The Bloch equation for spin dynamics in electron storage rings: Computational and theoretical aspects

Abstract

In this paper, we describe our work on spin polarization in high-energy electron storage rings which we base on the Full Bloch equation (FBE) for the polarization density and which aims towards the [Formula: see text] option of the proposed Future Circular Collider (FCC-ee) and the proposed Circular Electron Positron Collider (CEPC). The FBE takes into account non spin-flip and spin-flip effects due to synchrotron radiation including the spin-diffusion effects and the Sokolov–Ternov effect with its Baier–Katkov generalization as well as the kinetic-polarization effect. This mathematical model is an alternative to the standard mathematical model based on the Derbenev–Kondratenko formulas. For our numerical and analytical studies of the FBE, we develop an approximation to the latter to obtain an effective FBE. This is accomplished by finding a third mathematical model based on a system of stochastic differential equations (SDEs) underlying the FBE and by approximating that system via the method of averaging from perturbative ODE theory. We also give an overview of our algorithm for numerically integrating the effective FBE. This discretizes the phase space using spectral methods and discretizes time via the additive Runge–Kutta (ARK) method which is a high-order semi-implicit method. We also discuss the relevance of the third mathematical model for spin tracking.

Reevaluation of spin-orbit dynamics of polarized e+e− beams in high energy circular accelerators and storage rings: An approach based on a Bloch equation

“…6 from the laboratory frame to the beam frame. 33 However, since in this section we only deal with the RBE (not with the BE), ( 28) and ( 29) can also be found in older expositions on spin in high-energy electron storage rings, e.g., Ref. 34.…”

“…where P DK (∞) is given by (6) and where the rhs of ( 9) is the approximate equilibrium polarization vector.…”

“…We now sketch three approaches for computing P DK (∞) via the Derbenev-Kondratenko formulas. All three approaches use (6) DK via (11) where τ −1 dep is not computed via (10) but via Monte-Carlo spin tracking and where the remaining terms in (11) are approximated by using then 0 -axis. 4 (iii) Compute τ −1 0 via (8) and τ −1 DK via ( 7) by linear approximation in orbit and spin variables via the so-called SLIM formalism.…”

“…where P lab satisfies the Fokker-Planck equation corresponding to the system of SDEs in Ref. 6. These SDEs can be used as the basis for a Monte-Carlo spin tracking algorithm, i.e., for Method 2 mentioned in Section 1 above.…”

“…More details on Method 1 can be found in Ref. 6. The method of averaging we use is discussed in Refs.…”