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AlCapPSI/neutronsPW.tex

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+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}).