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{\Large \bf TRIUMF Experiment E614 \\}
{\Large \bf Technical Note \\}

\vspace{0.4cm}

{\Large \bf An Analysis of Energy Loss in the Proportional Chambers near the Target  \\}

\vspace{0.4cm}

\rm{\bf M. Grinder and D. H. Wright \\}

\vspace{0.4cm}

\rm{\bf 31 August 1998 \\}

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\begin{abstract}

  Monte Carlo simulations of muon energy loss recorded by the proportional 
chambers near the target are presented and analyzed.  These chambers were found
to be very useful in rejecting muon stops just upstream of the target and as a 
monitor of the muon stopping distribution in the target.  It was determined 
that both ADC's and TDC's are required for the proportional chambers near the 
target. 
\end{abstract}
\vspace{1.2cm}

{\large \bf Introduction \\} 

  Two issues vital to the success of E614 may be adressed by the use of four 
proportional chambers adjacent to the stopping target.  The first issue, which
affects the muon depolarization, is the removal of events in which muons stop 
near, but not in, the target. Nominally the trigger condition 
${\rm{PC(-2)}}\cdot{\rm{PC(-1)}}\cdot{\overline{\rm{PC(+1)}\cdot{\rm{PC(+2)}}}}$ 
removes such events.  The notation adopted here is that PC(-2), PC(-1) refer 
to the upstream proportional chambers while PC(+1), PC(+2) refer to the 
downstream chambers.

  However, this simple scheme does not account for those muons that could 
lose all their energy and stop in PC(-1), nor does it account for those 
muons that nearly stop in the target and lose a small amount of energy (too 
little to be detected) in PC(+1).  The addition of ADC's to the PC's 
may solve this first problem, allowing for a measurement of the energy loss.  
With this information, one can impose cuts that eliminate those muons stopping
in PC(-1). 

  The second issue is the shape of the muon stopping distribution in the 
target; a symmetric distribution is desired in order to remove the fore-aft 
asymmetries in the decay positron spectrum.  When the upstream mass of the 
detector is optimized so that the peak of the muon Bragg curve is centered on
the target, the stopping distribution is symmetric about $z=0$.  In this case 
the hit ratios $N_{PC(-1)}/N_{PC(+1)}$, $N_{PC(-2)}/N_{PC(+2)}$ have unique 
values.  If the stopping distribution becomes skewed upstream or downstream,
these ratios change significantly, providing a monitor of the shape of the 
stopping distribution. 
\vspace{1.0cm}

{\large \bf Monte Carlo Simulations \\}  

  Monte Carlo simulations were performed using version 1.0 of the E614 
GEANT simulation code.  The detector geometry was that adopted at the April 
1998 collaboration meeting and consists of, from the stopping target outward: 2
proportional chambers, 22 drift chambers, one plane of plastic scintillator of
thickness 340$\mu$m, 4 more proportional chambers, and 10 drift chambers for 
muon tracking. 

  In the simulation a beam of 29.7 MeV/c $\mu^{+}$, with a beam divergence of
15 mrad ($\sigma$) in $x$ and $y$ and a beamspot of 0.5 cm ($\sigma$) in $x$
and $y$, was started 120 cm upstream of the target.  Muons were allowed to
decay, but the resulting positrons were not tracked.  Finally, a uniform
solenoidal field of 2.2 T was oriented along the $z$-axis.  
  
  As muons entered and exited the PC's, their kinetic energy was recorded and
their energy loss in each chamber was histogrammed.

  The thickness of the plastic scintillator which produced a symmetric stopping
distribution in the target was 340 $\mu$m.  Varying this by ${\pm}20\mu$m 
caused the stopping distribution to be skewed upstream or downstream. 

\begin{figure}
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\centerline{\epsfbox{tn22fig1.ps}}                                        
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\caption{Muon kinetic energy loss in PC(-2),PC(-1),PC(+1), and PC(+2) chambers
for an initial 10,000 events.}
\end{figure}  
\vspace{0.8cm}

{\large \bf Results \\}

{\large \bf 1. Triggers from the Proportional Chambers \\}

  The mean energy loss per muon in the chambers was $\sim$0.1 MeV.  The 
distributions of energy loss in PC(-2), PC(-1) ,PC(+1), and PC(+2) are shown
in Fig.1.  

