\chapter{Data analysis and results}
\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 the right hand side of \cref{fig:al100_scan_rate}. With the 1.09
scale beam, the muons stopped \SI{28}{\um} off-centred to the right silicon arm.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.47\textwidth]{figs/al100_scan_rate}
    \includegraphics[width=0.47\textwidth]{figs/al100_mu_stop_mc}
  \caption{Number of X-rays per incoming muon as a function of momentum
    scaling factor (left); and muon stopping distributions from MC simulation
    (right). The depth of muons is measured normal to surface of the target
    facing the muon beam.}
  \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} of 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}}
  \exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta
        E}}\right]}
\end{equation}
where $\Delta E_{\textrm{meas.}}$ is energy deposition measured by 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 study. A threshold is set at \num{1E-4} to
extract protons, the resulted band of protons is shown in
(\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}

The cut efficiency in the energy range from \SIrange{2}{12}{\MeV} is estimated
by MC study. The fraction of protons that do not satisfy the probability cut
is 0.5\%. The number of other charged particles that are misidentified as
protons depends on the ratios between those species and protons. Assuming
a proton:deuteron:triton:alpha:muon ratio of 5:2:1:2:2, the number of
misidentified hits is 0.1\% of the number of protons.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}
The numbers of protons in the energy range from \SIrange{2.2}{8.5}{\MeV} after
applying the probability cut are: 
\begin{align}
  N_{\textrm{p meas. left}} = 1822\\% \pm 42.7\\
  N_{\textrm{p meas. right}} = 2373% \pm 48.7
\end{align}
The right arm received significantly more protons than the left arm did, which
is expected as in \cref{sub:momentum_scan_for_the_100_} where it is shown that
muons stopped off-centred to the right arm.

\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.
The unfolding, essentially, is finding a response function that relates proton's
true energy and measured value. This can be done in MC simulation by generating
protons with a spatial distribution as close as possible to the real
distribution of muons inside the target, then counting the number of protons
reaching the silicon detectors. Such response function conveniently includes
the geometrical acceptance.

For the 100-\si{\um} aluminium target and muons at the momentum scale of 1.09,
the parameters of the initial protons are:
\begin{itemize}
  \item horizontal distribution: Gaussian \SI{26}{\mm} FWHM, centred at the
    centre of the target;
  \item vertical distribution: Gaussian \SI{15}{\mm} FWHM, centred at the
    centre of the target;
  \item depth: Gaussian \SI{69.2}{\um} FWHM, centred at \SI{68.8}{\um}-deep from
    the upstream face of the target;
  \item energy: flatly distributed from \SIrange{1.5}{15}{\MeV}.
\end{itemize}
The resulting response matrices for the two arms are presented in
\cref{fig:al100_resp_matrices}. These are then used as MC truth to train  and
test the unfolding code. The code uses an existing ROOT package
called RooUnfold~\cite{Adye.2011} where the iterative Bayesian unfolding
method is implemented.
\begin{figure}[htb]
  \centering
  \includegraphics[width=0.85\textwidth]{./figs/al100_resp}
  \caption{Response functions for the two silicon arms.}
  \label{fig:al100_resp_matrices}
\end{figure}
%After training, the unfolding code is applied on the measured spectra from the
%left and right arms. The unfolded proton spectra in \cref{fig:al100_unfold}
%reasonably reflect the distribution of initial protons which is off-centred to
%the right arm. The path length to the left arm is longer so less protons at
%energy lower than \SI{5}{\MeV} could reach the detectors. The sharp low-energy
%cut off on the right arm is consistent with the Coulomb barrier for protons,
%which is \SI{4.1}{\MeV} for protons emitted from $^{27}$Mg. 
The unfolded spectra using the two observed spectra at the two arms as input
are shown in \cref{fig:al100_unfold}. The two unfolded spectra generally agree
with each other, except for a few first and last bins. The discrepancy and
large uncertainties at the low energy region are because of only a small
number of protons with those energies could reach the detectors. The jump on
the right arm at around \SI{9}{\MeV} can be explained as the punch-through
protons were counted as the proton veto counters were not used in this
analysis.
\begin{figure}[htb]
  \centering
  \includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr}
  \caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
  \label{fig:al100_unfold}
\end{figure}

