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@@ -402,10 +402,11 @@ correlation between detectors would be established in the analysis stage.
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At the beginning of each block, the time counter in each digitiser is reset to
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ensure time alignment across all modules. The period of 110~ms was chosen to be:
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{\em i} long enough compares to the time scale of several \si{\micro\second}\ of the
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physics of interest, {\em ii} short enough so that there is no timer rollover
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on any digitiser (a FADC runs at its maximum speed of \SI{170}{\mega\hertz} could
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handle up to about \SI{1.5}{\second} with its 28-bit time counter).
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{\em i} long enough compared to the time scale of several \si{\micro\second}\
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of the physics of interest, {\em ii} short enough so that there is no timer
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rollover on any digitiser (a FADC runs at its maximum speed of
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\SI{170}{\mega\hertz} could handle up to about \SI{1.5}{\second} with its
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28-bit time counter).
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To ease the task of handling data, the data collecting period was divided into
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short runs, each run stopped when the logger had recorded 2 GB of data.
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@@ -495,8 +496,8 @@ the recorded pulse height spectrum is shown in \cref{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 calculated. The peak centroids and areas were obtained by fitting a Gaussian
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peak on top of a first-order polynomial background. The only exception is the
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\SI{1085.84}{\keV} line because of the interference of \SI{1089.74}{\keV},
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the two were fitted with two Gaussian peaks on top of a first-order
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\SI{1085.84}{\keV} line because of the interference of the \SI{1089.74}{\keV}
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gamma, the two were fitted with two Gaussian peaks on top of a first-order
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polynomial background.
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The relation between pulse height in ADC value and energy is found to be:
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@@ -527,30 +528,86 @@ a little worse at 3.1~\si{\keV}~for the annihilation photons at
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\label{fig:ge_fwhm}
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\end{figure}
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The absolute efficiencies of the referenced points, and calculated
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efficiencies at the X-ray of interest are presented in
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\cref{tab:xray_eff}.
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%The absolute efficiencies for the $(2p-1s)$ lines of aluminium
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%(\SI{346.828}{\keV}) and silicon (\SI{400.177}{\keV})
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%are presented in \cref{tab:xray_eff}.
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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 \num{5.4d-6}, thanks to the far geometry of the calibration. The
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absorption in the source cover made of \SI{22}{\mg\per\cm^2}
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polyethylene is less than \num{4d-4} for a \SI{100}{\keV} photon.
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A Monte Carlo (MC) study on the acceptance of the germanium detector with two
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purposes:
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Following corrections for the peak areas are considered:
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\begin{enumerate}
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\item compare between measured and MC efficiencies: a point source made of
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$^152$Eu is placed at the target position
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\item estimate the uncertainty due to finite-size geometry: the source is
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made of silicon with the same dimensions as those of the thick silicon
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detector, namely \SI[product-units=power]{1.5 x 50 x 50}{\mm}; then the
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primary vertex of $^152$Eu is generated inside the source.
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\item Correction for counting loss due to finite response time of the
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detector system, where two gamma rays arrive at the detector within a time
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interval short compared to that response time. This correction is
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significant in our germanium system because of the current pulse
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information extracting method does not count the second pulse.
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\item Correction of counting time loss in the reset periods of the transistor
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reset preamplifier. A preamplifier of this type would reset itself after
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accumulating a predetermined amount of charge. During a reset, the
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preamplifier is insensitive so this can be counted as dead time.
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\item True coincidence summing correction: two cascade gamma rays hit the
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detector at the same time would cause loss of count under the two
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respective peaks and gain under the sum energy peak.
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\item Correction for self-absorption of a gamma ray by the source itself.
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\end{enumerate}
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The corrections for the first two mechanisms can be estimated by examining
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pulse length and intervals between two consecutive pulses in the germanium
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detector (\cref{fig:ge_cal_rate_pulselength}). The average pulse
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length is \SI{45.7}{\um}, the average count rate obtained from the decay rate
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of the interval spectrum is \SI{240}{\per\s}.
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The correction factor for the finite response time of the detector system is
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calculated as:
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\begin{align}
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k_{\textrm{finite response time}} &= e^{2\times \textrm{(pulse length)}
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\times \textrm{(count rate)}}\\
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&= e^{2\times 47.5\times10^{-6} \times 241} \nonumber\\
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&= 1.02 \label{eqn:finite_time_response}
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\end{align}
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The resets of the preamplifier show up as a peak around \SI{2}{\ms},
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consistent with specification of the manufacturer. Fitting the peak on top of
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an exponential background gives the actual reset pulse length of
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\SI{1947.34}{\us} and the number of resets during the calibration runs is
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2335.0. The total time loss for resetting is hence:
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$1947.34\times 10^{-6} \times 2335.0 = 4.55$ \si{\s}. That is a 0.14\% loss
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for a measuring time of \SI{3245.5}{\s}. This percentage loss is insignificant
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compared with the loss in \eqref{eqn:finite_time_response} and the statistical
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uncertainty of peak areas so correction for amplifier resets is not applied.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.95\textwidth]{figs/ge_cal_rate_pulselength}
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\caption{Germanium detector pulse length (left) and intervals between pulses
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on that detector (right). The peak around \SI{2}{\ms} corresponds to the
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resets of the preamplifier. The peak at \SI{250}{\us} is due to triggering
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by the timing channel which is on the same digitiser.}
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\label{fig:ge_cal_rate_pulselength}
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\end{figure}
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The true coincidence summing probability is estimated to be very small, about
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\num{5.4d-6}, thanks to the far geometry of the calibration. The absorption in
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the source cover made of \SI{22}{\mg\per\cm^2} polyethylene is less than
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\num{4d-4} for a \SI{100}{\keV} photon. Therefore these two corrections are
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omitted.
