\chapter{Data analysis and results} \label{cha:data_analysis} This chapter presents initial analysis on subsets of the collected data. 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 of proton emission from aluminium. \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Number of stopped muons normalisation} \label{sec:number_of_stopped_muons_normalisation} The active silicon target runs was used to check for the validity of the counting of number of stopped muons, where the number can be calculated by two methods: \begin{itemize} \item counting hits on the active target in coincidence with hits on the upstream scintillator counter; \item inferred from number of X-rays recorded by the germanium detector. \end{itemize} This analysis was done on a subset of the active target runs \numrange{2119}{2140} because of the problem of wrong clock frequency found in the data quality checking shown in \cref{fig:lldq}. The data set contains %\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events. \num{6.43E7} muon events. \subsection{Number of stopped muons from active target counting} \label{sub:event_selection} Because of the active target, a stopped muon would cause two coincident hits on the muon counter and the target. The energy of the muon hit on the active target is also well-defined as the narrow-momentum-spread beam was used. The correlation between the energy and timing of all the hits on the active target is shown in \cref{fig:sir2f_Et_corr}. The most intense spot at zero time and about \SI{5}{\MeV} energy corresponds to stopped muons in the thick target. The band below \SI{1}{\MeV} is due to electrons, either in the beam or from muon decay in orbits, or emitted during the cascading of muon to the muonic 1S state. The valley between time zero and 1200~ns shows the minimum distance in time between two pulses. It is the mentioned limitation of the current pulse parameter extraction method where no pile up or double pulses is accounted for. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_E_t_corr} \caption{Energy - timing correlation of hits on the active target.} \label{fig:sir2f_Et_corr} \end{figure} The hits on the silicon active target after 1200~ns are mainly secondary particles from the stopped muons: \begin{itemize} \item electrons from muon decay in the 1S orbit \item products emitted after nuclear muon capture, including: gamma, neutron, heavy charged particles and recoiled nucleus \end{itemize} It can be seen that there is a faint stripe of muons in the time larger than 1200~ns region, they are scattered muons by other materials without hitting the muon counter. The electrons in the beam caused the constant band below 1 MeV and $t > 5000$ ns (see \cref{fig:sir2_1us_slices}). \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_amp_1us_slices} \caption{Energy deposit on the active target in 1000 ns time slices from the muon hit. The peaks at about 800 keV in large delayed time are from the beam electrons.} \label{fig:sir2_1us_slices} \end{figure} From the energy-timing correlation above, the cuts to select stopped muons are: \begin{enumerate} \item has one hit on muon counter (where a threshold was set to reject MIPs), and the first hit on the silicon active target is in coincidence with that muon counter hit: \begin{equation} \lvert t_{\textrm{target}} - t_{\mu\textrm{ counter}}\rvert \le \SI{50}{\ns} \label{eqn:sir2_prompt_cut} \end{equation} \item the first hit on the target has energy of that of the muons: \begin{equation} \SI{3.4}{\MeV} \le E_{\textrm{target}} \le \SI{5.6}{\MeV} \label{eqn:sir2_muE_cut} \end{equation} \end{enumerate} The two cuts~\eqref{eqn:sir2_prompt_cut} and~\eqref{eqn:sir2_muE_cut} give a number of stopped muons counted by the active target: \begin{equation} N_{\mu \textrm{ active Si}} = 9.32 \times 10^6 \label{eqn:n_stopped_si_count} \end{equation} \subsection{Number of stopped muons from the number of X-rays} \label{sub:number_of_stopped_muons_from_the_number_of_x_rays} The number of nuclear captures, hence the number of stopped muons in the active silicon target, can be inferred from the number of emitted muonic X-rays. The reference energies and intensities of the most prominent lines of silicon and aluminium are listed in \cref{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-rays are emitted during the cascading of the muon to the muonic 1S state in the time scale of \SI{E-9}{\s}, so the hit caused by the X-rays must