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\title{GlueX Photon Source Characteristics as a 
Function of Electron Beam Energy}
\author{R.T. Jones}
\date{November 12, 2004}

\pagestyle{myheadings}
\markboth{}{GlueX Technical Note series}

\begin{document}

\maketitle
\thispagestyle{myheadings}

\begin{abstract}
The GlueX Design Report contains a detailed description of
the characteristics of the photon source assuming an electron beam
energy of 12 GeV.  The purpose of this note is to explore how these
characteristics change as the electron beam energy is varied.  This
is a supplement to the information contained in the design report,
and does not represent any updates in the basic design of the source
presented there.
\end{abstract}

\vspace*{2cm}

The final table in chapter 4 of the GlueX Design report v4 is
reproduced as Table~\ref{cdr_rates} below.
All four columns of numbers were obtained for the same beam conditions,
except that the crystal orientation was adjusted to align the coherent
intensity peak at the energy listed in row one.  The second row, labeled
$N_{\gamma}$, gives the integrated rate of beam photons in the coherent
peak downstream of the collimator.  Note the sharp decrease in the intensity
of the coherent peak as the energy approaches the end point.  By contrast,
the incoherent bremsstrahlung flux is approximately constant over this
range of energies.  The third and fourth row show the height and width of
the peak in the polarization spectrum of the beam.  Rows  five and six
report the height and width of the peak in the tagging efficiency spectrum.
The tagging efficiency is defined as the number of beam photons of a
particular energy reaching the target divided by the corresponding rate
in the tagging focal plane.  Large tagging efficiencies are required in
order to make effective use of tagging.  The width of the peak in the
tagging efficiency spectrum determines the width of the focal plane that
would be active when running with collimation.  The peak integral reported
in row two is summed within the f.w.h.m.\ tagging efficiency window.  Rows
seven and eight give the photon beam power that is incident on the
experimental target (and photon beam dump) and the photon collimator,
respectively.  

The last two rows in Table~\ref{cdr_rates} give the inclusive and tagged rates 
for hadronic triggers from a 30 cm liquid hydrogen target placed in
the beam following the collimator.  Note that the total hadronic rate is
dominated by background (\emph{i.e.} non-tagged) events associated with
the low-energy component of the beam.  This is why the total trigger rate
is essentially constant while the flux in the coherent peak varies with peak
energy over an order of magnitude.  This table illustrates the value of
having an electron beam energy well above the photon energy needed for
the experiment.

\begin{table}
\begin{center}
\caption[Operating parameters for an experiment]{\label{cdr_rates}
Operating parameters for an experiment using the coherent bremsstrahlung beam.
The calculation assumes a 12~GeV electron beam energy and a 
3.4~mm collimator 80~m downstream from a radiator of 
thickness $10^{-4}$ radiation lengths.  The electron beam current is taken to 
be 3~$\mu$A. The rates in the detector (last two rows) are calculated for 
a 30~cm liquid hydrogen target and an open hadronic trigger.}
\begin{tabular}{lcccc}
\hline\hline
$E$ of peak & 8 GeV & 9 GeV & 10 GeV & 11 GeV \\
$N_{\gamma}$ in peak & 185 M/s & 100 M/s & 45 M/s & 15 M/s \\
peak polarization & 0.54 & 0.41 & 0.27 & 0.11 \\
\multicolumn{1}{c}{(f.w.h.m.)} & (1140 MeV) & (900 MeV) & (600 MeV) & (240 MeV) \\
peak tagging efficiency & 0.55 & 0.50 & 0.45 & 0.29 \\
\multicolumn{1}{c}{(f.w.h.m.)} & (720 MeV) & (600 MeV) & (420 MeV) & (300 MeV) \\
power on collimator & 5.3 W & 4.7 W & 4.2 W & 3.8 W \\
power on target & 810 mW & 690 mW & 600 mW & 540 mW \\
total hadronic rate & 385 K/s & 365 K/s & 350 K/s & 345 K/s \\
tagged hadronic rate & 26 K/s & 14 K/s & 6.3 K/s & 2.1 K/s \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

