submitted to preevaluation committee
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@@ -475,21 +475,22 @@ The yields of protons from \SIrange{4}{8}{\MeV} are:
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\end{align}
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The number of emitted protons is taken as average of the two yields:
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\begin{equation}
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N_{\textrm{p unfold}} = (169.3 \pm 2.9) \times 10^3
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N_{\textrm{p unfold}} = (169.3 \pm 1.9) \times 10^3
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\end{equation}
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\subsection{Number of nuclear captures}
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\label{sub:number_of_nuclear_captures}
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
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\caption{X-ray spectrum from the aluminium target, the characteristic
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$(2p-1s)$ line shows up at 346.67~keV}
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\label{fig:al100_ge_spec}
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\end{figure}
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%\begin{figure}[!htb]
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%\centering
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%\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
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%\caption{X-ray spectrum from the aluminium target, the characteristic
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%$(2p-1s)$ line shows up at 346.67~keV}
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%\label{fig:al100_ge_spec}
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%\end{figure}
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The X-ray spectrum on the germanium detector is shown on
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\cref{fig:al100_ge_spec}. Fitting the double peaks on top of a first-order
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%The X-ray spectrum on the germanium detector is shown on
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%\cref{fig:al100_ge_spec}.
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Fitting the double peaks on top of a first-order
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polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
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procedure as in the case of the active target, the number stopped muons and
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the number of nuclear captures are:
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@@ -508,7 +509,15 @@ The proton emission rate in the range from \SIrange{4}{8}{\MeV} is therefore:
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\end{equation}
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The total proton emission rate can be estimated by assuming a spectrum shape
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with the same parameterisation as in \eqref{eqn:EH_pdf}. The fit parameters
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with the same parameterisation as in \eqref{eqn:EH_pdf}. The
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\eqref{eqn:EH_pdf} function has a power rising edge, and a exponential decay
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falling edge. The falling edge has only one decay component and is suitable to
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describe the proton spectrum with the equilibrium emission only assumption.
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The pre-equilibrium emission contribution should be small for low-$Z$ material,
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for aluminium the contribution of this component is 2.2\% according to
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Lifshitz and Singer~\cite{LifshitzSinger.1980}.
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The fitted results
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are shown in \cref{fig:al100_parameterisation} and \cref{tab:al100_fit_pars}.
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The average spectrum is obtained by taking the average of the two unfolded
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spectra from the left and right arms. The fitted parameters are compatible
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@@ -534,7 +543,7 @@ protons. The total proton emission rate is therefore estimated to be $3.5\times
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$A \times 10^{-5}$ & 2.0 \pm 0.7 & 1.3 \pm 0.1 & 1.5 \pm 0.3\\
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$T_{th}$ (\si{\keV}) & 1301 \pm 490 & 1966 \pm 68 & 1573 \pm 132\\
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$\alpha$ & 3.2 \pm 0.7 & 1.2 \pm 0.1 & 2.0 \pm 1.2\\
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$T_{th}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\
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$T_{0}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\
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\bottomrule
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\end{tabular}
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\end{center}
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@@ -597,7 +606,7 @@ presented in \cref{tab:al100_uncertainties_all}.
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\textbf{Item}& \textbf{Value} \\
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\midrule
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Number of muons & 3.2\% \\
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Statistical from measured spectra & 1.6\% \\
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Statistical from measured spectra & 1.1\% \\
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Systematic from unfolding & 5.0\% \\
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Systematic from PID & \textless0.5\% \\
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\midrule
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@@ -638,7 +647,7 @@ validated:
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\subsection{Proton emission rates and spectrum}
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\label{sub:proton_emission_rates_and_spectrum}
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The proton emission spectrum in \cref{sub:proton_emission_rate} peaks around
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\SI{4}{\MeV} which is comparable to the Coulomb barrier for proton of
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\SI{3.7}{\MeV} which is a little below the Coulomb barrier for proton of
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\SI{3.9}{\MeV} calculated using \eqref{eqn:classical_coulomb_barrier}. The
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spectrum has a decay constant of \SI{2.6}{\MeV} in higher energy region,
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makes the emission probability drop more quickly than silicon charged
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@@ -661,29 +670,48 @@ all energy is:
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R_{p \textrm{ total}} = (3.5 \pm 0.2)\%.
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\label{eqn:meas_total_rate}
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\end{equation}
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No direct comparison of this result to existing experimental or
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theoretical work is available. Indirectly, it is compatible with the figures
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calculated by Lifshitz and
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\subsubsection{Comparison to theoretical and other experimental results}
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\label{ssub:comparison_to_theoretical_and_other_experimental_results}
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There is no existing experimental or theoretical work that could be directly
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compared with the obtained proton emission rate. Indirectly, it is compatible
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with the figures calculated by Lifshitz and
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Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980} listed in
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\cref{tab:lifshitzsinger_cal_proton_rate}. It is significantly larger than
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the rate of 0.97\% for the $(\mu,\nu p)$ channel, and does not
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exceed the inclusion rate for all channels $\Sigma(\mu,\nu p(xn))$ at 4\%,
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leaving some room for other modes such as $(\mu,\nu d)$ or $(\mu,\nu p2n)$.
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Certainly, if the rate of deuterons can be extracted then the combined
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emission rate of protons and deuterons could be compared directly with the
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inclusive rate.
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leaving some room for other modes such as emission of deuterons or tritons.
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Certainly, when the full analysis is available, deuterons and tritons emission
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rates could be extracted and the combined emission rate could be compared
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directly with the inclusive rate.
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The result \eqref{eqn:meas_total_rate} is greater than the
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probability of the reaction $(\mu,\nu pn)$ measured by Wyttenbach et
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al.~\cite{WyttenbachBaertschi.etal.1978} at 2.8\%. It is expectable because
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the contribution from the $(\mu,\nu d)$ channel should be small since it
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needs to form a deuteron from a proton and a neutron.
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The rate of 3.5\% was estimated with an assumption that all protons are
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emitted in equilibrium. With the exponential constant of \SI{2.6}{\MeV}, the
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proton yield in the range from \SIrange{40}{70}{\MeV} is negligibly small
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($\sim\num{E-8}$). However, Krane and colleagues reported a significant yield
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of 0.1\% in that region~\cite{KraneSharma.etal.1979}. The energetic proton
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spectrum shape also has a different exponential constant of \SI{7.5}{\MeV}. One
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explanation for these protons is that they are emitted by other mechanisms,
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such as capture on two-nucleon cluster suggested by Singer~\cite{Singer.1961}
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(see \cref{sub:theoretical_models} and
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\cref{tab:lifshitzsinger_cal_proton_rate_1988}). Despite being sizeable, the
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yield of high energy protons is still small (3\%) in compared with the result
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in \eqref{eqn:meas_total_rate}.
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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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Compared with emission rate from silicon, the result
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\eqref{eqn:meas_total_rate} is indeed much smaller. It is even lower than
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the rate of the no-neutron reaction $(\mu,\nu p)$. This can be explained as
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\subsubsection{Comparison to the silicon result}
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\label{ssub:comparison_to_the_silicon_result}
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The probability of proton emission per nuclear capture of 3.5\% is indeed much
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smaller than that of silicon. It is even lower than the rate of the no-neutron
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reaction $(\mu,\nu p)$. This can be explained as
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the resulted nucleus from muon capture on silicon, $^{28}$Al, is an odd-odd
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nucleus and less stable than that from aluminium, $^{27}$Mg. The proton
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separation energy for $^{28}$Al is \SI{9.6}{\MeV}, which is significantly
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