diff --git a/thesis/chapters/chap6_analysis.tex b/thesis/chapters/chap6_analysis.tex index 3fa28dd..0bd4bae 100644 --- a/thesis/chapters/chap6_analysis.tex +++ b/thesis/chapters/chap6_analysis.tex @@ -337,13 +337,15 @@ 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}}} + \exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta + E}}\right]} \end{equation} -where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in the thin +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. A cut value of $3\sigma_{\Delta E}$, or -$p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}). +caused by the proton calculated by MC study. A threshold is set to extract +protons at 0.011 (equivalent to $3\sigma_{\Delta E}$), the band of protons is +shown in (\cref{fig:al100_protons}). \begin{figure}[htb] \centering \includegraphics[width=0.47\textwidth]{figs/al100_protons} @@ -354,6 +356,12 @@ $p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}). \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} @@ -365,25 +373,20 @@ 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: +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 + 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_} it is shown that -muons stopped off centre to the right arm. - -%%TODO -The uncertainties are statistical only. The systematic uncertainties due to -the cut on protons is estimated to be small compared to the statistical ones. +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 (also called unfolding, or -reconstruction) is necessary. +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 @@ -413,28 +416,65 @@ method is implemented. \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. +%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. -Comparing the reconstructed spectra from \SIrange{5}{8}{\MeV}, the protons -yields are consistent with each other: +%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 reco. left}} &= (110.9 \pm 2.0)\times 10^3\\ - N_{\textrm{p reco. right}} &= (110.2 \pm 2.3)\times 10^3 + 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} Therefore, the number of emitted protons is taken as average value: \begin{equation} - N_{\textrm{p reco.}} = (110.6 \pm 2.2) \times 10^3 + N_{\textrm{p unfold}} = (169.3 \pm 2.9) \times 10^3 \end{equation} \begin{figure}[htb] \centering - \includegraphics[width=0.85\textwidth]{figs/al100_unfold} + \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} @@ -458,29 +498,84 @@ the number of nuclear captures are: 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 proton emission rate in the range from \SIrange{5}{8}{\MeV} is therefore: + +\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{110.6\times 10^3}{9.57\times 10^6} = 1.16\times + R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.74\times 10^{-2} \label{eq:proton_rate_al} \end{equation} -%\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 protons: - %\begin{itemize} - %\item statistical uncertainty of the measured spectra; - %\item systematic uncertainty due to misidentification; - %\item systematic uncertainty from the unfolding - %\end{itemize} - %\item uncertainties in the number of nuclear captures: - %\begin{itemize} - %\item statistical uncertainty of the number of X-rays; - %\item uncertainty of the detector acceptance; - %\item uncertainty from the corrections: random summing and transistor - %reset amplifier - %\end{itemize} -%\end{itemize} +The total proton emission rate can be estimated by assuming a spectrum shape +with the same parameterisation as in \eqref{eqn:EH_pdf}. The fit parameters +are shown in . With such parameterisation, the integration in +range from \SIrange{4}{8}{\MeV} is 51\% of the total number of protons. The +total proton emission rate is therefore $3.5\times 10^{-2}$. +\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 & 1.6\% \\ + 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} + +The proton emission rate is then $(3.5 \pm 0.2)$\%. diff --git a/thesis/figs/al100_resp.pdf b/thesis/figs/al100_resp.pdf index d4728d2..963ea3a 100644 Binary files a/thesis/figs/al100_resp.pdf and b/thesis/figs/al100_resp.pdf differ diff --git a/thesis/thesis.bib b/thesis/thesis.bib index 154be37..6b34593 100644 --- a/thesis/thesis.bib +++ b/thesis/thesis.bib @@ -119,6 +119,7 @@ Year = {2003}, Pages = {250-303}, Volume = {A506}, + Collaboration = {GEANT4}, Doi = {10.1016/S0168-9002(03)01368-8}, Owner = {NT}, @@ -322,6 +323,23 @@ Timestamp = {2014-05-02} } +@Article{AudiWapstra.etal.2003, + Title = {The Ame2003 atomic mass evaluation: (II). Tables, graphs and references }, + Author = {G. Audi and A.H. Wapstra and C. Thibault}, + Journal = {Nuclear Physics A }, + Year = {2003}, + Note = {The 2003 \{NUBASE\} and Atomic Mass Evaluations }, + Number = {1}, + Pages = {337 - 676}, + Volume = {729}, + + Doi = {http://dx.doi.org/10.1016/j.nuclphysa.2003.11.003}, + ISSN = {0375-9474}, + Owner = {NT}, + Timestamp = {2014-10-26}, + Url = {http://www.sciencedirect.com/science/article/pii/S0375947403018098} +} + @Article{BadertscherBorer.etal.1982, Title = {A search for muon-electron and muon-positron conversion in sulfur}, Author = {Badertscher, A and Borer, K and Czapek, G and Fl{\"u}ckiger, A and H{\"a}nni, H and Hahn, B and Hugentobler, E and Markees, A and Marti, T and Moser, U and others}, @@ -485,6 +503,7 @@ Number = {1}, Pages = {154--197}, Volume = {562}, + Doi = {10.1016/j.nima.2006.03.009}, File = {Published version:Bichsel.2006.pdf:PDF}, Owner = {NT}, diff --git a/thesis/thesis.tex b/thesis/thesis.tex index d9df6d7..9aa2d7b 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -29,13 +29,13 @@ for the COMET experiment} \end{frontmatter} \mainmatter -\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/chap1_intro} +%\input{chapters/chap2_mu_e_conv} +%\input{chapters/chap3_comet} +%\input{chapters/chap4_alcap_phys} +%\input{chapters/chap5_alcap_setup} \input{chapters/chap6_analysis} -\input{chapters/chap7_results} +%\input{chapters/chap7_results} \begin{backmatter} \input{chapters/backmatter}