\section{Electric Field Correction}
\label{sec:efield}
The magnetic field $\bfB = B_z\bfhz$ and the electric field is in the radial direction
We assume no betatron oscillations so the particle circulates on its closed orbit.
The electric field is
\begin{equation}
\bfE = E_r(\sin\theta\ \bfhx +\cos\theta \bfhy).
\end{equation}
The velocity has the form
\begin{equation}
\bm\beta =\beta(\cos\omega t\ \bfhx-\sin\omega t \bfhy)
\end{equation}
Substitution into the Lorentz force law (Equation~\ref{eq:eom} gives us
\begin{eqnarray}
-\omega\beta(\sin\omega t\ \bfhx+\cos\omega t \bfhy)
&=&\frac{e}{mc\gamma}\left(E_r(\sin\omega t\ \bfhx+\cos\omega t \bfhy) - \beta B_z(\sin\omega t\ \bfhx+\cos\omega t \bfhy)\right)\nonumber\\
\rightarrow \omega&=&\frac{e}{mc\gamma}\left(\frac{E_r}{\beta} - B_z\right)\label{eq:omegae}
\end{eqnarray}
The effective magnetic field
\begin{equation}
{\bf\cal B}= \left[B_z (a_\mu+\frac{1}{\gamma}) -\left(a_\mu-\frac{1}{\gamma+1}\right)\beta E_r\right]\bfhz
\end{equation}
and the Hamiltonian
\begin{eqnarray*}
{\cal H}= -\frac{\hbar}{2}\begin{pmatrix}\omega_0 & 0 \cr 0 & -\omega_0\end{pmatrix}
\end{eqnarray*} where
\begin{eqnarray*}
\omega_0&=&\frac{e}{mc}\left(B_z (a_\mu+\frac{1}{\gamma}) -\left(a_\mu-\frac{1}{\gamma+1}\right)\beta E_r\right)
\end{eqnarray*}
In the rotating frame
\begin{equation}
{\cal H}_{rot}= -\frac{\hbar}{2}\begin{pmatrix}\omega_0-\omega & 0 \cr 0 &\omega-\omega_0\end{pmatrix}
\end{equation}
with $\omega$ given in Equation~\ref{eq:omegae}.
The precession frequency in the rotating frame
\begin{eqnarray*}
\omega^\prime &=&\omega_0-\omega\\
&=& \frac{e}{mc}\left(B_z a_\mu -\left(a_\mu-\frac{1}{\gamma+1}+\frac{1}{\beta^2\gamma}\right)\beta E_r\right)\\
&=& \frac{e}{mc}\left(B_z a_\mu -\left(a_\mu-\frac{1}{\gamma+1}+\frac{\gamma}{\gamma^2-1}\right)\beta E_r\right)\\
&=& \frac{e}{mc}\left(B_z a_\mu -\left(a_\mu-\frac{1}{\gamma^2-1}\right)\beta E_r\right)
\end{eqnarray*}
The contribution to the precession frequency in the rotating frame vanishes at the magic momentum.
