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Neutron scattering studies of K$_3$H(SO$_4)_2$ and K$_3$D(SO$_4)_2$: The particle-in-a-box model for the quantum phase transition

In the crystal of K$_3$H(SO$_4)_2$, or K$_3$D(SO$_4)_2$, dimers (SO$_4)...$H$...$(SO$_4)$, or (SO$_4)...$D$...$(SO$_4)$, are linked by strong centrosymmetric hydrogen or deuterium bonds whose O$...$O length is $\approx 2.50$ Å. We address two open questions. (i) Are H or D sites split or not? (ii) Is there any structural counterpart to the phase transition observed for K$_3$D(SO$_4)_2$ at $T_c \approx 85.5$ K, that does not exist for K$_3$H(SO$_4)_2$. Neutron diffraction by single-crystals at cryogenic or room temperature reveals no structural transition and no resolvable splitting of H or D sites. However, the width of the probability densities suggest unresolved splitting of the wavefunctions suggesting rigid entities H$_{L1/2}-$H$_{R1/2}$ or D$_{L1/2}-$D$_{R1/2}$ whose separation lengths are $l_\mathrm{H} \approx 0.16$ Å\ or $l_\mathrm{D} \approx 0.25$ Å. The vibrational eigenstates for the center of mass of H$_{L1/2}-$H$_{R1/2}$ revealed by inelastic neutron scattering are amenable to a square-well and we suppose the same potential holds for D$_{L1/2}-$D$_{R1/2}$. In order to explain dielectric and calorimetric measurements of mixed crystals K$_3$D$_{(1-ρ)}$H$_ρ$(SO$_4)_2$ ($0 \le ρ\le 1$), we replace the classical notion of order-disorder by the quantum notion of discernible (eg D$_{L1/2}-$D$_{R1/2}$) or indiscernible (eg H$_{L1/2}-$H$_{R1/2}$) components depending on the separation length of the split wavefunction. The discernible-indiscernible isostructural transition at finite temperatures is induced by a thermal pure quantum state or at 0 K by $ρ$.