Influence of vibrational motion on solid state line shapes and NMR relaxation

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Abstract
The influence of vibrational motion on bond lengths and quadrupole constants obtained from dipolar and quadrupolar solid state line shapes is considered. It is shown that such motions average both the magnitude and the orientation of the intrinsic interaction tensor. Explicit expressions for the effective coupling constants that can be conveniently evaluated using the results of a normal mode analysis are derived. When the vibrationally averaged interaction tensor is axially symmetric, it is shown that the effect of vibrational motion on relaxation can be rigorously incorporated into an effective coupling constant which is formally identical to the one that determines the line shape. Illustrative calculations for several alkanes, in both the gas and solid phases, are presented. The relative contributions of stretching and bending vibrations and the effect of anharmonicity on the effective C–H bond lengths and deuterium quadrupole constants are examined. The influence of vibrational averaging on the magnitude of the dipolar coupling is shown to be essentially independent of the nature of the molecule and its environment and to be quite small (i.e., for C–H bonds, the coupling constant decreases by 3% and hence the effective bond length increases by 1%). Vibrational averaging of the orientation of the dipolar interaction vector, on the other hand, depends on the size of the molecule and its environment because of the predominant role played by low frequency bending and torsional modes. The implication of these results for the value of the effective internuclear distance that should be used for the interpretation of the dipolar relaxation experiments is considered.