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\begin{document}
\begin{center}
\huge
\bf
CORRECTION OF $\nu_r=3/2$ RESONANCE IN THE TRIUMF CYCLOTRON
\end{center}
\author{T. Planche, R. Baartman, Y.-N. Rao}
\address{TRIUMF, Vancouver, Canada}
%\email{\{krab, raoyn\}@triumf.ca}
\makeheader
\begin{center}
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\paragraph{INTRODUCTION}
The TRIUMF cyclotron is 6-fold symmetric and ideally its magnetic field
will have only multiples of 6$^{\rm th}$ harmonic. The third harmonic in
the magnetic field gradient drives the $\nu_r=3/2$ resonance. This
results in a modulation of the current density versus radius observed
after the resonance crossing all the way to the extraction (480 MeV).
The cyclotron has sets of harmonic correction coils at different radii,
each set constituted of 6 pairs of coils placed in a 6-fold symmetrical
manner. The 6-fold symmetry of this layout cannot provide a third
harmonic of arbitrary phase. However, the outer two sets of harmonic
correction coils (number 12 and 13) are azimuthally displaced. We use
this fact to achieve a full correction of the resonance. This has the
effect of reducing intensity fluctuations in the extracted beams, when
there is beam sharing between two beamlines.
\paragraph{THIRD HARMONIC FIELD ERROR}
\vspace{5mm}
\begin{figure}
\begin{minipage}[t]{24cm}
\includegraphics[width=15cm,angle=90]{f1}
\caption{
Fourier coefficients $H_3$ and $G_3$ of the
third harmonic component in the base magnetic field,
in a region of radius between 288 and 306\,inch.
Also shown is its amplitude and phase angle.
The $\nu_r=3/2$ resonance is driven primarily by the
radial gradient of the third harmonic amplitude,
which is $\sim0.1$\,G/inch.
}
\label{ff1}
\end{minipage}
\end{figure}
}
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\paragraph{DENSITY MODULATION}
In $(r,P_r)$ phase space, the beam particles occupy an ellipse. From one
turn to the next, the particles move around the ellipse by an angle of
$2\pi\nu_r$, but the ellipse orientation remains constant. This ellipse
is called the ``matched ellipse''.
The third harmonic magnetic field gradient error causes perturbations
away from this matched ellipse, but in the general case, the particle
motion is out of step with this frequency and the perturbations average
to zero. However, when $\nu_r$ is precisely $3/2$ (428\,MeV), the
perturbations accumulate and the initially matched ellipse becomes
stretched.
As the beam is accelerated away from the resonance, the
ellipse, which is now mismatched: it no longer matches the ``matched
ellipse'', and starts to rotate as a propeller. Such a rotation leads to
a modulation of beam radial density.
\begin{figure}
\begin{minipage}[t]{24cm}
\includegraphics*[width=15cm, angle=90]{f2}\\
{\textcolor{white}{pp}} \includegraphics*[width=15cm, angle=90]{f3}
\caption{ Simulation results.
Top: turn-by-turn variation of the radial phase space
passing through the $\nu_r=3/2$ resonance.
Clearly, one can see the rotation (precession) of the ellipse.
Bottom: radial modulation of beam density, caused by the
precession of radial ellipses. Here we only illustrate the first
2 periods of precession and modulation; in fact, the precession and
density modulation persist to 480\,MeV extraction.
}
\label{ff2}
\end{minipage}
\end{figure}
\vspace{5mm}
The beam is extracted by placing a stripping foil at the radius
corresponding to the desired beam energy. When beam is shared at one
energy between two beam lines, the proportion received in each beamline
(the ``split ratio'') depends sensitively on the radial density. Thus,
when there are modulations in the radial density, small fluctuations in,
for example, the rf accelerating voltage result in relatively and
undesirably large fluctuations in the ``split ratio''. }
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\paragraph{RESONANCE CORRECTION}
The TRIUMF cyclotron is equipped with 13 sets of harmonic correction
coils; the outer-most two (numbers 12 and 13) can render the third
harmonic field gradient in the energy region of the 3/2 resonance.
However, with either set of coils, we cannot change the third harmonic
phase angle except for a switch of $180^{\circ}$, so cannot fully
compensate the resonance. But, using these two coils in combination, we
can change the phase angle. This is because these 2 sets of coils are
displaced azimuthally; the displacement is only $11^\circ$, but this is
sufficient.
The scheme is as follows. We wish to create a third harmonic field of amplitude $A$ and
phase $\phi$ from coils (U and V) which have fixed phase of zero and
$\delta=3\times 11^\circ$ respectively, and amplitudes $U$ and $V$
respectively. Then by the sine law, we have
\begin{equation}
\frac{U}{\sin(\delta-\phi)}=\frac{V}{\sin\phi}=\frac{A}{\sin\delta}
\end{equation}
Ideally, the desired field for arbitrary phase is most efficiently
obtained if $\delta=\pi/2$. Since in fact $\delta=33^\circ$, the coils
act partly in opposition to each other and their strengths are a factor
$\csc33^\circ=1.84$ higher than the ideal arrangement. Consequently, we have recently
upgraded the power supplies for harmonic coils 12 and 13 to higher
current. The results are shown below.
\vspace{5mm}
\begin{figure}
\begin{minipage}[t]{24cm}
\includegraphics*[angle=90, width=22cm]{f4}
\includegraphics*[width=24.cm]{f5}
\caption{ Current density radial modulation without correction, with
correction using HC\#13 only, and with correction using
HC\#12 and \#13 in combination.
Top: simulation result. Bottom: measured result.
Clearly, the full correction results in greatly reduced
density modulation.
}
\label{ff3}
\end{minipage}
\end{figure}
}
\end{center}
\end{document}
%===========================================
% HC12 spans from 272" to 297", while
% HC13 spans from 297" to 323".
%
% HC12 is centered at 282 degr,
% HC13 is centered at 293 degr.
%===========================================