1121 lines
54 KiB
TeX
1121 lines
54 KiB
TeX
\chapter{Data analysis}
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\label{cha:data_analysis}
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This chapter presents initial analysis on subsets of the collected data.
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Purposes of the analysis include:
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\begin{itemize}
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\item testing the analysis chain;
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\item verification of the experimental method, specifically the
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normalisation of number of stopped muons, and particle identification
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using specific energy loss;
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\item extracting a preliminary rate of proton emission from aluminium.
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\end{itemize}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Number of stopped muons normalisation}
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\label{sec:number_of_stopped_muons_normalisation}
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The active silicon target runs was used to check for the validity of the
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counting of number of stopped muons, where the number can be calculated by two
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methods:
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\begin{itemize}
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\item counting hits on the active target in coincidence with hits on the
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upstream scintillator counter;
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\item inferred from number of X-rays recorded by the germanium detector.
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\end{itemize}
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This analysis was done on a subset of the active target runs
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\numrange{2119}{2140} because of the problem of wrong clock frequency found in
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the data quality checking shown in \cref{fig:lldq}. The data set contains
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%\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events.
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\num{6.43E7} muon events.
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\subsection{Number of stopped muons from active target counting}
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\label{sub:event_selection}
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Because of the active target, a stopped muon would cause two coincident hits on
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the muon counter and the target. The energy of the muon hit on the active
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target is also well-defined as the narrow-momentum-spread beam was used. The
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correlation between the energy and timing of all the hits on the active target
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is shown in \cref{fig:sir2f_Et_corr}. The most intense spot at zero time
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and about \SI{5}{\MeV} energy corresponds to stopped muons in the thick target.
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The band below \SI{1}{\MeV} is due to electrons, either in the beam or from
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muon decay in orbits, or emitted during the cascading of muon to the muonic 1S
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state. The valley between time zero and 1200~ns shows the minimum distance in
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time between two pulses. It is the mentioned limitation of the current pulse
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parameter extraction method where no pile up or double pulses is accounted for.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_E_t_corr}
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\caption{Energy - timing correlation of hits on the active target.}
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\label{fig:sir2f_Et_corr}
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\end{figure}
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The hits on the silicon active target after 1200~ns are mainly secondary
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particles from the stopped muons:
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\begin{itemize}
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\item electrons from muon decay in the 1S orbit
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\item products emitted after nuclear muon capture, including: gamma, neutron,
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heavy charged particles and recoiled nucleus
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\end{itemize}
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It can be seen that there is a faint stripe of muons in the time larger than
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1200~ns region, they are scattered muons by other materials without hitting the
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muon counter. The electrons in the beam caused the constant band below 1 MeV and
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$t > 5000$ ns (see \cref{fig:sir2_1us_slices}).
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_amp_1us_slices}
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\caption{Energy deposit on the active target in 1000 ns time slices from the
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muon hit. The peaks at about 800 keV in large delayed time are from
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the beam electrons.}
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\label{fig:sir2_1us_slices}
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\end{figure}
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From the energy-timing correlation above, the cuts to select stopped muons are:
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\begin{enumerate}
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\item has one hit on muon counter (where a threshold was set to reject MIPs),
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and the first hit on the silicon active target is in coincidence with that
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muon counter hit:
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\begin{equation}
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\lvert t_{\textrm{target}} - t_{\mu\textrm{ counter}}\rvert \le \SI{50}{\ns}
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\label{eqn:sir2_prompt_cut}
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\end{equation}
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\item the first hit on the target has energy of that of the muons:
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\begin{equation}
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\SI{3.4}{\MeV} \le E_{\textrm{target}} \le \SI{5.6}{\MeV}
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\label{eqn:sir2_muE_cut}
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\end{equation}
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\end{enumerate}
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The two cuts~\eqref{eqn:sir2_prompt_cut} and~\eqref{eqn:sir2_muE_cut} give
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a number of stopped muons counted by the active target:
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\begin{equation}
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N_{\mu \textrm{ active Si}} = 9.32 \times 10^6
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\label{eqn:n_stopped_si_count}
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\end{equation}
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\subsection{Number of stopped muons from the number of X-rays}
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\label{sub:number_of_stopped_muons_from_the_number_of_x_rays}
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The number of nuclear captures, hence the number of stopped muons in the
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active silicon target, can be inferred from the number of emitted
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muonic X-rays. The reference energies and intensities of the most prominent
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lines of silicon and aluminium are listed in \cref{tab:mucap_pars}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{l l l}
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\toprule
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\textbf{Quantity} & \textbf{Aluminium} & \textbf{Silicon}\\
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\midrule
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Muonic mean lifetime (ns) & $864 \pm 2$ & $758 \pm 2$\\
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Nuclear capture probability (\%) & $60.9 $ & $65.8$\\
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$(2p-1s)$ X-ray energy (keV) & $346.828\pm0.002$ & $400.177\pm0.005$\\
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Intensity (\%) & $79.8\pm0.8$ & $80.3\pm0.8$\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Reference parameters of muon capture in aluminium and silicon taken
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from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday et
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al.~\cite{MeasdayStocki.etal.2007}.}
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\label{tab:mucap_pars}
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\end{table}
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The muonic X-rays are emitted during the cascading of the muon to the muonic 1S
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state in the time scale of \SI{E-9}{\s}, so the hit caused by the X-rays must
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be in coincidence with the muon hit on the active target. Therefore an
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additional timing cut is applied for the germanium detector hits:
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\begin{equation}
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\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns}
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\label{eqn:sir2_ge_cut}
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\end{equation}
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The germanium spectrum after three
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cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut}
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and~\eqref{eqn:sir2_ge_cut} is plotted in \cref{fig:sir2_xray}. The $(2p-1s)$
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line clearly showed up at \SI{400}{\keV} with very low background. A peak at
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\SI{476}{\keV} is identified as the $(3p-1s)$ transition. Higher transitions
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such as $(4p-1s)$, $(5p-1s)$ and $(6p-1s)$ can also be recognised at
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\SI{504}{\keV}, \SI{516}{\keV} and \SI{523}{\keV}, respectively.
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%The $(2p-1s)$
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%line is seen at 399.5~\si{\keV}, 0.7~\si{\keV} off from the reference value of
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%400.177~\si{\keV}. This discrepancy is within our detector's resolution,
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%and the calculated efficiency is $(4.549 \pm 0.108)\times 10^{-5}$ -- a 0.15\%
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%increasing from that of the 400.177~keV line, so no attempt for recalibration
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%or correction was made.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/sir2_xray_22}
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\caption{Prompt muonic X-rays spectrum from the active silicon target.
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}
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\label{fig:sir2_xray}
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\end{figure}
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The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian
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peak on top of a first-order polynomial from \SIrange{395}{405}{\keV}.
