198 lines
9.2 KiB
TeX
198 lines
9.2 KiB
TeX
A preliminary analysis has been done on half of the Al100 dataset. The analysis used information from silicon, germanium and
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upstream muon detectors. Pulse parameters were extracted from waveforms by the
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methods described above. Purposes of the analysis
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include:
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\begin{itemize}
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\item testing the analysis chain;
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\item verification of the experimental method, specifically the
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normalisation of number of stopped muons, and particle identification
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using specific energy loss;
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\item extracting a preliminary rate and spectrum of proton emission from
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aluminium.
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\end{itemize}
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\subsubsection{Event selection}
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\label{ssub:event_selection}
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As described earlier, the hits on all detectors are re-organised into
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muon-centred events, each event consists of one central muon and all hits
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within \SI{\pm 10}{\us} from the central muon. A pile-up protection mechanism
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is used to ensure only one muon appears in each event: if there are muon
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hits within \SI{\pm 10}{\us} of each other, both of them will be rejected. The
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dataset from runs \numrange{2808}{2873} contains \num{1.17E+9} such muon
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events.
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Selection of proton (and other heavy charged particles) events start from
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searching for a muon event that has at least one hit in the thick silicon. If there
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is a thin silicon hit within a coincidence window of $\pm 0.5$~\si{\us}\ around
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the thick silicon hit, the two hits are considered to belong to one particle.
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The thresholds for energy deposited in all silicon channels, except the thin
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silicon on the left arm, are set at \SI{100}{\keV} in this analysis. The
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threshold on the left $\Delta E$ counter was higher, at roughly
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\SI{400}{\keV}, in order to suppress higher noise in this channel that caused an excessive trigger rate.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/al100_dedx_overlay}
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\caption{Identification of charged particle bands: the dots are measured
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points, the histograms are the expected bands of protons (red), deuterons
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(green) and tritons (blue). The MC bands are calculated
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for a pair of 58-\si{\um}-thick and 1535-\si{\um}-thick silicon
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detectors. The error bars on MC bands show the standard deviation of
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$\Delta E$ in the respective bins of E.
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}
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\label{fig:pid_sim}
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\end{figure}
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\subsubsection{Charged particle identification}
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\label{ssub:charged_particle_identification}
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Charged particle identification is done using the energy deposition
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in the silicon detectors. \Cref{fig:al100_dedx} shows the energy
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deposited in the thin silicon detector as a function of the total energy
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deposited in both thin and thick detectors.
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With the aid of the MC simulation, the band of protons in \cref{fig:al100_dedx}
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can be identified as shown in \cref{fig:pid_sim}.
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A proton likelihood probability is defined as:
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\begin{equation}
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P_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
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\exp{\left[\dfrac{(\Delta E_\mathrm{meas} - \Delta E_i)^2} {2\sigma^2_{\Delta
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E}}\right]}\,,
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\end{equation}
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where $\Delta E_{\mathrm{meas}}$ is the measured energy deposition in
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the thin silicon detector; $\Delta E_i$ and
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$\sigma_{\Delta E}$ are the expected value and the standard deviation of the energy
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loss in the thin detector, of protons with summed energy $E_i$, calculated by the MC simulation.
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With a cut on proton-like probability of $P_{i}>\num{1E-4}$, the proton band
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can be extracted as shown in \cref{fig:al100_protons}. The numbers of protons
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observed in the
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two silicon arms in the energy range from \SIrange{2.2}{8}{\MeV} are:
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\begin{align}
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N_{\textrm{p right}} &= 2373\,,\\
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N_{\textrm{p left}} &= 1822\,.
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\end{align}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.47\textwidth]{figs/al100_protons}
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\includegraphics[width=0.47\textwidth]{figs/al100_protons_px_r}
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\caption{Protons (green) selected using the likelihood probability cut of
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\num{1.0E-4} (left). The proton spectrum (right) is obtained by projecting
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the proton band onto the total energy axis.}
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\label{fig:al100_protons}
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\end{figure}
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\subsubsection{Correction for energy loss in target and geometrical acceptance}
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\label{ssub:correction_for_energy_loss_in_target_and_geometrical_acceptance}
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The observed proton spectra are modified by the energy loss of protons travelling through the
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target, therefore correction (or unfolding) of the observed energy spectrum is needed to find the true spectrum. The iterative Bayesian
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unfolding method implemented in RooUnfold package~\cite{Adye.2011} was used.
