\documentclass{JAC2003}


\usepackage{graphicx,epsfig,amssymb,amsmath}


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%% RPPP028,
\begin{document}
\title{
Simulation of Wake Field Effects on High
Energy Particle Beams:
}

\author{ R. Barlow, G. Kourevlev and A. Mercer,  Manchester University, Manchester, UK and the
Cockcroft Institute
}

\maketitle

\begin{abstract}
 We present formulae for higher mode geometric intrabunch wakefields,
such as will be produced in the ILC. These have been incorporated in the 
MERLIN code, and thus validated against existing data, and simple
studies are done indicating the effect of wake fields in a typical ILC design.
\end{abstract}

\section{Introduction}

A bunch of charged particles induces currents
in the beam pipe, and these lead to electromagnetic
fields (wake fields) which affect the particles in the beam.
This may have important consequences for the ILC, as particle bunches will be
passing very close to collimators. This could lead to
further deflection of off-axis bunches, or emittance dilution,
and thus lowering
of the luminosity.

Deviations from perfect conductivity and
constant aperture lead to resistive and geometric wakefields.
We consider only geometric wakefields, as the spoiler elements are
relatively short.
We consider only the effects of wake fields within a 
bunch, as the time between bunches is believed to 
be long enough for the
wake currents to damp down.

For a bunch passing through an element the
detailed motion within the element is not relevant, and
we are interested in the deflection in angle of the
particles that emerge from the element. If the
bunch and the particle are on-axis then symmetry dictates that there is no effect: the effect can be expanded in powers of $r$.
We use  cylindrical  co-ordinates $r,\theta,z$
and consider the effect on a particle, a distance $r$ from the centre,
of a slice of thickness $dz$ of the bunch, preceding this particle by
a distance $z$.
Then~\cite{Yokoya,Stupakovone,Stupakovtwo} 
\begin{eqnarray*}
\overline F_\perp =& \\ -e &\sum_{m=1}^\infty  W_m(z) m r^{m-1}\Big[ \hat r\left(Q_m \cos m \theta + \tilde Q_m \sin m \theta \right)
\\ &- \hat\theta \left( Q_m \sin m \theta + \tilde Q_m \cos m \theta \right) 
\Big]
\end{eqnarray*}
where 
$$Q_m=\int_0^{2 \pi} d \theta' \int_0^\infty dr' (r')^{m+1} cos m 
\theta' \rho(r',\theta',z) $$
$$
\tilde Q_m=\int_0^{2 \pi} d \theta' \int_0^\infty dr' r'^{m+1} sin m \theta' \rho(r',\theta',z) $$
where $\rho(r,\theta,z)$ is the density of the bunch
We assume, as in \cite{Zagorodov}, that only $\theta'=0$ need be considered,
i.e. we consider deflection/distortion in a particular direction.  
This then simplifies to

$$F_{\perp,m}=-e W_m m r^{m-1} Q_m $$

$$Q_m=\int_0^{2\pi} d \theta' 
\int_0^\infty dr' {r'}^{m+1} \rho(r',\theta')=Q(z) \overline{r^m}$$
Where the moments $\overline{r^m}$ are known from the 
bunch distribution. $Q(z)$ is the charge
of the slice,
and the wake function for a steeply tapered collimator, moving from
aperture $b$ to aperture $a$, is given by(\cite{Raimondi})
$$W_m(z)=2 \left( {1 \over a^{2m}} - {1 \over b^{2m}} \right) e^{-mz/a} \Theta(z)$$
where $\Theta(z)$ is a unit step function.
The factor of 2 arises because one has to consider both sides of the collimator \cite{Zimmermann}.

In the SLAC tests \cite{SLACtests} the deflection of a 
particle bunch near a wall was measured. The effect of the wakefield is encompassed in the 
`kick factor' $K$
$$\Delta y' = {N r_e \over \gamma} K y$$ 

\section {Implementation}

MERLIN already contains a `wakefieldprocess' in collimator elements, 
however it only considers the first, linear ($m=1$) mode. We extend this to
$m=5$ and apply a deflection to a particle at radius $r_i$
 in bin $i$
\begin{equation}
\Delta y'_i = -{2 N r_e \over \gamma} \sum_{j=i+1}^{j=nbins}
\sum_{m=1}^5 {m r_i^{m-1} Q(z_j) \overline{r_j^m} \over a^{2m}} e^{-m(z_j-z_i)/a}
\end{equation}
where 
we have assumed $a<<b$. The code has been modified by the addition of
4 extra terms to the Wakefieldprocess.


\section {Validation}

We simulated the SLAC tests using a beam of energy 1.19 GeV, with $2\, 
10^{10} $ elctrons, a bunch with $\epsilon_x = 0.36 mm, \epsilon_y=0.16 mm$, and $\sigma_z=0.65 mm$. We considered a square aperture of gap half-width 1.9 mm. We took the lattice functions as $\beta_x=3\ m, \beta_y=10\ m$.

