938 lines
47 KiB
TeX
938 lines
47 KiB
TeX
\chapter{Data analysis and results}
|
|
\label{cha:data_analysis}
|
|
This chapter presents the first analysis on subsets of the collected data for
|
|
the aluminium 100-\si{\um}-thick target. The analysis use information from
|
|
silicon, germanium, and upstream muon detectors. Pulse parameters were
|
|
extracted from waveforms by the simplest method of peak sensing (as mentioned
|
|
in \cref{sub:offline_analyser}).
|
|
Purposes of the analysis include:
|
|
\begin{itemize}
|
|
\item testing the analysis chain;
|
|
\item verification of the experimental method, specifically the
|
|
normalisation of number of stopped muons, and particle identification
|
|
using specific energy loss;
|
|
\item extracting a preliminary rate and spectrum of proton emission from
|
|
aluminium.
|
|
\end{itemize}
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Number of stopped muons normalisation}
|
|
\label{sec:number_of_stopped_muons_normalisation}
|
|
The active silicon target runs was used to check for the validity of the
|
|
counting of number of stopped muons, where the number can be calculated by two
|
|
methods:
|
|
\begin{itemize}
|
|
\item counting hits on the active target in coincidence with hits on the
|
|
upstream scintillator counter;
|
|
\item inferred from number of X-rays recorded by the germanium detector.
|
|
\end{itemize}
|
|
This analysis was done on a subset of the active target runs
|
|
\numrange{2119}{2140}, which contains \num{6.43E7} muon events.
|
|
%\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events.
|
|
|
|
\subsection{Number of stopped muons from active target counting}
|
|
\label{sub:event_selection}
|
|
Because of the active target, a stopped muon would cause two coincident hits on
|
|
the muon counter and the target. The energy of the muon hit on the active
|
|
target is also well-defined as the narrow-momentum-spread beam was used. The
|
|
correlation between the energy and timing of all the hits on the active target
|
|
is shown in \cref{fig:sir2f_Et_corr}.
|
|
|
|
\begin{figure}[tbp]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_E_t_corr}
|
|
\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_amp_1us_slices}
|
|
\caption{Energy - timing correlation of hits on the active target (top),
|
|
and the projections onto the energy axis in 1000-\si{\ns}-long slices
|
|
from \SI{1500}{\ns} (bottom). The prompt peak at roughly \SI{5}{\MeV} in
|
|
the top plot is muon peak. In the delayed energy spectra, the Michel
|
|
electrons dominate at early time, then the beam electrons are more
|
|
clearly seen in longer delay.}
|
|
\label{fig:sir2f_Et_corr}
|
|
\end{figure}
|
|
|
|
The prompt hits on the active silicon detector are mainly beam particles:
|
|
muons and electrons. The most intense spot at time zero
|
|
and about \SI{5}{\MeV} energy corresponds to stopped muons in the thick target.
|
|
The band below \SI{1}{\MeV} is due to electrons, either in the beam or from
|
|
muon decay in orbits, or emitted during the cascading of muon to the muonic 1S
|
|
state. The valley between time zero and 1200~ns shows the minimum distance in
|
|
time between two pulses. It is the limitation of the current pulse
|
|
parameter extraction method where no pile up or double pulses is accounted for.
|
|
|
|
The delayed hits on the active target after 1200~ns are mainly secondary
|
|
particles from the stopped muons:
|
|
\begin{itemize}
|
|
\item electrons from muon decay in the 1S orbit,
|
|
\item products emitted after nuclear muon capture, including: gamma, neutron,
|
|
heavy charged particles and recoiled nucleus.
|
|
\end{itemize}
|
|
It can be seen that there is a faint stripe of muons in the time larger than
|
|
1200~ns region, they are scattered muons by other materials without hitting the
|
|
muon counter. The electrons in the beam caused the constant band below 1 MeV and
|
|
$t > 5000$ ns (see \cref{fig:sir2f_Et_corr} bottom).
|
|
%\begin{figure}[htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_amp_1us_slices}
|
|
%\caption{Energy deposit on the active target in 1000 ns time slices from the
|
|
%muon hit. The peaks at about 800 keV in large delayed time are from
|
|
%the beam electrons.}
|
|
%\label{fig:sir2_1us_slices}
|
|
%\end{figure}
|
|
|
|
From the energy-timing correlation above, the cuts to select stopped muons are:
|
|
\begin{enumerate}
|
|
\item has one hit on muon counter (where a threshold was set to reject MIPs),
|
|
and the first hit on the silicon active target is in coincidence with that
|
|
muon counter hit:
|
|
\begin{equation}
|
|
\lvert t_{\textrm{target}} - t_{\mu\textrm{ counter}}\rvert \le
|
|
\SI{50}{\ns}\,,
|
|
\label{eqn:sir2_prompt_cut}
|
|
\end{equation}
|
|
\item and the first hit on the target has energy of that of the muons:
|
|
\begin{equation}
|
|
\SI{3.4}{\MeV} \le E_{\textrm{target}} \le \SI{5.6}{\MeV}\,.
|
|
\label{eqn:sir2_muE_cut}
|
|
\end{equation}
|
|
\end{enumerate}
|
|
The two cuts~\eqref{eqn:sir2_prompt_cut} and~\eqref{eqn:sir2_muE_cut} give
|
|
a number of stopped muons counted by the active target:
|
|
\begin{equation}
|
|
N_{\mu \textrm{ active Si}} = 9.32 \times 10^6 \pm 3.0\times10^3\,.
