
\section{Electron Beam}

The performance of the photon source is dependent upon the
parameters of the electron beam in several important areas.
The assumptions regarding the electron beam that were used in
calculating the properties of the coherent bremsstrahlung source
reported in this chapter are collected together in Table~\ref{elecprop}.
Some of these parameters require further explanation.  Only circular
polarization can be transfered from the electron to the bremsstrahlung
photon beam.  In fact, helicity transfer from the electron is the only
means available for generating photons with circular polarization in
\hd.  Although circular polarization is less interesting than linear
polarization from the point of view of meson photoproduction, having
access to both kinds makes available a complete set of polarization
observables.  At the present a concrete design for the optics of the
transfer beam line to \hd\ does not exist, but when the design is made
it would be useful to check that it can deliver circularly polarized
photons to the hall.

\begin{table}
\begin{center}
\begin{tabular}{l@{\hspace{2cm}}c}
\hline\hline
energy & 12 \eunit \\
r.m.s.\ energy spread & $<10$ \Meunit \\
transverse $x$ emittance & 10\mmlunit $\cdot\mu$r \\
transverse $y$ emittance & 2.5\mmlunit$\cdot\mu$r \\
electron polarization & longitudinal when available \\
minimum useful current & 100 pA \\
maximum useful current & 3 $\mu$A \\
$x$ spot size at radiator & 1.7\mmlunit\ r.m.s. \\
$y$ spot size at radiator & 0.7\mmlunit\ r.m.s. \\
$x$ spot size at collimator & 0.5\mmlunit\ \\
$y$ spot size at collimator & 0.5 \mmlunit\ \\
position stability & $\pm$200 \micron \\
\hline\hline
\end{tabular}
\end{center}
\caption
[Assumed electron beam properties]
{Electron beam properties that were assumed in calculating
the properties of the coherent bremsstrahlung source.}
\label{elecprop}
\end{table}


\subsection{Beam emittance requirements}

The values for the electron beam emittances shown in Table~\ref{elecprop}
are estimates based upon the parameters of the current machine projected
to 12 \eunit.  They were provided by V. Lebedev of the CEBAF accelerator
division.  The definition of emittance used here is the product of the
r.m.s.\ widths of the beam in transverse position and divergence angle.
Because synchrotron radiation occurs mainly in the horizontal plane
inside the accelerator, the emittance values in $ x $ are generally
larger than those for $ y $.

The place where transverse emittance plays a critical role is at the
photon collimator.  For optimum effectiveness in collimation it is
important that the electron beam spot at the collimator position be
as small as possible.  The electron beam does not actually reach the
photon collimator, being bent into the dump shortly after the radiator
by the tagger magnet.  But considering the optics of the electron beam
as if the tagger dipole were switched off, the electron beam at the
radiator can be projected forward 80\mlunit\ to form a virtual image on the
collimator entrance plane. The position and size of this spot gives the
the definition of the $ 0^{\circ} $ emission angle for the photons.
If this spot is small compared to the collimator aperture and is centered
then the collimator does its job of removing photons with emission
angle beyond the collimator half-angle.  However, if the virtual spot is
larger than the aperture then the collimator serves only to reduce the
intensity of the beam and does not discriminate between coherent and
incoherent beam components.

Note that this analysis does not place any specific limits on the size
of the beam at the radiator. The beam spot can and should be larger
there to avoid burning a hole in the radiator.
For the SLAC coherent bremsstrahlung source the beam at the radiator
was 6\mmlunit\ in diameter and focused to image a 2\mmlunit\ 
spot at the collimator
positioned 91\mlunit\ downstream of the radiator.  The focus ensures
that photons emitted near zero degrees, even those far from the beam
axis at the radiator, will pass near to the axis at the collimator
position and so be seen at the experimental target.  The
virtual electron beam spot is not the same as the actual photon beam
spot on the collimator, which is concentric with the former but larger.
The collimator aperture is circular, so the choice was made to make the
electron beam come to a virtual focus there with both $x$ and $y$ spot
sizes equal there.  This is why the beam spot on the radiator is asymmetric.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=emittcuts.eps,width=\textwidth}}
\caption
[Coherent photon spectrum for three different values of the 
electron beam transverse emittance.]
{Coherent photon spectrum for three different values of the 
electron beam transverse emittance.  The horizontal (shown on the
plot) and vertical emittances are assumed to scale together.  A 3.4\mmlunit\
collimator located 80\mlunit\ from the radiator was used for this calculation.}
\label{emittcuts}
\end{center}
\end{figure}

