On the Statistical Theory of the Shape of Multiple Quantum NMR Spectra in Solids

The statistical model developed in this study allows us to calculate the shape of multiple quantum (MQ) NMR spectra (the dependence of the amplitudes of the corresponding MQ coherences on their orders) by decomposing the desired time-correlation functions (TCFs) over an infinite set of orthogonal operators and by using some well-known facts from the physics of traditional model systems. The resulting expression contains series of gradually increasing numbers of spins in clusters of correlated spins depending on time. The influence of the possible degradation of these clusters on the shape of the spectra is taken into account. Analytical and numerical calculations are performed for various parameter values included in the final expressions. The developed theory adequately describes the results of the numerical calculations of the MQ spectra performed by us and experiments: the transformation of the Gaussian profile into an exponential one, the asymptotics (wings) of the spectrum depending on the coherence order M, and the dependence of the relaxation rate of the MQ spectrum on M, as well as the narrowing and stabilization of the MQ spectrum under the influence of a perturbation.