
\section{Photon Source}

A horizontal plan view of the photon beam line is shown in 
Fig.~\ref{photon_line} with the major components labeled.
The electron beam enters the figure from below ground at the left and
is bent into the horizontal plane to enter the tagger building.  There
it passes through two small dipoles to impinge upon the bremsstrahlung
radiator.  After its exit from the radiator, the electron beam passes
into the tagger magnet where the primary beam is bent in the direction
of the electron beam dump.  The radiator crystal is thin enough that
most of the electrons lose less energy in traversing the radiator than
the intrinsic energy spread of the incident beam.
Those electrons which lose a significant fraction of their initial energy
inside the radiator do so by emitting a single bremsstrahlung photon.
These degraded electrons are bent out of the primary beam inside
the tagger magnet and exit the vacuum through a thin window,
passing through air for a short distance to strike the focal plane of
the spectrometer.  The primary electron beam is contained inside vacuum
all the way to the dump.  The two dipole magnets shown in the figure
downstream of the tagger may not be necessary if the deflection angle
of the primary beam inside the tagger magnet is deemed to be sufficient.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=photon_line.eps,width=\textwidth}}
\caption
[Schematic plan view of the photon beam line]
{Schematic plan view of the photon beam line, shown in the horizontal
plane as viewed from above.  The objects in this figure are not drawn
to scale.}
\label{photon_line}
\end{center}
\end{figure}

The photons that are produced in the radiator pass through a small hole
bored in the return yoke of the tagger magnet to exit the vacuum
through a thin window in the forward direction.  They then pass into
a transfer pipe, which may either be evacuated or filled with helium
to reduce photon beam degradation due to interactions, and travel to
the experimental hall.  Just before entering the hall the photon beam
passes through a system of collimators and sweeping magnets.  In the
figure they are shown in a separate enclosure for shielding purposes.
The primary collimator is first.  It defines the part of the photon beam
that is allowed to reach the target. Debris from interactions along
the inside surface of the collimator bore forms a halo around the
photon beam that exits the collimator.  The charged component of the
halo is deflected away from the beam axis by a dipole ``sweeping'' magnet
just downstream of the collimator.  A secondary collimator follows the
sweeping magnet to stop the deflected shower particles and block the
halo of secondary photons generated by the first collimator.  The
secondary collimator is of a larger diameter than the primary and so
sees a reduced rate of secondary interactions on the inner surface of
the hole.  What new showers are generated there are cleaned up by a
second sweeping magnet.  The beam then passes through a final
collimating aperture into the experimental hall.  This triple-collimation
system was copied from the setup developed at SLAC \cite{Kaune}.

The collimated photon beam, now only a few \mmlunit\ in diameter, is delivered
to the experimental target.  After passing through of order 3\% radiation
lengths of target, the photon beam passes through the detector and into
the photon beam dump at the back of the hall.  Based upon a design
upper limit of 60kW (5$\mu$A at 12 \eunit) being delivered to the electron
beam dump, the total power in the photon beam is at most 1.5W in the
experimental hall and at most 15W in the collimator enclosure.



\subsection{Essential features}

The adjective `coherent' in coherent bremsstrahlung does not indicate
that the photons in the beam are in a coherent state, as is light from
a laser.  Rather it refers to the coherent effect of multiple atoms in
a crystal lattice in absorbing the recoil momentum from a high energy
electron when it radiates a bremsstrahlung photon.  In X-ray spectroscopy
one encounters the same thing in the M\"ossbauer effect, except in that
case the chief physical consequence is the disappearance of the recoil
Doppler shift from the photoabsorption/emission spectrum.  Here the chief
consequence is the enhancement of bremsstrahlung at those particular
kinematics for which the recoil momentum matches one of the reciprocal
lattice vectors of the crystal.

Another useful way to view the process of coherent bremsstrahlung is
as virtual Compton scattering.  To the high energy electron, the atoms
in the radiator appear as clouds of virtual photons.  For a disoriented
radiator material, the virtual photon spectrum is given simply by the
atomic form factor, averaged over the different species in the material.
If the radiator is a single crystal, however, the atomic form factor
gets multiplied by the form factor of the crystal, which in the ideal
case looks like a series of delta-functions located at the sites of
the reciprocal lattice.  In effect, the crystal provides a set of virtual
laser beams, each one a standing wave tuned to a specific reciprocal
lattice vector.  In this view the process of hard bremsstrahlung is seen
to be the same as Compton back-scattering of laser light.  For a more
detailed discussion of the physics of coherent bremsstrahlung there are
a number of good references  \cite{Kaune,Bilokon,Diambrini,Timm}.

