981 lines
50 KiB
TeX
981 lines
50 KiB
TeX
\chapter
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[Proton emission following nuclear muon capture - The AlCap experiment]
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{Proton emission following \\nuclear muon capture \\and the AlCap experiment}
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\label{cha:alcap_phys}
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\thispagestyle{empty}
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As mentioned earlier, the emission rate of protons
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following nuclear muon capture on aluminium is of interest to the COMET Phase-I
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since protons could cause a very high hit rate on the proposed cylindrical drift
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chamber. Another \mueconv experiment, namely Mu2e at Fermilab, which aims at
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a similar goal sensitivity as that of the COMET, also shares the same interest
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on proton emission. Therefore, a joint COMET-Mu2e project was formed to carry
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out the measurement of proton, and other charged particles, emission. The
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experiment, so-called AlCap, has been proposed and approved to be carried out
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at PSI in 2013~\cite{AlCap.2013}. In addition to proton emission, the AlCap
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experiment will also measure:
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\begin{itemize}
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\item neutron emission, because neutrons could cause backgrounds on the other
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detectors and damage the front-end electronics; and
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\item photon emission to validate the normalisation number of stopped
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muons in the stopping target.
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\end{itemize}
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The emission of particles following muon capture in nuclei
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%Historically, the emission of protons, as well as other particles, has
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has been studied thoroughly for several nuclei in the context of ``intermediate
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energy nuclear physics'' where it is postulated that the weak interaction is
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well understood and muons are used as an additional probe to investigate the
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nuclear structure~\cite{Singer.1974, Measday.2001}.
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Unfortunately, the proton emission rate for aluminium in the energy range of
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interest has not been measured. This chapter reviews the current knowledge on
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emission of particles with emphasis on proton.
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%theoretically and experimentally, hence serves as the motivation for the AlCap
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%experiment.
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\begin{comment}
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\begin{itemize}
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%\item Motivation: why looked for protons in COMET, what is the status in
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%theory and experiment
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%\begin{itemize}
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%\item COMET Phase-I need
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%\item lack of experimental data
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%\item addition to protons: neutrons and photons
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%\end{itemize}
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\item Atomic capture of muon
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\begin{itemize}
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\item formation of the muonic atom
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\end{itemize}
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\item Nuclear muon capture
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\begin{itemize}
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\item physics: capture on proton
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\item energy
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\item de-excitation modes: mostly neutrons, other may occur
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\end{itemize}
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\item Charged particles/protons
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\begin{itemize}
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\item general
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\item alpha, protons
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\item
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\end{itemize}
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\item Plan and goals of the AlCap experiment
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\end{itemize}
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\end{comment}
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\section{Atomic capture of the negative muon}
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\label{sec:atomic_capture_of_the_negative_muon}
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Theoretically, the capturing process can be described in the following
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stages~\cite{FermiTeller.1947, WuWilets.1969}:
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\begin{enumerate}
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\item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity
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are greater than the velocity of the valence electrons of the atom. Slowing
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down process is similar to that of fast heavy charged particles. It takes
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about \SIrange{E-10}{E-9}{\s} to slow down from a relativistic
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\SI{E8}{\eV} energy to \SI{2000}{\eV} in condensed matter,
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and about 1000 times as long in air.
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\item Low energy to rest: in this phase, the muon velocity is less than that
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of the valence electrons, the muon is considered to be moving inside
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a degenerate electron gas. The muon rapidly comes to a stop either in
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condensed matters ($\simeq\SI{E-13}{\s}$) or in gases ($\simeq\SI{E-9}{\s}$).
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\item Atomic capture: when the muon has no kinetic energy, it is captured by
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a host atom into one of high orbital states, forming a muonic atom. The
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distribution of initial states is not well known. The details depend on
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whether the material is a solid or gas, insulator or metal.
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\item Electromagnetic cascade: since all muonic states are unoccupied, the
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muon cascades down to states of low energy. The transition is accompanied
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by the emission of Auger electrons or characteristic X-rays, or excitation
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of the nucleus. The time taken for the muon to enter the lowest possible
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state, 1S, from the instant of its atomic capture is
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$\sim$\sn{}{-14}\si{\second}.
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\item Muon disappearance: after reaching the 1S state, the muons either
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decays or gets captured by the nucleus. The possibility to be captured
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effectively shortens the mean lifetime of negative muons stopped in
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a material. In hydrogen, the capture to decay
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probability ratio is about \sn{4}{-4}. Around $Z=11$, the capture
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probability is roughly equal to the
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decay probability. In heavy nuclei ($Z\geq$), the ratio of capture to
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decay probabilities is about 25.
