1234 lines
56 KiB
TeX
1234 lines
56 KiB
TeX
\chapter{Data analysis}
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\label{cha:data_analysis}
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\section{Analysis modules}
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\label{sec:analysis_modules}
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A full offline analysis has not been completed yet, but initial analysis
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based on the existing modules (Table~\ref{tab:offline_modules}) is possible
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thanks to the modularity of the analysis framework.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{l p{8cm}}
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\toprule
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\textbf{Module name} & \textbf{Functions}\\
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\midrule
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MakeAnalysedPulses & make a pulse with parameters extracted from
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a waveform\\
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MaxBinAPGenerator & simplest algorithm to get pulse information\\
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TSimpleMuonEvent & sort pulses occur in a fixed time window around the
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muon hits\\
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ExportPulse \& PulseViewer & plot waveforms for diagnostics\\
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PlotAmplitude & plot pulse height spectra\\
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PlotAmpVsTdiff & plot pulse correlations in timing and amplitude\\
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EvdE & identify charged particles using dE/dx\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Available offline analysis modules.}
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\label{tab:offline_modules}
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\end{table}
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The MakeAnalysedPulses module takes a raw waveform, calculates the pedestal
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from a predefined number of first samples, subtracts this pedestal, takes
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pulse polarity into account, then calls another module to extract pulse
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parameters. At the moment, the simplest module, so-called MaxBinAPGenerator,
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for pulse information calculation is in use. The module looks for the
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sample that
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has the maximal deviation from the baseline, takes the deviation as pulse
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amplitude and the time stamp of the sample as pulse time. The procedure is
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illustrated on Figure~\ref{fig:tap_maxbin_algo}. This module could not detect
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pile up or double pulses in one \tpulseisland{} in
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Figure~\ref{fig:tap_maxbin_bad}.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/tap_maxbin_algo}
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\caption{Pulse parameters extraction with MaxBinAPGenerator.}
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\label{fig:tap_maxbin_algo}
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\end{figure}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/tap_maxbin_bad}
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\includegraphics[width=0.47\textwidth]{figs/tap_maxbin_bad2}
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\caption{Double pulse and pile up are taken as one single pulse by the
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MaxBinAPGenerator}
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\label{fig:tap_maxbin_bad}
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\end{figure}
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The TSimpleMuonEvent first picks a muon candidate, then loops through all
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pulses on all detector channels, and picks all pulses occur in
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a time window of $\pm 10$~\micro\second\ around each candidate to build a muon
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event. A muon candidates is a hit on the upstream plastic scintillator with
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an amplitude higher than a threshold which was chosen to reject minimum ionising
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particles (MIPs). The
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10~\micro\second\ is long enough compares to the mean life time of muons in the
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target materials (0.758~\micro\second\ for silicon, and 0.864~\micro\second\ for
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aluminium~\cite{SuzukiMeasday.etal.1987}) so practically all of emitted charged
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particles would be recorded in this time window.
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%\begin{figure}[htb]
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%\centering
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%\includegraphics[width=0.85\textwidth]{figs/tme_musc_threshold}
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%\caption{Pulse height spectrum of the $\mu$Sc scintillator}
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%\label{fig:tme_musc_threshold}
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%\end{figure}
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A pile-up protection mechanism is employed to reject multiple muons events: if
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there exists another muon hit in less than 15~\micro\second\ from the candidate
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then both the candidate and the other muon are discarded. This pile-up
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protection would cut out less than 11\% total number of events because the beam
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rate was generally less than 8~\kilo\hertz.
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%In runs with active silicon targets, another requirement is applied for the
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%candidate: a prompt hit on the target in $\pm 200$ \nano\second\ around the
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%time of the $\mu$Sc pulse. The number comes from the observation of the
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%time correlation between hits on the target and the $\mu$Sc
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%(Figure~\ref{fig:tme_sir_prompt_rational}).
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%\begin{figure}[htb]
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%\centering
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%\includegraphics[width=0.85\textwidth]{figs/tme_sir_prompt_rational}
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%\caption{Correlation in time between SiR2 hit and muon hit}
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%\label{fig:tme_sir_prompt_rational}
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%\end{figure}
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To make sure that we will analyse good data, a low level data quality checking
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was done on the whole data sets. The idea is plotting the variations of basic
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parameters, such as noise level, length of \tpulseisland{}, \tpulseisland{}
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rate, time correlation to hits on $\mu$Sc, \ldots on each channel during the
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data collecting period. Runs with significant difference from the nominal
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values were further checked for possible causes, and would be discarded if such
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discrepancy was too large or unaccounted for. Examples of such trend plots are
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shown in Figure~\ref{fig:lldq}.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/lldq_noise}
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\includegraphics[width=0.47\textwidth]{figs/lldq_tdiff}
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\caption{Example trend plots used in the low level data quality checking:
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noise level in FWHM (left) and time correlation with muon hits (right). The
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noise level was basically stable in in this data set, except for one
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channel. On the right hand side, this sanity check helped find out the
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sampling frequency was wrongly applied in the first tranche of the data
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set.}
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\label{fig:lldq}
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\end{figure}
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% section analysis_modules (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Detector calibration}
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\label{sec:detector_calibration}
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\subsection{Silicon detector}
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\label{sub:silicon_detector}
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The energy calibration for the silicon detectors were done routinely during the
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run, mainly by an
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$^{241}\textrm{Am}$ alpha source and a tail pulse generator. The source emits
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79.5 $\alpha\per\second$ in a 2$\pi$~\steradian~solid angle. The most
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prominent alpha particles have energies of 5.484~\mega\electronvolt\
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(85.2\%) and 5.442~\mega\electronvolt\ (12.5\%). A tail pulse with amplitude of
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66 \milli\volt~was used to simulate the response of the silicon detectors'
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preamplifiers to a particle with 1\mega\electronvolt~energy deposition.