The use of Compton back-scattering of laser light as a photon source
was earlier noted as ruled out by the limitation of high-power lasers and
cavities to wavelengths above 100 \micron.  The characteristic wavelength of
the crystal photons is a few Angstroms, three orders of magnitude shorter.
In this case, 180$^{\circ}$ scattering would result in essentially 100\%
of the electron beam momentum being transfered to the photon in the lab
frame.  However, the Compton cross section contains a factor
of $1/(\vec{q}\cdot\vec{p} )^2$ where $\vec{q}$ is the virtual
photon momentum and $\vec{p}$  is that of the electron, which
strongly favors incident photons with $\vec{q}$  nearly orthogonal
to $ \vec{p} $ .  With reciprocal lattice vectors pointing in almost
every direction, only those nearly perpendicular to the beam contribute
appreciably to the scattering rate.  This fact applies equally to
ordinary bremsstrahlung; in fact, to a first approximation the 
bremsstrahlung spectrum from a single crystal is the same as from a
disoriented radiator.  The reason is that, if the sum over crystal
momenta were replaced with a continuous integral, one would recover the
ordinary bremsstrahlung result for isolated atoms.  Beyond a few unit
cells from the origin in reciprocal lattice space, the atomic form
factor and kinematic factors become slowly varying on the scale of the
lattice spacing, and the sum becomes indistinguishable from the integral.
Besides that, the uncertainty principle requires that atoms localized
at the sites in a crystal undergo fluctuations about their mean position.
This has the effect of attenuating the discrete peaks in the crystal form
factor at progressively higher-order crystal momenta, eventually washing
them out and filling in the gaps between them, so that the sum deforms
smoothly into the integral at high momentum transfer.
Hence, the sum over crystal indices that yields the final
photon spectrum can be separated into two parts: a discrete sum over
a limited set of small crystal indices and an integral over the continuum
of momentum transfer values beyond.  The latter appears in the coherent
bremsstrahlung beam as an ordinary $ 1/k $  bremsstrahlung spectrum, while
the former appears as a set of peak structures superimposed upon it.
The $ 1/k $ continuum, referred to as the incoherent component, is
invariant as the crystal is rotated, whereas the coherent peaks change in
position and intensity, depending on crystal orientation.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=uncolspec.eps,width=\textwidth}}
\caption
[Coherent bremsstrahlung spectrum]
{Coherent bremsstrahlung spectrum, calculated for a diamond crystal
radiator $15\mu$m  thick and a 1$\mu$A electron beam of 12 \eunit\ energy.
Typical values were used for beam emittance and crystal mosaic spread.}
\label{uncolspec}
\end{center}
\end{figure}

A typical coherent bremsstrahlung spectrum is shown in
Fig.~\ref{uncolspec}.  The distinction between
incoherent and coherent components in the figure is artificial;
it is there to show the part of the spectrum that shifts as the
crystal is rotated.  The vertical scale in the figure gives the 
photon rate for the given beam current and crystal thickness.
Note that the intensity of the incoherent background is less than
what would be obtained with an amorphous carbon radiator of the
same thickness, because a part of the momentum transfer integral
in the Bethe-Heitler formula has been moved into the discrete
sum and contributes to the coherent part.  The radiation length
of diamond is actually an average over all orientations of the
crystal.  In the calculation for Fig.~\ref{uncolspec}
the leading 400 lattice sites were included in the discrete part
of the calculation, although it can be seen that only two or three
of them contribute with sufficient intensity to be individually
visible in the spectrum.