be in coincidence with the muon hit on the active target. Therefore an additional timing cut is applied for the germanium detector hits: \begin{equation} \lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns} \label{eqn:sir2_ge_cut} \end{equation} The germanium spectrum after three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and~\eqref{eqn:sir2_ge_cut} is plotted in \cref{fig:sir2_xray}. The $(2p-1s)$ line clearly showed up at \SI{400}{\keV} with very low background. A peak at \SI{476}{\keV} is identified as the $(3p-1s)$ transition. Higher transitions such as $(4p-1s)$, $(5p-1s)$ and $(6p-1s)$ can also be recognised at \SI{504}{\keV}, \SI{516}{\keV} and \SI{523}{\keV}, respectively. %The $(2p-1s)$ %line is seen at 399.5~\si{\keV}, 0.7~\si{\keV} off from the reference value of %400.177~\si{\keV}. 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_22} \caption{Prompt muonic X-rays spectrum from the active silicon target. } \label{fig:sir2_xray} \end{figure} The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian peak on top of a first-order polynomial from \SIrange{395}{405}{\keV}. Using the same procedure of correcting described in \cref{sub:germanium_detector}, and taking detector acceptance and X-ray intensity into account (see \cref{tab:sir2_xray_corr}), the number of muon stopped is: \begin{equation} N_{\mu \textrm{ stopped X-ray}} = (9.16 \pm 0.28)\times 10^6, \label{eqn:n_stopped_xray_count} \end{equation} which is consistent with the number of X-rays counted using the active target. \begin{table}[btp] \begin{center} \begin{tabular}{@{}llll@{}} \toprule \textbf{Measured X-rays} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule Gross integral & 3083 & & \\ Background & 101.5 & & \\ Net area $(2p-1s)$ & 2929.7 & 56.4 & 0.02 \\ \vspace{0.03em}\\ \toprule \textbf{Corrections} & \textbf{Value} & \multicolumn{2}{c}{\textbf{Details}}\\ \midrule Random summing & 1.06 & \multicolumn{2}{l}{average count rate \SI{491.4}{\Hz},}\\ & & \multicolumn{2}{l}{pulse length \SI{57}{\us}}\\ TRP reset & 1.03 & \multicolumn{2}{l}{\SI{298}{\s} loss during \SI{9327}{\s} run period}\\ Self-absorption & 1.008 & \multicolumn{2}{l}{silicon thickness \SI{750}{\um},}\\ & & \multicolumn{2}{l}{linear attenuation \SI{0.224}{\per\cm}}\\ True coincidence & 1 & \multicolumn{2}{l}{omitted} \\ \vspace{0.03em}\\ \toprule \textbf{Efficiency and intensity} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule Detector efficiency & \num{4.40E-4} & \num{0.978E-5} & 0.02 \\ X-ray intensity & 0.803 & 0.008 & 0.009 \\ \vspace{0.03em}\\ \toprule \textbf{Results} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule Number of X-rays emitted & \num{7.36E6} & \num{0.22E6} & 0.03 \\ Number of muons stopped & \num{9.16E6} & \num{0.28E6} & 0.03 \\ \bottomrule \end{tabular} \end{center} \caption{Corrections, efficiency and intensity used in calculating the number of X-rays from the active target.} \label{tab:sir2_xray_corr} \end{table} %In order to measure the charged particles after nuclear muon capture, one would %pick events where the emitted particles are well separated from the %muon stop time. The energy timing correlation plot suggests a timing window %starting from at least 1200~ns, therefore another cut is introduced: %\begin{enumerate} %\setcounter{enumi}{2} %\item there are at least two hits on the active target, the time %difference between the second hit on target (decay or capture product) and %the muon counter hit is at least 1300 ns: %\begin{equation} %t_{\textrm{target 2nd hit}} - t_{\mu\textrm{ counter}} \geq \SI{1300}{\ns} %\label{eqn:sir2_2ndhit_cut} %\end{equation} %\end{enumerate} %The three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and %~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Particle identification by specific energy loss} \label{sec:particle_identification_by_specific_energy_loss} In this analysis, a subset of runs from \numrange{2808}{2873} with the 100-\si{\um} aluminium target is used because of following advantages: \begin{itemize} \item it was easier to stop and adjust the muon stopping distribution in this thicker target; \item a thicker target means more stopped muons due to higher muon rate available at higher momentum and less scattering. \end{itemize} Muons momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of 1.09, normalised to \SI{28}{\MeV\per\cc}) were used for this target after a momentum scanning as described in the next