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Using the same procedure of correcting described in
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\cref{sub:germanium_detector}, and taking detector acceptance and X-ray
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intensity into account (see \cref{tab:sir2_xray_corr}), the number of muon
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stopped is:
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\begin{equation}
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N_{\mu \textrm{ stopped X-ray}} = (9.16 \pm 0.28)\times 10^6,
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\label{eqn:n_stopped_xray_count}
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\end{equation}
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which is consistent with the number of X-rays counted using the active target.
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\begin{table}[btp]
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\begin{center}
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\begin{tabular}{@{}llll@{}}
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\toprule
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\textbf{Measured X-rays} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule
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Gross integral & 3083 & & \\
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Background & 101.5 & & \\
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Net area $(2p-1s)$ & 2929.7 & 56.4 & 0.02 \\
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\vspace{0.03em}\\
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\toprule
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\textbf{Corrections} & \textbf{Value} & \multicolumn{2}{c}{\textbf{Details}}\\
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\midrule
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Random summing & 1.06 & \multicolumn{2}{l}{average count rate \SI{491.4}{\Hz},}\\
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& & \multicolumn{2}{l}{pulse length \SI{57}{\us}}\\
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TRP reset & 1.03 & \multicolumn{2}{l}{\SI{298}{\s} loss during \SI{9327}{\s} run period}\\
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Self-absorption & 1.008 & \multicolumn{2}{l}{silicon thickness \SI{750}{\um},}\\
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& & \multicolumn{2}{l}{linear attenuation \SI{0.224}{\per\cm}}\\
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True coincidence & 1 & \multicolumn{2}{l}{omitted} \\
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\vspace{0.03em}\\
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\toprule
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\textbf{Efficiency and intensity} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
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\midrule
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Detector efficiency & \num{4.40E-4} & \num{0.978E-5} & 0.02 \\
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X-ray intensity & 0.803 & 0.008 & 0.009 \\
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\vspace{0.03em}\\
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\toprule
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\textbf{Results} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
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\midrule
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Number of X-rays emitted & \num{7.36E6} & \num{0.22E6} & 0.03 \\
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Number of muons stopped & \num{9.16E6} & \num{0.28E6} & 0.03 \\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Corrections, efficiency and intensity used in calculating the number
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of X-rays from the active target.}
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\label{tab:sir2_xray_corr}
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\end{table}
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%In order to measure the charged particles after nuclear muon capture, one would
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%pick events where the emitted particles are well separated from the
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%muon stop time. The energy timing correlation plot suggests a timing window
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%starting from at least 1200~ns, therefore another cut is introduced:
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%\begin{enumerate}
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%\setcounter{enumi}{2}
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%\item there are at least two hits on the active target, the time
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%difference between the second hit on target (decay or capture product) and
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%the muon counter hit is at least 1300 ns:
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%\begin{equation}
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%t_{\textrm{target 2nd hit}} - t_{\mu\textrm{ counter}} \geq \SI{1300}{\ns}
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%\label{eqn:sir2_2ndhit_cut}
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%\end{equation}
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%\end{enumerate}
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%The three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and
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%~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Particle identification by specific energy loss}
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\label{sec:particle_identification_by_specific_energy_loss}
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In this analysis, a subset of runs from \numrange{2808}{2873} with the
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100-\si{\um} aluminium target is used because of following advantages:
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\begin{itemize}
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\item it was easier to stop and adjust the muon stopping distribution in
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this thicker target;
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\item a thicker target means more stopped muons due to higher muon rate
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available at higher momentum and less scattering.
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\end{itemize}
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Muons momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of
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1.09, normalised to \SI{28}{\MeV\per\cc}) were used for this target after
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a momentum scanning as described in the next subsection.
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\subsection{Momentum scan for the 100-\si{\um} aluminium target}
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\label{sub:momentum_scan_for_the_100_}
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Before deciding to use the momentum scaling factor of 1.09, we have scanned
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with momentum scales ranging from 1.04 to 1.12 to maximise the
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observed X-rays rate(and hence maximising the rate of stopped muons). The X-ray
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spectrum at each momentum point was accumulated in more than 30 minutes to
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assure a sufficient amount of counts. Details of the scanning runs are listed
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in \cref{tab:al100_scan}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{cccc}
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\toprule
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\textbf{Momentum (\si{\MeV\per\cc})} & \textbf{Scaling factor} & \textbf{Runs}
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& \textbf{Length (s)}\\
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\midrule
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29.12 & 1.04 & \numrange{2609}{2613} &2299\\
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29.68 & 1.06 & \numrange{2602}{2608} &2563\\
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29.96 & 1.07 & \numrange{2633}{2637} &2030\\
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30.24 & 1.08 & \numrange{2614}{2621} &3232\\
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30.52 & 1.09 & \numrange{2808}{2813} &2120\\
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30.80 & 1.10 & \numrange{2625}{2632} &3234\\
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31.36 & 1.12 & \numrange{2784}{2792} &2841\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Momentum scanning runs for the 100-\si{\um} aluminium target.}
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\label{tab:al100_scan}
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\end{table}
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The on-site quick analysis suggested the 1.09 scaling factor was the
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optimal value so it was chosen for all the runs on this aluminium target. But
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the offline analysis later showed that the actual optimal factor was 1.08.
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There were two reasons for the mistake:
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\begin{enumerate}
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\item the X-ray rates were normalised to run length, which is biased
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since there are more muons available at higher momentum;
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\item the $(2p-1s)$ peaks of aluminium at \SI{346.828}{\keV} were not
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fitted properly. The peak is interfered by a background peak at
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\SI{351}{\keV} from $^{214}$Pb, but the X-ray peak area was
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obtained simply by subtracting an automatically estimated background.
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\end{enumerate}
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In the offline analysis, the X-ray peak and the background peak are fitted by
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two Gaussian peaks on top of a first-order polynomial background. The X-ray peak
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area is then normalised to the number of muons hitting the upstream detector
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(\cref{fig:al100_xray_fit}).
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/al100_xray_fit}
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\includegraphics[width=0.47\textwidth]{figs/al100_xray_musc}
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\caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium and the background
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peak at \SI{351}{\keV} (left). The number of muons is integral of the
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upstream scintillator spectrum (right) from \numrange{400}{2000} ADC
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channels.}
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\label{fig:al100_xray_fit}
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\end{figure}
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The ratio between the number of X-rays and the number of muons as a function
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of momentum scaling factor is plotted on \cref{fig:al100_scan_rate}. The trend
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showed that muons penetrated deeper as the momentum increased, reaching the
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optimal value at the scale of 1.08, then decreased as punch through happened
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more often from 1.09. The distributions of stopped muons are illustrated by
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MC results on \cref{fig:al100_mu_stop_mc}. With the 1.09 scale beam, the muons
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stopped \SI{28}{\um} off-centre to the right silicon arm.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/al100_scan_rate}
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\caption{Number of X-rays per incoming muon as a function of momentum
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scaling factor.}
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\label{fig:al100_scan_rate}
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\end{figure}
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\subsection{Event selection for the passive targets}
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\label{sub:event_selection_for_the_passive_targets}
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As described in the \cref{sec:analysis_framework}, the hits on all detectors
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are re-organised into muon events: central muons; and all hits within
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\SI{\pm 10}{\us} from the central muons. The dataset from runs
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\numrange{2808}{2873} contains \num{1.17E+9} such muon events.