\subsection{Use of collimation}


The presence of the large incoherent flux in Fig.~\ref{uncolspec}
presents a significant handicap to
a photoproduction experiment.  Not only do the low energy photons
produce background in the detector, but they diminish the
polarization of the beam.  The entire beam polarization appears
in the coherent component; the underlying incoherent flux only
serves to dilute the polarization.  There is another difference
between the two components that allows them to be separated to
some extent.  The kinematics of bremsstrahlung confines most of the
intensity of the photon beam to forward angles within $ m/E $
radians of the incident electron direction.  In the lab this is
a small angle, but in the rest frame of the electron-photon system
it subtends all angles in the forward hemisphere.  A peak in the
coherent component that corresponds to a single reciprocal lattice
vector has two-body kinematics, so there is a well-defined relation
between the emission angle and the energy of the emitted photon in
the lab: emission at $ 0^{\circ} $ yields a maximum energy photon,
with energy decreasing with increasing angle. This accounts for the
shape of the coherent peaks in Fig.~\ref{uncolspec},
with the sharp right-hand edge corresponding to $ 0^{\circ} $ emission
and the tail to lower energies corresponding to scattering into 
higher angles.


The incoherent component, because it comes from a sum over momentum
transfers at all angles, has essentially no correlation between
photon energy and emission angle.  This means that collimating away
all photons beyond some angle 
$ \theta_{max} < m/E $ uniformly
attenuates the incoherent spectrum at all energies, whereas it preserves
all of the coherent photons from the maximum energy for the given peak
down to some cutoff.  The kinematic relations for coherent bremsstrahlung
are as follows,
\begin{equation}
\theta^2 + 1 = \left(\frac{1-x}{x} \right) 
\left( \frac{x_{max}}{1 - x_{max}} \right)
\label{rtj:eqn1}
\end{equation}
\begin{equation}
x_{max} = \frac{2 \vec{p}\cdot\vec{q} }
               {2 \vec{p}\cdot\vec{q} - m_e^2}
\label{rtj:eqn2}
\end{equation}
where $ x $ is the photon energy in units of the incident electron
energy and $ \theta $ is the lab emission angle
of the photon relative to the incident electron momentum axis, in units
of $ m/E $ .

\begin{figure}
\begin{center}
\mbox{\epsfig{file=collimcuts.eps,width=\textwidth}}
\caption
[Coherent bremsstrahlung spectrum after collimation.]
{Coherent bremsstrahlung spectrum, calculated under the same
conditions as in Fig.~\ref{uncolspec} , after collimation.  The upper
curve is the uncollimated spectrum from Fig.~\ref{uncolspec}.  The middle
curve corresponds to a 5mm diameter collimator placed 80m downstream of
the radiator, or approximately 0.75 $ m/E $ in collimator half-angle.  The
lower curve corresponds to a 2mm collimator in the same position,
approximately 0.30 $ m/E $}
\label{collimcuts}
\end{center}
\end{figure}

The effects of collimation are demonstrated in the calculated spectra
shown in Fig.~\ref{collimcuts}.  First, note that the
collimation angles are very small, which requires a long flight path
of order 100\mlunit\ in order that the collimator can be larger than the
intrinsic beam spot size, otherwise the collimator is cutting in
transverse coordinate instead of in angle.  This distance is, in fact,
a sensitive function of the electron beam emittance from the machine,
and must be increased in inverse proportion to the beam emittance if
the effectiveness of collimation is to be maintained.  This issue,
along with the associated demands placed on beam alignment and
position stability, are taken up in more detail in the following
section on the electron beam line.  


Second, note that the cut
imposed on the coherent peak by collimation does not produce a
perfectly sharp edge as would be expected from two-body kinematics.
This is because a collimator cuts on radius at some fixed position
which translates into a cut on emission angle only in an approximate
way, and is why the curves in Fig.~\ref{collimcuts}
are labeled by their collimator size and distance individually rather
than their ratio which is the nominal collimation angle.  
Multiple scattering by the electron in the radiator prior to emission
and beam spot size and divergence are the major contributors to the 
error involved in interpreting a collimator as a cut in emission
angle.  All of these effects have been incorporated into the analytical
calculation of the yields from a collimated coherent bremsstrahlung
source that has been used in preparing this report.  Crystal 
imperfections, which amount to an intrinsic spread in the direction
of the incoming virtual photon, are also accounted for in the
calculation. 

