
\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 a pair of steering dipoles and passes through the
bremsstrahlung radiator.  Immediately downstream from from the radiator,
the electron beam enters 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 predominantly 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 chamber through a thin window,
passing through air for a short distance before they reach the tagging
counters located near 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=photon_line.eps,width=6in}}
\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 Hall D, the photon beam
passes through a small collimator which blocks a large fraction
of the incident beam intensity, and selectively passes the coherent
bremsstrahlung component.  The primary collimator is housed in a
separate enclosure from the experimental hall for shielding purposes.
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 and 
stopped in a thick wall of shielding material.  Downstream of this
there is a second collimator whose purpose is to 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.  This two-stage collimation
system is similar to the setup at the SLAC coherent bremsstrahlung
beam line \cite{Kaune}.  The clean collimated beam is then sampled
in a pair spectrometer, which monitors the spectrum and flux of the
beam just before it enters the experimental target.

After passing through of order $3$\% radiation lengths of liquid hydrogen
in the \GX{} 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~kW (5~$\mu$A at 12~GeV) being delivered to the 
electron beam dump, the total power in the photon beam is not more than 
1.5~W in the experimental hall and not more than 15~W 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. In that
case, the chief physical consequence is the disappearance of the recoil
Doppler shift from the photoabsorption/emission spectrum.  In the case
of coherent bremsstrahlung, the chief
consequence is the enhancement of the cross section at those particular
kinematics for which the recoil momentum matches one of the reciprocal
lattice vectors of the crystal.

It is helpful to view the process of coherent bremsstrahlung
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 for an infinite
static crystal 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 limitations of high-power laser
cavities to wavelengths above 100~nm.  The characteristic wavelength of
the virtual photons in a crystal is a few Angstroms, three orders of
magnitude shorter than the limit for lasers.  At keV energies,
$180^{\circ}$ Compton scattering results 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.  This factor
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 that from a
disordered radiator.  The reason is that replacing the sum over crystal
momenta in the coherent bremsstrahlung cross section formula with a
continuous integral recovers the cross section for ordinary bremsstrahlung
from isolated atoms.

Furthermore, 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, so that the higher-order terms in the sum become
indistinguishable from the the high-$q$ part of the corresponding 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 the ordinary continuum bremsstrahlung spectrum,
while the former appears as a set of intensity peaks 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=5in}}
\caption[Coherent bremsstrahlung spectrum]{
\label{uncolspec}
Uncollimated coherent bremsstrahlung spectrum, calculated for a diamond
crystal radiator 20~microns thick and a 1~$\mu$A electron beam of
12~GeV energy.  The sharpness of the edge at 9~GeV is a result of the
excellent emittance of the 12~GeV electron beam.}
\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 distinguish the invariant part of the spectrum from
the part 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 appears as the coherent part\footnote{The typical figure of 12~cm
for the radiation length of diamond is actually an average over all
orientations of the crystal.}.  In the calculation used to produce
Fig.~\ref{uncolspec}, the leading 400 lattice sites were included in
the discrete part of the calculation, but only two or three
of them contribute with sufficient intensity to be identified with
individual peaks 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.  However there is a difference
between the angular distributions of 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\footnote{In the lab
frame this is a small angle, but in the rest frame of the
electron-photon system it subtends all angles in the forward
hemisphere.}. 
This is true both for the incoherent and coherent components.
The difference lies in the fact that a peak in the
coherent component corresponding to a single reciprocal lattice
vector has the two-body kinematics of Compton scattering,
so there is a well-defined relation
between the emission angle and the energy of the emitted photon in
the lab: for a given reciprocal lattice vector, emission at
$0^{\circ}$ yields a maximum photon energy, and energy decreases with 
increasing angle.  This accounts for the
shape of the coherent peaks in Fig.~\ref{uncolspec},
with the sharp right-hand edge of the peaks 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=collimcuts.eps,width=5in}}
\caption[Coherent bremsstrahlung spectrum after collimation.]{
\label{collimcuts}
Coherent bremsstrahlung spectrum, calculated under the same
conditions as in Fig.~\ref{uncolspec}, with varying amounts of
photon beam collimation.  Curves shown from top to bottom are
(1) the uncollimated spectrum, and collimated spectra with (2)
a 1~$m/E$ collimator, (3) a 0.5~$m/E$ collimator, and (4) a 
0.25~$m/E$ collimator.}
\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~m 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 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.
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 optimizing the design of the source.
Crystal imperfections, which amount to an intrinsic spread in the
direction of the incoming virtual photon, are also taken into account
in the model.