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During data taking period, electrons in the beam were were also used for energy
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calibration of thick silicon detectors where energy deposition is large enough.
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The muons at different momenta provided another mean of calibration in the beam
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tuning period.
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%Typical pulse height spectra of the silicon detectors are shown
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%in Figure~\ref{fig:si_eg_spectra}.
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According to Micron Semiconductor
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\footnote{\url{http://www.micronsemiconductor.co.uk/}}, the
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manufacturer of the silicon detectors, the nominal thickness of the dead layer on
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each side is 0.5~\micron. The alpha particles from the source would deposit
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about 66~keV in this layer, and the peak would appear at 5418~keV
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(Figure~\ref{fig:toyMC_alpha}).
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.6\textwidth]{figs/toyMC_alpha}
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\caption{Energy loss of the alpha particles after a dead layer of
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0.5~\micron.}
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\label{fig:toyMC_alpha}
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\end{figure}
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The calibration coefficients for the silicon channels are listed in
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Table~\ref{tab:cal_coeff}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{l c r}
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\toprule
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\textbf{Detector} & \textbf{Slope} & \textbf{Offset}\\
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\midrule
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SiL-2 & 7.86 & 14.14\\
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SiR-2 & 7.96 & 22.98\\
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\midrule
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SiL1-1 & 2.61 & 37.34\\
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SiL1-2 & 2.54 & -20.78\\
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SiL1-3 & 2.65 & 67.75\\
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SiL1-4 & 2.54 & -18.45\\
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\midrule
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SiR1-1 & 2.53 & 28.69\\
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SiR1-2 & 2.62 & 47.10\\
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SiR1-3 & 2.49 & 6.32\\
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SiR1-4 & 2.53 & 34.81\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Calibration coefficients of the silicon detector channels}
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\label{tab:cal_coeff}
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\end{table}
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% subsection silicon_detector (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Germanium detector}
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\label{sub:germanium_detector}
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The germanium detector was calibrated using a $^{152}\textrm{Eu}$
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source\footnote{Energies and intensities of gamma rays are taken from the
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X-ray and Gamma-ray Decay Data Standards for Detector Calibration and Other
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Applications, which is published by IAEA at \\
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\url{https://www-nds.iaea.org/xgamma_standards/}}, the
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recorded pulse height spectrum is shown in Figure~\ref{fig:ge_eu152_spec}. The
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source was placed at the target position so that the absolute efficiencies can
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be calibrated. The relation between pulse height in ADC count and energy is
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found to be:
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\begin{equation}
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\textrm{ E [keV]} = 0.1219 \times \textrm{ADC} + 1.1621
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\end{equation}
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The energy resolution (full width at half maximum) was better than
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2.6~\kilo\electronvolt\ for all the $^{152}\textrm{Eu}$ peaks. It was a little
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worse at 3.1~\kilo\electronvolt~for the annihilation photons at
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511.0~\kilo\electronvolt.
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The absolute efficiencies for the $(2p-1s)$ lines of aluminium
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(346.828~\kilo\electronvolt) and silicon (400.177~\kilo\electronvolt) are
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presented in Table~\ref{tab:xray_eff}. In the process of efficiency calibration,
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corrections for true coincidence summing and self-absorption were not applied.
