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

\vspace{0.4cm}

{\Large \bf Simulation of a Time Expansion Chamber as a Means of Correlating
 Initial Muon Track Direction and Polarization in the Target \\}

\vspace{0.4cm}

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

\vspace{0.4cm}

\rm{\bf 1 September 1998 \\}

\end{center}

\begin{abstract}

  In Monte Carlo simulations, a Time Expansion Chamber (TEC) was placed
upstream of the solenoidal fringe field.  The chamber was used to reconstruct
the incoming muon track and correlate its angle with the spin direction of the
muon as it stopped in the target.  A cut on the polar angle of the initial
muon track was found which retained $\sim$12\% of all muons stopping in the
target while reducing the mean depolarization $<1-P_\mu>$ of the retained
events to $\sim$2 $\times 10^{-4}$.  This result was obtained using a highly
convergent beam with a focus 160 cm upstream of the target.  A more parallel
beam was found to produce poorer results.  The spatial resolution of the TEC
had only a small effect on the results.  A comparison of the above results
to those obtained using additional drift chambers to track the muon, favors
the use of a TEC.

\end{abstract}

\vspace{1.0cm}

{\large \bf Introduction \\}  

  A central goal of the E614 experiment is to measure the product $P_\mu \xi$
to a precision of less than three parts in $10^4$.  The mean depolarization
($<1-P_{\mu}>$) must therefore also be less than three parts in $10^4$.  One
way to guarantee this is to exploit the near perfect polarization of the muon
beam in vacuum upstream of the solenoidal fringe field.  A very low mass
device, such as a Time Expansion Chamber (TEC), can be inserted to measure the
track direction and hence the polarization at this point.  Assuming that the
propagation of muons through the TEC and the rest of the E614 detector is
understood, the spin direction of the muons stopping in the target should
be correlated with the track direction reconstructed from the TEC.  A suitable
cut on the initial track angle should then reduce the mean depolarization of
the surviving events to desirable levels.

  The degree of correlation, and therefore the efficiency of the cut, depends
on the emittance of the muon beam, the position resolution of the TEC and the
degree to which the TEC disturbs the muon tracks.  Each of these factors is
discussed below.

\vspace{1.0cm}

\newpage

{\large \bf Monte Carlo Simulation \\}  

  Monte Carlo simulations were performed using version 1.0 of the GEANT E614
simulation code.  The geometry, as adopted at the April 1998 collaboration
meeting, was modified by removing the ten drift chambers (upstream and
downstream) used for muon tracking and replacing them with Time Expansion
Chambers (TECs) centered 174.25 cm upstream and downstream of the target.
The modified version thus includes, from the stopping target outward, two
proportional chambers, 22 drift chambers, one plane of plastic scintillator of
thickness 400 $\mu$m, four more proportional chambers, a helium-filled beamline
extending to 150 cm from the target and finally, the TECs.

  TECs were described and proposed for use in E614 in Technical Note 17
\cite{Vla}.  The TEC used in the present simulation consisted of a
rectangular box 29.5 $\times$ 3.6 $\times$ 3.6 cm$^3$ filled with DME gas at
a pressure of 20 torr.  At its upstream end the DME was separated from the beam
line vacuum by a 6 $\mu$m mylar foil.  At its downstream end it was separated
from a 1 atm helium-filled beam line by a 100 $\mu$m mylar foil.  The TEC was
divided into two rectangular drift volumes, one each to provide $xz$ and $yz$
muon track information.  The drift field is established (but not simulated in
this study) by a plane of 34 100 $\mu$m field wires located at the lower edge
of the $yz$ drift volume, out of the way of the muon beam.  Between the field
wires and the outer wall of the TEC, was a plane of 33 10 $\mu$m sense wires
with 1 mm spacing.  The $xz$ drift volume was identical, except that it was
rotated 90 degrees with respect to the $yz$ volume.  The $xz$ drift volume
was centered 177.25 cm from the target, while the $yz$ drift volume was
centered 171.75 cm from the target.

