\section{Tagging Spectrometer}
                         
The \hd\ tagging system will consist of a dipole magnet spectrometer with 
a set of plastic scintillation counters in the focal plane to tag photon
energies between 50\% and 95\% of the incident electron energy.

\subsection{Magnet}

The tagging spectrometer magnet envisioned for \hd\ is modeled closely
on the existing tagger magnet in Hall B.   Although the \hd\ tagger will
operate at a much higher energy, the combination of smaller deflection angle
and smaller dynamic range (50\% to 95\% of the incident energy instead of
20\% to 95\%) result in a device which is comparable to the Hall B tagger
in most of its dimensions, allowing the use of existing solutions to many
engineering problems.

The tagger magnet will have a length of about 6.5\mlunit\ and will deflect
12GeV electrons along a path with radius of curvature 40\mlunit, giving a
deflection of about $9.4^{\circ}$ (Fig.~\ref{tagger_plan}).  If necessary,
additional dipoles may be used downstream to guide the full-energy
electron beam to the beam dump.  The magnet gap width should be
approximately 2.5\cmlunit.  A preliminary estimate of power requirements (using
a coil configuration like that of the Hall B tagger) gives 500 amperes at
14 volts for 12 \eunit\ operation.

\begin{figure}
\begin{center}
\rotatebox{-90}{\mbox{\epsfig{file=tagger_plan.eps,height=\textwidth}}}
\caption
[A plan view of the tagging spectrometer]
{A plan view of the tagging spectrometer from above, showing the
path of the primary beam and the trajectory of post-bremsstrahlung
electrons of various recoil momenta.}
\label{tagger_plan}
\end{center}
\end{figure}

The magnet poles themselves will form part of the vacuum system, and will
be welded to a stainless steel vacuum chamber which terminates in a flange
for mounting a thin exit window at the focal plane.  The exit window, which
must be about 10\mlunit\ long by 10\cmlunit\
wide, can be built in the same manner
as the Hall B tagger window, using aluminized mylar backed by Kevlar cloth 
\cite{Matthews99}.

For the above dimensions, the magnetic field at 12 \eunit\ 
is 1.0 Tesla.  If the
field is increased to about 1.7 Tesla (the point at which saturation effects
become very large), the tagger can operate at a maximum beam energy of about
20.7 \eunit.
To assure adequate operation at a possible upgraded energy of 24 \eunit,
the dimensions and cost of the magnet would have to be scaled appropriately.

\subsection{Spectrometer Optics}
The bremsstrahlung electrons are deflected by angles varying from $25.4^{\circ}$
to $16.6^{\circ}$ over the range of tagged energies.  For a radiator position
located about 0.5\mlunit\ upstream of the magnet, a radial focus is obtained along
a nearly linear focal surface.  Some of the optical properties of
the proposed spectrometer are summarized in Table~\ref{tmagprops}.

Using the values in Table~\ref{tmagprops}, one can calculate the attainable
resolution of the tagging spectrometer, which is constrained by the following
assumed electron beam sizes:
\newcounter{li}
\begin{list}{\arabic{li}.}{\usecounter{li}}
\item $\Delta p/p_0$ = 0.8 $\times 10^{-3}$ RMS
\item $\Delta x$ = 1.7\mmlunit\ RMS at radiator (horizontal)
\item $\Delta y$ = 0.7\mmlunit\ RMS at radiator (vertical)
\end{list}

The horizontal and vertical sizes at the radiator are different because 
of the different emittances and the need for a symmetric virtual focus
at the collimator.  For a horizontally-bending tagger magnet, the value
x = 1.6\mmlunit\ would degrade the achievable energy resolution unless the
electron beam is rotated to put the smaller emittance in the vertical direction.  
Table~\ref{taggerres} indicates the effects of these values at the tagger
focal plane, comparing them to the counter width required for an energy
channel width of 0.2\% $E_0$.

\begin{table}
\begin{center}
\begin{tabular}{ccccccccc}
\hline\hline
k/$E_0$ & Bend & Drift &  Angle & \cmlunit/\%$E_0$ & \cmlunit/\%$E_0$ &
(x x) &	(y y) &	(y y') \\
        & (deg) & (\mlunit)  &  (deg)        &  along FP  & perp.to ray&
& & {\mmlunit/mr} \\
\hline
0.95 & 25.4 & 1.06 & 13.3 & 28.7 & 6.60 & -0.81 & 3.42 & 5.8 \\
0.90 & 21.2 & 1.78 & 9.8 & 24.7 & 4.19 & -0.93 & 3.49 & 8.7 \\
0.85 & 19.5 & 2.41 & 8.2 & 23.2 & 3.31 & -0.97 & 3.46 & 11.2 \\
0.80 & 18.5 & 3.00 & 7.3 & 22.4 & 2.85 & -0.99 & 3.42 & 13.5 \\
0.70 & 17.5 & 4.12 & 6.3 & 21.7 & 2.37 & -1.01 & 3.34 & 18.0 \\
0.60 & 16.9 & 5.21 & 5.7 & 21.3 & 2.13 & -1.01 & 3.28 & 22.4 \\
0.50 & 16.6 & 6.28 & 5.4 & 21.1 & 1.98 & -1.06 & 3.24 & 26.2 \\
\hline\hline
\end{tabular}
\end{center}
\caption
[Optical parameters of the tagging spectrometer.]
{Optical parameters of the tagging spectrometer:
Bend = deflection angle; Drift = distance from exit edge to focal plane;
Angle = angle between electron path and focal plane; 
cm/\%$E_0$ = dispersion in units of cm per percent of the incident energy;
(x x),(y y),(y y') = first-order transport matrix elements where x and y
are radial and transverse coordinates respectively; the focal plane is 
defined by (x x')=0.}
\label{tmagprops}
\end{table}