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The true coincidence summing probability is estimated to be very
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small, about \sn{5.4}{-6}, thanks to the far geometry of the calibration. The
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absorption in the source cover made of 22~\milli\gram\per\centi\meter$^2$
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polyethylene is less than \sn{4}{-4} for a 100~\kilo\electronvolt\ photon.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{c c c}
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\toprule
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\textbf{X-ray} & \textbf{Efficiency} & \textbf{Uncertainty}\\
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\midrule
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346.828 & $5.12 \times 10^{-4}$ & $0.14\times 10^{-4}$\\
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400.177 & $4.54 \times 10^{-4}$ & $0.11\times 10^{-4}$\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Calculated efficiencies at X-rays of interest}
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\label{tab:xray_eff}
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\end{table}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.70\textwidth]{figs/ge_eu152_spec}
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\caption{Energy spectrum of the $\rm^{152}\textrm{Eu}$ calibration source
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recorded by the germanium detector. The most prominent peaks of
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$^{152}\textrm{Eu}$ along with their energies are
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annotated in red; the 1460.82 \kilo\electronvolt~line is background from
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$^{40}\textrm{K}$; and the annihilation 511.0~\kilo\electronvolt~photons
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come both from the source and the surrounding environment.}
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\label{fig:ge_eu152_spec}
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\end{figure}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.89\textwidth]{figs/ge_ecal_fwhm}
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\caption{Germanium energy calibration and resolution.}
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\label{fig:ge_fwhm}
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\end{figure}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.80\textwidth]{figs/ge_ecal_eff}
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\caption{Absolute efficiency of the germanium detector, the fit was done with
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7 energy points from 244~keV because it is known that the linearity between
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$ln(\textrm{E})$ and $ln(\textrm{eff})$ holds better. The shaded area is
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95\% confidence interval of the fit.}
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\label{fig:ge_eff}
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\end{figure}
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% subsection germanium_detector (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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%\subsection{Beam tuning and muon momentum scanning}
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%\label{sub:muon_momentum_scanning}
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%Before taking any data, we carried out the muon momentum scanning to understand
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%the muon beam, as well as calibrate the detector system. The nominal muon
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%momentum used in the Run 2013 had been tuned to 28~MeV\cc\ before the run. By
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%changing simultaneously the strength of the key magnet elements in the $\pi$E1
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%beam line with the same factor, the muon beam momentum would be scaled with the
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%same factor.
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%The first study was on the range of muons in an active silicon target. The SiL2
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%detector was placed perpendicular to the nominal beam path, after an oval
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%collimator. The beam momentum scaling factor was scanned from 1.10 to 1.60,
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%muon momenta and energies in the measured points are listed in
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%Table~\ref{tab:mu_scales}.
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%\begin{table}[htbp]
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%\begin{center}
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%\begin{tabular}{c c c c}
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%\toprule
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%\textbf{Scaling} & \textbf{Momentum} & \textbf{Kinetic energy}
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%& \textbf{Momentum spread}\\
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%\textbf{factor} & \textbf{(MeV\per\cc)} & \textbf{(MeV)}
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%& \textbf{(MeV\per\cc, 3\% FWHM)}\\
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%\midrule
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%1.03 & 28.84 & 3.87& 0.87\\
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%1.05 & 29.40 & 4.01& 0.88\\
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%1.06 & 29.68 & 4.09& 0.89\\
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%1.07 & 29.96 & 4.17& 0.90\\
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%1.10 & 30.80 & 4.40& 0.92\\
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%1.15 & 32.20 & 4.80& 0.97\\
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%1.20 & 33.60 & 5.21& 1.01\\
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%1.30 & 36.40 & 6.09& 1.09\\
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%1.40 & 39.20 & 7.04& 1.18\\
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%1.43 & 40.04 & 7.33& 1.20\\
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%1.45 & 40.60 & 7.53& 1.22\\
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%1.47 & 41.16 & 7.73& 1.23\\
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%1.50 & 42.00 & 8.04& 1.26\\
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%\bottomrule
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%\end{tabular}
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%\end{center}
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%\caption{Muon beam scaling factors, energies and momenta.}
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%\label{tab:mu_scales}
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%\end{table}
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% subsection muon_momentum_scanning (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% section detector_calibration (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Charged particles following muon capture on a thick silicon target}
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\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
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This analysis was done on a subset of the active target runs 2119 -- 2140
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because of the problem of wrong clock frequency found in the data quality
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checking shown in Figure~\ref{fig:lldq}. The data set contains \sn{6.43}{7}
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muon events.
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%64293720
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Firstly, the number of charged particles emitted after nuclear muon capture on
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the active target is calculated. This number then is normalised to the number
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of nuclear muon capture to obtain an emission rate. Finally, the rate is
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compared with that from the literature.
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\subsection{Event selection}
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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 a 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 Figure~\ref{fig:sir2f_Et_corr}. The most intense spot at zero time
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and about 5 MeV energy corresponds to stopped muons in the thick target. The
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band below 1 MeV is due to electrons, either in the beam or from muon decay in
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orbits, or emitted during the cascading of muon to the muonic 1S state. The
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valley between time zero and 1200~ns shows the minimum distance in time between
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two pulses. It is the limitation of the current pulse parameter extraction
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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 Figure~\ref{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<50\textrm{ 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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3.4 \textrm{ MeV}<E_{\textrm{target}} < 5.6 \textrm{ 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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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 1300\textrm{
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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 muon events sample to the size of
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\sn{9.32}{6}.