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The K-shell muon will be $m_\mu/m_e \simeq 207$ times nearer the nucleus
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than a K-shell electron. The close proximity of the K-shell muon in the
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Coulomb field of a nuclear, together with its weak interaction with the
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nucleus, allows the muon to spend a significant fraction of time (\sn{}{-7}
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-- \sn{}{-6} \si{\second}) within the nucleus, serving as an ideal probe for the
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distribution of nuclear charge and nuclear moments.
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\end{enumerate}
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% section atomic_capture_of_the_negative_muon (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Nuclear capture of the negative muon}
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\label{sec:nuclear_muon_capture}
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The nuclear capture process is written as:
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\begin{equation}
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\mu^- + A(N, Z) \rightarrow A(N, Z-1) + \nu_\mu \,.
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\label{eq:mucap_general}
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\end{equation}
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The resulting nucleus can be either in its ground state or in an excited state.
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The reaction is manifestation of the elementary ordinary muon capture on the
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proton:
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\begin{equation}
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\mu^- + p \rightarrow n + \nu_\mu \,.
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\label{eq:mucap_proton}
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\end{equation}
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If the resulting nucleus at is in an excited state, it could cascade down to
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lower states by emitting light particles and gamma rays, leaving a residual
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heavy nucleus. The light particles are mostly neutrons and (or) photons.
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Neutrons can also be
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directly knocked out of the nucleus via the reaction~\eqref{eq:mucap_proton}.
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Charged particles are emitted with probabilities of a few percent, and are
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mainly protons, deuterons and alphas have been observed in still smaller
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probabilities. Because of the central interest on proton emission, it is
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discussed in a separated section.
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\subsection{Muon capture on the proton}
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\label{sub:muon_capture_on_proton}
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%It is theoretically
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%very important in understanding the structure of the Lagrangian for the
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%strangeness-preserving semileptonic weak interaction. But it is also the
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%hardest one experimentally. The first reason is the rate is small ($\sim$460
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%\reciprocal\second) compares to the decay rate
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%($\sim$\sn{455}{3}~\reciprocal\second)~\cite{Measday.2001}. Secondly, the
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%$\mu p$ atom is quite active, so it is likely to form muonic molecules like
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%$p\mu p$, $p\mu d$ and $p\mu t$, which complicate the study of weak
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%interaction.
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The underlying interaction in proton capture in~\eqref{eq:mucap_proton}
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at nucleon level and quark level
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are depicted in \cref{fig:feyn_protoncap}. The direction of time is
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from the left to the right hand side, as an incoming muon and an up quark
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exchange a virtual $W$ boson to produce a muon neutrino and a down quark, hence
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a proton transforms to a neutron.
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\begin{figure}[htb]
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\centering
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\includegraphics[width=0.4\textwidth]{figs/mucap_proton}
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\hspace{10mm}
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\includegraphics[width=0.4\textwidth]{figs/mucap_quark}
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\caption{A tree-level Feynman diagram of muon capture on the proton, at the
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nucleon-level (left), and at the quark-level (right).}
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\label{fig:feyn_protoncap}
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\end{figure}
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The four-momentum transfer in the interaction is fixed at
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\begin{equation}
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q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2\,.
