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thesis/chapters/chap5_alcap_setup.tex
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@@ -765,6 +765,7 @@ sets are shown in \cref{tb:stat}.
% section data_sets (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Analysis framework}
\label{sec:analysis_framework}
\subsection{Concept}
\label{sub:concept}
Since the AlCapDAQ is a trigger-less system, it stored all waveforms of the

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thesis/chapters/chap6_analysis.tex
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@@ -9,33 +9,37 @@ Purposes of the analysis include:
using specific energy loss;
\item extracting a preliminary rate of proton emission from aluminium.
\end{itemize}
\section{Charged particles following muon capture on a thick silicon target}
\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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.
Firstly, the number of charged particles emitted after nuclear muon capture on
the active target is calculated. The charged particles yield then is normalised
to the number of nuclear muon capture to obtain an emission rate.
%Finally, the
%rate is compared with that from the literature.
\subsection{Event selection}
\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 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 mentioned limitation of the current pulse parameter
extraction method where no pile up or double pulses is accounted for.
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
@@ -79,116 +83,19 @@ From the energy-timing correlation above, the cuts to select stopped muons are:
\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 \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}.
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 \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:
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}
\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns}
\label{eqn:sir2_ge_cut}
N_{\mu \textrm{ active Si}} = 9.32 \times 10^6
\label{eqn:n_stopped_si_count}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of charged particles with energy above \SI{3}{\MeV}}
\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation
in \cref{fig:sir2_mc_pdfs}, there are several components in
the energy spectrum recorded by the active target:
\begin{enumerate}
\item charged particles from nuclear muon capture, this is the signal of
interest;
\item beam electrons with a characteristic Landau peak around \SI{800}{\keV},
dominating at large delay (from \SI{6500}{\ns}), causing background at
energy lower than \SI{1}{\MeV} which drops sharply at energy larger than
\SI{3}{\MeV};
\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei
from neutron emitting muon captures, peak at
around \SI{300}{\keV}, dominate the region where energy smaller than
\SI{1}{\MeV} and delay less than \SI{3500}{\ns}. This component is
intrinsic background, having the same time structure as that of the signal;
\item stray muons scattered into the target, this component is small compares
to the charged particles yield so it is not considered further.
\end{enumerate}
\begin{figure}[htb]
\centering
\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
\caption{The observed spectrum in the timing window 1300 -- 10000~ns (left)
and its components from MC simulation (right). The charged particles
spectrum is obtained assuming the spectrum shape and emission rate from
Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scales between
components are arbitrarily chosen for the purpose of illustration.}
\label{fig:sir2_mc_pdfs}
\end{figure}
An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the
beam electrons at low energy. In order to obtain the yields of backgrounds
above \SI{2}{\MeV}, a binned maximum likelihood fit was done. The likelihood of
getting the measured spectrum is defined as:
\begin{equation}
L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
\label{eqn:llh_def}
\end{equation}
where $n$ is the total number of events observed, $\mu$ is the expected number
of events according to some linear combination of the signal and the
backgrounds shown in~\ref{fig:sir2_mc_pdfs}, namely:
\begin{align}
n &= n_{\textrm{sig}} + n_{\textrm{beam e}} + n_{\textrm{dio}}\\
\textrm{(sum pdf)} &= n_{\textrm{sig}}\times\textrm{(sig pdf)} +
n_{\textrm{beam e}}\times\textrm{(beam e pdf)} +
n_{\textrm{dio}}\times\textrm{(dio pdf)};
\label{eqn:sum_pdf}
\end{align}
and the $i$ index indicates the respective number of events in the $i$-th
bin.
