\chapter
[Proton emission following nuclear muon capture - The AlCap experiment]
{Proton emission following \\nuclear muon capture \\and the AlCap experiment}
\label{cha:alcap_phys}
\thispagestyle{empty}
As mentioned earlier, the emission rate of protons
following nuclear muon capture on aluminium is of interest to the COMET Phase-I
since protons can cause a very high hit rate on the proposed cylindrical drift
chamber. Another \mueconv experiment, namely Mu2e at Fermilab, which aims at
a similar goal sensitivity as that of the COMET, also shares the same interest
on proton emission. Therefore, a joint COMET-Mu2e project was formed to carry
out the measurement of proton, and other charged particles, emission. The
experiment, so-called AlCap, has been proposed and approved to be carried out
at PSI in 2013~\cite{AlCap.2013}. In addition to proton, the AlCap
experiment will also measure: 
\begin{itemize}
  \item neutrons, because they can cause backgrounds on other detectors and
    damage the front-end electronics; and
  \item photons, since they provide ways to normalise number of stopped muons
    in the stopping target.
\end{itemize}

The emission of particles following muon capture in nuclei 
%Historically, the emission of protons, as well as other particles, has
has been studied thoroughly for several nuclei in the context of ``intermediate
energy nuclear physics'' where it is postulated that the weak interaction is
well understood and muons are used as an additional probe to investigate the
nuclear structure~\cite{Singer.1974, Measday.2001}. 
Unfortunately, the proton emission rate for aluminium in the energy range of
interest is not available. This chapter reviews the current knowledge on
emission of particles with emphasis on proton.
%theoretically and experimentally, hence serves as the motivation for the AlCap
%experiment.

\begin{comment}
\begin{itemize}
  %\item Motivation: why looked for protons in COMET, what is the status in
    %theory and experiment
    %\begin{itemize}
      %\item COMET Phase-I need
      %\item lack of experimental data
      %\item addition to protons: neutrons and photons
    %\end{itemize}
  \item Atomic capture of muon
    \begin{itemize}
      \item formation of the muonic atom
    \end{itemize}
  \item Nuclear muon capture
    \begin{itemize}
      \item physics: capture on proton
      \item energy
      \item de-excitation modes: mostly neutrons, other may occur
    \end{itemize}
  \item Charged particles/protons 
    \begin{itemize}
      \item general
      \item alpha, protons
      \item 
    \end{itemize}
  \item Plan and goals of the AlCap experiment
\end{itemize}
\end{comment}

\section{Atomic capture of the negative muon}
\label{sec:atomic_capture_of_the_negative_muon}
Theoretically, the capturing process can be described in the following
stages~\cite{FermiTeller.1947, WuWilets.1969}: 
\begin{enumerate}
  \item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity are
    greater than the velocity of the valence electrons of the atom. Slowing
    down process is similar to that of fast heavy charged particles. It takes
    about \sn{}{-9} to \sn{}{-10} \si{\second}~to slow down from a relativistic
    \sn{}{8}~\si{\electronvolt}~energy to 2000~\si{\electronvolt}~in condensed matter, 
    and about 1000 times as long in air.  
  \item Low energy to rest: in this phase, the muon velocity is less than that
    of the valence electrons, the muon is considered to be moving inside
    a degenerate electron gas. The muon rapidly comes to a stop either in
    condensed matters ($\sim$\sn{}{-13}~\si{\second}) or in gases ($\sim$\sn{}{-9}
    \si{\second}).
  \item Atomic capture: the muon has no kinetic energy, it is captured by the
    host atom into one of high orbital states, forming a muonic atom. The
    distribution of initial states is not well known. The details depend on
    whether the material is a solid or gas, insulator or material
  \item Electromagnetic cascade: since all muonic states are unoccupied, the
    muon cascades down to states of low energy. The transition is accompanied
    by the emission of Auger electrons or characteristic X-rays, or excitation
    of the nucleus. The time taken for the muon to enter the lowest possible
    state, 1S, from the instant of its atomic capture is
    $\sim$\sn{}{-14}\si{\second}.
  \item Muon disappearance: after reaching the 1S state, the muons either
    decays with a half-life of \sn{2.2}{-6}~\si{\second}~or gets captured by the
    nucleus. In hydrogen, the capture to decay probability ratio is about
    \sn{4}{-4}. Around $Z=11$, the capture probability is roughly equal to the
    decay probability. In heavy nuclei ($Z\sim50$), the ratio of capture to
    decay probabilities is about 25.

    The K-shell muon will be $m_\mu/m_e \simeq 207$ times nearer the nucleus
    than a K-shell electron. The close proximity of the K-shell muon in the
    Coulomb field of a nuclear, together with its weak interaction with the
    nucleus, allows the muon to spend a significant fraction of time (\sn{}{-7}
    -- \sn{}{-6} \si{\second}) within the nucleus, serving as an ideal probe for the
    distribution of nuclear charge and nuclear moments.

\end{enumerate}
% section atomic_capture_of_the_negative_muon (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Nuclear capture of the negative muon}
\label{sec:nuclear_muon_capture}
The nuclear capture process is written as:
\begin{equation}
  \mu^- + A(N, Z) \rightarrow A(N, Z-1) + \nu_\mu
  \label{eq:mucap_general}
\end{equation}
The resulting nucleus can be either in its ground state or in an excited state.
The reaction is manifestation of the elementary ordinary muon capture on the
proton: 
\begin{equation}
  \mu^- + p \rightarrow n + \nu_\mu
  \label{eq:mucap_proton}
\end{equation}
If the resulting nucleus at is in an excited state, it could cascade to lower
states by emitting light particles and leaving a residual heavy nucleus. The
light particles are mostly neutrons and (or) photons. Neutrons can also be
directly knocked out of the nucleus via the reaction~\eqref{eq:mucap_proton}.
Charged particles are emitted with probabilities of a few percent, and are
mainly protons, deuterons and alphas have been observed in still smaller
probabilities. Because of the central interest on proton emission, it is covered
in a separated section.

