It has been known for a long time that the low temperature behavior shown by the dielectric constant of quantum paraelectric $SrTiO_{ 3}$ 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 $T\sim10K$ 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 $T\sim10K$ is unknown. In this work we show that the crossover between these two values of the saturation temperature at $T\sim10K$ can be taken as a direct consequence of (i) the quantum distribution of frequencies $g(\Omega)\propto\Omega^{ 2}$ 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 $T_{ tr}$.
Categories: cond-mat.mtrl-sci