
\chapter{Photon source}\label{chapter_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 tagging spectrometer
where the primary beam is bent in the direction
of the electron beam dump.  The radiator crystal is thin enough that
the average energy loss by the electrons in traversing the radiator is less
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.  


\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_photon_line.eps,width=\textwidth}}
\caption
[Schematic plan view of the photon beam line]{\label{photon_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.}
\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 in the forward direction.
They then pass into an evacuated photon beam pipe and travel to
the experimental hall.  Just before entering the hall the photon beam
passes through a system of collimators and sweeping magnets.
They are housed 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 primary 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 block of shielding
material into the experimental hall.  This triple-collimation
system is similar to the setup at the SLAC coherent bremsstrahlung
beam line \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 $60$\kpwunit{} ($5$\miamps{} at $12$\eunit) being delivered to the 
electron beam dump, the total power in the photon beam is not more than 
$1.5$\pwunit{} in the experimental hall and not more than $15$\pwunit{} in 
the collimator enclosure.



\section{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 disordered
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 transferred 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
disordered 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=ch4_uncolspec.eps,width=\textwidth}}
\caption
[Coherent bremsstrahlung spectrum]{\label{uncolspec}
Uncollimated coherent bremsstrahlung spectrum, calculated for a diamond
crystal radiator $15$\micron{}  thick and a $1$\miamps{} electron beam of
$12$\eunit{} energy.  Typical values are used for beam emittance and
crystal mosaic spread.}
\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.


\section{Use of collimation}


The presence of the large incoherent continuum in Fig.~\ref{uncolspec}
presents a significant handicap to
a photoproduction experiment.  Not only do the continuum 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.  This is true both for
the incoherent and coherent components.  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.  The difference
lies in the fact that a peak in the
coherent component corresponding 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 emission at
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=ch4_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 $5$\mmlunit\ diameter collimator placed 
$80$\mlunit{} downstream of the radiator, or approximately $0.75\, m/E$ in 
collimator half-angle.  The lower curve corresponds to a $3.4$\mmlunit{} 
collimator in the same position, approximately $0.50\, m/E $. For the 
$3.4$\mmlunit\ collimator there are approximately $3.3\times 10^{7} \gamma/s$ 
in the primary peak for a nominal electron beam current of $1$\miamps{} and 
crystal thickness of $15$\micron.}
\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 
held constant.  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 the collimator cuts on radius at some fixed 
distance
which translates into a cut on emission angle only in an approximate
way. Thus 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 translating a collimator
radius into a cut on 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 
taken into account 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. 

The $3.4$\mmlunit{} collimator diameter has been chosen for this
design because it provides for a maximum reduction in the incoherent
flux while transmitting $95$\% of the coherent flux at the peak.
Most of the total photon beam energy coming from the crystal
is absorbed by the collimator.
For this reason the collimator is located in a separate enclosure
outside the experimental hall, and must be surrounded by a considerable
amount of shielding.
The peak in Fig.~\ref{collimcuts} for a $3.4$\mmlunit\
collimator contains 33M photons/s for an electron beam
current of $1$\miamps , which will be increased by a factor
of  $3$ for full-intensity running of the \GX{} experiment in \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 full-intensity running at $10^8$ photons/s on target in the
coherent peak, Fig.~\ref{collimcuts} implies a rate of $240$\mhz{} in the
focal plane within a $600$\Meunit\ window around the peak.
Combining this rate with the beam pulse
spacing of 2ns leads to an accidental tagging rate of about $50$\% and
to a fraction of ambiguous tags of $40$\%.  Even with ideal electronics
the per-second yield of single-tag events is close to saturation at
this intensity.  The detector and tagging spectrometer design are based 
upon a maximum rate of $10^{8}$ photons/s on target and $400$\mhz{} per 
\eunit{} in the tagger.  A novel focal plane design is currently under 
study, to be discussed below in section~\ref{tagger}, which may enable the 
focal plane rate to be reduced by about a factor of two without any decrease 
in the collimated flux.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_pol_spectrum0.eps,width=\textwidth}}
\caption
[Linear polarization in the coherent bremsstrahlung peak]{\label{polspec0}
Linear polarization in the coherent bremsstrahlung peak as a function of
electron beam energy keeping the energy of the coherent peak fixed at 
$9$\eunit. The calculation is performed under the same conditions as in
Fig.~\ref{collimcuts}.}
\end{center}
\end{figure}


