\chapter{Discussions on the impact to the COMET Phase-I}
\label{cha:discussions_on_the_impact_to_the_comet_phase_i}
The measured proton emission rate of 3.5\% is about 5 times
smaller than the figure using to make the baseline design of the CDC in COMET
Phase-I. The spectrum shape
peaks around \SI{4}{\MeV} rather than at \SI{2.5}{\MeV}, and decays more
quickly in compared with the silicon spectrum(\cref{fig:sobottka_spec}).
Therefore CDC hit rate due to proton should be smaller than the current
estimation.

The CDC proton hit rate is calculated by a toy MC study. The dimensions of the
geometry shown in \cref{fig:cdc_toy_mc} are from \cref{ssub:CDC_configuration}. 
The inner wall of the CDC is \SI{0.5}{\mm} thick CFRP. 
A proton absorber made
of CFRP is placed \SI{5}{\cm} far from the inner wall of the CDC. The
absorber's thickness is varied from 0 (no absorber) to \SI{1}{\mm}. 
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.55\textwidth]{figs/cdc_toy_mc}
  \caption{Geometry of the toy MC study for hit rate study.}
  \label{fig:cdc_toy_mc}
\end{figure}

The protons with the energy spectrum shape as in
\cref{sub:proton_emission_rate} are generated inside the COMET's muon stopping
targets which are 17 200-\si{\um}-thick aluminium discs. The spatial
distribution of protons resembles the stopping distribution of muons inside the
target discs calculated from the full MC simulation of the COMET detectors
(\cref{fig:cdc_toy_mc_init_pos}).
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.65\textwidth]{figs/cdc_toy_mc_init_pos_xy}
    \includegraphics[width=0.60\textwidth]{figs/cdc_toy_mc_init_pos_z}
  \caption{Spatial distribution of the generated protons in X, Y (top) and
    Z (bottom). Z is the axis of the CDC, X, Y are the horizontal and vertical
    axes respectively.}
  \label{fig:cdc_toy_mc_init_pos}
\end{figure}

The protons are then tracked in a \SI{1}{\tesla} magnetic field. The protons
reaching the absorber, inner wall and the sensitive volume of the CDC are
recorded (see \cref{fig:cdc_toy_mc_p_spec_500um}). 
\begin{figure}[!htb]
  \centering
    \includegraphics[width=0.75\textwidth]{figs/cdc_toy_mc_p_spec_500um}
    \caption{Proton energy spectra at different stages from birth to the
      sensitive volume of the CDC. The baseline design of \SI{0.5}{\mm} thick
      absorber and \SI{0.5}{\mm} thick inner wall was used to produce this
      plot.} 
    \label{fig:cdc_toy_mc_p_spec_500um}
\end{figure}

A muon stopping rate of \SI{1.3E9}{\Hz} is assumed as in the COMET Phase I's
TDR. The number of proton emitted is then $\num{1.3E9} \times 0.609 \times
0.035 = \SI{2.8E7}{\Hz}$. The hit rates on a single cell in the inner most
layer due to these protons with 
different absorber thickness are shown in \cref{tab:proton_cdc_hitrate}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{S S S S S S}
      \toprule
      {\textbf{Absorber}} &{\textbf{Inner wall}} & {\textbf{Total CFRP}}&
      {\textbf{Proton}} & {\textbf{Momentum}} & {\textbf{Integrated charge}}\\
      {\textbf{thickness}} &{\textbf{thickness}} & {\textbf{thickness}}&
      {\textbf{hit rate}} &{\textbf{spread $\Delta p$}} &{\textbf{300 days}}\\
      {(\si{\mm})} & {(\si{\mm})} & {(\si{\mm})} & {(\si{\Hz})}
      & {(\si{\keV\per\cc)}} &{(mC/cm)}\\
      \midrule 
      1   &0.5&1.5 & 2    & 195 & 25\\
      0.5 &0.5&1.0 & 126  & 167 & 60\\
      0   &0.5&0.5 & 1436 & 133 & 160\\
      %0   &0.3&0.3 & 8281 & {-} & {-}\\
      %0   &0.1&0.1 & 15011& {-} & {-}\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{CDC proton hit rates at different configuration of proton absorber
    and inner wall. The momentum spreads for \SI{0.5}{\mm} thick inner wall are
    taken from \cref{tab:comet_absorber_impact}.}
  \label{tab:proton_cdc_hitrate}
\end{table}