  In the simulation, the TEC generated 33 $x$ and 33 $y$ distances from the
sense planes to the muon track.  An gaussian uncertainty of 200 $\mu$m
($\sigma$) was added to each recorded distance.  This represents a pessimistic
estimate of the TEC resolution \cite{Vla}.  A linear fit of the ($x,z$) and
($y,z$) coordinates returned the slopes of the muon track in the $xz$ and
$yz$ planes which were then combined to get the polar angle $\theta$ of the
track before it entered the strong solenoidal field.  $1 - cos$ of the muon
spin direction as it stopped in the target was then plotted versus $sin\theta$.
The TEC was placed far from the center of the target, where the solenoidal
field is weak, for three reasons:  1) such devices do not function in high
magnetic fields, 2) muon tracks are not significantly deflected and can be
assumed to be straight, and 3) the amount of matter the muons must pass through
before being analyzed is minimized. 

  Two muon beam tunes of differing emittance were studied in the simulation.
Both had a momentum of 29.8 MeV/c and were started 280 cm upstream of the
target.  It was assumed here that the last quadrupole magnet in the M13
beamline was centered 300 cm upstream of the target.  The nominal foci of both
tunes were set 160 cm upstream of the target, which was 14.5 cm downstream of
a point midway between the $xz$ and $yz$ drift volumes of the TEC.  At its
focus, the ``convergent'' M13 tune had a spot size of 0.94 mm (FWHM)
in $x$ and 5.0 mm (FWHM) in $y$.  Its divergence (FWHM) in $x$ and $y$ was
6.6 mrad and  100 mrad, respectively.  Its total emittance was therefore 6.2
mm-mrad in $x$ and 500 mm-mrad in $y$.  The ``parallel'' tune was a simulation
of the M15 beamline which had a spot size of 10 mm (FWHM) in $x$ and 8 mm
(FWHM) in $y$, and a divergence of 14 mrad (FWHM) in $x$ and 17 mrad (FWHM) in
$y$.  Its emittance was 140 mm-mrad in both $x$ and $y$.

  A solenoidal magnetic field aligned with the beam axis was used.  The value
of its longitudinal component at the origin was 2.2T and fell rapidly starting
around $z$ = 150 cm.  The radial field component peaked at around $z$ = 160 cm
and then fell off rapidly upstream and downstream of that point.  Both the
muon momentum and spin were propagated through the field.  Muons were allowed
to decay but the decay positrons were not tracked.

\vspace{1.0cm}

{\large \bf Results \\}

  Due to the small size in $x$ and $y$ (3.6 cm) of the TEC, and the large track
angles of the convergent beam tune, 0.5\% of the incident muons stopped in the
walls or wires of the TEC.  For the parallel tune, none of the incident muons
stopped in the TEC.  The surviving muons passed through the remainder of the
spectrometer with typically 78\% of them stopping in the target.  For each of
these the depolarization, $1 - cos\theta_s$, was histogrammed versus
$sin\theta$.  Here $\theta_s$ is the direction of the muon spin as it stopped
in the target and $\theta$ is the polar angle of the muon track as measured by
the TEC outside of the strong field region.  This histogram for the convergent
beam tune is shown in Fig. 1.

\begin{figure}
\epsfysize=15cm
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\centerline{\epsfbox{tn23fig1.ps}}
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\caption{Muon depolarization at the target vs. TEC muon angle for a convergent
beam.}
\end{figure}

There is a strong correlation between the depolarization and the angle of the
initial muon track.  Hence a cut on the measured muon track angle will reduce
the mean depolarization.  The effect of such a cut is shown in Table I.

\vspace{0.8cm}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Cut on TEC angle (rad) & \% $\mu$ surviving & mean depolarization($\times 10^{-4}$) \\ \hline  
$\theta$=0.03   & 52          & 6.9 \\ \hline
$\theta$=0.02   & 32          & 4.0 \\ \hline
$\theta$=0.01   & 12          & 2.2 \\ \hline
$\theta$=0.005  &  3          & 1.7 \\ \hline
\end{tabular}
\end{center}

\vspace{0.2cm}
Table I.  Percentage of muons surviving cut which also stop in the target, and
mean depolarization as a function of a cut on the initial muon polar angle
measured in the TEC.  These results were obtained with the convergent beam
tune. 
\vspace{0.2cm}

