From 582b15839eb8626c120b2094bec6d0a896001ae1 Mon Sep 17 00:00:00 2001 From: nam Date: Thu, 30 Oct 2014 01:27:55 +0900 Subject: [PATCH] prog saved --- thesis/chapters/chap4_alcap_phys.tex | 337 +++++++++++++++------------ thesis/thesis.tex | 4 +- 2 files changed, 196 insertions(+), 145 deletions(-) diff --git a/thesis/chapters/chap4_alcap_phys.tex b/thesis/chapters/chap4_alcap_phys.tex index 0501d5b..a95e682 100644 --- a/thesis/chapters/chap4_alcap_phys.tex +++ b/thesis/chapters/chap4_alcap_phys.tex @@ -5,19 +5,19 @@ \thispagestyle{empty} As mentioned earlier, the emission rate of protons following nuclear muon capture on aluminium is of interest to the COMET Phase-I -since protons can cause a very high hit rate on the proposed cylindrical drift +since protons could cause a very high hit rate on the proposed cylindrical drift chamber. Another \mueconv experiment, namely Mu2e at Fermilab, which aims at a similar goal sensitivity as that of the COMET, also shares the same interest on proton emission. Therefore, a joint COMET-Mu2e project was formed to carry out the measurement of proton, and other charged particles, emission. The experiment, so-called AlCap, has been proposed and approved to be carried out -at PSI in 2013~\cite{AlCap.2013}. In addition to proton, the AlCap +at PSI in 2013~\cite{AlCap.2013}. In addition to proton emission, the AlCap experiment will also measure: \begin{itemize} - \item neutrons, because they can cause backgrounds on other detectors and - damage the front-end electronics; and - \item photons, since they provide ways to normalise number of stopped muons - in the stopping target. + \item neutron emission, because neutrons could cause backgrounds on the other + detectors and damage the front-end electronics; and + \item photon emission to validate the normalisation number of stopped + muons in the stopping target. \end{itemize} The emission of particles following muon capture in nuclei @@ -27,7 +27,7 @@ energy nuclear physics'' where it is postulated that the weak interaction is well understood and muons are used as an additional probe to investigate the nuclear structure~\cite{Singer.1974, Measday.2001}. Unfortunately, the proton emission rate for aluminium in the energy range of -interest is not available. This chapter reviews the current knowledge on +interest has not been measured. This chapter reviews the current knowledge on emission of particles with emphasis on proton. %theoretically and experimentally, hence serves as the motivation for the AlCap %experiment. @@ -66,21 +66,20 @@ emission of particles with emphasis on proton. Theoretically, the capturing process can be described in the following stages~\cite{FermiTeller.1947, WuWilets.1969}: \begin{enumerate} - \item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity are - greater than the velocity of the valence electrons of the atom. Slowing + \item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity + are greater than the velocity of the valence electrons of the atom. Slowing down process is similar to that of fast heavy charged particles. It takes - about \sn{}{-9} to \sn{}{-10} \si{\second}~to slow down from a relativistic - \sn{}{8}~\si{\electronvolt}~energy to 2000~\si{\electronvolt}~in condensed matter, + about \SIrange{E-10}{E-9}{\s} to slow down from a relativistic + \SI{E8}{\eV} energy to \SI{2000}{\eV} in condensed matter, and about 1000 times as long in air. \item Low energy to rest: in this phase, the muon velocity is less than that of the valence electrons, the muon is considered to be moving inside a degenerate electron gas. The muon rapidly comes to a stop either in - condensed matters ($\sim$\sn{}{-13}~\si{\second}) or in gases ($\sim$\sn{}{-9} - \si{\second}). - \item Atomic capture: the muon has no kinetic energy, it is captured by the - host atom into one of high orbital states, forming a muonic atom. The + condensed matters ($\simeq\SI{E-13}{\s}$) or in gases ($\simeq\SI{E-9}{\s}$). + \item Atomic capture: when the muon has no kinetic energy, it is captured by + a host atom into one of high orbital states, forming a muonic atom. The distribution