Figure \ref{onemode} shows the predicted deflection when only the first ($m=1$)
 term is included.  It agrees with the data (Figure \ref{data}) for 
small displacements but, being linear, does not reproduce the 
non-linearity near the edge. However inclusion of 5 terms, as suggested by
~\cite{Zagorodov} on the basis of comparisons with the ECHO code, 
reproduces the data well
(Figure\ref{fivemodes}) 
even for beams close to the collimator wall.

\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{RPPP028f1.eps}
\caption{Predicted deflection using one term}
\label{onemode}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{RPPP028f2.eps}
\caption{Data  - from Ref\cite{SLACtests}}
\label{data}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{RPPP028f3.eps}
\caption{Predicted deflection using 5 terms}
\label{fivemodes}
\end{figure}

\section {Wake fields in the TESLA BDS}

We use the TESLA BDS to investigate wake fields at
a future linear collider, as
optics files are available. These are read into MERLIN by the MADinterface code, which
constructs the accelerator model. Wake potentials can be included at this stage.
The TESLA collimation system consists of four betatron collimation systems spaced $45^\circ$ apart, preceded by an  energy collimation system.
The collimation system is a spoiler/absorber combination, however wakefield
effects from the absorbers are generally
much smaller than those from the spoilers~\cite{Tenenbaum} so we consider only the spoiler wakefields.
In this simulation any particle striking the spoiler is simply removed.


We consider wake field effects on the core bunch, and also on the halo.
For the core bunch we take standard Gaussian distributions.
At present we only consider the first $m=1$ term, which is good for small
deviations from the axis. 


%To examine effects on the halo we consider a flat distribution covering a pessimistic
% $18 \sigma_x$ and $120 \sigma_y$, as in~\cite{Drozhdin}, though we consider a
%uniform distribution rather than a $1/x, 1/y$ shape. 
%This overlaps the collimation depth

We considered a standard bunch which was 0.01 mm off axis in $y$ (which is large: around $3\sigma$) at the exit from the
accelerator.  Figure~\ref{without} shows the distribution in $y$ at the IP
with no wakefields simulated, and Figure~\ref{with} shows the distribution with
wake fields included.  The difference is discernible but not large: the
rms spread is increased, the tail to negative $y$ is larger. The reduced number is caused by a few particles hitting a collimator.   Overall the 
effect appears small, and not a serious problem for the feasibility of the 
accelerator.

We used version 8.02 (June 26,2000) of the optics.  Preliminary examination of the more recent 8.05 gave effects which were even smaller.

\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{RPPP028f4.eps}
\caption{$y$ position at IP without wake fields}
\label{without}
\end{figure}

\begin{figure}[htb]
\centering
\includegraphics*[width=65mm]{RPPP028f5.eps}
\caption{$y$ position at IP with wake fields}
\label{with}
\end{figure}

These are early results, and need to be substantiated by more detailed studies
and cross-checks.

\section {Conclusions}

We have a formalism for including wake fields in Merlin which reproduces the 
existing data and can be used for Linear Collider studies: the first such
studies indicate that the effects are small.

%\section {Acknowledgements}



\begin{thebibliography}{9}   % Use for  1-9  references
%\begin{thebibliography}{99} % Use for 10-99 references
\addcontentsline{toc}{section}{References}

\bibitem{Yokoya} K. Yokoya, `Impedance of slowly tapered structures', CERN report SL/90-88 (AP) (1990)

\bibitem{Stupakovone} G. Stupakov, `Geometric Wake of a smooth taper', SLAC-PUB-95-7086 (1995)


\bibitem{Stupakovtwo} G. Stupakov, `Geometric Wake of a smooth flat collimator', SLAC-PUB-95-7167 (1996)
\bibitem{Zagorodov} I. Zagorodnov, T.l Weiland, M. Dohlus and M K\" orfer `Near-wall Wakefields for Optimize Geometry if te TTF-2 COllimator', TESLA report 2003-23

\bibitem{Raimondi} P. Raimondi, F. J. Decker, T. Useher and C. K. Ng. `Closed form expression for the Geometric effect of a Beam Scraper on the Transverse Beam Distribution',
PAC 97 (Vancouver) and SLAC-PUB-8552

\bibitem{Zimmermann} F. Zimmermann, K. L. F. Bane, C. K. Ng, `Collimator Wake Fields in the SLC Final Focus', EPAC 96 (BArcelona), SLAC-PUB-7137
 
\bibitem{SLACtests} D. Onoprienko, `Measurements of Transverse Wakefields of 
Tapered Collimators' in LC02, SLAC, February 2003


\bibitem{Tenenbaum} P. Tenenbaum, `Collimator Wakefield Calculation for ILC-TRC
Report', SLAC LCC-0101, (2002)

\bibitem{Drozhdin} A. Drozhdin et al, `Comparison of the TESLA, NLC and CLIC beam collimation system performace', FERMILAB-TM-2200, (2003)

\end{thebibliography}

\end{document}