Spin And Polarization

The spin vector of a relativistic particle moving through a magnetic field satisfies the BMT equation:

(1)
\begin{align} \frac{d\vec S}{dt} = \Omega \times \vec S \end{align}

The angular frequency is given by (Wolski, manual for SAMM)

(2)
\begin{align} \Omega = -\frac{q}{\gamma m}\left[(1+a\gamma)\vec B_\perp + (1+a)\vec B_{||}-\left(a+\frac{1}{1+\gamma}\right)\gamma \beta \frac{\vec E}{c}\right] \end{align}

The spin tune for a perfect machine is given by

(3)
\begin{align} \nu_{spin} = a\gamma \end{align}

The beam is polarized via the Sokolov Turnov effect. The maximum value is

(4)
\begin{align} P_{ST} = \frac{8}{5\sqrt{3}} \end{align}

and the polarization time is given by (Leeman, Master's thesis)

(5)
\begin{align} \tau_{ST}= \left(\frac{5\sqrt{3}}{8}\frac{e^2\hbar}{m^2 c^2}\right)^{-1}\frac{\rho^3}{\gamma^5} \end{align}
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