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\begin{document}
\title{SYNCHRO-BETATRON RESONANCE DUE TO GAP VOLTAGE ASYMMETRY}
\author{R. BAARTMAN\\
\it TRIUMF\\
\it Vancouver, B.C. Canada\\}
\maketitle
\begin{abstract}
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.
\end{abstract}
\section{Introduction}
Longitudinal-transverse coupling occurs when the energy gained by a
particle crossing an rf gap depends upon its transverse position. Since
the particle motion is symplectic, there is an associated transverse
deflection which depends upon rf phase. In proton machines, this
deflection is much more important than the longitudinal effect since the
longitudinal emittance is much larger than the transverse emittance.
\section{Theory}
We begin by using a Hamiltonian formalism to solve a well-known problem
in accelerator physics, namely, the calculation of the phase-dependent
focusing by an rf gap. Then we use the same formalism to find the
effect of an up-down asymmetryin an rf gap.
\subsection{Focusing by an RF Gap}
It is well-known that the variation of energy gain across an axially
symmetric rf gap can be described by the factor\cite{lapo}
\beq\label{fact}
{V(r)\over V_0}=I_0\left({\omrf r\over\beta\gamma c}\right).
\eeq
This is the ratio of energy gain at a distance $r$ from the beam axis to
the energy gain on-axis. For synchrotrons, the argument of the modified
Bessel function $I_0$ is very small so we write eq.\,\ref{fact} as
\beq\label{fact2}
{V(r)\over V_0}=1+\left({\omrf r\over 2\beta\gamma c}\right)^2=
1+\left({\pi r\over \gamma L}\right)^2,
\eeq
where $L$ is the distance between rf buckets (= circumference /
harmonic number).
The term in the Hamiltonian containing the rf gap is
\beq
H_V=-{\cos\phi\over\omrf}eV\dep (s-s_0),
\eeq
where $s$, the distance along the closed orbit, is the independent
variable, $s_0$ is the location of the gap, and $\dep$ is the periodic
$\delta$-function with period equal to the machine circumference. $V$
is the gap voltage and the phase $\phi$ of the particle with respect to
the rf is a canonical variable conjugate to $W=\Delta E/\omrf$.
Including in $V$ the factor (\ref{fact}) results in an rf term which
now depends also on transverse displacement. Differentiating $H_V$ with
respect to $y$, we find that there is a transverse kick at the gap of
\beq\label{kick}
\Delta p_y={\cos\phi\over\omrf}e{\partial V\over\partial y},
\eeq
which can be written as a phase-dependent focal length $f$ given by
\beq
{1\over f}=-{\pi\over L}{eV_0\over m_0c^2}{\cos\phi\over\beta^2\gamma^3}.
\eeq
For synchrotrons, this is never an important effect since for
low-$\gamma$ machines, both the gap voltage and the rf frequency are
small compared with those of linacs.
\subsection{Effect of Asymmetry}
At low energy ($\gamma\approx 1$) the difference in energy gain between
the on-axis particle and a particle near the beam pipe wall due to the
rf focusing effect (\ref{fact2}) is only $\sim 0.1$\% even for the
relatively high (for protons) rf frequency of 50\,MHz. In contrast, the
asymmetry in the rf cavity can cause a difference of a few percent in
the gap voltage from one side of the gap to the other, if the cavity is
not well-designed and constructed. Clearly then, the effect due to
asymmetry is the more important one.
To investigate this effect, we assume the first order (dipole) form
\beq
V=V_0(1+\xi y).
\eeq
Then the transverse kick (\ref{kick}) given to a particle of phase
$\phi$ is
\beq
\Delta p_y=eV_0\xi\cos\phi/\omrf.
\eeq
As the particle circulates, it receives such a kick every turn.
We are interested in whether or not there is a cumulative effect.
