Diffusion and spin correlation in fractal percolation clusters.

Abstract

We have performed Monte Carlo simulations and theoretical calculations of the time decay behavior of the dipolar and spin exchange autocorrelation functions ${G}_{N}$(\ensuremath{\tau}) over a range of concentrations p (0<p<1) of randomly diluted magnetic species diffusing in a lattice. For p below the percolation threshold ${p}_{c}$, ${G}_{N}$(\ensuremath{\tau}) decays exponentially, at long time, with an effective time constant \ensuremath{\tau}(p) increasing with the concentration. We show that this result is characteristic of a bounded diffusion in a single average cluster explored in the average time 〈\ensuremath{\tau}(p)〉.At the percolation threshold p=${p}_{c}$, ${G}_{N}$(\ensuremath{\tau}) tends asymptotically towards the same power law ${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}{d}_{\mathrm{eff}}({p}_{c})/2}$ either for a dipolar or a spin exchange interaction. Here ${d}_{\mathrm{eff}}$(${p}_{c}$) is an effective dimensionality, which appears to be lower than the Euclidean dimensionality d. A theoretical calculation of the kinetics of reencounters of two spins diffusing in a distribution of clusters leads to the relation ${d}_{\mathrm{eff}}$(${p}_{c}$)=2-${d}_{s}$(3-${\ensuremath{\tau}}_{a}$) in terms of the spectral density ${d}_{s}$ and the critical exponent ${\ensuremath{\tau}}_{a}$ for the distribution of clusters in the theory of percolation. This relation gives ${d}_{\mathrm{eff}}$(${p}_{c}$)=0.76, 0.95, and (4/3 for d=2, d=3, and d=6, respectively. The asymptotic expression found for ${G}_{N}$(\ensuremath{\tau}) is then coherent with a process of anomalous diffusion in a reduced spatial dimensionality. Finally for p above ${p}_{c}$, one finds for each value of p, a power law G(\ensuremath{\tau})\ensuremath{\propto}${\ensuremath{\tau}}^{\mathrm{\ensuremath{-}}{d}_{\mathrm{eff}}(p)/2}$. There exists a crossover for ${d}_{\mathrm{eff}}$(p) between ${d}_{\mathrm{eff}}$(${p}_{c}$) and d for times sufficiently long that the temperature-dependent diffusion length exceeds the concentration dependent correlation length. This solves the seeming contradiction appearing in recent interpretation of nuclear-magnetic-resonance (NMR) results in mixed paramagnetic compounds.