\subsection{Muon capture on the proton}
\label{sub:muon_capture_on_proton}
%It is theoretically
%very important in understanding the structure of the Lagrangian for the
%strangeness-preserving semileptonic weak interaction. But it is also the
%hardest one experimentally. The first reason is the rate is small ($\sim$460
%\reciprocal\second) compares to the decay rate
%($\sim$\sn{455}{3}~\reciprocal\second)~\cite{Measday.2001}. Secondly, the 
%$\mu p$ atom is quite active, so it is likely to form muonic molecules like
%$p\mu p$, $p\mu d$ and $p\mu t$, which complicate the study of weak
%interaction.
The underlying interaction in proton capture in Equation~\eqref{eq:mucap_proton}
at nucleon level and quark level
are depicted in the Figure~\ref{fig:feyn_protoncap}. The flow of time is from
the left to the right hand side, as an incoming muon and an up quark
exchange a virtual $W$ boson to produce a muon neutrino and a down quark, hence
a proton transforms to a neutron.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.4\textwidth]{figs/mucap_proton}
    \hspace{10mm}
    \includegraphics[width=0.4\textwidth]{figs/mucap_quark}
  \caption{A tree-level Feynman diagram of muon capture on the proton, at the
  nucleon-level (left), and at the quark-level (right).}
  \label{fig:feyn_protoncap}
\end{figure}

The four-momentum transfer in the interaction is fixed at 
$q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2$. The smallness of the momentum
transfer in comparison to the $W$ boson's mass makes it possible to treat the
interaction as a four-fermion interaction with Lorentz-invariant transition
amplitude:
\begin{equation}
  \mathcal{M} = \frac{G_F V_{ud}}{\sqrt{2}}J^\alpha j_\alpha
  \label{eq:4fermion_trans_amp}
\end{equation}
where $J$ is the nucleon current $p\rightarrow n$, and $j$ is the lepton
current $\mu \rightarrow \nu_\mu$, $G_F$ is the Fermi coupling constant, and
$V_{ud}$ is the  matrix element of the Cabibbo-Kobayashi-Maskawa
(CKM) matrix. The lepton current is expressed as a purely $V-A$ coupling of
lepton states:
\begin{equation}
  j_\alpha = i\bar{\psi}_\nu \gamma_\alpha (1 - \gamma_5) \psi_\mu
  \label{eq:weakcurrent_lepton}
\end{equation}
The weak current of individual quarks is similar to that of leptons with the
only modification is an appropriate element of the CKM matrix ($V_{ud}$, which
is factored out in Eq.~\eqref{eq:4fermion_trans_amp}):
\begin{equation}
  J^\alpha = i\bar{\psi}_d (1 - \gamma_5) \psi_u
  \label{eq:weakcurrent_ud}
\end{equation}
If the nucleon were point-like, the nucleon current would have the same form as
in Eq.~\eqref{eq:weakcurrent_ud} with suitable wavefunctions of the proton and
neutron. But that is not the case, in order to account for the complication of
the nucleon, the current must be modified by six real form factors 
$g_i(q^2), i = V, M, S, A, T, P$:
\begin{align}
  J_\alpha &= i\bar{\psi}_n(V^\alpha - A^\alpha)\psi_p,\\
  V^\alpha &= g_V (q^2) \gamma^\alpha + i \frac{g_M(q^2)}{2m_N}
  \sigma^{\alpha\beta} q_\beta + g_S(q^2)q^\alpha,\\
  A^\alpha &= g_A(q^2)\gamma^\alpha \gamma_5 + ig_T(q^2)
  \sigma^{\alpha\beta} q_\beta\gamma_5 + \frac{g_P(q^2)}{m_\mu}\gamma_5
  q^\alpha,
\end{align}
where the $V^\alpha$ and $A^\alpha$ are the vector and axial currents, $m_\mu$
and $m_N$ are the muon and nucleon mass, respectively. The scaling by the muon
and nucleon mass is by convention in Mukhopadhyay's
review~\cite{Mukhopadhyay.1977}. 

Among the six form factors, the so-called second class currents, $g_T$ and
$g_S$, vanish under the symmetry of G-parity, which is the product of charge
conjugation and isospin rotation. Experimental limits for non-zero $g_T$ and
$g_S$ are not very tight, but are negligible with respect to other
uncertainties in muon capture~\cite{Measday.2001}.  

The vector form factor $g_V$, and the weak-magnetic form factor $g_M$ are
equivalent to the electromagnetic form factors of the nucleon according the
conserved vector current (CVC) hypothesis. The values of these couplings are
determined from elastic electron-nucleon scattering experiments, then
extrapolated to the momentum transfer $q^2$. 

Using $\mu - e$ universality, the axial form factor $g_A$ in this case is
related to that of electron as: $(g_A/g_V)^\mu = (g_A/g_V)^e$ at zero momentum
transfer. This equality has been checked using results from muon decay and beta
decay experiments. The $q^2$-dependence of $g_A$ is deducted from neutrino
scattering experiments.