At the cut value of 0.01 rad, the mean depolarization falls below $3 \times
10^{-4}$.  Of those muons which stop in the target 12\% survive this cut.
Since 78\% of all incident muons stop in the target, the total percentage
of initial muons which can be used in an experiment thus becomes 0.78 $\times$
12 = 9.4.  Tighter cut values do not significantly improve the depolarization.

\begin{figure}
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\caption{Effect of multiple scattering on the correlation of TEC muon angle
and depolarization at the target.  Top: multiple scattering included;  bottom:
multiple scattering not included.  These results were obtained using the
convergent beam tune.}
\end{figure}

Attempts to increase the number of muons surviving this cut while keeping the
depolarization small depend on multiple scattering, the spatial resolution of
the TEC and the beam tune.  The effect of multiple scattering is shown in
Fig. 2.  Here the correlation of the TEC muon angle and the depolarization is
presented with multiple scattering turned on and off throughout the entire
detector.  As expected the correlation is better in the absence of multiple
scattering, as is the efficiency of the cut on the TEC muon track angle
(compare Tables I and II).  However, it is unlikely that the amount of
multiple scattering can be significantly reduced because the only scattering
centers that can perhaps be made thinner are the plastic scintillator disk and
the 100 $\mu$m mylar window on the downstream end of the TEC.

\begin{center}
\begin{tabular}{|c|c|c|} \hline
Cut on TEC angle (rad) & \% $\mu$ surviving & mean depolarization($\times 10^{-4}$) \\ \hline  
$\theta$=0.030   & 53          & 7.0 \\ \hline
$\theta$=0.020   & 35          & 3.6 \\ \hline
$\theta$=0.010   & 15          & 1.4 \\ \hline
$\theta$=0.005   &  5          & 0.83 \\ \hline
\end{tabular}
\end{center}

\vspace{0.2cm}
Table II.  Percentage of muons surviving cut which also stop in the target,
and mean depolarization as a function of a cut on the initial muon polar angle
measured in the TEC.  These results were obtained with the convergent beam
tune.
\vspace{0.2cm}

It is likely that the spatial resolution of the TEC can be improved from the
assumed 200$\mu$m.  The effect on the TEC angle - depolarization correlation is
shown in Fig. 3 for resolutions of 200, 100 and 50 $\mu$m.  The difference
between 200 and 50 $\mu$m is small, indicating that multiple scattering or
beam properties dominate over resolution effects.  A comparison of Tables III
and I shows that only a small improvement in the depolarization (from
2.2$\times 10^{-4}$ to 1.9$\times 10^{-4}$) is possible, and that there is no
significant improvement in cut efficiency.

\begin{figure}
\epsfysize=18cm
\epsfxsize=11cm
\begin{center}
%\epsffile{.ps}
\centerline{\epsfbox{tn23fig3.ps}}    
\end{center}
\caption{Effect of TEC resolution on the correlation between TEC muon angle and
depolarization.  The convergent beam tune was used and TEC resolutions of
200$\mu$m (top), 100 $\mu$m (middle) and 50 $\mu$m (bottom) were assumed.}
\end{figure}

 
\vspace{0.8cm}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Cut on TEC angle (rad) & \% $\mu$ surviving & mean depolarization($\times 10^{-4}$) \\ \hline  
$\theta$=0.030   & 53          & 6.8 \\ \hline
$\theta$=0.020   & 33          & 3.8 \\ \hline
$\theta$=0.010   & 12          & 1.9 \\ \hline
$\theta$=0.005   &  4          & 1.2 \\ \hline
\end{tabular}
\end{center}
\vspace{0.2cm}
Table III.  Percentage of muons surviving cut which also stop in the target,
and mean depolarization as a function of a cut on the initial muon polar angle
measured in the TEC.  These results were obtained with the convergent beam
tune and a TEC resolution of 50 $\mu$m ($\sigma$).
\vspace{0.2cm}

