The goal of the experiment is a measurement of the rates and energy spectra of
charged particle emission after muon capture in; 1) the favored
conversion target, 
Al, 2) in Si (as normalization and cross check),  and 3) Ti (as an alternative
conversion target material). Both the rate and energy spectrum 
would be measured at
5\% precision down to an energy of 2.5 MeV.

The basic requirements are summarized is Fig.~\ref{basic.fig}. As the emitted
charged particles deposit a significant amount of energy during their passage
through the target material, thin targets and thus excellent momentum
resolution of the low energy muon beam are critical for the experiment. This
is exactly the reason, why the older experiments, which were performed
with thick targets and less sophisticated beams, are unsuitable for providing
the required yield and spectral information at low energies. 
The observed energy spectrum $g(T_f)$ of protons emitted from the stopping
target is a convolution of the initial capture spectrum $f(T_i)$ with a
response function $k(T_f,T_i)$: 
\begin{equation} 
	g(T_f) = \int_0^\infty k(T_f,T_i) f(T_i) dT_i 
\end{equation} 
The response function can be readily calculated for a uniform muon stopping
distribution within the target depth.  The resulting distortion of the original
energy distribution taken from Equ.~\ref{eq:protons} is illustrated in
Fig.~\ref{response.fig}  for different target thickness.  

\begin{figure}[htb] 
	\begin{center}
		\includegraphics[width=\textwidth]{figs/protonrange.png}
		\caption{Left: Momentum vs. energy for p, d, $\alpha$,
		right: proton range 
		vs. energy in targets.} 
		\label{basic.fig} 
	\end{center} 
\end{figure}
%\vskip-5ex
\begin{figure}[tb] 
	\begin{center} 
		\includegraphics[width=0.6\textwidth]{figs/Si_emitted.pdf}
		\caption{Calculated proton emission spectrum as
		function of target 
		thickness (red: 0 $\mu m$, green: 50 $\mu m$, blue:
		100 $\mu m$, black :
		1000 $\mu m$).} 
		\label{response.fig} 
	\end{center} 
\end{figure}

\begin{figure}[tb] 
	\begin{center}
		\includegraphics[width=0.39\textwidth]{figs/expcad.png}\hspace{1mm}\includegraphics[width=0.59\textwidth]{figs/exp.png}  
		\caption{Left: CAD of layout, right: picture of vacuum
		vessel with detectors.} 
		\label{setup.fig} 
	\end{center} 
\end{figure}

A schematic layout of the experimental setup is shown in Fig.~\ref{setup.fig}.
It will be an improved version of a test experiment performed by part of this
collaboration at PSI in 2009.

Low energy muons will be detected by external beam counters (scintillator and
wire chamber, not shown). The muons then enter a vacuum vessel though
a thin mylar
window and a fraction stopped in passive Al and Ti foils of 25 to 100 $\mu m$
thickness.  These foils are aligned 45 degrees with respect to the
beam direction.  As a cross check, muons
will also be stopped in active Si detectors used as targets. Two packages
of charged particle detectors are positioned on opposite sides, perpendicular
to the target surface. The geometry is chosen so as to minimize the path length
of the emitted protons, and limit their direction to be nearly perpendicular to
the detectors.  This improves the PID resolution by dE/dx separation. The main
detector of the package is a 5$\times$5 cm$^2$ Si detector of 1500 $\mu m$
thickness (MSX), which will stop protons up to about 12 MeV. Plastic
scintillators positioned
behind this Si detector observe higher energy protons and veto
through-going electrons. To provide dE/dx information some data will be taken
with two 10x10 cm$^2$ thin Si detectors (65 $\mu m$, MSQ). These detectors are
4-fold segmented. Since their large capacitance deteriorates the overall
resolution, measurements with and without them are foreseen. The symmetry
between the left and right Si stations allows for a powerful monitor of
systematic effects. Differences between the detectors would indicate background
due to different stopping materials, non--uniform stopping distributions, or
differences due to muon scattering.  Careful shielding of direct or scattered
muons is required, as the stopping fraction is small and proton emission is a
rare capture branch. As shown, we are considering a geometry, where there is no
direct line of sight between any low Z material exposed to muons.  Thus, all
shielding uses lead. 