The pseudoscalar form factor $g_P$ is determined by measuring the capture rate
of the process in Eq.~\eqref{eq:mucap_proton}. However, because of the smallness
capture rate in comparison to muon decay rate, and other complications due to
muonic molecules $p\mu p$, $d\mu p$ and $t\mu p$,  $g_P$ is the least
well-defined form factor. Only recently, it is measured with a reasonable
precision~\cite{AndreevBanks.etal.2013a}.
The values of the six form factors at $q^2 = -0.88m^2_\mu$ are listed in
Table~\ref{tab:formfactors}. 
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l l l}
      \toprule
      \textbf{Form factor} & \textbf{Value at $-0.88m^2_\mu$}\\
      \midrule 
      $g_S$         & $0$\\
      $g_T$         & $0$\\
      $g_V$         & $0.976 \pm 0.001$\\
      $g_M$         & $3.583 \pm 0.003$\\
      $g_A$         & $1.247 \pm 0.004$\\
      $g_P$         & $8.06 \pm 0.55$\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Values of the weak form factors of the nucleon at $q^2
  = -0.88m^2_\mu$}
  \label{tab:formfactors}
\end{table}

%\hl{Radiative capture}
% subsection muon_capture_on_proton (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Total capture rate}
\label{sub:total_capture_rate}
The captured muon at the 1S state has only two choices, either to decay or to
be captured on the nucleus. Thus, the total capture rate for negative muon,
$\Lambda_t$ is given by: 
\begin{equation}
  \Lambda_t = \Lambda_c + Q \Lambda_d
  \label{eq:mu_total_capture_rate}
\end{equation}
where $\Lambda_c$ and $\Lambda_d$ are partial capture rate and  decay rate,
respectively, and $Q$ is the Huff factor, which is corrects for the fact that
muon decay rate in a bound state is reduced because of the binding energy
reduces the available energy. 
%The total capture rates for several selected
%elements are compiled by Measday~\cite{Measday.2001}, 
%and reproduced in
%Table~\ref{tab:total_capture_rate}.
%\begin{table}[htb]
  %\begin{center}
    %\begin{tabular}{l l r@{.}l r@{.}l@{$\pm$}l l}
      %\toprule
      %\textbf{$Z$ ($Z_{\textrm{eff}}$)} & 
      %\textbf{Element} & 
      %\multicolumn{2}{l}{\textbf{Mean lifetime}} & 
      %\multicolumn{3}{l}{\textbf{Capture rate}} & 
      %\textbf{Huff factor}\\
      %& & 
      %\multicolumn{2}{c}{\textbf{(\nano\second)}} & 
      %\multicolumn{3}{l}{\textbf{$\times 10^3$ (\reciprocal\second)}} &\\
      %\midrule 
      %1 (1.00) & $^1$H & 2194&90 $\pm$0.07 & 0&450 &0.020 & 1.00\\
               %& $^2$H & 2194&53 $\pm$0.11 & 0&470 &0.029 & \\
      %2 (1.98) & $^3$He & 2186&70 $\pm$0.10 & 2&15 &0.020 & 1.00\\
               %& $^4$He & 2195&31 $\pm$0.05 & 0&470&0.029 & \\
      %\bottomrule
    %\end{tabular}
  %\end{center}
  %\caption{Total capture rate of the muon in nuclei for several selected
  %elements, compiled by Measday~\cite{Measday.2001}}
  %\label{tab:total_capture_rate}
%\end{table}

Theoretically, it is assumed that the muon capture rate on a proton of the
nucleus depends only on the overlap of the muon with the nucleus. For light
nuclei where the point nucleus concept is applicable, there are $Z$ protons and
the radius of the muon orbital decreases as $Z^{-1}$, the probability of
finding the muon at the radius increases as $Z^3$, therefore the capture rate
increases as $Z^4$. Because the muon radius soon becomes comparable to that of
the nucleus, corrections are needed, so $Z_{\textrm{eff}}$ is used instead of
$Z$. 

The effect of the nucleus for higher $Z$ is more profound, there is no
theoretical model that provides a satisfied explanation for all experimental
data. One simple formula from Primakoff gives a reasonable,
and of course not perfect, description of the existing data~\cite{Measday.2001}:
\begin{equation}
  \Lambda_c(A,Z) = Z^4_{\textrm{eff}} X_1 \left[1
  - X_2\left(\frac{A-Z}{2A}\right)\right]
  \label{eq:primakoff_capture_rate}
\end{equation}
where $X_1 =$ \SI{170}{\second^{-1}}~is the muon capture rate for hydrogen, but
reduced because a smaller phase-space in the nuclear muon capture compares to
that of a nucleon; and $X_2 = 3.125$ takes into account the fact that it is
harder for protons to transforms into neutrons due to the Pauli exclusion
principle in heavy nuclei where there are more neutrons than protons.
% subsection total_capture_rate (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutron emission}
\label{sub:neutron_emission}
The average number of neutrons emitted per muon capture generally increases
with $Z$, but there are large deviations from the trend due to particular
nuclear structure effects. The trend is shown in Table~\ref{tab:avg_neutron}
and can be expressed by a simple empirical function 
$n_{avg} = (0.3 \pm 0.02)A^{1/3}$~\cite{Singer.1974}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{c c}
      \toprule
      \textbf{Elements} & \textbf{Average number of }\\
                        & \textbf{neutrons per capture}\\
      \midrule 
      Al & 1.262 $\pm$ 0.059\\
      Si & 0.864 $\pm$ 0.072\\
      Ca & 0.746 $\pm$ 0.032\\
      Fe & 1.125 $\pm$ 0.041\\
      Ag & 1.615 $\pm$ 0.060\\
      I  & 1.436 $\pm$ 0.056\\
      Au & 1.662 $\pm$ 0.044\\
      Pb & 1.709 $\pm$ 0.066\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Average number of neutrons emitted per muon capture compiled by
  Measday~\cite{Measday.2001}}
  \label{tab:avg_neutron}
\end{table}

