nam/writeup
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

prog saved

Browse Source
This commit is contained in:
nam
2014-10-30 01:27:55 +09:00
parent b21181a08a
commit 582b15839e
2 changed files with 196 additions and 145 deletions
Download Patch File Download Diff File Expand all files Collapse all files

337
thesis/chapters/chap4_alcap_phys.tex
View File

@@ -5,19 +5,19 @@
\thispagestyle{empty} \thispagestyle{empty}
As mentioned earlier, the emission rate of protons As mentioned earlier, the emission rate of protons
following nuclear muon capture on aluminium is of interest to the COMET Phase-I 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 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 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 on proton emission. Therefore, a joint COMET-Mu2e project was formed to carry
out the measurement of proton, and other charged particles, emission. The out the measurement of proton, and other charged particles, emission. The
experiment, so-called AlCap, has been proposed and approved to be carried out 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: experiment will also measure:
\begin{itemize} \begin{itemize}
\item neutrons, because they can cause backgrounds on other detectors and \item neutron emission, because neutrons could cause backgrounds on the other
damage the front-end electronics; and detectors and damage the front-end electronics; and
\item photons, since they provide ways to normalise number of stopped muons \item photon emission to validate the normalisation number of stopped
in the stopping target. muons in the stopping target.
\end{itemize} \end{itemize}
The emission of particles following muon capture in nuclei 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 well understood and muons are used as an additional probe to investigate the
nuclear structure~\cite{Singer.1974, Measday.2001}. nuclear structure~\cite{Singer.1974, Measday.2001}.
Unfortunately, the proton emission rate for aluminium in the energy range of 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. emission of particles with emphasis on proton.
%theoretically and experimentally, hence serves as the motivation for the AlCap %theoretically and experimentally, hence serves as the motivation for the AlCap
%experiment. %experiment.
@@ -66,21 +66,20 @@ emission of particles with emphasis on proton.
Theoretically, the capturing process can be described in the following Theoretically, the capturing process can be described in the following
stages~\cite{FermiTeller.1947, WuWilets.1969}: stages~\cite{FermiTeller.1947, WuWilets.1969}:
\begin{enumerate} \begin{enumerate}
\item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity are \item High to low (a few \si{\kilo\electronvolt}) energy: the muon velocity
greater than the velocity of the valence electrons of the atom. Slowing 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 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 about \SIrange{E-10}{E-9}{\s} to slow down from a relativistic
\sn{}{8}~\si{\electronvolt}~energy to 2000~\si{\electronvolt}~in condensed matter, \SI{E8}{\eV} energy to \SI{2000}{\eV} in condensed matter,
and about 1000 times as long in air. and about 1000 times as long in air.
\item Low energy to rest: in this phase, the muon velocity is less than that \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 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 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} condensed matters ($\simeq\SI{E-13}{\s}$) or in gases ($\simeq\SI{E-9}{\s}$).
\si{\second}). \item Atomic capture: when the muon has no kinetic energy, it is captured by
\item Atomic capture: the muon has no kinetic energy, it is captured by the a host atom into one of high orbital states, forming a muonic atom. The
host atom into one of high orbital states, forming a muonic atom. The
distribution of initial states is not well known. The details depend on 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 \item Electromagnetic cascade: since all muonic states are unoccupied, the
muon cascades down to states of low energy. The transition is accompanied muon cascades down to states of low energy. The transition is accompanied
by the emission of Auger electrons or characteristic X-rays, or excitation 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 state, 1S, from the instant of its atomic capture is
$\sim$\sn{}{-14}\si{\second}. $\sim$\sn{}{-14}\si{\second}.
\item Muon disappearance: after reaching the 1S state, the muons either \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 decays or gets captured by the nucleus. The possibility to be captured
nucleus. In hydrogen, the capture to decay probability ratio is about effectively shortens the mean lifetime of negative muons stopped in
\sn{4}{-4}. Around $Z=11$, the capture probability is roughly equal to the a material. In hydrogen, the capture to decay
decay probability. In heavy nuclei ($Z\sim50$), the ratio of capture to 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. decay probabilities is about 25.