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The number of stopped muons can also be calculated from the number of muonic
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X-rays recorded by the germanium detector. The X-rays are emitted during the
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cascading of the muon to the muonic 1S state in the time scale of \sn{}{-9}~s,
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so the hit caused by the X-rays must be in coincidence with the muon hit on the
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active target. Therefore an additional timing cut is applied for the germanium
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hits:
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\begin{equation}
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\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < 500\textrm{ ns}
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\label{eqn:sir2_ge_cut}
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\end{equation}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Number of charged particles with energy above 2~MeV}
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\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
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As shown in Figure~\ref{fig:sir2_1us_slices} and illustrated by MC simulation
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in Figure~\ref{fig:sir2_mc_pdfs}, there are several components in
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the energy spectrum recorded by the active target:
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\begin{enumerate}
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\item charged particles from nuclear muon capture, this is the signal we are
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interested in;
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\item beam electrons with a characteristic Landau peak around 800~keV,
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dominating at large timing (from 6500 ns), causing background at energy
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lower than 1~MeV which drops sharply at energy larger than 3~MeV;
|
|
\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei from
|
|
neutron emitting muon captures, peak at
|
|
around 300~keV, dominate the region where energy smaller than 1~MeV and
|
|
timing less than 3500~ns. This component is intrinsic background, having
|
|
the same time structure as that of the signal;
|
|
\item stray muons scattered into the target, this component is small compares
|
|
to the charged particles yield so it is not considered further.
|
|
\end{enumerate}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
|
|
\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
|
|
\caption{The observed spectrum in the timing window 1500 -- 9500~ns (left)
|
|
and its components from MC simulation (right). The charged particles
|
|
spectrum is obtained assuming the spectrum shape and emission rate from
|
|
Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scale between
|
|
components is arbitrarily chosen for the purpose of illustration.}
|
|
\label{fig:sir2_mc_pdfs}
|
|
\end{figure}
|
|
|
|
An energy cut at 2~MeV is introduced to reduce the domination of the beam
|
|
electrons. In order to obtain the yields of backgrounds above 2~MeV, a binned
|
|
maximum likelihood fit was done. The likelihood of getting the measured
|
|
spectrum is defined as:
|
|
\begin{equation}
|
|
L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
|
|
\label{eqn:llh_def}
|
|
\end{equation}
|
|
where $n$ is the total number of events observed, $\mu$ is the expected number
|
|
of events according to some linear combination of the signal and the
|
|
backgrounds shown in~\ref{fig:sir2_mc_pdfs}, namely:
|
|
\begin{align}
|
|
n &= n_{\textrm{sig}} + n_{\textrm{beam e}} + n_{\textrm{dio}}\\
|
|
\textrm{(sum pdf)} &= n_{\textrm{sig}}\times\textrm{(sig pdf)} +
|
|
n_{\textrm{beam e}}\times\textrm{(beam e pdf)} +
|
|
n_{\textrm{dio}}\times\textrm{(dio pdf)};
|
|
\label{eqn:sum_pdf}
|
|
\end{align}
|
|
and the $i$ index indicates the respective number of events in the $i$-th
|
|
bin.
|
|
|
|
The fit is done by the RooFit package~\cite{VerkerkeKirkby.2003} where the
|
|
negative log likelihood $-2\ln{L}$ is minimised. Fitting results are shown
|
|
in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
|
|
\begin{align}
|
|
n_{\textrm{beam e}} &= 23756 \pm 581\\
|
|
n_{\textrm{dio}} &= 111340 \pm 1245\\
|
|
n_{\textrm{sig}} &= 207201 \pm 856
|
|
\end{align}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nbkg}
|
|
\includegraphics[width=0.42\textwidth]{figs/sir2_mllfit_nebeam}
|
|
\includegraphics[width=0.84\textwidth]{figs/sir2_mllfit}
|
|
\caption{Results of the maximum likelihood fit of the energy spectrum on the
|
|
active target.}
|
|
\label{fig:sir2_mll_fit}
|
|
\end{figure}
|
|
|
|
The total number of charged particles from time zero is then calculated to be:
|
|
\begin{equation}
|
|
N_{\textrm{charged particles}} =(149.9\pm 0.6)\times 10^4
|
|
\label{eqn:sir2_Nchargedparticle}
|
|
\end{equation}
|
|
|
|
|
|
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\subsection{Number of nuclear muon captures}
|
|
\label{sub:number_of_stopped_muons}
|
|
The number of nuclear captures can be inferred from the number of recorded
|
|
muonic X-rays. The reference values of the parameters needed for the
|
|
calculation taken from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday
|
|
et al.~\cite{MeasdayStocki.etal.2007} are
|
|
listed in Table~\ref{tab:mucap_pars}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l l l}
|
|
\toprule
|
|
\textbf{Quantity} & \textbf{Aluminium} & \textbf{Silicon}\\
|
|
\midrule
|
|
Muonic mean lifetime (ns) & $864 \pm 2$ & $758 \pm 2$\\
|
|
Nuclear capture probability (\%) & $60.9 $ & $65.8$\\
|
|
$(2p-1s)$ X-ray energy (keV) & $346.828\pm0.002$ & $400.177\pm0.005$\\
|
|
Intensity (\%) & $79.8\pm0.8$ & $80.3\pm0.8$\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Reference parameters of muon capture in aluminium and silicon taken
|
|
from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday et
|
|
al.~\cite{MeasdayStocki.etal.2007}.}
|
|
\label{tab:mucap_pars}
|
|
\end{table}
|
|
|
|
The muonic X-ray spectrum emitted from the active target is shown in
|
|
Figure~\ref{fig:sir2_xray}. The $(2p-1s)$ line is seen at
|
|
399.5~\kilo\electronvolt, 0.7~\kilo\electronvolt\ off from the
|
|
reference value of 400.177~\kilo\electronvolt. This discrepancy is within our
|
|
detector's resolution, and the calculated efficiency is
|
|
$(4.549 \pm 0.108)\times 10^{-5}$ -- a 0.15\% increasing from that of the
|
|
400.177~keV line, so no attempt for recalibration or correction was made.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/sir2_xray}
|
|
\caption{Muonic X-rays spectrum from the active silicon target, the two major
|
|
lines $(2p-1s)$ and $(3p-1s)$ are clearly distinguishable at 400 and 476
|
|
keV, respectively. The $(5p-1s)$ line at 504 keV and $(6p-1s)$ line at 516
|
|
keV can also be seen.
