Matrix formalism for spin dynamics near a single depolarization resonance

A matrix formalism is developed to describe the spin dynamics in a synchrotron near a single depolarization resonance as the particle energy (and therefore its spin precession frequency) is varied in a prescribed pattern as a function of time such as during acceleration. This formalism is first applied to the case of crossing the resonance with a constant crossing speed and a finite total step size, and then applied also to other more involved cases when the single resonance is crossed repeatedly in a prescribed manner consisting of linear ramping segments or sudden jumps. How repeated crossings produce an interference behavior is discussed using the results obtained. For a polarized beam with finite energy spread, a spin echo experiment is suggested to explore this interference effect.