721 lines
36 KiB
TeX
721 lines
36 KiB
TeX
\chapter{Data analysis and results}
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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 the right hand side of \cref{fig:al100_scan_rate}. With the 1.09
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scale beam, the muons stopped \SI{28}{\um} off-centred to the right silicon arm.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/al100_scan_rate}
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\includegraphics[width=0.47\textwidth]{figs/al100_mu_stop_mc}
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\caption{Number of X-rays per incoming muon as a function of momentum
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scaling factor (left); and muon stopping distributions from MC simulation
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(right). The depth of muons is measured normal to surface of the target
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facing the muon beam.}
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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} of 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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\exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta
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E}}\right]}
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\end{equation}
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where $\Delta E_{\textrm{meas.}}$ is energy deposition measured by 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 study. A threshold is set at \num{1E-4} to
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extract protons, the resulted band of protons is shown in
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(\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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The cut efficiency in the energy range from \SIrange{2}{12}{\MeV} is estimated
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by MC study. The fraction of protons that do not satisfy the probability cut
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is 0.5\%. The number of other charged particles that are misidentified as
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protons depends on the ratios between those species and protons. Assuming
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a proton:deuteron:triton:alpha:muon ratio of 5:2:1:2:2, the number of
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misidentified hits is 0.1\% of the number of protons.
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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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The numbers of protons in the energy range from \SIrange{2.2}{8.5}{\MeV} after
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applying the probability cut are:
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\begin{align}
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N_{\textrm{p meas. left}} = 1822\\% \pm 42.7\\
|
|
N_{\textrm{p meas. right}} = 2373% \pm 48.7
|
|
\end{align}
|
|
The right arm received significantly more protons than the left arm did, which
|
|
is expected as in \cref{sub:momentum_scan_for_the_100_} where it is shown that
|
|
muons stopped off-centred to the right arm.
|
|
|
|
\subsection{Corrections for the number of protons}
|
|
\label{sub:corrections_for_the_number_of_protons}
|
|
The protons spectra observed by the silicon detectors have been modified by
|
|
the energy loss inside the target so correction (or unfolding) is necessary.
|
|
The unfolding, essentially, is finding a response function that relates proton's
|
|
true energy and measured value. This can be done in MC simulation by generating
|
|
protons with a spatial distribution as close as possible to the real
|
|
distribution of muons inside the target, then counting the number of protons
|
|
reaching the silicon detectors. Such response function conveniently includes
|
|
the geometrical acceptance.
|
|
|
|
For the 100-\si{\um} aluminium target and muons at the momentum scale of 1.09,
|
|
the parameters of the initial protons are:
|
|
\begin{itemize}
|
|
\item horizontal distribution: Gaussian \SI{26}{\mm} FWHM, centred at the
|
|
centre of the target;
|
|
\item vertical distribution: Gaussian \SI{15}{\mm} FWHM, centred at the
|
|
centre of the target;
|
|
\item depth: Gaussian \SI{69.2}{\um} FWHM, centred at \SI{68.8}{\um}-deep from
|
|
the upstream face of the target;
|
|
\item energy: flatly distributed from \SIrange{1.5}{15}{\MeV}.
|
|
\end{itemize}
|
|
The resulting response matrices for the two arms are presented in
|
|
\cref{fig:al100_resp_matrices}. These are then used as MC truth to train and
|
|
test the unfolding code. The code uses an existing ROOT package
|
|
called RooUnfold~\cite{Adye.2011} where the iterative Bayesian unfolding
|
|
method is implemented.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{./figs/al100_resp}
|
|
\caption{Response functions for the two silicon arms.}
|
|
\label{fig:al100_resp_matrices}
|
|
\end{figure}
|
|
%After training, the unfolding code is applied on the measured spectra from the
|
|
%left and right arms. The unfolded proton spectra in \cref{fig:al100_unfold}
|
|
%reasonably reflect the distribution of initial protons which is off-centred to
|
|
%the right arm. The path length to the left arm is longer so less protons at
|
|
%energy lower than \SI{5}{\MeV} could reach the detectors. The sharp low-energy
|
|
%cut off on the right arm is consistent with the Coulomb barrier for protons,
|
|
%which is \SI{4.1}{\MeV} for protons emitted from $^{27}$Mg.