|
|
\label{eqn:n_stopped_si_count}
|
|
\end{equation}
|
|
|
|
\subsection{Number of stopped muons from the number of X-rays}
|
|
\label{sub:number_of_stopped_muons_from_the_number_of_x_rays}
|
|
The number of nuclear captures, hence the number of stopped muons in the
|
|
active silicon target, can be inferred from the number of emitted
|
|
muonic X-rays. The reference energies and intensities of the most prominent
|
|
lines of silicon and aluminium are listed in \cref{tab:mucap_pars}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{l l l}
|
|
\toprule
|
|
\textbf{Quantity} & \textbf{Aluminium} & \textbf{Silicon}\\
|
|
\midrule
|
|
Muonic mean lifetime (ns) & $864 \pm 2$ & $758 \pm 2$\\
|
|
Nuclear capture probability (\%) & $60.9 $ & $65.8$\\
|
|
$(2p-1s)$ X-ray energy (keV) & $346.828\pm0.002$ & $400.177\pm0.005$\\
|
|
Intensity (\%) & $79.8\pm0.8$ & $80.3\pm0.8$\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Reference parameters of muon capture in aluminium and silicon taken
|
|
from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday et
|
|
al.~\cite{MeasdayStocki.etal.2007}.}
|
|
\label{tab:mucap_pars}
|
|
\end{table}
|
|
|
|
The muonic X-rays are emitted during the cascading of the muon to the muonic 1S
|
|
state in the time scale of \SI{E-9}{\s}, so the hit caused by the X-rays must
|
|
be in coincidence with the muon hit on the active target. Therefore an
|
|
additional timing cut is applied for the germanium detector hits:
|
|
\begin{equation}
|
|
\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns}
|
|
\label{eqn:sir2_ge_cut}
|
|
\end{equation}
|
|
\begin{figure}[!htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/sir2_xray_22}
|
|
\caption{Prompt muonic X-rays spectrum from the active silicon target. The
|
|
$(2p-1s)$ X-ray shows up at \SI{400}{\keV}; higher transitions can also
|
|
be identified.}
|
|
\label{fig:sir2_xray}
|
|
\end{figure}
|
|
|
|
The germanium spectrum after three
|
|
cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut}
|
|
and~\eqref{eqn:sir2_ge_cut} is plotted in \cref{fig:sir2_xray}. The $(2p-1s)$
|
|
line clearly showed up at \SI{400}{\keV} on a very low background. A peak at
|
|
\SI{476}{\keV} is identified as the $(3p-1s)$ transition. Higher transitions
|
|
such as $(4p-1s)$, $(5p-1s)$ and $(6p-1s)$ can also be recognised at
|
|
\SI{504}{\keV}, \SI{516}{\keV} and \SI{523}{\keV}, respectively.
|
|
%The $(2p-1s)$
|
|
%line is seen at 399.5~\si{\keV}, 0.7~\si{\keV} off from the reference value of
|
|
%400.177~\si{\keV}. This discrepancy is within our detector's resolution,
|
|
%and the calculated efficiency is $(4.549 \pm 0.108)\times 10^{-5}$ -- a 0.15\%
|
|
%increasing from that of the 400.177~keV line, so no attempt for recalibration
|
|
%or correction was made.
|
|
|
|
The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian
|
|
peak on top of a linear background from \SIrange{395}{405}{\keV}.
|
|
Using the same procedure of correcting described in
|
|
\cref{sub:germanium_detector}, and taking detector acceptance and X-ray
|
|
intensity into account (see \cref{tab:sir2_xray_corr}), the number of muon
|
|
stopped is:
|
|
\begin{equation}
|
|
N_{\mu \textrm{ stopped X-ray}} = (9.16 \pm 0.28)\times 10^6\,,
|
|
\label{eqn:n_stopped_xray_count}
|
|
\end{equation}
|
|
which is consistent with the number of X-rays counted using the active target.
|
|
|
|
The uncertainty of the number of muons inferred from the X-ray
|
|
has equal contributions from statistical uncertainty in peak
|
|
area and systematic uncertainty from efficiency calibration. The relative
|
|
uncertainty in number of muons is 3\%, good enough for the normalisation in
|
|
this measurement.
