diff --git a/thesis/chapters/chap5_alcap_setup.tex b/thesis/chapters/chap5_alcap_setup.tex index 0f8e3d0..df14625 100644 --- a/thesis/chapters/chap5_alcap_setup.tex +++ b/thesis/chapters/chap5_alcap_setup.tex @@ -765,6 +765,7 @@ sets are shown in \cref{tb:stat}. % section data_sets (end) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Analysis framework} +\label{sec:analysis_framework} \subsection{Concept} \label{sub:concept} Since the AlCapDAQ is a trigger-less system, it stored all waveforms of the diff --git a/thesis/chapters/chap6_analysis.tex b/thesis/chapters/chap6_analysis.tex index df4e407..b4eb45b 100644 --- a/thesis/chapters/chap6_analysis.tex +++ b/thesis/chapters/chap6_analysis.tex @@ -9,33 +9,37 @@ Purposes of the analysis include: using specific energy loss; \item extracting a preliminary rate of proton emission from aluminium. \end{itemize} -\section{Charged particles following muon capture on a thick silicon target} -\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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. -Firstly, the number of charged particles emitted after nuclear muon capture on -the active target is calculated. The charged particles yield then is normalised -to the number of nuclear muon capture to obtain an emission rate. -%Finally, the -%rate is compared with that from the literature. - -\subsection{Event selection} +\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 5 MeV energy corresponds to stopped muons in the thick target. The -band below 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. +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 @@ -79,116 +83,19 @@ From the energy-timing correlation above, the cuts to select stopped muons are: \label{eqn:sir2_muE_cut} \end{equation} \end{enumerate} -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}. - -The number of stopped muons can also be calculated from the number of muonic -X-rays recorded by the germanium detector. The 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: +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} - \lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns} - \label{eqn:sir2_ge_cut} + N_{\mu \textrm{ active Si}} = 9.32 \times 10^6 + \label{eqn:n_stopped_si_count} \end{equation} -%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -\subsection{Number of charged particles with energy above \SI{3}{\MeV}} -\label{sub:number_of_charged_particles_with_energy_from_8_10_mev} -As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation -in \cref{fig:sir2_mc_pdfs}, there are several components in -the energy spectrum recorded by the active target: -\begin{enumerate} - \item charged particles from nuclear muon capture, this is the signal of - interest; - \item beam electrons with a characteristic Landau peak around \SI{800}{\keV}, - dominating at large delay (from \SI{6500}{\ns}), causing background at - energy lower than \SI{1}{\MeV} which drops sharply at energy larger than - \SI{3}{\MeV}; - \item electrons from muon decay-in-orbit (DIO) and recoiled nuclei - from neutron emitting muon captures, peak at - around \SI{300}{\keV}, dominate the region where energy smaller than - \SI{1}{\MeV} and delay less than \SI{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 1300 -- 10000~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 scales between - components are arbitrarily chosen for the purpose of illustration.} - \label{fig:sir2_mc_pdfs} -\end{figure} - -An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the -beam electrons at low energy. In order to obtain the yields of backgrounds -above \SI{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}} &= 2.57 \pm 856 - \label{eqn:sir2_n_chargedparticles} -\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} - -% 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 \cref{tab:mucap_pars}. +\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} @@ -208,130 +115,452 @@ et al.~\cite{MeasdayStocki.etal.2007} are listed in \cref{tab:mucap_pars}. \label{tab:mucap_pars} \end{table} -The muonic X-ray spectrum emitted from the active target is shown in -\cref{fig:sir2_xray}. 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. +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} - \caption{Prompt 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. + \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 \cref{fig:al100_mu_stop_mc}. With the 1.09 scale beam, the muons +stopped \SI{28}{\um} off-centre to the right silicon arm. +\begin{figure}[htb] + \centering + \includegraphics[width=0.85\textwidth]{figs/al100_scan_rate} + \caption{Number of X-rays per incoming muon as a function of momentum + scaling factor.} + \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} 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}} + e^{\frac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}} +\end{equation} +where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in 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. A cut value of $3\sigma_{\Delta E}$, or +$p_i \ge 0.011$, was used to extract protons (\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} + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +\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} +From the particle identification above, number of protons having energy in the +range from \SIrange{2.2}{8.5}{\MeV} hitting the two arms are: +\begin{align} + N_{\textrm{p meas. left}} = 1789 \pm 42.3\\ + N_{\textrm{p meas. right}} = 2285 \pm 47.8 +\end{align} +The right arm received significantly more protons than the left arm did, which +is expected because in \cref{sub:momentum_scan_for_the_100_} it is shown that +muons stopped off centre to the right arm. + +The uncertainties are statistical only. The systematic uncertainties due to +the cut on protons is estimated to be small compared to the statistical ones. + +\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. +In the unfolding process, a response function that relates proton's true