The fit is done by the RooFit package~\cite{VerkerkeKirkby.2003} where the
negative log likelihood $-2\ln{L}$ is minimised. Fitting results are shown
in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
\begin{align}
n_{\textrm{beam e}} &= 23756 \pm 581\\
n_{\textrm{dio}} &= 111340 \pm 1245\\
n_{\textrm{sig}} &= 2.57 \pm 856
\label{eqn:sir2_n_chargedparticles}
\end{align}
\begin{figure}[htb]
\centering
\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nbkg}
\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nebeam}
\includegraphics[width=0.84\textwidth]{figs/sir2_mllfit}
\caption{Results of the maximum likelihood fit of the energy spectrum on the
active target.}
\label{fig:sir2_mll_fit}
\end{figure}
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of nuclear muon captures}
\label{sub:number_of_stopped_muons}
The 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}.
\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}
@@ -208,130 +115,452 @@ et al.~\cite{MeasdayStocki.etal.2007} are listed in \cref{tab:mucap_pars}.
\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{\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.
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}
\caption{Prompt 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.
\includegraphics[width=0.85\textwidth]{figs/sir2_xray_22}
\caption{Prompt muonic X-rays spectrum from the active silicon target.
}
\label{fig:sir2_xray}
\end{figure}
The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian
peak on top of a first-order polynomial from \SIrange{395}{405}{\keV}.
Using the same procedure of correcting described in
\cref{sub:germanium_detector}, and taking detector acceptance and X-ray
intensity into account (see \cref{tab:sir2_xray_corr}), the number of muon
stopped is:
\begin{equation}
N_{\mu \textrm{ stopped X-ray}} = (9.16 \pm 0.28)\times 10^6,
\label{eqn:n_stopped_xray_count}
\end{equation}
which is consistent with the number of X-rays counted using the active target.
\begin{table}[btp]
\begin{center}
\begin{tabular}{@{}llll@{}}
\toprule
\textbf{Measured X-rays} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule
Gross integral & 3083 & & \\
Background & 101.5 & & \\
Net area $(2p-1s)$ & 2929.7 & 56.4 & 0.02 \\
\vspace{0.03em}\\
\toprule
\textbf{Corrections} & \textbf{Value} & \multicolumn{2}{c}{\textbf{Details}}\\
\midrule
Random summing & 1.06 & \multicolumn{2}{l}{average count rate \SI{491.4}{\Hz},}\\
& & \multicolumn{2}{l}{pulse length \SI{57}{\us}}\\
TRP reset & 1.03 & \multicolumn{2}{l}{\SI{298}{\s} loss during \SI{9327}{\s} run period}\\
Self-absorption & 1.008 & \multicolumn{2}{l}{silicon thickness \SI{750}{\um},}\\
& & \multicolumn{2}{l}{linear attenuation \SI{0.224}{\per\cm}}\\
True coincidence & 1 & \multicolumn{2}{l}{omitted} \\
\vspace{0.03em}\\
\toprule
\textbf{Efficiency and intensity} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
\midrule
Detector efficiency & \num{4.40E-4} & \num{0.978E-5} & 0.02 \\
X-ray intensity & 0.803 & 0.008 & 0.009 \\
\vspace{0.03em}\\
\toprule
\textbf{Results} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
\midrule
Number of X-rays emitted & \num{7.36E6} & \num{0.22E6} & 0.03 \\
Number of muons stopped & \num{9.16E6} & \num{0.28E6} & 0.03 \\
\bottomrule
\end{tabular}
\end{center}
\caption{Corrections, efficiency and intensity used in calculating the number
of X-rays from the active target.}
\label{tab:sir2_xray_corr}
\end{table}
%In order to measure the charged particles after nuclear muon capture, one would
%pick events where the emitted particles are well separated from the
%muon stop time. The energy timing correlation plot suggests a timing window
%starting from at least 1200~ns, therefore another cut is introduced:
%\begin{enumerate}
%\setcounter{enumi}{2}
%\item there are at least two hits on the active target, the time
%difference between the second hit on target (decay or capture product) and
%the muon counter hit is at least 1300 ns:
%\begin{equation}
%t_{\textrm{target 2nd hit}} - t_{\mu\textrm{ counter}} \geq \SI{1300}{\ns}
%\label{eqn:sir2_2ndhit_cut}
%\end{equation}
%\end{enumerate}
%The three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and
%~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Particle identification by specific energy loss}
\label{sec:particle_identification_by_specific_energy_loss}
In this analysis, a subset of runs from \numrange{2808}{2873} with the
100-\si{\um} aluminium target is used because of following advantages:
\begin{itemize}
\item it was easier to stop and adjust the muon stopping distribution in
this thicker target;
\item a thicker target means more stopped muons due to higher muon rate
available at higher momentum and less scattering.