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\end{equation}
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The smallness of the momentum
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transfer in comparison to the $W$ boson's mass makes it possible to treat the
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interaction as a four-fermion interaction with Lorentz-invariant transition
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amplitude:
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\begin{equation}
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\mathcal{M} = \frac{G_F V_{ud}}{\sqrt{2}}J^\alpha j_\alpha
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\label{eq:4fermion_trans_amp}
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\end{equation}
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where $J$ is the nucleon current $p\rightarrow n$, and $j$ is the lepton
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current $\mu \rightarrow \nu_\mu$, $G_F$ is the Fermi coupling constant, and
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$V_{ud}$ is the matrix element of the Cabibbo-Kobayashi-Maskawa
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(CKM) matrix. The lepton current is expressed as a purely $V-A$ coupling of
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lepton states:
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\begin{equation}
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j_\alpha = i\bar{\psi}_\nu \gamma_\alpha (1 - \gamma_5) \psi_\mu
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\label{eq:weakcurrent_lepton}
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\end{equation}
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The weak current of individual quarks is similar to that of leptons with the
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only modification is an appropriate element of the CKM matrix ($V_{ud}$, which
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is factored out in Eq.~\eqref{eq:4fermion_trans_amp}):
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\begin{equation}
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J^\alpha = i\bar{\psi}_d (1 - \gamma_5) \psi_u
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\label{eq:weakcurrent_ud}
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\end{equation}
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If the nucleon were point-like, the nucleon current would have the same form as
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in \eqref{eq:weakcurrent_ud} with suitable wavefunctions of the proton and
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neutron. But that is not the case, in order to account for the complication of
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the nucleon, the current must be modified by six real form factors
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$g_i(q^2), i = V, M, S, A, T, P$:
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\begin{align}
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J_\alpha &= i\bar{\psi}_n(V^\alpha - A^\alpha)\psi_p\,,\\
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V^\alpha &= g_V (q^2) \gamma^\alpha + i \frac{g_M(q^2)}{2m_N}
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\sigma^{\alpha\beta} q_\beta + g_S(q^2)q^\alpha\,, \textrm{ and}\\
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A^\alpha &= g_A(q^2)\gamma^\alpha \gamma_5 + ig_T(q^2)
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\sigma^{\alpha\beta} q_\beta\gamma_5 + \frac{g_P(q^2)}{m_\mu}\gamma_5
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q^\alpha,
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\end{align}
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where the $V^\alpha$ and $A^\alpha$ are the vector and axial currents, $m_\mu$
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and $m_N$ are the muon and nucleon mass, respectively. The scaling by the muon
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and nucleon mass is by convention in Mukhopadhyay's
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review~\cite{Mukhopadhyay.1977}.
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Among the six form factors, the so-called second class currents, $g_T$ and
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$g_S$, vanish under the symmetry of G-parity, which is the product of charge
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conjugation and isospin rotation. Experimental limits for non-zero $g_T$ and
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$g_S$ are not very tight, but are negligible with respect to other
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uncertainties in muon capture~\cite{Measday.2001}.
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The vector form factor $g_V$, and the weak-magnetic form factor $g_M$ are
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equivalent to the electromagnetic form factors of the nucleon according the
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conserved vector current (CVC) hypothesis. The values of these couplings are
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determined from elastic electron-nucleon scattering experiments, then
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extrapolated to the momentum transfer $q^2$.
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Using $\mu - e$ universality, the axial form factor $g_A$ in this case is
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related to that of electron as: $(g_A/g_V)^\mu = (g_A/g_V)^e$ at zero momentum
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transfer. This equality has been checked using results from muon decay and beta
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decay experiments. The $q^2$-dependence of $g_A$ is deducted from neutrino
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scattering experiments.
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The pseudoscalar form factor $g_P$ is determined by measuring the capture rate
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of the process in Eq.~\eqref{eq:mucap_proton}. However, because of the smallness
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capture rate in comparison to muon decay rate, and other complications due to
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muonic molecules $p\mu p$, $d\mu p$ and $t\mu p$, $g_P$ is the least
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well-defined form factor. Only recently, it is measured with a reasonable
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precision~\cite{AndreevBanks.etal.2013a}.
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The values of the six form factors at $q^2 = -0.88m^2_\mu$ are listed in
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\cref{tab:formfactors}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{l l l}
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\toprule
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\textbf{Form factor} & \textbf{Value at $-0.88m^2_\mu$}\\
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\midrule
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$g_S$ & $0$\\
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$g_T$ & $0$\\
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$g_V$ & $0.976 \pm 0.001$\\
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$g_M$ & $3.583 \pm 0.003$\\
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$g_A$ & $1.247 \pm 0.004$\\
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$g_P$ & $8.06 \pm 0.55$\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Values of the weak form factors of the nucleon at $q^2
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= -0.88m^2_\mu$}
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\label{tab:formfactors}
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\end{table}
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%\hl{Radiative capture}
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% subsection muon_capture_on_proton (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Total capture rate}
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\label{sub:total_capture_rate}
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The captured muon at the 1S state has only two choices, either to decay or to
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be captured on the nucleus. Thus, the total capture rate for negative muon,
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$\Lambda_t$ is given by:
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\begin{equation}
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\Lambda_t = \Lambda_c + Q \Lambda_d
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\label{eq:mu_total_capture_rate}
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\end{equation}
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where $\Lambda_c$ and $\Lambda_d$ are partial capture rate and decay rate,
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respectively, and $Q$ is the Huff factor, which is corrects for the fact that
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muon decay rate in a bound state is reduced because of the binding energy
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reduces the available energy. The correction begins to be significant for
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$Z\geq 40$ as shown in \cref{tab:total_capture_rate}.