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Selection of proton (and other heavy charged particles) events starts from
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searching for muon event that has at least one hit on thick silicon. If there
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is a thin silicon hit within a coincidence window of $\pm 0.5$~\si{\us}\ around
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the thick silicon hit, the two hits are considered to belong to one particle.
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The specific energy loss spectra recorded by the two silicon arms are plotted
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on \cref{fig:al100_dedx}.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/al100_dedx}
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\caption{Energy loss in thin silicon detectors as a function of total energy
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recorded by both thin and thick detectors.}
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\label{fig:al100_dedx}
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\end{figure}
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With the aid from MC study (\cref{fig:pid_sim}), the banding on
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\cref{fig:al100_dedx} can be identified as follows:
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\begin{itemize}
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\item the densest spot at the lower left conner belonged to electron hits;
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\item the small blurry cloud just above the electron region was muon hits;
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\item the most intense band was due to proton hits;
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\item the less intense, upper band caused by deuteron hits;
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\item the highest band corresponded to alpha hits;
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\item the faint stripe above the deuteron band should be triton
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hits, which is consistent with a relatively low probability of emission of
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tritons.
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\end{itemize}
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The band of protons is then extracted by cut on likelihood probability
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calculated as:
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\begin{equation}
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p_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
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e^{\frac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}}
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\end{equation}
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where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in the thin
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silicon detector by a certain proton at energy $E_i$, $\Delta E_i$ and
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$\sigma_{\Delta E}$ are the expected and standard deviation of the energy loss
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caused by the proton calculated by MC. A cut value of $3\sigma_{\Delta E}$, or
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$p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}).
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/al100_protons}
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\includegraphics[width=0.47\textwidth]{figs/al100_protons_px_r}
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\caption{Protons (green) selected using the likelihood probability cut
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(left). The proton spectrum (right) is obtained by projecting the proton
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band onto the total energy axis.}
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\label{fig:al100_protons}
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\end{figure}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Proton emission rate from aluminium}
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\label{sec:proton_emission_rate_from_aluminium}
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The analysis is done on the same dataset used in
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\cref{sec:particle_identification_by_specific_energy_loss}. Firstly, the
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number of protons emitted is extracted using specific energy loss. Then
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correction for energy loss inside the target is applied. Finally, the number
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of protons is normalised to the number of nuclear muon captures.
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\subsection{Number of protons emitted}
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\label{sub:number_of_protons_emitted}
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From the particle identification above, number of protons having energy in the
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range from \SIrange{2.2}{8.5}{\MeV} hitting the two arms are:
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\begin{align}
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N_{\textrm{p meas. left}} = 1789 \pm 42.3\\
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N_{\textrm{p meas. right}} = 2285 \pm 47.8
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\end{align}
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The right arm received significantly more protons than the left arm did, which
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is expected because in \cref{sub:momentum_scan_for_the_100_} it is shown that
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muons stopped off centre to the right arm.
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The uncertainties are statistical only. The systematic uncertainties due to
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the cut on protons is estimated to be small compared to the statistical ones.
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\subsection{Corrections for the number of protons}
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\label{sub:corrections_for_the_number_of_protons}
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The protons spectra observed by the silicon detectors have been modified by
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the energy loss inside the target so correction (or unfolding) is necessary.
|
|
In the unfolding process, a response function that relates proton's true energy
|
|
and the measured one is needed.
|
|
|
|
The response function is
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\section{Charged particles following muon capture on a thick silicon target}
|
|
%\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
|
|
|
|
%Firstly, the number of charged particles emitted after nuclear muon capture on
|
|
%the active target is calculated. The charged particles yield then is normalised
|
|
%to the number of nuclear muon capture to obtain an emission rate.
|
|
%Finally, the
|
|
%rate is compared with that from the literature.
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\subsection{Number of charged particles with energy above \SI{3}{\MeV}}
|
|
%\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
|
|
%As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation
|
|
%in \cref{fig:sir2_mc_pdfs}, there are several components in
|
|
%the energy spectrum recorded by the active target:
|
|
%\begin{enumerate}
|
|
%\item charged particles from nuclear muon capture, this is the signal of
|
|
%interest;
|
|
%\item beam electrons with a characteristic Landau peak around \SI{800}{\keV},
|
|
%dominating at large delay (from \SI{6500}{\ns}), causing background at
|
|
%energy lower than \SI{1}{\MeV} which drops sharply at energy larger than
|
|
%\SI{3}{\MeV};
|
|
%\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei
|
|
%from neutron emitting muon captures, peak at
|
|
%around \SI{300}{\keV}, dominate the region where energy smaller than
|
|
%\SI{1}{\MeV} and delay less than \SI{3500}{\ns}. This component is
|
|
%intrinsic background, having the same time structure as that of the signal;
|
|
%\item stray muons scattered into the target, this component is small compares
|
|
%to the charged particles yield so it is not considered further.
|
|
%\end{enumerate}
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
|
|
%\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
|
|
%\caption{The observed spectrum in the timing window 1300 -- 10000~ns (left)
|
|
%and its components from MC simulation (right). The charged particles
|
|
%spectrum is obtained assuming the spectrum shape and emission rate from
|
|
%Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scales between
|
|
%components are arbitrarily chosen for the purpose of illustration.}
|
|
%\label{fig:sir2_mc_pdfs}
|
|
%\end{figure}
|
|
|
|
%An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the
|
|
%beam electrons at low energy. In order to obtain the yields of backgrounds
|
|
%above \SI{2}{\MeV}, a binned maximum likelihood fit was done. The likelihood of
|
|
%getting the measured spectrum is defined as:
|
|
%\begin{equation}
|
|
%L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
|
|
%\label{eqn:llh_def}
|
|
%\end{equation}
|
|
%where $n$ is the total number of events observed, $\mu$ is the expected number
|
|
%of events according to some linear combination of the signal and the
|
|
%backgrounds shown in~\ref{fig:sir2_mc_pdfs}, namely:
|
|
%\begin{align}
|
|
%n &= n_{\textrm{sig}} + n_{\textrm{beam e}} + n_{\textrm{dio}}\\
|
|
%\textrm{(sum pdf)} &= n_{\textrm{sig}}\times\textrm{(sig pdf)} +
|
|
%n_{\textrm{beam e}}\times\textrm{(beam e pdf)} +
|
|
%n_{\textrm{dio}}\times\textrm{(dio pdf)};
|
|
%\label{eqn:sum_pdf}
|
|
%\end{align}
|
|
%and the $i$ index indicates the respective number of events in the $i$-th
|
|
%bin.