|
|
\begin{table}[btp]
|
|
\begin{center}
|
|
\begin{tabular}{@{}llll@{}}
|
|
\toprule
|
|
\textbf{Measured X-rays} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\ \midrule
|
|
Gross integral & 3083 & & \\
|
|
Background & 101.5 & & \\
|
|
Net area $(2p-1s)$ & 2929.7 & 56.4 & 0.02 \\
|
|
\vspace{0.03em}\\
|
|
\toprule
|
|
\textbf{Corrections} & \textbf{Value} & \multicolumn{2}{c}{\textbf{Details}}\\
|
|
\midrule
|
|
Random summing & 1.06 & \multicolumn{2}{l}{average count rate \SI{491.4}{\Hz},}\\
|
|
& & \multicolumn{2}{l}{pulse length \SI{57}{\us}}\\
|
|
TRP reset & 1.03 & \multicolumn{2}{l}{\SI{298}{\s} loss during \SI{9327}{\s} run period}\\
|
|
Self-absorption & 1.008 & \multicolumn{2}{l}{silicon thickness \SI{750}{\um},}\\
|
|
& & \multicolumn{2}{l}{linear attenuation \SI{0.224}{\per\cm}}\\
|
|
True coincidence & 1 & \multicolumn{2}{l}{omitted} \\
|
|
\vspace{0.03em}\\
|
|
\toprule
|
|
\textbf{Efficiency and intensity} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
|
|
\midrule
|
|
Detector efficiency & \num{4.40E-4} & \num{0.10E-4} & 0.02 \\
|
|
X-ray intensity & 0.803 & 0.008 & 0.009 \\
|
|
\vspace{0.03em}\\
|
|
\toprule
|
|
\textbf{Results} & \textbf{Value} & \textbf{Absolute error} & \textbf{Relative error} \\
|
|
\midrule
|
|
Number of X-rays emitted & \num{7.36E6} & \num{0.22E6} & 0.03 \\
|
|
Number of muons stopped & \num{9.16E6} & \num{0.28E6} & 0.03 \\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Corrections, efficiency and intensity used in calculating the number
|
|
of X-rays from the active target.}
|
|
\label{tab:sir2_xray_corr}
|
|
\end{table}
|
|
|
|
%In order to measure the charged particles after nuclear muon capture, one would
|
|
%pick events where the emitted particles are well separated from the
|
|
%muon stop time. The energy timing correlation plot suggests a timing window
|
|
%starting from at least 1200~ns, therefore another cut is introduced:
|
|
%\begin{enumerate}
|
|
%\setcounter{enumi}{2}
|
|
%\item there are at least two hits on the active target, the time
|
|
%difference between the second hit on target (decay or capture product) and
|
|
%the muon counter hit is at least 1300 ns:
|
|
%\begin{equation}
|
|
%t_{\textrm{target 2nd hit}} - t_{\mu\textrm{ counter}} \geq \SI{1300}{\ns}
|
|
%\label{eqn:sir2_2ndhit_cut}
|
|
%\end{equation}
|
|
%\end{enumerate}
|
|
|
|
%The three cuts~\eqref{eqn:sir2_prompt_cut},~\eqref{eqn:sir2_muE_cut} and
|
|
%~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}.
|
|
|
|
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Particle identification by specific energy loss}
|
|
\label{sec:particle_identification_by_specific_energy_loss}
|
|
In this analysis, a subset of runs from \numrange{2808}{2873} with the
|
|
100-\si{\um} aluminium target is used because of following advantages:
|
|
\begin{itemize}
|
|
\item it was easier to stop and adjust the muon stopping distribution in
|
|
this thicker target;
|
|
\item a thicker target gives better statistics because of a higher
|
|
muon rate available at a higher momentum and less scattering.
|
|
\end{itemize}
|
|
Muons with momentum of \SI{30.52}{\MeV\per\hepclight}, 3\%-FWHM spread (scaling factor of
|
|
1.09, normalised to \SI{28}{\MeV\per\hepclight}) were used for this target after
|
|
a momentum scanning as described in the next subsection.
|
|
|
|
\subsection{Momentum scan for the 100-\si{\um} aluminium target}
|
|
\label{sub:momentum_scan_for_the_100_}
|
|
Before deciding to use the momentum scaling factor of 1.09, we have scanned
|
|
with momentum scales ranging from 1.04 to 1.12 to maximise the
|
|
observed X-rays rate (and maximising the rate of stopped muons). The X-ray
|
|
spectrum at each momentum point was accumulated in about 30 minutes to
|
|
assure a sufficient amount of counts. Details of the scanning runs are listed
|
|
in \cref{tab:al100_scan}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{cccc}
|
|
\toprule
|
|
\textbf{Momentum (\si{\MeV\per\hepclight})} & \textbf{Scaling factor} & \textbf{Runs}
|
|
& \textbf{Length (s)}\\
|
|
\midrule
|
|
29.12 & 1.04 & \numrange{2609}{2613} &2299\\
|
|
29.68 & 1.06 & \numrange{2602}{2608} &2563\\
|
|
29.96 & 1.07 & \numrange{2633}{2637} &2030\\
|
|
30.24 & 1.08 & \numrange{2614}{2621} &3232\\
|
|
30.52 & 1.09 & \numrange{2808}{2813} &2120\\
|
|
30.80 & 1.10 & \numrange{2625}{2632} &3234\\
|
|
31.36 & 1.12 & \numrange{2784}{2792} &2841\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Momentum scanning runs for the 100-\si{\um} aluminium target.}
|
|
\label{tab:al100_scan}
|
|
\end{table}
|
|
The on-site quick analysis suggested the 1.09 scaling factor was the
|
|
optimal value so it was chosen for all the runs on this aluminium target. But
|
|
the offline analysis later showed that the actual optimal factor was 1.08.
|
|
There were two reasons for the discrepancy:
|
|
\begin{enumerate}
|
|
\item the X-ray rates were normalised to run length, which is biased
|
|
since there are more muons available at higher momenta;
|
|
\item the $(2p-1s)$ peaks of aluminium at \SI{346.828}{\keV} were not
|
|
fitted properly. The peak is interfered by a background peak at
|
|
\SI{351}{\keV} from $^{214}$Pb, but the X-ray peak area was
|
|
obtained simply by subtracting an automatically estimated background.
|
|
\end{enumerate}
|
|
In the offline analysis, the X-ray peak and the background peak are fitted by
|
|
two Gaussian peaks on top of a linear background. The X-ray peak
|
|
area is then normalised to the number of muons hitting the upstream detector
|
|
(\cref{fig:al100_xray_fit}).