Third, note that the relatively weak collimation of 1 $m/E$ reduces
the incoherent background without significantly affecting the coherent
flux near the maximum, 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 0.5~$m/E$ collimator has been chosen for this
design because it provides for a maximum reduction in the incoherent
flux while transmitting more than 90\% of the coherent flux at the peak.
This is implemented by placing a tungsten collimator with a circular
aperture of diameter 3.4~mm at a distance 75~m downstream from the
cyrstal.

Most of the photon beam energy coming from the crystal is absorbed
by the collimator.  To prevent the radiation produced at the collimator
from producing background in the experimental, it is located in a
separate enclosure just upstream of the experimental hall, and
surrounded by a large amount of shielding.
The peak in Fig.~\ref{collimcuts} for the 0.5~$m/E$ collimator
contains $4.8\times 10^7$ photons/s in the primary coherent peak per
$\mu$A of electron beam current.  The \GX{} experiment is designed to
run at up to $10^8$ photons/s in the coherent peak region 8.4-9.0~GeV.
If the crystal is large enough to contain the entire electron beam
spot at the radiator, this corresponds to 2.1~$\mu$A of electron
beam current, safely below the design limit of 5~$\mu$A for the
Hall D beam dump.

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 250~MHz
in the focal plane within a 600~MeV window around the peak.
Combining this rate with the beam pulse
spacing of 2~ns 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 
GeV in the tagger.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=pol_spectrum0.eps,width=5in}}
\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~MeV. The 3.4~mm [5~mm] collimator represents a cut at 0.5 [0.75]~$m/E$.}
\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~GeV
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=pol_spectrum.eps,width=5in}}
\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~GeV.  The dashed lines indicate the 
trajectory of the peak polarization as the peak energy is swept
by rotating the crystal.  The 3.4~mm [5~mm] collimator represents a
cut at 0.5 [0.75]~$m/E$.}
\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~m.  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 by the tagger dipole
field 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 edge.  Taking a conservative estimate of 
$10^{-8}$m$\cdot$r for the 12~GeV electron beam emittance\footnote{
Simulations of the 12~GeV accelerator design indicate that the horizontal
emittance of the beam will be a factor 2 better than this, and a factor 4
bette in the vertical.} leads to a virtual spot size of 0.5~mm r.m.s.\
(1.2~mm f.w.h.m.).  Note that this argument does not assume any scale for
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 75~m 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~GeV. In this
calculation the collimator diameter is held constant at 3.4~mm to make
sure that the virtual beam spot of 1.2~mm 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 100~m separation distance the polarization enhancement
from collimation has saturated.  The design for Hall D calls for a
radiator-collimator distance of 75~m.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=radcoldist.eps,width=5in}}
\caption[Maximum polarization {\em vs} radiator-collimator distance.]{
\label{radcoldist}
Maximum polarization {\em vs} radiator-collimator distance
for a coherent peak at 9~GeV.  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.}
\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~nm apart, thus simulating the standing wave in a cavity driven
by a 15~eV 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 materials
containing substantial amounts of any elements 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.
The material shown with the highest figure of merit is the binary
crystal Be$_2$C.  In general, multi-element crystals are more sensitive
to radiation damage than single-element crystals because annealing
of dislocations is significantly less efficient when more than one
atomic species is involved.  This leaves diamond and beryllium as
the two alternatives.

Both diamond and beryllium have unusually high Debye temperatures.
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 $e^{-q^2/4M\theta_D}$
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 $2200^{\circ}$K, while that of beryllium is
$1400^{\circ}$K.

Considerable experience exists with both diamond and beryllium 
single crystals for use as monochromators within in the X-ray
diffraction community.  Single crystals of both can be produced
with diameters larger than 10~mm, which is sufficiently large for
use as a coherent bremsstrahlung radiator.  However there are a
number of features, besides the performance figure showin in
Table~\ref{radmat}, that make diamond a clear winner in this
application.  First of all, diamond has a thermal conductivity
that is a factor 10 larger than that of beryllium.  Although both
are excellent thermal conductors, the difference makes a difference
when the crystal is made very thin, while keeping the heat load
constant, which is the optimization scenario for a coherent
bremsstrahlung source.  As shown in a following section, the
thermal load on the radiator in the Hall~D source is enough to
heat the crystal to several hundred degrees at full operating
intensity.  Therefore a factor 10 higher thermal conductivity is a
significant advantage for diamond.  Combined with this, the thermal
expansion coefficient for beryllium is about a factor 10 higher than
that of diamond, which means that it will be much more subject to 
thermal stress and distortion in the beam than will diamond.  More
importantly, published rocking curves for beryllium single crystals
have widths greater than 300~$\mu$r f.w.h.m., as compared with less
than 20~$\mu$r f.w.h.m.\ for the best large-area diamonds
\cite{beryllium_mono94}.  For these reasons, diamond has been
adopted as the unique choice for the radiator material in the
Hall~D coherent bremsstrahlung source.

