\chapter{Data analysis}
\label{cha:data_analysis}

\section{Analysis modules}
\label{sec:analysis_modules}
A full analysis has not been completed yet, but initial analysis
based on the existing modules (\cref{tab:offline_modules}) is possible
thanks to the modularity of the analysis framework. 

\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l p{8cm}}
      \toprule
      \textbf{Module name} & \textbf{Functions}\\
      \midrule
      MakeAnalysedPulses  & make a pulse with parameters extracted from
      a waveform\\
      MaxBinAPGenerator   & simplest algorithm to get pulse information\\
      TSimpleMuonEvent    & sort pulses occur in a fixed time window around the
      muon hits\\
      ExportPulse \& PulseViewer & plot waveforms for diagnostics\\
      PlotAmplitude       & plot pulse height spectra\\
      PlotAmpVsTdiff      & plot pulse correlations in timing and amplitude\\
      EvdE                & identify charged particles using dE/dx\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Available offline analysis modules.}
  \label{tab:offline_modules}
\end{table}

The MakeAnalysedPulses module takes a raw waveform, calculates the pedestal
from a predefined number of first samples, subtracts this pedestal taking
pulse polarity into account, then calls another module to extract pulse
parameters. At the moment, the simplest module, so-called MaxBinAPGenerator,
for pulse information calculation is in use. The module looks for the
sample that
has the maximal deviation from the baseline, takes the deviation as pulse
amplitude and the time stamp of the sample as pulse time. The procedure is
illustrated on \cref{fig:tap_maxbin_algo}. This module could not detect
pile up or double pulses in one \tpulseisland{} in
\cref{fig:tap_maxbin_bad}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/tap_maxbin_algo}
  \caption{Pulse parameters extraction with MaxBinAPGenerator.}
  \label{fig:tap_maxbin_algo}
\end{figure}
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.47\textwidth]{figs/tap_maxbin_bad}
    \includegraphics[width=0.47\textwidth]{figs/tap_maxbin_bad2}
  \caption{Double pulse and pile up are taken as one single pulse by the
    MaxBinAPGenerator}
  \label{fig:tap_maxbin_bad}
\end{figure}

The TSimpleMuonEvent first picks a muon candidate, then loops through all
pulses on all detector channels, and picks all pulses occur in
a time window of \SI{\pm 10}{\si{\micro\second}} around each candidate to build
a muon event. A muon candidates is a hit on the upstream plastic scintillator
with an amplitude higher than a threshold which was chosen to reject minimum
ionising particles (MIPs). The period of \SI{10}{\si{\micro\second}} is long
enough compares to the mean life time of muons in the target materials
(\SI{0.758}{\si{\micro\second}} for silicon, and \SI{0.864}{\si{\micro\second}}
for aluminium~\cite{SuzukiMeasday.etal.1987}) so practically all of emitted
charged particles would be recorded in this time window.
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/tme_musc_threshold}
  %\caption{Pulse height spectrum of the $\mu$Sc scintillator}
  %\label{fig:tme_musc_threshold}
%\end{figure}

A pile-up protection mechanism is employed to reject multiple muons events: if
there exists another muon hit in less than \SI{15}{\si{\micro\second}} from the
candidate then both the candidate and the other muon are discarded. This
pile-up protection would cut out less than 11\% total number of events because
the beam rate was generally less than \SI{8}{\kilo\hertz}.

%In runs with active silicon targets, another requirement is applied for the
%candidate: a prompt hit on the target in $\pm 200$ \si{\nano\second}\ around the
%time of the $\mu$Sc pulse. The number comes from the observation of the
%time correlation between hits on the target and the $\mu$Sc
%(\cref{fig:tme_sir_prompt_rational}).
%\begin{figure}[htb]
  %\centering
    %\includegraphics[width=0.85\textwidth]{figs/tme_sir_prompt_rational}
  %\caption{Correlation in time between SiR2 hit and muon hit}
  %\label{fig:tme_sir_prompt_rational}
%\end{figure}

