Anisotropic muonium atoms: Energy levels and electron spin exchange

Abstract

The time evolution of the muon spin in fully anisotropic muonium $(\mathrm{Mu}={\ensuremath{\mu}}^{+}{+e}^{\ensuremath{-}})$ in the presence of Heisenberg spin exchange has been investigated theoretically. First, the energy levels of anisotropic Mu as a function of field are investigated analytically with a particular emphasis on the crossing and avoidance of energy levels at certain magnetic fields, which have important consequences in muon spin dynamics. Second, the knowledge of the energy levels is applied to investigate the muon spin depolarization due to electron spin exchange with spin-$\frac{1}{2}$ paramagnetic species, where the muon spin depolarization rate and the precession amplitude observed by the muon-spin-rotation (\ensuremath{\mu}SR) technique are explicitly expressed solely in terms of the matrix that diagonalizes the anisotropic Mu hyperfine Hamiltonian. The treatment presented here represents a special systematic and practical method that allows one to investigate the time evolution of the muon spin in anisotropic Mu in the presence of electron spin exchange. Several concrete examples are discussed in detail, including those in which all the \ensuremath{\mu}SR observables can be obtained analytically. The method developed in this work is used to explain the relaxation rate maximum in anisotropic Mu in semiconductors observed at the longitudinal fields at which two of the Mu energy levels avoid each other due to a strong level mixing or avoidance, where the present formalism takes the tensor nature of the anisotropic hyperfine interaction fully into account without invoking the convenient but not necessarily correct notion of an effective magnetic field in an anisotropic Mu. Also discussed is the possibility of observing additional relaxation maximum at a low-avoidance field, where the effective magnetic-field approximation completely breaks down. Observation of such a maximum will provide valuable information on the parameters characterizing the anisotropic Mu in question. The formalism presented here can also be applied to anisotropic positronium on surfaces, anisotropic Mu undergoing both charge exchange and spin exchange, and fast spin exchange.