subsection. \subsection{Momentum scan for the 100-\si{\um} aluminium target} \label{sub:momentum_scan_for_the_100_} Before deciding to use the momentum scaling factor of 1.09, we have scanned with momentum scales ranging from 1.04 to 1.12 to maximise the observed X-rays rate(and hence maximising the rate of stopped muons). The X-ray spectrum at each momentum point was accumulated in more than 30 minutes to assure a sufficient amount of counts. Details of the scanning runs are listed in \cref{tab:al100_scan}. \begin{table}[htb] \begin{center} \begin{tabular}{cccc} \toprule \textbf{Momentum (\si{\MeV\per\cc})} & \textbf{Scaling factor} & \textbf{Runs} & \textbf{Length (s)}\\ \midrule 29.12 & 1.04 & \numrange{2609}{2613} &2299\\ 29.68 & 1.06 & \numrange{2602}{2608} &2563\\ 29.96 & 1.07 & \numrange{2633}{2637} &2030\\ 30.24 & 1.08 & \numrange{2614}{2621} &3232\\ 30.52 & 1.09 & \numrange{2808}{2813} &2120\\ 30.80 & 1.10 & \numrange{2625}{2632} &3234\\ 31.36 & 1.12 & \numrange{2784}{2792} &2841\\ \bottomrule \end{tabular} \end{center} \caption{Momentum scanning runs for the 100-\si{\um} aluminium target.} \label{tab:al100_scan} \end{table} The on-site quick analysis suggested the 1.09 scaling factor was the optimal value so it was chosen for all the runs on this aluminium target. But the offline analysis later showed that the actual optimal factor was 1.08. There were two reasons for the mistake: \begin{enumerate} \item the X-ray rates were normalised to run length, which is biased since there are more muons available at higher momentum; \item the $(2p-1s)$ peaks of aluminium at \SI{346.828}{\keV} were not fitted properly. The peak is interfered by a background peak at \SI{351}{\keV} from $^{214}$Pb, but the X-ray peak area was obtained simply by subtracting an automatically estimated background. \end{enumerate} In the offline analysis, the X-ray peak and the background peak are fitted by two Gaussian peaks on top of a first-order polynomial background. The X-ray peak area is then normalised to the number of muons hitting the upstream detector (\cref{fig:al100_xray_fit}). \begin{figure}[htb] \centering \includegraphics[width=0.47\textwidth]{figs/al100_xray_fit} \includegraphics[width=0.47\textwidth]{figs/al100_xray_musc} \caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium and the background peak at \SI{351}{\keV} (left). The number of muons is integral of the upstream scintillator spectrum (right) from \numrange{400}{2000} ADC channels.} \label{fig:al100_xray_fit} \end{figure} The ratio between the number of X-rays and the number of muons as a function of momentum scaling factor is plotted on \cref{fig:al100_scan_rate}. The trend showed that muons penetrated deeper as the momentum increased, reaching the optimal value at the scale of 1.08, then decreased as punch through happened more often from 1.09. The distributions of stopped muons are illustrated by MC results on the right hand side of \cref{fig:al100_scan_rate}. With the 1.09 scale beam, the muons stopped \SI{28}{\um} off-centred to the right silicon arm. \begin{figure}[htb] \centering \includegraphics[width=0.47\textwidth]{figs/al100_scan_rate} \includegraphics[width=0.47\textwidth]{figs/al100_mu_stop_mc} \caption{Number of X-rays per incoming muon as a function of momentum scaling factor (left); and muon stopping distributions from MC simulation (right). The depth of muons is measured normal to surface of the target facing the muon beam.} \label{fig:al100_scan_rate} \end{figure} \subsection{Event selection for the passive targets} \label{sub:event_selection_for_the_passive_targets} As described in the \cref{sec:analysis_framework}, the hits on all detectors are re-organised into muon events: central muons; and all hits within \SI{\pm 10}{\us} from the central muons. The dataset from runs \numrange{2808}{2873} contains \num{1.17E+9} of such muon events. Selection of proton (and other heavy charged particles) events 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 $\pm 0.5$~\si{\us}\ around the thick silicon hit, the two hits are considered to belong to one particle. The specific energy loss spectra recorded by the two silicon arms are plotted on \cref{fig:al100_dedx}. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/al100_dedx} \caption{Energy loss in thin silicon detectors as a function of total energy recorded by both thin and thick detectors.