To make sure that we will analyse good data, a low level data quality checking
was done on the whole data sets. The idea is plotting the variations of basic
parameters, such as noise level, length of \tpulseisland{}, \tpulseisland{}
rate, time correlation to hits on $\mu$Sc, \ldots on each channel during the
data collecting period. Runs with significant difference from the nominal
values were further checked for possible causes, and would be discarded if such
discrepancy was too large or unaccounted for. Examples of such trend plots are
shown in \cref{fig:lldq}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.47\textwidth]{figs/lldq_noise}
    \includegraphics[width=0.47\textwidth]{figs/lldq_tdiff}
  \caption{Example trend plots used in the low level data quality checking:
    noise level in FWHM (left) and time correlation with muon hits (right). The
    noise level was basically stable in in this data set, except for one
    channel. On the right hand side, this sanity check helped find out the
    sampling frequency was wrongly applied in the first tranche of the data
    set.}
  \label{fig:lldq}
\end{figure}
% section analysis_modules (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Detector calibration}
\label{sec:detector_calibration}

\subsection{Silicon detector}
\label{sub:silicon_detector}
The energy calibration for the silicon detectors were done routinely during the
run, mainly by an
$^{241}\textrm{Am}$ alpha source and a tail pulse generator. The source emits
79.5 $\alpha$\si{\per\second} in a \SI{2\pi}{\steradian} solid angle. The most
prominent alpha particles have energies of \SI{5.484}{\si{\mega\electronvolt}}
(85.2\%) and \SI{5.442}{\si{\mega\electronvolt}} (12.5\%). A tail pulse with
amplitude of
\SI{66}{\milli\volt}~was used to simulate the response of the silicon detectors'
preamplifiers to a particle with \SI{1}{\si{\mega\electronvolt}} energy deposition.

During data taking period, electrons in the beam were were also used for energy
calibration of thick silicon detectors where energy deposition is large enough.
The muons at different momenta provided another mean of calibration in the beam
tuning period. 

The alpha particles from the source would deposit
about 66~keV in the \SI{0.5}{\micro\meter}-thick dead layer, and the peak would
appear at 5418~keV (\cref{fig:toyMC_alpha}). 
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.6\textwidth]{figs/toyMC_alpha}
  \caption{Energy loss of the alpha particles after a dead layer of
    0.5~\si{\micro\meter}.}
  \label{fig:toyMC_alpha}
\end{figure}

The calibration coefficients for the silicon channels are listed in
\cref{tab:cal_coeff}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l c r}
      \toprule
      \textbf{Detector} & \textbf{Slope} & \textbf{Offset}\\
      \midrule 
      SiL-2 & 7.86 & 14.14\\
      SiR-2 & 7.96 & 22.98\\
      \midrule
      SiL1-1 & 2.61 & 37.34\\
      SiL1-2 & 2.54 & -20.78\\
      SiL1-3 & 2.65 & 67.75\\
      SiL1-4 & 2.54 & -18.45\\
      \midrule
      SiR1-1 & 2.53 & 28.69\\
      SiR1-2 & 2.62 & 47.10\\
      SiR1-3 & 2.49 & 6.32\\
      SiR1-4 & 2.53 & 34.81\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Calibration coefficients of the silicon detector channels}
  \label{tab:cal_coeff}
\end{table}
% subsection silicon_detector (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Germanium detector}
\label{sub:germanium_detector}
The germanium detector was calibrated using a $^{152}\textrm{Eu}$
source\footnote{Energies and intensities of gamma rays are taken from the
  X-ray and Gamma-ray Decay Data Standards for Detector Calibration and Other
  Applications, which is published by IAEA at \\
  \url{https://www-nds.iaea.org/xgamma_standards/}}, the
recorded pulse height spectrum is shown in \cref{fig:ge_eu152_spec}. The
source was placed at the target position so that the absolute efficiencies can
be calibrated. The relation between pulse height in ADC count and energy is
found to be: 
\begin{equation}
  \textrm{ E [keV]} = 0.1219 \times \textrm{ADC} + 1.1621
\end{equation}
The energy resolution (full width at half maximum) was better than
2.6~\si{\kilo\electronvolt}\ for all the $^{152}\textrm{Eu}$ peaks. It was a little
worse at 3.1~\si{\kilo\electronvolt}~for the annihilation photons at
511.0~\si{\kilo\electronvolt}.

