Nuclear spin–lattice relaxation of coupled spin systems in liquids

Abstract

A new method based upon the Zwanzig–Mori projection operator technique has been introduced to treat the spin–lattice relaxation problem of coupled nuclear spin systems in liquids, with emphasis on high resolution NMR spectroscopy. The important quantities required for the use of the method are a set of orthogonal operators which can readily be constructed and given physical meaning in the eigenstate representation for any particular spin system. Some of the expectation values of these orthogonal operators may be related to the measurable quantity in high resolution NMR experiments. The time rate of change of each observable may then be formulated in terms of the trace of various operator functions, allowing the calculation to be carried out in any representation. The computational advantage of using simple spin-product functions is obvious. The prescription for constructing irreducible orthogonal tensor operators is given, and a variety of such tensor operators are explicitly given for several common spin systems. In the present method each orthogonal operator has a definite symmetry property under spin inversion which can be used to separate the coupled equations into a symmetric and an antisymmetric set. The problem of multiexponential decay in the time evaluation of the longitudinal magnetization, owing to effects of interferences between pairwise dipolar interactions, becomes clear in the present formalism. The present paper is a generalization of earlier work on the relaxation behavior of multiplet spectral structure in high resolution NMR.