The above comparison is obtained by keeping the electron beam parameters
fixed and simply rocking the crystal.  There were two factors which
were considered in deciding what the limits of the tagged window should
be: the polarization, and the tagging efficiency.  Both factors peak at
the coherent edge and decrease with decreasing photon energy.  Of the
two, the tagging efficiency is the more steep function, so it was used
to limit the size of the tagged window.  For the study of the dependence
of the source parameters on the electron beam energy, the F.W.H.M.\  of
the tagging efficiency listed in the second column of Table~\ref{cdr_rates}
has been used to set the tagging window to the range 8.4 - 9.0 GeV.  To
facilitate the comparison between collimated and uncollimated beams,
the electron beam current was adjusted to maintain a constant total
hadronic rate in the target of 370~KHz.

\begin{table}
\begin{center}
\caption[Dependence of source parameters on electron beam energy]{
\label{new_rates}
Operating parameters for the GlueX photon source under conditions of
varying electron beam energy.  The collimator distance and diameter
are kept constant at 80~m and 3.4~mm, respectively, and the radiator
thickness is $10^{-4}$ radiation lengths.
be 3~$\mu$A. The rates in the detector (last two rows) are calculated for 
a $30$~cm liquid hydrogen target and an open hadronic trigger.}
\begin{tabular}{lcccc}
\hline\hline
electron beam energy & 10 GeV & 11 GeV & 12 GeV & 13 GeV\ \\
electron beam current & 4.3 $\mu$A & 3.5 $\mu$A & 3.0 $\mu$A & 2.5 $\mu$A \\
$N_{\gamma}$ in peak & 32 M/s & 67 M/s & 100 M/s & 130 M/s \\
peak polarization & 0.14 & 0.28 & 0.41 & 0.48 \\
average polarization & 0.08 & 0.24 & 0.37 & 0.47 \\
peak tagging efficiency & 0.25 & 0.43 & 0.50 & 0.57 \\
average tagging efficiency & 0.15 & 0.29 & 0.41 & 0.51 \\
power on collimator & 4.4 W & 4.4 W & 4.5 W & 4.5 W \\
power on target & 510 mW & 610 mW & 730 mW & 850 mW \\
total hadronic rate & 370 K/s & 370 K/s & 370 K/s & 370 K/s \\
tagged hadronic rate & 5 K/s & 10 K/s & 16 K/s & 21 K/s \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

Numerical results are shown in Table~\ref{new_rates}.  The same results
are shown graphically in Figs.~\ref{polar10}-\ref{polar13} for polarization
and Figs.~\ref{flux10}-\ref{flux13} for flux.
Figs.~\ref{polar10},\ref{flux10} show that at
the large-angle setting chosen for the 10~GeV end-point case there is an
extra peak at 8.4~GeV that appears.  This comes from the accidental
overlap of the desired reciprocal lattice vector of the crystal with
another higher-order vector from another region of $q$-space.
The polarization in this peak lies in a different plane from the primary
peak at 9~GeV, so under actual operating conditions it would be desirable
to adjust one of the goniometer angles to rotate it away.  This is
possible to do, in general, without affecting the alignment of the
primary edge.  This peak
actually appears in the other spectra as well, but at energies above
the primary peak where it is interspersed among the higher-order
recurrences of the primary peak.
Its presence does not affect the results shown, so no
effort was made to eliminate it for the purposes of this study.

