Use of the Relaxation Function in the Nuclear Magnetic Resonance Analysis of Internal Motions in Solids

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Abstract
The motional narrowing of nuclear-magnetic-resonance spectra is examined from the point of view of Anderson. The relaxation function, which is the Fourier transform of the absorption spectrum, is used to connect the experimental data with the stochastic model of the motion. The interaction of the spins with their surroundings is treated as a stochastic process of the nature of a random telegraph signal with random amplitudes. The possibility of incomplete narrowing is easily included. Two theoretical relaxation functions are developed. The first function is derived from the assumption of a normal distribution for the phase deviation, and the second results from a uniformly distributed phase deviation. Ammonium chloride and titanium hydride are studied experimentally. The two theoretical relaxation functions are compared with the experimental relaxation function. The relaxation function based on a uniformly distributed phase deviation is in best agreement with experiment. Arrhenius activation energies are calculated and compared to those obtained from linewidth data. The apparent second moment is related to the frequency of motion. The relation has the advantage of internal consistency and is not contingent on an assumed line shape. The activation energies obtained from second-moment data are practically the same as those obtained by the relaxation-function method. The correction for the effect of modulation on the relaxation function is derived. This correction leads naturally to a relation between the observed line shape and the actual line shape. Partial experimental verification of the theory of modulation broadening is given.

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