Theory of muon-spin relaxation of simple gaseous free radicals

The relaxation of the muon spin in a free radical is complicated by the presence of several molecular degrees of freedom. In particular, for muonated ethylene, ${\mathrm{C}}_{2}$${\mathrm{H}}_{4}$Mu, which exemplifies the type of free radical that is being considered, the addition product has an unpaired electron, so the motion of the free radical in an external magnetic field is dominated by the precession of the electron spin. This paper formulates a theory for the muon-spin relaxation of such free radicals. It is based on the Boltzmann equation, so it has a rigorous basis by which both free motion (including all of the angular momentum couplings) and the collisional processes (which cause the decay to equilibrium) can be included. A multipole expansion of the density operator in terms of electron, muon, and rotational angular-momentum operators including terms up to quadratic in the rotational angular momentum is used to represent the state of the system. The multipole coefficients in the Hamiltonian are treated as fitting parameters to give a best interpretation of the experimental data for the longitudinal muon-spin relaxation rate of ${\mathrm{C}}_{2}$${\mathrm{H}}_{4}$Mu.