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.

Full Text
Published version (Free)

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call