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\begin{document}
\pagestyle{empty}
\LARGE
\begin{center}{\bf ABSTRACT}\end{center}
RF cavities for synchrotrons are not in general axially
symmetric. This can be due, for example, to the location of the input
power coupling loop. It can cause the voltage on one side of the
accelerating gap to be different from that on the other side.
Associated with this asymmetry is an rf magnetic field which deflects a
beam particle by an amount depending upon its rf phase. The deflection
can accumulate if the betatron tune is situated on a synchrotron
sideband of the integer resonance. We develop the theory for this
resonance and apply it to the KAON Factory Booster and to the SSC LEB.
We find that the upper limit on allowable voltage asymmetry across the
beam pipe is 0.1\% in both cases.
\newpage
At the first sideband $\nu_y\pm\nu_s=n$, emittance change per turn due
to voltage asymmetry at the rf gap:
\[{\delta\ey\over\ey}=
{\sin\ps\over\pi}\;
{e(\partial V/\partial y)(C/h)\over\beta^2\gamma(m_0c^2)}\;
\sqrt{\beta_{y\rm gap}\over\ey}\;
{\hat{\phi}\over 2}\]
As compared with emittance change per turn due to dispersion $D$ at the
rf gap:
\[{\delta\ey\over\ey}=
2\cos\ps\;
{eV_0\over\beta^2\gamma m_0c^2}\;
{D\over\sqrt{\beta_{y\rm gap}\ey}}\;
{\hat{\phi}\over 2}\]
Therefore, dispersion $D_V$ which gives same effect as gap voltage
asymmetry is
\[D_V={1\over V_0}\;
{\partial V\over\partial y}\;
{\beta_{y\rm gap}\over 2\pi}\;
{C\over h}\]
For both KAON Booster and SSC LEB, 2\% asymmetry across 15\,cm wide gap,
$\beta_{y\rm gap}=20$\,m, $C/h=5$\,m (50\,MHz rf), so
\[D_V=2.3\,{\rm m\,!}\]
\newpage
The perturbation term of the Hamiltonian can be expressed in the form
\[
\delta H_V=\\
{e\over\omrf}\,{\partial V\over\partial y}\,
\left[\sin\ps\,\Dp+{\cos\ps\over 2}(\Dp)^2\right]\,y\,\dep (s-s_0),
\]
for an rf gap at $s=s_0$.
\newpage
\begin{center}{\bf Examples}\end{center}
To illustrate the effect, we calculate the stopband widths
($\Delta e=\delta\ey/(2\pi\ey)$) in the single
cavity case for the TRIUMF KAON Factory Booster (racetrack design) and for the SSC Low
Energy Booster. These use similar 60\,kV, 50\,MHz rf cavity designs
whose uncorrected asymmetry has been found to be 2\% across the 15\,cm
beam pipe diameter. These
designs have similar injection energies (450 and 600\,MeV),
and relatively large synchrotron tunes (0.05 and 0.04) and quite
full buckets ($\hat{\phi}\approx\pi/2$) at these energies. In addition,
these machines have similar lattice designs where the rf cavities are
located in dispersionless straights with $\beta$-functions as large as
20\,m. The main difference between the two machines is in the rms
emittances: $\beta\gamma\pi\ey=15\pi$\,mm-mrad for KAON Booster and
$0.6\pi$\,mm-mrad for SSC LEB. The results are summarized in the
table.
\newpage
Stopband widths at the first and second sidebands for
\[{1\over V_0}{\partial V\over\partial y}=0.13\,{\rm m}^{-1}.\]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
Machine & $m=1$ & $m=2$ \\ \hline\hline
KAON Booster & 0.00045 & 0.00075 \\
SSC LEB & 0.00186 & 0.00095 \\ \hline
\end{tabular}
\end{center}
The maximum values of $\Delta e$ for $m=2$ occur right at injection.
The $m=1$ stopband, being proportional to $\sin\ps$, attains maximum
value somewhat after acceleration has started.
\end{document}