left(\frac{T}{T_c-1\mathrm{}}\right){\left|\ln \right(\frac{T}{T_c-1\mathrm{}}\left)\right|}^{-\frac{1}{3}}$ or the specific-heat singularity $C\propto {\left|\ln \right(\frac{T}{T_c-1\mathrm{}}\left)\right|}^{\frac{1}{3}}$. We show that recent experiments on tri-glycine sulfate by Ehses and Müser are in good agreement with our predictions. Moreover, we calculate the temperature dependence of the critical contribution to the Debye-Waller-factor exponent $W$ which corresponds to that of the electron-paramagnetic-resonance linewidth in the "slow-motion regime." We find $W_{\mathrm{crit}}\propto \left(\frac{T}{T_c-1\mathrm{}}\right)\times {\left|\ln \right(\frac{T}{T_c-1\mathrm{}}\left)\right|}^{\frac{1}{3}}$ above $T_c$ and