Dipolar broadening in magnetically diluted lattices

A method is presented for computing NMR line shapes at high temperatures for a system of magnetically diluted spins interacting via a dipolar Hamiltonian. We specialize to the case of a solid nonmagnetic host with spin-(1/2 impurities randomly placed on a simple-cubic lattice. For low concentrations, an iterative procedure is developed to separate the broadening into homogeneous and inhomogeneous components. We find that, even in the dilute limit, a significant fraction of the spins contribute to a continuum (homogeneously broadened) band. An information-theoretic maximum-entropy technique is used to reconstruct the homogeneous component of the line shape from moments, and a version of the statistical theory is then employed to obtain the inhomogeneous component. Implications for hole-burning experiments are briefly discussed. For high concentrations, theoretical line shapes are obtained by use of configuration-averaged moments and maximum entropy, including up to the sixth Van Vleck moment. To test our results, we also calculated a line shape by diagonalizing the secular dipolar Hamiltonian for several configurations containing up to nine spins. For low concentrations we find reasonable agreement with the moment analysis. All of these methods are readily extended to any well-defined lattice structure and to other spin problems with a 1/$r^n$ interaction.