\begin{table}
\begin{center}
\begin{tabular}{ccccc}
\hline\hline
k/$E_0$  &  Width & $p$ RMS & $x$ RMS & $y$ RMS \\
         &  (\mmlunit)  &  (\mmlunit)   &  (\mmlunit)   &  (\mmlunit)   \\
\hline
0.95 & 13.2 & 5.3 & 1.4 & 0.6 \\
0.90 & 8.4  & 3.4 & 1.6 & 0.7 \\
0.85 & 6.6  & 2.6 & 1.7 & 0.7 \\
0.80 & 5.7  & 2.3 & 1.7 & 0.7 \\
0.70 & 4.7  & 1.9 & 1.7 & 0.7 \\
0.60 & 4.3  & 1.7 & 1.7 & 0.7 \\
0.50 & 4.0  & 1.6 & 1.7 & 0.7 \\
\hline\hline
\end{tabular}
\end{center}
\caption
[Contributions to tagger energy resolution.]
{Contributions to tagger energy resolution measured in units of
length on the focal plane: Width = physical size of focal plane detector
for $\Delta E = 0.002 E_0$; $p$ RMS = size due to energy spread of 
electron beam $p=.0008 E_0$; $x$ RMS = size due to horizontal spot size on
the radiator $\sigma_x = $ 1.7\mmlunit; $y$ RMS = size due to vertical spot size
on the radiator $\sigma_y = $ 0.7\mmlunit.}
\label{taggerres}
\end{table}

For a vertical bend, where $y$ = 0.7\mmlunit\ applies, the resolution is
dominated by the 0.08\% momentum width of the beam.  If the tagger is
built with a horizontal deflection, then the $x$ resolution contribution
for $k/E_0 < 0.7$ is  comparable to the $p$ contribution, unless the beam 
is rotated.

\subsection{Focal-plane hodoscope}

Since it is desired to run the experiments at high rate, the tagger detectors
will not be incorporated into the trigger logic, but will instead be read out
by high-resolution time-to-digital converters (\tdc s) for off-line analysis.
Assuming that the time correlation with the main detector can be done with a
resolution of better than  3\nstunit, tagged photon rates of up to 100\mhz\ can be
utilized before accidental coincidences become a serious limit.

Electrons will be tagged in energy channels of width 0.002 $E_0$ (25 \Meunit\ at
$E_0$ = 12 \eunit). This will result in an RMS energy resolution of about 0.001 $E_0$,
which is consistent with the limits imposed by the beam as discussed in 
the previous section.  As shown in Table~\ref{taggerres}, the width of a counter
which defines such a channel varies from 13.2\mmlunit\ at $k/E_0=0.95$ to 4.0\mmlunit\
at $k/E_0 = 0.50$.  The spacing of the counters along the focal plane will 
vary between 4.2 and 5.7\cmlunit\ according to the data of Table~\ref{tmagprops}.

If each detector defines a single energy channel, some overlap with
adjacent counters is desirable to avoid gaps in the acceptance and edge
effects in the detectors.  Because of the close tolerances involved in
aligning such narrow counters for small fractional overlap, another
solution would be to increase the energy width of the counters and align
them so that adjacent counters overlap by 1/3 of their width.  In this case,
successive energy channels are defined by alternating coincidence and
anticoincidence between adjacent counters.  The principal disadvantage
of this solution is that the counting rate in each scintillator is 3 times
the value for the one-counter-per-channel design.  For collimated coherent
bremsstrahlung running, the coherent peak is confined to a very narrow
energy region, especially when the photon energy is close to the endpoint.
Under these conditions, individual counters may be required to sustain rates
as high as 10\mhz, which will require special bases and electronics,
and this rate will probably define the limiting rate for the experiments.
Increasing the overlap of the counters would then require a reduction in the
total event rate for coherent bremsstrahlung running, and thus is not an
attractive solution.

The length of the counters (in the transverse direction) is determined by
the desired efficiency for tagging uncollimated incoherent bremsstrahlung
events near $k = 0.95 E_0$.  Counters of length 6\cmlunit\ will be more than 97\%
efficient at the endpoint of a 12 \eunit\ beam, and even more efficient at other
photon energies.  Since the time shifts within a 6cm-long counter are
less than 200\pstunit, excellent timing can be obtained using a single
photomultiplier tube.

Thus the proposed configuration for tagging photons from 50\% to 95\% of
the endpoint energy consists of 225 scintillation counters, each subtending
an energy bin of about $0.002 E_0$,  with minimal (or no) overlap.  The
dimensions of each counter are approximately as follows: 
width = 4.0 to 13.2\mmlunit, length = 60\mmlunit, thickness   4\mmlunit.  Each counter is
read out by a single high-rate photomultiplier tube, permitting rates up
to 10\mhz\ per counter.  The counters are at least 4\cmlunit\ apart along the
focal plane, so 2.5\cmlunit\ \pmt s can be used without alternating directions. 