of initial states is not well known. The details depend on - whether the material is a solid or gas, insulator or material + whether the material is a solid or gas, insulator or metal. \item Electromagnetic cascade: since all muonic states are unoccupied, the muon cascades down to states of low energy. The transition is accompanied by the emission of Auger electrons or characteristic X-rays, or excitation @@ -88,10 +87,12 @@ stages~\cite{FermiTeller.1947, WuWilets.1969}: state, 1S, from the instant of its atomic capture is $\sim$\sn{}{-14}\si{\second}. \item Muon disappearance: after reaching the 1S state, the muons either - decays with a half-life of \sn{2.2}{-6}~\si{\second}~or gets captured by the - nucleus. In hydrogen, the capture to decay probability ratio is about - \sn{4}{-4}. Around $Z=11$, the capture probability is roughly equal to the - decay probability. In heavy nuclei ($Z\sim50$), the ratio of capture to + decays or gets captured by the nucleus. The possibility to be captured + effectively shortens the mean lifetime of negative muons stopped in + a material. In hydrogen, the capture to decay + probability ratio is about \sn{4}{-4}. Around $Z=11$, the capture + probability is roughly equal to the + decay probability. In heavy nuclei ($Z\geq$), the ratio of capture to decay probabilities is about 25. The K-shell muon will be $m_\mu/m_e \simeq 207$ times nearer the nucleus @@ -108,24 +109,25 @@ stages~\cite{FermiTeller.1947, WuWilets.1969}: \label{sec:nuclear_muon_capture} The nuclear capture process is written as: \begin{equation} - \mu^- + A(N, Z) \rightarrow A(N, Z-1) + \nu_\mu + \mu^- + A(N, Z) \rightarrow A(N, Z-1) + \nu_\mu \,. \label{eq:mucap_general} \end{equation} The resulting nucleus can be either in its ground state or in an excited state. The reaction is manifestation of the elementary ordinary muon capture on the proton: \begin{equation} - \mu^- + p \rightarrow n + \nu_\mu + \mu^- + p \rightarrow n + \nu_\mu \,. \label{eq:mucap_proton} \end{equation} -If the resulting nucleus at is in an excited state, it could cascade to lower -states by emitting light particles and leaving a residual heavy nucleus. The -light particles are mostly neutrons and (or) photons. Neutrons can also be +If the resulting nucleus at is in an excited state, it could cascade down to +lower states by emitting light particles and gamma rays, leaving a residual +heavy nucleus. The light particles are mostly neutrons and (or) photons. +Neutrons can also be directly knocked out of the nucleus via the reaction~\eqref{eq:mucap_proton}. Charged particles are emitted with probabilities of a few percent, and are mainly protons, deuterons and alphas have been observed in still smaller -probabilities. Because of the central interest on proton emission, it is covered -in a separated section. +probabilities. Because of the central interest on proton emission, it is +discussed in a separated section. \subsection{Muon capture on the proton} \label{sub:muon_capture_on_proton} @@ -138,10 +140,10 @@ in a separated section. %$\mu p$ atom is quite active, so it is likely to form muonic molecules like %$p\mu p$, $p\mu d$ and $p\mu t$, which complicate the study of weak %interaction. -The underlying interaction in proton capture in Equation~\eqref{eq:mucap_proton} +The underlying interaction in proton capture in~\eqref{eq:mucap_proton} at nucleon level and quark level -are depicted in the Figure~\ref{fig:feyn_protoncap}. The flow of time is from -the left to the right hand side, as an incoming muon and an up quark +are depicted in \cref{fig:feyn_protoncap}. The direction of time is +from the left to the right hand side, as an incoming muon and an up quark exchange a virtual $W$ boson to produce a muon neutrino and a down quark, hence a proton transforms to a neutron. @@ -156,7 +158,10 @@ a proton transforms to a neutron. \end{figure} The four-momentum transfer in the interaction is fixed at -$q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2$. The smallness of the momentum +\begin{equation} + q^2 = (q_n - q_p)^2 = -0.88m_\mu^2 \ll m_W^2\,. +\end{equation} +The smallness of the momentum transfer in comparison to the $W$ boson's mass makes it possible to treat the interaction as a four-fermion interaction with Lorentz-invariant transition amplitude: @@ -181,14 +186,14 @@ is factored out in Eq.