Dipolar interactions in magnetically very diluted spin networks

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Abstract
Dipolar interactions among nuclear magnetic moments embedded in rigid solids are examined. Substitutional disorder in lattices is shown to modify the statistical properties of the distributions of the absolute values of dipolar coupling constants. Numerical studies reveal that a crossover in the average of the coupling constants in dilute spin networks occurs as a function of the random fractional occupancy in the lattice. The origin of this crossover is related to an interplay between the statistical weights of the coupling constants and their values which are distributed over many orders of magnitude. Two different regimes of behavior are observed for the average of the coupling constants as a function of dilution; one regime is dominated by the most probable value of the distribution of coupling constants, while in the other regime, the arithmetic mean prevails. For a macroscopic sample of 1022 sites, the crossover between regimes is predicted to occur when 1011 sites are occupied. Averaging over the substitutional disorder is also investigated. It is suggested that the character of the fluctuations in the disorder average could possibly serve to establish when a measurement on a large system is equivalent to an ensemble average over the subsystems of a sample. An analogous dynamic crossover, originating from similar behavior in the statistical properties of distributions of products of dipolar coupling constants, is pointed out for multiple quantum nuclear magnetic resonance (NMR) spectroscopy of solids.