Toward Monte Carlo simulation of general cases of static muon spin relaxation in disordered magnetic materials: long-range magnetic order in alloys

Abstract

Monte Carlo simulations of zero-field (ZF) muon spin relaxation (μSR) functions generated by long-range-ordered states with disorder are presented, for the completely static limit. Understanding of this is necessary before Monte Carlo simulation of the effect of short-range magnetic ordering on μSR in spin glasses can begin. Alloy disorder, controlled by the magnetic ion concentration parameter f m , and partial ordering of each moment, controlled by the order parameter f o, are considered. Qualitatively different behavior is seen depending on whether the dense moment, perfect-order limit ( f m=1 , f o=1) field at the muon site is non-zero, or cancels (as can happen in high-symmetry materials). Around the edges of the two-dimensional ( f m,f o ) parameter space, four limit cases with qualitatively different behavior are identified: (A) f o→0, the random frozen spin glass for arbitrary magnetic ion concentration; (B) f o→1, nearly perfect magnetic ordering in a alloy of arbitrary magnetic ion concentration; (C) f m →0, magnetic order developing (as f o increases) in a dilute magnetic alloy; (D) f m →1, magnetic order developing (as f o increases) in a dense magnetic material. Case A was discussed in a previous publication. The results for case D answer the question of how the Gaussian Kubo–Toyabe relaxation function for perfect disorder develops into an oscillating function as magnetic order develops in a material. Case C indicates that the effects of magnetic ordering in the dilute moment limit produce only subtle effects in ZF-μSR spectra that would be difficult to unambiguously identify as due to ordering in a real-world experiment. Case B generates complicated multi-frequency behavior.