Spin-Exchange of Axially Symmetric Mu States in Polycrystalline Media J. Piroto Duarte a,b,∗ R. C.Vil˜ao b H. V. Alberto b J. M. Gil b and N. Ayres de Campos b a E.S.T.S.C., Polytechnic Institute of Coimbra, P-3040-854 Coimbra, Portugal b CEMDRX, Department of Physics, University of Coimbra, P-3004-516 Coimbra, Portugal Abstract A theoretical discussion of the features expected for the longitudinal-field relaxation signal of axially symmetric Mu states in polycrystalline samples is presented here. It is shown that the muon polarization may be described by the sum of an exponentially relaxed component and a non-relaxed one. Using this approximation, an analysis framework that allows extracting the spin-flip rate from polycrystalline data is also sketched. Key words: Muon Spin Rotation, Muonium, Spin Exchange 1. Introduction The scattering of a paramagnetic species and the bound electron in a Mu state strongly influences the time evolu- tion of the muon spin, constituting what is termed in μSR literature as Spin-Exchange (SE) dynamics. The effect of the scattering is conveyed to the muon via the hyperfine interaction, imparting a depolarization to the μSR signal which directly relates with the time structure of the physi- cal phenomena producing the scattering. In this way, μSR is used to study numerous dynamical effects, from charge transport in semiconductors to muon stopping ranges in gases. The μSR signal expected for a Mu state undergoing spin- exchange has been the subject of several theoretical stud- ies dating from the birth of μSR to our days (see e.g. [1– 3]), in which the transverse- and longitudinal-field signals in the slow and fast spin-flip regimes became well estab- lished using different theoretical approaches. Those results consider always that the Mu state either has an isotropic hyperfine interaction, or that when the hyperfine tensor is anisotropic, the three principal axis of the tensor have known orientation relative to the applied magnetic field and the direction along which the muon polarization is being observed. Recently, the investigation of local charge conduction in molecular organic semiconductors [4–6] forced the consid- eration of spin-exchange dynamics for axially symmetric * Tel: +351-239802430, Fax: +351-239-813395, email: piroto@ci.uc.pt Mu states in polycrystalline media, for which the known SE muon polarization functions cannot be directly applied. The need was prompted by the fact that these novel mate- rials cannot be grown in single crystals large enough to use in a μSR experiment, nor are soluble enough to use liquid samples where the anisotropy of the hyperfine interaction is averaged out. We present here a theoretical discussion of the features expected for the longitudinal-field signal of axially sym- metric Mu states in polycrystalline samples, and set-up an analysis framework that allows extracting the spin-flip rate from that type of data. 2. Spin exchange in polycrystalline media In longitudinal-field (LF) geometry, the non-oscillating time-dependent muon polarization of a single anisotropic Mu state undergoing random spin-flip events at a rate λ SF is an exponentially damped function [7] P (t)= a 0 e −λ L t , (1) with a relaxation rate given by [8] λ L = λ SF 4 ∑ n,m=1 ω 2 nm λ 2 SF + ω 2 nm a nm , (2) where ω nm are the Mu hyperfine transition frequencies, and a nm are the expected amplitudes of the corresponding precessions along the observation direction (see e.g. [9]). a 0 is the non-oscillating amplitude Preprint submitted to Elsevier 18 September 2008