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\begin{center}

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

\vspace{0.4cm}

{\Large \bf Muonium Formation in Helium Gas and the Choice of Whether or not
to Use Central Proportional Chambers \\}

\vspace{0.4cm}

\rm{\bf D.H. Wright, TRIUMF \\}

\vspace{0.4cm}

\rm{\bf 18 November 1997 \\}

\end{center}

\begin{abstract}

Two designs for the central region of the E614 spectrometer were re-evaluated
using new information on the polarization of muons in various materials.
New values of $P_\mu$ in mylar and CF$_4$-ISO gas were used.  It was also found
that $P_\mu$ in He gas was not the same as that in liquid, and had to be
replaced by a rough estimate based on muonium formation in gas.  The resulting
overall depolarizations strongly favor the use of central PCs unless precise
polarization measurements can be made before the time of the design decision.

\end{abstract}

In a previous Technical Note \cite{TN4}, two design options for the
determination of the muon stop location in the E614 spectrometer were
compared.  Based on estimates of the muon depolarization in the various media
near the stopping target, the option including proportional chambers (PCs)
next to the target was preferred to that with helium gaps next to the target.
The depolarization values used in that study are summarized in Table 1.  It
was also assumed that the proportional chamber inefficiency was 0.001.

\vspace{0.3cm}

\begin{center}
\begin{tabular}{| l | c |}  \hline
 Material         &   $1-P_\mu$  \\  \hline
 Al Target        &      0.00    \\
 He gas (STP)     &      0.05    \\
 DME (STP)        &      0.05    \\
 CF$_4$-ISO (STP) &      0.015   \\
 Mylar            &      1.00    \\  \hline
\end{tabular}

\vspace{0.2cm}
Table 1. Muon depolarization values used in previous analysis.
\end{center}

While it has recently been determined that 0.001 is a reasonable value for
chamber inefficiency, two of the depolarization values were found to be
incorrect.  From experience with MEGA \cite{Gag97}, it has been found that a
more reasonable estimate of the depolarization in mylar is 0.05.  Also, the
depolarization in helium gas was taken to be the same as that in liquid, an
assumption which ignores the large effect of low density on muon
depolarization.  To date there have been no measurements of muon depolarization
in helium gas in longitudinal magnetic fields.  There is however a measurement
at 1.18 atm in small transverse fields yielding a depolarization of 0.25 $\pm$
0.03 \cite{Fle87}. \\

{\large \bf Muonium Formation in Helium Gas \\}

An estimate of the polarization in strong longitudinal fields requires some
knowledge of muonium formation, which is the dominant depolarization mechanism
in helium.  A summary of work done in this field is given in \cite{Fle92}.
When a surface $\mu^+$ enters a gas it loses energy by Bethe-Bloch ionization.
In helium this process is dominant down to about 30 keV.  At 60 keV, when the
muon velocity begins to be comparable with that of the atomic electrons, the
first charge-exchange occurs with the formation of muonium (Mu).  From 60 keV
down to about 30 eV, cyclic charge exchange
\begin{equation}
   \mu^+ + He \leftrightarrow Mu + He^+
\end{equation}
occurs along with elastic energy loss.  Below 30 eV there is too little
energy to form many muonium atoms and the $\mu^+$ resides in a MuHe$^+$
molecule where it no longer suffers depolarization.  Very little depolarization
occurs while the muon is free, leaving muonium as the major cause.  When a 
100\% polarized muon captures an atomic electron, the muonium states
\begin{equation}
| 1 1 > ,\hspace*{0.5cm} \frac{1}{\sqrt{2}}| 1 0 > + \frac{1}{\sqrt{2}}| 0 0 >
\end{equation}
are formed with equal probability.  The first state is an eigenstate of the
hyperfine interaction while the second is not.  For each muonium formation
there is thus a 50\% chance that the muon will be in a state which flips its
spin and flips it back with a period of 0.224 ns.  Hence the longer the muon
stays in muonium, the more likely it will be depolarized.  The time spent in
muonium is governed by the incident muon energy and the density of the helium
gas it passes through.  The lower the density, the fewer charge exchange
cycles (Eq. 1) occur and the longer the muon spends in muonium.  This scenario
is confirmed by muon polarization measurements in small transverse fields.
As shown in Fig. 1, the depolarization increases as the helium pressure
decreases.