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Theoretically, it is assumed that the muon capture rate on a proton of the
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nucleus depends only on the overlap of the muon with the nucleus. For light
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nuclei where the point nucleus concept is applicable, there are $Z$ protons and
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the radius of the muon orbital decreases as $Z^{-1}$, the probability of
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finding the muon at the radius increases as $Z^3$, therefore the capture rate
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increases as $Z^4$. Because the muon radius soon becomes comparable to that of
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the nucleus, corrections are needed, so $Z_{\textrm{eff}}$ is used instead of
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$Z$.
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The effect of the nucleus for higher $Z$ is more profound, there is no
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theoretical model that provides a satisfied explanation for all experimental
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data. One simple formula from Primakoff gives a reasonable,
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and of course not perfect, description of the existing data~\cite{Measday.2001}:
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\begin{equation}
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\Lambda_c(A,Z) = Z^4_{\textrm{eff}} X_1 \left[1
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- X_2\left(\frac{A-Z}{2A}\right)\right]
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\label{eq:primakoff_capture_rate}
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\end{equation}
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where $X_1 =$ \SI{170}{\second^{-1}}~is the muon capture rate for hydrogen, but
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reduced because a smaller phase-space in the nuclear muon capture compares to
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that of a nucleon; and $X_2 = 3.125$ takes into account the fact that it is
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harder for protons to transforms into neutrons due to the Pauli exclusion
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principle in heavy nuclei where there are more neutrons than protons.
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The total capture rates for several selected elements are compiled by
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Measday~\cite{Measday.2001}, and reproduced in \cref{tab:total_capture_rate}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{r c S S S}
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\toprule
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$Z (Z_{eff})$ & \textbf{Element} & \textbf{Mean lifetime (\si{\ns})}
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& \textbf{Capture rate ($\times 10^{-3}$ \si{\ns})} & \textbf{Huff factor}\\
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%& & \textbf{(\si{\ns})} & \textbf{($\times 10^{-3} \si{\Hz}$)} &\\
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\midrule
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1 (1.00)& $^{1}$H & 2194.90 (7)& 0.450 (20)& 1.00 \\
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& $^{2}$H & 2194.53 (11)& 0.470 (29)& \\
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2 (1.98)& $^{3}$He & 2186.70 (10)& 2.15 (2)& 1.00\\
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& $^{4}$He & 2195.31 (5)& 0.356 (26)&\\
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3 (2.94)& $^{6}$Li & 2175.3 (4)& 4.68 (12)& 1.00 \\
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& $^{7}$Li & 2186.8 (4)& 2.26 (12)& \\
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4 (3.89)& $^{9}$Be & 2168 (3)& 6.1 (6)& 1.00 \\
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5 (4.81)& $^{10}$B & 2072 (3)& 27.5 (7)& 1.00 \\
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& $^{11}$B & 2089 (3)& 23.5 (7)& 1.00 \\
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6 (5.72)& $^{12}$C & 2028 (2)& 37.9 (5)& 1.00 \\
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& $^{13}$C & 2037 (8)& 35.0 (20)& \\
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7 (6.61)& $^{14}$N & 1919 (15)& 66 (4)& 1.00 \\
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8 (7.49)& $^{16}$O & 1796 (3)& 102.5 (10)& 0.998 \\
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& $^{18}$O & 1844 (5)& 88.0 (14)& \\
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9 (8.32)& $^{19}$F & 1463 (5)& 229 (1)& 0.998 \\
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13 (11.48)& $^{27}$Al& 864 (2)& 705 (3)& 0.993 \\
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14 (12.22)& $^{28}$Si& 758 (2)& 868 (3)& 0.992 \\
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20 (16.15)& Ca & 334 (2)& 2546 (20)& 0.985 \\
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40 (25.61)& Zr & 110.4 (10)& 8630 (80)& 0.940 \\
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82 (34.18)& Pb & 74.8 (4)& 12985 (70)& 0.844 \\
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83 (34.00)& Bi & 73.4 (4)& 13240 (70)& 0.840 \\
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90 (34.73)& Th & 77.3 (3)& 12560 (50)& 0.824 \\
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92 (34.94)& U & 77.0 (4)& 12610 (70)& 0.820 \\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Total nuclear capture rate for negative muon in several elements,
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compiled by Measday~\cite{Measday.2001}}
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\label{tab:total_capture_rate}
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\end{table}
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% subsection total_capture_rate (end)
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection{Neutron emission}
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\label{sub:neutron_emission}
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The average number of neutrons emitted per muon capture generally increases
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with $Z$, but there are large deviations from the trend due to particular
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nuclear structure effects. The trend is shown in \cref{tab:avg_neutron}
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and can be expressed by a simple empirical function
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$n_{avg} = (0.3 \pm 0.02)A^{1/3}$~\cite{Singer.1974}.
