\chapter{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 is softer than that of silicon,
peaks around \SI{4}{\MeV} rather than at \SI{2.5}{\MeV}
(\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 configurations are listed in
\cref{tab:proton_cdc_hitrate_comp}.

\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 in this study in comparison with the expected
    rates in COMET Phase-I's Technical Design Report~\cite{COMET.2014} at
    different configurations of proton absorber and inner wall.}
  \label{tab:proton_cdc_hitrate_comp}
\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\hepclight)}} &{(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{11}{\kHz}. Even without the
absorber, proton hit rate remains lower than that level at \SI{1.4}{\kHz}.
Therefore the absorber is not necessary as far as the hit rate is concerned.
%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\per\hepclight} to \SI{131}{\keV\per\hepclight} (\cref{tab:proton_cdc_hitrate}).
This is a small improvement since the momentum resolution is dominated by
intrinsic spread of \SI{197}{\keV\per\hepclight} due to multiple scattering in gas
and wires.

The last column of \cref{tab:proton_cdc_hitrate} shows the integrated charge
per unit length of a wire. The TDR deems an integrated charge level of
\SI{200}{\milli\coulomb\per\cm} safe. So even with the pessimistic estimation using
silicon rate and spectrum and without the proton absorber, the integrated
charge level in the CDC is still below the requirement.  Therefore removing the
absorber will not worsen the ageing process of the wires.  
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{S S S S S}
      \toprule
      {\textbf{Absorber}} &{\textbf{Inner wall}} & {\textbf{Total CFRP}}&
      {\textbf{Momentum}} & {\textbf{Integrated charge}}\\
      {\textbf{thickness}} &{\textbf{thickness}} & {\textbf{thickness}}&
      {\textbf{spread $\Delta p$}} &{\textbf{300 days}}\\
      {(\si{\mm})} & {(\si{\mm})} & {(\si{\mm})} 
      & {(\si{\keV\per\hepclight)}} &{(mC/cm)}\\
      \midrule 
      1   &0.5&1.5 &  195 & 25\\
      0.5 &0.5&1.0 &  167 & 60\\
      0   &0.5&0.5 &  133 & 160\\
      %0   &0.3&0.3 & 8281 & {-} & {-}\\
      %0   &0.1&0.1 & 15011& {-} & {-}\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Momentum spreads due to the inner wall and absorber, and integrated
    charge per unit length of wire as calculated in the COMET Phase-I's TDR.
    The momentum spreads were calculated for signal electrons at
    \SI{104.96}{\MeV\per\hepclight}. The integrated charge is estimated assuming 300
    days of operation.}
  \label{tab:proton_cdc_hitrate}
\end{table}

%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. 