  A scatterplot of energy losses in PC(-2) and PC(-1) is shown in Fig. 2 
and exhibits three main features: a dense locus of points with a positive 
slope, a less dense locus with a negative slope, and two sparse loci of points
in which a large energy loss in one chamber is paired with a small energy loss
in the other.  The dense locus with positive slope is due to muons stopping in
the target (Fig. 3, bottom left), and the low energy end of this locus
is due to those that pass through the target(Fig. 3, bottom right).  The 
less dense locus of negative slope is due to muons stopping in PC(-1).  The 
points which lie along either axis (those which have large energy loss in one
chamber, but small in the other) are due to muons scattering from the PC wires.
 
\begin{figure}
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\centerline{\epsfbox{tn22fig2.ps}}                                        
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\caption{Scatterplot of the kinetic energy loss in PC(-2) and PC(-1) for a
symmetric target stopping distribution.}
\end{figure}

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\centerline{\epsfbox{tn22fig3.ps}}                                        
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\caption{ Energy loss in PC(-2) and PC(-1) chambers for various muon stopping
conditions. Top left: no conditions;  top right: stop in PC(-1);  bottom left:
stop in target;  bottom right: stop in PC(+1) or PC(+2).}
\end{figure}


  As seen in Fig. 3, nearly all stops in PC(-1) have an energy loss in PC(-2) 
greater than $\sim$0.1 MeV, so a cut at that value would remove most events 
with stops in PC(-1).   A two dimensional cut along the lower edge of the locus
of PC(-1) stops would be more efficient, however.  The equation of the cut line
is approximately ${\Delta}E(-2)= -1.74\times {\Delta}E(-1) + 0.144$ MeV.  
A cut near this edge will eliminate all the muons stopping in PC(-1), but
still leave most of those stopping in the target.

  Table I shows the result of various cuts that attempt to minimize stops
in PC(-1), but maximize those that stop in the target.  The cuts differ only by
their x-intercepts.  The cut at x-intercept = 0.137 MeV eliminates nearly
all muons stopping in PC(-1), while leaving more than 97\% of the muons 
stopping in the target.  It also removes all events which have large energy 
loss in PC(-2) and small energy loss in PC(-1).  It is also desirable to cut 
out those muons losing a large amount of energy in PC(-1) due to hits on the 
sense wires.  These were removed by rejecting all events with energy losses in 
PC(-1) greater than 0.3 MeV and less than 0.03 MeV.  

\vspace{0.8cm}
\begin{tabular}{|c|c|c|} \hline
x-intercept of the cut line (MeV) & \% stopping in PC(-1) & \% stopping in target \\ \hline
0.151   & 7.3                   & 98.7 \\ \hline
0.144   & 2.3                   & 98.2  \\ \hline
0.137   & 0.6                   & 97.3  \\ \hline
\end{tabular}

\vspace{0.2cm}
Table I. $\mu^{+}$ stopping percentage in the target and PC(-1) as a function of x-intercept of 2-D cut.
\vspace{0.2cm}

  The performance of the proportional chambers, however, has not been 
simulated here, so it is unclear if the scattering off the wires will appear 
as shown.  The muon may appear to deposit more energy if it scatters 
off the wire at a large angle and creates more electron showers as it crosses 
the path of more wires.  Otherwise, in a low angle scatter, the energy it 
loses in the wire will not cause a shower, thus a lower energy signal will
be recorded.  Therefore, the same sparse loci should be seen, not 
precisely as shown here, but requiring similar cut values on the energy loss
in PC(-1).
  
  It is noted here that the energy threshold of the PC's is at most 
$\sim$1 keV$\cite{Vla}$.  From Figs. 1 and 3 (bottom right) it is seen that the
smallest energy loss recorded by PC(+1) is $\sim$20 keV.  In 10,000 entries, 
there were no muons that lost less than 1 keV of kinetic energy.  
Pessimistically assuming that the PC energy threshold is 25 keV, at least 2 in
$10^{4}$ events will be lost. Providing the chamber thresholds can be kept low
($\leq$ 5 keV), it is therefore unlikely that muons will pass through the 
target but not be recorded by PC(+1).
\vspace{0.8cm}

{\large \bf 2. Monitoring Stopping Distributions in the Target \\}

  A scatterplot of the muon energy losses in PC(-1) and PC(-2) was considered 
as a monitor of the stopping distribution in the target.  In Fig.4 three 
stopping distributions along with their corresponding energy loss scatterplots
are displayed.  A symmetric stopping distribution (Fig.4, middle) is produced
by placing a 340 $\mu$m thick scintillator in the muon path.  By decreasing the
thickness to 320 $\mu$m a stopping distribution skewed toward positive $z$ is 
produced (Fig.4, top).  A skew to negative $z$ is achieved by increasing 
the thickness to 360 $\mu$m (Fig.4, bottom).  These scatterplots are very 
similar and, except for a slight energy displacement, do not show any 
qualitative difference.  The energy displacements suggest mean energy loss as
a monitor of stopping distribution.  However, this would require precise
calibration of the detectors, since differing thresholds, gains, and
efficiencies could cause artificial shifts in the mean energy loss.