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.50\textwidth]{figs/al100_xray_fit}
|
|
\includegraphics[width=0.50\textwidth]{figs/al100_xray_musc}
|
|
\caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium (red) and the
|
|
interfered peak at \SI{351}{\keV} (brown) with a linear background (left).
|
|
The number of muons is integral of the upstream scintillator spectrum
|
|
(right) from \numrange{400}{2000} ADC channels.}
|
|
\label{fig:al100_xray_fit}
|
|
\end{figure}
|
|
|
|
The ratio between the number of X-rays and the number of muons as a function
|
|
of momentum scaling factor is plotted on \cref{fig:al100_scan_rate}. The trend
|
|
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 scales of 1.09 and above. The distributions of stopped muons
|
|
are illustrated by MC results on the bottom plot in
|
|
\cref{fig:al100_scan_rate}. At the 1.09
|
|
scale beam, the muons stopped \SI{18}{\um} off-centred to the right silicon
|
|
arm, the standard deviation of the depth distribution is \SI{29}{\um}.
|
|
\begin{figure}[!htb]
|
|
\centering
|
|
\includegraphics[width=0.77\textwidth]{figs/al100_scan_rate}
|
|
\includegraphics[width=0.77\textwidth]{figs/al100_mu_stop_mc}
|
|
\caption{Number of X-rays per incoming muon as a function of momentum
|
|
scaling factor (top); and muon stopping distributions with scaling factors
|
|
from 1.04 to 1.12 obtained by MC simulation
|
|
(bottom). The depth of muon stopping positions are measured normal to
|
|
the surface of the target facing the muon beam.}
|
|
\label{fig:al100_scan_rate}
|
|
\end{figure}
|
|
|
|
\subsection{Event selection for the passive targets}
|
|
\label{sub:event_selection_for_the_passive_targets}
|
|
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} of such muon events.
|
|
|
|
\subsubsection{Particle banding identification}
|
|
\label{ssub:particle_banding_identification}
|
|
|
|
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
|
|
is a thin silicon hit within a coincidence window of $\pm 0.5$~\si{\us}\ around
|
|
the thick silicon hit, the two hits are considered to belong to one particle.
|
|
The thresholds for energy deposited in all silicon channels, except the thin
|
|
silicon on the left arm, are set at \SI{100}{\keV} in this analysis. The
|
|
threshold on the left $\Delta E$ counter was higher, at roughly
|
|
\SI{400}{\keV}, due to higher noise in that channel and it was decided at the
|
|
run time to rise its threshold to reduce the triggering rate.
|
|
The specific energy loss as a function of total energy of the charged
|
|
particles are plotted on \cref{fig:al100_dedx}.
|
|
\begin{figure}[p]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_EdE_left}
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_EdE_right}
|
|
\caption{Energy loss in thin silicon detectors as a function of total energy
|
|
recorded by both thin and thick detectors on the left arm (top) and the
|
|
right arm (bottom).}
|
|
\label{fig:al100_dedx}
|
|
\end{figure}
|
|
With the aid from MC simulation (\cref{fig:pid_sim}), the banding on
|
|
\cref{fig:al100_dedx} can be identified as follows:
|
|
\begin{itemize}
|
|
\item the spot at the lower left conner belonged to electron hits;
|
|
\item the scattered muons formed the small blurry cloud just above the
|
|
electron region;
|
|
\item the most intense band was due to proton hits;
|
|
\item the less intense, upper band caused by deuteron hits;
|
|
\item the highest band corresponded to alpha hits;
|
|
\item the faint stripe above the deuteron band should be triton
|
|
hits, which is consistent with a relatively low probability of emission of
|
|
tritons.
|
|
\end{itemize}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_dedx_overlay}
|
|
\caption{Identifying of charged particles banding: the dots are measured
|
|
points, the histograms are expected bands of protons (red), deuterons
|
|
(green) and tritons (blue), respectively. The MC bands are calculated
|
|
for a pair of 58-\si{\um}-thick and 1535-\si{\um}-thick silicon
|
|
detectors. The error bars on MC bands show the standard deviation of
|
|
$\Delta E$ in E respective bins.
|
|
}
|
|
\label{fig:dummylabel}
|
|
\end{figure}
|
|
|
|
It is not clearly seen in the $\Delta E-E$ plots because of the rather high
|
|
thresholds on $\Delta E$, but protons with higher energy would punch through
|
|
both silicon detectors. Those events have low $\Delta E$ and $E$, making the
|
|
proton bands to go backward to the origin of the $\Delta E-E$ plots. For the
|
|
configuration of 58-\si{\um} thin, and 1535-\si{\um} thick detectors, the
|
|
effect shows up for protons with energy larger than \SI{16}{\MeV}. The
|
|
returning part of the proton band would make the cut described in
|
|
the next subsection to include protons with higher energy into lower energy
|
|
region. The effect of punch through protons could be eliminate using the veto
|
|
plastic scintillators at the back of each silicon arm. But in this initial
|
|
analysis, the veto information is not used, therefore the upper limit of
|
|
proton energy is set at \SI{8}{\MeV} where there is clear separation between
|
|
the protons at lower and higher energies with the same measured total energy
|
|
deposition $E$.
|
|
|
|
\subsubsection{Proton-like probability cut}
|
|
\label{ssub:proton_like_probability_cut}
|
|
Since protons of interested are at low kinetic energy, their $\Delta E$
|
|
distributions do not have long tails as that of the Landau distribution.
|
|
For a given $E$, the distribution of $\Delta E$ is more like a Gaussian, and
|
|
with slightly deformed high energy tail (see \cref{fig:dE_distribution}).
