\section{Monte Carlo Study of Photon Energy Resolution}
\label{mc:resolution}

%%%%%%%%%%%%%%%%%
% Graham's Results
%%%%%%%%%%%%%%%%%%

In this study the \textsc{genr8} program was used to generate the events. 
Four different exclusive reactions were studied, two with photons produced
at the baryon vertex:
\begin{eqnarray}
\label{reaction:1} \gamma p\rightarrow N^{*}(1500) {{\pi}^{{+}}} \rightarrow 
(n\eta) {{\pi}^{{+}}} \rightarrow n {{\pi}^{{+}}} {\gamma} {\gamma} \\
\label{reaction:2} \gamma p \rightarrow X^{+}(1600) {\Delta}^{0} \rightarrow 
({{\pi}^{{+}}} {{\pi}^{{+}}} {{\pi}^{{-}}}) (n \pi^{0}) \rightarrow 
{{\pi}^{{+}}} {{\pi}^{{+}}} {{\pi}^{{-}}} n {\gamma} {\gamma}
\end{eqnarray}
The $\Delta^{0}$ reaction (reaction \ref{reaction:2}) has a 3$\pi$-meson 
mass of $1.600$\munit{}, and a width of 300 MeV/$c^{2}$. The two meson 
vertex reactions are:
\begin{eqnarray}
\label{reaction:3} \gamma p \rightarrow X^{+}(1600) n \rightarrow (\eta
{{\pi}^{{+}}}) n \rightarrow n {{\pi}^{{+}}} {\gamma} {\gamma} \\
\label{reaction:4} \gamma p \rightarrow X(1600) p \rightarrow 
({{\pi}^{{+}}} {{\pi}^{{-}}} {{\pi}^{{o}}}) p \rightarrow 
p {{\pi}^{{+}}} {{\pi}^{{-}}} {\gamma} {\gamma}
\end{eqnarray}
In both reactions~\ref{reaction:3} and~\ref{reaction:4}, the meson 
systems were generated with a Breit--Wigner distribution of mass 
$1.6$\munit and a width of $0.3$\munit{}.
       
 
Each of the above reactions were simulated using a beam energy of $8$\eunit{}, and 
a $t$-channel slope of $5$\munit{}. The production and decay vertex was assumed to 
be at the center of the target. For each system, 10,000 events were generated. The 
direction and energy of the photons were recorded and analyzed.
 
\subsection{\label{sec:results:energy}Photon Detector Energy Resolution}
\hspace{1cm}
The photons produced in the above decays were traced into the Barrel Calorimeter
and the Lead Glass Detector. Figure~\ref{fig:Gram2} and~\ref{fig:Gram3} 
show the percentage of photons that would enter, but not be detected by the 
Barrel Calorimeter due to the minimum energy thresholds. 
\begin{figure}[h]\centering
\begin{tabular}{cc}
\mbox{\epsfig{file=ch10_3pi_energybcal2.eps,width=0.50\textwidth}} &
\mbox{\epsfig{file=ch10_delta_energybcal2.eps,width=0.50\textwidth}}
\end{tabular}
\vspace{-0.5cm}
\caption[Lost photons in the BCAL]{\label{fig:Gram2}
The percentage loss due to the energy threshold of the BCAL. Left is for 
reaction~\ref{reaction:4}, while the right figure is for the $\Delta^{0}$ decay from 
reaction~\ref{reaction:2}. The percent of the total photons entering the Barrel 
Calorimeter for reaction \ref{reaction:4} is 57\% and reaction \ref{reaction:2} 
is 87\%.}    
\end{figure}
\begin{figure}[h]\centering
\begin{tabular}{cc}
\mbox{\epsfig{file=ch10_etax_energybcal.eps,width=0.50\textwidth}} &
\mbox{\epsfig{file=ch10_etay_energybcal.eps,width=0.50\textwidth}}
\end{tabular}
\caption[Lost photons in the BCAL]{\label{fig:Gram3}
The percentage loss due to the energy threshold of the BCAL. The left plot is 
for reaction~\ref{reaction:3}, and right is for reaction \ref{reaction:1}. The 
percent of all the photons entering the Barrel Calorimeter for the 
$\eta$ X (reaction \ref{reaction:3}) and the $\eta$ $N^{*}$ (reaction~\ref{reaction:1}) 
are 55\% and 88\% respectively.}
\end{figure}                
 
Currently, the design calls for the energy sensitivity of 20 MeV for the
Barrel Calorimeter. One can see that this results in around a 1\% loss of 
photons which is quite acceptable. However, if this energy can not be
met, the percentage of photons lost rises rapidly with the increased energy  
threshold, especially for the $\Delta^{0}$ (reaction \ref{reaction:2}) decay. 
For example, if the threshold is 50 MeV, then 5\% of the 3$\pi$ reaction 
is lost, and 10\% of the $\Delta^{0}$ reaction is lost.  The situation for the $\eta$ 
reactions is not so severe, as would be expected from the higher energy photons in 
the $\eta$ decay (figure~\ref{fig:Gram3}).
 
 
The results for the Lead Glass Detector are similar, but the percentage rise is not 
so significant at higher energy thresholds. The only system with significant loss 
in the lead glass array is the 3$\pi$ (reaction \ref{reaction:4}) decay. At the 
sensitivity threshold of $100$ \Meunit{}, the lead glass detector will not see 0.718\%
of the photons. The design calls for a $150$\Meunit{} detection minimum in the LGD. At 
this energy, the detector will miss 1.86\% of the photons (figure \ref{fig:Gram4}).
\begin{figure}\centering
\begin{tabular}{c}
\epsfig{file=ch10_3pi_energylg.eps,width=0.60\textwidth}
\end{tabular}
\caption[Lost photons in the LGD]{\label{fig:Gram4}
The percentage of undetected photons for a given energy threshold of the lead glass 
detector. From reaction~\ref{reaction:4}.}
\end{figure}   \clearpage