|
|
}
|
|
\label{fig:sir2_xray}
|
|
\end{figure}
|
|
|
|
The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5 \pm 65.6$,
|
|
obtained by subtracting the background of 101.5 from the spectral integral of
|
|
2083 in the region from 396 to 402 keV. This number of X-rays needs to be
|
|
corrected for several effects:
|
|
\begin{itemize}
|
|
\item Self-absorption effect: the X-rays emitted could be absorbed by the
|
|
target itself, the probability of self-absorption becomes larger in case of
|
|
thick sample and low energy photons.
|
|
For this silicon target of 1500~\micron\ thick and the photon energy of
|
|
400~keV, and assuming a narrow muon stopping distribution at the centre of
|
|
the target, the self-absorption correction is estimated to be:
|
|
\begin{align}
|
|
k_{\textrm{self absorption}} &= \dfrac{\mu t}{1 - e^{-\mu t}} \nonumber\\
|
|
&= \dfrac {9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}}
|
|
{1 - e^{-9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}}}\nonumber \\
|
|
%&= \dfrac{1}{0.992} \nonumber\\
|
|
&= 1.008
|
|
\end{align}
|
|
where $t = 0.075\textrm{ cm}$ is the thickness of the target, and $\mu$ is the
|
|
linear attenuation coefficient of silicon for a photon of 400~keV. The
|
|
value of $\mu$ is calculated as product of the density of silicon
|
|
$\rho = 2.33 \textrm{ g/cm}^3$ and its mass attenuation coefficient
|
|
$\mu/\rho = 9.614\times 10^{-2} \textrm{ cm}^2/\textrm{g}$ taken
|
|
from the NIST's X-ray Mass Attenuation Coefficients
|
|
table~\footnote{\url{http://www.nist.gov/pml/data/xraycoef}}.
|
|
\item Dead time of the germanium detector system: there are two types of dead
|
|
time in our germanium detector, (a) the insensitive period due to long
|
|
pulse time, and (b) the reset pulses of the transistor reset preamplifier.
|
|
The effects of the two dead time could be calculated by examining the
|
|
interval between two consecutive pulses on the germanium detector in
|
|
Figure~\ref{fig:sir2_ge_deadtime}.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/sir2_ges_self_tdiff}
|
|
\caption{Interval between to consecutive pulses on the germanium
|
|
detector. The peak at 57~\micro\second\ indicates the pulse length, and
|
|
the bump at about 2000~\micro\second\ shows the width of the reset
|
|
pulses. The average count rate of this detector is extracted from the
|
|
decay constant of the time spectrum to be
|
|
$5.146 \times 10^{-7}\textrm{ ns}^{-1} = 514.6 \textrm{ s}^{-1}$}
|
|
\label{fig:sir2_ge_deadtime}
|
|
\end{figure}
|
|
|
|
The correction factor for the pulse length is calculated by the formula:
|
|
\begin{align}
|
|
k_{\textrm{pulse length}} &= e^{2\times \textrm{(pulse length)}
|
|
\times \textrm{(count rate)}} \nonumber\\
|
|
&= e^{2\times 57\times10^{-6} \times 514.6} \nonumber\\
|
|
&= 1.06
|
|
\end{align}
|
|
The 2-ms-long reset pulses effectively reduce the actual measurement time
|
|
compares to other channels, so the correction factor for the effect is:
|
|
\begin{align}
|
|
k_{\textrm{reset pulse}} &= \frac{\textrm{(measurement time)}}
|
|
{\textrm{(measurement time)}
|
|
- \textrm{(number of reset)}\times
|
|
\textrm{(reset pulse length)}}\nonumber\\
|
|
&= 1.033
|
|
\end{align}
|
|
|
|
\item The true coincidence summing is negligibly small due to the far
|
|
geometry as mentioned in the calibration process, so no correction is made.
|
|
%%TODO
|
|
\item The geometrical acceptance of the detector: the absolute efficiency
|
|
calibration was done with a point-like source, but the actual points of
|
|
origin of the X-rays have a finite spatial distribution. The correction
|
|
factor is estimated to be \ldots
|
|
\end{itemize}
|
|
|
|
The number of X-rays after applying all above corrections is 3293.5. The X-ray
|
|
intensity in Table~\ref{tab:mucap_pars} was normalised to the number of stopped
|
|
muons, so the number of stopped muons is:
|
|
|
|
\begin{align}
|
|
N_{\mu\textrm{ stopped}} &=
|
|
\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
|
|
&= \dfrac{3293.5}{4.54\times10^{-4} \times 0.803} \\
|
|
&= 9.03\times10^6 \nonumber
|
|
\end{align}
|
|
where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
|
|
detector for the 400.177~keV line in Table~\ref{tab:xray_eff}, and
|
|
$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
|
|
(80.3\% from Table~\ref{tab:mucap_pars}).