|
|
|
|
%The fit is done by the RooFit package~\cite{VerkerkeKirkby.2003} where the
|
|
%negative log likelihood $-2\ln{L}$ is minimised. Fitting results are shown
|
|
%in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
|
|
%\begin{align}
|
|
%n_{\textrm{beam e}} &= 23756 \pm 581\\
|
|
%n_{\textrm{dio}} &= 111340 \pm 1245\\
|
|
%n_{\textrm{sig}} &= 2.57 \pm 856
|
|
%\label{eqn:sir2_n_chargedparticles}
|
|
%\end{align}
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nbkg}
|
|
%\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nebeam}
|
|
%\includegraphics[width=0.84\textwidth]{figs/sir2_mllfit}
|
|
%\caption{Results of the maximum likelihood fit of the energy spectrum on the
|
|
%active target.}
|
|
%\label{fig:sir2_mll_fit}
|
|
%\end{figure}
|
|
|
|
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\subsection{Number of nuclear muon captures}
|
|
%\label{sub:number_of_stopped_muons}
|
|
|
|
%The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5 \pm 65.6$,
|
|
%obtained by subtracting the background of 101.5 from the spectral integral of
|
|
%2083 in the region from 396 to 402 keV.
|
|
%The area of the $(2p-1s)$ peak is $2929.7 \pm 56.4$ obtained by fitting
|
|
%a Gaussian peak on top of a first-order polynomial background to the spectrum
|
|
%in \cref{fgi:sir2_xray} in the region from \SIrange{395}{405}{\keV}.
|
|
|
|
%The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of
|
|
%stopped muons, so the number of stopped muons is:
|
|
|
|
%\begin{align}
|
|
%N_{\mu\textrm{ stopped}} &=
|
|
%\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
|
|
%&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\
|
|
%&= 1.05\times10^6 \nonumber
|
|
%\end{align}
|
|
%where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
|
|
%detector for the 400.177~keV line in \cref{tab:xray_eff}, and
|
|
%$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
|
|
%(80.3\% from \cref{tab:mucap_pars}).
|
|
|
|
%Taking the statistical uncertainty of the peak area, and systematic
|
|
%uncertainties from parameters of muon capture, the number of stopped muons
|
|
%calculated from the X-ray measurement is
|
|
%$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of
|
|
%stopped muons of $9.82\times10^5$ after the cuts described in the event
|
|
%selection process.
|
|
|
|
%The number of nuclear captured muons is:
|
|
%\begin{equation}
|
|
%N_{\mu\textrm{ nucl.capture}} =
|
|
%N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
|
|
%= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5
|
|
%\label{eqn:sir2_Ncapture}
|
|
%\end{equation}
|
|
%where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
|
|
%stopped muon from \cref{tab:mucap_pars}.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\subsection{Emission rate of charged particles}
|
|
%\label{sub:emission_rate_of_charged_particles}
|
|
%The emission rate of charged particles is calculated by taking the ratio of
|
|
%number of charged particles in ~\eqref{eqn:sir2_Nchargedparticle} and number of
|
|
%nuclear muon capture in~\eqref{eqn:sir2_Ncapture}:
|
|
%\begin{equation}
|
|
%R_{\textrm{Si}} = \frac{N_{\textrm{charged particle}}}{N_{\mu\textrm{ nucl.capture}}}
|
|
%= \frac{149.9\times10^4}{7.25\times10^6} = 0.252
|
|
%\end{equation}
|
|
%Uncertainties of this rate calculation are listed in
|
|
%\cref{tab:sir2_uncertainties}, including:
|
|
%\begin{itemize}
|
|
%\item uncertainties from number of charged particles, both statistical and
|
|
%systematic (from spectrum shape and fitting) ones are absorbed in the
|
|
%quoted value in~\eqref{sir2_Nchargedparticle};
|
|
%\item uncertainties from number of nuclear capture:
|
|
%\begin{itemize}
|
|
%\item statistical error of the peak area calculation,
|
|
%\item systematic errors from the efficiency calibration, and referenced
|
|
%values of X-ray intensity and capture probability.
|
|
%\end{itemize}
|
|
%\end{itemize}
|
|
%So, the emission rate is:
|
|
%\begin{equation}
|
|
%R_{\textrm{Si}} = 0.252 \pm 0.009
|
|
%\label{eqn:sir2_rate_cal}
|
|
%\end{equation}
|
|
|
|
%\begin{table}[htb]
|
|
%\begin{center}
|
|
%\begin{tabular}{l l l}
|
|
%\toprule
|
|
%Number of charged particles & &\\
|
|
%Statistical and systematic & &0.004\\
|
|
%\midrule
|
|
%Number of nuclear capture & &\\
|
|
%Statistical & Peak area calculation& 0.022\\
|
|
%Systematic & Efficiency calibration & 0.024\\
|
|
%& X-ray intensity & 0.009\\
|
|
%& Capture probability & 0\\
|
|
|
|
%\midrule
|
|
%Total relative error & & 0.035\\
|
|
%Total absolute error & & 0.009\\
|
|
|
|
%\bottomrule
|
|
%\end{tabular}
|
|
%\end{center}
|
|
%\caption{Uncertainties of the emission rate from the thick silicon target}
|
|
%\label{tab:sir2_uncertainties}
|
|
%\end{table}
|
|
|
|
% subsection partial_emission_rate_of_charged_particle_in_8_10_mev_range (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%TODO
|
|
%\subsection{Partial emission rate of charged particles from the literature}
|
|
%\label{sub:partial_emission_rate_of_charged_particles_from_the_literature}
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/sobottka_spec2}
|
|
%\caption{Reproduced charged particle spectrum from muon capture on silicon,
|
|
%measured by Sobottka and Wills. Integration region is shown in the green
|
|
%box.}
|
|
%\label{fig:sobottka_spec}
|
|
%\end{figure}
|
|
%The spectrum measured by Sobottka and Wills~\cite{SobottkaWills.1968} is
|
|
%reproduced in \cref{fig:sobottka_spec}, the spectral integral in the
|
|
%energy region from 8 to 10~\si{\MeV}\ is $2086.8 \pm 45.7$.
|
|
%The authors obtained the spectrum in a 4~\si{\us}\ gate period which began
|
|
%1~\si{\us}\ after a muon stopped, which would take 26.59\% of the emitted
|
|
%particles into account. The number of stopped muons was not explicitly stated,
|
|
%but can be inferred to be $55715/0.06 = 92858.3$.
|
|
|
|
%The partial rate of charged particle from 8 to 10~\si{\MeV}\ is then
|
|
%calculated to be:
|
|
%\begin{equation}
|
|
%R_{\textrm{8-10 MeV}}^{lit.} =
|
|
%\dfrac{2086.8}{0.2659 \times 92858.3 \times 0.658}
|
|
%= 1.28 \times 10^{-2}
|
|
%\end{equation}
|
|
%The authors did not mentioned how the uncertainties of their measurement was
|
|
%derived, but quoted the emission rate below 26~\si{\MeV}\ to be $0.15
|
|
%\pm 0.02$, which translates to a relative uncertainty of 0.133. The statistical
|
|
%uncertainty from the spectral integral and the number of stopped muons is:
|
|
%\begin{equation*}
|
|
%\dfrac{1}{\sqrt{25000}} + \dfrac{1}{\sqrt{92858.3}} = 0.9 \times 10^{-2}
|
|
%\end{equation*}
|
|
%Then their systematic uncertainty would be: $0.133 - 0.009 = 0.124$.