The neutron emission can be explained by several mechanisms:
\begin{enumerate}
  \item Direct emission follows reaction~\eqref{eq:mucap_proton}: these neutrons
    have fairly high energy, from a few \si{\mega\electronvolt}~to as high as 40--50
    \si{\mega\electronvolt}. 
  \item Indirect emission through an intermediate compound nucleus: the energy
    transferred to the neutron in the process~\eqref{eq:mucap_proton} is 5.2
    \si{\mega\electronvolt} if the initial proton is at rest, in nuclear
    environment, protons have a finite momentum distribution, therefore the
    mean excitation energy of the daughter nucleus is around 15 to 20
    \si{\mega\electronvolt}~\cite{Mukhopadhyay.1977}. This is above the nucleon
    emission threshold in all complex nuclei, thus the daughter nucleus can
    de-excite by emitting one or more neutrons. In some actinide nuclei, that
    excitation energy might trigger fission reactions. The energy of indirect
    neutrons are mainly in the lower range $E_n \le 10$ \si{\mega\electronvolt}
    with characteristically exponential shape of evaporation process. On top of
    that are prominent lines might appear where giant resonances occur.
\end{enumerate}
Experimental measurement of neutron energy spectrum is technically hard, and it
is difficult to interpret the results. Due to these difficulties, only a few
energy spectrum measurements were made, none of them covers the full energy
range and mostly at high energy region~\cite{Measday.2001}.
% subsection neutron_emission_after_muon_capture (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section[Proton emission]
{Proton emission}
\label{sec:proton_emission}
\subsection{Experimental status}
\label{sub:experimental_status}
The measurement of charged particle emission is quite difficult and
some early measurements with nuclear emulsion are still the best available
data. There are two reasons for that:
\begin{enumerate}
  \item The emission rate is small: the de-excitation of the nucleus through
    charged particle is possible, but occurs at very low rate compares to
    neutron emission. The rate is about 15\% for light nuclei and
		reduces to a few percent for medium and heavy nuclei.
		\item The charged particles are short ranged: the emitted protons,
		deuterons and alphas are typically low energy (2--20~\mega\electronvolt).
		But a relatively thick target is normally needed in order to achieve
		a reasonable muon stopping rate and charged particle statistics. Therefore,
		emulsion technique is particularly powerful.
\end{enumerate}
The first study was done by Morigana and Fry~\cite{MorinagaFry.1953} where
24,000 muon tracks were stopped in their nuclear emulsion which contains silver,
bromine, and other light elements, mainly nitrogen, carbon, hydrogen and
oxygen.  The authors identified a capture on a light element as it would leave
a recoil
track of the nucleus. They found that for silver bromide AgBr, $(2.2 \pm
0.2)\%$ of the captures produced protons and $(0.5 \pm 0.1)\%$ produced alphas.
For light elements, the emission rate for proton and alpha are respectively
$(9.5 \pm 1.1)\%$ and $(3.4 \pm 0.7)\%$.  Subsequently, Kotelchuk and
Tyler~\cite{KotelchuckTyler.1968} had a result which was about 3 times more
statistics and in fair agreement with Morigana and Fry
(Figure~\ref{fig:kotelchuk_proton_spectrum})
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.65\textwidth]{figs/kotelchuk_proton_spectrum}
    \caption{Early proton spectrum after muon capture in silver bromide AgBr
      recorded using nuclear emulsion. Image is taken from
      Ref.~\cite{KotelchuckTyler.1968}} 
    \label{fig:kotelchuk_proton_spectrum}
\end{figure}

Protons with higher energy are technically easier to measure, but because of
the much lower rate, they can only be studied at meson facilities.  Krane and
colleagues~\cite{KraneSharma.etal.1979} measured proton emission from
aluminium, copper and lead in the energy range above 40 \mega\electronvolt~and
found a consistent exponential shape in all targets.  The integrated yields
above 40 \mega\electronvolt~are in the \sn{}{-4}--\sn{}{-3} range (see
Table~\ref{tab:krane_proton_rate}), a minor contribution to total proton
emission rate. 
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{c c r@{$\pm$}l@{$\times$}r}
      \toprule
      \textbf{Target} & \textbf{Exponential constant}&
      \multicolumn{3}{c}{\textbf{Integrated yield}}\\
                      & \textbf{$E_0$ (MeV)}
      & \multicolumn{3}{c}{\textbf{$E_p\ge 40$ MeV}}\\
      \midrule 
      Al  & $7.5 \pm 0.4$  & (1.38&0.09)&\sn{}{-3}\\
      Cu  & $8.3 \pm 0.5$  & (1.96&0.12)&\sn{}{-3}\\
      Pb  & $9.9 \pm 1.1$  & (0.171&0.028)&\sn{}{-3}\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Proton integrated yields and exponential constants measured by Krane
    et al.~\cite{KraneSharma.etal.1979}. The yields are assumed to be
  proportional to exp($-E/E_0$).}
  \label{tab:krane_proton_rate}
\end{table}

Their result on aluminium, the only experimental data existing for this target,
is shown in Figure~\ref{fig:krane_proton_spec} in comparison with spectra from
neighbouring elements, namely silicon measured by Budyashov et
al.~\cite{BudyashovZinov.etal.1971} and magnesium measured Balandin et
al.~\cite{BalandinGrebenyuk.etal.1978}. The authors noted aluminium data and
silicon data are in reasonable agreement both in the yield and the energy
dependence, while magnesium data shows significant drop in intensity. They then
suggested the possibility of an interesting nuclear structure dependency that
might be at work in this mass range.

\begin{figure}[htb]
  \centering
    \includegraphics[width=0.65\textwidth]{figs/krane_proton_spec}
    \caption{Yield of charged particles following muon capture in aluminium
      target (closed circle) in the energy range above 40 MeV and an exponential
      fit. The open squares are silicon data from Budyashov et
      al.~\cite{BudyashovZinov.etal.1971}, the open triangles are magnesium data
      from Balandin et al.~\cite{BalandinGrebenyuk.etal.1978}.}
  \label{fig:krane_proton_spec}
\end{figure}

The aforementioned difficulties in charged particle measurements could be
solved using an active target, just like nuclear emulsion. Sobottka and
Wills~\cite{SobottkaWills.1968} took this approach when using a Si(Li) detector
to stop muons. They obtained a spectrum of charged particles up to 26
\mega\electronvolt~in Figure~\ref{fig:sobottka_spec}. The peak below 1.4
\mega\electronvolt~is due to the recoiling $^{27}$Al. The higher energy events
including protons, deuterons and alphas constitute $(15\pm 2)\%$ of capture
events, which is consistent with a rate of $(12.9\pm1.4)\%$ from gelatine
observed by Morigana and Fry. This part has an exponential
decay shape with a decay constant of 4.6 \mega\electronvolt. Measday
noted~\cite{Measday.2001} the fractions of events in
the 26--32 \mega\electronvolt~range being 0.3\%, and above 32
\mega\electronvolt~range being 0.15\%. This figure is in agreement with the
integrated yield above 40 \mega\electronvolt~from Krane et al.

