\chapter{Data analysis}
\label{cha:data_analysis}
This chapter presents initial analysis on subsets of the collected data.
Purposes of the analysis include:
\begin{itemize}
  \item testing the analysis chain;
  \item verification of the experimental method, specifically the
    normalisation of number of stopped muons, and particle identification
    using specific energy loss;
  \item extracting a preliminary rate of proton emission from aluminium.
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Number of stopped muons normalisation}
\label{sec:number_of_stopped_muons_normalisation}
The active silicon target runs was used to check for the validity of the
counting of number of stopped muons, where the number can be calculated by two
methods:
\begin{itemize}
  \item counting hits on the active target in coincidence with hits on the
    upstream scintillator counter;
  \item inferred from number of X-rays recorded by the germanium detector.
\end{itemize}
This analysis was done on a subset of the active target runs
\numrange{2119}{2140} because of the problem of wrong clock frequency found in
the data quality checking shown in \cref{fig:lldq}. The data set contains
%\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events.  
\num{6.43E7} muon events.  

\subsection{Number of stopped muons from active target counting}
\label{sub:event_selection}
Because of the active target, a stopped muon would cause two coincident hits on
the muon counter and the target. The energy of the muon hit on the active
target is also well-defined as the narrow-momentum-spread beam was used. The
correlation between the energy and timing of all the hits on the active target
is shown in \cref{fig:sir2f_Et_corr}. The most intense spot at zero time
and about \SI{5}{\MeV} energy corresponds to stopped muons in the thick target.
The band below \SI{1}{\MeV} is due to electrons, either in the beam or from
muon decay in orbits, or emitted during the cascading of muon to the muonic 1S
state.  The valley between time zero and 1200~ns shows the minimum distance in
time between two pulses. It is the mentioned limitation of the current pulse
parameter extraction method where no pile up or double pulses is accounted for. 

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_E_t_corr}
  \caption{Energy - timing correlation of hits on the active target.}
  \label{fig:sir2f_Et_corr}
\end{figure}

The hits on the silicon active target after 1200~ns are mainly secondary
particles from the stopped muons:
\begin{itemize}
  \item electrons from muon decay in the 1S orbit
  \item products emitted after nuclear muon capture, including: gamma, neutron,
    heavy charged particles and recoiled nucleus 
\end{itemize}
It can be seen that there is a faint stripe of muons in the time larger than
1200~ns region, they are scattered muons by other materials without hitting the
muon counter. The electrons in the beam caused the constant band below 1 MeV and 
$t > 5000$ ns (see \cref{fig:sir2_1us_slices}).
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_amp_1us_slices}
  \caption{Energy deposit on the active target in 1000 ns time slices from the
    muon hit. The peaks at about 800 keV in large delayed time are from
    the beam electrons.}
  \label{fig:sir2_1us_slices}
\end{figure}

From the energy-timing correlation above, the cuts to select stopped muons are:
\begin{enumerate}
  \item has one hit on muon counter (where a threshold was set to reject MIPs),
    and the first hit on the silicon active target is in coincidence with that
    muon counter hit: 
    \begin{equation}
      \lvert t_{\textrm{target}} - t_{\mu\textrm{ counter}}\rvert \le \SI{50}{\ns}
      \label{eqn:sir2_prompt_cut}
    \end{equation}
  \item the first hit on the target has energy of that of the muons: 
    \begin{equation}
      \SI{3.4}{\MeV} \le E_{\textrm{target}} \le  \SI{5.6}{\MeV}
      \label{eqn:sir2_muE_cut}
    \end{equation}
\end{enumerate}
The two cuts~\eqref{eqn:sir2_prompt_cut} and~\eqref{eqn:sir2_muE_cut} give
a number of stopped muons counted by the active target:
\begin{equation}
  N_{\mu \textrm{ active Si}} = 9.32 \times 10^6
  \label{eqn:n_stopped_si_count}
\end{equation}

\subsection{Number of stopped muons from the number of X-rays}
\label{sub:number_of_stopped_muons_from_the_number_of_x_rays}
The number of nuclear captures, hence the number of stopped muons in the
active silicon target, can be inferred from the number of emitted
muonic X-rays. The reference energies and intensities of the most prominent
lines of silicon and aluminium are listed in \cref{tab:mucap_pars}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l l l}
      \toprule
      \textbf{Quantity} & \textbf{Aluminium} & \textbf{Silicon}\\
      \midrule 
      Muonic mean lifetime (ns) & $864 \pm 2$ & $758 \pm 2$\\
      Nuclear capture probability (\%) & $60.9 $ & $65.8$\\
      $(2p-1s)$ X-ray energy (keV) & $346.828\pm0.002$ & $400.177\pm0.005$\\
      Intensity (\%) & $79.8\pm0.8$ & $80.3\pm0.8$\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Reference parameters of muon capture in aluminium and silicon taken
  from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday et
  al.~\cite{MeasdayStocki.etal.2007}.}
  \label{tab:mucap_pars}
\end{table}

The muonic X-rays are emitted during the cascading of the muon to the muonic 1S
state in the time scale of \SI{E-9}{\s}, so the hit caused by the X-rays must
be in coincidence with the muon hit on the active target. Therefore an
additional timing cut is applied for the germanium detector hits: 
\begin{equation}
  \lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns}
  \label{eqn:sir2_ge_cut}
\end{equation}

The germanium spectrum after three
cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut}
and~\eqref{eqn:sir2_ge_cut} is plotted in \cref{fig:sir2_xray}. The $(2p-1s)$
line clearly showed up at \SI{400}{\keV} with very low background. A peak at
\SI{476}{\keV} is identified as the $(3p-1s)$ transition. Higher transitions
such as $(4p-1s)$, $(5p-1s)$ and $(6p-1s)$ can also be recognised at
\SI{504}{\keV}, \SI{516}{\keV} and \SI{523}{\keV}, respectively.
%The $(2p-1s)$
%line is seen at 399.5~\si{\keV}, 0.7~\si{\keV} off from the reference value of
%400.177~\si{\keV}. This discrepancy is within our detector's resolution,
%and the calculated efficiency is $(4.549 \pm 0.108)\times 10^{-5}$ -- a 0.15\%
%increasing from that of the 400.177~keV line, so no attempt for recalibration
%or correction was made.  
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/sir2_xray_22}
  \caption{Prompt muonic X-rays spectrum from the active silicon target.
  }
  \label{fig:sir2_xray}
\end{figure}

