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thesis/chapters/chap6_analysis.tex
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@@ -37,9 +37,8 @@ for pulse information calculation is in use. The module looks for the
sample that
has the maximal deviation from the baseline, takes the deviation as pulse
amplitude and the time stamp of the sample as pulse time. The procedure is
illustrated on \cref{fig:tap_maxbin_algo}. This module could not detect
pile up or double pulses in one \tpulseisland{} in
\cref{fig:tap_maxbin_bad}.
illustrated on \cref{fig:tap_maxbin_algo}. This module could not account for
pile-up or double pulses in one \tpulseisland{} in \cref{fig:tap_maxbin_bad}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/tap_maxbin_algo}
@@ -57,12 +56,12 @@ pile up or double pulses in one \tpulseisland{} in
The TSimpleMuonEvent first picks a muon candidate, then loops through all
pulses on all detector channels, and picks all pulses occur in
a time window of \SI{\pm 10}{\si{\micro\second}} around each candidate to build
a time window of \SI{\pm 10}{\si{\us}} around each candidate to build
a muon event. A muon candidates is a hit on the upstream plastic scintillator
with an amplitude higher than a threshold which was chosen to reject minimum
ionising particles (MIPs). The period of \SI{10}{\si{\micro\second}} is long
enough compares to the mean life time of muons in the target materials
(\SI{0.758}{\si{\micro\second}} for silicon, and \SI{0.864}{\si{\micro\second}}
with an amplitude higher than a threshold which was chosen to reject MIPs. The
period of \SI{10}{\si{\us}} is long enough compares to the mean life time of
muons in the target materials
(\SI{0.758}{\si{\us}} for silicon, and \SI{0.864}{\si{\us}}
for aluminium~\cite{SuzukiMeasday.etal.1987}) so practically all of emitted
charged particles would be recorded in this time window.
%\begin{figure}[htb]
@@ -73,13 +72,13 @@ charged particles would be recorded in this time window.
%\end{figure}
A pile-up protection mechanism is employed to reject multiple muons events: if
there exists another muon hit in less than \SI{15}{\si{\micro\second}} from the
there exists another muon hit in less than \SI{15}{\si{\us}} from the
candidate then both the candidate and the other muon are discarded. This
pile-up protection would cut out less than 11\% total number of events because
the beam rate was generally less than \SI{8}{\kilo\hertz}.
%In runs with active silicon targets, another requirement is applied for the
%candidate: a prompt hit on the target in $\pm 200$ \si{\nano\second}\ around the
%candidate: a prompt hit on the target in $\pm 200$ \si{\ns}\ around the
%time of the $\mu$Sc pulse. The number comes from the observation of the
%time correlation between hits on the target and the $\mu$Sc
%(\cref{fig:tme_sir_prompt_rational}).
@@ -112,193 +111,13 @@ shown in \cref{fig:lldq}.
\end{figure}
% section analysis_modules (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Detector calibration}
\label{sec:detector_calibration}
\subsection{Silicon detector}
\label{sub:silicon_detector}
The energy calibration for the silicon detectors were done routinely during the
run, mainly by an
$^{241}\textrm{Am}$ alpha source and a tail pulse generator. The source emits
79.5 $\alpha$\si{\per\second} in a \SI{2\pi}{\steradian} solid angle. The most
prominent alpha particles have energies of \SI{5.484}{\si{\mega\electronvolt}}
(85.2\%) and \SI{5.442}{\si{\mega\electronvolt}} (12.5\%). A tail pulse with
amplitude of
\SI{66}{\milli\volt}~was used to simulate the response of the silicon detectors'
preamplifiers to a particle with \SI{1}{\si{\mega\electronvolt}} energy deposition.
During data taking period, electrons in the beam were were also used for energy
calibration of thick silicon detectors where energy deposition is large enough.
The muons at different momenta provided another mean of calibration in the beam
tuning period.
