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thesis/chapters/chap6_analysis.tex
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@@ -1,4 +1,4 @@
\chapter{Data analysis}
\chapter{Data analysis and results}
\label{cha:data_analysis}
This chapter presents initial analysis on subsets of the collected data.
Purposes of the analysis include:
@@ -288,7 +288,7 @@ showed that muons penetrated deeper as the momentum increased, reaching the
optimal value at the scale of 1.08, then decreased as punch through happened
more often from 1.09. The distributions of stopped muons are illustrated by
MC results on the right hand side of \cref{fig:al100_scan_rate}. With the 1.09
scale beam, the muons stopped \SI{28}{\um} off-centre to the right silicon arm.
scale beam, the muons stopped \SI{28}{\um} off-centred to the right silicon arm.
\begin{figure}[htb]
\centering
\includegraphics[width=0.47\textwidth]{figs/al100_scan_rate}
@@ -305,7 +305,7 @@ scale beam, the muons stopped \SI{28}{\um} off-centre to the right silicon arm.
As described in the \cref{sec:analysis_framework}, the hits on all detectors
are re-organised into muon events: central muons; and all hits within
\SI{\pm 10}{\us} from the central muons. The dataset from runs
\numrange{2808}{2873} contains \num{1.17E+9} such muon events.
\numrange{2808}{2873} contains \num{1.17E+9} of such muon events.
Selection of proton (and other heavy charged particles) events starts from
searching for muon event that has at least one hit on thick silicon. If there
@@ -431,6 +431,12 @@ number of protons with those energies could reach the detectors. The jump on
the right arm at around \SI{9}{\MeV} can be explained as the punch-through
protons were counted as the proton veto counters were not used in this
analysis.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr}
\caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
\label{fig:al100_unfold}
\end{figure}
%Several studies were conducted to assess the performance of the unfolding
%code, including:
@@ -467,18 +473,11 @@ The yields of protons from \SIrange{4}{8}{\MeV} are:
N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\\
N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3
\end{align}
Therefore, the number of emitted protons is taken as average value:
The number of emitted protons is taken as average of the two yields:
\begin{equation}
N_{\textrm{p unfold}} = (169.3 \pm 2.9) \times 10^3
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr}
\caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
\label{fig:al100_unfold}
\end{figure}
\subsection{Number of nuclear captures}
\label{sub:number_of_nuclear_captures}
\begin{figure}[htb]
@@ -503,16 +502,21 @@ the number of nuclear captures are:
\label{sub:proton_emission_rate}
The proton emission rate in the range from \SIrange{4}{8}{\MeV} is therefore:
\begin{equation}
R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.74\times
R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.7\times
10^{-2}
\label{eq:proton_rate_al}
\end{equation}
The total proton emission rate can be estimated by assuming a spectrum shape
with the same parameterisation as in \eqref{eqn:EH_pdf}. The fit parameters
are shown in \cref{fig:al100_parameterisation}. With such parameterisation, the
integration in range from \SIrange{4}{8}{\MeV} is 51\% of the total number of
protons. The total proton emission rate is therefore $3.5\times 10^{-2}$.
are shown in \cref{fig:al100_parameterisation} and \cref{tab:al100_fit_pars}.
The average spectrum is obtained by taking the average of the two unfolded
spectra from the left and right arms. The fitted parameters are compatible
with each other within their errors.
Using the fitted parameters of the average spectrum, the integration in range
from \SIrange{4}{8}{\MeV} is 51\% of the total number of
protons. The total proton emission rate is therefore estimated to be $3.5\times 10^{-2}$.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_parameterisation}
@@ -520,6 +524,27 @@ protons. The total proton emission rate is therefore $3.5\times 10^{-2}$.
\label{fig:al100_parameterisation}
\end{figure}
\begin{table}[htb]
\begin{center}
\begin{tabular}{l S[separate-uncertainty=true] S[separate-uncertainty=true]
S[separate-uncertainty=true]}
\toprule
\textbf{Parameter} &{\textbf{Left}} & {\textbf{Right}} & {\textbf{Average}}\\
\midrule
$A \times 10^{-5}$ & 2.0 \pm 0.7 & 1.3 \pm 0.1 & 1.5 \pm 0.3\\
$T_{th}$ (\si{\keV}) & 1301 \pm 490 & 1966 \pm 68 & 1573 \pm 132\\
$\alpha$ & 3.2 \pm 0.7 & 1.2 \pm 0.1 & 2.0 \pm 1.2\\
$T_{th}$ (\si{\keV}) & 2469 \pm 203 & 2641 \pm 106 & 2601 \pm 186\\
\bottomrule
\end{tabular}
\end{center}
\caption{Parameters of the fits on the unfolded spectra, the average spectrum
is obtained by taking average of the unfolded spectra from left and right
arms.}
\label{tab:al100_fit_pars}
\end{table}
\subsection{Uncertainties of the emission rate}
\label{sub:uncertainties_of_the_emission_rate}
The uncertainties of the emission rate come from:
@@ -584,4 +609,84 @@ presented in \cref{tab:al100_uncertainties_all}.
