Rotational tunneling in solids: Theory of neutron scattering. II.

Tedrahedral molecules or polyatomic ions (e.g., methane or the ammonium ion) in a solid experience a rotational potential with 12 equivalent minima. The rotational-energy spectrum shows a characteristic tunneling spectrum which, since the advent of high-resolution neutron scattering techniques, is accessible to the experiment. Tunneling frequencies have since been used to determine rotational potentials with the aim to assess the rather unknown anisotropic intermolecular forces. Further information is contained in the intensity of the tunneling lines, in their polarization for different directions of the momentum transfer and in their $\mathrm{}\left|Q\right|\mathrm{}$ dependence. To extract this information we have undertaken a detailed study of the tunneling wave functions including their librational width and the symmetry of the spin functions. For the properly symmetrized wave functions neutron scattering matrix elements have been calculated, and line intensities have been derived. The calculation has been performed for fully protonated and fully deuterated molecules ($X{\mathrm{H}}_4$ and $X{\mathrm{D}}_4$). The pocket-state formalism has been used, and the influence of finite-width pocket states on the line intensities has been studied in detail. Special attention has been attributed to molecules at low-symmetry lattice sites. On the one hand, the interpretation of the experimental data is particularly difficult for low site symmetries; on the other hand, there are fewer degeneracies and consequently more transition lines. The increased experimental information can be used to determine the symmetry elements and the strength of the rotational potential. The information contained in the intensities turns out to be extremely important.