\section{Tagging spectrometer}
\label{tagger}


\subsection{Specifications}

To satisfy the needs of the \hd\ physics program, the tagged photon
spectrometer should meet the following specifications:


\newcounter{li}
\begin{list}{\arabic{li}.}{\usecounter{li}}

 
\item Photon energy detection from 
70\% to 75\% of $E_0$ with energy resolution of about 0.1\% r.m.s.
Percentages refer to the primary beam energy $E_0$, i.e.
``0.1\%'' means 12~\Meunit\ energy resolution for a 12~\eunit\ beam.

\item A detector system which allows a counting rate of
at least
$5\times 10^6 $ electrons per second per 0.1\% over this range of
photon energies.     

\item
An additional capability for photon energy detection from 25\%
to 90\% of $E_0$, with less stringent resolution and count rate
requirements .

\item A quadrupole magnet between the radiator and dipole spectrometer
which images the beam spot on the radiator onto a line on the focal
plane.  This feature makes it possible to envision the use of focal
plane counters with two-dimensional readout, with which one could enhance
the tagging efficiency of the source.  Focal
plane detectors with two-dimensional readout are considered as a
possible upgrade beyond  the baseline design presented in this chapter.
Any improvements obtained using this technique would be over and above
the performance figures presented in this report.

%\item The capability to improve the tagging efficiency for collimated photons
%by transverse angle discrimination.  Since the angles of the outgoing photon
%and electron are highly correlated in coherent bremsstrahlung, coherent
%photons which pass through the small collimator aperture are correlated 
%with electrons which leave the target with small transverse angles.  
%Tagging only these electrons will improve the ratio of collimated photons 
%to tags (the ``tagging efficiency''), thus allowing a higher rate of photons
%on target before accidental coincidences make the tagger information
%useless.  Counting the bremsstrahlung electrons in only a narrow 
%region transverse to the tagger bend plane will have this effect 
%if the transverse position at the detector is strongly correlated 
%with the transverse angle at the radiator.
%This requires both a relatively large transverse defocusing and a small 
%transverse magnification, so that the effect of the transverse angle 
%is not masked by effects of the finite beam spot size on the radiator.
\end{list}

The system described below, based on a room-temperature design, meets
all of these criteria.  The option of a superconducting design was
also studied.  With a superconducting magnet, the spectrometer could
operate at much higher fields, offering the possibility of some
space savings in the size of the tagger focal plane array and larger
head-room for future possible energy upgrades beyond 12 \eunit\.
An iron yoke design was found which would clamp the 5 Tesla field 
sufficiently to make it possible to operate normal phototubes on the
nearby tagger focal plane.  However, as shown below, rate considerations
require a degree of segmentation in the tagging counters such that it is
impractical to increase the dispersion along the focal plane above
what is provided by a 1.5 Tesla room temperature magnet.  That being
the case, it was decided that considerations of upgrade margin and
electrical power alone do not justify the additional cost and
complexity of a superconducting magnet.

\subsection{Magnet}

%Ref: [D. I. Sober et al., Nucl. Instr. and  Meth. A 440, 263 (2000)]

\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_D15rtp.eps,width=\textwidth}}
\caption
[Plan view of 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 tagger magnet (Fig.~\ref{tagger_plan}) is an Elbek-type spectrometer
similar to the existing tagger magnet in Hall B of Jefferson Lab
\cite{Sober00}.
The higher energy of the \hd\ beam is largely compensated for by
going to smaller bend angles, so the sizes of the magnets are comparable,
and much of the experience from Hall B can be carried over.  
Unlike the Hall B tagger, which bends vertically, the
\hd\ tagger will deflect electrons in the horizontal plane, with both
the detector hall and the beam dump constructed above ground.