|
|
|
|
%For the partial spectrum from 8 to 10~\si{\MeV}, the statistical
|
|
%contribution to the uncertainty is:
|
|
%\begin{equation*}
|
|
%\dfrac{1}{\sqrt{2086.8}} + \dfrac{1}{\sqrt{92858.3}} = 2.5 \times 10^{-2}
|
|
%\end{equation*}
|
|
%So, the combined uncertainty of this partial rate calculation is: $0.124
|
|
%+ 0.025 = 0.150$. The partial rate of charged particles from 8 to
|
|
%10~\si{\MeV} per muon capture is:
|
|
%\begin{equation}
|
|
%R_{\textrm{8-10 MeV}}^{lit.} = (1.28 \pm 0.19) \times 10^{-2}
|
|
%\end{equation}
|
|
% subsection partial_emission_rate_of_charged_particles_from_the_literature
|
|
% (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Charged particles following muon capture on a thin silicon target}
|
|
\label{sec:charged_particles_following_muon_capture_on_a_thin_silicon_target}
|
|
In this measurement, a passive, 62-\si{\um}-thick silicon target was used as the
|
|
target. The silicon target is $5\times5$~\si{\centi\meter}$^2$ in area. The muon
|
|
momentum was chosen to be 1.06 after a scanning to maximise the stopping ratio.
|
|
The charged particles were measured by two arms of silicon detectors. The
|
|
plastic scintillators vetoing information were not used.
|
|
|
|
This data set consists of 66 runs, from 3474--3489 and 3491--3540.
|
|
Although there are a few issues found in the process of data quality
|
|
checking such as one very noisy timing channel, and several runs had
|
|
abnormally high rates, the whole data set is determined to be good. Without
|
|
an active target and veto, the muon signal is from the muon counter only. The
|
|
tree contains total $1.452 \times 10^8$ muon events. %145212698
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.49\textwidth]{figs/si16_lldq_noise}
|
|
\includegraphics[width=0.49\textwidth]{figs/si16_lldq_islandrate}
|
|
\caption{Oddities found in checking data quality: noise level on timing
|
|
output of the SiL1-2 was much higher than the other detectors, and some
|
|
runs show large pulse rate.}
|
|
\label{fig:si16_lldq}
|
|
\end{figure}
|
|
|
|
\subsection{Particle identification by dE/dx and proton selection}
|
|
\label{sub:particle_identification_by_de_dx}
|
|
%All silicon hits with energy deposition larger than
|
|
%200~\si{\keV}\ that happened within $\pm 10$~\si{\us}\ of the
|
|
%muon hit are then
|
|
%associated to the muon and stored in the muon event tree. The
|
|
%200~\si{\keV}\ cut effectively rejects all MIPs hits on thin silicon
|
|
%detectors of which the most probable value is about 40~\si{\keV}.
|
|
|
|
%In order to use dE/dx for particle identification, $\Delta$E and total E are
|
|
%needed.
|
|
The charged particle selection starts from searching for muon event
|
|
that has at least one hit on thick silicon. If there is a thin silicon hit
|
|
within a coincidence window of $\pm 0.5$~\si{\us}\ around the thick
|
|
silicon hit, the two hits are considered to belong to one particle with
|
|
$\Delta$E being the energy of the thin hit, and total E being the sum energy of
|
|
the two hits. Particle identification is done using correlation between
|
|
$\Delta$E and E. \cref{fig:si16p_dedx_nocut} shows clearly visible banding
|
|
structure. No cut on energy deposit or timing with respect to muon hit are
|
|
applied yet.
|
|
|
|
With the aid from MC study (\cref{fig:pid_sim}), the banding on the
|
|
$\Delta$E-E plots can be identified as follows:
|
|
\begin{itemize}
|
|
\item the densest spot at the lower left conner belonged to electron hits;
|
|
\item the small blurry cloud just above the electron region was muon hits;
|
|
\item the most intense band was due to proton hits;
|
|
\item the less intense, upper band caused by deuteron hits;
|
|
\item the highest band corresponded to alpha hits;
|
|
\item the faint stripe above the deuteron band should be triton
|
|
hits, which is consistent with a relatively low probability of emission of
|
|
tritons.
|
|
\end{itemize}
|
|
|
|
%The electrons either from Michel decay or from the beam are MIPs particles,
|
|
%which would deposit about 466~keV on the 1500-\si{\um}-thick silicon detector,
|
|
%and about 20~keV on the 65-\si{\um}-thick silicon detector. Therefore our thin
|
|
%silicon counters could not distinguish electrons from electronic
|
|
%noise. The brightest spots on the $\Delta$E-E plots are identified as electrons
|
|
%due to
|
|
%the total E of about 500~keV, and is the accidental coincidence between
|
|
%electron hits on the thick silicon and electronics noise on the thin silicon.
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.95\textwidth]{figs/si16p_dedx_nocut}
|
|
\caption{$\Delta$E as a function of E of particles from muon capture on the
|
|
thin silicon target.}
|
|
\label{fig:si16p_dedx_nocut}
|
|
\end{figure}
|
|
|
|
It is observed that the banding is more clearly visible in a log-log scale
|
|
plots like in \cref{fig:si16p_dedx_cut_explain}, this suggests
|
|
a geometrical cut on the logarithmic scale would be able to discriminate
|
|
protons from other particles. The protons and deuterons bands are nearly
|
|
parallel to the $\ln(\Delta \textrm{E [keV]}) + \ln(\textrm{E [keV]})$ line,
|
|
but have a slightly altered slope because $\ln(\textrm{E})$ is always greater
|
|
than $\ln(\Delta\textrm{E})$. The two parallel lines on
|
|
\cref{fig:si16p_dedx_cut_explain} suggest a check of
|
|
$\ln(\textrm{E}) + 0.85\times\ln(\Delta \textrm{E})$ could tell
|
|
protons from other particles.
|
|
|
|
Another feature of the $\Delta$E-E plots is their resolution power for protons
|
|
decrease as the energy E increases. The reason for this is the limited energy
|
|
resolution of the silicon detectors in use. The plots in logarithmic scale
|
|
show that this particle identification is good in the region where
|
|
$\ln(\textrm{E}) < 9$, which corresponds to $\textrm{E} < 8$~MeV.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.95\textwidth]{figs/si16p_dedx_cut}
|
|
\caption{$\Delta$E-E plots in the logarithmic scale and the geometrical cuts
|
|
for protons.}
|
|
\label{fig:si16p_dedx_nocut_log}
|
|
\end{figure}
|
|
|
|
The cut of $\ln(\textrm{E}) < 9$ is applied first, then
|
|
$\ln(\textrm{E})+ 0.85\times\ln(\Delta \textrm{E}) $ is plotted as
|
|
\cref{fig:si16p_loge+logde}. The protons make a clear peak in the region
|
|
between 14 and 14.8, the next peak at 15 corresponds to deuteron.
|
|
Imposing the
|
|
$14<\ln(\textrm{E})+ 0.85\times\ln(\Delta \textrm{E})<14.8$ cut,
|
|
the remaining proton band is shown on \cref{fig:si16p_proton_after_ecut}.
