Spatial Effects in the Theory of the Phonon Avalanche

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Abstract
A set of coupled, nonlinear integro-differential equations is proposed to describe the space-time variation of the spin and phonon excitations in a paramagnetic crystal under conditions appropriate to a phonon avalanche. An effective phonon lifetime is derived and found to exceed the simple time-of-flight lifetime by orders of magnitude. For the case of an infinite slab, the theory is shown to be capable of remedying the inability of earlier models to describe the time evolution of the average magnetization.