Third, note that the relatively weak collimation at 5\mmlunit\ reduces the
incoherent background without significantly affecting the coherent
flux near the maximum, and thereby almost doubling the polarization
of the beam at the peak relative to the uncollimated case.  Further
reducing the collimator diameter continues to narrow the peak and
reduce the incoherent flux relative to the peak, albeit at some cost
in peak intensity.  In the case of the 2\mmlunit\ collimator, the peak
polarization is essentially that of the coherent component alone.
For this collimator diameter most of the photon beam energy is being
absorbed at the collimator.  However the necessary flux in the
collimated beam can be made up by increasing the electron beam current.
provided there is adequate shielding around the collimator.
The peak in Fig.~\ref{collimcuts} for a 2\mmlunit\ collimator contains
20M photons/s for an electron beam current of 1$\mu$A, which could be
increased by a factor of up to 3 in the facility being planned for \hd.

Fourth, note that the rate seen in the focal plane of the tagging
spectrometer corresponds to the upper curve in Fig.~\ref{collimcuts},
regardless of the collimation.  This means that collimating the bremsstrahlung
beam increases the rate in the tagger focal plane relative to what
is seen at the detector.  For example, running with the 2mm collimator
and $10^7$  photons/s in the main coherent peak implies according to
Fig.~\ref{collimcuts}  a rate of 70\mhz\ per \eunit\ in the focal plane.
Under these conditions, only a small part of the focal plane is in
coincidence with the events of interest, which limits the rate of
accidental tags to a reasonable level.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=pol_spectrum.eps,width=\textwidth}}
\caption
[Plane polarization of the coherent bremsstrahlung beam]
{Plane polarization of the coherent bremsstrahlung beam, calculated
under the same conditions as in Fig.~\ref{collimcuts}.  The dashed lines
indicate the trajectory of the peak polarization as the peak energy is swept
across the focal plane by rotating the crystal.}
\label{polspec}
\end{center}
\end{figure}


The net polarization of the beam under different collimation
conditions is shown in Fig.~\ref{polspec}.  The
dashed curves show how the maximum polarization in the peak varies
as the peak energy is changed by rotating the crystal.  The
polarization in all cases is zero at the endpoint.  Without
collimation it rises as
$(k-E)^2$ , one power
coming from the intensity of the coherent peak relative to the
incoherent component going to zero linearly at the endpoint, and the
other from the intrinsic polarization of the coherent photons also
behaving like $(k-E)$ near the endpoint.  Collimation allows one
to essentially isolate the coherent component, so that the polarization
available to the experiment rises from zero at the endpoint in a
linear fashion.  The dashed curves in Fig.~\ref{polspec} 
demonstrate this point.

In order to obtain the full polarization enhancement from collimation, it
is necessary to have a distance between the radiator and collimator on
the order of 100\mlunit.  This distance scale is set by the requirement that
the collimator aperture must be large compared to the virtual electron
beam spot on the collimator but small compared to the actual photon spot
size.  Fig.~\ref{radcoldist} shows the maximum polarization as a function
of radiator-collimator distance for a coherent peak at 9 \eunit.  The 
collimator diameter is adjusted in this calculation to keep the
collimation half-angle at 0.5 $m/E$.  At zero
distance the collimator has no effect except to attenuate the beam, and
so the uncollimated polarization from coherent bremsstrahlung is obtained.
At 100m separation distance the polarization enhancement has saturated.
The design for \hd\ calls for a radiator-collimator distance of 80\mlunit.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=radcoldist.eps,width=\textwidth}}
\caption
[Maximum polarization {\em vs} radiator-collimator distance.]
{Maximum polarization {\em vs} radiator-collimator distance
for a coherent peak at 9 \eunit.  The collimator diameter is varied in
this calculation to keep the collimation half-angle at 0.5 $m/E$.}
\label{radcoldist}
\end{center}
\end{figure}


\subsection{Choice of radiator}


The ideal radiator would be a layered structure with a strong
transverse fields that alternate between layers spaced about
50\nmlunit\ apart, thus simulating the standing wave in a cavity driven
by a 15 \eeunit\ laser.  While it is possible to construct ordered
materials with unit cells as large as this, the self-shielding
of atoms means that beyond the atomic length scale the residual
fields are comparatively weak.  Hence heterogeneous structures
are not viable for use as a coherent radiator.  Since the strong
fields inside a solid are revealed at the atomic scale, the first
requirement for a good radiator is that the unit cell be compact
and closely packed.  The best radiators are those with the smallest
unit cells because these provide the best match between the atomic
and the crystal form factors.  This match is best for the light
elements, and essentially prohibits the effectiveness of any materials
heavier than carbon.  An extensive survey of possible radiator
materials is presented in Ref. \cite{Bilokon}.
In Table~\ref{radmat} is shown the figure of merit that
those authors report for favored crystalline materials.  The figure
of merit is the product of the atomic times the crystal form factor
evaluated at the leading peak, normalized to the value for diamond.