In principle, the active target technique could be applied to other material
such as germanium, sodium iodine, caesium iodine, and other scintillation
materials. The weak point of this method is that there is no particle
identification like in nuclear emulsion, the best one can achieve after all
corrections is a sum of all charged particles. It should be noted here
deuterons can contribute significantly, Budyashov et
al.~\cite{BudyashovZinov.etal.1971} found deuteron components to be
$(34\pm2)\%$ of the charged particle yield above 18 \mega\electronvolt~in
silicon, and $(17\pm4)\%$ in copper. 
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.75\textwidth]{figs/sobottka_spec}
  \caption{Charged particle spectrum from muon capture in a silicon detector,
  image taken from Sobottka and Wills~\cite{SobottkaWills.1968}.}
  \label{fig:sobottka_spec}
\end{figure}

Another technique had been used to study proton emission is the activation
method where the residual nucleus is identified by its radioactivity. This
method can provide the rate of charged particles emission by adding up the
figures from all channels such as $(\mu^-,\nu p)$, $(\mu^-,\nu p(xn))$, 
$(\mu^-, \nu \alpha)$, $(\mu^-, \nu \alpha(xn))$. The number of elements that
can be studied using this method is limited by several requirements: (a)
mono-isotopic element is preferable; (b) the radioactive daughter should emit
gamma-rays with a reasonable half-life; (c) the $(\mu^-,\nu xn)$ reactions
should lead to either stable daughters, or daughters with very short
half-lives. The last condition is important in ensuring the dominating neutron
emission processes do not interfere with counting of the much less frequent
proton emission reactions.

Vil'gel'mova et al.~\cite{VilgelmovaEvseev.etal.1971} found the single proton
(unaccompanied by any neutron)
emission rates in the $^{28}\textrm{Si}(\mu^-,\nu p)^{27}\textrm{Mg}$ and 
$^{39}\textrm{K}(\mu^-,\nu p)^{38}\textrm{Cl}$ reactions are $(5.3 \pm 1.0)$\%
and $(3.2 \pm 0.6)$\%, respectively. 
Singer~\cite{Singer.1974} compared the figure for silicon and the result from
active target measurement and found that the reaction 
$^{28}\textrm{Si}(\mu^-,\nu pn)^{26}\textrm{Mg}$ could occur at a similar rate
to that of the $^{28}\textrm{Si}(\mu^-,\nu p)^{27}\textrm{Mg}$. That also
indicates that the deuterons and alphas might constitute a fair amount in the
spectrum in Figure~\ref{fig:sobottka_spec}.

Wyttenbach et al.~\cite{WyttenbachBaertschi.etal.1978} studied $(\mu^-,\nu p)$,
$(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$, $(\mu^-,\nu p3n)$ and $(\mu^-,\nu\alpha)$
in a wide range of 18 elements from sodium to bismuth.Their results plotted
against the Coulomb barrier for the outgoing protons are given in
Figure~\ref{fig:wyttenbach_rate_1p}, ~\ref{fig:wyttenbach_rate_23p}. The
classical Coulomb barrier $V$ they used are given by:
\begin{equation}
  V = \frac{zZe^2}{r_0A^{\frac{1}{3}} + \rho},
  \label{eqn:classical_coulomb_barrier}
\end{equation}
where $z$ and $Z$ are the charges of the outgoing particle and of the residual
nucleus, values $r_0 = 1.35 \textrm{ fm}$, and $\rho = 0 \textrm{ fm}$ for
protons were taken.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_1p}
    \caption{Activation results from Wyttenbach et
    al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$ and
  $(\mu^-,\nu pn)$ reactions.}
  \label{fig:wyttenbach_rate_1p}
\end{figure}
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_23p}
    \caption{Activation results from Wyttenbach et
    al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p2n)$ and
  $(\mu^-,\nu p3n)$ reactions.}
  \label{fig:wyttenbach_rate_23p}
\end{figure}

Wyttenbach et al.\ saw that the cross section of each reaction decreases
exponentially with increasing Coulomb barrier. The decay constant for all
$(\mu^-,\nu pxn)$ is about 1.5 per \mega\electronvolt~of Coulomb barrier. They
also commented a ratio for different de-excitation channels: 
\begin{equation}
  (\mu^-,\nu p):(\mu^-,\nu pn):(\mu^-,\nu p2n):(\mu^-,\nu p3n) = 1:6:4:4,
  \label{eqn:wyttenbach_ratio}
\end{equation}
The authors  compared their results with many preceded works and rejected
the results from Vil'gel'mova et al.~\cite{VilgelmovaEvseev.etal.1971} as being
too high, but Measday~\cite{Measday.2001} noted it it is not
necessarily true since  there has been suggestion from other experiments that
$(\mu^-, \nu p)$ reactions might become more important for light nuclei. 
Measday also commented that the ratio~\eqref{eqn:wyttenbach_ratio} holds over
a broad range of mass, but below $A=40$ the $(\mu^-,\nu p)$ reaction can vary
significantly from nucleus to nucleus.
% subsection experimental_status (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Theoretical models}
\label{sub:theoretical_models}

The first attempt to explain the result of Morigana and Fry was done by
Ishii~\cite{Ishii.1959}. He assumed a two-step scenario: firstly a compound
nucleus is formed, and then it releases energy by statistical emission of
various particles. Three models for  momentum distribution of protons in the
nucleus were used: (I) the Chew-Goldberger distribution 
$\rho(p) \sim A/(B^2 + p^2)^2$; (II) Fermi gas at zero temperature; and (III)
Fermi gas at a finite temperature ($kT = 9$ \mega\electronvolt).