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\begin{table}[htb]
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\begin{center}
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\begin{tabular}{c c}
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\toprule
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\textbf{Elements} & \textbf{Average number of }\\
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& \textbf{neutrons per capture}\\
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\midrule
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Al & 1.262 $\pm$ 0.059\\
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Si & 0.864 $\pm$ 0.072\\
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Ca & 0.746 $\pm$ 0.032\\
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Fe & 1.125 $\pm$ 0.041\\
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Ag & 1.615 $\pm$ 0.060\\
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I & 1.436 $\pm$ 0.056\\
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Au & 1.662 $\pm$ 0.044\\
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Pb & 1.709 $\pm$ 0.066\\
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\bottomrule
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\end{tabular}
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\end{center}
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\caption{Average number of neutrons emitted per muon capture compiled by
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Measday~\cite{Measday.2001}}
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\label{tab:avg_neutron}
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\end{table}
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The neutron emission can be explained by several mechanisms:
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\begin{enumerate}
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\item Direct emission follows reaction~\eqref{eq:mucap_proton}: these neutrons
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have fairly high energy, from a few \si{\si{\MeV}}~to as high as 40--50
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\si{\si{\MeV}}.
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\item Indirect emission through an intermediate compound nucleus: the energy
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transferred to the neutron in the process~\eqref{eq:mucap_proton} is 5.2
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\si{\si{\MeV}} if the initial proton is at rest, in nuclear
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environment, protons have a finite momentum distribution, therefore the
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mean excitation energy of the daughter nucleus is around 15 to 20
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\si{\si{\MeV}}~\cite{Mukhopadhyay.1977}. This is above the nucleon
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emission threshold in all complex nuclei, thus the daughter nucleus can
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de-excite by emitting one or more neutrons. In some actinide nuclei, that
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excitation energy might trigger fission reactions. The energy of indirect
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neutrons are mainly in the lower range $E_n \le 10$ \si{\si{\MeV}}
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with characteristically exponential shape of evaporation process. On top of
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that are prominent lines might appear where giant resonances occur.
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\end{enumerate}
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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 ( \SIrange{2}{20}{\MeV}).
|
|
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 AgBr, 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, $(2.2 \pm
|
|
0.2)\%$ 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
|
|
(\cref{fig:kotelchuk_proton_spectrum})
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.65\textwidth]{figs/kotelchuk_proton_spectrum}
|
|
\caption{Proton spectrum after muon capture in silver bromide AgBr in
|
|
early experiments recorded using nuclear emulsion. The closed circles
|
|
are data points from Morigana and Fry~\cite{MorinagaFry.1953}, the
|
|
histogram is measurement result of Kotelchuk and
|
|
Tyler~\cite{KotelchuckTyler.1968}. Reprinted figure from
|
|
reference~\cite{KotelchuckTyler.1968}. Copyright 1968 by the American
|
|
Physical Society.}
|
|
\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 \SI{40}{\MeV} and
|
|
found a consistent exponential shape in all targets. The integrated yields
|
|
above \SI{40}{\MeV} are in the \sn{}{-4}--\sn{}{-3} range (see
|
|
\cref{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 \cref{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}. Reprinted
|
|
figure from reference~\cite{KraneSharma.etal.1979}. Copyright 1979 by
|
|
the American Physical Society.}
|
|
\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 \SI{26}{\MeV}
|
|
in \cref{fig:sobottka_spec}. The peak below \SI{1.4}{\MeV}
|
|
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 \si{\MeV}. Measday
|
|
noted~\cite{Measday.2001} the fractions of events in
|
|
the 26--32 \si{\MeV}~range being 0.3\%, and above 32
|
|
\si{\MeV}~range being 0.15\%. This figure is in agreement with the
|
|
integrated yield above 40 \si{\MeV}~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 \si{\MeV}~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,
|
|
measured by Sobottka and Wills~\cite{SobottkaWills.1968}. The plot is
|
|
reproduced from the original figure in reference~\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 \cref{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
|
|
\cref{fig:wyttenbach_rate_1p}.