%\begin{table}[htb]
  %\begin{center}
    %\begin{tabular}{S S S S S}
      %\toprule
      %{\textbf{Absorber}} &{\textbf{Inner wall}} & {\textbf{Total CFRP}}&
      %{\textbf{Proton hit rate}} & {\textbf{Proton hit rate}}\\  
      %{\textbf{thickness}} &{\textbf{thickness}} & {\textbf{thickness}}&
      %{\textbf{Phase-I TDR}} & {\textbf{New estimation}}\\
      %{(\si{\mm})} & {(\si{\mm})} & {(\si{\mm})} & {(\si{\Hz})}& {(\si{\Hz})}\\
      %\midrule 
      %1   &0.5&1.5 & 4E+3 & 2     \\
      %0.5 &0.5&1.0 & 11E+3& 126   \\
      %0   &0.5&0.5 & 30E+3& 1436 \\
      %\bottomrule
    %\end{tabular}
  %\end{center}
  %\caption{CDC proton hit rates at different configuration of proton absorber
    %and inner wall. The momentum spreads for \SI{0.5}{\mm} thick inner wall are
    %taken from \cref{tab:comet_absorber_impact}.}
  %\label{tab:proton_cdc_hitrate}
%\end{table}

%\begin{table}[htb]
  %\begin{center}
    %\begin{tabular}{S S S S S S}
      %\toprule
      %{\textbf{Absorber}} &{\textbf{Inner wall}} & {\textbf{Total CFRP}}& {\textbf{Proton}} & {\textbf{Momentum}} & {\textbf{Integrated charge}}\\
      %{\textbf{thickness}} &{\textbf{thickness}} & {\textbf{thickness}}& {\textbf{hit rate}} &{\textbf{spread $\Delta p$}} &{\textbf{300 days}}\\
      %{(\si{\mm})} & {(\si{\mm})} & {(\si{\mm})} & {(\si{\Hz})} & {(\si{\keV\per\cc)}} &{(mC/cm)}\\
      %\midrule 
      %1   &0.5&1.5 & 2    & 195 & 25\\
      %0.5 &0.5&1.0 & 126  & 167 & 60\\
      %0   &0.5&0.5 & 1436 & 133 & 160\\
      %\bottomrule
    %\end{tabular}
  %\end{center}
  %\caption{CDC proton hit rates at different configuration of proton absorber
    %and inner wall. The momentum spreads for \SI{0.5}{\mm} thick inner wall are
    %taken from \cref{tab:comet_absorber_impact}.}
%\end{table}
At the baseline design of \SI{0.5}{\mm}, the hit rate is only \SI{126}{\Hz},
much smaller than the current estimation at \SI{34}{\kHz}. Even without the
absorber, proton hit rate remains low at \SI{1.4}{\kHz}. 
%Therefore a proton
%absorber is not needed for the COMET Phase I's CDC.

If the proton absorber is not used, the momentum spread of the signal electron
reduces from \SI{167}{\keV} to \SI{131}{\keV}. In case a lower momentum spread
is desired, it is possible to reduce the thickness of the inner wall. The last
two rows of \cref{tab:proton_cdc_hitrate} show that even with thinner walls at
\SI{0.3}{\mm} and \SI{0.1}{\mm} the hit rate by protons are still at
manageable levels. However, reducing the wall thickness would be governed by
other requirements such as mechanical structure and gas-tightness.

In summary, the toy MC study with the preliminary proton rate and spectrum
shows that a proton absorber is not needed. It confirms the known fact that the
estimation used in COMET Phase-I is conservative, and provides a solid
prediction of the hit rate caused by protons.