The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian
peak on top of a first-order polynomial from \SIrange{395}{405}{\keV}. 
Using the same procedure of correcting described in
\cref{sub:germanium_detector}, and taking detector acceptance and X-ray
intensity into account (see \cref{tab:sir2_xray_corr}), the number of muon
stopped is: 
\begin{equation}
  N_{\mu \textrm{ stopped X-ray}} = (9.16 \pm 0.28)\times 10^6,
  \label{eqn:n_stopped_xray_count}
\end{equation}
which is consistent with the number of X-rays counted using the active target.
\begin{table}[btp]
  \begin{center}
    \begin{tabular}{@{}llll@{}}
      \toprule
      \textbf{Measured X-rays}          & \textbf{Value} & \textbf{Absolute error}                                    & \textbf{Relative error}                                   \\ \midrule
      Gross integral                    & 3083           &                                                            &                                                           \\
      Background                        & 101.5          &                                                            &                                                           \\
      Net area $(2p-1s)$                & 2929.7         & 56.4                                                       & 0.02                                                      \\ 
      \vspace{0.03em}\\
      \toprule
      \textbf{Corrections} & \textbf{Value} & \multicolumn{2}{c}{\textbf{Details}}\\
      \midrule
      Random summing                    & 1.06  & \multicolumn{2}{l}{average count rate \SI{491.4}{\Hz},}\\ 
                                        &       & \multicolumn{2}{l}{pulse length \SI{57}{\us}}\\
      TRP reset                         & 1.03 & \multicolumn{2}{l}{\SI{298}{\s} loss during \SI{9327}{\s} run period}\\
      Self-absorption                   & 1.008 & \multicolumn{2}{l}{silicon thickness \SI{750}{\um},}\\ 
                                        &       & \multicolumn{2}{l}{linear attenuation \SI{0.224}{\per\cm}}\\
      True coincidence                  & 1              & \multicolumn{2}{l}{omitted}                                                                                            \\ 
      \vspace{0.03em}\\
      \toprule
      \textbf{Efficiency and intensity} & \textbf{Value} & \textbf{Absolute error}                                    & \textbf{Relative error}                                   \\
      \midrule
      Detector efficiency               & \num{4.40E-4}        & \num{0.978E-5}                                                   & 0.02                                                     \\
      X-ray intensity                   & 0.803          & 0.008                                                      & 0.009                                                     \\ 
      \vspace{0.03em}\\
      \toprule
      \textbf{Results}                  & \textbf{Value} & \textbf{Absolute error}                                    & \textbf{Relative error}                                   \\
      \midrule
      Number of X-rays emitted          & \num{7.36E6}         & \num{0.22E6}                                                     & 0.03                                                      \\
      Number of muons stopped           & \num{9.16E6}         & \num{0.28E6}                                                     & 0.03                                                      \\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Corrections, efficiency and intensity used in calculating the number
    of X-rays from the active target.} 
  \label{tab:sir2_xray_corr}
\end{table}

%In order to measure the charged particles after nuclear muon capture, one would
%pick events where the emitted particles are well separated from the
%muon stop time. The energy timing correlation plot suggests a timing window
%starting from at least 1200~ns, therefore another cut is introduced:
%\begin{enumerate}
    %\setcounter{enumi}{2}
  %\item there are at least two hits on the active target, the time
    %difference between the second hit on target (decay or capture product) and
    %the muon counter hit is at least 1300 ns:
    %\begin{equation}
      %t_{\textrm{target 2nd hit}} - t_{\mu\textrm{ counter}} \geq \SI{1300}{\ns}
      %\label{eqn:sir2_2ndhit_cut}
    %\end{equation}
%\end{enumerate}

%The three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and
%~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Particle identification by specific energy loss}
\label{sec:particle_identification_by_specific_energy_loss}
In this analysis, a subset of runs from \numrange{2808}{2873} with  the
100-\si{\um} aluminium target is used because of following advantages:
\begin{itemize}
  \item it was easier to stop and adjust the muon stopping distribution in
    this thicker target;
  \item a thicker target means more stopped muons due to higher muon rate
    available at higher momentum and less scattering.
\end{itemize}
Muons momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of
1.09, normalised to \SI{28}{\MeV\per\cc}) were used for this target after
a momentum scanning as described in the next subsection.

\subsection{Momentum scan for the 100-\si{\um} aluminium target}
\label{sub:momentum_scan_for_the_100_}
Before deciding to use the momentum scaling factor of 1.09, we have scanned
with momentum scales ranging from 1.04 to 1.12 to maximise the 
observed X-rays rate(and hence maximising the rate of stopped muons). The X-ray
spectrum at each momentum point was accumulated in more than 30 minutes to 
assure a sufficient amount of counts. Details of the scanning runs are listed
in \cref{tab:al100_scan}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{cccc}
      \toprule
      \textbf{Momentum (\si{\MeV\per\cc})} & \textbf{Scaling factor} & \textbf{Runs}
      & \textbf{Length (s)}\\
      \midrule 
      29.12 & 1.04 & \numrange{2609}{2613} &2299\\
      29.68 & 1.06 & \numrange{2602}{2608} &2563\\
      29.96 & 1.07 & \numrange{2633}{2637} &2030\\
      30.24 & 1.08 & \numrange{2614}{2621} &3232\\
      30.52 & 1.09 & \numrange{2808}{2813} &2120\\
      30.80 & 1.10 & \numrange{2625}{2632} &3234\\
      31.36 & 1.12 & \numrange{2784}{2792} &2841\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Momentum scanning runs for the 100-\si{\um} aluminium target.}
  \label{tab:al100_scan}
\end{table}
The on-site quick analysis suggested the 1.09 scaling factor was the
optimal value so it was chosen for all the runs on this aluminium target. But
the offline analysis later showed that the actual optimal factor was 1.08.
There were two reasons for the mistake:
\begin{enumerate}
  \item the X-ray rates were normalised to run length, which is biased
    since there are more muons available at higher momentum;
  \item the $(2p-1s)$ peaks of aluminium at \SI{346.828}{\keV} were not
    fitted properly. The peak is interfered by a background peak at
    \SI{351}{\keV} from $^{214}$Pb, but the X-ray peak area was
    obtained simply by subtracting an automatically estimated background.
\end{enumerate}
In the offline analysis, the X-ray peak and the background peak are fitted by
two Gaussian peaks on top of a first-order polynomial background. The X-ray peak
area is then normalised to the number of muons hitting the upstream detector
(\cref{fig:al100_xray_fit}).
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.47\textwidth]{figs/al100_xray_fit}
    \includegraphics[width=0.47\textwidth]{figs/al100_xray_musc}
  \caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium and the background
    peak at \SI{351}{\keV} (left). The number of muons is integral of the
    upstream scintillator spectrum (right) from \numrange{400}{2000} ADC
    channels.} 
  \label{fig:al100_xray_fit}
\end{figure}
The ratio between the number of X-rays and the number of muons as a function
of momentum scaling factor is plotted on \cref{fig:al100_scan_rate}. The trend
showed that muons penetrated deeper as the momentum increased, reaching the
optimal value at the scale of 1.08, then decreased as punch through happened
more often from 1.09. The distributions of stopped muons are illustrated by
MC results on \cref{fig:al100_mu_stop_mc}. With the 1.09 scale beam, the muons
stopped \SI{28}{\um} off-centre to the right silicon arm.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/al100_scan_rate}
  \caption{Number of X-rays per incoming muon as a function of momentum
    scaling factor.}
  \label{fig:al100_scan_rate}
\end{figure}