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~\cite{Element_Six_talk_at_Diamond_2008,Sumitomo_talk_at_Diamond_2008}.
Although the details of the crystal growth process are typically
treated in the highly competitive diamond business as sensitive
corporate information, researchers both in Europe and Japan have
been able to obtain large-area crystals from the firms Element Six
and Sumitomo Electric Industries and demonstrate that they have
X-ray rocking curves that are very close to the theoretical ideal
for a perfect crystal~\cite{NIM_diamond_assessment_article,Spring-8_diamond_assessment_article}.
Within the \GX{}
collaboration, the University of Glasgow group has been able
to obtain a significant number of high-quality crystals from
Element Six, cut along the desired crystal direction and polished
down to a desired thickness.  The techniques used for 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.  The only known
way to obtain large high-quality Ib diamonds is through the process
of synthetic crystal growth.  The primary impurity in these synthetics
is nitrogen, which is artificially introduced as a growth catalyst.
At present type Ib diamond mono-crystals can be obtained with nitrogen
concentrations as low as 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.
Synthetics from the HPHT process are not uniform in their crystal
quality, but it is not rare to find large regions of a crystal that
approach the theoretical limit in the X-ray rocking curve width.
Among the high-quality crystals obtained by the Glasgow group from
Element Six, one of them was polished down to a thickness less than
18~microns, demonstrating the feasibility of producing the 20~micron
diamonds needed for \GX{}.


\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 whose local normal unit vectors
are slightly misaligned with one another.
The angular scale of the deviations between the local normals
across the face of a single crystal plane is termed the
{\em mosaic spread} of the crystal.  Because crystal lattice
distortions affect all of the planes in that region of the crystal,
mosaic spread tends to have the same scale for all sets of planes
in a given region, and is characterized by a single parameter.
In coherent bremsstrahlung, 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 new planes 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 the accummulation of small imperfections that occur
during the growth process tend to build up there, forming recognizable
patterns of concentrated defects known as {\em{growth boundaries}}.
If the stresses grow too large then new strain regions 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
produced, typically only one or two are of sufficient quality for use
as a coherent bremsstrahlung radiator for.  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 thinner.

To produce a coherent bremsstrahlung radiator from a synthetic diamond,
the ingot from the synthetic process was sliced into sections along the
(1,0,0) axis using a diamond saw.  This cut was made by the vendor at
the laboratory where they are produced.  Sample slices from different
regions along the axis were provided to the Glasgow group for assessment.
The samples were first examined under a microscope with polarized light.
Crystals which exhibit large plastic deformation were discovered and
eliminated at this stage.  Those which appeared clear and featureless
under polarized light where then taken to a synchrotron light source
and examined using X-ray diffraction.  Two types of X-ray measurements 
were performed, topographs and rocking curves. 

\begin{figure}
\begin{center}
\mbox{\epsfig{file=SRSsetup.eps,width=6in}}
\caption[SRS setup]{
\label{SRSsetup}
Experimental setup for assessment of diamond crystals at the Daresbury
Synchrotron Light Source beam line, configured for topograph measurements
(a), and rocking curves (b).}
\end{center}
\end{figure}

\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 Daresbury 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.  If the crystal is a single crystal then the X-ray
image formed on the plane of the detector is a real-space image
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 used at the Daresbury SRS 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 scattering peak 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 and the crystal plane.  The natural width
of the (2,2,0) planes in 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 known as the rocking curve width.
Rocking curves widths, for a selected set of crystal planes,
measure quantitatively how defects distort the crystal lattice.
By adjusting the slits it was possible to examine
the rocking curve in a local region of the crystal or to measure
the entire crystal at once.  From the rocking curves it is straight
forward to determine how close to ideal the lattice structure of
the diamond is for coherent bremsstrahlung.
\end{enumerate}