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.75\textwidth]{figs/dE_distribution}
|
|
\caption{Distributions of $\Delta E$ of protons in a 58-\si{\um}-thick
|
|
silicon detector for given $E$ in various energy ranges.}
|
|
\label{fig:dE_distribution}
|
|
\end{figure}
|
|
|
|
%In order to select protons, a proton likelihood probability is defined as:
|
|
%The band of protons is therefore by cut on likelihood probability
|
|
%calculated as:
|
|
For a measured event, a proton likelihood probability is defined as:
|
|
\begin{equation}
|
|
P_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
|
|
\exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta
|
|
E}}\right]}\,,
|
|
\end{equation}
|
|
where $\Delta E_{\textrm{meas.}}$ and $E_i$ are measured energy deposition in
|
|
thin silicon detector and in both detectors, respectively; $\Delta E_i$ and
|
|
$\sigma_{\Delta E}$ are the expected value and standard deviation of the energy
|
|
loss in the thin detector of protons with energy $E$, calculated by the
|
|
MC simulation. A measured event with higher $P_i$ is more likely to be
|
|
a proton event.
|
|
|
|
The lower threshold of proton-like probability, the more protons will be
|
|
selected, but also more contamination from other charged particles would be
|
|
classified as protons. The number of protons on the left and right arms at
|
|
different cuts on $P_i$ are listed in \cref{tab:nprotons_vs_pcut}. The proton
|
|
yields are integrated in the energy range from \SIrange{2.2}{8}{\MeV}. The
|
|
lower limit comes from the requirement of having at least one hit on the thick
|
|
counter. The upper limit is to avoid the inclusion of punch through particles
|
|
as explained in the previous session.
|
|
|
|
The cut efficiency depends on actual shape of the proton spectrum, other
|
|
charged particles spectra, relative ratio between the yields of different
|
|
particle species. The fraction of protons missed out, and the fraction of
|
|
contamination from other charged particles at different probability
|
|
thresholds, with two different assumptions on spectrum shape: flat distribution
|
|
and Sobottka and Wills silicon shape (see \eqref{eqn:EH_pdf}),
|
|
are listed in the four last columns of \cref{tab:nprotons_vs_pcut}. The
|
|
relative ratio of proton:deuteron:triton:alpha:muon is assumed to be
|
|
5:2:1:2:2. The probability threshold is therefore chosen to be \num{1.0E-4} in
|
|
in order to have both relatively low missing and contamination fractions
|
|
compare to the statistical uncertainty of the measurement. The resulted band of
|
|
protons is shown in (\cref{fig:al100_protons}).
|
|
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{c c c c S S S S}
|
|
\toprule
|
|
\textbf{$P_i$} & \textbf{Equiv.} & \multirow{2}{*}{\textbf{Left}} &
|
|
\multirow{2}{*}{\textbf{Right}} & {\textbf{Miss.}} & \textbf{Contam.}
|
|
& {\textbf{Miss.}} & \textbf{Contam.}\\
|
|
|
|
\textbf{threshold} & \textbf{$\sigma$} & &
|
|
&{\textbf{flat}} & {\textbf{flat}}
|
|
&{\textbf{expo.}} & {\textbf{expo.}}\\
|
|
\midrule
|
|
\num{4.5E-2} & 2 & 1720 & 2214 & 1.9 \%& 0.03 \%&6.1 \%& 0.06 \%\\
|
|
\num{2.7E-3} & 3 & 1801 & 2340 & 0.7 \%& 0.05 \%&2.8 \%& 0.1 \%\\
|
|
\num{1.0E-4} & 3.89 & 1822 & 2373 & 0.5 \%& 0.1 \%&1.2 \%& 0.3 \%\\
|
|
\num{5.7E-7} & 5 & 1867 & 2421 & 0.4 \%& 0.7 \%&0.7 \%& 0.9 \%\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Proton yields in energy range from \SIrange{2.2}{8}{\MeV} on the two
|
|
silicon arms with different thresholds on proton-like probability $P_i$,
|
|
and the MC calculated missing fractions and contamination levels with two
|
|
different assumptions on spectrum shape: flatly distributed, and
|
|
exponential decay spectrum (see \eqref{eqn:EH_pdf}).}
|
|
\label{tab:nprotons_vs_pcut}
|
|
\end{table}
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.47\textwidth]{figs/al100_protons}
|
|
\includegraphics[width=0.47\textwidth]{figs/al100_protons_px_r}
|
|
\caption{Protons (green) selected using the likelihood probability cut of
|
|
\num{1.0E-4} (left). The proton spectrum (right) is obtained by projecting
|
|
the proton band onto the total energy axis.}
|
|
\label{fig:al100_protons}
|
|
\end{figure}
|
|
|
|
\subsubsection{Possible backgrounds}
|
|
\label{ssub:possible_backgrounds}
|
|
|
|
There are several sources of potential backgrounds in this proton measurement:
|
|
\begin{enumerate}
|
|
\item Protons emitted after capture of scattered muons in the lead
|
|
shield: the incoming muons could be scattered to other materials
|
|
surrounding the target, emitting protons to the silicon detectors. In
|
|
order to avoid complication of estimating this background, we used lead
|
|
sheets to collimate and shield around the target and detectors. If
|
|
a scattered muon is captured by the lead shielding, the proton from lead
|
|
would be emitted shortly after the muon hit because of the short average
|
|
lifetime of muons in lead (\SI{78.4}{\ns}~\cite{Measday.2001}). In
|
|
comparison, average lifetime of muons in aluminium is
|
|
\SI{864}{\ns}~\cite{Measday.2001}, therefore a simple cut in timing could
|
|
eliminate background of this kind.\\
|
|
The timing of events classified as protons are plotted in
|
|
\cref{fig:al100_proton_timing}. The spectra show no significant fast
|
|
decaying component, which should show up if the background from lead
|
|
shielding were sizeable. A fit of an exponential function on top of a flat
|
|
background gives the average lifetimes of muons as:
|
|
\begin{align}
|
|
\tau_{\textrm{left}} &= \SI{870 \pm 25}{\ns} \,,\\
|
|
\tau_{\textrm{right}} &= \SI{868 \pm 21}{\ns} \,.