\begin{table}
\begin{center}
\begin{tabular*}{4.00in}{@{\extracolsep{.25in}}ccc} \\ \hline \hline
crystal    & best reciprocal vector & $ {\rm{P/P_{diamond}}} $ \\ \hline
diamond   & 2,-2,0 & 1.00 \\ \hline
beryllium & 0,0,2  & 0.86 \\ \hline
boron     & 2,0,8  & 0.38 \\ \hline
silicon   & 2,-2,0 & 0.19 \\ \hline
${\rm{Be_2C}}$ & 2,2,0 & 1.10 \\ \hline \hline
\end{tabular*}
\caption
[Figure of merit of various materials that might be used as a
coherent bremsstrahlung radiator.]
{Figure of merit of various materials that might be used as a
coherent bremsstrahlung radiator.  This table is reproduced from
Table 2 in  H. Bilokon {\em{ et al.}}
\label{radmat}}
\end{center}
\end{table}

Table \ref{radmat} shows that the list of viable materials for a crystal
radiator is relatively short.  Silicon would be an excellent choice
from the point of view of price and fabrication, but unfortunately
it is far inferior in terms of performance.  Beryllium carbide is
not a material that is familiar to the crystal growth industry,
and nothing is known at present concerning its suitability for the
growth of single crystals of large area.  In general compound
materials are more susceptible to radiation damage than are pure
elements, which would argue in favor of diamond and beryllium metal.
These two materials are comparable in terms of their performance.


Most of the experience to date with coherent bremsstrahlung has
been with diamond radiators.  Extensive expertise for handling
large diamond crystals exists within the gem industry.  High
quality monocrystals of  6\mmlunit\ $\times$ 6\mmlunit\ area
were obtained by SLAC \cite{Kaune} some time ago from a jeweller,
who was able to cleave a thin wafer from the face of a large
high-quality gem.  Those wafers that passed quality checks were
then milled down to less than 100~\micron\ using chemical erosion
in an atomic oxygen discharge.  Now three decades later, it is not
clear whether this same route can be travelled again, as it owed
its success in a large measure to a friendly contact within the gem
business at that time.  The cost of the entire gems from which the
shavings were taken would have been prohibitive.  Today other
methods exist for growing single crystals which might be capable
of producing diamonds of sufficient size and quality for this
application.  A variety of potential sources for single crystals 
is being pursued.


Beryllium is another material that might be used as a crystal radiator.
Beryllium metal is widely used in industry, being preferred for its
high strength-to-weight ratio and robustness, in addition to its
transparency to X-rays.  Thin films of high-purity beryllium are
routinely produced for vacuum window applications, which use some
of the same vacuum deposition techniques that would be used for the
growth of single crystals.  As a radiator material, beryllium is
distinguished as the metal with the highest Debye temperature,
around $ 1400^{\circ} $K .  The Debye temperature measures the temperature at
which the thermal motion of the atoms in the lattice reaches the
level of the zero-point motion due to their confinement in the
lattice, as dictated by the uncertainty principle.  A high Debye
temperature indicates a stiff crystal lattice, in which the atoms
have little liberty to move and so have large momentum fluctuations,
as dictated by the uncertainty principle. 


A high Debye temperature is important for a bremsstrahlung radiator
material for three reasons.  First, the cross section for coherent
bremsstrahlung from a discrete crystal momentum vector $ \vec{q} $
contains a factor $ e^{-q^2 / 4 M \theta_D } $
which reflects the fact that position fluctuations of atoms in the
lattice diminish the coherent effect.  This factor is near unity for 
the low-order crystal momenta provided that 
$ \theta $ is
sufficiently large.  Second, the Debye temperature is, roughly speaking,
a measure of the stability of the crystal structure and hence its
readiness to form as a single crystal and to survive significant doses
of radiation.  Third, the radiator material will inevitably be heated
by the beam, and will normally operate in vacuum well above the ambient
temperature.  A high Debye temperature means that there is a large
range of temperatures over which the material may operate without
degraded performance as a crystal radiator.  The Debye temperature
of diamond is about $ 2200^{\circ} $ K.  Past experience has shown that diamond
meets all of the requirements for a good crystal radiator.  There
is good reason to believe that beryllium would function almost as
well at substantially less cost, but this remains to be proved.