\end{itemize}
Muons momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of
1.09, normalised to \SI{28}{\MeV\per\cc}) were used for this target after
a momentum scanning as described in the next subsection.
\subsection{Momentum scan for the 100-\si{\um} aluminium target}
\label{sub:momentum_scan_for_the_100_}
Before deciding to use the momentum scaling factor of 1.09, we have scanned
with momentum scales ranging from 1.04 to 1.12 to maximise the
observed X-rays rate(and hence maximising the rate of stopped muons). The X-ray
spectrum at each momentum point was accumulated in more than 30 minutes to
assure a sufficient amount of counts. Details of the scanning runs are listed
in \cref{tab:al100_scan}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{cccc}
\toprule
\textbf{Momentum (\si{\MeV\per\cc})} & \textbf{Scaling factor} & \textbf{Runs}
& \textbf{Length (s)}\\
\midrule
29.12 & 1.04 & \numrange{2609}{2613} &2299\\
29.68 & 1.06 & \numrange{2602}{2608} &2563\\
29.96 & 1.07 & \numrange{2633}{2637} &2030\\
30.24 & 1.08 & \numrange{2614}{2621} &3232\\
30.52 & 1.09 & \numrange{2808}{2813} &2120\\
30.80 & 1.10 & \numrange{2625}{2632} &3234\\
31.36 & 1.12 & \numrange{2784}{2792} &2841\\
\bottomrule
\end{tabular}
\end{center}
\caption{Momentum scanning runs for the 100-\si{\um} aluminium target.}
\label{tab:al100_scan}
\end{table}
The on-site quick analysis suggested the 1.09 scaling factor was the
optimal value so it was chosen for all the runs on this aluminium target. But
the offline analysis later showed that the actual optimal factor was 1.08.
There were two reasons for the mistake:
\begin{enumerate}
\item the X-ray rates were normalised to run length, which is biased
since there are more muons available at higher momentum;
\item the $(2p-1s)$ peaks of aluminium at \SI{346.828}{\keV} were not
fitted properly. The peak is interfered by a background peak at
\SI{351}{\keV} from $^{214}$Pb, but the X-ray peak area was
obtained simply by subtracting an automatically estimated background.
\end{enumerate}
In the offline analysis, the X-ray peak and the background peak are fitted by
two Gaussian peaks on top of a first-order polynomial background. The X-ray peak
area is then normalised to the number of muons hitting the upstream detector
(\cref{fig:al100_xray_fit}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.47\textwidth]{figs/al100_xray_fit}
\includegraphics[width=0.47\textwidth]{figs/al100_xray_musc}
\caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium and the background
peak at \SI{351}{\keV} (left). The number of muons is integral of the
upstream scintillator spectrum (right) from \numrange{400}{2000} ADC
channels.}
\label{fig:al100_xray_fit}
\end{figure}
The ratio between the number of X-rays and the number of muons as a function
of momentum scaling factor is plotted on \cref{fig:al100_scan_rate}. The trend
showed that muons penetrated deeper as the momentum increased, reaching the
optimal value at the scale of 1.08, then decreased as punch through happened
more often from 1.09. The distributions of stopped muons are illustrated by
MC results on \cref{fig:al100_mu_stop_mc}. With the 1.09 scale beam, the muons
stopped \SI{28}{\um} off-centre to the right silicon arm.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_scan_rate}
\caption{Number of X-rays per incoming muon as a function of momentum
scaling factor.}
\label{fig:al100_scan_rate}
\end{figure}
\subsection{Event selection for the passive targets}
\label{sub:event_selection_for_the_passive_targets}
As described in the \cref{sec:analysis_framework}, the hits on all detectors
are re-organised into muon events: central muons; and all hits within
\SI{\pm 10}{\us} from the central muons. The dataset from runs
\numrange{2808}{2873} contains \num{1.17E+9} such muon events.