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/si16p_dedx_loge+logde}
|
|
\caption{Rationale for the cut on $\ln(\textrm{E})$ and $\ln(\Delta
|
|
\textrm{E})$}
|
|
\label{fig:si16p_loge+logde}
|
|
\end{figure}
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/si16p_proton_after_ecut}
|
|
\caption{Proton bands after cuts on energy}
|
|
\label{fig:si16p_proton_after_ecut}
|
|
\end{figure}
|
|
|
|
% subsection particle_identification_by_de_dx (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Number of muon captures}
|
|
\label{sub:number_stopped_muons}
|
|
The X-ray spectrum from this silicon target on \cref{fig:si16_xray} is
|
|
significantly noisier than the previous data set of SiR2, suffers from both
|
|
lower statistics and a more relaxed muon definition. The peak of $(2p-1s)$
|
|
X-ray at 400.177~keV can still be recognised but on a very high background. The
|
|
same timing requirement for the hit timing on the germanium detector as
|
|
in~\eqref{eqn:sir2_ge_cut}.
|
|
|
|
The double peaks of muonic X-rays from the lead shield at 431 and 438~keV are
|
|
very intense, reflects the fact that the low momentum muon beam of
|
|
29.68~MeV\cc\ (scaling factor 1.06) was strongly scattered by the upstream
|
|
counters. After a prompt cut that requires the photon
|
|
hit occured in $\pm 1$~\si{\us}\ around the muon hit, the peaks from lead
|
|
remain prominent which is an expected result because of all the lead shield
|
|
inside the chamber was to capture stray muons. The cut shows its effect on
|
|
reducing the background level under the 400.177 keV peak by about one third.
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.98\textwidth]{figs/si16p_xray}
|
|
\caption{X-ray spectrum from the passive 62-\si{\um}-thick silicon target with
|
|
and with out timing cut.}
|
|
\label{fig:si16_xray}
|
|
\end{figure}
|
|
|
|
Using the same procedure on the region from 396 to 402 keV (without
|
|
self-absorption correction since this is a thin target), the number of
|
|
X-rays recorded and the number of captures are shown in
|
|
\cref{tab:si16p_ncapture_cal}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l l c c c}
|
|
\toprule
|
|
\textbf{Source}& \textbf{Quantity}& \textbf{Value} & \textbf{Absolute}
|
|
& \textbf{Relative}\\
|
|
& & & \textbf{error} & \textbf{error}\\
|
|
\midrule
|
|
Measured & $(2p-1s)$ peak area & 2613 & 145.5 & 0.056\\
|
|
\midrule
|
|
Calibration & X-ray efficiency & \sn{4.54}{-4} & \sn{1.11}{-5}
|
|
& 0.024\\
|
|
\midrule
|
|
Reference & X-ray intensity & 0.803 & 0.008 & \sn{9.9}{-3}\\
|
|
& Capture probability & 0.658 & 0 & 0 \\
|
|
\midrule
|
|
Corrections& Self absorption & 1 & 0 & 0\\
|
|
& True coincidence summing & 1 &0 & 0\\
|
|
& TRP reset time & 1.01 & 0 & 0 \\
|
|
& Dead time & 1.041& 0 & 0\\
|
|
\midrule
|
|
Results & Number of X-rays & \sn{6.05}{6} & \sn{0.37}{6} & 0.06\\
|
|
& Number of $\mu$ stopped & \sn{7.54}{6} & \sn{0.46}{6}&0.06\\
|
|
& Number of captures& \sn{4.96}{6} & \sn{0.31}{6} & 0.06\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Number of X-rays and muon captures in the passive silicon runs.}
|
|
\label{tab:si16p_ncapture_cal}
|
|
\end{table}
|
|
% subsection number_stopped_muons (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\subsection{Lifetime measurement}
|
|
\label{sub:lifetime_measurement}w
|
|
To check the origin of the protons recorded, lifetime measurements were made by
|
|
cutting on time difference between a hit on one thick silicon and the muon
|
|
hit. Applying the time cut in 0.5~\si{\us}\ time steps on the proton
|
|
events in \cref{fig:si16p_proton_after_ecut}, the number of surviving
|
|
protons on each arm are plotted on \cref{fig:si16p_proton_lifetime}. The
|
|
curves show decay constants of $762.9 \pm 13.7$~\si{\ns}\ and $754.6 \pm
|
|
11.9$,
|
|
which are consistent with the each other, and with mean life time of muons in
|
|
silicon in the literatures of $758 \pm 2$~\cite{}. This is the confirmation
|
|
that the protons seen by the silicon detectors were indeed from the silicon
|
|
target.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.75\textwidth]{figs/si16p_proton_lifetime}
|
|
\caption{Lifetime measurement of protons seen on the silicon detectors.}
|
|
\label{fig:si16p_proton_lifetime}
|
|
\end{figure}
|
|
|
|
The fits are consistent with lifetime of muons in silicon in from after 500~ns,
|
|
before that, the time constants are shorter ($655.9\pm 9.9$ and $731.1\pm8.9$)
|
|
indicates the contamination from muon captured on material with higher $Z$.
|
|
Therefore a timing cut from 500~ns is used to select good silicon events, the
|
|
remaining protons are shown in \cref{fig:si16p_proton_ecut_500nstcut}.
|
|
The spectra have a low energy cut off at 2.5~MeV because protons with energy:
|
|
lower than that could not pass through the thin silicon to make the cuts as the
|
|
range of 2.5~MeV protons in silicon is about 68~\si{\um}.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/si16p_proton_ecut_500nstcut}
|
|
\caption{Proton spectrum after energy and timing cuts}
|
|
\label{fig:si16p_proton_ecut_500nstcut}
|
|
\end{figure}
|
|
|
|
% subsection lifetime_measurement (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Proton emission rate from the silicon target}
|
|
\label{sub:proton_emission_rate_from_the_silicon_target}
|
|
The number of protons in \cref{fig:si16p_proton_ecut_500nstcut} is
|
|
counted from 500~ns after the muon event, where the survival rate is
|
|
$e^{-500/758} = 0.517$.
