Rotational tunneling in solids: The theory of neutron scattering

The wave functions of tunneling four proton groups (e.g., C${\mathrm{H}}_4$, N${\mathrm{H}}_4^{+}$) in a crystal field of low symmetry are obtained from products of pocket states and proton spin states. The results are applied to C${\mathrm{H}}_4$ in phase II of solid methane, and to N${\mathrm{H}}_4^{+}$ in ${\left(\mathrm{N}{\mathrm{H}}_4^{+}\right)}_2$Sn${\mathrm{Cl}}_6^{--}$ and m (N${\mathrm{H}}_4^{+}$)Cl${\mathrm{O}}_4^{-}$, where the tunneling spectra are known. It is shown that the observed degeneracy of two $T$ levels in ammonium perchlorate is accidental. Neutron scattering can only supply an upper bound of 25 MHz for the difference of the two levels. It is suggested to enlarge the splitting by hydrostatic pressure or to determine its value at zero pressure by NMR techniques. The theory of neutron scattering from tunnelling molecules is developed on the basis of the pocket-state formalism. Symmetry correlates the nuclear spin state and the rotational part of the wave functions in the tunneling levels. Therefore, the scattering from the four portons of one molecule is coherent, in contrast with the usual spin incoherence in protonated samples. Predictions about the angular and radial dependence of the scattering are made and experiments to examine these predictions are suggested. Angular averages of the cross section agree very well with the existing data.