The K-shell muon will be $m_\mu/m_e \simeq 207$ times nearer the nucleus 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} \label{sec:nuclear_muon_capture}
The nuclear capture process is written as: The nuclear capture process is written as:
\begin{equation} \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} \label{eq:mucap_general}
\end{equation} \end{equation}
The resulting nucleus can be either in its ground state or in an excited state. 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 The reaction is manifestation of the elementary ordinary muon capture on the
proton: proton:
\begin{equation} \begin{equation}
\mu^- + p \rightarrow n + \nu_\mu \mu^- + p \rightarrow n + \nu_\mu \,.
\label{eq:mucap_proton} \label{eq:mucap_proton}
\end{equation} \end{equation}
If the resulting nucleus at is in an excited state, it could cascade to lower If the resulting nucleus at is in an excited state, it could cascade down to
states by emitting light particles and leaving a residual heavy nucleus. The lower states by emitting light particles and gamma rays, leaving a residual
light particles are mostly neutrons and (or) photons. Neutrons can also be 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}. 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 Charged particles are emitted with probabilities of a few percent, and are
mainly protons, deuterons and alphas have been observed in still smaller mainly protons, deuterons and alphas have been observed in still smaller
probabilities. Because of the central interest on proton emission, it is covered probabilities. Because of the central interest on proton emission, it is
in a separated section. discussed in a separated section.
\subsection{Muon capture on the proton} \subsection{Muon capture on the proton}
\label{sub:muon_capture_on_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 %$\mu p$ 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 %$p\mu p$, $p\mu d$ and $p\mu t$, which complicate the study of weak
%interaction. %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 at nucleon level and quark level
are depicted in the Figure~\ref{fig:feyn_protoncap}. The flow of time is from are depicted in \cref{fig:feyn_protoncap}. The direction of time is
the left to the right hand side, as an incoming muon and an up quark 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 exchange a virtual $W$ boson to produce a muon neutrino and a down quark, hence
a proton transforms to a neutron. a proton transforms to a neutron.
@@ -156,7 +158,10 @@ a proton transforms to a neutron.
\end{figure} \end{figure}
The four-momentum transfer in the interaction is fixed at 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 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 interaction as a four-fermion interaction with Lorentz-invariant transition
amplitude: amplitude:
@@ -181,14 +186,14 @@ is factored out in Eq.~\eqref{eq:4fermion_trans_amp}):
\label{eq:weakcurrent_ud} \label{eq:weakcurrent_ud}
\end{equation} \end{equation}
If the nucleon were point-like, the nucleon current would have the same form as 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 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 the nucleon, the current must be modified by six real form factors
$g_i(q^2), i = V, M, S, A, T, P$: $g_i(q^2), i = V, M, S, A, T, P$:
\begin{align} \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} 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) 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 \sigma^{\alpha\beta} q_\beta\gamma_5 + \frac{g_P(q^2)}{m_\mu}\gamma_5
q^\alpha, 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 well-defined form factor. Only recently, it is measured with a reasonable
precision~\cite{AndreevBanks.etal.2013a}. precision~\cite{AndreevBanks.etal.2013a}.
The values of the six form factors at $q^2 = -0.88m^2_\mu$ are listed in 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{table}[htb]
\begin{center} \begin{center}
\begin{tabular}{l l l} \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, 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 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 muon decay rate in a bound state is reduced because of the binding energy
reduces the available energy. reduces the available energy. The correction begins to be significant for
%The total capture rates for several selected $Z\geq 40$ as shown in \cref{tab:total_capture_rate}.
%elements are compiled by Measday~\cite{Measday.2001},
%and reproduced in
%Table~\ref{tab:total_capture_rate}.