\subsection{Event selection for the passive targets}
\label{sub:event_selection_for_the_passive_targets}
As described in the \cref{sec:analysis_framework}, the hits on all detectors
are re-organised into muon events: central muons; and all hits within
\SI{\pm 10}{\us} from the central muons. The dataset from runs
\numrange{2808}{2873} contains \num{1.17E+9} such muon events.

Selection of proton (and other heavy charged particles) events starts from
searching for muon event that has at least one hit on thick silicon. If there
is a thin silicon hit within a coincidence window of $\pm 0.5$~\si{\us}\ around
the thick silicon hit, the two hits are considered to belong to one particle.
The specific energy loss spectra recorded by the two silicon arms are plotted
on \cref{fig:al100_dedx}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/al100_dedx}
  \caption{Energy loss in thin silicon detectors as a function of total energy
    recorded by both thin and thick detectors.}
  \label{fig:al100_dedx}
\end{figure}
With the aid from MC study (\cref{fig:pid_sim}), the banding on
\cref{fig:al100_dedx} can be identified as follows: 
\begin{itemize}
  \item the densest spot at the lower left conner belonged to electron hits; 
  \item the small blurry cloud just above the electron region was muon hits;
  \item the most intense band was due to proton hits; 
  \item the less intense, upper band caused by deuteron hits; 
  \item the highest band corresponded to alpha hits; 
  \item the faint stripe above the deuteron band should be triton
    hits, which is consistent with a relatively low probability of emission of
    tritons.  
\end{itemize}

The band of protons is then extracted by cut on likelihood probability
calculated as:
\begin{equation}
  p_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
  e^{\frac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}}
\end{equation}
where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in the thin
silicon detector by a certain  proton at energy $E_i$, $\Delta E_i$  and
$\sigma_{\Delta E}$ are the expected and standard deviation of the energy loss
caused by the proton calculated by MC. A cut value of $3\sigma_{\Delta E}$, or
$p_i \ge 0.011$,  was used to extract protons (\cref{fig:al100_protons}).
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.47\textwidth]{figs/al100_protons}
    \includegraphics[width=0.47\textwidth]{figs/al100_protons_px_r}
  \caption{Protons (green) selected using the likelihood probability cut
    (left). The proton spectrum (right) is obtained by projecting the proton
    band onto the total energy axis.}
  \label{fig:al100_protons}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proton emission rate from aluminium}
\label{sec:proton_emission_rate_from_aluminium}
The analysis is done on the same dataset used in
\cref{sec:particle_identification_by_specific_energy_loss}. Firstly, the
number of protons emitted is extracted using specific energy loss. Then
correction for energy loss inside the target is applied. Finally, the number
of protons is normalised to the number of nuclear muon captures.

\subsection{Number of protons emitted}
\label{sub:number_of_protons_emitted}
From the particle identification above, number of protons having energy in the
range from \SIrange{2.2}{8.5}{\MeV} hitting the two arms are:
\begin{align}
  N_{\textrm{p meas. left}} = 1789 \pm 42.3\\
  N_{\textrm{p meas. right}} = 2285 \pm 47.8
\end{align}
The right arm received significantly more protons than the left arm did, which
is expected because in \cref{sub:momentum_scan_for_the_100_} it is shown that
muons stopped off centre to the right arm.

The uncertainties are statistical only. The systematic uncertainties due to
the cut on protons is estimated to be small compared to the statistical ones.

\subsection{Corrections for the number of protons}
\label{sub:corrections_for_the_number_of_protons}
The protons spectra observed by the silicon detectors have been modified by
the energy loss inside the target so correction (or unfolding) is necessary.
In the unfolding process, a response function that relates proton's true energy
and the measured one is needed. 

The response function is 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Charged particles following muon capture on a thick silicon target}
%\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}

%Firstly, the number of charged particles emitted after nuclear muon capture on
%the active target is calculated. The charged particles yield then is normalised
%to the number of nuclear muon capture to obtain an emission rate. 
%Finally, the
%rate is compared with that from the literature.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Number of charged particles with energy above \SI{3}{\MeV}}
%\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
%As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation
%in \cref{fig:sir2_mc_pdfs}, there are several components in
%the energy spectrum recorded by the active target:
%\begin{enumerate}
  %\item charged particles from nuclear muon capture, this is the signal of 
    %interest;
  %\item beam electrons with a characteristic Landau peak around \SI{800}{\keV},
    %dominating at large delay (from \SI{6500}{\ns}), causing background at
    %energy lower than \SI{1}{\MeV} which drops sharply at energy larger than
    %\SI{3}{\MeV}; 
  %\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei
    %from neutron emitting muon captures, peak at
    %around \SI{300}{\keV}, dominate the region where energy smaller than
    %\SI{1}{\MeV} and delay less than \SI{3500}{\ns}. This component is
    %intrinsic background, having the same time structure as that of the signal; 
  %\item stray muons scattered into the target, this component is small compares
    %to the charged particles yield so it is not considered further.
%\end{enumerate}
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
    %\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
    %\caption{The observed spectrum in the timing window 1300 -- 10000~ns (left)
      %and its components from  MC simulation (right). The charged particles
      %spectrum is obtained assuming the spectrum shape and emission rate from
      %Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scales between
      %components are arbitrarily chosen for the purpose of illustration.}
  %\label{fig:sir2_mc_pdfs}
%\end{figure}