energy +and the measured one is needed. + +The response function is +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\section{Charged particles following muon capture on a thick silicon target} +%\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon} + +%Firstly, the number of charged particles emitted after nuclear muon capture on +%the active target is calculated. The charged particles yield then is normalised +%to the number of nuclear muon capture to obtain an emission rate. +%Finally, the +%rate is compared with that from the literature. + + +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%\subsection{Number of charged particles with energy above \SI{3}{\MeV}} +%\label{sub:number_of_charged_particles_with_energy_from_8_10_mev} +%As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation +%in \cref{fig:sir2_mc_pdfs}, there are several components in +%the energy spectrum recorded by the active target: +%\begin{enumerate} + %\item charged particles from nuclear muon capture, this is the signal of + %interest; + %\item beam electrons with a characteristic Landau peak around \SI{800}{\keV}, + %dominating at large delay (from \SI{6500}{\ns}), causing background at + %energy lower than \SI{1}{\MeV} which drops sharply at energy larger than + %\SI{3}{\MeV}; + %\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei + %from neutron emitting muon captures, peak at + %around \SI{300}{\keV}, dominate the region where energy smaller than + %\SI{1}{\MeV} and delay less than \SI{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 1300 -- 10000~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 scales between + %components are arbitrarily chosen for the purpose of illustration.} + %\label{fig:sir2_mc_pdfs} +%\end{figure} + +%An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the +%beam electrons at low energy. In order to obtain the yields of backgrounds +%above \SI{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}} &= 2.57 \pm 856 + %\label{eqn:sir2_n_chargedparticles} +%\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} + +% 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 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. %The area of the $(2p-1s)$ peak is $2929.7 \pm 56.4$ obtained by fitting %a Gaussian peak on top of a first-order polynomial background to the spectrum %in \cref{fgi:sir2_xray} in the region from \SIrange{395}{405}{\keV}. -Using the same procedure of fitting and correcting described in -\cref{sub:germanium_detector}, the number of X-rays is calculated to be 370. -Details of the correction factors are given in \cref{tab:sir2_xray_corr}. -\begin{table}[htb] - \begin{center} - \begin{tabular}{l} - \toprule - \textbf{Col1}\\ - \midrule - item1\\ - \bottomrule - \end{tabular} - \end{center} - \caption{Corrections for the number of X-rays from the active target.} - \label{tab:sir2_xray_corr} -\end{table} -The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of -stopped muons, so the number of stopped muons is: +%The X-ray intensity in \cref{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{370}{4.38\times10^{-4} \times 0.803} \\ - &= 1.05\times10^6 \nonumber -\end{align} -where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the -detector for the 400.177~keV line in \cref{tab:xray_eff}, and -$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon -(80.3\% from \cref{tab:mucap_pars}). +%\begin{align} + %N_{\mu\textrm{ stopped}} &= + %\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\ + %&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\ + %&= 1.05\times10^6 \nonumber +%\end{align} +%where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the +%detector for the 400.177~keV line in \cref{tab:xray_eff}, and +%$I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon +%(80.3\% from \cref{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 -$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of -stopped muons of $9.82\times10^5$ after the cuts described in the event -selection process. +%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 +%$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of +%stopped muons of $9.82\times10^5$ 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}} - = 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5 - \label{eqn:sir2_Ncapture} -\end{equation} -where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per -stopped muon from \cref{tab:mucap_pars}. +%The number of nuclear captured muons is: +%\begin{equation} + %N_{\mu\textrm{ nucl.capture}} = + %N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}} + %= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5 + %\label{eqn:sir2_Ncapture} +%\end{equation} +%where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per +%stopped muon from \cref{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 -\cref{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} +%\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 +%\cref{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\\ +%\begin{table}[htb] + %\begin{center} + %\begin{tabular}{l l l} + %\toprule + %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\\ + %\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} + %\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) %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% diff --git a/thesis/figs/sir2_ges_self_tdiff.png b/thesis/figs/sir2_ges_self_tdiff.png new file mode 100644 index 0000000..ff592b8 Binary files /dev/null and b/thesis/figs/sir2_ges_self_tdiff.png differ diff --git a/thesis/thesis.tex b/thesis/thesis.tex index ac73cc3..9aa2d7b 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -32,8 +32,8 @@ for the COMET experiment} %\input{chapters/chap1_intro} %\input{chapters/chap2_mu_e_conv} %\input{chapters/chap3_comet} -\input{chapters/chap4_alcap_phys} -\input{chapters/chap5_alcap_setup} +%\input{chapters/chap4_alcap_phys} +%\input{chapters/chap5_alcap_setup} \input{chapters/chap6_analysis} %\input{chapters/chap7_results}