Selection of proton (and other heavy charged particles) events starts from
searching for muon event that has at least one hit on thick silicon. If there
is a thin silicon hit within a coincidence window of $\pm 0.5$~\si{\us}\ around
the thick silicon hit, the two hits are considered to belong to one particle.
The specific energy loss spectra recorded by the two silicon arms are plotted
on \cref{fig:al100_dedx}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_dedx}
\caption{Energy loss in thin silicon detectors as a function of total energy
recorded by both thin and thick detectors.}
\label{fig:al100_dedx}
\end{figure}
With the aid from MC study (\cref{fig:pid_sim}), the banding on
\cref{fig:al100_dedx} can be identified as follows:
\begin{itemize}
\item the densest spot at the lower left conner belonged to electron hits;
\item the small blurry cloud just above the electron region was muon hits;
\item the most intense band was due to proton hits;
\item the less intense, upper band caused by deuteron hits;
\item the highest band corresponded to alpha hits;
\item the faint stripe above the deuteron band should be triton
hits, which is consistent with a relatively low probability of emission of
tritons.
\end{itemize}
The band of protons is then extracted by cut on likelihood probability
calculated as:
\begin{equation}
p_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
e^{\frac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}}
\end{equation}
where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in the thin
silicon detector by a certain proton at energy $E_i$, $\Delta E_i$ and
$\sigma_{\Delta E}$ are the expected and standard deviation of the energy loss
caused by the proton calculated by MC. A cut value of $3\sigma_{\Delta E}$, or
$p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.47\textwidth]{figs/al100_protons}
\includegraphics[width=0.47\textwidth]{figs/al100_protons_px_r}
\caption{Protons (green) selected using the likelihood probability cut
(left). The proton spectrum (right) is obtained by projecting the proton
band onto the total energy axis.}
\label{fig:al100_protons}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proton emission rate from aluminium}
\label{sec:proton_emission_rate_from_aluminium}
The analysis is done on the same dataset used in
\cref{sec:particle_identification_by_specific_energy_loss}. Firstly, the
number of protons emitted is extracted using specific energy loss. Then
correction for energy loss inside the target is applied. Finally, the number
of protons is normalised to the number of nuclear muon captures.
\subsection{Number of protons emitted}
\label{sub:number_of_protons_emitted}
From the particle identification above, number of protons having energy in the
range from \SIrange{2.2}{8.5}{\MeV} hitting the two arms are:
\begin{align}
N_{\textrm{p meas. left}} = 1789 \pm 42.3\\
N_{\textrm{p meas. right}} = 2285 \pm 47.8
\end{align}
The right arm received significantly more protons than the left arm did, which
is expected because in \cref{sub:momentum_scan_for_the_100_} it is shown that
muons stopped off centre to the right arm.
The uncertainties are statistical only. The systematic uncertainties due to
the cut on protons is estimated to be small compared to the statistical ones.
\subsection{Corrections for the number of protons}
\label{sub:corrections_for_the_number_of_protons}
The protons spectra observed by the silicon detectors have been modified by
the energy loss inside the target so correction (or unfolding) is necessary.
In the unfolding process, a response function that relates proton's true energy
and the measured one is needed.
The response function is
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Charged particles following muon capture on a thick silicon target}
%\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
%Firstly, the number of charged particles emitted after nuclear muon capture on
%the active target is calculated. The charged particles yield then is normalised
%to the number of nuclear muon capture to obtain an emission rate.
