The quantum paraelectric behavior of SrTiO_{3} revisited: relevance of the structural phase transition temperature

It has been known for a long time that the low temperature behavior shown by the dielectric constant of quantum paraelectric $SrTi{O}_3$ can not be fitted properly by Barrett’s formula using a single zero point energy or saturation temperature ($T_1$). As it was originally shown [K. A. M"{u}ller and H. Burkard, Phys. Rev. B {\bf 19}, 3593 (1979)] a crossover between two different saturation temperatures ($T_{1l}$=77.8K and $T_{1h}$=80K) at $T\sim 10K$ is needed to explain the low and high temperature behavior of the dielectric constant. However, the physical reason for the crossover between these two particular values of the saturation temperature at $T\sim 10K$ is unknown. In this work we show that the crossover between these two values of the saturation temperature at $T\sim 10K$ can be taken as a direct consequence of (i) the quantum distribution of frequencies $g(\mathrm{\Omega })\propto {\mathrm{\Omega }}^2$ associated with the complete set of low-lying modes and (ii) the existence of a definite maximum phonon frequency given by the structural transition critical temperature $T_{tr}$.