Mechanism of quantum effects in hydrogen-bonded crystals of the ${\text{K}}_3\text{H}{\left({\text{SO}}_4\right)}_2$ group

The complete isotope effect and quantum paraelectricity in hydrogen-bonded crystals of the ${\text{K}}_3\text{H}{\left({\text{SO}}_4\right)}_2$ group are usually attributed to the proton tunneling and geometric isotope effect. However, a mechanism of the quantum effects is not unambiguously determined because neutron-scattering experiments indicate different forms of hydrogen-bond proton potential. The quantum effects and contradictory experimental results are accounted for by the model where the proton potential is strongly affected by a low-frequency heavy-atom mode.

Figure 1

(a) The energy levels of proton and deuteron (dashed lines) and their ground-state distributions $f_0\left(x\right)$ (thick dashed lines) in the double Morse potentials $V_{\text{DM}}\left(x\right)$ (full lines) of KHSe and DKHSe. The distributions $f\left(x\right)$ of proton and deuteron at $T_c$ in the dipole model are shown by thick full lines. (b) The energy levels of dipole (dashed lines) and its distributions at $T_{c}f\left(u\right)$ (thick full lines) in the potentials $V_d\left(u\right)$ (full lines) of KHSe and DKHSe in the dipole model.

Figure 2

The dependencies of the critical temperature $T_c$ (full line), the critical temperature for classical dipoles $T_0$ (dotted line), and the temperature corresponding to the half of dipole-excitation energy ${\Delta }_u/2k$ (dashed line) on the hydrogen-bond length $R$ of the protonated (h) and deuterated (d) systems in the dipole model. The full and open circles represent the experimental $R$ and $T_c$ from Table for protonated and deuterated crystals, respectively, where $T_c=0\text{K}$ is taken if the phase transition is absent.