%An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the
%beam electrons at low energy. In order to obtain the yields of backgrounds
%above \SI{2}{\MeV}, a binned maximum likelihood fit was done. The likelihood of
%getting the measured spectrum is defined as:
%\begin{equation}
  %L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
  %\label{eqn:llh_def}
%\end{equation}
%where $n$ is the total number of events observed, $\mu$ is the expected number
%of events according to some linear combination of the signal and the
%backgrounds shown in~\ref{fig:sir2_mc_pdfs}, namely: 
%\begin{align}
  %n &= n_{\textrm{sig}} + n_{\textrm{beam e}} + n_{\textrm{dio}}\\
  %\textrm{(sum pdf)} &= n_{\textrm{sig}}\times\textrm{(sig pdf)} + 
  %n_{\textrm{beam e}}\times\textrm{(beam e pdf)} + 
  %n_{\textrm{dio}}\times\textrm{(dio pdf)};
  %\label{eqn:sum_pdf}
%\end{align}
%and the $i$ index indicates the respective number of events in the $i$-th
%bin. 

%The fit is done by the RooFit package~\cite{VerkerkeKirkby.2003} where the
%negative log likelihood $-2\ln{L}$ is minimised. Fitting results are shown
%in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
%\begin{align}
  %n_{\textrm{beam e}} &= 23756 \pm 581\\
  %n_{\textrm{dio}} &= 111340 \pm 1245\\
  %n_{\textrm{sig}} &= 2.57 \pm 856
  %\label{eqn:sir2_n_chargedparticles}
%\end{align}
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nbkg}
    %\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nebeam}
    %\includegraphics[width=0.84\textwidth]{figs/sir2_mllfit}
  %\caption{Results of the maximum likelihood fit of the energy spectrum on the
  %active target.}
  %\label{fig:sir2_mll_fit}
%\end{figure}

% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Number of nuclear muon captures}
%\label{sub:number_of_stopped_muons}

%The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5  \pm 65.6$,
%obtained by subtracting the background of 101.5 from the spectral integral of
%2083 in the region from 396 to 402 keV. 
%The area of the $(2p-1s)$ peak is $2929.7 \pm 56.4$ obtained by fitting
%a Gaussian peak on top of a first-order polynomial background to the spectrum
%in \cref{fgi:sir2_xray} in the region from \SIrange{395}{405}{\keV}. 

%The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of
%stopped muons, so the number of stopped muons is: 

%\begin{align}
  %N_{\mu\textrm{ stopped}} &= 
  %\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
  %&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\
  %&= 1.05\times10^6 \nonumber
%\end{align}
%where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
%detector for the 400.177~keV line in \cref{tab:xray_eff}, and
%$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
%(80.3\% from \cref{tab:mucap_pars}). 

%Taking the statistical uncertainty of the peak area, and systematic
%uncertainties from parameters of muon capture, the number of stopped muons
%calculated from the X-ray measurement is 
%$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of
%stopped muons of $9.82\times10^5$ after the cuts described in the event
%selection process.

%The number of nuclear captured muons is:
%\begin{equation}
  %N_{\mu\textrm{ nucl.capture}} = 
  %N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
  %= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5
  %\label{eqn:sir2_Ncapture}
%\end{equation}
%where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
%stopped muon from \cref{tab:mucap_pars}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Emission rate of charged particles}
%\label{sub:emission_rate_of_charged_particles}
%The emission rate of charged particles is calculated by taking the ratio of
%number of charged particles in ~\eqref{eqn:sir2_Nchargedparticle} and number of
%nuclear muon capture in~\eqref{eqn:sir2_Ncapture}:
%\begin{equation}
  %R_{\textrm{Si}} = \frac{N_{\textrm{charged particle}}}{N_{\mu\textrm{ nucl.capture}}}
  %= \frac{149.9\times10^4}{7.25\times10^6} = 0.252
%\end{equation}
%Uncertainties of this rate calculation are listed in
%\cref{tab:sir2_uncertainties}, including: 
%\begin{itemize}
  %\item uncertainties from number of charged particles, both statistical and
    %systematic (from spectrum shape and fitting) ones are absorbed in the
    %quoted value in~\eqref{sir2_Nchargedparticle};
  %\item uncertainties from number of nuclear capture:
    %\begin{itemize}
      %\item statistical error of the peak area calculation,
      %\item systematic errors from the efficiency calibration, and referenced
        %values of X-ray intensity and capture probability. 
    %\end{itemize}
%\end{itemize}
%So, the emission rate is:
%\begin{equation}
  %R_{\textrm{Si}} =  0.252 \pm 0.009
  %\label{eqn:sir2_rate_cal}
%\end{equation}

%\begin{table}[htb]
  %\begin{center}
    %\begin{tabular}{l l l}
      %\toprule
      %Number of charged particles & &\\
      %Statistical and systematic & &0.004\\
      %\midrule 
      %Number of nuclear capture & &\\
      %Statistical & Peak area calculation& 0.022\\
      %Systematic  & Efficiency calibration & 0.024\\
                  %& X-ray intensity & 0.009\\
                  %& Capture probability & 0\\
                  
      %\midrule
      %Total relative error & & 0.035\\
      %Total absolute error & & 0.009\\

      %\bottomrule
    %\end{tabular}
  %\end{center}
  %\caption{Uncertainties of the emission rate from the thick silicon target}
  %\label{tab:sir2_uncertainties}
%\end{table}

% subsection partial_emission_rate_of_charged_particle_in_8_10_mev_range (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%TODO
%\subsection{Partial emission rate of charged particles from the literature}
%\label{sub:partial_emission_rate_of_charged_particles_from_the_literature}
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/sobottka_spec2}
  %\caption{Reproduced charged particle spectrum from muon capture on silicon,
    %measured by Sobottka and Wills. Integration region is shown in the green
    %box.} 
  %\label{fig:sobottka_spec}
%\end{figure}
%The spectrum measured by Sobottka and Wills~\cite{SobottkaWills.1968} is
%reproduced in \cref{fig:sobottka_spec}, the spectral integral in the
%energy region from 8 to 10~\si{\MeV}\ is $2086.8 \pm 45.7$. 
%The authors obtained the spectrum in a 4~\si{\us}\ gate period which began
%1~\si{\us}\ after a muon stopped, which would take 26.59\% of the emitted
%particles into account. The number of stopped muons was not explicitly stated,
%but can be inferred to be $55715/0.06 = 92858.3$.

