Configurational averaging of dipolar interactions in magnetically diluted spin networks

Some analytical results, concerning the disorder average of magnetic dipolar interactions in dilute spin networks and valid in the thermodynamic limit, are derived within a mean field approximation. In particular, it is shown that the quenched substitutional disorder average of the free induction decay of the nuclear magnetization in such a homonuclear spin system yields, [G(t)](c) similar to exp[-rho(gamma(2)ht)(D/3)], where rho is the concentration of magnetic nuclei embedded in a D-dimensional space and gamma the nuclear magnetogyric ratio. Furthermore, the relative fluctuations about the disorder average, {[(G(2)(t)](c)-[G(t)](2)(c)](1/2)/[G(t))(c)}, scale as [rho(1/2)(gamma(2)ht)(D/6)/N-1/2], when rho(gamma(2)ht)(D/3) much less than 1 and as N--1/2 exp[rho(gamma(2)ht)(D/3)][1 + rho(-3)(gamma(2)ht)(-D)](1/2) when rho(gamma(2)ht)(D/3) much greater than 1; N is the number of spins in the lattice. The distinction between the role of the arithmetic mean and the most probable value of the distribution of dipolar coupling constants in the configurational averages of the free induction decay and its relative fluctuations is pointed out. Implications for nuclear magnetic resonance (NMR) experiments in magnetically diluted solids are also examined. (C) 1996 American Institute of Physics.