18-22 October 2021
Matsue, Shimane Prefecture, Japan
Asia/Tokyo timezone

Spin depolarization in high energy electron storage rings: stochastic differential equations, the reduced Bloch equation and the invariant-spin-field approximation

22 Oct 2021, 08:40
Room 305-306 (Kunibiki Messe)

Room 305-306

Kunibiki Messe

Parallel Session Presentation Acceleration, Storage and Polarimetry of polarized Beams Acceleration, Storage and Polarimetry of polarized Beams


Dr Oleksii Beznosov (University of New Mexico (UNM), Department of Department of Mathematics and Statistics)


A spin distribution in a high energy $e^{+}$ or $e^{-}$ storage ring is governed by a $(6+1)$ dimensional linear Fokker-Planck-type equation known as the full Bloch equation. Its reduction, the reduced Bloch equation (RBE), is the spin-orbit diffusion equation which takes into account just those terms in the spin-orbit dynamics that suffice for calculating the spin-depolarization rate. While the RBE gives the full picture of the depolarization phenomenon, it has limitations, both in terms of analysis and numerical simulations. Here we introduce a framework for studying the Bloch equation based on a stochastic differential equation (SDE). The framework combines SDEs, modelling the spin-orbit motion of the electrons(positrons), Fokker-Planck type equations for orbit and spin-orbit density functions, and approximations for studying them. The SDEs in the framework are derived from the Lorenz force law coupled with Thomas-Bargmann-Michel-Telegdi spin precession and, as the name suggests, include a stochastic process modelling the effect of synchrotron radiation. This SDE framework reproduces the known equations modelling the depolarization phenomenon, like the Bloch equation and the Derbenev-Kondratenko (DK) formulas. It can also be used, as we do here, either to generalize the DK formulas and the Bloch equation, or to simplify the latter to a form that makes it amenable for solution by numerical methods.

The Bloch equation models the spin distribution as a three-scale time dependent process. The first scale comes from the rapidly varying transverse motion, governed by accelerator optics; the second scale arises from the classical spin precession, with rates varying with reference energy of the electron bunch; and finally the synchrotron motion, which is slowly varying yields the last scale. Coupling between the synchrotron motion and transverse betatron motion introduces spin-orbit resonances that have a devastating effect on spin-polarization in most optical settings and reference energies. Multiple scales involved require hybrid approaches. We will present three approaches to approximate solutions of the Bloch equation that can be hybridized within the SDE framework. The first approach uses perturbation theory and delivers the results consistent with formalism previously derived from QED. The second approach is built on averaging analysis of the slowly varying form of the SDEs and brings in the effective models to study spin-depolarization. The third approach is the numerical integration of the Bloch equations with a spectral method based on Fourier-Chebyshev expansions.

Finally, at the end of talk we use a simple model that contains most of the difficulties present in equations for a realistic accelerator. We will demonstrate the efficacy of the SDE framework and the approximations. We finish by presenting an approach in which coefficients in these equations are averaged to produce a so-called effective Bloch equation and then present a scheme for the numerical integration of the latter.

Primary authors

Dr Oleksii Beznosov (University of New Mexico (UNM), Department of Department of Mathematics and Statistics) Dr Desmond Barber (Deutsches Elektronen-Synchrotron (DESY)) Prof. James Ellison (University of New Mexico (UNM), Department of Department of Mathematics and Statistics) Prof. Klaus Heinamann (University of New Mexico (UNM), Department of Department of Mathematics and Statistics)

Presentation Materials