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\chapter{Results and discussions}
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\label{cha:results_and_discussions}
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\section{Verification of the experimental method}
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\label{sec:verification_of_the_experimental_method}
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\subsection{Number of stopped muons calculation}
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\label{sub:number_of_stopped_muons_normalisation}
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The number of stopped muons calculated from the muonic X-ray spectrum is shown
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to be consistent with that calculated from the active target spectrum. This
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proves the validity of normalisation using muon X-ray measurement.
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\subsection{Particle identification and unfolding}
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\label{sub:particle_identification_and_unfolding}
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The particle identification using specific energy loss using cut on
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likelihood probability is shown in
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\cref{sub:event_selection_for_the_passive_targets}. Since the distribution of
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$\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do not
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make the cut is 0.5\%, much larger than the threshold at \num{1E-4}. However,
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that missing fraction is small compared to the statistical uncertainty of the
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measurement (2.3\%) so the threshold is sufficient.
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\chapter{Discussions on the impact to the COMET Phase-I}
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\label{cha:discussions_on_the_impact_to_the_comet_phase_i}
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The measured proton emission rate of 3.5\% is about 5 times
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smaller than the figure using to make the baseline design of the CDC in COMET
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Phase-I. The spectrum shape
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peaks around \SI{4}{\MeV} rather than at \SI{2.5}{\MeV}, and decays more
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quickly in compared with the silicon spectrum(\cref{fig:sobottka_spec}).
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Therefore CDC hit rate due to proton should be smaller than the current
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estimation.
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The observed spectra on the two silicon arms reflect the muon stopping
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distribution discussed in \cref{sub:momentum_scan_for_the_100_} where more
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muons stopped at the downstream side of the target. The proton yields
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calculated from two arms are consistent with each other, and show that the muon
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stopping distribution used to generate the response matrices is reasonable.
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The CDC proton hit rate is calculated by a toy MC study. The dimensions of the
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geometry shown in \cref{fig:cdc_toy_mc} are from \cref{ssub:CDC_configuration}.
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The inner wall of the CDC is \SI{0.5}{\mm} thick CFRP.
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A proton absorber made
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of CFRP is placed \SI{5}{\cm} far from the inner wall of the CDC. The
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absorber's thickness is varied from 0 (no absorber) to \SI{1}{\mm}.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.55\textwidth]{figs/cdc_toy_mc}
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\caption{Geometry of the toy MC study for hit rate study.}
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\label{fig:cdc_toy_mc}
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\end{figure}
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\section{Emission rate of protons and the COMET Phase I's CDC}
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\label{sec:emission_rate_of_protons_and_the_comet_phase_i_s_cdc}
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The proton emission rate from the 100-\si{\um} aluminium target is
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$(3.5 \pm 0.2)$\%. This rate is significantly larger than the calculation rate
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of 0.97\% by Lifshitz and Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980}.
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The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
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\eqref{eqn:wyttenbach_ratio}.
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The rate smaller that the proton emission rate from silicon of
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5.3\%~\cite{Measday.2001} which is expected since an odd-odd nucleus as
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$^{28}$Al is less stable than an even-odd one.
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The protons with the energy spectrum shape as in
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\cref{sub:proton_emission_rate} are generated inside the COMET's muon stopping
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targets which are 17 200-\si{\um}-thick aluminium discs. The spatial
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distribution of protons resembles the stopping distribution of muons inside the
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target discs calculated from the full MC simulation of the COMET detectors
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(\cref{fig:cdc_toy_mc_init_pos}).
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.65\textwidth]{figs/cdc_toy_mc_init_pos_xy}
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\includegraphics[width=0.60\textwidth]{figs/cdc_toy_mc_init_pos_z}
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\caption{Spatial distribution of the generated protons in X, Y (top) and
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Z (bottom). Z is the axis of the CDC, X, Y are the horizontal and vertical
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axes respectively.}
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\label{fig:cdc_toy_mc_init_pos}
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\end{figure}
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For the COMET Phase I experiment, the emission rate of 3.5\% is about 5 times
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smaller than the figure using to design the CDC. The measured spectrum shape
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peaks around \SI{4}{\MeV} rather than \SI{2.5}{\MeV} in the silicon
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spectrum(\cref{fig:sobottka_spec}). Therefore the proton hit rate on the CDC
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should be smaller than the current estimation.
