Inverse Laplace Transformation Analysis of Stretched Exponential Relaxation

Abstract

We investigate the effectiveness of the Inverse Laplace Transform (ILT) analysis method to extract the distribution of relaxation rates from nuclear magnetic resonance data with stretched exponential relaxation. Stretched-relaxation is a hallmark of a distribution of relaxation rates, and an analytical expression exists for this distribution for the case of a spin-1/2 nucleus. We compare this theoretical distribution with those extracted via the ILT method for several values of the stretching exponent and at different levels of experimental noise. The ILT accurately captures the distributions for $\beta \lesssim 0.7$, and for signal to noise ratios greater than $\sim 40$; however the ILT distributions tend to introduce artificial oscillatory components. We further use the ILT approach to analyze stretched relaxation for spin $I>1/2$ and find that the distributions are accurately captured by the theoretical expression for $I=1/2$. Our results provide a solid foundation to interpret distributions of relaxation rates for general spin $I$ in terms of stretched exponential fits.