%Several studies were conducted to assess the performance of the unfolding
%code, including:
%\begin{itemize}
  %\item stability against cut-off energy;
  %\item comparison between the two arms;
  %\item and unfolding of a MC-generated spectrum.
%\end{itemize}
The stability of the unfolding code is tested by varying the lower cut-off
energy of the input spectrum. \cref{fig:al100_cutoff_study} show that the
shapes of the unfolded spectra are stable. The lower cut-off energy of the
output increases as that of the input increases, and the shape is generally
unchanged after a few bins.
\begin{figure}[htb]
  \centering
  \includegraphics[width=0.85\textwidth]{figs/al100_cutoff_study}
  \caption{Unfolded spectra with different cut-off energies.}
  \label{fig:al100_cutoff_study}
\end{figure}
The proton yields calculated from observed spectra in two arms are compared in
\cref{fig:al100_integral_comparison} where the upper limit of the integrals
is fixed at \SI{8}{\MeV}, and the lower limit is varied in \SI{400}{\keV} step.
The difference is large at cut-off energies less than \SI{4}{\MeV} due to
large uncertainties at the first bins. Above \SI{4}{\MeV}, the two arms show
consistent numbers of protons.
\begin{figure}[htb]
  \centering
  \includegraphics[width=0.85\textwidth]{figs/al100_integral_comparison}
  \caption{Proton yields calculated from two arms.}
  \label{fig:al100_integral_comparison}
\end{figure}
The yields of protons from \SIrange{4}{8}{\MeV} are:
\begin{align}
  N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\\
  N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3
\end{align}
The number of emitted protons is taken as average of the two yields:
\begin{equation}
  N_{\textrm{p unfold}} = (169.3 \pm 2.9) \times 10^3
\end{equation}

\subsection{Number of nuclear captures}
\label{sub:number_of_nuclear_captures}
\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 X-ray spectrum on the germanium detector is shown on
\cref{fig:al100_ge_spec}. Fitting the double peaks on top of a first-order
polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
procedure as in the case of the active target, the number stopped muons and
the number of nuclear captures are:
\begin{align}
  N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\\
  N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6
\end{align}

\subsection{Proton emission rate}
\label{sub:proton_emission_rate}
The proton emission rate in the range from \SIrange{4}{8}{\MeV} is therefore:
\begin{equation}
  R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.7\times
  10^{-2}
  \label{eq:proton_rate_al}
\end{equation}

The total proton emission rate can be estimated by assuming a spectrum shape
with the same parameterisation as in \eqref{eqn:EH_pdf}. The fit parameters
are shown in \cref{fig:al100_parameterisation} and \cref{tab:al100_fit_pars}.
The average spectrum is obtained by taking the average of the two unfolded
spectra from the left and right arms. The fitted parameters are compatible
with each other within their errors.

Using the fitted parameters of the average spectrum, the integration in range
from \SIrange{4}{8}{\MeV} is 51\% of the total number of
protons. The total proton emission rate is therefore estimated to be $3.5\times 10^{-2}$.
\begin{figure}[htb]
  \centering
  \includegraphics[width=0.85\textwidth]{figs/al100_parameterisation}
  \caption{Fitting of the unfolded spectra.}
  \label{fig:al100_parameterisation}
\end{figure}

\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l S[separate-uncertainty=true] S[separate-uncertainty=true] 
        S[separate-uncertainty=true]}
      \toprule
      \textbf{Parameter} &{\textbf{Left}} & {\textbf{Right}} & {\textbf{Average}}\\
      \midrule 
      $A \times 10^{-5}$ & 2.0 \pm 0.7 & 1.3 \pm 0.1 & 1.5 \pm 0.3\\
      $T_{th}$ (\si{\keV}) & 1301 \pm 490 & 1966 \pm 68 & 1573 \pm 132\\
      $\alpha$             & 3.2 \pm 0.7 & 1.2 \pm 0.1 & 2.0 \pm 1.2\\
      $T_{th}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Parameters of the fits on the unfolded spectra, the average spectrum
    is obtained by taking average of the unfolded spectra from left and right
    arms.} 
  \label{tab:al100_fit_pars}
\end{table}


\subsection{Uncertainties of the emission rate}
\label{sub:uncertainties_of_the_emission_rate}
The uncertainties of the emission rate come from:
\begin{itemize}
  \item uncertainties in the number of nuclear captures: these were discussed
    in \cref{sub:number_of_stopped_muons_from_the_number_of_x_rays};
  \item uncertainties in the number of protons:
    \begin{itemize}
      \item statistical uncertainties of the measured spectra which are
        propagated during the unfolding process;
      \item systematic uncertainties due to misidentification: this number is
        small compared to other uncertainties as discussed in
        \cref{sub:event_selection_for_the_passive_targets};
      \item systematic uncertainty from the unfolding
    \end{itemize}
\end{itemize}
The last item is studied by MC method using the parameterisation in
\cref{sub:proton_emission_rate}:
\begin{itemize}
  \item protons with energy distribution obeying the parameterisation are
    generated inside the target. The spatial distribution is the same as that
    of in \cref{sub:corrections_for_the_number_of_protons}. MC truth including
    initial energies and positions are recorded;
  \item the number of protons reaching the silicon detectors are counted,
    the proton yield is set to be the same as the measured yield to make the
    statistical uncertainties comparable;
  \item the unfolding is applied on the observed proton spectra, and the
    results are compared with the MC truth.
\end{itemize}
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.48\textwidth]{figs/al100_MCvsUnfold}
    \includegraphics[width=0.48\textwidth]{figs/al100_unfold_truth_ratio}
  \caption{Comparison between an unfolded spectrum and MC truth: spectra
    (left), and yields (right). The ratio is defined as $\textrm{(Unfold - MC
      truth)/(MC truth)}$}
  \label{fig:al100_MCvsUnfold}
\end{figure}
\Cref{fig:al100_MCvsUnfold} shows that the yield obtained after unfolding is
in agreement with that from the MC truth. The difference is less than 5\% if
the integration is taken in the range from \SIrange{4}{8}{\MeV}. Therefore
a systematic uncertainty of 5\% is assigned for the unfolding routine.