Writing the phase as $\ps+\Dp$, $\cos\phi$ can be expanded in
a power series in $\Dp$. The constant term $\cos\ps$ only contributes a
closed orbit distortion. The $m^{\rm th}$ term causes an accumulation
of transverse kicks if the betatron and synchrotron tunes
satisfy\footnote{In principle, there is a difference between the upper
and lower signs in this resonance condition: in one case transverse
emittance grows at the expense of longitudinal emittance, while in the
other case both grow without limit. In practice, however, the
longitudinal emittance is usually so much larger than the transverse
emittance, that this distinction is unimportant.}
\beq\label{res}
\nu_y\pm m\nu_s=n,
\eeq
where $n$ is an integer. We expand $\cos\phi$ only up to second order
\beq
\cos\phi-\cos\ps\approx -\sin\ps\Dp-{\cos\ps\over 2}\left(\Dp\right)^2,
\eeq
In this way we ignore non-linear synchrotron motion which drives the
sidebands $m>2$. These higher order resonances can in principle also be
analyzed but they are quite complicated because we must for
self-consistency include the fact that $\Dp$ does not oscillate
harmonically\cite{baa1}.
The perturbation term of the Hamiltonian can be expressed in the form
\beq
\delta H_V={eV_0\xi\over\omrf}
\left(\alpha_1\Dp+\alpha_2(\Dp)^2\right)y\dep (s-s_0),
\eeq
where $\alpha_1=\sin\ps$ and $\alpha_2=\cos\ps/2$. Following
Suzuki \cite{suzu}, we Fourier expand $\dep$ retaining only the $n^{\rm
th}$ term, substitute action-angle variables for $\Dp$ and $y$, and
differentiate with respect to the betatron angle to find the rate of
change of the betatron action. The result in terms of the fractional
change per turn in betatron emittance $\ey$ is
\beq\label{growth}
{\delta\ey\over\ey}=
{\alpha_m\over\pi}{eV_0\xi L\over\beta^2\gamma m_0c^2}
\sqrt{\beta_{y0}\over\ey}\left({\hat{\phi}\over 2}\right)^m
\sin(2\pi\epsilon s/C+\psi_0),
\eeq
where $\epsilon=\nu_y\pm m\nu_s-n$, $C$ is the machine circumference
and $\psi_0$ is a phase angle (containing betatron and synchrotron
angles). Here we have introduced $\hat{\phi}$ as the phase oscillation
amplitude and $\beta_{y0}$ as the $\beta$-function at the rf gap. One
can obtain the stopband width from eq.\,\ref{growth} by dropping the sine
and dividing by $2\pi$.
So far we have looked only at the single cavity effect. The
generalization to many cavities is straight-forward \cite{suzu}: the
main effect is that the betatron phase advance between cavities becomes
important. The result is that $eV_0\xi\sqrt{\beta_{y0}}$ in
eq.\,\ref{growth} is replaced by
\beq
\left[
\left(\sum_jeV_j\xi_j\sqrt{\beta_{yj}}\cos n\phi_{yj}\right)^2+
\left(\sum_jeV_j\xi_j\sqrt{\beta_{yj}}\sin n\phi_{yj}\right)^2
\right]^{1/2}\,,
\eeq
where the index $j$ applies to the $j^{\rm th}$ rf gap. The angle
$\phi_{yj}$ is the betatron oscillation angle divided by the tune,
making it approximately equal to the angular location of the $j^{\rm
th}$ gap: i.e.,
\beq
\phi_{yj}=\int_0^{s_j}ds/\beta_y/\nu_y\approx 2\pi s_j/C.
\eeq
As pointed out in previous work \cite{suzu}, it is possible to cancel
the synchro-betatron effect by a proper arrangement of the rf cavities.
For example, if all cavities have identical asymmetry, a good
arrangement is to place them in pairs either together
right-side-up-upside-down, or right-side-up and an odd number of half
betatron oscillations apart. Other arrangements in which the tune is
not equal to an integer multiple of the rf-superperiodicity are also
effective, however, pairwise arrangement is preferable because then if
one cavity fails, only one other cavity needs to be turned off.
Unavoidably, the different cavities will not have exactly the same
voltages so there will always be some remanent effect.
\section{Examples}
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\cite{roger}; i.e., $\xi=0.13\,{\rm
m}^{-1}$. These designs have similar injection energies (resp.\ 450\,MeV
and 600\,MeV), and relatively large synchrotron tunes (resp.\ 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 the KAON Booster and
$0.6\pi$\,mm-mrad for the LEB. The results are summarized in Table 1.
