Theory of two maxima in the temperature dependence of the spin–lattice relaxation rate of rotating three-spin systems in solids

Abstract

The spin–lattice relaxation of a system of three identical spin‐1/2 nuclei at the corners of an equilateral triangle is calculated on the assumptions that the relaxation is due to the dipole–dipole interactions between the nuclei, and that the motion of the triangle is a hindered rotation about an axis perpendicular to the plane of the triangle, the hindered rotation being described by a ’’stochastic six‐well model.’’ In this model there are six equilibrium orientations about the fixed axis, separated alternately by angles Δ and (2π/3)−Δ. The system rotates by jumping randomly from one of these orientations to an adjacent orientation. There are in general four different jump probabilities per unit time: ν′1 for a jump of +Δ, ν′−1 for a jump of −Δ, ν1 for a jump of (2π/3)−Δ, and ν−1 for a jump of −(2π/3)+Δ. This model is plausible for some cases in which the potential for the rotational motion includes a term of period π/3 in addition to a term of period 2π/3. If cross correlations of the three different dipole–dipole interactions are omitted in the calculation, the relaxation is a simple exponential decay with relaxation timeT 1. Expressions for 1/T 1, and for the average of 1/T 1 over all orientations of the axis of hindered rotation, are obtained for the general stochastic six‐well model. For the special case in which ν1=ν−1≪ν′1 =ν′−1, it is shown that 1/T 1 and (1/T 1)av can have two distinct maxima as a function of temperature. The nonexponential relaxation which results when cross correlations of the different dipole–dipole interactions are included in the calculation is also considered, and it is shown for the case ν1=ν−1≪ν′1=ν′−1 that there can be two distinct maxima in the relaxation rate as a function of temperature, and that the form of the nonexponential relaxation in the temperature ranges of the two maxima can be obtained from a previous calculation.