Fictitious spin-(1/2 operators and multitransition nuclear relaxation in solids: General theory.

Abstract

The fictitious spin-(1/2 operator formalism is used to describe the nuclear relaxation of spins Ig(1/2 in solids for single or mixed relaxation processes. This permits us to describe the time evolution of the nuclear magnetization with a macroscopic kinetic equation extremely convenient for analyzing NMR data. Applications to the spin-spin and spin-lattice relaxations are given for quadrupolar (I=(3/2) and/or dipolar fluctuations for like (I=(3/2) and unlike (I=(3/2, S\ensuremath{\ge}(1/2) spins. In these calculations we have purposely chosen an exponential correlation function for all these fluctuating interactions to clearly point out the quantum behavior of the spin system. For quadrupolar relaxation, we predict a narrowing of the central line and a very large broadening of the satellite lines at sufficiently low temperature (${\ensuremath{\tau}}_{c}$\ensuremath{\gg}1/${\ensuremath{\omega}}_{0}$). We explain this seemingly paradoxical result by the nonexistence of the adiabatic part in the ${T}_{2}$ process for the central line of a half-integer spin Ig(1/2.Our treatment takes into account the second-order dynamical shifts, which are, in some temperature range, of the same order of magnitude and even much greater than the homogeneous linewidths. Consequently, in the central line, exists an inherent structure or differential line shifts in the lines, respectively, in the absence or in the presence of a residual static quadrupolar interaction. We find a nonexponential time evolution of the longitudinal magnetization coming from a cross relaxation between different spin states. For dipolar relaxation (unlike spins I,S) we predict that such cross relaxation dominates largely, at low temperature, over the direct relaxation and enhances the ${T}_{1}^{\mathrm{\ensuremath{-}}1}$ by a factor ${\ensuremath{\omega}}_{I}$/(${\ensuremath{\omega}}_{I}$-${\ensuremath{\omega}}_{S}$). We have also studied the general case of quadrupolar and dipolar (like and unlike spins) relaxations. The occurrence of several distinct maxima in the temperature variation of the longitudinal relaxation rates comes from the quantum characteristic frequencies of the spin system (i.e., ${\ensuremath{\omega}}_{I}$, ${\ensuremath{\omega}}_{S}$, ${\ensuremath{\omega}}_{I}$-${\ensuremath{\omega}}_{S}$, ${\ensuremath{\omega}}_{Q}$,. . . ) rather than by using different ad hoc correlation times. The objective of the general theory proposed is indeed to interpret, with a single calculation, all the phenomena induced by the relaxation, including the residual nonaveraged static interaction always present in anisotropic motions. The effects of a residual quadrupolar interaction have been considered explicitly for quadrupolar and dipolar fluctuations.Finally, we have considered the quadrupolar relaxation of the double- and triple-quantum spectra. We show that the double-quantum spectra are composed of two lines whose temperature behavior is similar to that of the satellites in the monoquantum spectra, while the triple-quantum spectra has a temperature behavior similar to the central line of the single-quantum spectra. The theoretical results appear to be directly applicable for analyzing relaxation data for quadrupolar nuclear spins involving anisotropic motions in solids.