The linear polarization of the photons in the coherent peak is shown
in Fig.~\ref{polspec0} as a function of the energy of the electron beam.
This figure demonstrates why it is essential to have electrons of as high
energy as possible, even though photon energies of no more than $9$\eunit{}
are required.  The intensity of the coherent peak, not shown in the
figure, has a similar dependence on the electron beam energy in this
region.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_pol_spectrum.eps,width=\textwidth}}
\caption
[Linear polarization of the coherent bremsstrahlung beam]{\label{polspec}
Linear polarization of the coherent bremsstrahlung beam
for a fixed electron beam energy of $12$\eunit , 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.}
\end{center}
\end{figure}

Shown in Fig.~\ref{polspec}
is the linear polarization of the photon beam {\em vs} photon energy for
fixed electron beam energy.
The dashed curves show how the maximum polarization in the 
primary peak varies
as the peak energy is changed by rotating the crystal.  The
polarization in all cases is zero at the end-point.  Without
collimation it rises as $(E_0-k)^{2}$ , one power
coming from the intensity of the coherent peak relative to the
incoherent component, and the other from the intrinsic polarization
of the coherent photons.  Collimation allows one
to essentially isolate the coherent component, so that the polarization
available to the experiment rises from zero at the end-point 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.  The virtual electron beam spot is defined as the profile that the
electron beam would have at the entrance to the collimator if it were
allowed to propagate freely instead of being bent into the beam dump.

The size of the virtual spot at the collimator is determined by the
beam emittance combined with an upper limit of $20\mu$r on the angular
spread of the electron beam at the radiator.  The latter value was chosen
to match the spread in the beam incidence angle to the mosaic spread of the
crystal because it is the combination of the two that limits the definition
of the coherent peak.  Taking this value together with an emittance of
$10^{-8}$m$\cdot$r, which has been projected for the \cebaf{} beam at 
$12$\eunit\
leads to a virtual spot size of $0.5$\mmlunit{} r.m.s.\ ($1.2$\mmlunit{} 
f.w.h.m.). \ \ Note that this scale does not
depend on the radiator-collimator distance.  The size of the real photon
spot is given by one characteristic angle $m/E$ which defines a circle on
the collimator containing approximately $50$\% of the total photon intensity.
The real spot size is proportional to the radiator-collimator distance.
At a distance of 80m the ratio of spot sizes is $6$, sufficient to allow 
collimator apertures that satisfy both of the above inequalities.

Fig.~\ref{radcoldist} shows the peak polarization of the beam as a function
of radiator-collimator distance for a coherent peak at $9$ \eunit. In this
calculation the collimator diameter is held constant at $3.4$\mmlunit\ to make
sure that the virtual beam spot of $1.2$\mmlunit\ f.w.h.m.\ is well-contained
within the aperture, which is the main condition for effective collimation.
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
from collimation has saturated.  The design for \hd\ calls for a
radiator-collimator distance of approximately 80\mlunit.
However from the figure one can see that the performance of the
photon source is not a very sensitive function of this variable.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_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
held fixed in this calculation to keep a constant ratio
between the sizes of the virtual electron spot and the collimator.}
\label{radcoldist}
\end{center}
\end{figure}


\section{Choice of radiator}
\label{radiators}

The ideal radiator would be a layered structure with  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 \\
beryllium & 0,0,2  & 0.86 \\
boron     & 2,0,8  & 0.38 \\
silicon   & 2,-2,0 & 0.19 \\
${\rm{Be_2C}}$ & 2,2,0 & 1.10 \\ \hline \hline
\end{tabular*}
\caption
[Figure of merit for various materials]
{Figure of merit for various materials that might be used as a
coherent bremsstrahlung radiator.  This table is reproduced from
Table 2 in  Ref.~\cite{Bilokon}.}
\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 with large
diamond crystals, such as would be required for the production of
coherent bremsstrahlung radiators, already exists within the gem industry.
However such capabilities are typically treated in that highly competitive
business as sensitive corporate information, particularly as they
pertain to the creation of large gem-quality synthetics.
Researchers at the University of Glasgow have established contacts
within the gem industry for procuring single-crystal diamonds of
high quality and large surface area \cite{Glasgow}.
The techniques used for selecting and assessing the quality of the
diamonds are discussed in the next section.