|
|
\end{align}
|
|
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_proton_timing}
|
|
\caption{Timing of protons relative to muon hit. The spectra show the
|
|
characteristic one-component decay shape.}
|
|
\label{fig:al100_proton_timing}
|
|
\end{figure}
|
|
The consistency between fitted lifetimes and the reference value of average
|
|
lifetime of muons in aluminium at \SI{864\pm 2}{\ns} suggests the background
|
|
from the lead shielding is negligible. This smallness can be explained as
|
|
a combination of the two facts: (i) only
|
|
a minority fraction of muons punched through the target and reached the
|
|
downstream lead shield as illustrated in
|
|
\cref{fig:al100_scan_rate}; and (ii) the probability of emitting protons from
|
|
lead is very low compare to that of aluminium, about 0.4\% per
|
|
capture (see \cref{tab:lifshitzsinger_cal_proton_rate}).
|
|
|
|
\item The protons emitted after scattered muons stopped at the surface of
|
|
the thin silicon detectors: these protons could mimic the signal if they
|
|
appear within \SI{1}{\us} around the time muon hit the upstream counter.
|
|
The $\Delta E$ and $E$ in this case would be sum of energy of a muon and
|
|
energy of the resulted proton. The average energy of scattered muons can be
|
|
seen in \cref{fig:al100_dedx} to be about \SI{1.4}{\MeV}. The measured
|
|
$\Delta E$ and $E$ then would be shifted by \SI{1.4}{\MeV}, makes the
|
|
measured data point move far away from the expected proton band. Therefore
|
|
this kind of background should be small with the current probability cut.
|
|
|
|
\item The random background: this kind of background can be
|
|
examined by the same timing spectrum in
|
|
\cref{fig:al100_proton_timing}. The random events show up at negative time
|
|
difference and large delay time regions and give a negligible contribution
|
|
to the total number of protons.
|
|
\end{enumerate}
|
|
|
|
It is concluded from above arguments that the backgrounds of this proton
|
|
measurement is negligibly small.
|
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
|
|
\section{Proton emission rate from aluminium}
|
|
\label{sec:proton_emission_rate_from_aluminium}
|
|
The analysis is done on the same dataset used in
|
|
\cref{sec:particle_identification_by_specific_energy_loss}. Firstly, the
|
|
number of protons emitted is extracted using specific energy loss. Then
|
|
correction for energy loss inside the target is applied. Finally, the number
|
|
of protons is normalised to the number of nuclear muon captures.
|
|
|
|
\subsection{Number of protons emitted}
|
|
\label{sub:number_of_protons_emitted}
|
|
The numbers of protons in the energy range from \SIrange{2.2}{8.5}{\MeV} after
|
|
applying the probability cut are:
|
|
\begin{align}
|
|
N_{\textrm{p meas. left}} = 1822 \pm 42.7 \,,\\
|
|
N_{\textrm{p meas. right}} = 2373 \pm 48.7 \,.
|
|
\end{align}
|
|
The right arm received significantly more protons than the left arm did, which
|
|
is expected as in \cref{sub:momentum_scan_for_the_100_} where it is shown that
|
|
muons stopped off-centred to the right arm.
|
|
|
|
\subsection{Corrections for the number of protons}
|
|
\label{sub:corrections_for_the_number_of_protons}
|
|
The protons spectra observed by the silicon detectors have been modified by
|
|
the energy loss inside the target so correction (or unfolding) is necessary.
|
|
The unfolding, essentially, is finding a response function that relates proton's
|
|
true energy and measured value. This can be done in MC simulation by generating
|
|
protons with a spatial distribution as close as possible to the real
|
|
distribution of muons inside the target, then counting the number of protons
|
|
reaching the silicon detectors. Such response function conveniently includes
|
|
the geometrical acceptance.
|
|
|
|
For the 100-\si{\um} aluminium target and muons at the momentum scale of 1.09,
|
|
the parameters of the initial protons are:
|
|
\begin{itemize}
|
|
\item horizontal distribution: Gaussian \SI{26}{\mm} FWHM, centred at the
|
|
centre of the target;
|
|
\item vertical distribution: Gaussian \SI{15}{\mm} FWHM, centred at the
|
|
centre of the target;
|
|
\item depth: Gaussian \SI{69.2}{\um} FWHM, centred at \SI{68.8}{\um}-deep from
|
|
the upstream face of the target;
|
|
\item energy: flatly distributed from \SIrange{1.5}{15}{\MeV}.
|
|
\end{itemize}
|
|
The calculated response matrices for the two arms are presented in
|
|
\cref{fig:al100_resp_matrices}. The different path lengths inside the target
|
|
to the two silicon arms causes the difference in the two matrices. The
|
|
response matrices are then used as MC truth to train and test the unfolding
|
|
code. The code uses an existing ROOT package called RooUnfold~\cite{Adye.2011}
|
|
where the iterative Bayesian unfolding method is implemented.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.99\textwidth]{./figs/al100_resp}
|
|
\caption{Response functions for the two silicon arms, showing the relation
|
|
between protons energy at birth and as detected by the silicon detector
|
|
arms.}
|
|
\label{fig:al100_resp_matrices}
|
|
\end{figure}
|
|
%After training, the unfolding code is applied on the measured spectra from the
|
|
%left and right arms. The unfolded proton spectra in \cref{fig:al100_unfold}
|
|
%reasonably reflect the distribution of initial protons which is off-centred to
|
|
%the right arm. The path length to the left arm is longer so less protons at
|
|
%energy lower than \SI{5}{\MeV} could reach the detectors. The sharp low-energy
|
|
%cut off on the right arm is consistent with the Coulomb barrier for protons,
|
|
%which is \SI{4.1}{\MeV} for protons emitted from $^{27}$Mg.