%\begin{table}[htb]
%\begin{center}
%\begin{tabular}{l l r@{.}l r@{.}l@{$\pm$}l l}
%\toprule
%\textbf{$Z$ ($Z_{\textrm{eff}}$)} &
%\textbf{Element} &
%\multicolumn{2}{l}{\textbf{Mean lifetime}} &
%\multicolumn{3}{l}{\textbf{Capture rate}} &
%\textbf{Huff factor}\\
%& &
%\multicolumn{2}{c}{\textbf{(\nano\second)}} &
%\multicolumn{3}{l}{\textbf{$\times 10^3$ (\reciprocal\second)}} &\\
%\midrule
%1 (1.00) & $^1$H & 2194&90 $\pm$0.07 & 0&450 &0.020 & 1.00\\
%& $^2$H & 2194&53 $\pm$0.11 & 0&470 &0.029 & \\
%2 (1.98) & $^3$He & 2186&70 $\pm$0.10 & 2&15 &0.020 & 1.00\\
%& $^4$He & 2195&31 $\pm$0.05 & 0&470&0.029 & \\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{Total capture rate of the muon in nuclei for several selected
%elements, compiled by Measday~\cite{Measday.2001}}
%\label{tab:total_capture_rate}
%\end{table}
Theoretically, it is assumed that the muon capture rate on a proton of the 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 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 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 harder for protons to transforms into neutrons due to the Pauli exclusion
principle in heavy nuclei where there are more neutrons than protons. 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 total_capture_rate (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Neutron emission} \subsection{Neutron emission}
\label{sub:neutron_emission} \label{sub:neutron_emission}
The average number of neutrons emitted per muon capture generally increases The average number of neutrons emitted per muon capture generally increases
with $Z$, but there are large deviations from the trend due to particular 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 and can be expressed by a simple empirical function
$n_{avg} = (0.3 \pm 0.02)A^{1/3}$~\cite{Singer.1974}. $n_{avg} = (0.3 \pm 0.02)A^{1/3}$~\cite{Singer.1974}.
\begin{table}[htb] \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 $(9.5 \pm 1.1)\%$ and $(3.4 \pm 0.7)\%$. Subsequently, Kotelchuk and
Tyler~\cite{KotelchuckTyler.1968} had a result which was about 3 times more Tyler~\cite{KotelchuckTyler.1968} had a result which was about 3 times more
statistics and in fair agreement with Morigana and Fry statistics and in fair agreement with Morigana and Fry
(Figure~\ref{fig:kotelchuk_proton_spectrum}) (\cref{fig:kotelchuk_proton_spectrum})
\begin{figure}[htb] \begin{figure}[htb]
\centering \centering
\includegraphics[width=0.65\textwidth]{figs/kotelchuk_proton_spectrum} \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 aluminium, copper and lead in the energy range above \SI{40}{\MeV} and
found a consistent exponential shape in all targets. The integrated yields found a consistent exponential shape in all targets. The integrated yields
above \SI{40}{\MeV} are in the \sn{}{-4}--\sn{}{-3} range (see 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. emission rate.
\begin{table}[htb] \begin{table}[htb]
\begin{center} \begin{center}
@@ -438,7 +459,7 @@ emission rate.
\end{table} \end{table}
Their result on aluminium, the only experimental data existing for this target, 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 neighbouring elements, namely silicon measured by Budyashov et
al.~\cite{BudyashovZinov.etal.1971} and magnesium measured Balandin et al.~\cite{BudyashovZinov.etal.1971} and magnesium measured Balandin et
al.~\cite{BalandinGrebenyuk.etal.1978}. The authors noted aluminium data and 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 solved using an active target, just like nuclear emulsion. Sobottka and
Wills~\cite{SobottkaWills.1968} took this approach when using a Si(Li) detector 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 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 \si{\MeV}~is due to the recoiling $^{27}$Al. The higher energy events
including protons, deuterons and alphas constitute $(15\pm 2)\%$ of capture 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 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 $^{28}\textrm{Si}(\mu^-,\nu pn)^{26}\textrm{Mg}$ could occur at a similar rate
to that of the $^{28}\textrm{Si}(\mu^-,\nu p)^{27}\textrm{Mg}$. That also 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 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 $(\mu^-,\nu p)$, Wyttenbach et al.~\cite{WyttenbachBaertschi.etal.1978} studied $(\mu^-,\nu p)$,
$(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$, $(\mu^-,\nu p3n)$ and $(\mu^-,\nu\alpha)$ $(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$, $(\mu^-,\nu p3n)$ and $(\mu^-,\nu\alpha)$
in a wide range of 18 elements from sodium to bismuth.Their results plotted 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 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: classical Coulomb barrier $V$ they used are given by:
\begin{equation} \begin{equation}
V = \frac{zZe^2}{r_0A^{\frac{1}{3}} + \rho}, V = \frac{zZe^2}{r_0A^{\frac{1}{3}} + \rho},
\label{eqn:classical_coulomb_barrier} \label{eqn:classical_coulomb_barrier}
\end{equation} \end{equation}
where $z$ and $Z$ are the charges of the outgoing particle and of the residual 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. protons were taken.