The absolute efficiencies for the $(2p-1s)$ lines of aluminium
(346.828~\si{\kilo\electronvolt}) and silicon (400.177~\si{\kilo\electronvolt}) are
presented in \cref{tab:xray_eff}. In the process of efficiency calibration,
corrections for true coincidence summing and self-absorption were not applied.
The true coincidence summing probability is estimated to be very
small, about \sn{5.4}{-6}, thanks to the far geometry of the calibration. The
absorption in the source cover made of 22~\si{\milli\gram\per\si{\centi\meter}^2}
polyethylene is less than \sn{4}{-4} for a 100~\si{\kilo\electronvolt}\ photon.

\begin{table}[htb]
  \begin{center}
    \begin{tabular}{c c c}
      \toprule
      \textbf{X-ray} & \textbf{Efficiency} & \textbf{Uncertainty}\\
      \midrule 
      346.828 & $5.12 \times 10^{-4}$ & $0.14\times 10^{-4}$\\
      400.177 & $4.54 \times 10^{-4}$ & $0.11\times 10^{-4}$\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Calculated efficiencies at X-rays of interest}
  \label{tab:xray_eff}
\end{table}

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.70\textwidth]{figs/ge_eu152_spec}
  \caption{Energy spectrum of the $\rm^{152}\textrm{Eu}$ calibration source
    recorded by the germanium detector. The most prominent peaks of
    $^{152}\textrm{Eu}$ along with their energies are
    annotated in red; the 1460.82 \si{\kilo\electronvolt}~line is background from
    $^{40}\textrm{K}$; and the annihilation 511.0~\si{\kilo\electronvolt}~photons
    come both from the source and the surrounding environment.}
  \label{fig:ge_eu152_spec}
\end{figure}

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.89\textwidth]{figs/ge_ecal_fwhm}
  \caption{Germanium energy calibration and resolution.}
  \label{fig:ge_fwhm}
\end{figure}

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.80\textwidth]{figs/ge_ecal_eff}
  \caption{Absolute efficiency of the germanium detector, the fit was done with
  7 energy points from 244~keV because it is known that the linearity between
    $ln(\textrm{E})$ and $ln(\textrm{eff})$ holds better. The shaded area is
    95\% confidence interval of the fit.}
  \label{fig:ge_eff}
\end{figure}

% subsection germanium_detector (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Beam tuning and muon momentum scanning}
%\label{sub:muon_momentum_scanning}
%Before taking any data, we carried out the muon momentum scanning to understand
%the muon beam, as well as calibrate the detector system. The nominal muon
%momentum used in the Run 2013 had been tuned to 28~MeV\cc\ before the run. By
%changing simultaneously the strength of the key magnet elements in the $\pi$E1
%beam line with the same factor, the muon beam momentum would be scaled with the
%same factor.

%The first study was on the range of muons in an active silicon target. The SiL2
%detector was placed perpendicular to the nominal beam path, after an oval
%collimator. The beam momentum scaling factor was scanned from 1.10 to 1.60,
%muon momenta and energies in the measured points are listed in
%\cref{tab:mu_scales}.
%\begin{table}[htbp]
  %\begin{center}
    %\begin{tabular}{c c c c}
      %\toprule
      %\textbf{Scaling} & \textbf{Momentum} & \textbf{Kinetic energy}
      %& \textbf{Momentum spread}\\ 
      %\textbf{factor} & \textbf{(MeV\per\cc)} & \textbf{(MeV)}
      %& \textbf{(MeV\per\cc, 3\% FWHM)}\\ 
      %\midrule 
      %1.03 &   28.84 &     3.87&    0.87\\
      %1.05 &   29.40 &     4.01&    0.88\\
      %1.06 &   29.68 &     4.09&    0.89\\
      %1.07 &   29.96 &     4.17&    0.90\\
      %1.10 &   30.80 &     4.40&    0.92\\
      %1.15 &   32.20 &     4.80&    0.97\\
      %1.20 &   33.60 &     5.21&    1.01\\
      %1.30 &   36.40 &     6.09&    1.09\\
      %1.40 &   39.20 &     7.04&    1.18\\
      %1.43 &   40.04 &     7.33&    1.20\\
      %1.45 &   40.60 &     7.53&    1.22\\
      %1.47 &   41.16 &     7.73&    1.23\\
      %1.50 &   42.00 &     8.04&    1.26\\
      %\bottomrule
    %\end{tabular}
  %\end{center}
  %\caption{Muon beam scaling factors, energies and momenta.}
  %\label{tab:mu_scales}
%\end{table}