A very good agreement with the experimental result for the alpha emission was
obtained with distribution (III), both in the absolute percentage and the energy
distribution (curve (III) in the left hand side of
Figure~\ref{fig:ishii_cal_result}). However, the calculated emission of protons
at the same temperature falls short by about 10
times compares to the data. The author also found that the distribution
(I) is unlikely to be suitable for proton emission, and using that distribution
for alpha emission resulted in a rate 15 times larger than observed.

\begin{figure}[htb]
  \centering
    \includegraphics[width=.49\textwidth]{figs/ishii_cal_alpha}
    %\hspace{10mm}
    \includegraphics[width=.49\textwidth]{figs/ishii_cal_proton}
    \caption{Alpha spectrum (left) and proton spectrum (right) from Ishii's
    calculation~\cite{Ishii.1959} in comparison with experimental data from
    Morigana and Fry. Image is taken from Ishii's paper.} 
  \label{fig:ishii_cal_result}
\end{figure}
Singer~\cite{Singer.1974} noted that by assuming a reduced effective mass for
the nucleon, the average excitation energy will increase, but the proton
emission rate does not significantly improve and still could not explain the
large discrepancy. He concluded that the evaporation mechanism can account
for only a small fraction of emitted protons. Moreover, the high energy protons
of 25--50 \mega\electronvolt~cannot be explained by the evaporation mechanism.
He and Lifshitz~\cite{LifshitzSinger.1978, LifshitzSinger.1980} proposed two
major corrections to Ishii's model: 
\begin{enumerate}
  \item A new description of the nucleon momentum in the nucleus with more high
    momentum components. This helps explaining the high momentum part of the
    proton spectrum.
  \item Pre-equilibrium emission of proton is included: both pre-equilibrium
    and statistical emission were taken into account. The equilibrium state is
    achieved through a series of intermediate states, and at each state there
    is possibility for particles to escape from the nucleus. 
\end{enumerate}
With these improvements, the calculated proton spectrum agreed reasonably with
data from Morigana and Fry in the energy range $E_p \le 30$ \mega\electronvolt.
Lifshitz and Singer noted the pre-equilibrium emission is more important for
heavy nuclei. Its contribution in light nuclei is about a few percent,
increasing to several tens of percent for $100<A<180$, then completely
dominating in very heavy nuclei. This trend is also seen in other nuclear
reactions at similar excitation energies. The pre-equilibrium emission also
dominates the higher-energy part, although it falls short at energies higher
than 30 \mega\electronvolt. The comparison between the calculated proton
spectrum and experimental data is shown in
Fig.~\ref{fig:lifshitzsinger_cal_proton}.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/lifshitzsinger_cal_proton}
    \caption{Proton energy spectrum from muon capture in AgBr, the data in
      histogram is from Morigana and Fry, calculation by Lifshitz and
      Singer~\cite{LifshitzSinger.1978} showed contributions from the
      pre-equilibrium emission and the equilibrium emission.}
    \label{fig:lifshitzsinger_cal_proton}
\end{figure}

The authors found their corrections accounts well for the observed data in
a wide range of elements $23 \le A \le 209$. They calculated both the single
proton emission rate $(\mu^-, \nu p)$ and the inclusive emission rate:
\begin{align*}
  \sum(\mu^-, \nu p) = &(\mu^-, \nu p) + (\mu^-, \nu pn) + (\mu^-, \nu p2n)\\
                       &+ \ldots + (\mu^-, \nu d) + (\mu^-, \nu dn)) + \ldots
\end{align*}
The deuteron emission channels are included to comparisons with activation
data where there is no distinguish between $(\mu^-, \nu pn)$ and $(\mu^-,d)$,
\ldots Their calculated emission rates together with available experimental
data is reproduced in Table~\ref{tab:lifshitzsinger_cal_proton_rate}.
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{c c c c c}
      \toprule
		Target nucleus & Calculation & Experiment & Estimate & Comments \\ 
      %\textbf{Col1}\\
      \midrule 
		$^{27}_{13}$Al 	& 40 	& $>28 \pm 4$ & (70) 	& 7.5 for $T>40$ MeV \\
		$^{28}_{14}$Si 	& 144 & $150\pm30$ 	& 			& 3.1 and 0.34 $d$ for $T>18$ MeV \\ 
		$^{31}_{15}$P 	& 35 	& $>61\pm6$ 	& (91) 	& \\ 
		$^{46}_{22}$Ti 	& 		& 						& 			& \\
		$^{51}_{23}$V 	& 25 	& $>20\pm1.8$ & (32) 	& \\ 
      %item1\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Calculated of the single proton emission rate and the inclusive
  proton emission rate. The experimental data are mostly from Wyttenbach et
al.\cite{WyttenbachBaertschi.etal.1978}}
  \label{tab:lifshitzsinger_cal_proton_rate}
\end{table}
A generally good agreement between calculation and experiment can be seen from
Table~\ref{tab:lifshitzsinger_cal_proton_rate}. The rate of $(\mu^-,\nu p)$
reactions for $^{28}\textrm{Al}$ and $^{39}\textrm{K}$ are found to be indeed
higher than average, though not as high as Vil'gel'mora et
al.~\cite{VilgelmovaEvseev.etal.1971} observed. 