\begin{figure}[htb] \begin{figure}[htb]
\centering \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 \caption{Activation results from Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$ and al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p)$,
$(\mu^-,\nu pn)$ reactions.} $(\mu^-,\nu pn)$, $(\mu^-,\nu p2n)$ and $(\mu^-,\nu p3n)$ reactions.}
\label{fig:wyttenbach_rate_1p} \label{fig:wyttenbach_rate_1p}
\end{figure} \end{figure}
\begin{figure}[htb] %\begin{figure}[htb]
\centering %\centering
\includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_23p} %\includegraphics[width=0.85\textwidth]{figs/wyttenbach_rate_23p}
\caption{Activation results from Wyttenbach et %\caption{Activation results from Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p2n)$ and %al.~\cite{WyttenbachBaertschi.etal.1978} for the $(\mu^-,\nu p2n)$ and
$(\mu^-,\nu p3n)$ reactions.} %$(\mu^-,\nu p3n)$ reactions.}
\label{fig:wyttenbach_rate_23p} %\label{fig:wyttenbach_rate_23p}
\end{figure} %\end{figure}
Wyttenbach et al.\ saw that the cross section of each reaction decreases Wyttenbach et al.\ saw that the cross section of each reaction decreases
exponentially with increasing Coulomb barrier. The decay constant for all 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 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 obtained with distribution (III), both in the absolute percentage and the energy
distribution (curve (III) in the left hand side of 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 at the same temperature falls short by about 10
times compares to the data. The author also found that the distribution 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 (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} \label{fig:ishii_cal_result}
\end{figure} \end{figure}
Singer~\cite{Singer.1974} noted that by assuming a reduced effective mass for Singer~\cite{Singer.1974} noted that by assuming a reduced effective mass for
the nucleon, the average excitation energy will increase, but the proton the nucleon, the average excitation energy increases, but the proton
emission rate does not significantly improve and still could not explain the emission rate is not significantly improved and still could not explain the
large discrepancy. He concluded that the evaporation mechanism can account large discrepancy. He concluded that the evaporation mechanism can account
for only a small fraction of emitted protons. Moreover, the high energy protons 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. 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 dominates the higher-energy part, although it falls short at energies higher
than 30 \si{\MeV}. The comparison between the calculated proton than 30 \si{\MeV}. The comparison between the calculated proton
spectrum and experimental data is shown in spectrum and experimental data is shown in
Fig.~\ref{fig:lifshitzsinger_cal_proton}. \cref{fig:lifshitzsinger_cal_proton}.
\begin{figure}[htb] \begin{figure}[htb]
\centering \centering
\includegraphics[width=0.85\textwidth]{figs/lifshitzsinger_cal_proton} \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 The deuteron emission channels are included to comparisons with activation
data where there is no distinguish between $(\mu^-, \nu pn)$ and $(\mu^-,d)$, data where there is no distinguish between $(\mu^-, \nu pn)$ and $(\mu^-,d)$,
\ldots Their calculated emission rates together with available experimental \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{table}[htb]
\begin{center} \begin{center}
\begin{tabular}{c c c c c} \begin{tabular}{l S S[separate-uncertainty=true]
S S[separate-uncertainty=true] c}
\toprule \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}\\ %\textbf{Col1}\\
\midrule \midrule
$^{27}_{13}$Al & 40 & $>28 \pm 4$ & (70) & 7.5 for $T>40$ MeV \\ $^{27}_{13}$Al & 9.7 & {(4.7)} & 40 & {$> 28 \pm 4$} &(70)\\