\begin{figure}
\begin{center}
\begin{minipage}{1.5in}
\mbox{\epsfig{file=1482a3.eps,width=1.5in}}
\end{minipage}
\mbox{\epsfig{file=r1482a3.eps,width=4in}}
\caption[Rocking curve 1]{
\label{rockingcurve1}
Experimental data collected using highly-parallel X-rays from the
Daresbury 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.}
\end{center}
\end{figure}

\begin{figure}
\begin{center}
\begin{minipage}{1.5in}
\mbox{\epsfig{file=1482a2.eps,width=1.5in}}
\end{minipage}
\mbox{\epsfig{file=r1482a2.eps,width=4in}}
\caption[Rocking curve 2]{
\label{rockingcurve2}
Experimental data collected using highly-parallel X-rays from the
Daresbury 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.}
\end{center}
\end{figure}

Figs.~\ref{rockingcurve1}-\ref{rockingcurve2} show some of the
results that were obtained at the SRS in January, 2002.  At the 
left of the figures is shown
a projection topograph taken with the (0,4,0) planes.
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 within the boundaries
of the crystal 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.  This is expected because the crystal is thick enough to
scatter essentially 100\% of the beam photons that fall within the
peak region in the rocking curve, so the wider is the rocking curve,
the wider the energy bite of the broad-band beam that is scattered.
It should be recalled that both crystals appeared clear and featureless
under polarized light at visible wavelengths.  Only X-rays can
reveal the significant defect structure of these crystals.

The specification for a diamond radiator for use as a coherent
bremsstrahlung radiator in \hd\ is that the rocking curve width
be no greater than 20~$\mu$r r.m.s.  The conclusion drawn from the
rocking curve measurements 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 would have been possible
to request that the manufacturer cut several wafers from the same
region of the original stone, and expect that the quality of the
new slices will be similar.  This was not done because at that time
the demand for new crystals was one or two every few years.  In
the case of \hd\ it will be important to take advantage of such
efficiencies in order to obtain a steady supply of new radiators
to replace those damaged by radiation.

\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 rate of radiation damage that increases for thinner crystals,
as the electron beam current is increased to maintain the desired
beam intensity.

There is a second and more fundamental lower bound
imposed by the fact that the crystal must have some minimum thickness
in order to achieve the full coherent gain.  The calculation of the
coherent bremsstrahlung cross section begins with the assumption of
an infinite crystal, but practically this means 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 initial-state virtual photon momentum component along the incident
electron axis.   This means that the electron travels an average
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\hbar E(1-x)}{x m^2 c}
\label{rtj:eqn3}
\end{equation}
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 12~GeV electron beam energy and a 9~GeV
coherent photon the coherence length is 6~nm, or about 17 unit 
cells for diamond.  This shows that the coherence length does not impose a 
practical limit on how thin the radiator should be for \GX{}.

\begin{figure}
\begin{center}
\mbox{\epsfig{file=radthick.eps,width=5in}}
\caption[Collimated coherent bremsstrahlung spectrum]{
\label{radthick}
Collimated coherent bremsstrahlung spectrum from a 12~GeV electron
beam using diamond radiators of different thicknesses.  The beam
currents have been renormalized to keep the rates in the tagger
constant.  The calculation assumes a  3.4~mm collimator
located 75~m from the radiator, and standard values for beam
emittance and crystal mosaic spread.}
\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 diamond
radiators of thickness 10, 20, 50, and 100 microns.  The electron
beam current in each case is rescaled to keep the rates in the
tagger constant.  The loss in normalized intensity with the
thicker radiator and as the broadening of the left edge of the peak
are due to the smearing out of the photon beam spot on the collimator
face due to multiple scattering of the electron beam in the crystal
prior to radiation.  The plot shows that improvements in the
coherent / incoherent flux ratio with decreasing radiator thickness
saturate around 20 microns.  The design for the \GX{} photon source  
specifies diamond radiators of thickness 20 microns.