A summary of uncertainties in the measurement of proton emission rate is
presented in \cref{tab:al100_uncertainties_all}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{@{}ll@{}}
      \toprule
      \textbf{Item}& \textbf{Value}          \\ 
      \midrule
      Number of muons                   & 3.2\%          \\
      Statistical from measured spectra & 1.6\%          \\
      Systematic from unfolding         & 5.0\%            \\
      Systematic from PID               & \textless0.5\% \\ 
      \midrule
      Total                             & 6.1\%\\              
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Uncertainties of the proton emission rate.}
  \label{tab:al100_uncertainties_all}
\end{table}

\section{Results of the initial analysis}
\label{sec:results_of_the_initial_analysis}
\subsection{Verification of the experimental method}
\label{sub:verification_of_the_experimental_method}
The experimental method described in \cref{sub:experimental_method} has been
validated:
\begin{itemize}
  \item Number of muon capture normalisation: the number of stopped muons
    calculated from the muonic X-ray spectrum is shown to be consistent with
    that calculated from the active target spectrum. 
  \item Particle identification: the particle identification by specific
    energy loss has been demonstrated. The banding of different particle
    species is clearly visible. The proton extraction method using cut on
    likelihood probability has been established. Since the distribution of
    $\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do
    not make the cut is 0.5\%, much larger than the threshold at \num{1E-4}.
    The fraction of other charged particles being misidentified as protons is
    smaller than 0.1\%. These uncertainties from particle identification are
    still small in compared with the
    uncertainty of the measurement (2.3\%).
  \item Unfolding of the proton spectrum: the unfolded spectra inferred from
    two measurements at the two silicon arms show good agreement with each
    other, and with the muon stopping distribution obtained in the momentum
    scanning analysis.
\end{itemize}

\subsection{Proton emission rates and spectrum}
\label{sub:proton_emission_rates_and_spectrum}
The proton emission spectrum in \cref{sub:proton_emission_rate} peaks around
\SI{4}{\MeV} which is comparable to the Coulomb barrier for proton of
\SI{3.9}{\MeV} calculated using \eqref{eqn:classical_coulomb_barrier}. The
spectrum has a decay constant of \SI{2.6}{\MeV} in higher energy region, 
makes the emission probability drop more quickly than silicon charged
particles spectrum of Sobottka and Wills~\cite{SobottkaWills.1968} where the
decay constant was \SI{4.6}{\MeV}. This can be explained as the silicon
spectrum includes other heavier particles which have higher Coulomb barriers,
hence contribute more in the higher energy bins, effectively reduces the decay
rate.

The partial emission rate measured in the energy range from
\SIrange{4}{8}{\MeV} is: 
\begin{equation}
  R_{p \textrm{ meas. }} = (1.7 \pm 0.1)\%. 
  \label{eqn:meas_partial_rate}
\end{equation}

The total emission rate from aluminium assuming the spectrum shape holds for
all energy is:
\begin{equation}
  R_{p \textrm{ total}} = (3.5 \pm 0.2)\%. 
  \label{eqn:meas_total_rate}
\end{equation}
No direct comparison of this result to existing experimental or
theoretical work is available. Indirectly, it is compatible with the figures
calculated by Lifshitz and
Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980} listed in
\cref{tab:lifshitzsinger_cal_proton_rate}.  It is significantly larger than
the rate of 0.97\% for the $(\mu,\nu p)$ channel, and does not
exceed the inclusion rate for all channels $\Sigma(\mu,\nu p(xn))$ at 4\%,
leaving some room for other modes such as $(\mu,\nu d)$ or $(\mu,\nu p2n)$.
Certainly, if the rate of deuterons can be extracted then the combined
emission rate of protons and deuterons could be compared directly with the
inclusive rate.

The result \eqref{eqn:meas_total_rate} is greater than the 
probability of the reaction $(\mu,\nu pn)$ measured by Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} at 2.8\%. It is expectable because
the contribution from the $(\mu,\nu d)$ channel should be small since it
needs to form a deuteron from a proton and a neutron. 
%The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
%\eqref{eqn:wyttenbach_ratio}.

Compared with emission rate from silicon, the result
\eqref{eqn:meas_total_rate} is indeed much smaller. It is even lower than
the rate of the no-neutron reaction $(\mu,\nu p)$. This can be explained as
the resulted nucleus from muon capture on silicon, $^{28}$Al, is an odd-odd
nucleus and less stable than that from aluminium, $^{27}$Mg. The proton
separation energy for $^{28}$Al is \SI{9.6}{\MeV}, which is significantly
lower than that of $^{27}$Mg at \SI{15.0}{\MeV}~\cite{AudiWapstra.etal.2003}. 


