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thesis/chapters/chap6_analysis.tex
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@@ -1,13 +1,18 @@
\chapter{Data analysis and results}
\label{cha:data_analysis}
This chapter presents initial analysis on subsets of the collected data.
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 of proton emission from aluminium.
\item extracting a preliminary rate and spectrum of proton emission from
aluminium.
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
@@ -22,10 +27,8 @@ methods:
\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} because of the problem of wrong clock frequency found in
the data quality checking shown in \cref{fig:lldq}. The data set contains
\numrange{2119}{2140}, which contains \num{6.43E7} muon events.
%\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events.
\num{6.43E7} muon events.
\subsection{Number of stopped muons from active target counting}
\label{sub:event_selection}
@@ -33,40 +36,49 @@ 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}. The most intense spot at zero time
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 mentioned limitation of the current pulse
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.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/sir2_sir2f_E_t_corr}
\caption{Energy - timing correlation of hits on the active target.}
\label{fig:sir2f_Et_corr}
\end{figure}
The hits on the silicon active target after 1200~ns are mainly secondary
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 electrons from muon decay in the 1S orbit,
\item products emitted after nuclear muon capture, including: gamma, neutron,
heavy charged particles and recoiled nucleus
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:sir2_1us_slices}).
\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}
$t > 5000$ 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}
@@ -74,19 +86,20 @@ From the energy-timing correlation above, the cuts to select stopped muons are:
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}
\lvert t_{\textrm{target}} - t_{\mu\textrm{ counter}}\rvert \le
\SI{50}{\ns}\,,
\label{eqn:sir2_prompt_cut}
\end{equation}
\item the first hit on the target has energy of that of the muons:
\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}
\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
N_{\mu \textrm{ active Si}} = 9.32 \times 10^6 \pm 3.0\times10^3\,.
\label{eqn:n_stopped_si_count}
\end{equation}
@@ -123,11 +136,19 @@ additional timing cut is applied for the germanium detector hits:
\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} with very low background. A peak at
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.
@@ -137,25 +158,24 @@ such as $(4p-1s)$, $(5p-1s)$ and $(6p-1s)$ can also be recognised at
%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.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/sir2_xray_22}
\caption{Prompt muonic X-rays spectrum from the active silicon target.
}
\label{fig:sir2_xray}
\end{figure}
The net area of the $(2p-1s)$ is found to be 2929.7 by fitting a Gaussian
peak on top of a first-order polynomial from \SIrange{395}{405}{\keV}.
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,
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@{}}
@@ -221,10 +241,10 @@ In this analysis, a subset of runs from \numrange{2808}{2873} with the
\begin{itemize}
\item it was easier to stop and adjust the muon stopping distribution in
this thicker target;
\item a thicker target means more stopped muons due to higher muon rate
available at higher momentum and less scattering.
\item a thicker target gives better statistics because of a higher
muon rate available at a higher momentum and less scattering.
\end{itemize}
Muons momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of
Muons with momentum of \SI{30.52}{\MeV\per\cc}, 3\%-FWHM spread (scaling factor of
1.09, normalised to \SI{28}{\MeV\per\cc}) were used for this target after
a momentum scanning as described in the next subsection.
@@ -232,8 +252,8 @@ a momentum scanning as described in the next subsection.
\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 hence maximising the rate of stopped muons). The X-ray
spectrum at each momentum point was accumulated in more than 30 minutes to
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]
@@ -259,44 +279,48 @@ in \cref{tab:al100_scan}.
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 mistake:
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 momentum;
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 first-order polynomial background. The X-ray peak
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=0.47\textwidth]{figs/al100_xray_fit}
\includegraphics[width=0.47\textwidth]{figs/al100_xray_musc}
\caption{Fitting of the $(2p-1s)$ muonic X-ray of aluminium and the background
peak at \SI{351}{\keV} (left). The number of muons is integral of the
upstream scintillator spectrum (right) from \numrange{400}{2000} ADC
channels.}
\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 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-centred to the right silicon arm.