|
|
|
|
The geometry acceptance of each silicon arm is estimated to be \sn{2.64}{-2}
|
|
using a toy MC study where geantinos are generated within the image of the
|
|
collimator on the target, and the number of hits on each silicon package was
|
|
counted. Taking the geometry acceptance into account, the number of protons
|
|
with energy from 2.5 to 8~MeV emitted is:
|
|
\begin{equation}
|
|
N_{p \textrm{eff.}} = \dfrac{1927 + 1656}{0.517\times2.64\times10^{-2}}
|
|
= 2.625 \times 10^5
|
|
\end{equation}
|
|
The emission rate per muon capture is:
|
|
\begin{align}
|
|
R_{2.5-8\textrm{ MeV}}^{\textrm{eff.}} &= \dfrac{N_{p \textrm{eff.}}}
|
|
{N_{\mu \textrm{ captured}}^{\textrm{Si16p}}}\nonumber\\
|
|
&= \dfrac{2.625 \times 10^5}{6.256\times10^6} \nonumber\\
|
|
&= 4.20\times10^{-2}\nonumber
|
|
\end{align}
|
|
The proton spectra on the \cref{fig:si16p_proton_ecut_500nstcut} and the
|
|
emission rate are only effective ones, since the energy of protons are modified
|
|
by energy loss in the target, and low energy protons could not escape the
|
|
target. Therefore further corrections are needed for both rate and spectrum of
|
|
protons.
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Proton emission rate uncertainties}
|
|
\label{sub:proton_emission_rate_and_uncertainties_estimation}
|
|
The uncertainty of the emission rate could come from several sources:
|
|
\begin{enumerate}
|
|
\item number of captures $\pm0.562\times10^6$, or 9\%, mainly from the
|
|
background under the X-ray peak (5.5\%) and the efficiency calibration
|
|
\item number of protons: efficiency of the cuts in energy, impacts of the
|
|
timing resolution on timing cut. The energy cuts' contribution should be
|
|
small since it can be seen from \cref{fig:si16p_loge+logde}, the peak
|
|
of protons is strong and well separated from others. The uncertainty in
|
|
timing contribution is significant because all the timing done in this
|
|
analysis was on the peak of the slow signals. As it is clear from the
|
|
\cref{fig:tme_sir_prompt_rational}, the timing resolution of the
|
|
silicon detector could not be better than 100~ns. Putting $\pm100$~ns into
|
|
the timing cut could change the survival rate of proton by about
|
|
$1-e^{-100/758} \simeq 13\%$. Also, the low statistics contributes a few
|
|
percent to the uncertainty budget.
|
|
\item acceptance of the silicon packages: muon stopping distribution,
|
|
imperfect alignment, efficiency of the detectors, different response to
|
|
different species. The muon stopping distribution is important in unfolding
|
|
the initial proton spectrum and also greatly affects the rate of protons.
|
|
By the end of the run, we found that the target was displaced from the
|
|
previously aligned position by 10~mm. Whether this misalignment is serious
|
|
or not depends on the spatial distribution of the muons after the
|
|
collimator. In the worst case when the muon beam is flatly distributed,
|
|
that displacement could change the acceptance of the silicon detectors by
|
|
12\%. Although no measurement was done to determine the efficiency of the
|
|
silicon detectors, it would have small effect compared to other factors.
|
|
\end{enumerate}
|
|
|
|
The combined uncertainty from known sources above therefore could be as large
|
|
as 35\%, and the effective proton emission rate in the 2.5--8~MeV could be
|
|
written as:
|
|
\begin{equation}
|
|
R_{2.5-8\textrm{ MeV}}^{\textrm{eff.}} = (4.20\pm1.47)\times 10^{-2}
|
|
\end{equation}
|
|
|
|
\subsection{Ratio of protons to other heavy charged particles}
|
|
\label{sub:heavy_charged_particles_emission_rate}
|
|
By using only the lower limit on
|
|
$\ln(\textrm{E}) + 0.85\times\ln(\Delta \textrm{E})$, the heavy charged
|
|
particles can be selected. These particles also show a lifetime that is
|
|
consistent with that of muons in silicon
|
|
(\cref{fig:si16p_allparticle_lifetime}).
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/si16p_allparticle_lifetime}
|
|
\caption{Lifetime of heavy charged particles}
|
|
\label{fig:si16p_allparticle_lifetime}
|
|
\end{figure}
|
|
The ratio between the number of protons and other particles at 500~ns is $(1927
|
|
+ 1656)/(2202 + 1909) \simeq 0.87$.
|
|
|
|
% subsection heavy_charged_particles_emission_rate (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%I have started the initial study on the correction ()
|
|
% subsection proton_emission_rate_from_the_silicon_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\subsection{Rate and spectrum correction}
|
|
%\label{sub:proton_spectrum_deconvolution}
|
|
%The proton spectra on the \cref{fig:si16p_proton_ecut_500nstcut} and the
|
|
%emission rate are only effective ones, since the energy of protons are modified
|
|
%by energy loss in the target, and low energy protons could not escape the
|
|
%target. Therefore corrections are needed for both rate and spectrum of protons.
|
|
|
|
%To solve the unfolding problem, one needs to supply a response function that
|
|
%relates the observed energy to the initial energy of protons. This response
|
|
%function can be obtained from Monte Carlo simulation where protons with an
|
|
%assumed initial spatial distribution inside the target, and a uniform energy
|
|
%distribution are generated, then their modified energy spectrum is recorded.
|
|
%The initial spatial distribution of protons is inferred from the muon beam
|
|
%momentum using Monte Carlo simulation, and available measured data in momentum
|
|
%scanning runs. The response function for this thin silicon target is shown in
|
|
%\cref{fig:si16p_toyMC}.
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/si16p_toyMC}
|
|
%\caption{An example of response function between the observed energy and
|
|
%initial energy of protons in a 62-\si{\um}-target.}
|
|
%\label{fig:si16p_toyMC}
|
|
%\end{figure}
|
|
|
|
%The response function is then used to train the unfolding program, which is
|
|
%based on the RooUnfold package. The package supports several unfolding methods,
|
|
%and I adopted the so-called Bayesian unfolding method~\cite{DAgostini.1995a}.
|
|
%The Bayesian method is chosen because it tends to be fast, typical number of
|
|
%iterations is from 4--8.
|
|
|
|
%\cref{fig:si16p_unfold_train} presented results of two tests unfolding with
|
|
%two distributions of initial energy, a Gaussian distribution and
|
|
%a parameterized function in~\eqref{eqn:EH_pdf}. The numbers of protons obtained
|
|
%from the tests show agreement with the generated numbers.
|
|
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/si16p_unfold_train}
|
|
%\caption{Bayesian unfolding tests with two different initial proton energy
|
|
%distributions: Gaussian (left) and parameterized function of Sobottka and
|
|
%Wills's proton spectrum (right).}
|
|
%\label{fig:si16p_unfold_train}
|
|
%\end{figure}
|
|
|
|
%Finally, the unfolding is applied on the spectra in
|
|
%\cref{si16p_proton_spec}, the results are shown in
|
|
%\cref{si16p_unfold_meas}.