\begin{figure}
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%\epsffile{el+1+2.ps}                                        
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\caption{ Stopping distribution in the target for a scintillator of thickness
320 $\mu$m (top graphs), 340 $\mu$m (middle), and 360 $\mu$m (bottom).
Corresponding scatterplots show energy loss in PC(-1) and PC(-2).}
\end{figure}

  A simpler method is merely to count the number of muons passing through the 
proportional chambers.  The ratio of the number of muons recorded by PC(+2) to
those recorded by PC(-2), and the ratio of PC(+1) to PC(-1) should change with 
deviations from a symmetric stopping distribution.  These ratios should not be 
affected by factors such as PC efficiency (unless each chamber has different 
efficiency) or muons stopping in PC(-1).  Other ratios change as well, but 
$N_{PC(+1)}/N_{PC(-1)}$, and $N_{PC(+2)}/N_{PC(-2)}$ were found to be the most 
sensitive to changes in the stopping distribution.  

\vspace{0.8cm}
\begin{tabular}{|l|l|l|l|l|l|l|l|} \hline
\multicolumn{1}{|c}{stopping distribution} & \multicolumn{1}{|c|}{mean in z ($\mu$m)}& \multicolumn{4}{|c|}{number in PC chamber} & \multicolumn{2}{|c|}{Ratios}  \\
           &       & -2   & -1   & +1  & +2  & +2/-2   & +1/-1  \\ \cline{3-8}
symmetric  &  0.18 & 8999 & 8581 & 464 & 156 & 0.0173 $\pm$0.001 & 0.054$\pm$0.003 \\ \hline
right skew &  4.9  & 9327 & 9021 & 986 & 443 & 0.0475 $\pm$0.002 & 0.109$\pm$0.004 \\ \hline
left skew  & -5.6  & 8533 & 7910 & 176 & 39  & 0.0046 $\pm$0.001 & 0.022$\pm$0.002 \\ \hline
\end{tabular}

\vspace{0.2cm}
Table II. Ratios of number of muons recorded by the proportional chambers as a
function of stopping distribution mean. 
 
  For a symmetric stopping distribution, table II shows that the ratios, 
$N_{PC(+1)}/N_{PC(-1)}$, and $N_{PC(+2)}/N_{PC(-2)}$ were 0.054 and 0.0173, 
respectively.  Deviations of $\pm$20$\mu$m in scintillator thickness produced 
a significant change in ratios.  These ratios are statistically robust.  For
example, the ratio $N_{PC(+2)}/N_{PC(-2)}$ for the right-skewed distribution is
16$\sigma$ different from the same ratio in the symmetric case.  These ratios 
are therefore very sensitive to differences in the stopping distribution, and 
are shown graphically in Fig.5.  


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\begin{figure}
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%\centerline{\epsfbox{ratio.ps}}                                           
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\caption{Graphical representation of PC ratios as a function of stopping
distribution mean. }
\end{figure}

\vspace{0.8cm}
{\large \bf Conclusion \\}

  Using ADC's to read out the wires in PC(-2), PC(-1), PC(+1) and PC(+2) 
provides a means of rejecting events in which muons stop in PC(-1) instead of 
the target.  Making a two dimensional cut on the energy losses in PC(-1) and
PC(-2) removes all but 0.6\% of such events.  The remaining 0.6\%, which are
due to scattering in the chamber wires, can be reduced by additional cuts.  
It seems unlikely that muons passing through the target with too little energy
to be recorded by PC(+1) will cause a significant error.

  The central PC's provide a reliable monitor of stopping distribution when the
ratios of the number of muons recorded by PC(+2) and PC(-2) and PC(+1) and 
PC(-1) are examined.  

  The above results are largely independent of detector configuration, as long
as mass can be added or removed to center the peak of the Bragg curve around 
$z=0$.  

  In sum, the addition of ADC's to the PC chambers, though not required to 
monitor the stopping distribution in the target, will be a useful means of 
rejecting false stops near the target.  

\begin{thebibliography}{10}

\bibitem{Vla}{V.I.Selivanov, private communication, 1998.}  

\end{thebibliography}

\end{document}