%Finally, the
%rate is compared with that from the literature.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Number of charged particles with energy above \SI{3}{\MeV}}
%\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
%As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation
%in \cref{fig:sir2_mc_pdfs}, there are several components in
%the energy spectrum recorded by the active target:
%\begin{enumerate}
%\item charged particles from nuclear muon capture, this is the signal of
%interest;
%\item beam electrons with a characteristic Landau peak around \SI{800}{\keV},
%dominating at large delay (from \SI{6500}{\ns}), causing background at
%energy lower than \SI{1}{\MeV} which drops sharply at energy larger than
%\SI{3}{\MeV};
%\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei
%from neutron emitting muon captures, peak at
%around \SI{300}{\keV}, dominate the region where energy smaller than
%\SI{1}{\MeV} and delay less than \SI{3500}{\ns}. This component is
%intrinsic background, having the same time structure as that of the signal;
%\item stray muons scattered into the target, this component is small compares
%to the charged particles yield so it is not considered further.
%\end{enumerate}
%\begin{figure}[htb]
%\centering
%\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
%\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
%\caption{The observed spectrum in the timing window 1300 -- 10000~ns (left)
%and its components from MC simulation (right). The charged particles
%spectrum is obtained assuming the spectrum shape and emission rate from
%Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scales between
%components are arbitrarily chosen for the purpose of illustration.}
%\label{fig:sir2_mc_pdfs}
%\end{figure}
%An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the
%beam electrons at low energy. In order to obtain the yields of backgrounds
%above \SI{2}{\MeV}, a binned maximum likelihood fit was done. The likelihood of
%getting the measured spectrum is defined as:
%\begin{equation}
%L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
%\label{eqn:llh_def}
%\end{equation}
%where $n$ is the total number of events observed, $\mu$ is the expected number
%of events according to some linear combination of the signal and the
%backgrounds shown in~\ref{fig:sir2_mc_pdfs}, namely:
%\begin{align}
%n &= n_{\textrm{sig}} + n_{\textrm{beam e}} + n_{\textrm{dio}}\\
%\textrm{(sum pdf)} &= n_{\textrm{sig}}\times\textrm{(sig pdf)} +
%n_{\textrm{beam e}}\times\textrm{(beam e pdf)} +
%n_{\textrm{dio}}\times\textrm{(dio pdf)};
%\label{eqn:sum_pdf}
%\end{align}
%and the $i$ index indicates the respective number of events in the $i$-th
%bin.
%The fit is done by the RooFit package~\cite{VerkerkeKirkby.2003} where the
%negative log likelihood $-2\ln{L}$ is minimised. Fitting results are shown
%in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
%\begin{align}
%n_{\textrm{beam e}} &= 23756 \pm 581\\
%n_{\textrm{dio}} &= 111340 \pm 1245\\
%n_{\textrm{sig}} &= 2.57 \pm 856
%\label{eqn:sir2_n_chargedparticles}
%\end{align}
%\begin{figure}[htb]
%\centering
%\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nbkg}
%\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nebeam}
%\includegraphics[width=0.84\textwidth]{figs/sir2_mllfit}
%\caption{Results of the maximum likelihood fit of the energy spectrum on the
%active target.}
%\label{fig:sir2_mll_fit}
%\end{figure}
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Number of nuclear muon captures}
%\label{sub:number_of_stopped_muons}
%The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5 \pm 65.6$,
%obtained by subtracting the background of 101.5 from the spectral integral of
%2083 in the region from 396 to 402 keV.
%The area of the $(2p-1s)$ peak is $2929.7 \pm 56.4$ obtained by fitting
%a Gaussian peak on top of a first-order polynomial background to the spectrum
%in \cref{fgi:sir2_xray} in the region from \SIrange{395}{405}{\keV}.
Using the same procedure of fitting and correcting described in
\cref{sub:germanium_detector}, the number of X-rays is calculated to be 370.