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/si16p_unfold_meas}
|
|
%\caption{Unfolded spectrum from a thin silicon target}
|
|
%\label{fig:si16p_unfold_meas}
|
|
%\end{figure}
|
|
% subsection proton_spectrum_deconvolution (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%\subsection{Proton emission rate and uncertainties estimation}
|
|
%\label{sub:proton_emission_rate_and_uncertainties_estimation}
|
|
|
|
%The rate of proton emission from 2.5--10~\si{\MeV} is:
|
|
%\begin{equation}
|
|
%R =
|
|
%\end{equation}
|
|
%\begin{equation}
|
|
%R =
|
|
%\end{equation}
|
|
%The uncertainties are:
|
|
|
|
% subsection proton_emission_rate_and_uncertainties_estimation (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%The uncertainties are:
|
|
|
|
% subsection proton_emission_rate_and_uncertainties_estimation (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\section{Proton emission following muon capture on an aluminium target}
|
|
\label{sec:proton_emission_following_muon_capture_on_an_aluminium_target}
|
|
The aluminium is the main object of the AlCap experiment, in this preliminary
|
|
analysis I chose one target, Al100 the 100-\si{\um}-thick target, on
|
|
a sub-range of the data set runs 2808--2873, as a demonstration.
|
|
Because this is a passive target, the same procedure and cuts used in the
|
|
passive silicon runs were applied.
|
|
\subsection{The number of stopped muons}
|
|
\label{sub:the_number_of_stopped_muons}
|
|
The X-ray spectrum on the germanium detector is shown on
|
|
\cref{fig:al100_ge_spec}.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
|
|
\caption{X-ray spectrum from the aluminium target, the characteristic
|
|
$(2p-1s)$ line shows up at 346.67~keV}
|
|
\label{fig:al100_ge_spec}
|
|
\end{figure}
|
|
|
|
The area of the $(2p-1s)$ line of aluminium and the number of captured in this
|
|
target are:
|
|
\begin{align}
|
|
N_{(2p-1s)\textrm{Al}} &= 3800.0 \pm 179.4 \nonumber\\
|
|
N_{\mu \textrm{ captured}}^{\textrm{Al100}}
|
|
&= \dfrac{N_{(2p-1s)\textrm{Al}}}
|
|
{\epsilon_{(2p-1s)\textrm{Al}} \times I_{(2p-1s)\textrm{Al}}}
|
|
\times f_{\textrm{capture-Al}} \nonumber \\
|
|
&= \dfrac{3800.0} {5.12\times 10^{-4} \times 0.798} \times 0.609 \nonumber \\
|
|
&= (5.664 \pm 0.479) \times 10^6
|
|
\end{align}
|
|
% subsection the_number_of_stopped_muons (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Particle identification}
|
|
\label{sub:particle_identification}
|
|
Using the same charged particle selection
|
|
procedure and the cuts on $\ln(\textrm{E})$ and $\ln(\Delta\textrm{E})$, the
|
|
proton energy spectrum is shown in \cref{fig:al100_proton_spec}.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=1\textwidth]{figs/al100_selection}
|
|
\caption{Selection of protons from the Al100 target: coincidence cut (top),
|
|
cuts on energy (middle) and the results (bottom).}
|
|
\label{fig:al100_selection}
|
|
\end{figure}
|
|
|
|
The lifetime of these protons are shown in
|
|
\cref{fig:al100_proton_lifetime}, the fitted decay constant on the right
|
|
arm is consistent with the reference value of $864 \pm 2$~\si{\ns}~\cite{}.
|
|
But the left arm gives $918 \pm 16.1$~\si{\ns}, significantly larger than
|
|
the reference value.
|
|
%The longer lifetime suggested some contributions from
|
|
%other lighter materials, one possible source is from muons captured on the back
|
|
%side of the collimator (\cref{fig:alcap_setup_detailed}).
|
|
%For this reason, the emission rate calculated from the left arm will be taken as upper
|
|
%limit only.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_proton_lifetime}
|
|
\caption{Lifetime of protons from the aluminium Al100 target}
|
|
\label{fig:al100_proton_lifetime}
|
|
\end{figure}
|
|
Further investigation of the problem of longer lifetime was made and the first
|
|
channel on the thin silicon detector on that channel was the offender. The
|
|
lifetime measurement with out that SiL1-1 channel gives a reasonable result,
|
|
and the decay constant on the SiL1-1 alone was nearly about 1000~\si{\us}.
|
|
The reason for this behaviour is not known yet. For this emission rate
|
|
calculation, this channel is discarded and the rate on the left arm is scaled
|
|
with a factor of 4/3. The proton spectrum from the aluminium target is plotted
|
|
on \cref{fig:al100_proton_spec_wosil11}.
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.40\textwidth]{figs/al100_proton_lifetime_wosil11}
|
|
\includegraphics[width=0.40\textwidth]{figs/al100_proton_lifetime_sil11}
|
|
\caption{Lifetime of protons without channel SiL1-1 (right) and of the
|
|
channel SiL1-1 alone (left).}
|
|
\label{fig:al100_proton_lifetime_sil11}
|
|
\end{figure}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_proton_spec_wosil11}
|
|
\caption{Spectrum of protons from the Al100 target after cuts on energy and
|
|
time, without channel SiL1-1}
|
|
\label{fig:al100_proton_spec_wosil11}
|
|
\end{figure}
|
|
% subsection particle_identification (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Proton emission rate}
|
|
\label{sub:proton_emission_rate_and_corrections}
|
|
The proton rate is calculated as:
|
|
\begin{equation}
|
|
N_{p \textrm{eff.}} = \dfrac{1132\times \frac{4}{3} + 2034}
|
|
{e^{-500/864}\times2.64\times10^{-2}}
|
|
= 1.34 \times 10^5
|
|
\end{equation}
|
|
\begin{equation}
|
|
R_{2.5-8\textrm{ MeV}}^{\textrm{Al eff.}} = \dfrac{N_{p \textrm{eff.}}}
|
|
{N_{\mu \textrm{ captured}}^{\textrm{Al100}}}
|
|
= \dfrac{1.34 \times 10^5}{5.664\times10^6}
|
|
= 2.37\times10^{-2}
|
|
\end{equation}
|
|
|
|
The uncertainty of the emission rates will be smaller than that of the rate
|
|
from silicon because of a longer lifetime of muons in aluminium and a higher
|
|
momentum beam made the misalignment of the target, if any, less important. To
|
|
be conservative, I take to 35\% above as this calculation uncertainty, and the
|
|
rates will be:
|
|
\begin{equation}
|
|
R_{2.5-8\textrm{ MeV}}^{\textrm{Al eff.}}=(2.37\pm0.83)\times10^{-2}
|
|
\end{equation}
|
|
% subsection proton_emission_rate_and_corrections (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section proton_emission_following_muon_capture_on_an_aluminium_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% chapter data_analysis (end)
|