Incoherent elastic-neutron-scattering study of the vibrational dynamics and spin-related symmetry of protons in theKHCO3crystal

Abstract

The potassium hydrogen carbonate crystal $({\mathrm{KHCO}}_{3})$ and its deuterated analog $({\mathrm{KDCO}}_{3})$ contain centrosymmetric dimer entities $({\mathrm{HCO}}_{3}^{\mathrm{\ensuremath{-}}}{)}_{2},$ or $({\mathrm{DCO}}_{3}^{\mathrm{\ensuremath{-}}}{)}_{2},$ linked by moderately strong hydrogen or deuterium bonds. The OH/OD bonds are virtually parallel to each other throughout the crystal. The incoherent elastic-neutron-scattering functions (IENSF) ${S(Q}_{x}{,Q}_{y}{,Q}_{z}=0,\ensuremath{\omega}=0)$ and ${S(Q}_{x}{,Q}_{y}=0,{Q}_{z},\ensuremath{\omega}=0)$ were measured over the momentum transfer range from \ensuremath{\sim}0 to \ensuremath{\sim}30 ${\mathrm{\AA{}}}^{\mathrm{\ensuremath{-}}1}$, with resolution of $\ensuremath{\Delta}|\mathbf{Q}|\ensuremath{\sim}1{\AA{}}^{\ensuremath{-}1}.$ The components of the momentum transfer ${Q}_{x},$ ${Q}_{y},$ and ${Q}_{z}$ were parallel to the mean directions of the stretching, in-plane, and out-of-plane bending modes of protons or deuterons, respectively. For ${\mathrm{KHCO}}_{3},$ the broad incoherent scattering peaks, centered at $\mathbf{Q}=0,$ are not amenable to the IENSF calculated with various models: the single harmonic oscillator, the double minimum potential, and the pair of coupled harmonic oscillators. Then, the degenerate ground state of a pair of coupled protons is regarded as composed of indistinguishable fermions. According to the Pauli principle, the symmetric and antisymmetric coordinates are singlet and triplet states, respectively. The IENSF gives quantum interferences that compare favorably to the observations. The estimated mean-square amplitudes in the ground state are amenable to isolated harmonic proton oscillators. Incoherent scattering for ${\mathrm{KDCO}}_{3}$ is amenable to Gaussian profiles for bosons. There is no evidence for quantum interference. The profile narrowing reveals an increase of the mean-square amplitudes attributed to a rather large coupling of the deuteron dynamics to the crystal lattice.