$^{28}_{14}$Si & 144 & $150\pm30$ & & 3.1 and 0.34 $d$ for $T>18$ MeV \\ $^{28}_{14}$Si & 32 & 53 \pm 10 & 144 & 150 \pm 30 & \\
$^{31}_{15}$P & 35 & $>61\pm6$ & (91) & \\ $^{31}_{15}$P & 6.7 & {(6.3)} & 35 & {$> 61 \pm 6$}&(91) \\
$^{46}_{22}$Ti & & & & \\ $^{39}_{19}$K & 19 & 32 \pm 6 & 67 & {} \\
$^{51}_{23}$V & 25 & $>20\pm1.8$ & (32) & \\ $^{41}_{19}$K & 5.1 & {(4.7)} & 30 & {$> 28 \pm 4$} &(70)\\
%item1\\ $^{51 }_{23}$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 \bottomrule
\end{tabular} \end{tabular}
\end{center} \end{center}
\caption{Calculated of the single proton emission rate and the inclusive \caption{Probabilities in units of \num{E-3} per muon capture for the
proton emission rate. The experimental data are mostly from Wyttenbach et reaction $^A_Z X (\mu,\nu p) ^{A-1}_{Z-2}Y$ and for inclusive proton
al.\cite{WyttenbachBaertschi.etal.1978}} 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} \label{tab:lifshitzsinger_cal_proton_rate}
\end{table} \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 For protons with higher energies in the range of
40--90 \si{\MeV}~observed in the emulsion data as well as in later 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 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 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 smaller in cases of Al and Cu, and about 10 times higher in case of AgBr
(Table~\ref{tab:lifshitzsinger_cal_proton_rate_1988}). (\cref{tab:lifshitzsinger_cal_proton_rate_1988}).
\begin{table}[htb] \begin{table}[!ht]
\begin{center} \begin{center}
\begin{tabular}{l l c} \begin{tabular}{l l c}
\toprule \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} \subsection{Summary on proton emission from aluminium}
\label{sub: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 There is no direct measurement of proton emission following
muon capture in the relevant energy for the COMET Phase-I of 2.5--10 muon capture in the relevant energy for the COMET Phase-I of 2.5--10
\si{\MeV}: \si{\MeV}:
\begin{enumerate} \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 for energies above 40 \si{\MeV}~is available from Krane et
al.~\cite{KraneSharma.etal.1979}, al.~\cite{KraneSharma.etal.1979},
where an exponential decay shape with a decay constant of 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 $7.5 \pm 0.4$~\si{\MeV}. At low energy range, the best one can get is
the charged particle spectrum, which includes protons, deuterons and alphas, 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 This charged particle spectrum peaks around 2.5 \si{\MeV}~and
reduces exponentially with a decay constant of 4.6 \si{\MeV}. reduces exponentially with a decay constant of 4.6 \si{\MeV}.
\item The activation data from Wyttenbach et \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} \label{sub:motivation_of_the_alcap_experiment}
As mentioned, protons from muon capture on aluminium might cause a very high 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 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}. with momenta in the range of \SIrange{75}{120}{\MeV\per\cc}.
Figure~\ref{fig:proton_impact_CDC} shows that protons with kinetic energies of \cref{fig:proton_impact_CDC} shows that protons with kinetic energies larger
2.5--8 \si{\MeV}~will hit the CDC. Such events are troublesome due to than \SI{2.5}{\MeV} could hit the CDC. Such events are troublesome due to
their large energy deposition. Deuterons and alphas at that momentum range is their large energy deposition. Deuterons and alphas at the same momentum are
not of concern because they have lower kinetic energy and higher stopping not of concern because they have lower kinetic energy compared with protons and
power, thus are harder to escape the muon stopping target. higher stopping power, thus are harder to escape the muon stopping target.
