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\begin{center}               
 {\bf \Large   TRIUMF Experiment E614 \\
    Technical Note \#47\\ }
\vspace{1cm}                      
{\bf \Large 
Monte Carlo Simulation of Muon Stop Distributions in E614 Targets \\}
\vspace{1cm}                      
{\bf V.I.Selivanov, M.A.Vasiliev \\
 14 September 2000 } 
\end{center}



Surface muon distributions in a stopping target of E614 detector has been
simulated with GEANT/E614/1.2 package for different target materials. 
A minimal target 
thickness was selected to provide 70-75\% muons stop rate for Li, Be, C, Al,
Fe and W. The muon distributions were calculated versus magnetic field value,
beam divergence, and momentum spread value.

All elements of E614 detector  were involved in the simulation: muon counter (TEC),
(Ar +He) degrader, scintillator, PC2 and PDC4 chambers. Scintillator thickness was changed on
$50\mu$ maximum to have a symmetrical muons stop distribution along Z axis inside target.
Magnetic field maps~\cite{ram} was also used in simulations.  

\section*{Minimal Target Thickness}

A minimal target thickness is needed for E614 in order to decrease an outgoing
positron energy loss inside the stopping target, and to weaken therefore its
influence on reconstructed Michel parameters.	     

Surface muon beam with $\sigma_{\theta_x} = \sigma_{\theta_y} = 6mr$ 
and $\sigma_x = \sigma_y= 6mm$ was used for the 
simulation (beam \#1). The beam focus was placed
at $Z =-140cm$ (before the muon counter). Original muon momentum 29.7MeV/c 
with spread of $\frac{\Delta p}{p} = 0.5\%$ 
has been selected. Magnetic field  -  $B=2.2T$.
The simulation results are shown in Table~\ref{tab1}. 
Typical results of the simulation
are shown in Fig.1 
% ~\ref{fig:f1}. 


 
\begin{table}[ht]
\caption{Selected target thickness, muons stop rate, and positron energy losses}
\label{tab1}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
Name & Z & $\rho$,  & 
\multicolumn{2}{|c|} {Target thickness} &
Muons & 
\multicolumn{6}{|c|} {Positron energy loss}   \\
\cline{4-5}
 & & $g/cm^3$ &($mg/cm^2$) &($\mu$) & stop & 
\multicolumn{6}{|c|}{(KeV/target thickness)} \\ 
\cline{7-12} 
 &  &  &  &  &  rate,  &
\multicolumn{3}{|c|}{$T_e=20$MeV}&
\multicolumn{3}{|c|}{$T_e=50$MeV} \\
\cline{7-12}
 & & & & & (\%)  & Coll. & Rad. & Total & Coll. & Rad. & Total \\
\hline
Li & 3  & 0.534 & 16   & 300 & 71 & 26.6 & 2.7  & 29.3 & 27.7 & 7.8  & 35.5 \\
\hline
Be & 4  & 1.848 & 16.6 & 90  & 72 & 26.9 & 3.6  & 30.5 & 28.1 & 10.3 & 38.4 \\
\hline
C  & 6  & 2.265 & 15.9 & 70  & 73 & 28.6 & 5.4  & 34.0 & 29.9 & 15.3 & 45.2 \\
\hline
Al & 13 & 2.70  & 20   & 75  & 76 & 34.0 & 12.8 & 46.8 & 35.8 & 35.2 & 71.0 \\
\hline
Fe & 26 & 7.87  & 23.6 & 30  & 75 & 36.8 & 27.1 & 63.9 & 38.9 & 73.9 & 112.8\\
\hline
W  & 79 & 19.30 & 38.6 & 20  & 78 & 49.0 & 97.3 & 146.3& 52.5 & 261.3& 313.8\\
\hline
\end{tabular}
\end{center}
\end{table}
  
\begin{figure}
\label{fig:f1}
\begin{center}
\includegraphics[height=15.5cm]{/home/vasiliev/tex/e614/target/fig1.eps}
\caption{Muon stop distributions for $90\mu$ of Be stopping target at B=2.2T
(beam \#1). Total 10000 events has been simulated.}
\end{center}
\end{figure}
 

% (be_90mk_140_05_2t). 




 
One can see that using of the Be target instead of the Al 
target allows to decrease 
the total positron energy loss almost in 2 times at 50MeV. The radiative loss
is decreased in 3.5 times. On the contrary the total positron energy loss for
W target is in 4.5 times more than for Al target.

