Abstract
$\mathrm{Sr}\mathrm{Ti}{\mathrm{O}}_{3}$ (STO) displays a huge isotope effect. Replacing $^{16}\mathrm{O}$ with $^{18}\mathrm{O}$ changes the incipient ferroelectric crystal into a ferroelectric crystal with a transition temperature of $24.5\phantom{\rule{0.3em}{0ex}}\mathrm{K}$. We present a model that takes into account the soft mode, the NMR observed spontaneous Ti off-center displacements present in both $\mathrm{ST}^{16}\mathrm{O}$ and $\mathrm{ST}^{18}\mathrm{O}$, and their interaction with each other. Replacing $^{16}\mathrm{O}$ with $^{18}\mathrm{O}$ increases the $\mathrm{Ti}{\mathrm{O}}_{3}$ reduced mass by 5.5%. This is the only effect taken into account, and it affects only the soft mode leaving the Ti off-center displacements and their interaction with the soft mode constant. We show that the model accounts quantitatively for the ferroelectric phase transition, the Raman and neutron scattering observed soft modes, the temperature and $^{18}\mathrm{O}$ concentration dependence of the dielectric constant, and the $^{18}\mathrm{O}$ concentration and pressure dependence of the phase transition temperature.
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