\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.  Eq.~\ref{rtj:eqn2} shows that this condition
is equivalent to requiring that the dot product of the crystal momentum
vector corresponding to the coherent peak and the beam momentum vector
be of order $m^{2}$.  For $p = 12$~GeV/$c$ and $q\sim 10$~keV, this
amounts to requiring that $p$ and $q$ be mutually perpendicular to
within about a degree, and within one degree of variation 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, incidence angle adjustments are
made by changing the orientation of the crystal. This is achieved
with a precision goniometer (shown schematically in Fig.~\ref{goni.fig}).
A goniometer with three rotation axes and two translation axes gives
the necessary control over both where the beam spot is located on
the crystal and what its angle is with respect to the beam direction.
Rotation about the azimuthal axis $\phi$ sets the orientation of
the polarization plane.  Rotations about the $\theta_{v},\theta_{h}$
axes select the energy of the coherent peak and provides an additional
handle for eliminating extra peaks in the spectrum from unwanted
lattice vectors.  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=goni.eps,width=0.7\textwidth}}
\caption[Schematic illustration of crystal mounted in goniometer]{
\label{goni.fig}
Schematic illustration of crystal mounted in goniometer}
\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~mm -- +50~mm & 0.01~mm \\
y & vertical translation& -20~mm -- +20~mm & 0.01~mm \\
$\theta_{v}$ & vertical rotation& -100~mr -- +100~mr& 10~$\mu$r\\
$\theta_{h}$ & horizontal rotation& -100~mr -- +100~mr& 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}

It is foreseen that the crystal goniometer will hold a target ladder 
in which several targets are mounted, which can be interchanged under
remote control.  The minimum requirement for the ladder is
a diamond crystal, an amorphous radiator, a blank position, and a
fluorescent screen to show the position and shape of the beam spot.
A second diamond radiator would also be useful, in case the first
one is damaged in some way.  A goniometer with the required precision
can be obtained commercially.  Remote control of all stages of the
goniometer is a part of the slow controls plan for the \GX{}
experiment.

\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 vector normal to
its inner plate (O). Ideally, at its zero setting $\theta_{v}=\theta_{h}=0$
this normal vector would coincide with the electron beam direction (B),
but in practice there are small offsets $\theta_{vb},\theta_{hb}$ which
vary slightly with the tune of the electron beam.  There are also two
other corresponding offsets that specify the direction of the (1,0,0)
axis of the crystal with respect to the inner plate normal vector.
These arise both from imperfections in the way that the crystal is
mounted in its holder, and from imperfections in the way that the
crystal was originally cut.  These corrections can be parameterized
in terms of the tilt $\theta_t$ of the (1,0,0) crystal axis from
the O direction, and also its azimuthal vector $\phi_t$.  A third
offset angle $\phi_0$ is also needed to specify the azimuthal plane
containing the (0,2,2) axis direction in the coordinate system of the
goniometer mount.  Once the goniometer $\theta_v$ and $\theta_h$
have been adjusted to align the (1,0,0) crystal axis with the beam
direction B then the (0,2,2) direction is normal to the beam and the
gonimeter azimuthal setting $\phi$ - $\phi_0$ determines the plane
of polarization of the beam relative to the horizontal.

Each time a new crystal is installed, the offsets $\theta_{v0}$,
$\theta_{h0}$, and $\phi_{0}$ must be determined empirically by
systematically rotating the crystal while monitoring the positions of
the coherent peaks in the coherent bremsstrahlung spectrum seen in
the rates measured by the tagger focal plane counters.  A collimated
photon spectrum is not required for the alignment procedure, but it
is essential to sample low enough in the photon energy spectrum to
see the peaks at low $x$.  For this reason, the tagger focal plane is
instrumented with a broad-band counter array that covers the full
energy spectrum from 3~GeV up to 11.7~GeV.  Simply counting the
singles rates in the broad-band array with scalers and plotting
them in a two-dimensional histogram versus the wobble angle of the
crystal provides the fast feedback that allows the alignment procedure
to be completed in a relatively short period of time.  During normal
running after the alignment has been carried out, the broad-band tagging
counters provide essential online diagnostics to monitor drifts in
angles caused by changes in the beam tune, thermal effects in the crystal 
mount, and radiation damage.  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 angles within predefined limits. 


\section{Crystal lifetime}

There are no published results that give precise information about
the kinds and densities of crystal defects in diamond as a function
of dose.  The best estimates for crystal lifetime in a coherent
bremsstrahlung source are found in an unpublished SLAC 
report~\cite{diamond_radiation_damage_reference} which states that
``serious degradation'' of the coherent bremsstrahlung spectrum
was observed after 2-5 Coulombs of electrons had passed through
the crystal.  The SLAC beam energy was 19.7~GeV, which should not
be much different from 12~GeV for these purposes.  The SLAC beam spot
was large enough to fill the entire crystal, of approximate area
1~cm$^2$, although this is not to say that the beam intensity was 
uniform over that area.  From these results can be derived a useful
upper bound of about 0.25~Coulomb/mm$^2$ on the integrated current that
can pass through a diamond before it must be replaced.  In the same
report claims are made that it was possible to recover acceptable
performance from a damaged diamond by annealing it in a high-temperature
oven.  The annealing procedure was found to work over several use
cycles, before the accumulated damage was so severe that the diamond
could no longer be used.