For protons with higher energies in the range of
40--90 \mega\electronvolt~observed in the emulsion data as well as in later
experiments~\cite{BudyashovZinov.etal.1971,BalandinGrebenyuk.etal.1978,
KraneSharma.etal.1979}, Lifshitz and Singer~\cite{LifshitzSinger.1988}
suggested another contribution from capturing on correlated two-nucleon
cluster, an idea that had been proposed earlier by Singer~\cite{Singer.1961}.
In this calculation, the authors considered the captures on cluster in which
two nucleons interact with each other via meson exchange current. There is
experimental evidence that the nuclear surface is reach in nucleon clusters,
and it had been shown that the meson exchange current increases the total
capture rate in deuterons by 6\%. The result of this model was a mix, it
accounted well for Si, Mg and Pb data, but predicted rates about 4 times
smaller in cases of Al and Cu, and about 10 times higher in case of AgBr
(Table~\ref{tab:lifshitzsinger_cal_proton_rate_1988}).
\begin{table}[htb]
  \begin{center}
    \begin{tabular}{l l c}
      \toprule
      \textbf{Nucleus} & \textbf{Exp.$\times 10^3$} & \textbf{MEC cal.$\times
    10^3$}\\
      \midrule 
      Al & $1.38 \pm 0.09$ & 0.3\\
      Si & $0.87 \pm 0.14$ & 0.5\\
      Mg & $0.17 \pm 0.05$ & 0.2\\
      Cu & $1.96 \pm 0.12$ & 0.5\\
      AgBr & $(4.7 \pm 1.1)\times 10^{-2}$ & 0.4\\
      Pb & $0.17 \pm 0.03$ & 0.3\\
      \bottomrule
    \end{tabular}
  \end{center}
  \caption{Probability of proton emission with $E_p \ge 40$
    \mega\electronvolt~as calculated by Lifshitz and
    Singer~\cite{LifshitzSinger.1988} in comparison with available data.}
  \label{tab:lifshitzsinger_cal_proton_rate_1988}
\end{table}
% subsection theoretical_models (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Summary on proton emission from aluminium}
\label{sub:summary_on_proton_emission_from_aluminium}
There is no direct measurement of proton emission following
muon capture in the relevant energy for the COMET Phase-I of 2.5--10
\mega\electronvolt: 
\begin{enumerate}
  \item Spectrum wise, only one energy spectrum (Figure~\ref{fig:krane_proton_spec})
    for energies above 40 \mega\electronvolt~is available from Krane et
    al.~\cite{KraneSharma.etal.1979},
    where an exponential decay shape with a decay constant of 
    $7.5 \pm 0.4$~\mega\electronvolt. At low energy range, the best one can get is
    the charged particle spectrum, which includes protons, deuterons and alphas,
    from the neighbouring element silicon (Figure~\ref{fig:sobottka_spec}).
    This charged particle spectrum peaks around 2.5 \mega\electronvolt~and
    reduces exponentially with a decay constant of 4.6 \mega\electronvolt.
  \item The activation data from Wyttenbach et
    al.~\cite{WyttenbachBaertschi.etal.1978} only gives rate of
    $^{27}\textrm{Al}(\mu^-,\nu pn)^{25}\textrm{Na}$ reaction, and set a lower
		limit for proton emission rate at $(2.8 \pm 0.4)\%$ per muon capture.  If
		the ratio~\eqref{eqn:wyttenbach_ratio} holds true for aluminium, then the
		inclusive proton rate would be $7\%$, higher than the calculated rate of 
    $4\%$ by Lifshitz and Singer~\cite{LifshitzSinger.1980}.
    Both activation technique and inclusive rate calculation do not distinguish
    between different channels that give the same final state, such as between
    $^{27}\textrm{Al}(\mu^-,\nu pn)^{25}\textrm{Na}$ and 
    $^{27}\textrm{Al}(\mu^-,\nu d)^{25}\textrm{Na}$ reactions.
\end{enumerate}

In short, the knowledge on proton emission from aluminium at low energy is
limited. The rate estimation does not separate protons from deuterons,
and experimentally, there is a lower limit of $(2.8\pm0.4)\%$ per muon capture.
A spectrum shape at this energy range is not available.
% subsection summary_on_proton_emission_from_aluminium (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section proton_emission (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The AlCap experiment}
\label{sec:the_alcap_experiment}
\subsection{Motivation of the AlCap experiment}
\label{sub:motivation_of_the_alcap_experiment}
As mentioned, protons from muon capture on aluminium might cause a very high
rate in the COMET Phase-I CDC. The detector is designed to accept particles
with momenta in the range of 75--120 \mega\electronvolt\per\cc.
Figure~\ref{fig:proton_impact_CDC} shows that protons with kinetic energies of
2.5--8 \mega\electronvolt~will hit the CDC. Such events are troublesome due to
their large energy deposition. Deuterons and alphas at that momentum range is
not of concern because they have lower kinetic energy and higher stopping
power, thus are harder to escape the muon stopping target.
\begin{figure}[htb]
  \centering
    \includegraphics[width=0.85\textwidth]{figs/proton_impact_CDC}
  \caption{Momentum-kinetic energy relation of protons, deuterons and alphas
  below 10\mega\electronvolt. Shaded area is the acceptance of the COMET
Phase-I's CDC. Protons with energies in the range of 2.5--8
\mega\electronvolt~are in the acceptance of the CDC. Deuterons and alphas at
low energies should be stopped inside the muon stopping target.}
  \label{fig:proton_impact_CDC}
\end{figure}

The COMET plans to introduce a thin, low-$Z$ proton absorber in between the
target and the CDC to produce proton hit rate. The absorber will be effective
in removing low energy protons. The high energy protons that are moderated by
the absorber will fall into the acceptance range of the CDC, but because of the
exponential decay shape of the proton spectrum, the hit rate caused by these
protons should be affordable. 

The proton absorber solves the problem of hit rate, but it degrades the
reconstructed momentum resolution. Therefore its thickness and geometry should
be carefully designed. The limited information available makes it difficult to
arrive at a conclusive detector design. The proton emission rate could be 4\%
as calculated by Lifshitz and Singer~\cite{LifshitzSinger.1980}; or 7\% as
estimated from the $(\mu^-,\nu pn)$ activation data and the ratio
\eqref{eqn:wyttenbach_ratio}~\cite{WyttenbachBaertschi.etal.1978}; or as high
as 15-20\% from silicon and neon.