|
|
%and \cref{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 respectively, $e^2 = 1.44 \si{\MeV}\cdot\textrm{fm}$, $r_0 = 1.35
|
|
\textrm{ fm}$, and $\rho = 0 \textrm{ fm}$ for protons were taken.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.48\textwidth]{figs/wyttenbach_rate_1p}
|
|
\includegraphics[width=0.505\textwidth]{figs/wyttenbach_rate_23p}
|
|
\caption{Activation results from Wyttenbach and
|
|
colleagues~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$,
|
|
$(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$ and $(\mu^-,\nu p3n)$ reactions. The
|
|
cross section of each individual channels decreases exponentially as the
|
|
Coulomb barrier for proton emission increases.
|
|
Reprinted figure from reference~\cite{WyttenbachBaertschi.etal.1978} with
|
|
permission from Elsevier.}
|
|
\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 and colleagues 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 \si{\MeV}~of Coulomb barrier. They
|
|
also observed 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 noted 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 ($kT = 9$ \si{\MeV}).
|
|
|
|
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
|
|
%\cref{fig:ishii_cal_result}).
|
|
However, the calculated emission rate of protons at the same temperature was 10
|
|
times smaller the experimental results from Morigana and Fry. The author
|
|
found 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 the observed rate.
|
|
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=.49\textwidth]{figs/ishii_cal_alpha}
|
|
%\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 increases, but the proton
|
|
emission rate is not significantly improved 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 \si{\MeV}~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 $E_p \le 30$ \si{\MeV}.
|
|
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 \si{\MeV}. The comparison between the calculated proton
|
|
spectrum and experimental data is shown in
|
|
\cref{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. Reprinted figure
|
|
from reference~\cite{LifshitzSinger.1978}. Copyright 1978 by the American
|
|
Physical Society.}
|
|
\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 \cref{tab:lifshitzsinger_cal_proton_rate} where
|
|
a generally good agreement between calculation and experiment can be seen from.
|
|
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.
|
|
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l S S[separate-uncertainty=true]
|
|
S S[separate-uncertainty=true] c}
|
|
\toprule
|
|
{Capturing} & {$(\mu,\nu p)$} & {$(\mu,\nu p)$}&
|
|
{$\Sigma(\mu,\nu p(xn))$}&
|
|
{$\Sigma(\mu,\nu p(xn))$} & {Est.}\\
|
|
{nucleus} & {calculation} & {experiment} & {calculation} & {experiment}
|
|
&{}\\
|
|
%nucleus & calculation & experiment & calculation & experiment \\
|
|
%\textbf{Col1}\\
|
|
\midrule
|
|
$^{27}_{13}$Al & 9.7 & {(4.7)} & 40 & {$> 28 $} &(70)\\
|
|
$^{28}_{14}$Si & 32 & 53 \pm 10 & 144 & 150 \pm 30 & \\
|
|
$^{31}_{15}$P & 6.7 & {(6.3)} & 35 & {$> 61$}&(91) \\
|
|
$^{39}_{19}$K & 19 & 32 \pm 6 & 67 & {} \\
|
|
$^{41}_{19}$K & 5.1 & {(4.7)} & 30 & {$> 28$} &(70)\\
|
|
$^{51 }_{23}$V &3.7 &2.9 \pm 0.4 &25 &{$>20 \pm 1.8$}& (32)\\
|
|
$^{55 }_{25}$Mn &2.4 &2.8 \pm 0.4 &16 &{$>26 \pm 2.5$}& (35)\\
|
|
$^{59 }_{27}$Co &3.3 &1.9 \pm 0.2 &21 &{$>37 \pm 3.4$}& (50)\\
|
|
$^{60 }_{28}$Ni &8.9 &21.4 \pm 2.3 &49 &40 \pm 5&\\
|
|
$^{63 }_{29}$Cu &4.0 &2.9 \pm 0.6 &25 &{$>17 \pm 3 $}& (36)\\
|
|
$^{65 }_{29}$Cu &1.2 &{(2.3)} &11 &{$>35 \pm 4.5$}& (36)\\
|
|
$^{75 }_{33}$As &1.5 &1.4 \pm 0.2 &14 &{$>14 \pm 1.3$}& (19)\\
|
|
$^{79 }_{35}$Br &2.7 &{} &22 & &\\
|
|
$^{107}_{47}$Ag &2.3 &{} &18 & &\\
|
|
$^{115}_{49}$In &0.63 &{(0.77)} &7.2 &{$>11 \pm 1$} &(12)\\
|
|
$^{133}_{55}$Cs &0.75 &0.48 \pm 0.07 &8.7 &{$>4.9 \pm 0.5$} &(6.7)\\
