The measurement of neutron emission after muon capture proposes to use
an Al target of sufficient width and depth to capture and stop all
muons from the low momentum beam incident on the target. The emitted
neutrons are to be detected with counters using pulse shape discrimination,
as described below, with detector readout is triggered by muon entry
into the target. The number of captured muons is given
by counting the muonium x-rays, as described previously. A beam rate
of a few kHz prevents signal overlap in the detector(s) and
provides a sufficient statistical sample in a few days. 

\subsection{Simulation}

 A particle emission simulation was obtained using the FLUKA simulation
 code, version FLUKA2011.2.. 
The model uses a thick, cylindrical target of pure Al. The incident 
low energy muon is completely stopped in the target, and  is captured in an atomic orbit.  The captured muons 
are then allowed to decay in orbit (DIO) or capture in the Al nucleus 
with nucleon emission, as well as photons and a muon neutrino.  
%All
%particles are counted as they are produced, and lepton flavor and
%energy are conserved. 
The simulation of the energies of the neutron, proton, and gamma
particles emitted  after $\mu$ capture in Al is shown in Figure
 ~\ref{part_rates}.  Emission from a Si target is similar.
The Si target does have approximately 25\% more gamma
emission, with the excess gammas at very low energies. The simulation 
produces a ratio, 0.57, of gammas 
above 0.5 Mev per $\mu$ capture, and a ratio, 0.72, of gammas
per emitted neutron.  The correlation between neutron and gamma
emission is shown in a correlation plot of neutron vs gamma energy in
Figure ~\ref{n_gamma_corr}.  In this plot the highest neutron energy is
plotted against the highest gamma energy, so multiplicities are
not counted. The simulated spectrum only includes prompt
photons. \\

\begin{figure}[htb!]
\begin{center}
\begin{minipage}{7.cm}
\includegraphics[width=6.5cm] {figs/part_rates.eps}
  \caption{\label{part_rates} 
The FLUKA simulated spectrum for proton(red), neutron(blue), and
gamma(black) emission per $\mu$ stop after $\mu$ capture on Al}
\end{minipage}
\parbox{0.3 cm}{ }
\begin{minipage}{8. cm}
\includegraphics[width=7.5 cm, angle=0]{figs/n_gamma_corr.eps}
\caption{\label{n_gamma_corr} A FLUKA simulation of the energy
correlation between neutron (vertical) and prompt gamma (horizontal)
  emission after $\mu$   capture on Al} 
\end{minipage}
\end{center}
\end{figure}

\subsection{Determination of the Neutron Spectrum}

While we are still evaluating the possibility of
the neutron TOF measurement to determine the neutron energy distribution, we 
propose the use of neutron spectrum unfolding techniques
\cite{KoohiFayegh2001391}. The information used in this method requires the
measured pulse energy for each detector hit and a detector response
function, $R(E, E')$. For a neutron energy 
spectrum $\phi(E)$, the measured detector response $N(E')$ is given
by: 
\begin{equation}
N(E') = \int_E R(E, E') \cdot \phi(E)\, \textrm{d}E,
\end{equation}

If $R(E,E')$ is well known, the neutron energy spectrum can be
obtained by unfolding the measured energy distribution
with $R(E, E')$. In this method, the TOF is not used but only the
pulse integral to obtain $N(E')$ of the neutrons coming from the
target. Therefore, the detector can be moved closer to the muon
stopping target when compared to the TOF method.\\ 

Response function, $R(E, E')$, measurements with known neutron energy
distributions spanning the entire energy range of interest, have to be
obtained. This can be achieved with a combination of different neutron
sources, specific reactions with emission of
mono-energetic neutrons, or measurements at facilities with neutrons of
known energy distribution. We will explore the optimal choice such
input measurements over the next weeks in order to calibrate $R(E,
E')$ prior to mounting the experiment at PSI.  We have had
initial
discussions with the TUNL facility on this matter. While it would 
be advantageous to
measure $R(E, E')$ ahead of running the experiment, we could still 
proceed with the
measurements at PSI if $R(E, E')$ was not fully quantified.\\

Over the course of the next weeks, we intend to test existing unfolding
codes \cite{KoohiFayegh2001391} with Monte Carlo generated input test
distributions $\phi(E)$ and typical detector
response functions $R(E, E')$. We also intend to study the influence of the
knowledge of $R(E,E')$ on the precision with which the neutron energy
spectrum can be extracted. 

\subsubsection{Neutron detectors and readout}

We propose to use at least one of the six identical
neutron counters from the MuSun
experiment\footnote{http://muon.npl.washington.edu/exp/MuSun/}. These
counters are cylindrical cells of 13~cm diameter by 13~cm depth and
contain approximately 1.2~liters of BC501A organic scintillator. The
cell is coupled to a 13~cm diameter photo-multiplier tube. For
comparison, we might also employ one of the two home made neutron
detectors which were built by Regis University. While they are similar
in size to the six BC501A ones, these detectors are filled with the
EJ-301 and EJ-309 liquid scintillator, respectively. However, there
are no major differences in the three types of available detectors. \\ 

Any of these detectors would use 12-bit, 170 MHz custom-built waveform
digitizers, and an eight channel board from the MuSun experiment is
available. Each board can sustain data rates of a few MB/s before loss
of data packages occurs. While the expected neutron rates are well below
this limit, additional background
rates in the experimental hall can be suppressed by sufficient
shielding around the detector. Fig. \ref{fig:neutronFADC} shows a
typical, digitized signal from one of the BC501A neutron detectors with
5.88\,ns binning (170 MHz). The full digitization of each signal
allows separation of neutrons from gammas by means of pulse shape
discrimination (PSD). The two dimensional plot in
Fig.~\ref{fig:neutronPSD} of the so-called slow integral (the sum of
the bins 5 to 20 to the right of the signal peak in
Fig.~\ref{fig:neutronFADC}) versus the total integral reveals two
distinct bands. The lower band are $\gamma$'s mainly from the
background in experimental hall whereas the upper band is composed of
the neutrons. Both integrals are expressed in terms of the
electron-equivalent energy which were obtained from calibrations with
$^{60}$Co and $^{137}$Cs sources. 

\begin{figure}[htb!]
\subfigure[\label{fig:neutronFADC}]{\includegraphics[width=0.49\textwidth]{figs/NeutronFADC.png}}\hfill
\subfigure[\label{fig:neutronPSD}]{\includegraphics[width=0.49\textwidth]{figs/nd14_peak.pdf}}  
\caption{a) Digitized signal from a BC501A neutron detector (x-axis in
ns). b) Neutron-gamma separation via pulse shape discrimination. The
slow integral corresponds to the sum of the bins 5 to 20 to the right
of the peak of the digitized signal. The lower band contains $\gamma$s and
the upper band the neutrons.} 
\vspace{-2mm}
\end{figure}

The PSD analysis of the fully digitized neutron signals has been
successfully employed in the MuSun experiment. The distance between
the neutron and the $\gamma$ peaks divided by the sum of their FWHM
defines the figure of merit $M$. A higher value of $M$ indicates a
better performance. It should be mentioned that the waveform
digitizer board was optimized for these neutron detectors by fine
tuning a low-pass filter on the analog input. This led to a
significant improvement of the figure of merit $M$. Currently $M=1$ is
achieved at an electron-equivalent energy of 200 keV corresponding to
a neutron energy of about 0.7\,MeV (MuSun analysis, see also
\cite{Nakao1995454}).