~\eqref{eq:4fermion_trans_amp}): \label{eq:weakcurrent_ud} \end{equation} If the nucleon were point-like, the nucleon current would have the same form as -in Eq.~\eqref{eq:weakcurrent_ud} with suitable wavefunctions of the proton and +in \eqref{eq:weakcurrent_ud} with suitable wavefunctions of the proton and neutron. But that is not the case, in order to account for the complication of the nucleon, the current must be modified by six real form factors $g_i(q^2), i = V, M, S, A, T, P$: \begin{align} - J_\alpha &= i\bar{\psi}_n(V^\alpha - A^\alpha)\psi_p,\\ + J_\alpha &= i\bar{\psi}_n(V^\alpha - A^\alpha)\psi_p\,,\\ V^\alpha &= g_V (q^2) \gamma^\alpha + i \frac{g_M(q^2)}{2m_N} - \sigma^{\alpha\beta} q_\beta + g_S(q^2)q^\alpha,\\ + \sigma^{\alpha\beta} q_\beta + g_S(q^2)q^\alpha\,, \textrm{ and}\\ A^\alpha &= g_A(q^2)\gamma^\alpha \gamma_5 + ig_T(q^2) \sigma^{\alpha\beta} q_\beta\gamma_5 + \frac{g_P(q^2)}{m_\mu}\gamma_5 q^\alpha, @@ -223,7 +228,7 @@ muonic molecules $p\mu p$, $d\mu p$ and $t\mu p$, $g_P$ is the least well-defined form factor. Only recently, it is measured with a reasonable precision~\cite{AndreevBanks.etal.2013a}. The values of the six form factors at $q^2 = -0.88m^2_\mu$ are listed in -Table~\ref{tab:formfactors}. +\cref{tab:formfactors}. \begin{table}[htb] \begin{center} \begin{tabular}{l l l} @@ -259,35 +264,8 @@ $\Lambda_t$ is given by: where $\Lambda_c$ and $\Lambda_d$ are partial capture rate and decay rate, respectively, and $Q$ is the Huff factor, which is corrects for the fact that muon decay rate in a bound state is reduced because of the binding energy -reduces the available energy. -%The total capture rates for several selected -%elements are compiled by Measday~\cite{Measday.2001}, -%and reproduced in -%Table~\ref{tab:total_capture_rate}. -%\begin{table}[htb] - %\begin{center} - %\begin{tabular}{l l r@{.}l r@{.}l@{$\pm$}l l} - %\toprule - %\textbf{$Z$ ($Z_{\textrm{eff}}$)} & - %\textbf{Element} & - %\multicolumn{2}{l}{\textbf{Mean lifetime}} & - %\multicolumn{3}{l}{\textbf{Capture rate}} & - %\textbf{Huff factor}\\ - %& & - %\multicolumn{2}{c}{\textbf{(\nano\second)}} & - %\multicolumn{3}{l}{\textbf{$\times 10^3$ (\reciprocal\second)}} &\\ - %\midrule - %1 (1.00) & $^1$H & 2194&90 $\pm$0.07 & 0&450 &0.020 & 1.00\\ - %& $^2$H & 2194&53 $\pm$0.11 & 0&470 &0.029 & \\ - %2 (1.98) & $^3$He & 2186&70 $\pm$0.10 & 2&15 &0.020 & 1.00\\ - %& $^4$He & 2195&31 $\pm$0.05 & 0&470&0.029 & \\ - %\bottomrule - %\end{tabular} - %\end{center} - %\caption{Total capture rate of the muon in nuclei for several selected - %elements, compiled by Measday~\cite{Measday.2001}} - %\label{tab:total_capture_rate} -%\end{table} +reduces the available energy. The correction begins to be significant for +$Z\geq 40$ as shown in \cref{tab:total_capture_rate}. 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 @@ -312,13 +290,56 @@ 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. + +The total capture rates for several selected elements are compiled by +Measday~\cite{Measday.2001}, and reproduced in \cref{tab:total_capture_rate}. +\begin{table}[htb] + \begin{center} + \begin{tabular}{r c S S S} + \toprule + $Z (Z_{eff})$ & \textbf{Element} & \textbf{Mean lifetime (\si{\ns})} + & \textbf{Capture rate ($\times 10^{-3}$ \si{\ns})} & \textbf{Huff factor}\\ + + %& & \textbf{(\si{\ns})} & \textbf{($\times 10^{-3} \si{\Hz}$)} &\\ + \midrule + 1 (1.00)& $^{1}$H & 2194.90 (7)& 0.450 (20)& 1.00 \\ + & $^{2}$H & 2194.53 (11)& 0.470 (29)& \\ + 2 (1.98)& $^{3}$He & 2186.70 (10)& 2.15 (2)& 1.00\\ + & $^{4}$He & 2195.31 (5)& 0.356 (26)&\\ + 3 (2.94)& $^{6}$Li & 2175.3 (4)& 4.68 (12)& 1.00 \\ + & $^{7}$Li & 2186.8 (4)& 2.26 (12)& \\ + 4 (3.89)& $^{9}$Be & 2168 (3)& 6.1 (6)& 1.00 \\ + 5 (4.81)& $^{10}$B & 2072 (3)& 27.5 (7)& 1.00 \\ + & $^{11}$B & 2089 (3)& 23.5 (7)& 1.00 \\ + 6 (5.72)& $^{12}$C & 2028 (2)& 37.9 (5)& 1.00 \\ + & $^{13}$C & 2037 (8)& 35.0 (20)& \\ + 7 (6.61)& $^{14}$N & 1919 (15)& 66 (4)& 1.00 \\ + 8 (7.49)& $^{16}$O & 1796 (3)& 102.5 (10)& 0.998 \\ + & $^{18}$O & 1844 (5)& 88.0 (14)& \\ + 9 (8.32)& $^{19}$F & 1463 (5)& 229 (1)& 0.998 \\ + 13 (11.48)& $^{27}$Al& 864 (2)& 705 (3)& 0.993 \\ + 14 (12.22)& $^{28}$Si& 758 (2)& 868 (3)& 0.992 \\ + 20 (16.15)& Ca & 334 (2)& 2546 (20)& 0.985 \\ + 40 (25.61)& Zr & 110.4 (10)& 8630 (80)& 0.940 \\ + 82 (34.18)& Pb & 74.8 (4)& 12985 (70)& 0.844 \\ + 83 (34.00)& Bi & 73.4 (4)& 13240 (70)& 0.840 \\ + 90 (34.73)& Th & 77.3 (3)& 12560 (50)& 0.824 \\ + 92 (34.94)& U & 77.0 (4)& 12610 (70)& 0.820 \\ + \bottomrule + \end{tabular} + \end{center} + \caption{Total nuclear capture rate for negative muon in several elements, + compiled by Measday~\cite{Measday.2001}} + \label{tab:total_capture_rate} +\end{table} + % 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} +nuclear structure effects. The trend is shown in \cref{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] @@ -398,7 +419,7 @@ 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}) +(\cref{fig:kotelchuk_proton_spectrum}) \begin{figure}[htb] \centering \includegraphics[width=0.65\textwidth]{figs/kotelchuk_proton_spectrum} @@ -414,7 +435,7 @@ 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 -Table~\ref{tab:krane_proton_rate}), a minor contribution to total proton +\cref{tab:krane_proton_rate}), a minor contribution to total proton emission rate. \begin{table}[htb] \begin{center} @@ -438,7 +459,7 @@ emission 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 +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 @@ -462,7 +483,7 @@ 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 -\si{\MeV}~in Figure~\ref{fig:sobottka_spec}. The peak below 1.4 +\si{\MeV}~in \cref{fig:sobottka_spec}. The peak below 1.4 \si{\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 @@ -513,37 +534,38 @@ 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}. +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 -Figure~\ref{fig:wyttenbach_rate_1p}, ~\ref{fig:wyttenbach_rate_23p}. The +\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, values $r_0 = 1.35 \textrm{ fm}$, and $\rho = 0 \textrm{ fm}$ for +nucleus respectively, $r_0 = 1.35 \textrm{ fm}$, and $\rho = 0 \textrm{ fm}$ for protons were taken. \begin{figure}[htb] \centering - \includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_1p} + \includegraphics[width=0.48\textwidth]{figs/wyttenbach_rate_1p} + \includegraphics[width=0.505\textwidth]{figs/wyttenbach_rate_23p} \caption{Activation results from Wyttenbach et - al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$ and - $(\mu^-,\nu pn)$ reactions.} + al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$, + $(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$ and $(\mu^-,\nu p3n)$ 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} +%\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 @@ -577,7 +599,7 @@ 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 -Figure~\ref{fig:ishii_cal_result}). However, the calculated emission of protons +\cref{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 @@ -594,8 +616,8 @@ for alpha emission resulted in a rate 15 times larger than observed. \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 +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. @@ -620,7 +642,7 @@ 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 -Fig.~\ref{fig:lifshitzsinger_cal_proton}. +\cref{fig:lifshitzsinger_cal_proton}. \begin{figure}[htb] \centering \includegraphics[width=0.85\textwidth]{figs/lifshitzsinger_cal_proton} @@ -641,33 +663,59 @@ proton emission rate $(\mu^-, \nu p)$ and the inclusive emission rate: 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}. +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}{c c c c c} + \begin{tabular}{l S S[separate-uncertainty=true] + S S[separate-uncertainty=true] c} \toprule - Target nucleus & Calculation & Experiment & Estimate & Comments \\ + {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 & 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\\ + $^{27}_{13}$Al & 9.7 & {(4.7)} & 40 & {$> 28 \pm 4$} &(70)\\ + $^{28}_{14}$Si & 32 & 53 \pm 10 & 144 & 150 \pm 30 & \\ + $^{31}_{15}$P & 6.7 & {(6.3)} & 35 & {$> 61 \pm 6$}&(91) \\ + $^{39}_{19}$K & 19 & 32 \pm 6 & 67 & {} \\ + $^{41}_{19}$K & 5.1 & {(4.7)} & 30 & {$> 28 \pm 4$} &(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{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}} + \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 et al.