The best quantitative information on crystal degradation from
radiation damage 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\cite{NIM_diamond_assessment_article}.
The electron beam on the Mainz crystal had a full width of about 
100~microns.  It was estimated that 5-10 Coulombs of electrons had
passed through the diamond during its use in the source.  There
was a small greenish black spot visible where the beam had passed
through the diamond.  This small beam spot means that the
exposed region of this crystal had seen three orders of magnitude
more integrated charge than allowed by the upper limit estimated
above based on the experience reported by the SLAC group.
Indeed, X-ray rocking curve measurements with a very small
X-ray beam showed that in the center of the beam spot the rocking
curve was split into many peaks, and that the full width was 
several mr.  However 2~mm away from the damage center, a single
narrow peak was seen in the rocking curve, with essentially the
same width as had been observed for the pristine crystal.  This
confirms that the lifetime of a crystal can be extended by
occasionally moving the beam spot on the face of the crystal.

The area of the beam spot on the damaged MAMI radiator is two orders
of magnitude smaller than what is being planned for \hd.  A larger
spot means a longer crystal lifetime before radiation damage substantially
degrades the crystal properties.  Appropriately scaled, the exposure of
the Mainz crystal in the center of the beam spot corresponds to more
than 10 years of running in \hd\ at full intensity without a spot move.
Plans for the \hd\ source are to keep the local dose three orders of
magnitude less than this.  Based on the estimated upper limit stated
above, the \hd\ source can run at a full intensity of 2~$\mu$A for 100
hours before it is necessary to move the spot on the crystal.  If it
had no bad zones, a square crystal of area $5\times 5$~mm$^2$ would
accommodate 5 spot moves before the crystal would need to be replaced.

Measurements of crystal radiation damage rates will be made during
the first two years of \GX{} running.  During those two years, the
source will operate at 10\% of design intensity, permitting a single
crystal to last for an entire year of running.  These same estimates
suggest that as many as 5 diamonds per year will be required to
run the \hd\ source at full intensity. It may or may not turn out to
be economically advantageous to try annealing damaged crystals,
depending on the availability and cost of new diamonds at that time.


\begin{figure}
\begin{center}
\mbox{\epsfig{file=temper.eps,width=5in}}
\caption[Temperature profile of crystal at full current]{
\label{temper}
Calculated temperature profile of diamond crystal with a 12~GeV beam
at the maximum electron beam current of 2~$\mu$A.  The crystal dimensions
are 5~mm $\times$ 5~mm $\times$ 20~microns.
The ambient room temperature was taken to be $27^{\circ}$C (300$^{\circ}$K).
The azimuthal asymmetry is caused by the elliptical shape of the
electron beam spot on the radiator.}
\end{center}
\end{figure}


Another issue related to crystal degradation is that of heat dissipation
for very thin crystals.  The heat from the ionization energy loss of
the beam as it passes through the crystal must be dissipated either
via conduction through the crystal mount or via thermal radiation.
Although the ionization energy loss is small compared to that
from bremsstrahlung, it is not entirely negligible at beam currents
planned for \hd.  It can be calculated using the restricted
energy loss formula, which yields 21~mW for a 20~micron crystal
at a current of 2.1~$\mu$A.  This is not much power, but the
crystal is very thin.  Diamond has a very high melting point; at 
low pressures it sublimates at about $4027^{\circ}$C.  However 
it begins to transform into graphite above $707^{\circ}$C, at a rate 
that increase with temperature.  It is essential 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~mm $\times$ 5~mm $\times$ 20~microns and a beam 
current of 2~$\mu$A.  This calculation shows that the conductivity of
diamond is sufficient to prevent significant temperature gradients
across the crystal even for very thin wafers.  It also shows that
radiative cooling alone is sufficient to dissipate the heat being
generated by the beam passing through the crystal and keep the
entire crystal well below the graphite transition temperature.
This calculation includes only radiative cooling, and shows that
the crystal mount does not need to be designed to dissipate heat
from the crystal.  It does indicate, however, that the materials
used to attach the diamond the mount must either be capable of
maintaining their mechanical properties at 500$^{\circ}$C or be
sufficiently thermally conductive themselves to allow 20~mW of
heat to be removed from the diamond by conduction to the mount.