|
|
The unfolded spectra using the two observed spectra at the two arms as input
|
|
are shown in \cref{fig:al100_unfold}. The two unfolded spectra generally agree
|
|
with each other, except for a few first and last bins.
|
|
In the lower energy region, there is a small probability for such protons to
|
|
escape and reach the detectors, therefore the unfolding is generally unstable
|
|
and the uncertainties are large.
|
|
At the higher end, 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. The lower threshold on the thin silicon
|
|
detector at the right arm compared with that at the left arm makes this
|
|
misidentification worse.
|
|
\begin{figure}[!htb]
|
|
\centering
|
|
\includegraphics[width=0.80\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:
|
|
%\begin{itemize}
|
|
%\item stability against cut-off energy;
|
|
%\item comparison between the two arms;
|
|
%\item and unfolding of a MC-generated spectrum.
|
|
%\end{itemize}
|
|
The stability of the unfolding code is tested by varying the lower and upper
|
|
cut-off energies of the input spectrum. Plots in \cref{fig:al100_cutoff_study}
|
|
show that the shapes of the unfolded spectra are stable after a few first or
|
|
last bins.
|
|
%The
|
|
%lower cut-off energy of the
|
|
%output increases as that of the input increases, and the shape is generally
|
|
%unchanged after a few bins.
|
|
\begin{figure}[!htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_cutoff_study}
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_up_cut_off_reco}
|
|
\caption{Unfolded spectra with different lower (top) and upper (bottom)
|
|
cut-off energies.}
|
|
\label{fig:al100_cutoff_study}
|
|
\end{figure}
|
|
|
|
The proton yields calculated from observed spectra in two arms are compared in
|
|
\cref{fig:al100_integral_comparison} where the upper limit of the integrals
|
|
is fixed at \SI{8}{\MeV}, and the lower limit is varied in \SI{400}{\keV}
|
|
step. The upper limit was chosen to avoid the effects of punched through
|
|
protons. The difference is large at cut-off energies less than \SI{4}{\MeV}
|
|
due to large uncertainties at the first bins. Above \SI{4}{\MeV}, the two arms
|
|
show consistent numbers of protons.
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_integral_comparison}
|
|
\caption{Proton yields calculated from two arms. The upper limit of
|
|
integrations is fixed at \SI{8}{\MeV}, the horizontal axis is the lower
|
|
limit of the integrations. The proton yields on the two arm agree well
|
|
with each other from above \SI{4}{\MeV}.}
|
|
\label{fig:al100_integral_comparison}
|
|
\end{figure}
|
|
The yields of protons from \SIrange{4}{8}{\MeV} are:
|
|
\begin{align}
|
|
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}
|
|
The number of emitted protons is taken as average of the two yields:
|
|
\begin{equation}
|
|
N_{\textrm{p unfold}} = (169.3 \pm 1.9) \times 10^3
|
|
\end{equation}
|
|
|
|
\subsection{Number of nuclear captures}
|
|
\label{sub:number_of_nuclear_captures}
|
|
%\begin{figure}[!htb]
|
|
%\centering
|
|
%\includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
|
|
%\caption{X-ray spectrum from the aluminium target, the characteristic
|
|
%$(2p-1s)$ line shows up at 346.67~keV}
|
|
%\label{fig:al100_ge_spec}
|
|
%\end{figure}
|
|
|
|
%The X-ray spectrum on the germanium detector is shown on
|
|
%\cref{fig:al100_ge_spec}.
|
|
Fitting the double peaks on top of a linear background
|
|
gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
|
|
procedure as in the case of the active target, the number stopped muons and
|
|
the number of nuclear captures are:
|
|
\begin{align}
|
|
N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\,,\\
|
|
N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6\,.
|
|
\end{align}
|
|
|
|
\subsection{Proton emission rate}
|
|
\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} = 1.7\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
|
|
\eqref{eqn:EH_pdf} function has a power rising edge, and a exponential decay
|
|
falling edge. The falling edge has only one decay component and is suitable to
|
|
describe the proton spectrum with the equilibrium emission only assumption.
|
|
The pre-equilibrium emission contribution should be small for low-$Z$ material,
|
|
for aluminium the contribution of this component is 2.2\% of total number of
|
|
protons according to Lifshitz and Singer~\cite{LifshitzSinger.1980}.
|
|
%%TODO: draw the function and integral
|
|
The fitted results
|
|
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}[!p]
|
|
\centering
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_parameterisation}
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_fit_avgspec}
|
|
\includegraphics[width=0.85\textwidth]{figs/al100_fitted_func_integral}
|
|
\caption{Fitting of the unfolded spectra on the left and right arms (top),
|
|
and on the average spectrum (middle). The bottom plot shows the fitted
|
|
function of the average spectrum in the energy range from
|
|
\SIrange{1}{50}{\MeV}. The proton yield in the region from
|
|
\SIrange{4}{8}{\MeV} (shaded) is 51\% of the whole spectral integral.}
|
|
\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\\
|
|
$T_{0}$ (\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:
|
|
\begin{itemize}
|
|
\item uncertainties in the number of nuclear captures: these were discussed
|
|
in \cref{sub:number_of_stopped_muons_from_the_number_of_x_rays};
|
|
\item uncertainties in the number of protons:
|
|
\begin{itemize}
|
|
\item statistical uncertainties of the measured spectra which are
|
|
propagated during the unfolding process;
|
|
\item systematic uncertainties due to misidentification: this number is
|
|
small compared to other uncertainties as discussed in
|
|
\cref{sub:event_selection_for_the_passive_targets};
|
|
\item systematic uncertainty from the unfolding
|
|
\end{itemize}
|
|
\end{itemize}
|
|
The last item is studied by MC method using the parameterisation in
|
|
\cref{sub:proton_emission_rate}:
|
|
\begin{itemize}
|
|
\item protons with energy distribution obeying the parameterisation are
|
|
generated inside the target. The spatial distribution is the same as that
|
|
of in \cref{sub:corrections_for_the_number_of_protons}. MC truth including
|
|
initial energies and positions are recorded;
|
|
\item the number of protons reaching the silicon detectors are counted,
|
|
the proton yield is set to be the same as the measured yield to make the
|
|
statistical uncertainties comparable;
|
|
\item the unfolding is applied on the observed proton spectra, and the
|
|
results are compared with the MC truth.
