  The ferroelectric phase transition temperature of a single crystal $SrTiO_3$
cylindrical puck inserted into a cylindrical copper cavity has been measured
around $51~K$ using quasi $TE_{m,1,1}$ and quasi $TM_{m,1,1}$ electromagnetic
modes. The observed results of such dielectric spectroscopy may be explained by
implementing a quasiharmonic approximation of phonons in the incipient
ferroelectric crystal $SrTiO_3$ lattice. The resonant modes act as very
sensitive probe by implementing measurements of $Q$-factor, transmission or
frequency shift when cooling down to low temperatures around a few Kelvin. The
resonator dielectric losses reveals different crystal unit structure states and
ferroelectric phase transition in continuous cooling. The ferroelectric phase
transition initiates a spontaneous polarization $(P_s)$ forming a double-well
potential in each crystal unit causing a two-level systems, which creates extra
loss and changes the transmission characteristics and Q-factors of the modes.
This process is able to describe resonator properties over a broad range of
frequency. Landau's theory of correlation length supports the reported
observations as the implication of phonon wave vector $q_c=0$ of central ion in
crystal unit symmetry regarding soft-mode.