\subsection{Crystal thickness}

The range of permissible thicknesses for a crystal radiator is bounded
both from above and below.  It is bounded from above by multiple
scattering of the electron beam as it passes through the radiator,
which causes the divergence of the incident beam to grow, thereby
enlarging the photon beam spot on the collimator face and degrading
the degree to which collimation discriminates against the incoherent
component in favor of the coherent part.  It is bounded from below
by the fact that the crystal must be of some minimum thickness in order
to achieve the full coherent gain.  In the calculation of the coherent
bremsstrahlung process one begins by assuming an infinite crystal,
although practically it is presumed to mean only that the crystal is
large compared to the some characteristic scale.  It is important to
identify what the characteristic scale is in this problem in order to
know how thin one can make the crystal without hurting performance.
In the analogous case of the M\"ossbauer effect, one can estimate the
number of atoms participating in the collective absorption by looking
at the emission time of the photon (lifetime of the radiating transition)
and asking how many nuclei lie within the envelope of the photon wave
packet.  In the coherent bremsstrahlung process, the lifetime of the
radiating system is given in the lab system by the uncertainty
principle and by how far the electron energy deviates from its on-shell
value between absorbing the virtual photon and emitting the real one.
The latter quantity is almost exactly given by $ q_z$ ,
the virtual photon momentum component along the incident electron axis,
which means that the electron travels a distance
$ \lambda = \hbar c / q_z $
during the interaction.
For a given coherent peak at normalized energy $ x $ in the photon
spectrum, the coherence length is given by
\begin{equation}
\lambda = \frac{2 e (1-x) }{x m^2}
\label{rtj:eqn3}
\end{equation}
in units of $ \hbar c $ .  From this simple
argument one sees that the coherent gain goes linearly to zero at the
endpoint, a result that is borne out by the full QED calculation.
One also sees that the lower limit on crystal thickness imposed by
the coherence length depends upon both the electron beam energy and
the photon energy.  For a 12 \eunit\ beam energy and a 6 \eunit\
coherent photon
the coherence length is 18\nmlunit, or about 50 unit cells for diamond.  This
shows that the coherence length does not impose a practical limit on
how thin the radiator should be.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=radthick.eps,width=\textwidth}}
\caption
[Collimated coherent bremsstrahlung spectrum ]
{Collimated coherent bremsstrahlung spectrum from a 1$\mu$A electron
beam at 12 \eunit\ using diamond radiators of two different thicknesses.
The calculation assumes a  2\mmlunit\ collimator located 80\mlunit\
from the radiator,
and typical values for beam emittance and crystal quality.}
\label{radthick}
\end{center}
\end{figure}

The effects of multiple scattering are best presented by showing
the calculated spectra for various radiator thicknesses.
In Fig.~\ref{radthick} is shown the photon spectrum for a 20 \micron\
and a 100 \micron\ radiator to demonstrate the effect.  The 100 \micron\
spectrum is scaled down by a factor of 5 to facilitate the comparison.
The calculation assumed a 2\mmlunit\ collimator located 80\mlunit\ 
downstream of the radiator.  The loss in normalized intensity with the
thicker radiator, as well as the broadening of the left edge of the peak,
is due to the enlarging of the photon beam spot on the collimator face
from multiple scattering of the electron beam in the crystal prior to
radiation.  A 100 \micron\ diamond crystal is roughly $10^{-3}$ radiation
lengths thick.  It is reasonable to run with as thick a radiator as is
permitted by multiple scattering considerations, in order to keep the beam
current to a minimum and prolong as much as possible the lifetime of the
crystal.