|
|
The unfolded spectra using the two observed spectra at the two arms as input
|
|
are shown in \cref{fig:al100_unfold}. The two unfolded spectra generally agree
|
|
with each other, except for a few first and last bins. The discrepancy and
|
|
large uncertainties at the low energy region are because of only a small
|
|
number of protons with those energies could reach the detectors. The jump on
|
|
the right arm at around \SI{9}{\MeV} can be explained as the punch-through
|
|
protons were counted as the proton veto counters were not used in this
|
|
analysis.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr}
|
|
\caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
|
|
\label{fig:al100_unfold}
|
|
\end{figure}
|
|
|
|
%Several studies were conducted to assess the performance of the unfolding
|
|
%code, including:
|
|
%\begin{itemize}
|
|
%\item stability against cut-off energy;
|
|
%\item comparison between the two arms;
|
|
%\item and unfolding of a MC-generated spectrum.
|
|
%\end{itemize}
|
|
The stability of the unfolding code is tested by varying the lower cut-off
|
|
energy of the input spectrum. \cref{fig:al100_cutoff_study} show that the
|
|
shapes of the unfolded spectra are stable. The lower cut-off energy of the
|
|
output increases as that of the input increases, and the shape is generally
|
|
unchanged after a few bins.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_cutoff_study}
|
|
\caption{Unfolded spectra with different cut-off energies.}
|
|
\label{fig:al100_cutoff_study}
|
|
\end{figure}
|
|
The proton yields calculated from observed spectra in two arms are compared in
|
|
\cref{fig:al100_integral_comparison} where the upper limit of the integrals
|
|
is fixed at \SI{8}{\MeV}, and the lower limit is varied in \SI{400}{\keV} step.
|
|
The difference is large at cut-off energies less than \SI{4}{\MeV} due to
|
|
large uncertainties at the first bins. Above \SI{4}{\MeV}, the two arms show
|
|
consistent numbers of protons.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_integral_comparison}
|
|
\caption{Proton yields calculated from two arms.}
|
|
\label{fig:al100_integral_comparison}
|
|
\end{figure}
|
|
The yields of protons from \SIrange{4}{8}{\MeV} are:
|
|
\begin{align}
|
|
N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\\
|
|
N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3
|
|
\end{align}
|
|
The number of emitted protons is taken as average of the two yields:
|
|
\begin{equation}
|
|
N_{\textrm{p unfold}} = (169.3 \pm 1.9) \times 10^3
|
|
\end{equation}
|
|
|
|
\subsection{Number of nuclear captures}
|
|
\label{sub:number_of_nuclear_captures}
|
|
%\begin{figure}[!htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
|
|
%\caption{X-ray spectrum from the aluminium target, the characteristic
|
|
%$(2p-1s)$ line shows up at 346.67~keV}
|
|
%\label{fig:al100_ge_spec}
|
|
%\end{figure}
|
|
|
|
%The X-ray spectrum on the germanium detector is shown on
|
|
%\cref{fig:al100_ge_spec}.
|
|
Fitting the double peaks on top of a first-order
|
|
polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
|
|
procedure as in the case of the active target, the number stopped muons and
|
|
the number of nuclear captures are:
|
|
\begin{align}
|
|
N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\\
|
|
N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6
|
|
\end{align}
|
|
|
|
\subsection{Proton emission rate}
|
|
\label{sub:proton_emission_rate}
|
|
The proton emission rate in the range from \SIrange{4}{8}{\MeV} is therefore:
|
|
\begin{equation}
|
|
R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.7\times
|
|
10^{-2}
|
|
\label{eq:proton_rate_al}
|
|
\end{equation}
|
|
|
|
The total proton emission rate can be estimated by assuming a spectrum shape
|
|
with the same parameterisation as in \eqref{eqn:EH_pdf}. The
|
|
\eqref{eqn:EH_pdf} function has a power rising edge, and a exponential decay
|
|
falling edge. The falling edge has only one decay component and is suitable to
|
|
describe the proton spectrum with the equilibrium emission only assumption.