\begin{table}
\begin{center}
\begin{tabular}{lc}
\hline\hline
    Radius of curvature	    & 26.7 m \\
    Full-energy deflection  & 13.4$^{\mbox{\rm o}}$ \\
    Field at 12 \eunit	    & 1.5 Tesla \\
    Gap width		    & 2.0 cm \\
    Length of pole	    & 6.1 m \\
    Weight		    & 100 tons \\
    Length of focal plane (25\% to 90\% of $E_0$) & ~8.7 m \\
    Coil power	    	    & 800 A at 22 V \\
\hline\hline
\end{tabular}
\end{center}
\caption[Tagging spectrometer parameters]
{Design parameters for the tagging spectrometer dipole magnet}
\label{magnetpars}
\end{table}

The parameters of the magnet are given in Table~\ref{magnetpars}.
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 thin window along the length of the focal plane.

The detector package is divided into two parts: a set of 128 fixed
scintillation counters spanning the full energy range from 25\% to
92\%, and a movable ``microscope'' of 64 narrow counters that is
optimized for normal operation spanning the energy range from 70\%
to 75\%.

The fixed array provides access to the full tagged photon spectrum,
albeit at a modest energy resolution of 0.5\% and reduced photon
spectral intensity.  These detectors are well suited for running with
a broadband incoherent bremsstrahlung source.  They enable experiments
to be performed with the highest photon energies possible with the
source.  When running with a coherent source they play an essential
role in the crystal alignment procedure, and provide a continuous
monitor of the performance of the source.  The microscope is needed
in order to run the source in coherent mode at the highest polarization
and intensities, and whenever energy resolution better than 0.5\% is
required.  Using the microscope, the source is capable of producing
photon spectral intensities in excess of $2\times 10^8$ photons/\eunit,
although accidental tagging rates will limit normal operation to
somewhat less than this.


\subsection{Spectrometer optics}

Table~\ref{tmagprops} and Table~\ref{taggerres} give some of the beam
optics parameters as functions of the photon energy.  The energy
resolution and transverse position resolution were calculated using 
the following assumptions for the beam at the radiator:


\begin{center}
\begin{tabular}{c}
     $\Delta E/E_0$ = 0.080\% r.m.s. \\
     $\Delta x$  =   1.7\mmlunit\ r.m.s. \\
     $\Delta y$  =   0.5\mmlunit\ r.m.s. \\
     $\Delta x'$ =  .020 mr r.m.s. \\
     $\Delta y'$ =  .005 mr r.m.s. \\
\end{tabular}
\end{center}


The intrinsic energy resolution (i.e. the energy resolution
independent of detector size) is limited in most cases by the 0.08\% 
energy spread of the primary electron beam.


\begin{table}
\caption[Geometry of the tagger]{Geometrical parameters of the tagging spectrometer for 
$E_0 = 12$~\eunit :
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}
\begin{center}
\begin{tabular}{cccccc}
\hline\hline
   k   & Bend & Drift &  Angle & cm/\%$E_0$ & cm/\%$E_0$ \\
 (\eunit) & (deg) & (m)  &  (deg) &  perp.to ray
& along FP \\
\hline
   6  &  17.270 & 3.7790 &    6.035 &  1.467 & 13.956 \\
   7  &  17.664 & 3.2039 &    6.428 &  1.568 & 14.008 \\
   8  &  18.28 & 2.6276 &    6.992 &  1.716 & 14.096 \\
   9  &  19.108 & 2.0485 &    7.872 &  1.954 & 14.264 \\
  10  &  20.695 & 1.4626 &    9.459 &  2.407 & 14.644 \\
  11  &  24.608 & 0.8560 &   13.372 &  3.668 & 15.860 \\
\hline\hline
\end{tabular}
\end{center}
\label{tmagprops}
\end{table}