% subsection muon_momentum_scanning (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section detector_calibration (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charged particles following muon capture on a thick silicon target}
\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
This analysis was done on a subset of the active target runs 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 \sn{6.43}{7}
muon events.  
%64293720

Firstly, the number of charged particles emitted after nuclear muon capture on
the active target is calculated. This number 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{Event selection}
\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 a 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 5 MeV energy corresponds to stopped muons in the thick target. The
band below 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 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<50\textrm{ ns}
      \label{eqn:sir2_prompt_cut}
    \end{equation}
  \item the first hit on the target has energy of that of the muons: 
    \begin{equation}
      3.4 \textrm{ MeV}<E_{\textrm{target}} < 5.6 \textrm{ MeV}
      \label{eqn:sir2_muE_cut}
    \end{equation}
\end{enumerate}
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 1300\textrm{
        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 muon events sample to the size of
\sn{9.32}{6}.

The number of stopped muons can also be calculated from the number of muonic
X-rays recorded by the germanium detector. The X-rays are emitted during the
cascading of the muon to the muonic 1S state in the time scale of \sn{}{-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
hits: 
\begin{equation}
  \lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < 500\textrm{ ns}
  \label{eqn:sir2_ge_cut}
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of charged particles with energy above 2~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 we are
    interested in;
  \item beam electrons with a characteristic Landau peak around 800~keV,
    dominating at large timing (from 6500 ns), causing background at energy
    lower than 1~MeV which drops sharply at energy larger than 3~MeV;
  \item electrons from muon decay-in-orbit (DIO) and recoiled nuclei from
    neutron emitting muon captures, peak at
    around 300~keV, dominate the region where energy smaller than 1~MeV and
    timing less than 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 1500 -- 9500~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 scale between
      components is arbitrarily chosen for the purpose of illustration.}
  \label{fig:sir2_mc_pdfs}
\end{figure}

An energy cut at 2~MeV is introduced to reduce the domination of the beam
electrons. In order to obtain the yields of backgrounds above 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}} &= 207201 \pm 856
\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}

The total number of charged particles from time zero is then calculated to be:
\begin{equation}
N_{\textrm{charged particles}} =(149.9\pm 0.6)\times 10^4 
\label{eqn:sir2_Nchargedparticle} 
\end{equation}


% 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 number of nuclear captures can be inferred from the number of recorded
muonic X-rays. The reference values of the parameters needed for the
calculation taken from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday
et al.~\cite{MeasdayStocki.etal.2007} 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-ray spectrum emitted from the active target is shown in
\cref{fig:sir2_xray}. The $(2p-1s)$ line is seen at
399.5~\si{\kilo\electronvolt}, 0.7~\si{\kilo\electronvolt}\ off from the
reference value of 400.177~\si{\kilo\electronvolt}. 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}
  \caption{Muonic X-rays spectrum from the active silicon target, the two major
    lines $(2p-1s)$ and $(3p-1s)$ are clearly distinguishable at 400 and 476
    keV, respectively. The $(5p-1s)$ line at 504 keV and $(6p-1s)$ line at 516
    keV can also be seen.
  }
  \label{fig:sir2_xray}
\end{figure}