For the moment, design decisions in the COMET Phase-I are made based on
conservative assumptions: emission rate of 15\% and an exponential decay shape
are adopted follow the silicon data from Sobottka and Will
~\cite{SobottkaWills.1968}. The spectrum shape is fitted with an empirical
function given by:
\begin{equation}
  p(T) = A\left(1-\frac{T_{th}}{T}\right)^\alpha e^{-(T/T_0)},
  \label{eqn:EH_pdf}
\end{equation}
where $T$ is the kinetic energy of the proton, and the fitted parameters are
$A=0.105\textrm{ MeV}^{-1}$, $T_{th} = 1.4\textrm{ MeV}$, $\alpha = 1.328$ and
$T_0 = 3.1\textrm{ MeV}$. The baseline
design of the absorber is 1.0 \milli\meter~thick
carbon-fibre-reinforced-polymer (CFRP) which contributes
195~\kilo\electronvolt\per\cc~to the momentum resolution. The absorber also
down shifts the conversion peak by 0.7 \mega\electronvolt. This is an issue as
it pushes the signal closer to the DIO background region. For those reasons,
a measurement of the rate and spectrum of proton emission after muon capture is
required in order to optimise the CDC design.
% subsection motivation_of_the_alcap_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Experimental method for proton measurement}
\label{sub:experimental_method}
We planned to use a low energy, narrow momentum spread available at PSI to
fight the aforementioned difficulties in measuring protons. The beam momentum
is tunable from 28 to 45~\mega\electronvolt\ so that targets at different
thickness from 25 to 100 \micro\meter\ can be studied. The $\pi$E1 beam line
could provide about \sn{}{3} muons\per\second\ at 1\% momentum spread, and
\sn{}{4} muons\per\second\ at 3\% momentum spread. With this tunable beam, the
stopping distribution of the muons is well-defined. 

The principle of the particle identification used in the AlCap experiment is
that for each species, the function describes the relationship between energy
loss per unit length (dE/dx) and the particle energy E is uniquely defined.
With a simple system of two detectors, dE/dx can be obtained by
measuring energy deposit $\Delta$E in one detector of known thickness
$\Delta$x, and E is the sum of energy deposit in both detector if the particle
is fully stopped.

In the AlCap, we realise the idea with a pair of silicon detectors: one thin
detector of 65~\micron\ serves as the $\Delta$E counter, and one thick detector
of 1500~\micron\ that can fully stop protons up to about 12~MeV. Since the
$\Delta \textrm{d}=65$~\micron\ is known, the function relates dE/dx to
E reduces to a function between $\Delta$E and E. Figure~\ref{fig:pid_sim} shows
that the function of protons can be clearly distinguished from other charged
particles in the energy range of interest. 
\begin{figure}[htbp]
  \centering
  \includegraphics[width=0.75\textwidth]{figs/pid_sim}
  \caption{Simulation study of PID using a pair of silicon detectors}
  \label{fig:pid_sim}
\end{figure}

The AlCap uses two pairs of detector with large area, placed symmetrically with
respect to the target provide a mean to check for muon stopping distribution.
The absolute number of stopped muons are inferred
from the number of muonic X-rays recorded by a germanium detector. For
aluminium, the $(2p-1s)$ line is at 346 \kilo\electronvolt. The acceptances of
detectors will be assessed by detailed Monte Carlo study using Geant4. 

% subsection experimental_method (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Goals and plan of the experiment}
\label{sub:goals_of_the_experiment}

Our experimental program is organised in three distinct work packages (WP),
directed by different team leaders, given in parentheses.

\begin{itemize} 
\item[WP1:] (Kammel (Seattle), Kuno(Osaka)) \textbf{Charged
Particle Emission after Muon Capture.}\\ Protons emitted after nuclear muon
capture in the stopping target dominate the single-hit rates in the tracking
chambers for both the Mu2e and COMET Phase-I experiments. We plan to measure
both the total rate and the energy spectrum to a precision of 5\% down to
proton energies of 2.5 MeV.  
\item[WP2:] (Lynn(PNNL), Miller(BU))
\textbf{Gamma and X-ray Emission after Muon Capture.}\\ A Ge detector will
be used to measure X-rays from the muonic atomic cascade, in order to provide
the muon-capture normalization for WP1, and is essential for very thin
stopping targets. It is also the primary method proposed for calibrating the
number of muon stops in the Mu2e and COMET experiments. Two additional
calibration techniques  will also be explored; (1) detection of delayed gamma
rays from nuclei activated during nuclear muon capture, and (2) measurement
of the rate of photons produced in radiative muon decay. The first of these
would use a Ge detector and the second a NaI detector.   The NaI
calorimeter will measure the rate of high energy  photons from radiative muon
capture (RMC), electrons from muon decays in orbit (DIO), and photons from
radiative muon decay (RMD), as potential background sources for the
conversion measurement. As these rates are expected to be extremely low near
the conversion electron energy, only data at energies well below 100 MeV will
be obtained.   
\item[WP3:] (Hungerford(UH), Winter(ANL)) \textbf{Neutron
Emission after Muon Capture.}\\  Neutron rates and spectra after capture in
Al and Ti are not well known. In particular, the low energy region below 10
MeV is important for determining backgrounds in the Mu2e/COMET detectors and
 veto counters as well as evaluating the radiation damage to electronic
components. Carefully calibrated  liquid scintillation detectors, employing
neutron-gamma discrimination and spectrum unfolding techniques, will measure
these spectra. The measurement will attempt to obtain spectra as low or lower
than 1 MeV up to 10 MeV. \\ 
\end{itemize}   

WP1 is the most developed
project in this program. Most of the associated apparatus has been built and
optimized. We are ready to start this experiment in 2013, while preparing and
completing test measurements and simulations to undertake WP2 and WP3. 

The measurement of proton has been carried out in November and December 2013,
the details are described in following chapters.
% subsection goals_of_the_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section the_alcap_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% section nuclear_muon_capture (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% chapter alcap_phys (end)