Dielectric susceptibility in quantum ferroelectrics

Dielectric susceptibilities are reported for $\mathrm{K}{\mathrm{Ta}}_{1-x}{\mathrm{Nb}}_x{\mathrm{O}}_3$ and ${\mathrm{K}}_{1-y}{\mathrm{Na}}_y\mathrm{Ta}{\mathrm{O}}_3$ mixed-crystal ferroelectrics with a Curie temperature $T_C$ close to absolute zero. According to the temperature range, the two following formulas are fitted to the experimental data: $\epsilon =A+\frac{B}{[\frac{1}{2}{T}_1\times coth(\frac{1}{2}\frac{T_1}{T})-T_0]}$ for $T>\mathrm{}\frac{1}{2}{T}_1$ and $\epsilon =A+B{T}^{-2\mathrm{}}$ for $T$, $T_C<~\frac{1}{2}{T}_1$. This latter condition defines the quantum limit of a ferroelectric. Both formulas are obtained as limiting cases of a renormalized harmonic approximation of a model involving nonlinear polarizability of the oxygen shell. The constants $A$, $B$, $T_1$, and $T_0$ are explained in terms of model parameters. The leading term of the divergence of the susceptibility at the quantum limit coincides with the one obtained by the renormalization-group approach. Although renormalized harmonic approximation in general is bound to fail close to $T_C$, it gives valid results for the isolated (Gaussian) fixed point $T_C=0\mathrm{}$.