Characterizing Nonexponential Spin-Lattice Relaxation in Solid-State NMR by Fitting to the Stretched Exponential

Abstract

Spin-lattice relaxation in solids often deviates from the usual exponential law, for example, in dilute spin-12 solids like silicon carbide, when relaxation arises from fixed paramagnetic impurities. We examine nonexponential spin-lattice relaxation from the point of view of nonlinear regression to the stretched-exponential function M(τ) = a − bexp[−(τ/T′)n], where n is typically 1 or 0.5 but can have intermediate values. Criteria for error estimation as well as an optimal choice of delay times τ are examined in detail and tested on typical data from our studies on silicon carbide. Two methods are outlined for the estimation of confidence intervals which are typically found to be somewhat larger than the standard errors in the parameters. The typical experiment for determination of spin-lattice relaxation time is found to be robust even when relaxation is nonexponential. The determinant of the curvature matrix is a good experimental design criterion and correlates well with the estimated errors. While any reasonable choice of points yields good results, an optimal choice of delay times τ can be made by following the guidelines presented here.