} \label{fig:al100_dedx} \end{figure} With the aid from MC study (\cref{fig:pid_sim}), the banding on \cref{fig:al100_dedx} 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 band of protons is then extracted by cut on likelihood probability calculated as: \begin{equation} p_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}} \exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}\right]} \end{equation} where $\Delta E_{\textrm{meas.}}$ is energy deposition measured by the thin silicon detector by a certain proton at energy $E_i$, $\Delta E_i$ and $\sigma_{\Delta E}$ are the expected and standard deviation of the energy loss caused by the proton calculated by MC study. A threshold is set at \num{1E-4} to extract protons, the resulted band of protons is shown in (\cref{fig:al100_protons}). \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 (left). The proton spectrum (right) is obtained by projecting the proton band onto the total energy axis.} \label{fig:al100_protons} \end{figure} The cut efficiency in the energy range from \SIrange{2}{12}{\MeV} is estimated by MC study. The fraction of protons that do not satisfy the probability cut is 0.5\%. The number of other charged particles that are misidentified as protons depends on the ratios between those species and protons. Assuming a proton:deuteron:triton:alpha:muon ratio of 5:2:1:2:2, the number of misidentified hits is 0.1\% of the number of protons. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Proton emission rate from aluminium} \label{sec:proton_emission_rate_from_aluminium} The analysis is done on the same dataset used in \cref{sec:particle_identification_by_specific_energy_loss}. Firstly, the number of protons emitted is extracted using specific energy loss. Then correction for energy loss inside the target is applied. Finally, the number of protons is normalised to the number of nuclear muon captures. \subsection{Number of protons emitted} \label{sub:number_of_protons_emitted} The numbers of protons in the energy range from \SIrange{2.2}{8.5}{\MeV} after applying the probability cut are: \begin{align} N_{\textrm{p meas. left}} = 1822\\% \pm 42.7\\ N_{\textrm{p meas. right}} = 2373% \pm 48.7 \end{align} The right arm received significantly more protons than the left arm did, which is expected as in \cref{sub:momentum_scan_for_the_100_} where it is shown that muons stopped off-centred to the right arm. \subsection{Corrections for the number of protons} \label{sub:corrections_for_the_number_of_protons} The protons spectra observed by the silicon detectors have been modified by the energy loss inside the target so correction (or unfolding) is necessary. The unfolding, essentially, is finding a response function that relates proton's true energy and measured value. This can be done in MC simulation by generating protons with a spatial distribution as close as possible to the real distribution of muons inside the target, then counting the number of protons reaching the silicon detectors. Such response function conveniently includes the geometrical acceptance. For the 100-\si{\um} aluminium target and muons at the momentum scale of 1.09, the parameters of the initial protons are: \begin{itemize} \item horizontal distribution: Gaussian \SI{26}{\mm} FWHM, centred at the centre of the target; \item vertical distribution: Gaussian \SI{15}{\mm} FWHM, centred at the centre of the target; \item depth: Gaussian \SI{69.2}{\um} FWHM, centred at \SI{68.8}{\um}-deep from the upstream face of the target; \item energy: flatly distributed from \SIrange{1.5}{15}{\MeV}. \end{itemize} The resulting response matrices for the two arms are presented in \cref{fig:al100_resp_matrices}. These are then used as MC truth to train and test the unfolding code. The code uses an existing ROOT package called RooUnfold~\cite{Adye.2011} where the iterative Bayesian unfolding method is implemented. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{./figs/al100_resp} \caption{Response functions for the two silicon arms.