%The partial rate of charged particle from 8 to 10~\si{\MeV}\ is then
%calculated to be:
%\begin{equation}
  %R_{\textrm{8-10 MeV}}^{lit.} = 
  %\dfrac{2086.8}{0.2659 \times 92858.3 \times 0.658} 
  %= 1.28 \times 10^{-2}
%\end{equation}
%The authors did not mentioned how the uncertainties of their measurement was
%derived, but quoted the emission rate below 26~\si{\MeV}\ to be $0.15
%\pm 0.02$, which translates to a relative uncertainty of 0.133. The statistical
%uncertainty from the spectral integral and the number of stopped muons is:
%\begin{equation*}
  %\dfrac{1}{\sqrt{25000}} + \dfrac{1}{\sqrt{92858.3}} = 0.9 \times 10^{-2}
%\end{equation*}
%Then their systematic uncertainty would be: $0.133 - 0.009 = 0.124$.

%For the partial spectrum from 8 to 10~\si{\MeV}, the statistical
%contribution to the uncertainty is:
%\begin{equation*}
  %\dfrac{1}{\sqrt{2086.8}} + \dfrac{1}{\sqrt{92858.3}} = 2.5 \times 10^{-2}
%\end{equation*}
%So, the combined uncertainty of this partial rate calculation is: $0.124
%+ 0.025 = 0.150$. The partial rate of charged particles from 8 to
%10~\si{\MeV} per muon capture is:
%\begin{equation}
  %R_{\textrm{8-10 MeV}}^{lit.} = (1.28 \pm 0.19) \times 10^{-2}
%\end{equation}
% subsection partial_emission_rate_of_charged_particles_from_the_literature
% (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charged particles following muon capture on a thin silicon target}
\label{sec:charged_particles_following_muon_capture_on_a_thin_silicon_target}
In this measurement, a passive, 62-\si{\um}-thick silicon target was used as the
target. The silicon target is $5\times5$~\si{\centi\meter}$^2$ in area. The muon
momentum was chosen to be 1.06 after a scanning to maximise the stopping ratio.
The charged particles were measured by two arms of silicon detectors. The
plastic scintillators vetoing information were not used.

This data set consists of 66 runs, from 3474--3489 and 3491--3540.
Although there are a few issues found in the process of data quality
checking such as one very noisy timing channel, and several runs had 
abnormally high rates, the whole data set is determined to be good. Without
an active target and veto, the muon signal is from the muon counter only. The
tree contains total $1.452 \times 10^8$ muon events.  %145212698
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.49\textwidth]{figs/si16_lldq_noise}
    \includegraphics[width=0.49\textwidth]{figs/si16_lldq_islandrate}
  \caption{Oddities found in checking data quality: noise level on timing
    output of the SiL1-2 was much higher than the other detectors, and some
    runs show large pulse rate.}
  \label{fig:si16_lldq}
\end{figure}

\subsection{Particle identification by dE/dx and proton selection}
\label{sub:particle_identification_by_de_dx}
%All silicon hits with energy deposition larger than
%200~\si{\keV}\ that happened within $\pm 10$~\si{\us}\ of the
%muon hit are then
%associated to the muon and stored in the muon event tree. The
%200~\si{\keV}\ cut effectively rejects all MIPs hits on thin silicon
%detectors of which the most probable value is about 40~\si{\keV}. 

%In order to use dE/dx for particle identification, $\Delta$E and total E are
%needed. 
The charged particle selection starts from searching for muon event
that has at least one hit on thick silicon. If there is a thin silicon hit
within a coincidence window of $\pm 0.5$~\si{\us}\ around the thick
silicon hit, the two hits are considered to belong to one particle with
$\Delta$E being the energy of the thin hit, and total E being the sum energy of
the two hits. Particle identification is done using correlation between
$\Delta$E and E. \cref{fig:si16p_dedx_nocut} shows clearly visible banding
structure. No cut on energy deposit or timing with respect to muon hit are
applied yet.

With the aid from MC study (\cref{fig:pid_sim}), the banding on the
$\Delta$E-E plots can be identified as follows: 
\begin{itemize}
  \item the densest spot at the lower left conner belonged to electron hits; 
  \item the small blurry cloud just above the electron region was muon hits;
  \item the most intense band was due to proton hits; 
  \item the less intense, upper band caused by deuteron hits; 
  \item the highest band corresponded to alpha hits; 
  \item the faint stripe above the deuteron band should be triton
    hits, which is consistent with a relatively low probability of emission of
    tritons.  
\end{itemize}

%The electrons either from Michel decay or from the beam are MIPs particles,
%which would deposit about 466~keV on the 1500-\si{\um}-thick silicon detector,
%and about 20~keV on the 65-\si{\um}-thick silicon detector. Therefore our thin
%silicon counters could not distinguish electrons from electronic
%noise. The brightest spots on the $\Delta$E-E plots are identified as electrons
%due to
%the total E of about 500~keV, and is the accidental coincidence between
%electron hits on the thick silicon and electronics noise on the thin silicon.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.95\textwidth]{figs/si16p_dedx_nocut}
  \caption{$\Delta$E as a function of E of particles from muon capture on the
    thin silicon target.}
  \label{fig:si16p_dedx_nocut}
\end{figure}

It is observed that the banding is more clearly visible in a log-log scale
plots like in \cref{fig:si16p_dedx_cut_explain}, this suggests
a geometrical cut on the logarithmic scale would be able to discriminate
protons from other particles. The protons and deuterons bands are nearly
parallel to the $\ln(\Delta \textrm{E [keV]}) + \ln(\textrm{E [keV]})$ line,
but have a slightly altered slope because $\ln(\textrm{E})$ is always greater
than $\ln(\Delta\textrm{E})$. The two parallel lines on
\cref{fig:si16p_dedx_cut_explain} suggest a check of 
$\ln(\textrm{E}) + 0.85\times\ln(\Delta \textrm{E})$  could tell
protons from other particles.

