NMR in Hydrate Crystals: Structural Information from Second Moments

The effects of fast 180° flip motion of the water molecules in hydrate crystals on the first and second moment of the proton magnetic resonance spectrum are explored theoretically. When the spectrum consists of two well-separated line components, it is proved that the first moment of the spectrum about the midpoint is the correct measure of the interaction responsible for the main splitting. Earlier this has been proved for a system of water molecules rigidly bound in the lattice. A formula for the second moment, consisting of an intramolecular term, an intermolecular term, and a term due to unlike spins, is derived. The first and the last of these terms are shown to be independent of the shape of the spectrum. The intermolecular term, however, is proportional to a factor which is 5/6 when the spectrum consists of two well-separated line components and 1 when the spectrum shows no sign of a fine structure. This shape dependence of the intermolecular term is due to the fact that some of the intermolecular spin exchange transitions are forbidden to first order when the spectrum consists of two well-split lines. It is pointed out that the partial quenching of some transitions when the spectrum shows signs of a fine structure will in general lead to an uncertainty in the theoretical second moment. It is concluded that to get accurate structural information from second moments, the fine structure has to be completely resolved or the spectrum should show no signs of a fine structure. The obtained formula for the second moment has been used to calculate the second moments for a particular set of orientations of a single crystal of gypsum. The experimentally measured second moments show fair agreement with the calculated values.