In order to normalize the  number of muon-stops in the aluminum target, a
Germanium detector will be used to measure muonic X-rays from muons
that stop in the aluminum 
target. We also will have telescopes to detect electrons from
muons for an additional normalization of muon-stops. 

The main systematic issues are as follows. 

\begin{itemize}
\item Deconvolute the original proton spectrum $f(T_i)$. An optimal
	cloud muon beam is requested for the experiment. Then  an active
	Si target allows an experimental calibration of the response function,
	because both $T_i$ and $T_f$ are accessible with an active target.

\item Absolute calibration. The number of muon stops will be determined with
	the Ge detector. Again, the use of an active Si target allows a cross
	calibration. The proton detection efficiency will be simulated
	by Geant and calibrated with the active Si target.

\item PID. The PID of emitted charged particles will be determined by dE/dx.
	The use of time of flight will be investigated.

\item Background. Electron background will be determined with $\mu^+$, neutron
	recoils by absorbing the proton component before the Si detectors. A
	dangerous background are muons stops in walls and scattered into the Si
	detector.\footnote{If the low energy target scattering is too
          high, we are also studying a configuration, with the two Si
          detectors positioned  symmetrically and perpendicular to the beam.}


\end{itemize}

A realistic Geant4 simulation is being developed. It will serve as an important
tool to optimize the geometry, in particular the investigation of the
background and response 
function. Currently the geometry of the PSI test run is being implemented for a
realistic check of the simulation.

\begin{figure}[htb] 
	\begin{center}
	\includegraphics[width=0.8\textwidth]{figs/dEdx_new.pdf} 
	\caption{Two dimensional plots of energy deposit in 
	Si1 (65 $\mu$m) vs sum of that in Si1 (65 $\mu$m) and Si2 (1500$\mu$m)
	(horizontal) counters. The protons can be distinguished from
	deuterons and 
	tritons in the range of 2.5 $-$ 10
	MeV. The low energy particles near zero point are electrons
	and photons.}  
	\label{fg:dedx}
\end{center} 
\end{figure}

\begin{table}[htb]
	\begin{center}
		\caption{
	\label{tb:rates}  Estimated event rates for various targets of different
		thickness. Incoming $10^{4}$ muons/sec and proton
		emission rate of 
		0.15 per muon capture are assumed. The efficiency of
		Si detector of 
		100 \% is also assumed. \label{tb:rates} } 
		\vskip1ex
		%\scalebox{0.75}{
		\begin{tabular}{|c|c|c|c|c|}
			\hline
			Target 						& Muon momentum &\% Stopping & Event rate (Hz) & Event rate (Hz) \\  
			thickness ($\mu$m)& (MeV/c) &in target & All particles
		& Protons \\  
			\hline
			50  & 26 & 22.2   & 34.8 	& 4.6   \\ 
			100 & 27 & 32.9 	& 48.5 	& 5.4  \\ 
			150 & 28 & 38.5  	& 54.5 & 4.8  \\ 
			200 & 28 & 51.2 	& 47.7 & 4.5  \\ 

			%50  & 26 & 22.2   & 14.8 	& 2.3   \\ 
			%100 & 27 & 32.9 	& 18.5 	& 2.1  \\ 
			%150 & 28 & 38.5  	& 16.6 & 1.7  \\ 
			%200 & 28 & 51.2 	& 19.8 & 2.0  \\ 
			\hline
		\end{tabular}
		%}
		\end{center}
	\end{table}

Figure~\ref{fg:dedx} shows Monte Carlo simulation studies of two-dimensional
plots of energy in the MSQ counter (dE/dX) vs. energy of the MSX counter. From
Fig.~\ref{fg:dedx}, it is clearly seen that we can discriminate protons,
deuterons, scattered muons, and electrons up to 10 MeV. 

The event rates are estimated based on Monte Carlo simulation.  Preliminary
results are summarized in Table~\ref{tb:rates}. They will be updated once we
have full information about the PSI beam properties. As seen in
Table~\ref{tb:rates}, proton rates are not large. Additional collimation,
if required to obtain background reduction, and the low muon momenta,
which might require the use of degraders, may further reduce the rates
 below these estimates.
A muon beam
of low and well defined momentum is of critical importance.