\epsfysize=9cm
\begin{figure}
\begin{center}
\epsffile{tn5fig1.ps}
\end{center}
\caption{Muon polarization in He in small transverse magnetic fields.
The calculation (solid curve) and data points are from \cite{Fle87}.}
\end{figure}

In large longitudinal magnetic fields the depolarization per collision
(charge exchange) is significantly reduced, but not by enough to compensate
for the relatively small number of collisions in the gas.  Per muonium
formation the muon depolarization is given by \cite{Sen97}
\begin{equation}
   P_\mu = \frac{1}{2}[ 1 + \frac{X^2}{1+X^2} +
\frac{cos(\omega_{HF}\Delta t)}{1+X^2} ]
\end{equation}
where $\omega_{HF}$ = 4463 MHz is the hyperfine oscillation frequency,
$\Delta t$ is the time the muon spends in muonium and
\begin{equation}
 X = \frac{B}{B_o} = \frac{22000}{1585} = 13.88
\end{equation}
for a longitudinal field of 2.2 T.  Due to the low density in helium gas at 1
atm, $\Delta t$ is of order 0.2 ns or greater so that
$cos(\omega_{HF} \Delta t)$ averages to zero.  The last term in Eq. 3 can then
be dropped to give 
\begin{equation}
 P_\mu /{\rm collision} = 0.9974 .
\end{equation}
There are roughly 100 $\pm$ 50 collisions in the muonium regime, so that an
estimate of the total polarization before stopping is
\begin{equation}
 P_\mu = 0.9974^{100 \pm 50} = 0.77 \pm 0.10 .
\end{equation}
The depolarization is therefore large;  more importantly the uncertainty in the
depolarization is large.  In addition, the case of small transverse magnetic
fields (Fig. 1) shows a large deviation of measurement from calculation,
decreasing confidence in the estimate for longitudinal fields. \\

{\large \bf No Proportional Chambers Adjacent to Target \\}

  An earlier Monte Carlo study (see Table 1 of \cite{TN4}) of the spectrometer
with no proportional chambers next to the target determined an average
depolarization for muons stopping in and around the target of
2.3$\times 10^{-3}$.  This method tracked both the positron and incoming muon
in order to determine the decay vertex.  It also arranged for a symmetric
stopping distribution of muons in the target.  At first glance, better values
of muon polarization greatly improved the situation.  Taking the muon
depolarization in mylar to be 0.05 instead of 1.0, and taking the
depolarization in He gas to be 0.002 (Strovink, liquid) instead of 0.05,
lowered the overall depolarization to 1.5$\times 10^{-4}$.  However, the above
estimate for polarization in He gas 0.77$\pm$0.10 raises the overall
depolarization and its error to an unacceptable level, 0.011$\pm$0.001 .

The kinetic energies of muons entering the helium gaps just upstream and
downstream of the target are shown in Fig. 2, compared with the energy range
over which muonium formation occurs.  All but a few muons entering the helium
gaps have energies above that of the first muonium formation and none have
energies below that of the last muonium formation.  Hence any muon stopping in
the gaps must pass through the entire muonium regime, the last few cycles of
which are highly depolarizing.

\begin{figure}
\begin{center}
\epsfig{file=tn5fig2.eps, height=5.0in,width=5.0in}
\end{center}
\caption{Kinetic energy of muons entering helium gaps upstream (top) and
downstream (bottom) of the target.}
\end{figure}

The roughness of the above estimate and the lack of muon polarization
data in helium gas in longitudinal fields make a measurement of this parameter
imperative if the no-PC option is to work.  Even if such a measurement were
performed the following problems arise with the no-PC option:

1) If the measured polarization in He is much different from 1.0 (such as
0.77), a sizable correction to the measured value of $P_\mu \xi$ will be
required.  The precision of the polarization in He must then be very high
($\pm$ 0.0004) in order to attain an uncertainty of 3$\times 10^{-5}$ in the
corrected $P_\mu \xi$.

2) Muonium is much more likely to form in gases with low ionization potentials
such as hydrocarbons and heavy noble elements.  The He gas likely to be used
in the gaps will have total impurities of about 50 ppm much of which could be
hydrocarbons.  Studies of muons stopping in He-Xe mixtures show that complete
depolarization can occur when the Xe concentration is only a few ppm
\cite{Sen97}.  Methane, for example, which has nearly the same ionization
potential as Xe, could pose a serious problem unless the He gas is first
passed through a cold trap to remove hydrocarbons. \\

{\Large \bf Proportional Chambers Adjacent to Target \\}

  The Monte Carlo simulation of the PC option was repeated with two additional
pieces of information:

1) The muon polarization measured for the PC gas
(CF$_4$-isobutane) was $ > $ 0.985.  In the present calculation it was assumed
that the measured value of $P_\mu \xi$ will be corrected for the central
value of the depolarization in CF$_4$-ISO, with only the error bars
contributing to the total error in $P_\mu \xi$.  Therefore $P_{\mu CF4-ISO}$
was taken to be 0.992$\pm$0.008. 