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. This number of X-rays needs to be
corrected for several effects:
\begin{itemize}
  \item Self-absorption effect: the X-rays emitted could be absorbed  by the
    target itself, the probability of self-absorption becomes larger in case of
    thick sample and low energy photons.
    For this silicon target of 1500~\si{\micro\meter}\ thick and the photon energy of
    400~keV, and assuming a narrow muon stopping distribution at the centre of
    the target, the self-absorption correction is estimated to be:
    \begin{align}
      k_{\textrm{self absorption}} &= \dfrac{\mu t}{1 - e^{-\mu t}} \nonumber\\
      &= \dfrac {9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}} 
      {1 - e^{-9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}}}\nonumber \\
      %&= \dfrac{1}{0.992} \nonumber\\
      &= 1.008
    \end{align}
    where  $t = 0.075\textrm{ cm}$ is the thickness of the target, and $\mu$ is the
    linear attenuation coefficient of silicon for a photon of 400~keV. The
    value of $\mu$ is calculated as product of the density of silicon 
    $\rho = 2.33 \textrm{ g/cm}^3$ and its mass attenuation coefficient 
    $\mu/\rho = 9.614\times 10^{-2} \textrm{ cm}^2/\textrm{g}$ taken
    from the NIST's X-ray Mass Attenuation Coefficients
    table~\footnote{\url{http://www.nist.gov/pml/data/xraycoef}}.
  \item Dead time of the germanium detector system: there are two types of dead
    time in our germanium detector, (a) the insensitive period due to long
    pulse time, and (b) the reset pulses of the transistor reset preamplifier.
    The effects of the two dead time could be calculated by examining the
    interval between two consecutive pulses on the germanium detector in
    \cref{fig:sir2_ge_deadtime}.
    \begin{figure}[htb]
      \centering
        \includegraphics[width=0.85\textwidth]{figs/sir2_ges_self_tdiff}
      \caption{Interval between to consecutive pulses on the germanium
        detector. The peak at 57~\si{\micro\second}\ indicates the pulse length, and
        the bump at about 2000~\si{\micro\second}\ shows the width of the reset
        pulses. The average count rate of this detector is extracted from the
        decay constant of the time spectrum to be 
        $5.146 \times 10^{-7}\textrm{ ns}^{-1} = 514.6 \textrm{ s}^{-1}$}
      \label{fig:sir2_ge_deadtime}
    \end{figure}

    The correction factor for the pulse length is calculated by the formula:
    \begin{align}
      k_{\textrm{pulse length}} &= e^{2\times \textrm{(pulse length)} 
        \times \textrm{(count rate)}} \nonumber\\
      &= e^{2\times 57\times10^{-6} \times 514.6} \nonumber\\
      &= 1.06
    \end{align}
    The 2-ms-long reset pulses effectively reduce the actual measurement time
    compares to other channels, so the correction factor for the effect is:
    \begin{align}
      k_{\textrm{reset pulse}} &= \frac{\textrm{(measurement time)}}
      {\textrm{(measurement time)}
        - \textrm{(number of reset)}\times
        \textrm{(reset pulse length)}}\nonumber\\
      &= 1.033
    \end{align}

  \item The true coincidence summing is negligibly small due to the far
    geometry as mentioned in the calibration process, so no correction is made. 
    %%TODO
  \item The geometrical acceptance of the detector: the absolute efficiency
    calibration was done with a point-like source, but the actual points of
    origin of the X-rays have a finite spatial distribution. The correction
    factor is estimated to be \ldots
\end{itemize}

The number of X-rays after applying all above corrections is 3293.5. 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{3293.5}{4.54\times10^{-4} \times 0.803} \\
  &= 9.03\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 
$(9.03 \pm 0.31)\times 10^6$. This figure is consistent with the number of
stopped muons of $9.32\times10^6$ 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}}
  = 9.03 \times 10^6 \times 0.658 = 7.25 \times 10^6
  \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
      %\textbf{Source} & \textbf{Type} & \textbf{Relative error}\\
      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{\mega\electronvolt}\ is $2086.8 \pm 45.7$. 
%The authors obtained the spectrum in a 4~\si{\micro\second}\ gate period which began
%1~\si{\micro\second}\ 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{\mega\electronvolt}\ 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{\mega\electronvolt}\ 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{\mega\electronvolt}, 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{\mega\electronvolt} 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{\micro\meter}-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{\kilo\electronvolt}\ that happened within $\pm 10$~\si{\micro\second}\ of the
%muon hit are then
%associated to the muon and stored in the muon event tree. The
%200~\si{\kilo\electronvolt}\ cut effectively rejects all MIPs hits on thin silicon
%detectors of which the most probable value is about 40~\si{\kilo\electronvolt}. 

%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{\micro\second}\ 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{\micro\meter}-thick silicon detector,
%and about 20~keV on the 65-\si{\micro\meter}-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{\micro\second}\ 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{\micro\meter}-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}
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{\micro\second}\ 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{\nano\second}\ 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{\micro\meter}.  
\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 compare 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{\micro\meter}-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{\mega\electronvolt} 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{\micro\meter}-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{\nano\second}~\cite{}.
But the left arm gives $918 \pm 16.1$~\si{\nano\second}, 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{\micro\second}.
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)