  The convergent beam tune had a spot size elongated in $y$ and very narrow in
$x$.  It was also highly convergent in $y$ and nearly parallel in $x$.  For
these reasons, it was thought that the $y$ and $\theta_y$ values measured
in the TEC may provide a better correlation with depolarization than the polar
angle used previously.  The correlations with $y$ and $\theta_y$ are shown in
Fig. 4.  In this case two cuts are required to reduce the mean depolarization:
one on the reconstructed $y$ coordinate and another on $\theta_y$.  However,
the results are not much different from those obtained with the single cut on
the polar angle.  Accepting only events for which $ 0.3 < y < 0.7$ cm and
$-0.02 < \theta_y < 0.02 $ rad, the mean depolarization was reduced to 3.7
$\times 10^{-4}$.  Of those muons stopping in the target, 29.5\% survived the
cuts.  This result is very similar to the polar angle cut at 0.02 rad from
Table I.

\begin{figure}
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\epsfxsize=11cm
\begin{center}
%\epsffile{.ps}
\centerline{\epsfbox{tn23fig4.ps}}    
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\caption{Muon depolarization at the target vs. reconstructed $\theta_y$ (top)
and $y$ (bottom).}
\end{figure}

  All of the above results were obtained with a highly convergent M13 tune.
In order to quantify the effects of beam tune on the TEC angle - depolarization
correlation, a parallel tune simulating the M15 beamline was also examined.
Using this tune, the correlation of the muon TEC polar angle with the
depolarization was plotted as shown in Fig. 5.  A comparison of Figs. 5 and 1
shows less correlation for the parallel tune than for the convergent tune.
As a result the cut on muon TEC angle is much less efficient.  As shown in
Table IV, not even the tightest cuts can achieve depolarizations of less than
$3 \times 10^{-4}$. 

\begin{figure}
\epsfysize=15cm
\epsfxsize=11cm
\begin{center}
\centerline{\epsfbox{tn23fig5.ps}}    
\end{center}
\caption{Muon depolarization at the target vs. reconstructed TEC muon angle
for a nearly parallel beam tune.}
\end{figure}
  
\vspace{0.8cm}
\begin{center}
\begin{tabular}{|c|c|c|} \hline
Cut on TEC angle (rad) & \% $\mu$ surviving & mean depolarization($\times 10^{-4}$) \\ \hline  
$\theta$=0.030   & 90          & 19.7 \\ \hline
$\theta$=0.020   & 67          & 15.0 \\ \hline
$\theta$=0.010   & 26          &  8.9 \\ \hline
$\theta$=0.005   &  8          &  6.6 \\ \hline
\end{tabular}
\end{center}
\vspace{0.2cm}
Table IV.  Percentage of muons surviving cut which also stop in the target and
mean depolarization as a function of cut on polar $\theta$ for a parallel beam.
\vspace{0.2cm}

{\large \bf Discussion and Conclusions \\}

  In the above simulations the TEC replaced the five pairs of drift chambers
which were to be used to track the incoming muon.  Previous simulations of
these chambers \cite{Tn14} showed that a cut on the radius of the fitted
muon helix could be used to reduce the mean depolarization at the target to
$5 \times 10^{-4}$ (see Fig. 14 of Technical Note \# 14) while keeping 6\% of
the muons which stopped in the target.  Using the same converging beam tune,
the TEC allowed depolarizations of $2.2 \times 10^{-4}$ while keeping 12\% of
the muons which stopped in the target.  The TEC results are superior to those
from the muon tracking drift chambers because of the smaller mass intercepted
by the muon beam and because the TEC is able to measure the beam polarization
before it encounters large magnetic fields.

  Simulations were performed using beam tunes intermediate between the highly
convergent beam and the nearly parallel beam.  However the depolarization -
TEC angle correlation became worse as the beam became more parallel.  It is
therefore recommended that instead of additional drift chambers for muon
tracking, a TEC should be used in conjunction with a highly convergent beam
tune.

\begin{thebibliography}{10}

\bibitem{Vla} Yu.I.Davyydov, V.I.Selivanov, V.D.Torokov, E614 Technical Note
 \#17, May 1998.

\bibitem{Tn14} A.R. Parikh, E614 Technical Note \# 14, April 1998.

\end{thebibliography}


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