2) ADCs were added to the wires of the central proportional chambers.  Muons
stopping in the PC just upstream of the target look as if they have stopped in
the target since they satisfy the trigger condition
\begin{equation}
PC(-2)\cdot PC(-1) \cdot \overline{PC(+1) \cdot PC(+2)}
\end{equation}
and hence increase the error in the overall polarization.  Using ADCs to
distinguish between muon and positron pulse heights, the additional condition
\begin{equation}
PC(-1)_\mu \cdot \overline{PC(+1)_\mu} \cdot \overline{PC(-1)_e} \cdot PC(+1)_e
\end{equation}
removes events in which a muon stops in the upstream chamber (PC(-1)) and has
a decay positron traveling downstream.  In addition, tracking of the decay
positron in PC(-1) will allow some events to be removed in which the positron
travels upstream with initial angles $> 45^\circ$.  Together, these two
methods can remove 65\% of the muon stops in PC(-1), according to Monte Carlo
estimates.

  A Monte Carlo study of the PC option was reported in an earlier Technical
Note \cite{TN4}, the results of which are reproduced in Table 2 along with
better depolarization values.

\begin{center}
\begin{tabular}{| c | c | c |}  \hline
  Material           &  $N_{stops}$ &  $1-P_\mu$  \\  \hline
 Al Target           &   7212    &   0.000    \\
 CF$_4$-ISO (PC(-1)) &    269    &   0.008   \\
 CF$_4$-ISO (PC(+1)) &    338    &   0.008   \\
 1st Mylar foil downstream &    172    &  0.05    \\  \hline
\end{tabular}

\vspace{0.2cm}
Table 2. Number of Monte Carlo stops in the region $-0.4 < z < 0.4$.
\end{center}
That study performed 10,000 trials and assumed a symmetric distribution of
muons stopping in the target.  With the use of ADCs on PC wires for
positron-muon discrimination and tracking, the number of muons stopping in
PC(-1) which contribute to depolarization can be reduced by 65\%, from 269 to
94.  Using the values of Table 2 and Eq. 4 of \cite{TN4}, the overall
depolarization and error in $P_\mu \xi$ is 
\begin{equation}
 \overline{1-P_\mu} = \frac{94(0.008) + 0.001[338(0.008) + 172(0.05)]}
{7212 + 94 + 0.001(338 + 172)} = 1.04 \times 10^{-4}.
\end{equation}
Using a muon stopping distribution which is asymmetric in the target and
shifted downstream reduces the above number to 7$\times 10^{-5}$.  A better
measurement of the lower limit of the depolarization in CF$_4$-ISO will reduce
this value further. \\

{\Large \bf Conclusion \\}

Based on Monte Carlo studies and improved muon depolarization values, the
spectrometer design option which includes central PCs will be sufficient to
limit the systematic error in $P_\mu \xi$ due to depolarization in the PCs to
a part in $10^4$ or less.  Before the no-PC option can be seriously
considered, a very precise measurement of the depolarization of muons in He in
a longitudinal magnetic field must be done.  In addition, given the grade of He
gas to be used in the gaps, it is also likely that impurities will contribute
significant depolarization, and therefore additional error, to $P_\mu \xi$.

\begin{thebibliography}{10}

\bibitem{TN4} D.H. Wright, E614 Technical Note 4 (1997).
\bibitem{Gag97} C. Gagliardi, private communication (1997).
\bibitem{Fle87} D.G. Fleming and M. Senba, ``$\mu^+$ Charge Exchange, Muonium
Formation and Depolarization in Gases'', {\it Atomic Physics with Positrons ed.
Humbertson and Armour}, Plenum, 343 (1987).
\bibitem{Fle92} D.G. Fleming and M. Senba, ``Recent results in gas phase
$\mu$SR and muonium chemistry at TRIUMF'',{\it Perspectives of Meson Science
ed. Yamazaki, Nakai and Nagamine}, Elsevier, 219 (1992).
\bibitem{Sen97} M. Senba, private communication (1997).

\end{thebibliography}

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