|
|
|
|
Taking the statistical uncertainty of the peak area, and systematic
|
|
uncertainties from parameters of muon capture, the number of stopped muons
|
|
calculated from the X-ray measurement is
|
|
$(9.03 \pm 0.31)\times 10^6$. This figure is consistent with the number of
|
|
stopped muons of $9.32\times10^6$ after the cuts described in the event
|
|
selection process.
|
|
|
|
The number of nuclear captured muons is:
|
|
\begin{equation}
|
|
N_{\mu\textrm{ nucl.capture}} =
|
|
N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
|
|
= 9.03 \times 10^6 \times 0.658 = 7.25 \times 10^6
|
|
\label{eqn:sir2_Ncapture}
|
|
\end{equation}
|
|
where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
|
|
stopped muon from Table~\ref{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
|
|
Table~\ref{tab:sir2_uncertainties}, including:
|
|
\begin{itemize}
|
|
\item uncertainties from number of charged particles, both statistical and
|
|
systematic (from spectrum shape and fitting) ones are absorbed in the
|
|
quoted value in~\eqref{sir2_Nchargedparticle};
|
|
\item uncertainties from number of nuclear capture:
|
|
\begin{itemize}
|
|
\item statistical error of the peak area calculation,
|
|
\item systematic errors from the efficiency calibration, and referenced
|
|
values of X-ray intensity and capture probability.
|
|
\end{itemize}
|
|
\end{itemize}
|
|
So, the emission rate is:
|
|
\begin{equation}
|
|
R_{\textrm{Si}} = 0.252 \pm 0.009
|
|
\label{eqn:sir2_rate_cal}
|
|
\end{equation}
|
|
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l l l}
|
|
\toprule
|
|
%\textbf{Source} & \textbf{Type} & \textbf{Relative error}\\
|
|
Number of charged particles & &\\
|
|
Statistical and systematic & &0.004\\
|
|
\midrule
|
|
Number of nuclear capture & &\\
|
|
Statistical & Peak area calculation& 0.022\\
|
|
Systematic & Efficiency calibration & 0.024\\
|
|
& X-ray intensity & 0.009\\
|
|
& Capture probability & 0\\
|
|
|
|
\midrule
|
|
Total relative error & & 0.035\\
|
|
Total absolute error & & 0.009\\
|
|
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Uncertainties of the emission rate from the thick silicon target}
|
|
\label{tab:sir2_uncertainties}
|
|
\end{table}
|
|
|
|
% subsection partial_emission_rate_of_charged_particle_in_8_10_mev_range (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%TODO
|
|
%\subsection{Partial emission rate of charged particles from the literature}
|
|
%\label{sub:partial_emission_rate_of_charged_particles_from_the_literature}
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/sobottka_spec2}
|
|
%\caption{Reproduced charged particle spectrum from muon capture on silicon,
|
|
%measured by Sobottka and Wills. Integration region is shown in the green
|
|
%box.}
|
|
%\label{fig:sobottka_spec}
|
|
%\end{figure}
|
|
%The spectrum measured by Sobottka and Wills~\cite{SobottkaWills.1968} is
|
|
%reproduced in Figure~\ref{fig:sobottka_spec}, the spectral integral in the
|
|
%energy region from 8 to 10~\mega\electronvolt\ is $2086.8 \pm 45.7$.
|
|
%The authors obtained the spectrum in a 4~\micro\second\ gate period which began
|
|
%1~\micro\second\ after a muon stopped, which would take 26.59\% of the emitted
|
|
%particles into account. The number of stopped muons was not explicitly stated,
|
|
%but can be inferred to be $55715/0.06 = 92858.3$.
|
|
|
|
%The partial rate of charged particle from 8 to 10~\mega\electronvolt\ is then
|
|
%calculated to be:
|
|
%\begin{equation}
|
|
%R_{\textrm{8-10 MeV}}^{lit.} =
|
|
%\dfrac{2086.8}{0.2659 \times 92858.3 \times 0.658}
|
|
%= 1.28 \times 10^{-2}
|
|
%\end{equation}
|
|
%The authors did not mentioned how the uncertainties of their measurement was
|
|
%derived, but quoted the emission rate below 26~\mega\electronvolt\ to be $0.15
|
|
%\pm 0.02$, which translates to a relative uncertainty of 0.133. The statistical
|
|
%uncertainty from the spectral integral and the number of stopped muons is:
|
|
%\begin{equation*}
|
|
%\dfrac{1}{\sqrt{25000}} + \dfrac{1}{\sqrt{92858.3}} = 0.9 \times 10^{-2}
|
|
%\end{equation*}
|
|
%Then their systematic uncertainty would be: $0.133 - 0.009 = 0.124$.