\subsection{Crystal mount}

It has already been shown that in order to achieve appreciable coherent
gain the crystal must be oriented so that the coherent peaks appear
well below the end point.  Eqn.~\ref{rtj:eqn2} then implies that
the orientation must be such that the crystal momentum dotted with the
beam momentum be of order $m^2$ .
Given a $ p $  of 12 \eunit\ and $ q $ of 10\keunit, this requires that the
two vectors must be within 100$\mu$r of perpendicular to each other and
that, within a range of angles of that order, the coherent peak sweeps out
nearly the full range in x from 0 to 1.  Hence both the incident beam
angle and the angle of the crystal radiator must be stable at the level
of a few $\mu$r  in order to have a stable photon beam for running.  At
least one of the two must be adjustable in steps that are small compared
to 100$\mu$r  in order to have the ability to position the coherent peak 
at the desired energy.  These requirements are most simply achieved using
a precision two-axis goniometer.  Commercial devices of this type exist
with step sizes as small as required for this application.  With two-axis
remote control the crystal can be reoriented in a matter of minutes to
produce a coherent beam of any desired energy and in either vertical or
horizontal plane polarization modes.


Thermal effects in the crystal mount and small variations in the beam
tune from the accelerator make long term stability and reproducibility
more difficult to achieve than adequate controls for the initial setup.
For this reason it is important that there be sufficient diagnostics
present to monitor the crystal orientation relative to the beam.  This
diagnostic is provided by the rates in the tagger focal plane counters.
From Fig.~\ref{uncolspec} it is clear that the critical
angle between the beam and the crystal, as well as the quality of the
crystal, are accurately reflected in the position and sharpness of the
coherent edge in the focal plane rate profile.  Better than simply
monitoring the critical angle, the tagger profile provides a direct
look at the photon beam spectrum prior to collimation.  This means that
there is no need for an absolute alignment of the crystal and goniometer
to this level of precision, because only relative accuracy is required.
The second angle is non critical, typically of order a degree, and does
not need to be monitored.  If conditions require it, a feedback system
could be put in place to correct for drifts seen in the tagger focal
plane profile by periodically adjusting the goniometer setting within
predefined limits.  This would involve a slow control system, and does
not need to be designed in advance.

\subsection{Crystal quality and lifetime}

There are two considerations that fall under the heading of crystal
quality.  The first respects the quality of the original crystal,
and the second is with regard to degradation from radiation damage.
In the calculation of the coherent bremsstrahlung spectrum a perfect
single crystal was assumed, but even the very best crystals have some
dislocations and other defects.  Besides locally disrupting the
regularity of the crystal, these defects impose stresses which
produce small ripples in the crystal planes.  If these ripples were
amplified, the surface of a crystal would appear like a mosaic of
planar regions with approximately parallel surfaces.  The scale of
deviations from planarity across the face of a single crystal is
termed the {\em{mosaic spread}} of the crystal.  The mosaic spread
contributes in the same way as beam divergence to the smearing in the
exact energy-angle relation for coherent photons.

The mosaic spread of a crystal can be measured by means of X-ray
diffraction.  At the time when the SLAC group was preparing their
samples, a laboratory at the National Bureau of Standards had a
state-of-the-art facility for these measurements.  Their source of
high-quality natural diamonds cleaved several samples and sent them
to the NBS for rocking-curve measurements.  Most of the crystals
failed the tests, but among them were a few of sufficiently low
mosaic spread to be acceptable.  Ref.~\cite{Kaune} does not report
a value for the mosaic spread of the radiator, but from the sharpness
of the coherent peaks in the beam spectra shown in the paper, one 
can infer that crystals with less than 50 $\mu$r mosaic spread were
obtained in this way.  Other coherent bremsstrahlung facilities that
have operated at lower energy are less sensitive to mosaic spread
because the characteristic angle scales like $m/E$.  The calculations
for this report assumed a mosaic spread of 50 $\mu$r for the crystal
radiator.


The SLAC group found that the performance of their diamonds had
degraded considerably after a total charge of 3 Coulombs had been
accumulated over a surface of 6\mmlunit\ $\times$  6\mmlunit\  
in area.  They were able to
recover a better performance for the damaged crystals by putting them
through an annealing process.  This translates to five weeks of running,
or about two months of real time at 1$\mu$A of current, before the
crystal would need to be removed for treatment.  A small set of
radiators of different thicknesses would accommodate a run plan
that includes some running with maximum polarization, i.e. narrow
collimation, a thin radiator and microamps of beam current, and running
the rest of the time with a thicker radiator, less beam current and more
modest collimation.