\section*{Muon Stop Distributions vs Beam Divergency.}

Proposal exists to make a measurement of 
$\rho$  
and $\eta$ parameters first of
all using a nonpolarised muon beam. Such kind of beam could be provided with
so called "mixed" beam consisting surface muons with $P_{\mu}=-1$ and cloud muons
with $P_{\mu}=+0.4$ in corresponding proportion. 
Disadvantage of the proposal is
a small enough admixture 
(about 2\%, \cite{oram}) of the cloud muons around 
$p_{\mu}=29.8MeV/c$. 
Possible way to increase the beam intensity is an opening of
slits placed between Q2 and B1 (F0VJ (vertical jaw) and F0HSL (horizontal 
slit)). According to Fig.12 of the reference \cite{oram} beam flux can be 
increased
in 6-7 times by opening of the horizontal slit only. It means a total gain
of flux is about 30-50 times with the jaw and the slit open. Another way is 
possible for the estimation. Particle flux at $p_{\mu}=30 MeV/c$ 
is about $3000 s^{-1}$ 
at $100\mu A$ with all slits and jaws fully open (\cite{oram}, fig.7) 
using a 1.45mm 
carbon target. It means the flux about $20000 s^{-1}$ 
with a 10mm carbon target.
Maximal momentum acceptance of the M13 is about 7\% with F1 horizontal slit 
width equals to 15cm (\cite{oram}, Fig.10). 
We need to put the F1 horizontal slit width
equals to 0.5cm in order to have a beam momentum acceptance equals to 0.5\%.
It corresponds to a flux decreasing it 10 times (\cite{oram},Fig.13). 
So, we can have
cloud muon flux about $2000s^{-1}$ at 
$\frac{\Delta p}{p}=0.5\%$, in result. Another 
disadvantage of the "mixed" beam is a high rate of positrons 
(about $10^5 s^{-1}$)
from beam line. It means we will see nearly always both a positron
from beam line and a decaying positron.


Muon stop distributions in a stopping target for a "mixed" beam with a big
divergency has been simulated. Simulation result is shown in Fig.2
% ~\ref{fig2} 
at
$\sigma_{\theta_x} = \sigma_{\theta_y} = 80 mr$ 

\begin{figure}
\begin{center}
\includegraphics[height=15.5cm]{/home/vasiliev/tex/e614/target/fig2.eps}
\caption{Muon stop distributions for $75\mu$  Al stopping target at the beam
divergency equals to 80mr (beam \#2, focus plane at $Z=-140cm$, B=2.2T). Total
10000 events has been simulated.
}
\end{center}
\label{fig2}
\end{figure}


% (al_75mk_140_05_2t_80mr). 
One can see that the
X and Y stop distributions are acceptable even for the 80mr 
beam divergency.
 

\section*{Muon Stop Distributions vs Momentum Spread}

Additional possibility to gain the "mixed" muon flux is increasing of beam 
momentum spread. The results of GEANT simulation are shown in Fig.3. 
% ~\ref{fig3}
% (total al_75mk_140_05-6_2t). 
One can see that the results are not so good 
because muon rate drops rapidly with the momentum spread growth.

\begin{figure}
\begin{center}
\includegraphics[height=15.5cm]{/home/vasiliev/tex/e614/target/fig3.eps}
\caption{Muon stop distributions for $75\mu$  Al stopping target at different
muon momentum spread $\Delta p/p$ (beam \#1, focus plane at $Z = -140cm$,
B=2.2T). 
Total 10000 events has been simulated. 	   
} 
\end{center}
\label{fig3}
\end{figure}


\begin{figure}
\begin{center}
\includegraphics[height=15.5cm]{/home/vasiliev/tex/e614/target/fig4.eps}
\caption{Muon stop distributions for $75\mu$  
Al stopping target (beam \#2, beam 
divergency is 19mr, focus plane at Z=0, B=0.6T). Total 10000 events has been 
simulated.                       
} 
\end{center}
\label{fig4}
\end{figure}

\section*{Muon Stop Distributions at B=0.6T}

Measurement of $\eta$ parameter could request of a smaller magnetic field
value because the $\eta$ is more sensitive to small positron energies. Results
of simulation are shown in  Fig.4.
%~\ref{fig4}. 
% (al_75mk_0_05_06t_19mr). 
 

Focus plane of the
beam was placed at Z=0 (inside the stopping target) to decrease a beam 
scattering by chambers placed before the stopping target. One can see that the
muon stop rate is 55\% only, and 
$\sigma_x = \sigma_y = 1.66cm$ at B = 0.6T instead 
of $\sigma = 0.75cm$ at B=2.2T (Fig.1,
%~\ref{fig1} 
for example). 
The above distributions are 
not change actually for beam focus position at $Z =-140cm$. It means the 
B = 0.6T is too small value to compress beam during a muon scattering by E614 
detector.



\begin{thebibliography}{999}  
  
\bibitem{ram}
R.Ramadanovic, D.H.Wright, TN-31, 17 May 1999  

\bibitem{oram}
C.J.Oram, J.B.Warren, G.Marshall, J.Doornbos, D.Ottewell, TRIUMF preprint
TRI-80-1, May 1980

\end{thebibliography}  
                    
	                       
	                       
\end{document}


FIGURES CAPTION.
Table 1. Selected target thickness, muons stop rate, and positron energy losses.

Fig.1. 
Muons stop distributions for 90mk of Be stopping target at B=2.2T
(beam #1). Total 10000 events has been simulated.
Fig.2. 
Muons stop distributions for 75mk of Al stopping target at the beam
divergency equals to 80mr (beam #2, focus plane at Z=-140cm, B=2.2T). Total
10000 events has been simulated.
Fig.3. 
Muons stop distributions for 75mk of Al stopping target at different
muon momentum spread deltap/p (beam #1, focus plane at Z = -140cm ,B=2.2T). 
Total 10000 evetns has been simulated. 	   
Fig.4.
Muons stop distributions for 75mk of Al stopping target (beam #2, beam 
divergency is 19mr, focus plane at Z=0, B=0.6T). Total 10000 events has been 
simulated.