|
|
$^{165}_{67}$Ho &0.26 &0.30 \pm 0.04 &4.1 &{$>3.4 \pm 0.3$} &(4.6)\\
|
|
$^{181}_{73}$Ta &0.15 &0.26 \pm 0.04 &2.8 &{$>0.7 \pm 0.1$} &(3.0)\\
|
|
$^{208}_{82}$Pb &0.14 &0.13 \pm 0.02 &1.1 &{$>3.0 \pm 0.8$} &(4.1)\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Probabilities in units of \num{E-3} per muon capture for the
|
|
reaction $^A_Z X (\mu,\nu p) ^{A-1}_{Z-2}Y$ and for inclusive proton
|
|
emission compiled by Measday~\cite{Measday.2001}. The calculated values
|
|
are from Lifshitz and Singer. The experimental data are mostly from
|
|
Wyttenbach and colleagues~\cite{WyttenbachBaertschi.etal.1978}. The
|
|
inclusive emission the experimental figures are lower limits because only
|
|
a few decay channels could be studied. The figures in crescent parentheses
|
|
are estimates for the total inclusive rate derived from the measured
|
|
exclusive channels by the use of ratio in \eqref{eqn:wyttenbach_ratio}.}
|
|
\label{tab:lifshitzsinger_cal_proton_rate}
|
|
\end{table}
|
|
|
|
For protons with higher energies in the range of
|
|
40--90 \si{\MeV}~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
|
|
(\cref{tab:lifshitzsinger_cal_proton_rate_1988}).
|
|
\begin{table}[!ht]
|
|
\begin{center}
|
|
\begin{tabular}{l l c}
|
|
\toprule
|
|
\textbf{Nucleus} & \textbf{Experiment$\times 10^3$} & \textbf{Calculation$\times
|
|
10^3$}\\
|
|
\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 $E_p \ge \SI{40}{\MeV}$
|
|
calculated by Lifshitz and
|
|
Singer~\cite{LifshitzSinger.1988} with the two-nucleon capture hypothesis
|
|
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}
|
|
%%TODO equations, products as in Sobottkas'
|
|
There is no direct measurement of proton emission following
|
|
muon capture in the relevant energy for the COMET Phase-I of 2.5--10
|
|
\si{\MeV}:
|
|
\begin{enumerate}
|
|
\item Spectrum wise, only one energy spectrum (\cref{fig:krane_proton_spec})
|
|
for energies above 40 \si{\MeV}~is available from Krane et
|
|
al.~\cite{KraneSharma.etal.1979},
|
|
where an exponential decay shape with a decay constant of
|
|
$7.5 \pm 0.4$~\si{\MeV}. 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 (\cref{fig:sobottka_spec}).
|
|
This charged particle spectrum peaks around 2.5 \si{\MeV}~and
|
|
reduces exponentially with a decay constant of 4.6 \si{\MeV}.
|
|
\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 \SIrange{75}{120}{\MeV\per\hepclight}.
|
|
\cref{fig:proton_impact_CDC} shows that protons with kinetic energies larger
|
|
than \SI{2.5}{\MeV} could hit the CDC. Such events are troublesome due to
|
|
their large energy deposition. Deuterons and alphas at the same momentum are
|
|
not of concern because they have lower kinetic energy compared with protons and
|
|
higher stopping power, thus are harder to escape the muon stopping target.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/alcap_proton_vs_acceptance}
|
|
\caption{Momentum - kinetic energy relation of protons, deuterons and alphas
|
|
at low energy region below 20\si{\MeV}. Charged particles in the shaded
|
|
area could reach the COMET Phase-I's CDC, for protons that corresponds
|
|
kinetic energies higher than \SI{2.5}{\MeV}. 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 reduce 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 optimised. 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 in
|
|
\eqref{eqn:wyttenbach_ratio}; 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
|
|
\exp{\left(-\frac{T}{T_0}\right)},
|
|
\label{eqn:EH_pdf}
|
|
\end{equation}
|
|
where $T$ is the kinetic energy of the proton in \si{\MeV}, 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 function rises from the
|
|
cut-off value of $T_{th}$, its rising edge is governed by the parameter
|
|
$\alpha$. The exponential decay component dominates at higher energy.
