TUNNEL EFFECT IN MAGNETIC SYSTEMS - FROM THE MICROSCOPIC DESCRIPTION TO THE MASTER EQUATION

Spin relaxation in a solid depends sometimes on tunneling between two excited crystal field states \a) and \b). The calculation of the tunne ling frequency omega(ab) at resonance in the absence of damping is a c lassical problem. In the presence of damping, one can deduce the relax ation time from master equations in which tunneling appears through th e transition probabilty Gamma(ab) between \a] and \b]. We derive the r esult Gamma(ab) = 2 omega(ab)(2) tau(ab)/1 + tau(ab)(2)(E-a-E-b)(2)/(H ) over bar(2)], where E-a-E-b is a linear function of the magnetic fie ld, tau(ab) = tau(a) tau(b)/(tau(a) + tau(b)) and tau(a) and tau(b) ar e the respective lifetimes of \a] and \b]) which depend on the spin-ph onon interaction and can be calculated in the absence of tunneling. A mechanical analogy provides an intuitive picture of the phenomenon. Th e master equations are identical to the equations which describe the d ischarge of condensators through an electric network.