\hyphenation{quad-ru-poles}
\section{Intensity Limitations in Cyclotrons}
\subsection*{Introduction}
The 500\,MeV TRIUMF cyclotron's limitation on peak intensity is due to
repulsive space charge forces overpowering the vertical focusing forces
in the central region where beam energy is lowest. For upgrading to
higher intensity, it is necessary to understand the limitation.
%\subsection*{Timeline}
The vertical section of the injection line is to be re-built. The study
will determine optimum matching conditions into the cyclotron, and so
must be completed this year (2008), so that the new injection line can
be completed within the current 5 year plan.
\subsection*{Results \& Progress}
The dynamics in an isochronous accelerator is non-intuitive as the space
charge forces cannot relieve themselves by lengthening the bunches: the
particles act in a sense as if they have infinite mass. One can model
the dynamics using individual macro-particles, but the optimization is
not transparent. There exists a much more efficient formalism which is
statistical in nature; the 21 six-dimensional second moments of the
particle bunch distributions are followed rather than the motion of
millions of individual particles. This formalism has never been used to
study the effects of space charge in the first few turns of a cyclotron.
Besides transport in 6-dimensional phase space, acceleration must also
be correctly handled. An error has been found in one standard code
(TRACE-3D).
At sufficiently high intensity, the bunches become vortices; circular
when observed from above the median plane. A new discovery is that this
tendency alleviates the harmful influence of the ``gap-crossing
resonance'' which causes a stretching of the beam in the median plane.
This aspect is currently under study; it may result in higher charge per
bunch than envisaged when the cyclotron was originally designed.
\begin{itemize}\footnotesize
\item Richard Baartman, Yi-Nong Rao (TRIUMF, Vancouver),
Tianjue Zhang (CIAE, Beijing), Hongjuan Yao (CIAE, Beijing; TUB,
Beijing), Yuzheng Lin (TUB, Beijing)
{\em Gap-Crossing Resonance in CYCIAE-100 Cyclotron\/},
\href{http://lin12.triumf.ca/text/2007Cyc/gapcross.pdf}
{Proc.\ 2007 Conference on Cyclotrons and their Applications}.
\end{itemize}
\subsection*{List of Institutes}
There is a collaboration with the Chinese Institute of Atomic Energy
(CIAE), as they are designing a 100\,MeV cyclotron that would be
affected by similar intensity limitations.
\subsection*{TRIUMF Role}
This research is critical to our upgrade plans, so we play the key lead role.