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\begin{slide}
\large
\raggedright
\begin{center}
\Large
Isochronous and Scaling FFAGs\\
\large
\vspace{2cm}
\color{blue}
R. Baartman\\
TRIUMF\\
\vspace{2cm}
\normalsize
\color{red}
April, 2002\\
\vspace{2cm}
\end{center}\small
\underline{Abstract}: If an FFAG can be made isochronous, acceleration can
occur without ramping the rf frequency. But this is of course simply a
cyclotron, of which many examples exist worldwide: 2 of the largest are PSI
(580\,MeV p, up to 1\,MW beam power), TRIUMF (500\,MeV H$^-$, up to 0.1\,MW
beam power). How do the beam dynamics change if an FFAG is isochronous?
E.g.\ can we still achieve large acceptance?
\end{slide}
\begin{slide}\raggedright
\large
\underline{\color{red}Isochronism}
\small
Let us define $\theta$ to be the angle of the reference particle momentum
w.r.t.\ the lab frame. Orbit length $L$ is given by speed and orbit period
$T$:
\begin{equation*}
L=\oint ds=\oint\rho d\theta=\beta cT.
\end{equation*}
The local curvature $\rho=\rho(s)$ can vary and for reversed-field bends
even changes sign. (Along an orbit, $ds=\rho d\theta>0$ so $d\theta$ is also
negative in reversed-field bends.) Of course on one orbit, we always have
\begin{equation*}
\oint d\theta=2\pi.
\end{equation*}
What is the magnetic field averaged over the orbit?
\begin{equation*}
\overline{B}={\oint Bds\over\oint ds}={\oint B\rho\,d\theta\over\beta cT}.
\end{equation*}
But $B\rho$ is constant and in fact is $\beta\gamma m_0c/q$. Therefore
\begin{equation*}
\overline{B}={2\pi\over T}{m_0\over q}\,\gamma\,\,\,\,\equiv B_{\rm c}\,\gamma={B_{\rm c}\over\sqrt{1-\beta^2}}.
\end{equation*}
Remember, $\beta$ is related to the orbit length: $\beta=L/(cT)=2\pi
R/(cT)\equiv R/R_\infty$. So \color{blue}
\begin{equation*}
\overline{B}={B_{\rm c}\over\sqrt{1-(R/R_\infty)^2}}.
\end{equation*}
\end{slide}
\begin{slide}\raggedright
\begin{center}
\includegraphics[width=4in]{iso.pdf}
\end{center}
Of course, this means the field index is
$ k={R\over B}{dB\over dR}={\beta\over\gamma}{d\gamma\over d\beta}
=\beta^2\gamma^2\color{red}\neq\mbox{constant}$.
So \color{red}isochronism $\Rightarrow$ non-scaling.
\end{slide}
\begin{slide}\raggedright
\large
\underline{\color{red}Focusing (flat field)}
\small
For the moment, let us imagine that there are no sectors, no azimuthal
field variation, just radial variation with field index
$k=\beta^2\gamma^2$.
We know the transfer matrix in such a dipole, and can write directly:
$\color{blue}\begin{array}{rclcrcl}
\nu_r^2&=&1+k&{\hspace{4cm}}&\nu_z^2&=&-k\\
&=&1+\beta^2\gamma^2&{\hspace{4cm}}&&=&-\beta^2\gamma^2\\
&=&\gamma^2&{\hspace{4cm}}&&<&0\\ \\
\nu_r&=&\gamma&{\hspace{4cm}}&\nu_z&=&\mbox{\color{red}imaginary}
\end{array}$
So why do such cyclotrons work at all? Lawrence's first cyclotrons, and
those built before the early 50s, had no sector focusing. To obtain
vertical focusing, $k$ was made slightly negative. This resulted in phase
slippage, so the rf voltage was made as large as possible to achieve the
final energy before an accumulated phase slip of $\pi/2$. In addition, some
vertical focusing was achieved by accelerating on the falling side of the
rf voltage. In this way, such cyclotrons achieved maximum energies of 20 to
30 MeV (protons).
To reach higher energy, it was necessary to release
isochronism, ramp the rf frequency, and pulse the machine. Unfortunately,
this meant that the maximum intensity was 2 or 3 orders of magnitude lower
than for the (cw) cyclotron.
\end{slide}
\begin{slide}\raggedright
\large
\underline{\color{red}Focusing (AVF or FFAG, regular lattice)}
\small Strong focusing was invented/developed in the 50s. This had
implications for
\begin{itemize}
\item \textbf{\color{blue}cyclotrons}: they could now be isochronous AND vertically
focusing
\item \textbf{\color{blue}synchro-cyclotrons}: tunes need be no longer near 1, so beams
were smaller, space charge limits higher
\item and \textbf{\color{blue}synchrotrons}: same advantages as for synchro-cyclotrons
\end{itemize} The application was by far simplest for synchrotrons. For FF
machines, the extended nature of the field was a calculational
headache. That is why synchrotrons developed rapidly in the 50s (when large
computing power was unavailable), while synchro-cyclotrons of the FFAG type
did not. Nowadays, large computing power is freely available.