In general terms,
diamonds are classified as type I or type II, where type II have
been subjected to greater stresses during their formation than type
I. Commonly, type II exhibit substantial plastic deformation.
Diamonds are also classified according to the form in which nitrogen
atoms are present in the crystal lattice. In type {\em a} the nitrogen is
aggregated into clusters of atoms, whereas in type {\em b} the nitrogen is
almost uniformly distributed throughout the crystal. For coherent
bremsstrahlung  radiators, type Ib diamonds are the most suitable.
Unfortunately, type Ib natural diamonds are very rare and probably 
the most reliable source of Ib diamonds will be synthetics. At present
synthetic diamond mono-crystals typically have nitrogen
concentrations around $100$ ppm.

Synthetic diamonds are made using either vapor deposition (CVD) or
high pressure high temperature (HPHT) techniques. CVD diamonds have
an extensive mosaic and are unsuitable for
coherent bremsstrahlung.  However HPHT synthetics
look very promising, and the Glasgow group have recently acquired a
$5$\mmlunit\ $\times 5$\mmlunit\ synthetic diamond less than $18$\micron{}
thick which has a [100] orientation. It produces a
very good coherent bremsstrahlung spectrum
and X-ray measurements show it has rocking curve widths of less than
$10\mu$r, quite close to the ideal value for diamond.

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.  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 the Debye temperature
$ \theta_D $ is sufficiently large.
Second, the Debye temperature is, roughly speaking,
a measure of the stability of the crystal structure and hence its
capacity 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.
Beryllium remains a second choice, to be investigated
further in the case that affordable sources of large-area diamond
crystals at some point are no longer available.

\section{Crystal quality}

In the calculation of the coherent bremsstrahlung spectrum it is
necessary to take into account the fact that 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 electron beam divergence to the
blurring of the exact energy-angle relation for coherent photons.

Besides dislocations, there are other kinds of crystal defects.
The presence of foreign atomic species during the crystal growth
process can result in the substitution of impurities at some lattice
sites, or the formation of voids where impurities tend to collect in
clusters of several atoms.  In the growth of diamond crystals under
conditions of high pressure and temperature, the growth rate is 
greatly enhanced by the presence of a small amount of nitrogen.  Thus
it is normal that small amounts of nitrogen impurities should exist
even in the best natural stones, as well as in the synthetics created
by the HPHT process.

The ideal conditions for growth of a perfect synthetic
crystal require pre-existing mono-crystalline diamond with clean planar
facets cleaved along the major crystal planes, upon which new layers
of carbon are deposited in succession.  If conditions are right, the
registry of the atoms with the original crystal is preserved over
millions of deposited layers, starting from the original seed.  In
principle, the expansion of the regular lattice should continue to
match up perfectly at the boundaries between the different growth
surfaces that originated on the facets of the seed, but in practice
the strains from small imperfections that occur during the growth
process tend to accumulate there, forming recognizable patterns of
concentrated defects known as {\em{growth boundaries}}.  If the
stresses grow too large then new strain regions may develop, leading
to a more pronounced mosaic pattern in the subsequent layers.

Unfortunately the growth process has proved difficult to control in
a reproducible fashion.  As a result, out of several dozen stones
examined, only one or two may be of sufficient quality for use as a
coherent bremsstrahlung radiator for \hd.  The selection process
described below was formerly developed by the Glasgow group to supply
crystals for the coherent bremsstrahlung source at Mainz, Germany
and subsequently for the Hall B source at Jefferson Lab.  The
requirements for \hd\ are very similar to those of Mainz and Hall B,
except that the electron beam current will be higher by about an
order of magnitude and the crystals will be cut much thinner.

The diamond ingots from the synthetic process are sliced into sections
at the laboratory where they are produced.  From these, thin wafers
of about $100$\micron{} thickness are cleaved along the (1,0,0) axis
and provided to the Glasgow group for assessment.  The samples are
first examined under a microscope with polarized light.  Many of the
stresses in the crystal lattice can be revealed in this way,
particularly those which exhibit plastic deformation.  If the diamond
appears clear and featureless under polarized light then it is
examined with X-rays.  Two types of X-ray measurements 
are performed. 

