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