Polarization of Electron Beams

As first mentioned by Ternov, Loskutov and Korovina in 1961 electrons gradually polarize in storage rings due to sustained transverse acceleration while orbiting. The mechanism is the emission of spin-flip synchrotron radiation: While being accelerated, electrons radiate electromagnetic waves in quanta of photons which carry a spin.

Therefore two cases must be distinguished: After emitting the synchrotron photon the electron spin stays in its initial state or flips over. It has been shown [6] that only an extremely small fraction ($\sim10^{-11}$) of the emitted power is due to spin-flip radiation, the large fraction of other synchrotron emissions has no influence on the electron's spin. Nevertheless, the process of spin-flip radiation is crucial in order to understand the meaning of beam polarization.

The transition rates for the two possible spin-flips have been calculated [6] [7] to be:

$$\ W_{\uparrow \downarrow}=\frac{5\sqrt{3}}{16}\cdot\frac{e^2... ...r}{{m_{e}}^2c^2\rho^3}\cdot\left(1+\frac{8}{5 \sqrt{3}}\right)$$ (1)

$$\ W_{\downarrow \uparrow}=\frac{5\sqrt{3}}{16}\cdot\frac{e^2... ...r}{{m_{e}}^2c^2\rho^3}\cdot\left(1-\frac{8}{5 \sqrt{3}}\right)$$ (2)

where $\gamma$ is the Lorentz factor and $\rho$ is the instantaneous bending radius. The symbol $\uparrow$ denotes the spin along the guiding dipole field, whereas the symbol $\downarrow$ denotes a spin against the guiding dipole field.

The difference between these two rates causes an injected electron beam to get polarized anti-parallel with respect to the guiding dipole field (for positrons the accumulated polarization would be along the guiding dipole field). The maximum achievable polarization level in a planar ring without imperfections is given by:

$$\ P_{ST}=\frac{W_{\uparrow \downarrow}-W_{\downarrow \uparro... ...narrow}+W_{\downarrow \uparrow}}=\frac{8}{5 \sqrt{3}}= 92.38\%$$ (3)

where $P_{ST}$ is the Sokolov-Ternov Level of polarization [8]. The time constant of the exponential build-up process of this equilibrium polarization by the initially unpolarized beam is:

$$\tau_p={\left(W_{\uparrow \downarrow}+W_{\downarrow \uparrow... ...2\hbar}{{m_{e}}^2c^2}\right)}^{-1}\cdot\frac{\rho^3}{\gamma^5}$$ (4)

In case of the SLS storage ring $\tau_p$ is roughly 31 minutes. So finally, we would expect polarization build-up to be described by:

$$P(t)=P_{ST}\left(1-\exp\left(-\frac{t}{\tau_p}\right)\right)$$ (5)

Meaning that an unpolarized beam can be injected into the SLS storage ring and after roughly an hour over 85% of the beam will be fully polarized anti-parallel to the guiding dipole field!

However we have (until now) neglected diffusion effects on spin orientation. Similar to the fact that radiation damping does not lead to dimensionless beam size due to quantum emission, spin-flip radiation is accompanied by the depolarizing effect of spin diffusion. Thus the measured beam polarization is an equilibrium state and shall be examined in the next chapters.