\begin{figure}[htb] \begin{figure}[htb]
\centering \centering
\includegraphics[width=0.85\textwidth]{figs/proton_impact_CDC} \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 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 Phase-I's CDC. Protons with energies in higher than \SI{2.5}{\MeV} are in the
\si{\MeV}~are in the acceptance of the CDC. Deuterons and alphas at acceptance of the CDC. Deuterons and alphas at low energies should be stopped
low energies should be stopped inside the muon stopping target.} inside the muon stopping target.}
\label{fig:proton_impact_CDC} \label{fig:proton_impact_CDC}
\end{figure} \end{figure}
%%TODO replace a figure without upper limit
The COMET plans to introduce a thin, low-$Z$ proton absorber in between the 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 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 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 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 The proton absorber solves the problem of hit rate, but it degrades the
reconstructed momentum resolution. Therefore its thickness and geometry should reconstructed momentum resolution. Therefore its thickness and geometry should
be carefully designed. The limited information available makes it difficult to be carefully optimised. The limited information available makes it difficult to
arrive at a conclusive detector design. The proton emission rate could be 4\% 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 as calculated by Lifshitz and Singer~\cite{LifshitzSinger.1980}; or 7\% as
estimated from the $(\mu^-,\nu pn)$ activation data and the ratio estimated from the $(\mu^-,\nu pn)$ activation data and the ratio in
\eqref{eqn:wyttenbach_ratio}~\cite{WyttenbachBaertschi.etal.1978}; or as high \eqref{eqn:wyttenbach_ratio}; or as high as 15-20\% from silicon and neon.
as 15-20\% from silicon and neon.
For the moment, design decisions in the COMET Phase-I are made based on 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 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 ~\cite{SobottkaWills.1968}. The spectrum shape is fitted with an empirical
function given by: function given by:
\begin{equation} \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} \label{eqn:EH_pdf}
\end{equation} \end{equation}
where $T$ is the kinetic energy of the proton, and the fitted parameters are where $T$ is the kinetic energy of the proton in \si{\MeV}, and the fitted
$A=0.105\textrm{ MeV}^{-1}$, $T_{th} = 1.4\textrm{ MeV}$, $\alpha = 1.328$ and parameters are $A=0.105\textrm{ MeV}^{-1}$, $T_{th} = 1.4\textrm{ MeV}$,
$T_0 = 3.1\textrm{ MeV}$. The baseline $\alpha = 1.328$ and $T_0 = 3.1\textrm{ MeV}$. The function rises from the
design of the absorber is 1.0 \si{\mm}~thick cut-off value of $T_{th}$, its rising edge is governed by the parameter
carbon-fibre-reinforced-polymer (CFRP) which contributes $\alpha$. The exponential decay component dominates at higher energy.
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 The baseline design of the proton absorber for the COMET Phase-I based on
it pushes the signal closer to the DIO background region. For those reasons, above assumptions is a 1-\si{\mm}-thick CFRP layer as has been described in
a measurement of the rate and spectrum of proton emission after muon capture is \cref{ssub:hit_rate_on_the_cdc}. The hit rate estimation is
required in order to optimise the CDC design. 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 motivation_of_the_alcap_experiment (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Experimental method for proton measurement} \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 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 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 \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 Emitting charged particles from nuclear muon capture will be identified by the
specific energy loss. The specific energy loss is calculated as energy loss specific energy loss.
per unit path length \sdEdx at a certain energy $E$. The quantity is uniquely %The specific energy loss is calculated as energy loss
defined for each particle species. %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 Experimentally, the specific energy loss is measured in the AlCap using a pair
detectors: a \SI{65}{\um}-thick detector, and a \SI{1500}{\um}-thick detector. of silicon detectors: a \SI{65}{\um}-thick detector, and a \SI{1500}{\um}-thick
Each detector is $5\times5$ \si{\cm^2} in area. detector. Each detector is $5\times5$ \si{\cm^2} in area.
The thinner one provides $\mathop{dE}$ information, while the sum energy 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 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 detectors pair could help distinguish protons from other charged particles from

4
thesis/thesis.tex
View File

@@ -31,9 +31,9 @@ for the COMET experiment}
\mainmatter \mainmatter
%\input{chapters/chap1_intro} %\input{chapters/chap1_intro}
%\input{chapters/chap2_mu_e_conv} %\input{chapters/chap2_mu_e_conv}
\input{chapters/chap3_comet} %\input{chapters/chap3_comet}
%\input{chapters/chap4_alcap_phys} %\input{chapters/chap4_alcap_phys}
%\input{chapters/chap5_alcap_setup} \input{chapters/chap5_alcap_setup}
%\input{chapters/chap6_analysis} %\input{chapters/chap6_analysis}
%\input{chapters/chap7_results} %\input{chapters/chap7_results}
%\input{chapters/chap8_conclusions} %\input{chapters/chap8_conclusions}