\label{tab:al100_uncertainties_all}
\end{table}
The proton emission rate is then $(3.5 \pm 0.2)$\%.
\section{Results of the initial analysis}
\label{sec:results_of_the_initial_analysis}
\subsection{Verification of the experimental method}
\label{sub:verification_of_the_experimental_method}
The experimental method described in \cref{sub:experimental_method} has been
validated:
\begin{itemize}
\item Number of muon capture normalisation: the number of stopped muons
calculated from the muonic X-ray spectrum is shown to be consistent with
that calculated from the active target spectrum.
\item Particle identification: the particle identification by specific
energy loss has been demonstrated. The banding of different particle
species is clearly visible. The proton extraction method using cut on
likelihood probability has been established. Since the distribution of
$\Delta E$ at a given $E$ is not Gaussian, the fraction of protons that do
not make the cut is 0.5\%, much larger than the threshold at \num{1E-4}.
The fraction of other charged particles being misidentified as protons is
smaller than 0.1\%. These uncertainties from particle identification are
still small in compared with the
uncertainty of the measurement (2.3\%).
\item Unfolding of the proton spectrum: the unfolded spectra inferred from
two measurements at the two silicon arms show good agreement with each
other, and with the muon stopping distribution obtained in the momentum
scanning analysis.
\end{itemize}
\subsection{Proton emission rates and spectrum}
\label{sub:proton_emission_rates_and_spectrum}
The proton emission spectrum in \cref{sub:proton_emission_rate} peaks around
\SI{4}{\MeV} which is comparable to the Coulomb barrier for proton of
\SI{3.9}{\MeV} calculated using \eqref{eqn:classical_coulomb_barrier}. The
spectrum has a decay constant of \SI{2.6}{\MeV} in higher energy region,
makes the emission probability drop more quickly than silicon charged
particles spectrum of Sobottka and Wills~\cite{SobottkaWills.1968} where the
decay constant was \SI{4.6}{\MeV}. This can be explained as the silicon
spectrum includes other heavier particles which have higher Coulomb barriers,
hence contribute more in the higher energy bins, effectively reduces the decay
rate.
The partial emission rate measured in the energy range from
\SIrange{4}{8}{\MeV} is:
\begin{equation}
R_{p \textrm{ meas. }} = (1.7 \pm 0.1)\%.
\label{eqn:meas_partial_rate}
\end{equation}
The total emission rate from aluminium assuming the spectrum shape holds for
all energy is:
\begin{equation}
R_{p \textrm{ total}} = (3.5 \pm 0.2)\%.
\label{eqn:meas_total_rate}
\end{equation}
No direct comparison of this result to existing experimental or
theoretical work is available. Indirectly, it is compatible with the figures
calculated by Lifshitz and
Singer~\cite{LifshitzSinger.1978, LifshitzSinger.1980} listed in
\cref{tab:lifshitzsinger_cal_proton_rate}. It is significantly larger than
the rate of 0.97\% for the $(\mu,\nu p)$ channel, and does not
exceed the inclusion rate for all channels $\Sigma(\mu,\nu p(xn))$ at 4\%,
leaving some room for other modes such as $(\mu,\nu d)$ or $(\mu,\nu p2n)$.
Certainly, if the rate of deuterons can be extracted then the combined
emission rate of protons and deuterons could be compared directly with the
inclusive rate.
The result \eqref{eqn:meas_total_rate} is greater than the
probability of the reaction $(\mu,\nu pn)$ measured by Wyttenbach et
al.~\cite{WyttenbachBaertschi.etal.1978} at 2.8\%. It is expectable because
the contribution from the $(\mu,\nu d)$ channel should be small since it
needs to form a deuteron from a proton and a neutron.
%The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
%\eqref{eqn:wyttenbach_ratio}.
Compared with emission rate from silicon, the result
\eqref{eqn:meas_total_rate} is indeed much smaller. It is even lower than
the rate of the no-neutron reaction $(\mu,\nu p)$. This can be explained as
the resulted nucleus from muon capture on silicon, $^{28}$Al, is an odd-odd
nucleus and less stable than that from aluminium, $^{27}$Mg. The proton
separation energy for $^{28}$Al is \SI{9.6}{\MeV}, which is significantly
lower than that of $^{27}$Mg at \SI{15.0}{\MeV}~\cite{AudiWapstra.etal.2003}.