|
|
The pre-equilibrium emission contribution should be small for low-$Z$ material,
|
|
for aluminium the contribution of this component is 2.2\% according to
|
|
Lifshitz and Singer~\cite{LifshitzSinger.1980}.
|
|
|
|
The fitted results
|
|
are shown in \cref{fig:al100_parameterisation} and \cref{tab:al100_fit_pars}.
|
|
The average spectrum is obtained by taking the average of the two unfolded
|
|
spectra from the left and right arms. The fitted parameters are compatible
|
|
with each other within their errors.
|
|
|
|
Using the fitted parameters of the average spectrum, the integration in range
|
|
from \SIrange{4}{8}{\MeV} is 51\% of the total number of
|
|
protons. The total proton emission rate is therefore estimated to be $3.5\times 10^{-2}$.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_parameterisation}
|
|
\caption{Fitting of the unfolded spectra.}
|
|
\label{fig:al100_parameterisation}
|
|
\end{figure}
|
|
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l S[separate-uncertainty=true] S[separate-uncertainty=true]
|
|
S[separate-uncertainty=true]}
|
|
\toprule
|
|
\textbf{Parameter} &{\textbf{Left}} & {\textbf{Right}} & {\textbf{Average}}\\
|
|
\midrule
|
|
$A \times 10^{-5}$ & 2.0 \pm 0.7 & 1.3 \pm 0.1 & 1.5 \pm 0.3\\
|
|
$T_{th}$ (\si{\keV}) & 1301 \pm 490 & 1966 \pm 68 & 1573 \pm 132\\
|
|
$\alpha$ & 3.2 \pm 0.7 & 1.2 \pm 0.1 & 2.0 \pm 1.2\\
|
|
$T_{0}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Parameters of the fits on the unfolded spectra, the average spectrum
|
|
is obtained by taking average of the unfolded spectra from left and right
|
|
arms.}
|
|
\label{tab:al100_fit_pars}
|
|
\end{table}
|
|
|
|
|
|
\subsection{Uncertainties of the emission rate}
|
|
\label{sub:uncertainties_of_the_emission_rate}
|
|
The uncertainties of the emission rate come from:
|
|
\begin{itemize}
|
|
\item uncertainties in the number of nuclear captures: these were discussed
|
|
in \cref{sub:number_of_stopped_muons_from_the_number_of_x_rays};
|
|
\item uncertainties in the number of protons:
|
|
\begin{itemize}
|
|
\item statistical uncertainties of the measured spectra which are
|
|
propagated during the unfolding process;
|
|
\item systematic uncertainties due to misidentification: this number is
|
|
small compared to other uncertainties as discussed in
|
|
\cref{sub:event_selection_for_the_passive_targets};
|
|
\item systematic uncertainty from the unfolding
|
|
\end{itemize}
|
|
\end{itemize}
|
|
The last item is studied by MC method using the parameterisation in
|
|
\cref{sub:proton_emission_rate}:
|
|
\begin{itemize}
|
|
\item protons with energy distribution obeying the parameterisation are
|
|
generated inside the target. The spatial distribution is the same as that
|
|
of in \cref{sub:corrections_for_the_number_of_protons}. MC truth including
|
|
initial energies and positions are recorded;
|
|
\item the number of protons reaching the silicon detectors are counted,
|
|
the proton yield is set to be the same as the measured yield to make the
|
|
statistical uncertainties comparable;
|
|
\item the unfolding is applied on the observed proton spectra, and the
|
|
results are compared with the MC truth.
