Radiative Spin Polarization in High Energy Storage Rings

TL;DR

The paper surveys radiative spin polarization in high-energy storage rings, highlighting that real machines with inhomogeneous fields and varied orbital distributions exhibit depolarizing spin resonances beyond the idealized uniform-field model. It develops a semiclassical QED framework and introduces the Derbenev–Kondratenko formula, linking spin-flip dynamics to phase-space averages and the gradient of the invariant spin field, $P_{ m DK}$, and its associated buildup time $ au_{ m DK}$. It then delves into both perturbative and nonperturbative methods to compute the invariant spin field $m{n}$ and the amplitude-dependent spin tune, including SLIM, SMILE, SODOM2, MILES, and stroboscopic averaging, and discusses spin matching, harmonic techniques, and topological phases that influence polarization. The work also connects to broader contexts such as Unruh-like effects, synchrotron-sideband resonances, and topological spin phenomena, emphasizing practical implications for energy calibration, beam polarization control, and accelerator design. Together, these insights provide a comprehensive toolkit for predicting and optimizing radiative polarization in real storage rings.

Abstract

The usual theoretical model for synchrotron radiation in circular accelerators (synchrotrons and storage rings) is to treat a single electron moving in a horizontal circle in a uniform vertical magnetic field, but the true situation in real storage rings is more complicated and exhibits much richer physics. The magnetic fields are inhomogeneous, and there is a bunch of many particles and they traverse a distribution of orbits (hence they encounter different magnetic fields). This results in so-called ``depolarizing spin resonances'' (which do not appear in a simple model of a uniform vertical magnetic field). The calculation of the equilibrium electron spin polarization requires a much more careful analysis. For example, a key insight is that, for motion in inhomogeneous magnetic fields, ``spin flip'' is in general \emph{not} a $180^\circ$ reversal of the spin orientation. The physics of radiative spin polarization involves a mix of many disciplines, and provides a good example of cross-disciplinary thinking. We shall also briefly note the connection to astrophysics. The astrophysics literature mainly treats electron motion in very strong magnetic fields, stronger than the Schwinger critical field (for example a neutron star). It is a problem of ongoing interest in astrophysics to study the radiation by electrons circulating in such strong magnetic fields. This article aims to provide the reader with a survey of the basic physics principles of radiative spin polarization, omitting low-level mathematical algebra as much as possible. Such details can be found in the literature, and are not relevant here.

Figures (4)

• Figure 1: Theoretical fit of the SPEAR polarization data. From ManeSPEARfit.
• Figure 2: Graph of the asymptotic polarization $P_{eq}$ as a function of $g$, for a model of circular motion in a uniform vertical magnetic field. For $0 < g \le 1.2$, the polarization is negative, i.e. the naıvely higher-energy spin state is preferentially populated. The range is indicated by the horizontal arrow.
• Figure 3: Graph of the polarization buildup time $\tau$ (actually the ratio $\tau/\tau_{\rm ST}$) as a function of $g$, for a model of circular motion in a uniform vertical magnetic field.
• Figure 4: Top panel: graph of $\nu-Q$ vs. $\nu_0-Q$, where $\nu$ is the spin tune, $\nu_0$ is the unperturbed spin tune and $Q$ is the tune of the orbital mode. The arrow indicates the jump in the value of the spin tune at the spin resonance $\nu_0=Q$. The dashed curve plots the value of $\nu_0-Q$. Bottom panel: graph of $n_3$, the vertical component of the spin quantization axis $\bm{n}$, vs. $\nu_0-Q$.