|
|
\end{itemize}
|
|
\begin{figure}[htb]
|
|
\centering
|
|
\includegraphics[width=0.48\textwidth]{figs/al100_MCvsUnfold}
|
|
\includegraphics[width=0.48\textwidth]{figs/al100_unfold_truth_ratio}
|
|
\caption{Comparison between an unfolded spectrum and MC truth. On the left
|
|
hand side, the solid, red line is MC truth, the blue histogram is the
|
|
unfoldede spectrum. The ratio between the two yields is compared in the
|
|
right hand side plot with the upper end of integration is fixed at
|
|
\SI{8}{\MeV}, and a moving lower end of integration. The discrepancy
|
|
is genenerally smaller than 5\% if the lower end energy is smaller than
|
|
\SI{6}{\MeV}.}
|
|
\label{fig:al100_MCvsUnfold}
|
|
\end{figure}
|
|
\Cref{fig:al100_MCvsUnfold} shows that the yield obtained after unfolding is
|
|
in agreement with that from the MC truth. The difference is less than 5\% if
|
|
the integration is taken in the range from \SIrange{4}{8}{\MeV}. Therefore
|
|
a systematic uncertainty of 5\% is assigned for the unfolding routine.
|
|
|
|
A summary of uncertainties in the measurement of proton emission rate is
|
|
presented in \cref{tab:al100_uncertainties_all}.
|
|
\begin{table}[htb]
|
|
\begin{center}
|
|
\begin{tabular}{@{}ll@{}}
|
|
\toprule
|
|
\textbf{Item}& \textbf{Value} \\
|
|
\midrule
|
|
Number of muons & 3.2\% \\
|
|
Statistical from measured spectra & 1.1\% \\
|
|
Systematic from unfolding & 5.0\% \\
|
|
Systematic from PID & \textless1.0\% \\
|
|
\midrule
|
|
Total & 6.1\%\\
|
|
\bottomrule
|
|
\end{tabular}
|
|
\end{center}
|
|
\caption{Uncertainties of the proton emission rate.}
|
|
\label{tab:al100_uncertainties_all}
|
|
\end{table}
|
|
|
|
\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{3.7}{\MeV} which is a little below 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.
|
|
|
|
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}
|
|
|
|
\subsubsection{Comparison to theoretical and other experimental results}
|
|
\label{ssub:comparison_to_theoretical_and_other_experimental_results}
|
|
There is no existing experimental or theoretical work that could be directly
|
|
compared with the obtained proton emission rate. 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 emission of deuterons or tritons.
|
|
Certainly, when the full analysis is available, deuterons and tritons emission
|
|
rates could be extracted and the combined emission rate 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 rate of 3.5\% was estimated with an assumption that all protons are
|
|
emitted in equilibrium. With the exponential constant of \SI{2.6}{\MeV}, the
|
|
proton yield in the range from \SIrange{40}{70}{\MeV} is negligibly small
|
|
($\sim\num{E-8}$). However, Krane and colleagues reported a significant yield
|
|
of 0.1\% in that region~\cite{KraneSharma.etal.1979}. The energetic proton
|
|
spectrum shape also has a different exponential constant of \SI{7.5}{\MeV}. One
|
|
explanation for these protons is that they are emitted by other mechanisms,
|
|
such as capture on two-nucleon cluster suggested by Singer~\cite{Singer.1961}
|
|
(see \cref{sub:theoretical_models} and
|
|
\cref{tab:lifshitzsinger_cal_proton_rate_1988}). Despite being sizeable, the
|
|
yield of high energy protons is still small (3\%) in compared with the result
|
|
in \eqref{eqn:meas_total_rate}.
|
|
|
|
%The $(\mu^-,\nu p):(\mu^-,\nu pn)$ ratio is then roughly 1:1, not 1:6 as in
|
|
%\eqref{eqn:wyttenbach_ratio}.
|
|
|
|
\subsubsection{Comparison to the silicon result}
|
|
\label{ssub:comparison_to_the_silicon_result}
|
|
The probability of proton emission per nuclear capture of 3.5\% is indeed much
|
|
smaller than that of silicon. 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}.
|
|
|
|
The proton spectrum from aluminium is softer than silicon charged
|
|
particles spectrum of Sobottka and Wills~\cite{SobottkaWills.1968} where the
|
|
decay constant was \SI{4.6}{\MeV}. Two possible reasons can explain this
|
|
difference in shape:
|
|
\begin{enumerate}
|
|
\item The higher proton separation energy of $^{27}$Mg gives less
|
|
phase space for protons at higher energies than that in the case of
|
|
$^{28}$Al if the excitation energies of the two compound nuclei are similar.
|
|
\item 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.
|
|
\end{enumerate}
|
|
|