\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{polar10.eps}
\caption{\label{polar10}
Degree of linear polarization {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 10 GeV electron beam, with collimation
(solid curve) and without (dashed curve).}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{polar11.eps}
\caption{\label{polar11}
Degree of linear polarization {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 11 GeV electron beam, with collimation
(solid curve) and without (dashed curve).}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{polar12.eps}
\caption{\label{polar12}
Degree of linear polarization {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 12 GeV electron beam, with collimation
(solid curve) and without (dashed curve).}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{polar13.eps}
\caption{\label{polar13}
Degree of linear polarization {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 13 GeV electron beam, with collimation
(solid curve) and without (dashed curve).}
\end{figure}

\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{flux10.eps}
\caption{\label{flux10}
Photon beam rates {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 10 GeV electron beam, with collimation
(solid curve) and without (dashed curve).  Rates are normalized as
shown in Table~\ref{new_rates}.}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{flux11.eps}
\caption{\label{flux11}
Photon beam rates {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 11 GeV electron beam, with collimation
(solid curve) and without (dashed curve).  Rates are normalized as
shown in Table~\ref{new_rates}.}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{flux12.eps}
\caption{\label{flux12}
Photon beam rates {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 12 GeV electron beam, with collimation
(solid curve) and without (dashed curve).  Rates are normalized as
shown in Table~\ref{new_rates}.}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{flux13.eps}
\caption{\label{flux13}
Photon beam rates {\it vs} photon energy for the GlueX coherent
bremsstrahlung source for a 13 GeV electron beam, with collimation
(solid curve) and without (dashed curve).  Rates are normalized as
shown in Table~\ref{new_rates}.}
\end{figure}

\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{fom.eps}
\caption{\label{fomvsD}
Photon beam figure of merit, defined by Eq.~\ref{fom}, plotted as
a function of the radiator-collimator distance, for a variety of
collimator apertures (colored curves) and for no collimator (black curve).
The vertical scale is defined by setting the FOM for the uncollimated case
to unity.}
\end{figure}
\begin{figure}\centering
\includegraphics[width=1.0\textwidth]{fom_E.eps}
\caption{\label{fomvsE}
Photon beam figure of merit, defined by Eq.~\ref{fom}, plotted as
a function of the electron beam energy, under conditions of constant
total hadronic background rates in the detector.  The green curve is
for a 3.4 mm collimator at 80 m from the radiator, and the black curve
is for an uncollimated beam.}
\end{figure}

An overall figure of merit for the combination of flux enhancement and
polarization for the GlueX photon source is defined by Eq.~\ref{fom}.
\begin{equation}
\mbox{\rm FOM} = P^2 F
\label{fom}
\end{equation}
where $P$ is the degree of linear polarization averaged over the tagged
window, and $F$ is the total photon flux on target within the same window.
Thus defined, the FOM is a relative measure of the sensitivity of an
experiment to a small signal in a polarization observable.  The overall
scale for the beam flux factor $F$ is determined by the maximum beam
intensity at which the detector can operate effectively.  The GlueX it
is expected that the first-level trigger rate will be the limiting factor
in setting the beam current.  The first-level trigger rate is determined
by the total hadronic rate in the target.  Therefore the FOM is defined
as the product of mean-squared polarization in the tagged peak and the
flux at fixed total hadronic rate in the detector, which was chosen to
be 370 KHz, as shown in Table~\ref{new_rates}.

In Fig.~\ref{fomvsD} is shown the dependence of the FOM on the distance
between the radiator and collimator, under constant electron beam conditions
and hadronic background rates in the detector.  The limit of a large
collimator aperture is shown by the black line as a normalization reference.
In Fig.~\ref{fomvsE} is shown the dependence of the FOM on the electron
beam energy, under constant conditions of collimation and hadronic
background rates in the detector.  All electron beam properties (emittance,
spot size, energy resolution) were held constant in this study, as the
beam energy was varied.

One concern that is not explicit in the FOM is the tagging efficiency,
which determines the effectiveness of the tagging coincidence for
determining the photon energy for any given event.  The tagging efficiency
is implicitly incorporated into this analysis by its role in setting the
size of the window of tagged photons which is subtended by the active
tagging counters.  It also played a role in the choice of the collimator
diameter, by limiting the extent to which further collimation can be
exploited to improve polarization and flux conditions for the beam.
This is why the curves for 2~mm, 3.4~mm, and 5~mm collimators cannot
be extended indefinitely in Fig.~\ref{fomvsD}.  For the purposes of
this figure, the curves terminate at the point where the peak tagging
efficiency drops below 30\%.

\end{document}
