Nuclear magnetic resonance line shapes of methyl groups undergoing tunnelling rotation

Abstract

Nuclear magnetic resonance spectra are calculated for an equilateral triangle of protons tunnelling between equivalent orientations. It is shown that the effect of the motion may be described by introducing an interproton exchange interaction into the nuclear spin Hamiltonian, the coefficient J being related to the height of the potential barrier to rotation. This term has matrix elements equivalent to those of the rotational energy operator. Spectra are obtained for arbitrary J, giving a quantum-mechanical solution of a motional narrowing problem and demonstrating the formal identity of the phenomenon, in this case with exchange narrowing. The limit of large J is in agreement with the classical result of Andrew and Bersohn. Isotropic averages of the spectra with Gaussian broadening to take account of inter-methyl-group interactions are computed for comparison with measurements on powders at low temperatures.