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The protons are then tracked in a \SI{1}{\tesla} magnetic field. The protons
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reaching the absorber, inner wall and the sensitive volume of the CDC are
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recorded (see \cref{fig:cdc_toy_mc_p_spec_500um}).
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\begin{figure}[!htb]
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\centering
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\includegraphics[width=0.75\textwidth]{figs/cdc_toy_mc_p_spec_500um}
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\caption{Toy MC study of the CDC hit rate due to protons. The absorber
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thickness was set at \SI{0.5}{\mm} in this plot.}
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\label{fig:cdc_toy_mc_p_spec_500um}
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\end{figure}
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The CDC proton hit rate is calculated by a toy MC study. The protons with the
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energy spectrum as the parameterisation in \cref{sub:proton_emission_rate} are
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generated inside the COMET's muon stopping targets which are 17
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200-\si{\um}-thick aluminium discs. A proton absorber made of CFRP is placed
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\SI{5}{\cm} far from the inner wall of the CDC.
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A muon stopping rate of \SI{1.3E9}{\Hz} is assumed as in the COMET Phase I's
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TDR. The number of proton emitted is then $\num{1.3E9} \times 0.609 \times
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0.035 = \SI{2.8E7}{\Hz}$. The hit rates on a single cell in the inner most
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@@ -52,13 +55,19 @@ layer due to these protons with
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different absorber thickness are shown in \cref{tab:proton_cdc_hitrate}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{l r}
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\begin{tabular}{S S S S}
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\toprule
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\textbf{Absorber thickness} & \textbf{Hit rate}\\
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{\textbf{Absorber}} &{\textbf{Inner wall}} & {\textbf{Total CFRP}}&
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{\textbf{Proton}}\\
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{\textbf{thickness}} &{\textbf{thickness}} & {\textbf{thickness}}&
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{\textbf{hit rate}}\\
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{(\si{\mm})} & {(\si{\mm})} & {(\si{\mm})} & {(\si{\Hz})}\\
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\midrule
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\SI{1}{\mm} & \SI{2}{\Hz}\\
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\SI{0.5}{\mm} & \SI{126}{\Hz}\\
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\SI{0}{\mm} & \SI{1436}{\Hz}\\
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1 &0.5&1.5 & 2\\
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0.5 &0.5&1.0 & 126\\
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0 &0.5&0.5 & 1436\\
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0 &0.3&0.3 & 8281\\
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0 &0.1&0.1 & 15011\\
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\bottomrule
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\end{tabular}
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\end{center}
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@@ -66,7 +75,14 @@ different absorber thickness are shown in \cref{tab:proton_cdc_hitrate}.
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\label{tab:proton_cdc_hitrate}
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\end{table}
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The proton hit rate even without the absorber is only \SI{1.4}{\kHz}, much
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smaller than the current estimation of \SI{11}{\kHz} (using 1-mm-thick
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absorber). Therefore a proton absorber is not needed for the COMET Phase I's
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CDC.
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At the baseline design of \SI{0.5}{\mm}, the hit rate is only \SI{126}{\Hz},
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much smaller than the current estimation at \SI{34}{\kHz}. Even without the
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absorber, proton hit rate remains low at \SI{1.4}{\kHz}. Therefore a proton
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absorber is not needed for the COMET Phase I's CDC.
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Without the proton absorber, the momentum spread of the signal electron
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reduces from \SI{167}{\keV} to \SI{131}{\keV}. If a lower momentum spread is
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desired, it is possible to reduce the thickness of the inner wall. The last
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two rows of \cref{tab:proton_cdc_hitrate} show that even with thinner walls at
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\SI{0.3}{\mm} and \SI{0.1}{\mm} the hit rate by protons are still at
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manageable levels.
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