Figure \ref{emittcuts} shows how the collimated photon spectrum
depends upon the transverse emittance of the electron beam.  To generate
this plot the increases in emittance were simply translated into an
increased virtual spot size on the collimator.  This was done because
it was assumed that the spot size of the electron beam on the radiator,
already close to 2\mmlunit\ 
r.m.s., cannot be further inflated and stay contained
within the limits of the crystal.  When the virtual spot size becomes
comparable with the collimator aperture then the collimation is rendered
ineffective, and the photon spectrum and polarization revert to their 
uncollimated values.  There is another connection between focal spot
size and beam emittance that is connected with the requirement that
all electrons enter the radiator at the same incidence angle with
respect to the planes of the crystal.  Practically, the divergence does
not broaden the coherent peak provided that it is kept below the mosaic
spread of the crystal.  A conservative
value for the allowable angular divergence $ \delta $
in the electron beam at the radiator would then be  20$\mu$r .  Taken
together with a 500 \micron\ r.m.s.\ spot size at the focus, this leads to
an emittance of 10\mmlunit $ \cdot \mu $r  at 12 \eunit.  
This corresponds to the
upper curve in Fig.~\ref{emittcuts}.

\subsection{Electron beam line optics}

Taking the electron beam out of the machine and focusing it onto a
500 \micron\ spot requires some optics in the beam line leading up to the
radiator.  Translating the beam emittance into a r.m.s.\ values for the
beam radius and divergence requires the knowledge of the $ \beta $
parameter, defined as the ratio of the r.m.s.\ beam size to its angular
divergence.  When it comes out of the accelerator, $ \beta $
is large, of order 1km, corresponding a very highly parallel beam.  In
order to focus a beam of emittance 10\mmlunit $\cdot 
\mu$r down to a r.m.s.\ size
of $ 500 \mu r$ , the $ \beta $ must
be reduced from 1km down to 25\mlunit\ by the use of focusing elements.  The
focusing must be done in both $ x $ and $ y $ axes separately to
achieve the desired result, and must be accomplished by elements located
along the transfer line from the accelerator vault to the surface.
The focusing requirements are very specific for this type of source,
and will require careful study in the design of the electron beam line.

Not only must the virtual electron spot be small enough to fit within
the collimator aperture, but it must also be centered on the aperture
and stable.  In order to maintain a stable beam position on the
collimator, the SLAC experiment \cite{Kaune} instrumented the 
collimator with a secondary-emission detector (SEQ).  The detector was
of the ``pin-cushion'' design and was installed as a segment of the
collimator near the position of the shower maximum.  The readout was
divided into four quadrants, which read equal currents when the beam
was properly aligned on the collimator.  The SEQ was connected via a
feedback loop to the last steering elements on the electron beam line
prior to the radiator.  From that distance a bend of only $ 10 \mu r$
results in a shift of 1\mmlunit\ at the collimator position.  Hence
the beam alignment feedback system is effectively decoupled from the 
crystal orientation controls.  Small deflections that are necessary to
keep the beam centered on the collimator do not produce appreciable walk
in the beam-crystal angle, and by using a goniometer to control the latter,
the former can be operated in a feedback circuit that operates
independent of the photon source energy controls.  The experimental
program in parity violation at Jefferson Lab has already designed a
position stabilization circuit that is able to keep the beam position
steady to within $ 20 \mu m $ over a 20\mlunit\ lever arm.  
A less sophisticated version of this circuit should be a part of the
collimated photon source for \hd.

\subsection{Electron beam dump}

The electron beam is dumped in the horizontal plane, as shown in
figure \ref{photon_line}.  The horizontal bend offers several
advantages over dumping the beam into the ground.  The tagger magnet
is easier to support if it sits in the horizontal position.  It is also
easier to mount and service the focal plane in this position.  The
dump itself is also more accessible in case it needs to be serviced.
An above-ground dump also affords the possibility of running parasitic
beam dump experiments that do not interfere with the operation of the
experimental hall.

The primary design requirement for the electron beam dump is that it
have a sufficiently high capacity to handle beams of the highest
intensities foreseen for \hd. A 60KW design would provide a
a healthy margin for operation of a 12 \eunit\ beam at 3$\mu$A and sufficient
capacity to handle 3$\mu$A at 20 \eunit\ in the case of a further upgrade.