\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_SRSsetup.eps,width=\textwidth}}
\caption
[SRS setup]
{Experimental setup for assessment of diamond crystals at the Synchrotron
Light Source beam line, configured for topograph measurements (a), and
rocking curves (b).}
\label{SRSsetup}
\end{center}
\end{figure}

%\newcounter{i}
%\begin{list}{\roman{i}. }{\usecounter{i}}
\begin{enumerate}
\item Topographs\\
A topograph is a real-space image of a diamond formed from X-rays
that Bragg-scatter from a particular set of planes in the crystal,
as shown in Fig.~\ref{SRSsetup}a.  Using the highly-parallel X-ray
beam from the Synchrotron Light Source (SRS) and
setting the detector at twice the Bragg angle for a known set of
planes for diamond, X-rays of the appropriate wavelength to satisfy
the Bragg condition are scattered at a precise angle $ \theta $ into
the detector.  The X-ray image formed on the plane of the detector
is a simple real-space projection of the crystal,
called a {\em{projection topograph}}.  
If the vertical slits defining the X-ray beam are narrowed forming
the incident beam into a thin ribbon a few $ \mu m$ wide, then the
image at the detector reveals a slice though the crystal, called a
{\em{section topograph}}.
Projection topographs reveal any large-scale imperfections
in the crystal.  Section topographs can be used to examine the depth
profile of imperfections.  Topographs sample the whole volume of the
crystal.  Hence, by measuring projection and section topographs, a 
3-dimensional picture of the diamond can be obtained. It is also 
possible to differentiate between screw and edge dislocations.
The topograph image reveals dislocations, growth 
boundaries and any feature which suppresses or enhances Bragg scattering
at the selected angle.  In principle, topographs taken at different
angles provide independent views of the crystal structure.  In
in practice, however, the imperfections that are revealed with one
set of planes appear in a similar fashion when viewed from other
orientations.

\item Rocking curves\\
A rocking curve is a plot of Bragg-scattering intensity {\em{vs}}
angle between the incident X-ray beam and the normal to the crystal
planes.  A diagram of the setup is shown in Fig.~\ref{SRSsetup}b.
First the broad-band X-ray beam from the SRS is monochromated by
scattering at a known fixed angle from a reference crystal, in this
case silicon.  This beam is then directed at the diamond crystal
under study, from which it scatters a second time and is detected.
The scattering is appreciable only when the diamond is at just
the right angle with respect to the incident beam such that the
Bragg condition is satisfied at both crystals.  The variation in
the scattering intensity with angle as the diamond wafer is rotated
through the resonance is called the {\em{rocking curve}} for
that diamond.  A perfect crystal exhibits a rocking curve consisting
of a single peak whose width is called the {\em{natural width}}
and depends on the material.  The natural width for diamond is
about $5\, \mu r$.  Instead of a single peak, for actual crystals
one typically sees a number of peaks spread out over a region in
angle over known as the rocking curve width.
Rocking curves widths, for a selected set of crystal planes,
measure quantitatively how any defects or dislocations distort the
crystal lattice.  By adjusting the slits it is possible to examine
the rocking curve of a region of the crystal or to examine the
entire crystal at once.  Using rocking curves it is possible to
measure how close to ideal is the lattice structure of the diamond
being investigated.  
\end{enumerate}
%\end{list}

\begin{figure}
\begin{center}
\begin{minipage}{1.5in}
\mbox{\epsfig{file=ch4_1482a3.eps,width=1.5in}}
\end{minipage}
\mbox{\epsfig{file=ch4_r1482a3.eps,width=\textwidth}}
\caption
[Rocking curve 1]
{Experimental data collected using highly-parallel X-rays from the
SRS light source for stone 1482A slice 3 (close to the seed).
At the top is shown a projection topograph of the wafer taken
using the broad-band X-ray beam and a Polaroid film placed at the
angle for reflection from the (0,4,0) planes.  The image is a
magnified by a factor of $5$.  The graph shows the rocking curve
for the same set of planes, taken using a NaI counter and
1 \AA\ X-rays monochromated by a silicon crystal.}
\label{rockingcurve1}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}{1.5in}
\mbox{\epsfig{file=ch4_1482a2.eps,width=1.5in}}
\end{minipage}
\mbox{\epsfig{file=ch4_r1482a2.eps,width=\textwidth}}
\caption
[Rocking curve 2]
{Experimental data collected using highly-parallel X-rays from the
SRS light source for stone 1482A slice 2 (further from the seed).
At the top is shown a projection topograph of the wafer taken
using the broad-band X-ray beam and a Polaroid film placed at the
angle for reflection from the (0,4,0) planes.  The image is
magnified by a factor of $5$.  The graph shows the
the rocking curve for the same set of planes, taken using a NaI
counter and 1 \AA\ X-rays monochromated by a silicon crystal.}
\label{rockingcurve2}
\end{center}
\end{figure}