\end{slide}
\begin{slide}\raggedright\small
For Azimuthally-Varying Field (AVF)
or the special case of Alternating Gradient (FFAG), let us use {\bf
Mathematica} to calculate the tunes. To make it transparent, let us
consider all identical dipoles and drifts; no reverse bends. We have
drifts $d$, dipoles with index $k$, radius $\rho$, bend angle $\phi$, and
edge angles $\phi-\theta$:
\includegraphics[width=6in]{ffagsector.pdf}
In addition, imagine that the edges are inclined by an extra angle
$\xi$. This is called the ``spiral angle'' (hard to draw).
In this hard-edged case, the ``flutter''
$F^2\equiv \langle(B-\overline{B})^2\rangle/\overline{B}^2=R/\rho-1$.
Aside: Notice that the particle trajectory ({\color{blue} blue curve}) does
not coincide with a contour of constant $B$ (dashed curves). This has large
implications for using existing transport codes to describe FFAGs.
\end{slide}
\begin{slide}\raggedright
\includegraphics[height=11in]{ffagtrig3.pdf}
\end{slide}
\begin{slide}\raggedright \small
These expressions are identical to those originally derived by Symon et
al.\ in the original 1956 Phys.\ Rev.\ paper about FFAGs.
But beware! Since there is now a distinction between
local curvature ($\rho$) and global ($R$), the definition of field index is
ambiguous. The local index, used in the dipole transfer matrix, is
\begin{equation*}
k={\rho\over B}{dB\over d\rho},
\end{equation*}
while the Symon formula uses
\begin{equation*}
\kappa={R\over B}{dB\over dR}\approx k\,{R\over\rho}
\end{equation*} As we proved, it is in fact this latter quantity which must
be equal to $\beta^2\gamma^2$ for isochronism. We therefore still have
\begin{equation*}\color{blue}
\nu_r=\gamma \mbox{\hspace{4cm}(isochronous)}
\end{equation*}
But
\begin{equation*}\color{blue}
\nu_z^2=-\beta^2\gamma^2+F^2\left(1+2\tan^2\xi\right) \mbox{\hspace{1cm}(isochronous)}
\end{equation*}
Aside: We could have {\bf Mathematica} print out the next higher order, but it
would be in error because the trajectories are not circular arcs.
\end{slide}
\begin{slide}\raggedright
\underline{\color{red}Example: TRIUMF cyclotron}
\small
\includegraphics[width=6.5in]{cyclotron-1972sm.png}
\begin{tabular}{|c|c|c|c|c|c|c|}\hline
Energy&$R$&$\beta\gamma$&$\xi$&$1+2\tan^2\xi$&$F^2$&$\nu_z$\\\hline\hline
100\,MeV&175\,in.&0.47& 0$^\circ$& 0.0&0.30 &0.28\\
250\,MeV&251\,in.&0.78&47$^\circ$& 3.3&0.20 &0.24\\
505\,MeV&311\,in.&1.17&72$^\circ$&\color{red}20.0&0.07 &0.24\\\hline
\end{tabular}
\end{slide}
\begin{slide}\raggedright
\underline{\color{red}Irregular FFAG cyclotrons}
\small
By adjusting the flutter $F$ and spiral angle $\xi$ as functions of $R$, we
can arrange to make $\nu_z$ constant.
But what about $\nu_r$? Can we change it by departing from the regular
$N$-cell lattice? Perhaps, but no one has ever tried it.
In synchrotron language, isochronism means $\gamma=\gamma_{\rm t}$. But we
know that for a ``regular'' lattice, $\gamma_{\rm t}=\nu_r$. Hence,
$\nu_r=\gamma$. For an irregular lattice,
\begin{equation*}
{1\over\gamma_{\rm t}^2}={\nu_r^3\over R}\sum_n{|a_n|^2\over\nu_r^2-n^2}
\end{equation*}
where $a_n$ is the Fourier transform of $\beta_x^{3/2}/\rho$. For $N$
identical cells where $N\gg\nu_r$, this gives $\gamma_{\rm t}=\nu_r$
because the $n=0$ term gives by far the largest contribution to the sum.
If the lattice is arranged to have a superperiodicity $n=\pm S$ where
$S\sim\nu_r$, $\gamma_{\rm t}$ can be moved away from $\nu_r$:
\begin{equation*}
{1\over\gamma_{\rm t}^2}={1\over\nu_r^2}+{2|a_S|^2\over R}{\nu_r^3\over\nu_r^2-S^2}
\end{equation*}
This is the approach used to make $\gamma_{\rm t}$ imaginary in e.g.\ the
JHF main ring. Whether this can be used to fix $\nu_r$ in a cyclotron where
necessarily $\gamma_{\rm t}=\gamma$, is not clear, since it prescribes a
peculiar variation of $a_S$ with $R$.
\end{slide}
\begin{slide}\raggedright
\underline{\color{red}Conclusions}:
The chief virtue that FFAG scaling machines have over non-scaling
isochronous machines is the constancy of the tunes. This results in huge
acceptance ($>1000\pi$\,mm-mrad). But because they cannot be isochronous,
they must be pulsed.
The chief virtue of isochronous FFAGs is that they can run cw and reach
very high intensity. However, because $\nu_r=\gamma$, many resonances must
be traversed to reach high energy. This reduces acceptance to only a few
$\pi$\,mm-mrad.
\end{slide}
\end{document}