~\cite{WyttenbachBaertschi.etal.1978}. For inclusive emission + the experimental figures are lower limits, determined from the + actually measured channels. 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} -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 \si{\MeV}~observed in the emulsion data as well as in later @@ -682,8 +730,8 @@ 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] +(\cref{tab:lifshitzsinger_cal_proton_rate_1988}). +\begin{table}[!ht] \begin{center} \begin{tabular}{l l c} \toprule @@ -708,17 +756,18 @@ smaller in cases of Al and Cu, and about 10 times higher in case of AgBr %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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 (Figure~\ref{fig:krane_proton_spec}) + \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 (Figure~\ref{fig:sobottka_spec}). + 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 @@ -748,25 +797,26 @@ A spectrum shape at this energy range is not available. \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 \si{\MeV\per\cc}. -Figure~\ref{fig:proton_impact_CDC} shows that protons with kinetic energies of -2.5--8 \si{\MeV}~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. +with momenta in the range of \SIrange{75}{120}{\MeV\per\cc}. +\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/proton_impact_CDC} - \caption{Momentum-kinetic energy relation of protons, deuterons and alphas + \caption{Momentum kinetic energy relation of protons, deuterons and alphas below 10\si{\MeV}. Shaded area is the acceptance of the COMET -Phase-I's CDC. Protons with energies in the range of 2.5--8 -\si{\MeV}~are in the acceptance of the CDC. Deuterons and alphas at -low energies should be stopped inside the muon stopping target.} + Phase-I's CDC. Protons with energies in higher than \SI{2.5}{\MeV} 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} +%%TODO replace a figure without upper limit 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 +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 @@ -774,12 +824,11 @@ 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\% +be carefully optimised. The limited information available makes it difficult to +arrive at a conclusive detector design. The proton emission rate could be 0.97\% 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. +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 @@ -787,19 +836,21 @@ 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)}, + p(T) = A\left(1-\frac{T_{th}}{T}\right)^\alpha \exp{-\frac{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 \si{\mm}~thick -carbon-fibre-reinforced-polymer (CFRP) which contributes -195~\si{\keV\per\cc}~to the momentum resolution. The absorber also -down shifts the conversion peak by 0.7 \si{\MeV}. 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. +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 1-\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} @@ -810,16 +861,16 @@ 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-identified using this excellent beam. +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. - -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. +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 diff --git a/thesis/thesis.tex b/thesis/thesis.tex index 6ae020a..b980c80 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -31,9 +31,9 @@ for the COMET experiment} \mainmatter %\input{chapters/chap1_intro} %\input{chapters/chap2_mu_e_conv} -\input{chapters/chap3_comet} +%\input{chapters/chap3_comet} %\input{chapters/chap4_alcap_phys} -%\input{chapters/chap5_alcap_setup} +\input{chapters/chap5_alcap_setup} %\input{chapters/chap6_analysis} %\input{chapters/chap7_results} %\input{chapters/chap8_conclusions}