Figs.~\ref{rockingcurve1}-\ref{rockingcurve2} show some of the
results that were obtained at the SRS laboratory in Daresbury,
England in January, 2002.  At the left of the figures is shown
a projection topograph taken using the (0,4,0) planes,
the second harmonic of the (0,2,0) planes used for coherent
bremsstrahlung.  At the right is shown the corresponding rocking
curve taken in combination with a silicon crystal set to reflect
from the (3,3,3) planes at a wavelength of $1$\AA.  The two diamond
wafers had been cut from the same original type Ib stone, with 
Fig.~\ref{rockingcurve1} coming from the end close to the seed,
and Fig.~\ref{rockingcurve2} coming from near the middle of
the ingot.  The topographs are negatives, meaning that the image
is dark in regions where the X-ray intensity was largest.

The first thing to notice from the topographs is that both wafers
are mono-crystalline; there are no regions where X-rays do not scatter.
Even so, there are important differences between the two samples.
The growth boundaries (the picture-frame pattern) which are visible
in Fig.~\ref{rockingcurve1} spread out and become less pronounced in
slice 2 which was taken further from the seed.  It is interesting that
the strain pattern appears mostly as dark regions rather than light,
which indicates stronger scattering in the defects than in the ordered
regions, the opposite from what one might naively expect.  It
should be recalled that both crystals appeared clear and featureless
under polarized light at visible wavelengths.  The requirement for
a diamond radiator useful for \hd\ is that the rocking curve width
be of the same order of magnitude as the divergence of the electron
beam at the radiator, which when folded with multiple-scattering is
about $25\ \mu r$ r.m.s.  The conclusion is that slice 2 is a good
candidate for use in the \GX{} experiment, and that slice 3 is not.  
Having confirmed the quality of slice 2, it should now be possible for 
the manufacturer to cut a dozen or more wafers of similar quality from 
that region of the original stone.

\section{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 have 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 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
end-point, 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=ch4_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  $3.4$\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 
$10^{-4}$ and a
$10^{-3}$ radiation-lengths
radiator to demonstrate the effect.  The
$10^{-3}$ radiator
spectrum is scaled down by a factor of 10 to facilitate the comparison.
The calculation assumes a $3.4$\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 $10^{-4}$ diamond radiator is $15$\micron{} thick.  The goal for \GX{} 
is to run with crystals of thickness in the range $10$\micron{} to 
$20$\micron .

\section{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.  Equation~\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, to have a stable photon beam with the
coherent peak positioned at the right energy, the angle between the incident
electron beam and the crystal radiator must be adjustable in steps of
a few $\mu$r and remain stable at this level. Since the angle of the incident
beam is fixed by the beamline optics and the position of the photon
collimator, all adjustments must be made by changing the orientation of
the crystal. This is achieved with a precision goniometer (shown schematically
in Fig.~\ref{goni.fig}) which should provide motion on at least 5 axes.
Rotation about the azimuthal axis $\phi$
sets the orientation of the polarization plane,
rotations about the $\theta_{v},\theta_{h}$ axes set the angle of the
crystal relative to the beam, and $x,y$ translations
select the position of the beam spot on the crystal. Estimates
of the approximate range and step size for each of the axes are given
in Table~\ref{goni.tab}.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_goni.eps,width=0.7\textwidth}}
\caption
[Schematic illustration of crystal mounted in goniometer]
{Schematic illustration of crystal mounted in goniometer}
\label{goni.fig}
\end{center}
\end{figure}