} \label{fig:al100_resp_matrices} \end{figure} %After training, the unfolding code is applied on the measured spectra from the %left and right arms. The unfolded proton spectra in \cref{fig:al100_unfold} %reasonably reflect the distribution of initial protons which is off-centred to %the right arm. The path length to the left arm is longer so less protons at %energy lower than \SI{5}{\MeV} could reach the detectors. The sharp low-energy %cut off on the right arm is consistent with the Coulomb barrier for protons, %which is \SI{4.1}{\MeV} for protons emitted from $^{27}$Mg. The unfolded spectra using the two observed spectra at the two arms as input are shown in \cref{fig:al100_unfold}. The two unfolded spectra generally agree with each other, except for a few first and last bins. The discrepancy and large uncertainties at the low energy region are because of only a small number of protons with those energies could reach the detectors. The jump on the right arm at around \SI{9}{\MeV} can be explained as the punch-through protons were counted as the proton veto counters were not used in this analysis. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr} \caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.} \label{fig:al100_unfold} \end{figure} %Several studies were conducted to assess the performance of the unfolding %code, including: %\begin{itemize} %\item stability against cut-off energy; %\item comparison between the two arms; %\item and unfolding of a MC-generated spectrum. %\end{itemize} The stability of the unfolding code is tested by varying the lower cut-off energy of the input spectrum. \cref{fig:al100_cutoff_study} show that the shapes of the unfolded spectra are stable. The lower cut-off energy of the output increases as that of the input increases, and the shape is generally unchanged after a few bins. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/al100_cutoff_study} \caption{Unfolded spectra with different cut-off energies.} \label{fig:al100_cutoff_study} \end{figure} The proton yields calculated from observed spectra in two arms are compared in \cref{fig:al100_integral_comparison} where the upper limit of the integrals is fixed at \SI{8}{\MeV}, and the lower limit is varied in \SI{400}{\keV} step. The difference is large at cut-off energies less than \SI{4}{\MeV} due to large uncertainties at the first bins. Above \SI{4}{\MeV}, the two arms show consistent numbers of protons. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/al100_integral_comparison} \caption{Proton yields calculated from two arms.} \label{fig:al100_integral_comparison} \end{figure} The yields of protons from \SIrange{4}{8}{\MeV} are: \begin{align} N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\\ N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3 \end{align} The number of emitted protons is taken as average of the two yields: \begin{equation} N_{\textrm{p unfold}} = (169.3 \pm 1.9) \times 10^3 \end{equation} \subsection{Number of nuclear captures} \label{sub:number_of_nuclear_captures} %\begin{figure}[!htb] %\centering %\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec} %\caption{X-ray spectrum from the aluminium target, the characteristic %$(2p-1s)$ line shows up at 346.67~keV} %\label{fig:al100_ge_spec} %\end{figure} %The X-ray spectrum on the germanium detector is shown on %\cref{fig:al100_ge_spec}. Fitting the double peaks on top of a first-order polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same procedure as in the case of the active target, the number stopped muons and the number of nuclear captures are: \begin{align} N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\\ N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6 \end{align} \subsection{Proton emission rate} \label{sub:proton_emission_rate} The proton emission rate in the range from \SIrange{4}{8}{\MeV} is therefore: \begin{equation} R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.7\times 10^{-2} \label{eq:proton_rate_al} \end{equation} The total proton emission rate can be estimated by assuming a spectrum shape with the same parameterisation as in \eqref{eqn:EH_pdf}. The \eqref{eqn:EH_pdf} function has a power rising edge, and a exponential decay falling edge. The falling edge has only one decay component and is suitable to describe the proton spectrum with the equilibrium emission only assumption. The pre-equilibrium emission contribution should be small for low-$Z$ material, for aluminium the contribution of this component is 2.2\% according to Lifshitz and Singer~\cite{LifshitzSinger.1980}. The fitted results are shown in \cref{fig:al100_parameterisation} and \cref{tab:al100_fit_pars}. The average spectrum is obtained by taking the average of the two unfolded spectra from the left and right arms. The fitted parameters are compatible with each other within their errors. Using the fitted parameters of the average spectrum, the integration in range from \SIrange{4}{8}{\MeV} is 51\% of the total number of protons. The total proton emission rate is therefore estimated to be $3.5\times 10^{-2}$. