Interpreting nuclear magnetic resonance spectra of disordered materials: direct inversion of powder patterns

The nuclear magnetic resonance (NMR) spectra of disordered materials are often interpreted by assuming distributions of the interaction parameters and fitting the spectra under these assumptions. Here we illustrate methods to extract the distributions directly from the spectra, making no such prior assumptions. The inhomogeneously broadened powder pattern observed in the NMR spectrum of a disordered solid is expressed as an integral over the powder patterns for individual sites weighted by the population distribution of the different sites. The resulting integral equation is solved for the underlying probability distribution of sites, both by singular value decomposition and by a regularization method. Results are shown for model and real one-dimensional and two-dimensional NMR experiments, with and without noise.