\begin{table}
\begin{center}
\begin{tabular}{clcc} \\ \hline\hline
\textbf{Axis}& \textbf{Motion}& \textbf{Range}& \textbf{Step size}\\ \hline
x & horizontal translation& -50\mmlunit\ -- +50\mmlunit\ & 0.01\mmlunit\ \\
y & vertical translation& -20\mmlunit\ -- + 20\mmlunit& 0.01\mmlunit\\
$\theta_{v}$ & vertical rotation& -100mr -- +100mr& 10$\mu$r\\
$\theta_{h}$ & horizontal rotation& -100mr -- +100mr& 10$\mu$r\\
$\phi$ & azimuthal rotation& -100$^{\circ}$ -- +100$^{\circ}$& 0.01$^{\circ}$\\
\hline\hline
\end{tabular}
\caption
[Requirements for goniometer axes]
{Requirements for goniometer axes}
\label{goni.tab}
\end{center}
\end{table}

In practice several targets need to be mounted in the goniometer. The
minimum requirement is a diamond crystal, an amorphous radiator, and a blank.
It is also desirable to have a screen to show the position of the beam spot
and a spare diamond. This means either mounting some targets off-axis on the
azimuthal plate (as in the Mainz setup), or having a sixth
axis to allow a target ladder to sit inside the azimuthal plate
(as in the Jlab Hall B setup). A goniometer with the required precision
can be obtained commercially, and would be controlled with the slow controls
system.

\section{Crystal alignment and monitoring}
\label{alignment}

As can be seen in Fig.~\ref{goni.fig} the goniometer setting
$\theta_{v}$,$\theta_{h}$ defines the direction of the normal to its inner
plate (O). Ideally at its zero setting $\theta_{v}=\theta_{h}=0$ this would
coincide with the electron beam direction (B), but in practice there are
small offsets $\theta_{vb},\theta_{hb}$ which may vary according to the
stability of the electron beam. There will also be a misalignment of the
crystal lattice with respect to the inner plate due to imperfections in
the mounting and in the cutting from the original stone.
The 100 axis (C) will be tilted with respect to the inner plate at an angle
$\theta_{t}$ with this maximum tilt occurring at an azimuthal angle $\phi_{t}$.
In addition, the 022 vector will be offset by $\phi_{0}$ with respect
to the horizontal. Any motion about the azimuthal axis $\phi$ changes
the angle of the 100 axis (C) relative to the beam. The angle of the
polarization plane is set by adjusting the
azimuthal angle of the crystal $\phi$.
Hence when a new crystal is installed, the default value $\phi_{0}$
needs to be measured. Furthermore, to position the coherent peak at the
required photon energy, the angle (or {\em offsets}) between the beam and 100 
crystal axis (C) at the chosen value of $\phi$ must also be established.  

Feedback on the relative angle between the crystal and the beam is obtained
from a photon energy spectrum derived from the tagger focal plane counters,
either via scalers or a TDC hit pattern. The scaler spectrum does not show the
effect of collimation (unless the scalers are gated with a downstream photon
detector), but can be obtained very quickly since it does not require a 
triggered data acquisition system. The scaler readout is essential for the 
alignment process, where the offsets are measured by carrying out a series of 
scans in which 2d histograms of photon energy vs. crystal angle are built up 
by moving the goniometer in a sequence of small angular steps and reading the 
tagger scalers. In addition to providing the feedback required for alignment, 
the focal plane counters provide essential online diagnostics to monitor 
drifts in angles caused by the beam tuning, or thermal effects in the crystal 
mount. If necessary a feedback system could be implemented via the slow 
control system, where any drift in the position of the coherent peak could be 
corrected by periodically adjusting the goniometer within predefined limits. 

The spectrum obtained from the tagger focal plane can also provide online
monitoring of the photon polarization to within $5$\% by fitting with an
analytic bremsstrahlung code.  A more detailed discussion of polarimetry
appears in the following sections.


\section{Crystal lifetime}

The best information regarding crystal degradation comes from X-ray
studies performed by the Glasgow group of a diamond which had been
used in the MAMI coherent bremsstrahlung source at Mainz for several years.
The electron beam on the Mainz crystal had a diameter of about $100$\micron{}
and it was estimated that around $10^{20}$ electrons had passed through 
the diamond during its use in the source.  There was a small greenish
black spot where the beam had hit the diamond.