|
|
\end{itemize}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.48\textwidth]{figs/al100_MCvsUnfold}
|
|
\includegraphics[width=0.48\textwidth]{figs/al100_unfold_truth_ratio}
|
|
\caption{Comparison between an unfolded spectrum and MC truth: spectra
|
|
(left), and yields (right). The ratio is defined as $\textrm{(Unfold - MC
|
|
truth)/(MC truth)}$}
|
|
\label{fig:al100_MCvsUnfold}
|
|
\end{figure}
|
|
\Cref{fig:al100_MCvsUnfold} shows that the yield obtained after unfolding is
|
|
in agreement with that from the MC truth. The difference is less than 5\% if
|
|
the integration is taken in the range from \SIrange{4}{8}{\MeV}. Therefore
|
|
a systematic uncertainty of 5\% is assigned for the unfolding routine.
|
|
|
|
A summary of uncertainties in the measurement of proton emission rate is
|
|
presented in \cref{tab:al100_uncertainties_all}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{@{}ll@{}}
|
|
\toprule
|
|
\textbf{Item}& \textbf{Value} \\
|
|
\midrule
|
|
Number of muons & 3.2\% \\
|
|
Statistical from measured spectra & 1.1\% \\
|
|
Systematic from unfolding & 5.0\% \\
|
|
Systematic from PID & \textless0.5\% \\
|
|
\midrule
|
|
Total & 6.1\%\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Uncertainties of the proton emission rate.}
|
|
\label{tab:al100_uncertainties_all}
|
|
\end{table}
|
|
|
|
\section{Results of the initial analysis}
|
|
\label{sec:results_of_the_initial_analysis}
|
|
\subsection{Verification of the experimental method}
|
|
\label{sub:verification_of_the_experimental_method}
|
|
The experimental method described in \cref{sub:experimental_method} has been
|
|
validated:
|
|
\begin{itemize}
|
|
\item Number of muon capture normalisation: the number of stopped muons
|
|
calculated from the muonic X-ray spectrum is shown to be consistent with
|
|
that calculated from the active target spectrum.
|
|
\item Particle identification: the particle identification by specific
|
|
energy loss has been demonstrated. The banding of different particle
|
|
species is clearly visible. The proton extraction method using cut on
|
|
likelihood probability has been established. Since the distribution of
|
|
$\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do
|
|
not make the cut is 0.5\%, much larger than the threshold at \num{1E-4}.
|
|
The fraction of other charged particles being misidentified as protons is
|
|
smaller than 0.1\%. These uncertainties from particle identification are
|
|
still small in compared with the
|
|
uncertainty of the measurement (2.3\%).
|
|
\item Unfolding of the proton spectrum: the unfolded spectra inferred from
|
|
two measurements at the two silicon arms show good agreement with each
|
|
other, and with the muon stopping distribution obtained in the momentum
|
|
scanning analysis.
|
|
\end{itemize}
|
|
|
|
\subsection{Proton emission rates and spectrum}
|
|
\label{sub:proton_emission_rates_and_spectrum}
|
|
The proton emission spectrum in \cref{sub:proton_emission_rate} peaks around
|
|
\SI{3.7}{\MeV} which is a little below the Coulomb barrier for proton of
|
|
\SI{3.9}{\MeV} calculated using \eqref{eqn:classical_coulomb_barrier}. The
|
|
spectrum has a decay constant of \SI{2.6}{\MeV} in higher energy region,
|
|
makes the emission probability drop more quickly than silicon charged
|
|
particles spectrum of Sobottka and Wills~\cite{SobottkaWills.1968} where the
|
|
decay constant was \SI{4.6}{\MeV}. This can be explained as the silicon
|
|
spectrum includes other heavier particles which have higher Coulomb barriers,
|
|
hence contribute more in the higher energy bins, effectively reduces the decay
|
|
rate.