Another feature of the $\Delta$E-E plots is their resolution power for protons
decrease as the energy E increases. The reason for this is the limited energy
resolution of the silicon detectors in use. The plots in logarithmic scale
show that this particle identification is good in the region where 
$\ln(\textrm{E}) < 9$, which corresponds to $\textrm{E} < 8$~MeV.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.95\textwidth]{figs/si16p_dedx_cut}
  \caption{$\Delta$E-E plots in the logarithmic scale and the geometrical cuts
  for protons.}
  \label{fig:si16p_dedx_nocut_log}
\end{figure}

The cut of $\ln(\textrm{E}) < 9$ is applied first, then 
$\ln(\textrm{E})+ 0.85\times\ln(\Delta \textrm{E}) $ is plotted as
\cref{fig:si16p_loge+logde}. The protons make a clear peak in the region
between 14 and 14.8, the next peak at 15 corresponds to deuteron. 
Imposing the 
$14<\ln(\textrm{E})+ 0.85\times\ln(\Delta \textrm{E})<14.8$ cut,
the remaining proton band is shown on \cref{fig:si16p_proton_after_ecut}.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/si16p_dedx_loge+logde}
  \caption{Rationale for the cut on $\ln(\textrm{E})$ and $\ln(\Delta
    \textrm{E})$}
  \label{fig:si16p_loge+logde}
\end{figure}

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/si16p_proton_after_ecut}
  \caption{Proton bands after cuts on energy}
  \label{fig:si16p_proton_after_ecut}
\end{figure}

% subsection particle_identification_by_de_dx (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of muon captures}
\label{sub:number_stopped_muons}
The X-ray spectrum from this silicon target on \cref{fig:si16_xray} is
significantly noisier than the previous data set of SiR2, suffers from both
lower statistics and a more relaxed muon definition. The peak of $(2p-1s)$
X-ray at 400.177~keV can still be recognised but on a very high background. The
same timing requirement for the hit timing on the germanium detector as
in~\eqref{eqn:sir2_ge_cut}.

The double peaks of muonic X-rays from the lead shield at 431 and 438~keV are
very intense, reflects the fact that the low momentum muon beam of
29.68~MeV\cc\ (scaling factor 1.06) was strongly scattered by the upstream
counters. After a prompt cut that requires the photon
hit occured in $\pm 1$~\si{\us}\ around the muon hit, the peaks from lead
remain prominent which is an expected result because of all the lead shield
inside the chamber was to capture stray muons. The cut shows its effect on
reducing the background level under the 400.177 keV peak by about one third.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.98\textwidth]{figs/si16p_xray}
  \caption{X-ray spectrum from the passive 62-\si{\um}-thick silicon target with
    and with out timing cut.}
  \label{fig:si16_xray}
\end{figure}

Using the same procedure  on the region from 396 to 402 keV (without
self-absorption correction since this is a thin target), the number of
X-rays recorded and the number of captures are shown in
\cref{tab:si16p_ncapture_cal}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l l c c c}
      \toprule
      \textbf{Source}& \textbf{Quantity}& \textbf{Value} & \textbf{Absolute}
      & \textbf{Relative}\\ 
      & & & \textbf{error} & \textbf{error}\\
      \midrule 
      Measured & $(2p-1s)$ peak area  & 2613 & 145.5 & 0.056\\
      \midrule
      Calibration & X-ray efficiency & \sn{4.54}{-4} & \sn{1.11}{-5}
      & 0.024\\
      \midrule
      Reference & X-ray intensity & 0.803 & 0.008 & \sn{9.9}{-3}\\
                & Capture probability & 0.658 & 0 & 0 \\
      \midrule
      Corrections& Self absorption & 1 & 0 & 0\\
                 & True coincidence summing & 1 &0 & 0\\
                 & TRP reset time & 1.01 & 0 & 0 \\
                 & Dead time      & 1.041& 0 & 0\\
      \midrule
      Results   & Number of X-rays & \sn{6.05}{6} & \sn{0.37}{6} & 0.06\\
                & Number of $\mu$ stopped & \sn{7.54}{6} & \sn{0.46}{6}&0.06\\
                & Number of captures& \sn{4.96}{6} & \sn{0.31}{6} & 0.06\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Number of X-rays and muon captures in the passive silicon runs.}
  \label{tab:si16p_ncapture_cal}
\end{table}
% subsection number_stopped_muons (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\subsection{Lifetime measurement}
\label{sub:lifetime_measurement}w
To check the origin of the protons recorded, lifetime measurements were made by
cutting on time difference between a hit on one thick silicon and the muon
hit. Applying the time cut in 0.5~\si{\us}\ time steps on the proton
events in \cref{fig:si16p_proton_after_ecut}, the number of surviving
protons on each arm are plotted on \cref{fig:si16p_proton_lifetime}. The
curves show decay constants of $762.9 \pm 13.7$~\si{\ns}\ and $754.6 \pm
11.9$,
which are consistent with the each other, and with mean life time of muons in
silicon in the literatures of $758 \pm 2$~\cite{}. This is the confirmation
that the protons seen by the silicon detectors were indeed from the silicon
target.  
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.75\textwidth]{figs/si16p_proton_lifetime}
  \caption{Lifetime measurement of protons seen on the silicon detectors.}
  \label{fig:si16p_proton_lifetime}
\end{figure}

The fits are consistent with lifetime of muons in silicon in from after 500~ns,
before that, the time constants are shorter ($655.9\pm 9.9$ and $731.1\pm8.9$)
indicates the contamination from muon captured on material with higher $Z$.
Therefore a timing cut from 500~ns is used to select good silicon events, the
remaining protons are shown in \cref{fig:si16p_proton_ecut_500nstcut}.
The spectra have a low energy cut off at 2.5~MeV because protons with energy:
lower than that could not pass through the thin silicon to make the cuts as the
range of 2.5~MeV protons in silicon is about 68~\si{\um}.  
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/si16p_proton_ecut_500nstcut}
  \caption{Proton spectrum after energy and timing cuts}
  \label{fig:si16p_proton_ecut_500nstcut}
\end{figure}

% subsection lifetime_measurement (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proton emission rate from the silicon target}
\label{sub:proton_emission_rate_from_the_silicon_target}
The number of protons in \cref{fig:si16p_proton_ecut_500nstcut} is
counted from 500~ns after the muon event, where the survival rate is
$e^{-500/758} = 0.517$. 