The X-ray rocking curve measurements showed that considerable damage
had occurred to the integrity of the crystal structure in the center
of the beam spot.  However $2$\mmlunit\ away from the damage center the width
of the diffraction peak was the same as it had been for the pristine
crystal, which indicates that the lifetime of the crystal could be
extended by occasionally moving the beam spot on the face of the
crystal.

The area of the MAMI beam spot on the radiator is two orders
of magnitude smaller than what is being planned for \GX{} in \hd.  A larger
spot means a longer crystal lifetime before radiation damage substantially
degrades its crystal properties.  Appropriately scaled, the exposure of
the Mainz crystal would correspond to $15$ years of running in
\hd\ at the full intensity of $3$\miamps{} without a spot move.  Plans for
the \hd\ source are to keep the exposure about three orders of
magnitude less than this.  At the SLAC coherent bremsstrahlung beam
line it was found that the performance of their diamond radiators had
degraded noticeably after a total charge of $3$ Coulombs had been
accumulated over a spot of size roughly $2$\mmlunit\ r.m.s.,
leading to a limit of about $0.25$ Coulomb/\mmlunit$^2$ \cite{CSinclair98}.
Taking this as a conservative estimate for the allowed exposure, the
source can run at a full intensity of $3$\miamps{} for $60$ hours before it
is necessary to move the spot on the crystal.  If it had no bad zones,
a square crystal of $5$\mmlunit\ $\times 5$\mmlunit\ would accommodate $5$
spot moves before the crystal would need to be replaced.
SLAC researchers were able to recover a good performance for the
damaged crystals by putting them through an annealing process.
Further research and development will be required to determine
whether crystal recovery through annealing is an effective way to
reduce the operating costs of the \hd\ source.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=ch4_temper.eps,width=\textwidth}}
\caption
[Temperature profile of crystal at full current]
{Calculated temperature profile of diamond crystal with a $12$ \eunit\ beam
at $3$\miamps .  The crystal dimensions are $5$\mmlunit\ $\times 5$\mmlunit\
$\times 15$\miamps .  The ambient room temperature was taken to be $27^{o}\ C$
($300$ K).  The X-ray asymmetry is caused by the elliptical shape of the
electron beam spot on the radiator.}
\label{temper}
\end{center}
\end{figure}


Another issue related to crystal degradation is that of heat dissipation
for very thin crystals.  The heat comes from the ionization energy loss of
the beam as it passes through the crystal.  Although this is small compared
to the bremsstrahlung energy loss, it is not entirely negligible at
these beam currents.  It can be calculated using the restricted
energy loss formula, which yields $21$\mpwunit{} for a $15$\micron{} 
($10^{-4}$ radiation lengths) crystal at a current of $3$\miamps . This 
is not much power, but the crystal is very thin.  Heat dissipation is 
through radiation and conduction.  Diamond has a very high melting point; at 
low pressures it sublimates at about $4027^{o}\ C$.  However at normal 
pressures it begins to transform into graphite above $707^{o}\ C$, at a rate 
that depends on temperature.  It is therefore important that the crystal at 
the center of the beam spot stay well below this limit.

The diffusion equation including a heating term and one for radiative
cooling can be written as
\[
C_P\,a\;\frac{dT}{dt} = h(x,y) - 2\sigma \left( T^4 - T_0^4\right) 
+\kappa\, a \nabla^2 T
\]
where the heating term $h(x,y)$ has units of power/area, $\sigma$ is
the Stefan-Boltzmann constant, $C_{P}$ is the heat capacity and $\kappa$
the coefficient of conduction for diamond, and $a$ is the thickness
of the crystal.  $T_{0}$ is the ambient temperature of the environment
and $T$ is the local crystal temperature, a function of space and time
coordinates.  After a certain time, $T$ converges to the steady-state
solution shown in Fig.~\ref{temper}.  The calculation used a crystal
of dimensions $5$\mmlunit\ $\times 5$\mmlunit\ $\times 15$\micron{} and a beam 
current of $3$\miamps .  This calculation shows that the conductivity of
diamond is sufficient to prevent significant temperature gradients across the
crystal even for very thin wafers, and that radiative cooling alone is
sufficient to dissipate the heat being generated by the beam passing
through the crystal so that the crystal mount does not need to act
as a heat sink.