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The absolute efficiencies of the reference gamma rays show agreement with those
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obtained from a Monte Carlo (MC) study where a point source made of $^{152}$Eu
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is placed at the target position (see \cref{fig:ge_eff_cal}). A comparison
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between efficiencies in case of the point-like source and a finite-size
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source is also done by MC simulation. As shown in \cref{fig:ge_eff_cal}, the
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differences are in line with the uncertainties of the measured efficiencies.
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%The dimensions of the latter are set to
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%resemble the distribution of muons inside the target: Gaussian spreading
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%\SI{11}{\mm} vertically, \SI{13}{\mm} horizontally, and \SI{127}{\um} in
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.40\textwidth]{figs/ge_eff_cal}
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\includegraphics[width=0.40\textwidth]{figs/ge_eff_mc_finitesize_vs_pointlike_root}
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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, the shaded area is
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95\% confidence interval of the fit.}
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%because it is known that the linearity between
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%$ln(\textrm{E})$ and $ln(\textrm{eff})$ holds better.
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\label{fig:ge_eff_cal}
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\end{figure}
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The absolute efficiencies of the referenced points, and calculated efficiencies
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at X-rays of interest are listed in \cref{tab:xray_eff}.
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\begin{table}[htb]
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\begin{center}
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\pgfplotstabletypeset[
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@@ -601,18 +658,6 @@ purposes:
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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=0.40\textwidth]{figs/ge_eff_cal}
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\includegraphics[width=0.40\textwidth]{figs/ge_eff_mc_finitesize_vs_pointlike_root}
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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, the shaded area is
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95\% confidence interval of the fit.}
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%because it is known that the linearity between
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%$ln(\textrm{E})$ and $ln(\textrm{eff})$ holds better.
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\label{fig:ge_eff_cal}
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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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@@ -684,12 +729,12 @@ different targets were carried out for silicon targets:
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As the emitted protons deposit a significant amount of energy in the target
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material, thin targets and thus excellent momentum resolution of the low energy
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muon beam are critical. Aluminium targets of 50-\si{\micro\meter}\ and
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100~\si{\micro\meter}\ thick were used. Although a beam with low momentum spread of
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1\% is preferable, it was used for only a small portion of the run due to the
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low beam rate (see \cref{fig:Rates}). The beam momentum for each target
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was chosen to maximise the number of stopped muons. The collected data sets are
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shown in \cref{tb:stat}.
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muon beam are critical, aluminium targets of 50-\si{\micro\meter}\ and
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100-\si{\micro\meter}\ thick were used. Although a beam with low momentum
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spread of 1\% is preferable, it was used for only a small portion of the run
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due to the low beam rate (see \cref{fig:Rates}). The beam momentum for each
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target was chosen to maximise the number of stopped muons. The collected data
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sets are shown in \cref{tb:stat}.
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\begin{table}[btp!]
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\begin{center}
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@@ -865,20 +910,8 @@ update the plots to reflect real-time status of the detector system.
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Some offline analysis modules has been developed during the beam time and could
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provide quick feedback in confirming and guiding the decisions at the time. For
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example, the X-ray spectrum analysis was done to confirm that we could observe
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the muon capture process (\cref{fig:muX}), and to help in choosing optimal
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momenta which maximised the number of stopped muons.
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\begin{figure}[btp]
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\centering
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\includegraphics[width=0.7\textwidth]{figs/muX.png}
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\caption{Germanium
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detector spectra in the range of 300 - 450 keV with different setups: no
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target, 62-\si{\micro\meter}-thick silicon target, and
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100-\si{\micro\meter}-thick aluminium target. The ($2p-1s$) lines from
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aluminium (346.828 keV) and silicon (400.177 keV) are clearly visible,
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the double peaks at 431 and 438 keV are from the lead shield, the peak at
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351~keV is a background gamma ray from $^{211}$Bi.}
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\label{fig:muX}
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\end{figure}
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the muon capture process and to help in choosing optimal momenta which
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maximised the number of stopped muons.
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Although the offline analyser is still not fully developed yet, several modules
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are ready. They are described in detailed in the next chapter.
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