Spin decoherence in electron storage rings --- more from a simple model

This is an addendum to the paper "Some models of spin coherence and decoherence in storage rings" by one of the authors [1] in which spin diffusion in simple electron storage rings is studied. In particular, we illustrate in a compact way, a key implication in the Epilogue of [1], namely that the exact formalism of [1] delivers a rate of depolarisation which can differ from that obtained by the conventional treatments of spin diffusion which rely on the use of the derivative $\partial \hat n/\partial\eta$ [2,3,4]. As a vehicle we consider a ring with a Siberian Snake and electron polarisation in the plane of the ring (Machine II in [1]). For this simple setup with its one-dimensional spin motion, we avoid having to deal directly with the Bloch equation [5,6] for the polarisation density. Our treatment, which is deliberately pedagogical, shows that the use of $\partial \hat n/\partial\eta$ provides a very good approximation to the rate of spin depolarisation in the model considered. But it then shows that the exact rate of depolarisation can be obtained by replacing $\partial \hat n/\partial\eta$ by another derivative as suggested in the Epilogue of [1], while giving a heuristic justification for the new derivative.