|
|
|
|
%For the partial spectrum from 8 to 10~\mega\electronvolt, the statistical
|
|
%contribution to the uncertainty is:
|
|
%\begin{equation*}
|
|
%\dfrac{1}{\sqrt{2086.8}} + \dfrac{1}{\sqrt{92858.3}} = 2.5 \times 10^{-2}
|
|
%\end{equation*}
|
|
%So, the combined uncertainty of this partial rate calculation is: $0.124
|
|
%+ 0.025 = 0.150$. The partial rate of charged particles from 8 to
|
|
%10~\mega\electronvolt per muon capture is:
|
|
%\begin{equation}
|
|
%R_{\textrm{8-10 MeV}}^{lit.} = (1.28 \pm 0.19) \times 10^{-2}
|
|
%\end{equation}
|
|
% subsection partial_emission_rate_of_charged_particles_from_the_literature
|
|
% (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Charged particles following muon capture on a thin silicon target}
|
|
\label{sec:charged_particles_following_muon_capture_on_a_thin_silicon_target}
|
|
In this measurement, a passive, 62-\micron-thick silicon target was used as the
|
|
target. The silicon target is $5\times5$~\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~\kilo\electronvolt\ that happened within $\pm 10$~\micro\second\ of the
|
|
%muon hit are then
|
|
%associated to the muon and stored in the muon event tree. The
|
|
%200~\kilo\electronvolt\ cut effectively rejects all MIPs hits on thin silicon
|
|
%detectors of which the most probable value is about 40~\kilo\electronvolt.
|
|
|
|
%In order to use dE/dx for particle identification, $\Delta$E and total E are
|
|
%needed.
|
|
The charged particle selection starts from searching for muon event
|
|
that has at least one hit on thick silicon. If there is a thin silicon hit
|
|
within a coincidence window of $\pm 0.5$~\micro\second\ around the thick
|
|
silicon hit, the two hits are considered to belong to one particle with
|
|
$\Delta$E being the energy of the thin hit, and total E being the sum energy of
|
|
the two hits. Particle identification is done using correlation between
|
|
$\Delta$E and E. Figure~\ref{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 (Figure~\ref{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-\micron-thick silicon detector,
|
|
%and about 20~keV on the 65-\micron-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 Figure~\ref{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
|
|
Figure~\ref{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
|
|
Figure~\ref{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 Figure~\ref{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 Figure~\ref{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 $\pm 1$~\micro\second\ around the muon hit, the peaks from lead
|
|
remain prominent which is an expected result because of all the lead shield
|
|
inside the chamber was to capture stray muons. The cut shows its effect on
|
|
reducing the background level under the 400.177 keV peak by about one third.
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.98\textwidth]{figs/si16p_xray}
|
|
\caption{X-ray spectrum from the passive 62-\micron-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
|
|
Table~\ref{tab:si16p_ncapture_cal}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l l c c c}
|
|
\toprule
|
|
\textbf{Source}& \textbf{Quantity}& \textbf{Value} & \textbf{Absolute}
|
|
& \textbf{Relative}\\
|
|
& & & \textbf{error} & \textbf{error}\\
|
|
\midrule
|
|
Measured & $(2p-1s)$ peak area & 2613 & 145.5 & 0.056\\
|
|
\midrule
|
|
Calibration & X-ray efficiency & \sn{4.54}{-4} & \sn{1.11}{-5}
|
|
& 0.024\\
|
|
\midrule
|
|
Reference & X-ray intensity & 0.803 & 0.008 & \sn{9.9}{-3}\\
|
|
& Capture probability & 0.658 & 0 & 0 \\
|
|
\midrule
|
|
Corrections& Self absorption & 1 & 0 & 0\\
|
|
& True coincidence summing & 1 &0 & 0\\
|
|
& TRP reset time & 1.01 & 0 & 0 \\
|
|
& Dead time & 1.041& 0 & 0\\
|
|
\midrule
|
|
Results & Number of X-rays & \sn{6.05}{6} & \sn{0.37}{6} & 0.06\\
|
|
& Number of $\mu$ stopped & \sn{7.54}{6} & \sn{0.46}{6}&0.06\\
|
|
& Number of captures& \sn{4.96}{6} & \sn{0.31}{6} & 0.06\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Number of X-rays and muon captures in the passive silicon runs.}
|
|
\label{tab:si16p_ncapture_cal}
|
|
\end{table}
|
|
% subsection number_stopped_muons (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\subsection{Lifetime measurement}
|
|
\label{sub:lifetime_measurement}
|
|
To check the origin of the protons recorded, lifetime measurements were made by
|
|
cutting on time difference between a hit on one thick silicon and the muon
|
|
hit. Applying the time cut in 0.5~\micro\second\ time steps on the proton
|
|
events in Figure~\ref{fig:si16p_proton_after_ecut}, the number of surviving
|
|
protons on each arm are plotted on Figure~\ref{fig:si16p_proton_lifetime}. The
|
|
curves show decay constants of $762.9 \pm 13.7$~\nano\second\ and $754.6 \pm
|
|
11.9$,
|
|
which are consistent with the each other, and with mean life time of muons in
|
|
silicon in the literatures of $758 \pm 2$~\cite{}. This is the confirmation
|
|
that the protons seen by the silicon detectors were indeed from the silicon
|
|
target.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.75\textwidth]{figs/si16p_proton_lifetime}
|
|
\caption{Lifetime measurement of protons seen on the silicon detectors.}
|
|
\label{fig:si16p_proton_lifetime}
|
|
\end{figure}
|
|
|
|
The fits are consistent with lifetime of muons in silicon in from after 500~ns,
|
|
before that, the time constants are shorter ($655.9\pm 9.9$ and $731.1\pm8.9$)
|
|
indicates the contamination from muon captured on material with higher $Z$.