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The unfolding code was trained by a MC-generated proton spectra. The unfolded
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results are shown in \cref{fig:al100_unfold}. The proton yields observed in the
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range \SIrange{4}{8}{\MeV} by the two silicon arms are:
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\begin{align}
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N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\,,\\
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N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3\,.
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\end{align}
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The average proton yield is then:
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\begin{equation}
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N_{\textrm{p unfold avg}} = (169.3 \pm 1.9) \times 10^3
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\end{equation}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.80\textwidth]{figs/al100_unfolded_lr}
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\caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
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\label{fig:al100_unfold}
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\end{figure}
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\subsubsection{Normalisation to the number of nuclear muon captures}
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\label{ssub:number_of_stopped_muons}
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The number of stopped muons in the target is inferred from the number of
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X-rays recorded. The number of \atrn{2p}{1s} transitions and the number of nuclear
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captures are calculated to be:
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\begin{align}
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N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\,,\\
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N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6\,.
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\end{align}
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The emission rate of protons in the energy range of \SIrange{4}{8}{\MeV} is
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then:
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\begin{equation}
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R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.7\times
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10^{-2}\,.
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\label{eq:proton_rate_al}
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\end{equation}
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Uncertainty of the rate in Equation~\eqref{eq:proton_rate_al} is estimated to be 6.1\%,
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where the dominant sources are from the unfolding process (5\%), and
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from the number of nuclear captures (3.2\%). We are studying the consistency
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between the data sets to check for any overlooked systematic uncertainty.
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%There are two sources of uncertainties in the emission
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%rate~\eqref{eq:proton_rate_al}:
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%\begin{itemize}
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%\item from the number of nuclear captures, including the statistical
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%uncertainty of the peak area determination and
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%\item uncertainties in the number of protons:
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%\begin{itemize}
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%\item statistical uncertainties of the measured spectra which are
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%propagated during the unfolding process;
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%\item systematic uncertainties due to misidentification: this number is
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%small compared to other uncertainties as discussed in
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%\cref{sub:event_selection_for_the_passive_targets};
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%\item systematic uncertainty from the unfolding
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%\end{itemize}
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%\end{itemize}
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%The last item is studied by MC method using the parameterisation in
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%\cref{sub:proton_emission_rate}:
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%\begin{itemize}
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%\item protons with energy distribution obeying the parameterisation are
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%generated inside the target. The spatial distribution is the same as that
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%of in \cref{sub:corrections_for_the_number_of_protons}. MC truth including
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%initial energies and positions are recorded;
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%\item the number of protons reaching the silicon detectors are counted,
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%the proton yield is set to be the same as the measured yield to make the
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%statistical uncertainties comparable;
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%\item the unfolding is applied on the observed proton spectra, and the
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%results are compared with the MC truth.
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%\end{itemize}
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%\begin{figure}[htb]
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%\centering
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%\includegraphics[width=0.48\textwidth]{figs/al100_MCvsUnfold}
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%\includegraphics[width=0.48\textwidth]{figs/al100_unfold_truth_ratio}
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%\caption{Comparison between an unfolded spectrum and MC truth. On the left
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%hand side, the solid, red line is MC truth, the blue histogram is the
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%unfoldede spectrum. The ratio between the two yields is compared in the
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%right hand side plot with the upper end of integration is fixed at
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%\SI{8}{\MeV}, and a moving lower end of integration. The discrepancy
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%is genenerally smaller than 5\% if the lower end energy is smaller than
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%\SI{6}{\MeV}.}
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%\label{fig:al100_MCvsUnfold}
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%\end{figure}
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%\Cref{fig:al100_MCvsUnfold} shows that the yield obtained after unfolding is
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%in agreement with that from the MC truth. The difference is less than 5\% if
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%the integration is taken in the range from \SIrange{4}{8}{\MeV}. Therefore
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%a systematic uncertainty of 5\% is assigned for the unfolding routine.
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%A summary of uncertainties in the measurement of proton emission rate is
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%presented in \cref{tab:al100_uncertainties_all}.
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%\begin{table}[htb]
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%\begin{center}
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%\begin{tabular}{@{}ll@{}}
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%\toprule
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%\textbf{Item}& \textbf{Value} \\
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%\midrule
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%Number of muons & 3.2\% \\
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%Statistical from measured spectra & 1.1\% \\
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%Systematic from unfolding & 5.0\% \\
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%Systematic from PID & \textless1.0\% \\
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%\midrule
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%Total & 6.1\%\\
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%\bottomrule
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%\end{tabular}
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%\end{center}
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%\caption{Uncertainties of the proton emission rate.}
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%\label{tab:al100_uncertainties_all}
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%\end{table}
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