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/al100_parameterisation} \caption{Fitting of the unfolded spectra.} \label{fig:al100_parameterisation} \end{figure} \begin{table}[htb] \begin{center} \begin{tabular}{l S[separate-uncertainty=true] S[separate-uncertainty=true] S[separate-uncertainty=true]} \toprule \textbf{Parameter} &{\textbf{Left}} & {\textbf{Right}} & {\textbf{Average}}\\ \midrule $A \times 10^{-5}$ & 2.0 \pm 0.7 & 1.3 \pm 0.1 & 1.5 \pm 0.3\\ $T_{th}$ (\si{\keV}) & 1301 \pm 490 & 1966 \pm 68 & 1573 \pm 132\\ $\alpha$ & 3.2 \pm 0.7 & 1.2 \pm 0.1 & 2.0 \pm 1.2\\ $T_{0}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\ \bottomrule \end{tabular} \end{center} \caption{Parameters of the fits on the unfolded spectra, the average spectrum is obtained by taking average of the unfolded spectra from left and right arms.} \label{tab:al100_fit_pars} \end{table} \subsection{Uncertainties of the emission rate} \label{sub:uncertainties_of_the_emission_rate} The uncertainties of the emission rate come from: \begin{itemize} \item uncertainties in the number of nuclear captures: these were discussed in \cref{sub:number_of_stopped_muons_from_the_number_of_x_rays}; \item uncertainties in the number of protons: \begin{itemize} \item statistical uncertainties of the measured spectra which are propagated during the unfolding process; \item systematic uncertainties due to misidentification: this number is small compared to other uncertainties as discussed in \cref{sub:event_selection_for_the_passive_targets}; \item systematic uncertainty from the unfolding \end{itemize} \end{itemize} The last item is studied by MC method using the parameterisation in \cref{sub:proton_emission_rate}: \begin{itemize} \item protons with energy distribution obeying the parameterisation are generated inside the target. The spatial distribution is the same as that of in \cref{sub:corrections_for_the_number_of_protons}. MC truth including initial energies and positions are recorded; \item the number of protons reaching the silicon detectors are counted, the proton yield is set to be the same as the measured yield to make the statistical uncertainties comparable; \item the unfolding is applied on the observed proton spectra, and the results are compared with the MC truth. \end{itemize} \begin{figure}[htb] \centering \includegraphics[width=0.48\textwidth]{figs/al100_MCvsUnfold} \includegraphics[width=0.48\textwidth]{figs/al100_unfold_truth_ratio} \caption{Comparison between an unfolded spectrum and MC truth: spectra (left), and yields (right). The ratio is defined as $\textrm{(Unfold - MC truth)/(MC truth)}$} \label{fig:al100_MCvsUnfold} \end{figure} \Cref{fig:al100_MCvsUnfold} shows that the yield obtained after unfolding is in agreement with that from the MC truth. The difference is less than 5\% if the integration is taken in the range from \SIrange{4}{8}{\MeV}. Therefore a systematic uncertainty of 5\% is assigned for the unfolding routine. A summary of uncertainties in the measurement of proton emission rate is presented in \cref{tab:al100_uncertainties_all}. \begin{table}[htb] \begin{center} \begin{tabular}{@{}ll@{}} \toprule \textbf{Item}& \textbf{Value} \\ \midrule Number of muons & 3.2\% \\ Statistical from measured spectra & 1.1\% \\ Systematic from unfolding & 5.0\% \\ Systematic from PID & \textless0.5\% \\ \midrule Total & 6.1\%\\ \bottomrule \end{tabular} \end{center} \caption{Uncertainties of the proton emission rate.} \label{tab:al100_uncertainties_all} \end{table} \section{Results of the initial analysis} \label{sec:results_of_the_initial_analysis} \subsection{Verification of the experimental method} \label{sub:verification_of_the_experimental_method} The experimental method described in \cref{sub:experimental_method} has been validated: \begin{itemize} \item Number of muon capture normalisation: the number of stopped muons calculated from the muonic X-ray spectrum is shown to be consistent with that calculated from the active target spectrum. \item Particle identification: the particle identification by specific energy loss has been demonstrated. The banding of different particle species is clearly visible. The proton extraction method using cut on likelihood probability has been established. Since the distribution of $\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do