|
|
Therefore a timing cut from 500~ns is used to select good silicon events, the
|
|
remaining protons are shown in Figure~\ref{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~\micron.
|
|
\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 Figure~\ref{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 Figure~\ref{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 Figure~\ref{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
|
|
Figure~\ref{fig:tme_sir_prompt_rational}, the timing resolution of the
|
|
silicon detector could not be better than 100~ns. Putting $\pm100$~ns into
|
|
the timing cut could change the survival rate of proton by about
|
|
$1-e^{-100/758} \simeq 13\%$. Also, the low statistics contributes a few
|
|
percent to the uncertainty budget.
|
|
\item acceptance of the silicon packages: muon stopping distribution,
|
|
imperfect alignment, efficiency of the detectors, different response to
|
|
different species. The muon stopping distribution is important in unfolding
|
|
the initial proton spectrum and also greatly affects the rate of protons.
|
|
By the end of the run, we found that the target was displaced from the
|
|
previously aligned position by 10~mm. Whether this misalignment is serious
|
|
or not depends on the spatial distribution of the muons after the
|
|
collimator. In the worst case when the muon beam is flatly distributed,
|
|
that displacement could change the acceptance of the silicon detectors by
|
|
12\%. Although no measurement was done to determine the efficiency of the
|
|
silicon detectors, it would have small effect compare to other factors.
|
|
\end{enumerate}
|
|
|
|
The combined uncertainty from known sources above therefore could be as large
|
|
as 35\%, and the effective proton emission rate in the 2.5--8~MeV could be
|
|
written as:
|
|
\begin{equation}
|
|
R_{2.5-8\textrm{ MeV}}^{\textrm{eff.}} = (4.20\pm1.47)\times 10^{-2}
|
|
\end{equation}
|
|
|
|
\subsection{Ratio of protons to other heavy charged particles}
|
|
\label{sub:heavy_charged_particles_emission_rate}
|
|
By using only the lower limit on
|
|
$\ln(\textrm{E}) + 0.85\times\ln(\Delta \textrm{E})$, the heavy charged
|
|
particles can be selected. These particles also show a lifetime that is
|
|
consistent with that of muons in silicon
|
|
(Figure~\ref{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 Figure~\ref{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
|
|
%Figure~\ref{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-\micron-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.
|
|
|
|
%Figure~\ref{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
|
|
%Figure~\ref{si16p_proton_spec}, the results are shown in
|
|
%Figure~\ref{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~\mega\electronvolt is:
|
|
%\begin{equation}
|
|
%R =
|
|
%\end{equation}
|
|
%\begin{equation}
|
|
%R =
|
|
%\end{equation}
|
|
%The uncertainties are:
|
|
|
|
% subsection proton_emission_rate_and_uncertainties_estimation (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%The uncertainties are:
|
|
|
|
% subsection proton_emission_rate_and_uncertainties_estimation (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section charged_particles_following_muon_capture_on_a_thin_silicon_target (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
\section{Proton emission following muon capture on an aluminium target}
|
|
\label{sec:proton_emission_following_muon_capture_on_an_aluminium_target}
|
|
The aluminium is the main object of the AlCap experiment, in this preliminary
|
|
analysis I chose one target, Al100 the 100-\micron-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
|
|
Figure~\ref{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 Figure~\ref{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
|
|
Figure~\ref{fig:al100_proton_lifetime}, the fitted decay constant on the right
|
|
arm is consistent with the reference value of $864 \pm 2$~\nano\second~\cite{}.
|
|
But the left arm gives $918 \pm 16.1$~\nano\second, significantly larger than
|
|
the reference value.
|
|
%The longer lifetime suggested some contributions from
|
|
%other lighter materials, one possible source is from muons captured on the back
|
|
%side of the collimator (Figure~\ref{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~\micro\second.
|
|
The reason for this behaviour is not known yet. For this emission rate
|
|
calculation, this channel is discarded and the rate on the left arm is scaled
|
|
with a factor of 4/3. The proton spectrum from the aluminium target is plotted
|
|
on Figure~\ref{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
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|
\includegraphics[width=0.85\textwidth]{figs/al100_proton_spec_wosil11}
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|
\caption{Spectrum of protons from the Al100 target after cuts on energy and
|
|
time, without channel SiL1-1}
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|
\label{fig:al100_proton_spec_wosil11}
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|
\end{figure}
|
|
% subsection particle_identification (end)
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|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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|
\subsection{Proton emission rate}
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|
\label{sub:proton_emission_rate_and_corrections}
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|
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)
|