The geometry acceptance of each silicon arm is estimated to be \sn{2.64}{-2}
using a toy MC study where geantinos are generated within the image of the
collimator on the target, and the number of hits on each silicon package was
counted. Taking the geometry acceptance into account, the number of protons
with energy from 2.5 to 8~MeV emitted is:
\begin{equation}
  N_{p \textrm{eff.}} = \dfrac{1927 + 1656}{0.517\times2.64\times10^{-2}} 
  = 2.625 \times 10^5
\end{equation}
The emission rate per muon capture is:
\begin{align}
  R_{2.5-8\textrm{ MeV}}^{\textrm{eff.}}  &= \dfrac{N_{p \textrm{eff.}}}
  {N_{\mu \textrm{ captured}}^{\textrm{Si16p}}}\nonumber\\
  &= \dfrac{2.625 \times 10^5}{6.256\times10^6} \nonumber\\
  &= 4.20\times10^{-2}\nonumber
\end{align}
The proton spectra on the \cref{fig:si16p_proton_ecut_500nstcut} and  the
emission rate are only effective ones, since the energy of protons are modified
by energy loss in the target, and low energy protons could not escape the
target. Therefore further corrections are needed for both rate and spectrum of
protons.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proton emission rate uncertainties}
\label{sub:proton_emission_rate_and_uncertainties_estimation}
The uncertainty of the emission rate could come from several sources:
\begin{enumerate}
  \item number of captures $\pm0.562\times10^6$, or 9\%, mainly from the
    background under the X-ray peak (5.5\%) and the efficiency calibration
  \item number of protons: efficiency of the cuts in energy, impacts of the
    timing resolution on timing cut. The energy cuts' contribution should be
    small since it can be seen from \cref{fig:si16p_loge+logde}, the peak
    of protons is strong and well separated from others. The uncertainty in
    timing contribution is significant because all the timing done in this
    analysis was on the peak of the slow signals. As it is clear from the
    \cref{fig:tme_sir_prompt_rational}, the timing resolution of the
    silicon detector could not be better than 100~ns. Putting $\pm100$~ns into
    the timing cut could change the survival rate of proton by about
    $1-e^{-100/758} \simeq 13\%$. Also, the low statistics contributes a few
    percent to the uncertainty budget.
  \item acceptance of the silicon packages: muon stopping distribution,
    imperfect alignment, efficiency of the detectors, different response to
    different species. The muon stopping distribution is important in unfolding
    the initial proton spectrum and also greatly affects the rate of protons.
    By the end of the run, we found that the target was displaced from the
    previously aligned position by 10~mm. Whether this misalignment is serious
    or not depends on the spatial distribution of the muons after the
    collimator. In the worst case when the muon beam is flatly distributed,
    that displacement could change the acceptance of the silicon detectors by
    12\%. Although no measurement was done to determine the efficiency of the
    silicon detectors, it would have small effect compared to other factors.
\end{enumerate}

The combined uncertainty from known sources above therefore could be as large
as 35\%, and the effective proton emission rate in the 2.5--8~MeV could be
written as:
\begin{equation}
  R_{2.5-8\textrm{ MeV}}^{\textrm{eff.}} = (4.20\pm1.47)\times 10^{-2}
\end{equation}

\subsection{Ratio of protons to other heavy charged particles}
\label{sub:heavy_charged_particles_emission_rate}
By using only the lower limit on 
$\ln(\textrm{E}) + 0.85\times\ln(\Delta \textrm{E})$, the heavy charged
particles can be selected. These particles also show a lifetime that is
consistent with that of muons in silicon
(\cref{fig:si16p_allparticle_lifetime}).
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/si16p_allparticle_lifetime}
  \caption{Lifetime of heavy charged particles}
  \label{fig:si16p_allparticle_lifetime}
\end{figure}
The ratio between the number of protons and other particles at 500~ns is $(1927
+ 1656)/(2202 + 1909) \simeq 0.87$. 

% subsection heavy_charged_particles_emission_rate (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%I have started the initial study on the correction ()
% subsection proton_emission_rate_from_the_silicon_target (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Rate and spectrum correction}
%\label{sub:proton_spectrum_deconvolution}
%The proton spectra on the \cref{fig:si16p_proton_ecut_500nstcut} and  the
%emission rate are only effective ones, since the energy of protons are modified
%by energy loss in the target, and low energy protons could not escape the
%target. Therefore corrections are needed for both rate and spectrum of protons.

%To solve the unfolding problem, one needs to supply a response function that
%relates the observed energy to the initial energy of protons. This response
%function can be obtained from Monte Carlo simulation where protons with an
%assumed initial spatial distribution inside the target, and a uniform energy
%distribution are generated, then their modified energy spectrum is recorded.
%The initial spatial distribution of protons is inferred from the muon beam
%momentum using Monte Carlo simulation, and available measured data in momentum
%scanning runs. The response function for this thin silicon target is shown in
%\cref{fig:si16p_toyMC}.
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/si16p_toyMC}
  %\caption{An example of response function between the observed energy and
    %initial energy of protons in a 62-\si{\um}-target.}
  %\label{fig:si16p_toyMC}
%\end{figure}

%The response function is then used to train the unfolding program, which is
%based on the RooUnfold package. The package supports several unfolding methods,
%and I adopted the so-called Bayesian unfolding method~\cite{DAgostini.1995a}.
%The Bayesian method is chosen because it tends to be fast, typical number of
%iterations is from 4--8.

%\cref{fig:si16p_unfold_train} presented results of two tests unfolding with
%two distributions of initial energy, a Gaussian distribution and
%a parameterized function in~\eqref{eqn:EH_pdf}. The numbers of protons obtained
%from the tests show agreement with the generated numbers.

