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thesis2/chapters/chap4_alcap_phys.tex
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\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 can 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, the AlCap
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experiment will also measure:
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\begin{itemize}
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\item neutrons, because they can cause backgrounds on other detectors and
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damage the front-end electronics; and
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\item photons, since they provide ways to normalise number of stopped muons
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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 is not available. 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 \kilo\electronvolt) energy: the muon velocity are
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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 \sn{}{-9} to \sn{}{-10} \second~to slow down from a relativistic
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\sn{}{8} \electronvolt~energy to 2000 \electronvolt~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 ($\sim$\sn{}{-13} \second) or in gases ($\sim$\sn{}{-9}
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\second).
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\item Atomic capture: the muon has no kinetic energy, it is captured by the
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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 material
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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}\second.
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\item Muon disappearance: after reaching the 1S state, the muons either
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decays with a half-life of \sn{2.2}{-6} \second~or gets captured by the
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nucleus. In hydrogen, the capture to decay probability ratio is about
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\sn{4}{-4}. Around $Z=11$, the capture probability is roughly equal to the
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decay probability. In heavy nuclei ($Z\sim50$), 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} \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 to lower
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states by emitting light particles and leaving a residual heavy nucleus. The
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light particles are mostly neutrons and (or) photons. 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 covered
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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 Equation~\eqref{eq:mucap_proton}
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at nucleon level and quark level
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are depicted in the Figure~\ref{fig:feyn_protoncap}. The flow of time is from
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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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$q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2$. 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 Eq.~\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,\\
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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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|
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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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Table~\ref{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.
|
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%The total capture rates for several selected
|
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%elements are compiled by Measday~\cite{Measday.2001},
|
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%and reproduced in
|
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%Table~\ref{tab:total_capture_rate}.
|
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%\begin{table}[htb]
|
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%\begin{center}
|
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%\begin{tabular}{l l r@{.}l r@{.}l@{$\pm$}l l}
|
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%\toprule
|
||||
%\textbf{$Z$ ($Z_{\textrm{eff}}$)} &
|
||||
%\textbf{Element} &
|
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%\multicolumn{2}{l}{\textbf{Mean lifetime}} &
|
||||
%\multicolumn{3}{l}{\textbf{Capture rate}} &
|
||||
%\textbf{Huff factor}\\
|
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%& &
|
||||
%\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 $X_1 = 170$ \reciprocal\second~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 \mega\electronvolt~to as high as 40--50
|
||||
\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
|
||||
\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
|
||||
\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 $E_n \le 10$ \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)
|
||||
Reference in New Issue
Block a user