\begin{figure}[htb]
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=0.47\textwidth]{figs/al100_scan_rate}
\includegraphics[width=0.47\textwidth]{figs/al100_mu_stop_mc}
\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 (left); and muon stopping distributions from MC simulation
(right). The depth of muons is measured normal to surface of the target
facing the muon beam.}
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}
@@ -307,24 +331,35 @@ 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 specific energy loss spectra recorded by the two silicon arms are plotted
on \cref{fig:al100_dedx}.
\begin{figure}[htb]
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_dedx}
\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.}
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 study (\cref{fig:pid_sim}), the banding on
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 densest spot at the lower left conner belonged to electron hits;
\item the small blurry cloud just above the electron region was muon hits;
\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;
@@ -332,36 +367,186 @@ With the aid from MC study (\cref{fig:pid_sim}), the banding on
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}
The band of protons is then extracted by cut on likelihood probability
calculated as:
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]}
E}}\right]}\,,
\end{equation}
where $\Delta E_{\textrm{meas.}}$ is energy deposition measured by the thin
silicon detector by a certain proton at energy $E_i$, $\Delta E_i$ and
$\sigma_{\Delta E}$ are the expected and standard deviation of the energy loss
caused by the proton calculated by MC study. A threshold is set at \num{1E-4} to
extract protons, the resulted band of protons is shown in
(\cref{fig:al100_protons}).
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
(left). The proton spectrum (right) is obtained by projecting the proton
band onto the total energy axis.}
\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}
The cut efficiency in the energy range from \SIrange{2}{12}{\MeV} is estimated
by MC study. The fraction of protons that do not satisfy the probability cut
is 0.5\%. The number of other charged particles that are misidentified as
protons depends on the ratios between those species and protons. Assuming
a proton:deuteron:triton:alpha:muon ratio of 5:2:1:2:2, the number of
misidentified hits is 0.1\% of the number of protons.
\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}
@@ -376,8 +561,8 @@ of protons is normalised to the number of nuclear muon captures.
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
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
@@ -405,15 +590,18 @@ the parameters of the initial protons are:
the upstream face of the target;
\item energy: flatly distributed from \SIrange{1.5}{15}{\MeV}.
\end{itemize}
The resulting response matrices for the two arms are presented in
\cref{fig:al100_resp_matrices}. These 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.
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=0.85\textwidth]{./figs/al100_resp}
\caption{Response functions for the two silicon arms.}
\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
@@ -425,15 +613,18 @@ method is implemented.
%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. The discrepancy and
large uncertainties at the low energy region are because of only a small
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]
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=0.85\textwidth]{figs/al100_unfolded_lr}
\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}
@@ -445,27 +636,37 @@ analysis.
%\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 cut-off
energy of the input spectrum. \cref{fig:al100_cutoff_study} show that the
shapes of the unfolded spectra are stable. 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]
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}
\caption{Unfolded spectra with different cut-off energies.}
\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 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.
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.}
\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:
@@ -490,13 +691,13 @@ The number of emitted protons is taken as average of the two yields:
%The X-ray spectrum on the germanium detector is shown on
%\cref{fig:al100_ge_spec}.
Fitting the double peaks on top of a first-order
polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
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
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}
@@ -504,7 +705,7 @@ the number of nuclear captures are:
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}
10^{-2}\,.
\label{eq:proton_rate_al}
\end{equation}
@@ -514,9 +715,9 @@ with the same parameterisation as in \eqref{eqn:EH_pdf}. The
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\% according to
Lifshitz and Singer~\cite{LifshitzSinger.1980}.
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
@@ -526,10 +727,16 @@ 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]
\begin{figure}[!p]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_parameterisation}
\caption{Fitting of the unfolded spectra.}
\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}
@@ -587,9 +794,13 @@ The last item is studied by MC method using the parameterisation in
\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: spectra
(left), and yields (right). The ratio is defined as $\textrm{(Unfold - MC
truth)/(MC truth)}$}
\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
@@ -608,7 +819,7 @@ presented in \cref{tab:al100_uncertainties_all}.
Number of muons & 3.2\% \\
Statistical from measured spectra & 1.1\% \\
Systematic from unfolding & 5.0\% \\
Systematic from PID & \textless0.5\% \\
Systematic from PID & \textless1.0\% \\
\midrule
Total & 6.1\%\\
\bottomrule
@@ -649,13 +860,7 @@ validated:
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,
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.
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:
@@ -717,4 +922,16 @@ 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}