%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/si16p_unfold_train}
  %\caption{Bayesian unfolding tests with two different initial proton energy
    %distributions: Gaussian (left) and parameterized function of Sobottka and
    %Wills's proton spectrum (right).}
  %\label{fig:si16p_unfold_train}
%\end{figure}

%Finally, the unfolding is applied on the spectra in
%\cref{si16p_proton_spec}, the results are shown in
%\cref{si16p_unfold_meas}.
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/si16p_unfold_meas}
  %\caption{Unfolded spectrum from a thin silicon target}
  %\label{fig:si16p_unfold_meas}
%\end{figure}
% subsection proton_spectrum_deconvolution (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Proton emission rate and uncertainties estimation}
%\label{sub:proton_emission_rate_and_uncertainties_estimation}

%The rate of proton emission from 2.5--10~\si{\MeV} is:
%\begin{equation}
  %R = 
%\end{equation}
%\begin{equation}
  %R = 
%\end{equation}
%The uncertainties are:

% subsection proton_emission_rate_and_uncertainties_estimation (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%The uncertainties are:

% subsection proton_emission_rate_and_uncertainties_estimation (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\section{Proton emission following muon capture on an aluminium target}
\label{sec:proton_emission_following_muon_capture_on_an_aluminium_target}
The aluminium is the main object of the AlCap experiment, in this preliminary
analysis I chose one target, Al100 the 100-\si{\um}-thick target, on
a sub-range of the data set runs 2808--2873, as a demonstration.
Because this is a passive target, the same procedure and cuts used in the
passive silicon runs were applied.
\subsection{The number of stopped muons}
\label{sub:the_number_of_stopped_muons}
The X-ray spectrum on the germanium detector is shown on
\cref{fig:al100_ge_spec}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
  \caption{X-ray spectrum from the aluminium target, the characteristic
    $(2p-1s)$ line shows up at 346.67~keV} 
  \label{fig:al100_ge_spec}
\end{figure}

The area of the $(2p-1s)$ line of aluminium and the number of captured in this
target are:
\begin{align}
  N_{(2p-1s)\textrm{Al}} &= 3800.0  \pm 179.4 \nonumber\\
  N_{\mu \textrm{ captured}}^{\textrm{Al100}}
  &= \dfrac{N_{(2p-1s)\textrm{Al}}}
  {\epsilon_{(2p-1s)\textrm{Al}} \times I_{(2p-1s)\textrm{Al}}}
  \times f_{\textrm{capture-Al}} \nonumber \\ 
  &= \dfrac{3800.0} {5.12\times 10^{-4} \times 0.798} \times 0.609 \nonumber \\
  &= (5.664 \pm 0.479) \times 10^6
\end{align}
% subsection the_number_of_stopped_muons (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Particle identification}
\label{sub:particle_identification}
Using the same charged particle selection
procedure and the cuts on $\ln(\textrm{E})$ and $\ln(\Delta\textrm{E})$, the
proton energy spectrum is shown in \cref{fig:al100_proton_spec}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=1\textwidth]{figs/al100_selection}
  \caption{Selection of protons from the Al100 target: coincidence cut (top),
    cuts on energy (middle) and the results (bottom).}
  \label{fig:al100_selection}
\end{figure}

The lifetime of these protons are shown in
\cref{fig:al100_proton_lifetime}, the fitted decay constant on the right
arm is consistent with the reference value of $864 \pm 2$~\si{\ns}~\cite{}.
But the left arm gives $918 \pm 16.1$~\si{\ns}, significantly larger than
the reference value. 
%The longer lifetime suggested some contributions from
%other lighter materials, one possible source is from muons captured on the back
%side of the collimator (\cref{fig:alcap_setup_detailed}). 
%For this reason, the emission rate calculated from  the left arm will be taken as upper
%limit only.  
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/al100_proton_lifetime}
  \caption{Lifetime of protons from the aluminium Al100 target}
  \label{fig:al100_proton_lifetime}
\end{figure}
Further investigation of the problem of longer lifetime was made and the first
channel on the thin silicon detector on that channel was the offender. The
lifetime measurement with out that SiL1-1 channel gives a reasonable result,
and the decay constant on the SiL1-1 alone was nearly about 1000~\si{\us}.
The reason for this behaviour is not known yet. For this emission rate
calculation, this channel is discarded and the rate on the left arm is scaled
with a factor of 4/3. The proton spectrum from the aluminium target is plotted
on \cref{fig:al100_proton_spec_wosil11}.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.40\textwidth]{figs/al100_proton_lifetime_wosil11}
    \includegraphics[width=0.40\textwidth]{figs/al100_proton_lifetime_sil11}
  \caption{Lifetime of protons without channel SiL1-1 (right) and of the
    channel SiL1-1 alone (left).}
  \label{fig:al100_proton_lifetime_sil11}
\end{figure}
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/al100_proton_spec_wosil11}
  \caption{Spectrum of protons from the Al100 target after cuts on energy and
    time, without channel SiL1-1}
  \label{fig:al100_proton_spec_wosil11}
\end{figure}
% subsection particle_identification (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Proton emission rate}
\label{sub:proton_emission_rate_and_corrections}
The proton rate is calculated as:
\begin{equation}
  N_{p \textrm{eff.}} = \dfrac{1132\times \frac{4}{3} + 2034}
  {e^{-500/864}\times2.64\times10^{-2}} 
  = 1.34 \times 10^5
\end{equation}
\begin{equation}
  R_{2.5-8\textrm{ MeV}}^{\textrm{Al eff.}}  = \dfrac{N_{p \textrm{eff.}}}
  {N_{\mu \textrm{ captured}}^{\textrm{Al100}}}
  = \dfrac{1.34 \times 10^5}{5.664\times10^6} 
  = 2.37\times10^{-2}
\end{equation}

The uncertainty of the emission rates will be smaller than that of the rate
from silicon because of a longer lifetime of muons in aluminium and a higher
momentum beam made the misalignment of the target, if any, less important. To
be conservative, I take to 35\% above as this calculation uncertainty, and the
rates will be:
\begin{equation}
  R_{2.5-8\textrm{ MeV}}^{\textrm{Al eff.}}=(2.37\pm0.83)\times10^{-2}
\end{equation}
% subsection proton_emission_rate_and_corrections (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section proton_emission_following_muon_capture_on_an_aluminium_target (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% chapter data_analysis (end)