|
|
|
|
The baseline design of the proton absorber for the COMET Phase-I based on
|
|
above assumptions is a 0.5-\si{\mm}-thick CFRP layer as has been described in
|
|
\cref{ssub:hit_rate_on_the_cdc}. The hit rate estimation is
|
|
conservative and the contribution of the absorber to the momentum resolution
|
|
is not negligible, further optimisation is desirable. Therefore a measurement
|
|
of the rate and spectrum of proton emission after muon capture is required.
|
|
% 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 \SIrange{28}{45}{\MeV} so that targets at different
|
|
thickness from \SIrange{25}{100}{\um} can be studied. The $\pi$E1 beam line
|
|
could deliver \sn{}{3} muons/\si{\s} at 1\% momentum spread, and
|
|
\sn{}{4} muons/\si{\s} at 3\% momentum spread. The muon stopping distribution
|
|
of the muons could be well-tuned using this excellent beam.
|
|
|
|
Emitting charged particles from nuclear muon capture will be identified by the
|
|
specific energy loss.
|
|
%The specific energy loss is calculated as energy loss
|
|
%per unit path length \sdEdx at a certain energy $E$. The quantity is uniquely
|
|
%defined for each particle species.
|
|
Experimentally, the specific energy loss is measured in the AlCap using a pair
|
|
of silicon detectors: a \SI{65}{\um}-thick detector, and a \SI{1500}{\um}-thick
|
|
detector. Each detector is $5\times5$ \si{\cm^2} in area.
|
|
The thinner one provides $\mathop{dE}$ information, while the sum energy
|
|
deposition in the two gives $E$, if the particle is fully stopped. The silicon
|
|
detectors pair could help distinguish protons from other charged particles from
|
|
\SIrange{2.5}{12}{\MeV} as shown in \cref{fig:pid_sim}.
|
|
\begin{figure}[htbp]
|
|
\centering
|
|
\includegraphics[width=0.75\textwidth]{figs/pid_sim}
|
|
\caption{Simulation study of PID using a pair of silicon detectors. The
|
|
detector resolutions follow the calibration results provided by the
|
|
manufacturer.}
|
|
\label{fig:pid_sim}
|
|
\end{figure}
|
|
|
|
Two pairs of detectors, placed symmetrically with
|
|
respect to the target, provide a mean to check for muon stopping distribution
|
|
inside the target. The absolute number of stopped muons is calculated
|
|
from the number of muonic X-rays recorded by a germanium detector. For
|
|
aluminium, the $(2p-1s)$ line is at \SI{346.828}{\keV}. 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}
|
|
|
|
The goal of the experiment is measure protons following nuclear muon capture
|
|
on aluminium:
|
|
\begin{enumerate}
|
|
\item emission rate,
|
|
\item and spectrum shape in the lower energy region down to \SI{2.5}{\MeV},
|
|
\item with a precision of about 5\%.
|
|
\end{enumerate}
|
|
The measured proton spectrum and rate will be used to assess the hit rate on
|
|
the tracking drift chamber of the COMET Phase-I.
|
|
|
|
The measurement of protons itself is part of the AlCap, where
|
|
experimental program is organised in three distinct work packages (WP),
|
|
directed by different team leaders, given in parentheses.
|
|
|
|
\begin{itemize}
|
|
\item[WP1:] (P. Kammel (University of Washington), Y. Kuno(Osaka University))
|
|
\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 \SI{2.5}{\MeV}.
|
|
\item[WP2:] (J. Miller(Boston University))
|
|
\textbf{Gamma and X-ray Emission after Muon Capture.}\\ A germanium detector
|
|
will be used to measure X-rays from the muonic atomic cascade, in order to
|
|
provide
|
|
the muon-capture normalisation 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 germanium detector and the second a sodium iodine detector.
|
|
The sodium iodine
|
|
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:] (E. Hungerford (University of Houston), P. Winter(Argonne
|
|
National Laboratory)) \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 was the most developed project in this program with most of the associated
|
|
apparatus had been built and optimised. Therefore the measurement of proton has
|
|
been carried out in November and December 2013, while preparing and completing
|
|
test measurements and simulations to undertake WP2 and WP3.
|
|
|
|
% subsection goals_of_the_experiment (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% section the_alcap_experiment (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
|
|
% section nuclear_muon_capture (end)
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
% chapter alcap_phys (end)
|