Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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 $\approx
2.50$ {\AA}. 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 $l_\mathrm{H} \approx
0.16$ {\AA}\ or $l_\mathrm{D} \approx 0.25$ {\AA}. 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$_{(1-\rho)}$H$_\rho$(SO$_4)_2$ ($0 \le \rho \le 1$), 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 $\rho$.

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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an
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The categories of this paper are: dg-ga
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The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: solv-int
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The categories of this paper are: hep-ex
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The categories of this paper are: cmp-lg
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The categories of this paper are: math
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The categories of this paper are: adap-org
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The categories of this paper are: atom-ph
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The categories of this paper are: nlin
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The categories of this paper are: eess