Details of the correction factors are given in \cref{tab:sir2_xray_corr}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l}
\toprule
\textbf{Col1}\\
\midrule
item1\\
\bottomrule
\end{tabular}
\end{center}
\caption{Corrections for the number of X-rays from the active target.}
\label{tab:sir2_xray_corr}
\end{table}
The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of
stopped muons, so the number of stopped muons is:
%The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of
%stopped muons, so the number of stopped muons is:
\begin{align}
N_{\mu\textrm{ stopped}} &=
\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\
&= 1.05\times10^6 \nonumber
\end{align}
where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
detector for the 400.177~keV line in \cref{tab:xray_eff}, and
$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
(80.3\% from \cref{tab:mucap_pars}).
%\begin{align}
%N_{\mu\textrm{ stopped}} &=
%\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
%&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\
%&= 1.05\times10^6 \nonumber
%\end{align}
%where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
%detector for the 400.177~keV line in \cref{tab:xray_eff}, and
%$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
%(80.3\% from \cref{tab:mucap_pars}).
Taking the statistical uncertainty of the peak area, and systematic
uncertainties from parameters of muon capture, the number of stopped muons
calculated from the X-ray measurement is
$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of
stopped muons of $9.82\times10^5$ after the cuts described in the event
selection process.
%Taking the statistical uncertainty of the peak area, and systematic
%uncertainties from parameters of muon capture, the number of stopped muons
%calculated from the X-ray measurement is
%$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of
%stopped muons of $9.82\times10^5$ after the cuts described in the event
%selection process.
The number of nuclear captured muons is:
\begin{equation}
N_{\mu\textrm{ nucl.capture}} =
N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5
\label{eqn:sir2_Ncapture}
\end{equation}
where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
stopped muon from \cref{tab:mucap_pars}.
%The number of nuclear captured muons is:
%\begin{equation}
%N_{\mu\textrm{ nucl.capture}} =
%N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
%= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5
%\label{eqn:sir2_Ncapture}
%\end{equation}
%where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
%stopped muon from \cref{tab:mucap_pars}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Emission rate of charged particles}
\label{sub:emission_rate_of_charged_particles}
The emission rate of charged particles is calculated by taking the ratio of
number of charged particles in ~\eqref{eqn:sir2_Nchargedparticle} and number of
nuclear muon capture in~\eqref{eqn:sir2_Ncapture}:
\begin{equation}
R_{\textrm{Si}} = \frac{N_{\textrm{charged particle}}}{N_{\mu\textrm{ nucl.capture}}}
= \frac{149.9\times10^4}{7.25\times10^6} = 0.252
\end{equation}
Uncertainties of this rate calculation are listed in
\cref{tab:sir2_uncertainties}, including:
\begin{itemize}
\item uncertainties from number of charged particles, both statistical and
systematic (from spectrum shape and fitting) ones are absorbed in the
quoted value in~\eqref{sir2_Nchargedparticle};
\item uncertainties from number of nuclear capture:
\begin{itemize}
\item statistical error of the peak area calculation,
\item systematic errors from the efficiency calibration, and referenced
values of X-ray intensity and capture probability.
\end{itemize}
\end{itemize}
So, the emission rate is:
\begin{equation}
R_{\textrm{Si}} = 0.252 \pm 0.009
\label{eqn:sir2_rate_cal}
\end{equation}
%\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\\
%\begin{table}[htb]
%\begin{center}
%\begin{tabular}{l l l}
%\toprule
%Number of charged particles & &\\
%Statistical and systematic & &0.004\\
%\midrule
%Number of nuclear capture & &\\
%Statistical & Peak area calculation& 0.022\\
%Systematic & Efficiency calibration & 0.024\\
%& X-ray intensity & 0.009\\
%& Capture probability & 0\\
\midrule
Total relative error & & 0.035\\
Total absolute error & & 0.009\\
%\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}
%\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)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%