|
|
|
|
The partial emission rate measured in the energy range from
|
|
\SIrange{4}{8}{\MeV} is:
|
|
\begin{equation}
|
|
R_{p \textrm{ meas. }} = (1.7 \pm 0.1)\%.
|
|
\label{eqn:meas_partial_rate}
|
|
\end{equation}
|
|
|
|
The total emission rate from aluminium assuming the spectrum shape holds for
|
|
all energy is:
|
|
\begin{equation}
|
|
R_{p \textrm{ total}} = (3.5 \pm 0.2)\%.
|
|
\label{eqn:meas_total_rate}
|
|
\end{equation}
|
|
|
|
\subsubsection{Comparison to theoretical and other experimental results}
|
|
\label{ssub:comparison_to_theoretical_and_other_experimental_results}
|
|
There is no existing experimental or theoretical work that could be directly
|
|
compared with the obtained proton emission rate. Indirectly, it is compatible
|
|
with the figures calculated by Lifshitz and
|
|
Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980} listed in
|
|
\cref{tab:lifshitzsinger_cal_proton_rate}. It is significantly larger than
|
|
the rate of 0.97\% for the $(\mu,\nu p)$ channel, and does not
|
|
exceed the inclusion rate for all channels $\Sigma(\mu,\nu p(xn))$ at 4\%,
|
|
leaving some room for other modes such as emission of deuterons or tritons.
|
|
Certainly, when the full analysis is available, deuterons and tritons emission
|
|
rates could be extracted and the combined emission rate could be compared
|
|
directly with the inclusive rate.
|
|
|
|
The result \eqref{eqn:meas_total_rate} is greater than the
|
|
probability of the reaction $(\mu,\nu pn)$ measured by Wyttenbach et
|
|
al.~\cite{WyttenbachBaertschi.etal.1978} at 2.8\%. It is expectable because
|
|
the contribution from the $(\mu,\nu d)$ channel should be small since it
|
|
needs to form a deuteron from a proton and a neutron.
|
|
|
|
The rate of 3.5\% was estimated with an assumption that all protons are
|
|
emitted in equilibrium. With the exponential constant of \SI{2.6}{\MeV}, the
|
|
proton yield in the range from \SIrange{40}{70}{\MeV} is negligibly small
|
|
($\sim\num{E-8}$). However, Krane and colleagues reported a significant yield
|
|
of 0.1\% in that region~\cite{KraneSharma.etal.1979}. The energetic proton
|
|
spectrum shape also has a different exponential constant of \SI{7.5}{\MeV}. One
|
|
explanation for these protons is that they are emitted by other mechanisms,
|
|
such as capture on two-nucleon cluster suggested by Singer~\cite{Singer.1961}
|
|
(see \cref{sub:theoretical_models} and
|
|
\cref{tab:lifshitzsinger_cal_proton_rate_1988}). Despite being sizeable, the
|
|
yield of high energy protons is still small (3\%) in compared with the result
|
|
in \eqref{eqn:meas_total_rate}.
|
|
|
|
%The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
|
|
%\eqref{eqn:wyttenbach_ratio}.
|
|
|
|
\subsubsection{Comparison to the silicon result}
|
|
\label{ssub:comparison_to_the_silicon_result}
|
|
The probability of proton emission per nuclear capture of 3.5\% is indeed much
|
|
smaller than that of silicon. It is even lower than the rate of the no-neutron
|
|
reaction $(\mu,\nu p)$. This can be explained as
|
|
the resulted nucleus from muon capture on silicon, $^{28}$Al, is an odd-odd
|
|
nucleus and less stable than that from aluminium, $^{27}$Mg. The proton
|
|
separation energy for $^{28}$Al is \SI{9.6}{\MeV}, which is significantly
|
|
lower than that of $^{27}$Mg at \SI{15.0}{\MeV}~\cite{AudiWapstra.etal.2003}.
|
|
|
|
|