Spin relaxation for random walks on disordered lattices

Abstract

We consider the free induction decay (FID) (equivalent to spin depolarisation in muon spin resonance experiments) of a particle that performs a random walk on a d-dimensional lattice. With each lattice site n a random rotation frequency ω n for the particle spin is associated. We investigate this problem using analytical as well as numerical methods. The analytical calculations are carried out for arbitrary dimensions using Mori's projection formalism. In one dimension we find a decay that is purely exponential F(t) = exp(-λ t) (λ− σ 4 3 /γ 1 3 )in the asymptotic limit t→-∞, but the experimentally relevant time behaviour is governed by a F(t) = exp[-(λ t) 3 2 ] law. This non-Debye relaxation behaviour determines the decay of F(t) down to 10 −1 −10 −2 of its initial value. These results are in agreement with our numerical simulations which were carried out using a vectorized form of continued fractions ( d = 1) and multigrid methods ( d = 2 and 3). Furthermore, we note that in one dimension the decay time T 2 = ∫ 0 ∞d t F(t) is given by T 2 ∼ γ 1 3 /σ 4 3 in the motional narrowing regime. In addition, we find that the ESR spectra are self-averaging quantities, i.e. they do not depend sensitively on the realization of disorder. In the motional narrowing regime the spectra are nearly independent of the disorder distribution and converge to a universal function in the limit γ→∞.