\begin{table}
\caption[Optical properties of the tagger]
{Optical properties and resolutions of the tagging spectrometer
at the focal plane, for $E_0 = 12$~\eunit :
(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.; 
$\Delta k_{beam}$ = r.m.s.\ energy resolution due to beam energy uncertainty;
$\Delta k_{spot}$ = r.m.s.\ energy resolution due to spot size on radiator;
$\Delta k_{tot}$ = total r.m.s.\ energy resolution excluding detector size;
$\Delta y_{tot}$ = transverse r.m.s.\ position resolution due to spot size
on radiator;
$y_{char}$ = transverse size corresponding to one characteristic electron 
angle $\theta_{Ce} = (m/E_0)(k/(E_0 - k))$.
}
\begin{center}
{\small
\begin{tabular}{ccccccccc}
\hline\hline
   k  &  (x x) & (y y) & (y y') & $\Delta k_{beam}$ & $\Delta k_{spot}$ &
 $\Delta k_{tot}$ & $\Delta y_{tot}$ & $y_{char}$ \\
{\scriptsize (\eunit)} &
{\scriptsize  (mm/mm)} &
{\scriptsize  (mm/mm)} &
{\scriptsize  (mm/mr)} &
{\scriptsize  (\%$E_0$)} & 
{\scriptsize  (\%$E_0$)} &
{\scriptsize  (\%$E_0$)} &
{\scriptsize  (mm)} &
{\scriptsize  (mm)} \\
\hline
\multicolumn{9}{l}{Without quadrupole:} \\
\hline
   6 & -0.701 & 2.737 & 18.882 & 0.080 &  0.081 &  0.114 &  1.372 &  0.804 \\
   7 & -0.667 & 2.708 & 16.538 & 0.080 &  0.072 &  0.108 &  1.357 &  0.986 \\
   8 & -0.625 & 2.670 & 14.178 & 0.080 &  0.062 &  0.101 &  1.337 &  1.207 \\
   9 & -0.569 & 2.617 & 11.788 & 0.080 &  0.050 &  0.094 &  1.310 &  1.506 \\
  10 & -0.494 & 2.539 & 9.341 & 0.080 &  0.035 &  0.087 &  1.270 &  1.989 \\
  11 & -0.389 & 2.402 &  6.745 & 0.080 &  0.018 &  0.082 &  1.201 &  3.159 \\
\hline
\multicolumn{9}{l}{With quadrupole:  (length = 50 cm, gradient = -0.4 kGauss/cm)}\\
\hline
   6 & -0.628 &  0.451 & 17.622 & 0.080 &  0.073 &  0.108 &  0.242 & 0.750 \\
   7 & -0.583 &  0.348 & 15.121 & 0.080 &  0.063 &  0.102 &  0.190 & 0.901 \\
   8 & -0.526 &  0.202 & 12.535 & 0.080 &  0.052 &  0.095 &  0.119 & 1.068 \\
   9 & -0.449 &  -0.024 & 9.792 & 0.080 &  0.039 &  0.089 &  0.050 & 1.251 \\
  10 & -0.338 & -0.427 &  6.699 & 0.080 &  0.024 &  0.083 &  0.216 & 1.426 \\
  11 & -0.162 & -0.416 &  2.474 & 0.080 &  0.008 &  0.080 &  0.708 & 1.159 \\
\hline\hline
\end{tabular}
}
\end{center}
\label{taggerres}
\end{table}



At the focal plane, the characteristic bremsstrahlung angle corresponds to
a few millimeters of transverse displacement.  The vertical beam spot
size at the radiator (0.5\mmlunit\ r.m.s.) contributes a
comparable amount because of the large transverse magnification in the
dipole transport matrix.  However, placing a quadrupole magnet between
the radiator and the tagger dipole magnet reduces this magnification
nearly to zero over a substantial range of photon energies without
substantially changing the other optical properties.  Including the
quadrupole in the design makes possible a future upgrade of the photon
source to employ tagging detectors with two-dimensional readout.