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.
\begin{table}[hb]
\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}
\caption{Stopband widths at the first and second sidebands for
$\xi=0.13\,{\rm m}^{-1}$.}
\end{center}
\end{table}
This is clearly a dangerous effect, especially so considering that the
calculation was done for only one of the 12 cavities. Moreover, the
effect at $m>2$ sidebands will also be large, not only because of
nonlinear synchrotron oscillations, but more importantly because of
betatron tune modulation\cite{baa2}.
Given the sensitivity of the effect to cavity distribution, tune
modulation, and that parameters like tune-spread and synchrotron tune
change continuously during acceleration, it is difficult to make
quantitative predictions in terms of beam quality. Nevertheless, it is
simple to make a quantitative comparison with the synchro-betatron
effect due to dispersion at the locations of the rf gaps. This is so
because the two effects are formally very similar. The formula for
emittance growth due to dispersion ($D$ and its derivative $D'$)
is\cite{suzu}
\beq\label{dgrowth}
{\delta\ex\over\ex}=
2\alpha_m^*{eV_0\over\beta^2\gamma m_0c^2}
{A\over\sqrt{\beta_{x0}\ex}}\left({\hat{\phi}\over 2}\right)^m
\sin(2\pi\epsilon s/C+\psi_0),
\eeq
where $A=\sqrt{D^2+(\alpha_xD+\beta_xD')^2}$ and $\alpha_1^*=\cos\ps$
and $\alpha_2^*=\sin\ps/2$. This is to be compared with
eq.\,\ref{growth}. The main difference between the two formulas is that
in the dispersion synchro-betatron effect growth rates at odd sidebands
are proportional to $\cos\ps$ and at even sidebands to $\sin\ps$,
while in the voltage asymmetry case it is the other way around. This
arises because the transverse kick due to dispersion is proportional to
$\sin\phi-\sin\ps$ whereas that due to voltage asymmetry is
proportional to $\cos\phi$.
In an approximate comparison between the two effects, we ignore the
difference in the dependence upon $\ps$. Then we find that the
dispersion $D_\xi$ at a cavity which gives the same effect as the
voltage asymmetry is
\beq
D_\xi={\xi\beta_{y0}L\over 2\pi}.
\eeq
Using $\xi=0.13\,{\rm m}^{-1}$, $\beta_{y0}=20\,$m, and $L=5\,$m,
as appropriate for both the KAON Booster and the SSC LEB, we find
\beq
D_\xi=2.3\,{\rm m}.
\eeq
Ideally one would like to reduce synchrobetatron effects to the level
where there is no constraint on the distribution of rf cavities and
their voltages and no constraint to keep the betatron tune well away
from an integer. To meet this goal in the TRIUMF rings and in the SSC
LEB, an effort was made in the lattice designs to keep the dispersion at
the rf cavities below 0.1\,m. Since a 2\% voltage asymmetry gives the
same effect as a dispersion of 2.3\,m, consistency demands that
the asymmetry be reduced to below 0.1\%.
\begin{thebibliography}
\noindent
\bibitem{lapo}P.\,Lapostolle {\it Introduction \`{a} la Th\'{e}orie des
Ac\-c\'{e}l\-\'{e}r\-a\-teurs Lin\'{e}aires} CERN 87-09.
\bibitem{baa1}R.\,Baartman {\it Synchrobetatron Resonance Driven by
Dispersion in RF Cavities: A Revised Theory} TRIUMF Internal Report
TRI-DN-89-K40.
\bibitem{suzu}T.\,Suzuki {\it Synchrobetatron Resonance Driven by
Dispersion in RF Cavities} Particle Accelerators {\bf 18}, pp.\ 115-128
(1985).
\bibitem{roger}R.\,Poirier, Private Communication (1992).
\bibitem{baa2}R.\,Baartman and U.\,Wienands {\it Synchro-Betatron
Resonances in the Presence of Tune Modulation} Proc.\ EPAC 90, pp.\
1627-1629.
\end{thebibliography}
\end{document}