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

TL;DR

This paper demonstrates that an exact formalism for spin depolarisation in electron storage rings can differ from conventional methods, and shows how a modified derivative provides a more accurate rate in a simplified, pedagogical model.

Contribution

It introduces a simple, pedagogical model to compare the exact formalism with conventional spin diffusion treatments, highlighting the improved accuracy of a modified derivative for depolarisation rates.

Findings

The exact formalism can differ from conventional treatments.

Replacing the derivative with a modified one yields a more accurate depolarisation rate.

The simplified model confirms the heuristic suggested in previous work.

Abstract

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.