\subsection{Tagger detectors}


\subsubsection{Focal plane microscope detectors}

The design energy resolution of 0.1\% r.m.s.\ (see Table~\ref{taggerres})
could be met by non-overlapping detectors which span an energy range
of 0.2\% each (the r.m.s.\ deviation of a flat distribution of
width $W$ is $W/\sqrt{12} = 0.29 W$.) The principal limitation on detector
size is imposed by the design goal of tagging collimated photons at rates
up to 100 MHz over the coherent peak.  The nominal
collimated coherent peak has its highest intensity between about 8.5
and 9 \eunit\ (see Fig.~\ref{collimcuts}). 
However, the the tagger sees both collimated and uncollimated photons,
and the total tagging rate in this region is about twice the collimated
rate (see Fig.~\ref{collimcuts}), about 200 MHz.
If a single phototube can count reliably at 5 MHz,
then the energy range detected by a single scintillation counter should be
5 MHz / 200 MHz * 600 \Meunit\ = 15 \Meunit = 0.12\% of $E_0$.  Thus a
detector width of 0.1\% would satisfy the requirements of both resolution
and rate.  
According to Table~\ref{tmagprops}, such a detector would have a width 
of 2.5 \mmlunit\ perpendicular to the electron direction, and detectors for
individual channels would be spaced about 17 \mmlunit\ apart along the
focal plane.







\subsubsection{Fixed focal plane array}

Tagging of photons over the full range from 25\% to 90\% of $E_0$
is not required as part of the physics program here proposed for \hd,
but is desirable for two separate reasons.  First, it will increase the
flexibility of the source by allowing other possible experiments using
highly polarized photons below 8 \eunit, or incoherent bremsstrahlung
up to 11 \eunit.  This capability is not available elsewhere at JLab,
since the Hall B tagger will not handle beam energies above 6 \eunit\
and its photon polarization will be quite low for photon energies above
about 4.5 \eunit.  Second, the process of aligning the crystal radiator
for coherent bremsstrahlung requires rotation about several axes and
rapid observation of the resulting energy spectra, as described in
section ~\ref{alignment}.
The low-energy portion of the spectrum, between about
25\% and 50\% of $E_0$, is most sensitive to
these rotations, and experience with the coherent bremsstrahlung beam
at Mainz \cite{MainzTS}-\cite{MainzFP} indicates that the
alignment process would be severely compromised if photon energies
below 0.5 $E_0$ were not measurable.

A high energy resolution in the tagger is not deemed necessary for 
these purposes.  A counter width spanning 0.5\% of $E_0$ is considered
sufficient in most cases.  For operations with an amorphous radiator,
these counters would be capable of running a broad-band photon beam
with the highest intensities compatible with tagging.  Crystal alignment
procedures are not normally carried out at full source intensity, so
rate capacity is not a limitation in that application.  If a need
arose to operate the source in collimated-coherent mode at lower
photon energies, for example to obtain increased polarization, then
full-intensity operation would always be possible by repositioning the
microscope on the focal plane.  In this case, the fixed array would
be useful in the energy calibration of the movable segment.
When used as the primary tagging detectors, the fixed array
would be capable of pre-collimated intensities up to 80 MHz/\eunit.


\subsection{Beam dump optics} 

Although the full-energy beam leaving the tagger magnet is diverging in 
both directions, the range of angles is small enough that the 
beam does not blow up rapidly. For a dump distance of 30 m the r.m.s.\
beam size is 6.3\mmlunit\ horizontal
(dominated by the 0.08\% beam energy spread) and 0.7\mmlunit\
vertical (combination of vertical spot size and
multiple scattering in a $10^{-4}$ radiation length radiator.)\ \ 
These values scale
approximately linearly with distance from the magnet to the dump, and
are not very sensitive either to the quadrupole or to small rotations of 
the exit edge of the tagger magnet.  Thus the beam dump design is quite
insensitive to the beam optics, and depends only on the lateral and 
longitudinal spread of the shower in the absorber.