not make the cut is 0.5\%, much larger than the threshold at \num{1E-4}. The fraction of other charged particles being misidentified as protons is smaller than 0.1\%. These uncertainties from particle identification are still small in compared with the uncertainty of the measurement (2.3\%). \item Unfolding of the proton spectrum: the unfolded spectra inferred from two measurements at the two silicon arms show good agreement with each other, and with the muon stopping distribution obtained in the momentum scanning analysis. \end{itemize} \subsection{Proton emission rates and spectrum} \label{sub:proton_emission_rates_and_spectrum} The proton emission spectrum in \cref{sub:proton_emission_rate} peaks around \SI{3.7}{\MeV} which is a little below the Coulomb barrier for proton of \SI{3.9}{\MeV} calculated using \eqref{eqn:classical_coulomb_barrier}. The spectrum has a decay constant of \SI{2.6}{\MeV} in higher energy region, makes the emission probability drop more quickly than silicon charged particles spectrum of Sobottka and Wills~\cite{SobottkaWills.1968} where the decay constant was \SI{4.6}{\MeV}. This can be explained as the silicon spectrum includes other heavier particles which have higher Coulomb barriers, hence contribute more in the higher energy bins, effectively reduces the decay rate. The partial emission rate measured in the energy range from \SIrange{4}{8}{\MeV} is: \begin{equation} R_{p \textrm{ meas. }} = (1.7 \pm 0.1)\%. \label{eqn:meas_partial_rate} \end{equation} The total emission rate from aluminium assuming the spectrum shape holds for all energy is: \begin{equation} R_{p \textrm{ total}} = (3.5 \pm 0.2)\%. \label{eqn:meas_total_rate} \end{equation} \subsubsection{Comparison to theoretical and other experimental results} \label{ssub:comparison_to_theoretical_and_other_experimental_results} There is no existing experimental or theoretical work that could be directly compared with the obtained proton emission rate. Indirectly, it is compatible with the figures calculated by Lifshitz and Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980} listed in \cref{tab:lifshitzsinger_cal_proton_rate}. It is significantly larger than the rate of 0.97\% for the $(\mu,\nu p)$ channel, and does not exceed the inclusion rate for all channels $\Sigma(\mu,\nu p(xn))$ at 4\%, leaving some room for other modes such as emission of deuterons or tritons. Certainly, when the full analysis is available, deuterons and tritons emission rates could be extracted and the combined emission rate could be compared directly with the inclusive rate. The result \eqref{eqn:meas_total_rate} is greater than the probability of the reaction $(\mu,\nu pn)$ measured by Wyttenbach et al.~\cite{WyttenbachBaertschi.etal.1978} at 2.8\%. It is expectable because the contribution from the $(\mu,\nu d)$ channel should be small since it needs to form a deuteron from a proton and a neutron. The rate of 3.5\% was estimated with an assumption that all protons are emitted in equilibrium. With the exponential constant of \SI{2.6}{\MeV}, the proton yield in the range from \SIrange{40}{70}{\MeV} is negligibly small ($\sim\num{E-8}$). However, Krane and colleagues reported a significant yield of 0.1\% in that region~\cite{KraneSharma.etal.1979}. The energetic proton spectrum shape also has a different exponential constant of \SI{7.5}{\MeV}. One explanation for these protons is that they are emitted by other mechanisms, such as capture on two-nucleon cluster suggested by Singer~\cite{Singer.1961} (see \cref{sub:theoretical_models} and \cref{tab:lifshitzsinger_cal_proton_rate_1988}). Despite being sizeable, the yield of high energy protons is still small (3\%) in compared with the result in \eqref{eqn:meas_total_rate}. %The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in %\eqref{eqn:wyttenbach_ratio}. \subsubsection{Comparison to the silicon result} \label{ssub:comparison_to_the_silicon_result} The probability of proton emission per nuclear capture of 3.5\% is indeed much smaller than that of silicon. It is even lower than the rate of the no-neutron reaction $(\mu,\nu p)$. This can be explained as the resulted nucleus from muon capture on silicon, $^{28}$Al, is an odd-odd nucleus and less stable than that from aluminium, $^{27}$Mg. The proton separation energy for $^{28}$Al is \SI{9.6}{\MeV}, which is significantly lower than that of $^{27}$Mg at \SI{15.0}{\MeV}~\cite{AudiWapstra.etal.2003}.