The alpha particles from the source would deposit
about 66~keV in the \SI{0.5}{\micro\meter}-thick dead layer, and the peak would
appear at 5418~keV (\cref{fig:toyMC_alpha}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.6\textwidth]{figs/toyMC_alpha}
\caption{Energy loss of the alpha particles after a dead layer of
0.5~\si{\micro\meter}.}
\label{fig:toyMC_alpha}
\end{figure}
The calibration coefficients for the silicon channels are listed in
\cref{tab:cal_coeff}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l c r}
\toprule
\textbf{Detector} & \textbf{Slope} & \textbf{Offset}\\
\midrule
SiL-2 & 7.86 & 14.14\\
SiR-2 & 7.96 & 22.98\\
\midrule
SiL1-1 & 2.61 & 37.34\\
SiL1-2 & 2.54 & -20.78\\
SiL1-3 & 2.65 & 67.75\\
SiL1-4 & 2.54 & -18.45\\
\midrule
SiR1-1 & 2.53 & 28.69\\
SiR1-2 & 2.62 & 47.10\\
SiR1-3 & 2.49 & 6.32\\
SiR1-4 & 2.53 & 34.81\\
\bottomrule
\end{tabular}
\end{center}
\caption{Calibration coefficients of the silicon detector channels}
\label{tab:cal_coeff}
\end{table}
% subsection silicon_detector (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Germanium detector}
\label{sub:germanium_detector}
The germanium detector was calibrated using a $^{152}\textrm{Eu}$
source\footnote{Energies and intensities of gamma rays are taken from the
X-ray and Gamma-ray Decay Data Standards for Detector Calibration and Other
Applications, which is published by IAEA at \\
\url{https://www-nds.iaea.org/xgamma_standards/}}, the
recorded pulse height spectrum is shown in \cref{fig:ge_eu152_spec}. The
source was placed at the target position so that the absolute efficiencies can
be calibrated. The relation between pulse height in ADC count and energy is
found to be:
\begin{equation}
\textrm{ E [keV]} = 0.1219 \times \textrm{ADC} + 1.1621
\end{equation}
The energy resolution (full width at half maximum) was better than
2.6~\si{\kilo\electronvolt}\ for all the $^{152}\textrm{Eu}$ peaks. It was a little
worse at 3.1~\si{\kilo\electronvolt}~for the annihilation photons at
511.0~\si{\kilo\electronvolt}.
The absolute efficiencies for the $(2p-1s)$ lines of aluminium
(346.828~\si{\kilo\electronvolt}) and silicon (400.177~\si{\kilo\electronvolt}) are
presented in \cref{tab:xray_eff}. In the process of efficiency calibration,
corrections for true coincidence summing and self-absorption were not applied.
The true coincidence summing probability is estimated to be very
small, about \sn{5.4}{-6}, thanks to the far geometry of the calibration. The
absorption in the source cover made of 22~\si{\milli\gram\per\si{\centi\meter}^2}
polyethylene is less than \sn{4}{-4} for a 100~\si{\kilo\electronvolt}\ photon.
\begin{table}[htb]
\begin{center}
\begin{tabular}{c c c}
\toprule
\textbf{X-ray} & \textbf{Efficiency} & \textbf{Uncertainty}\\
\midrule
346.828 & $5.12 \times 10^{-4}$ & $0.14\times 10^{-4}$\\
400.177 & $4.54 \times 10^{-4}$ & $0.11\times 10^{-4}$\\
\bottomrule
\end{tabular}
\end{center}
\caption{Calculated efficiencies at X-rays of interest}
\label{tab:xray_eff}
\end{table}
\begin{figure}[htb]
\centering
\includegraphics[width=0.70\textwidth]{figs/ge_eu152_spec}
\caption{Energy spectrum of the $\rm^{152}\textrm{Eu}$ calibration source
recorded by the germanium detector. The most prominent peaks of
$^{152}\textrm{Eu}$ along with their energies are
annotated in red; the 1460.82 \si{\kilo\electronvolt}~line is background from
$^{40}\textrm{K}$; and the annihilation 511.0~\si{\kilo\electronvolt}~photons
come both from the source and the surrounding environment.}
\label{fig:ge_eu152_spec}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.89\textwidth]{figs/ge_ecal_fwhm}
\caption{Germanium energy calibration and resolution.}
\label{fig:ge_fwhm}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.80\textwidth]{figs/ge_ecal_eff}
\caption{Absolute efficiency of the germanium detector, the fit was done with
7 energy points from 244~keV because it is known that the linearity between
$ln(\textrm{E})$ and $ln(\textrm{eff})$ holds better. The shaded area is
95\% confidence interval of the fit.}
\label{fig:ge_eff}
\end{figure}
% subsection germanium_detector (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Beam tuning and muon momentum scanning}
%\label{sub:muon_momentum_scanning}
%Before taking any data, we carried out the muon momentum scanning to understand
%the muon beam, as well as calibrate the detector system. The nominal muon
%momentum used in the Run 2013 had been tuned to 28~MeV\cc\ before the run. By
%changing simultaneously the strength of the key magnet elements in the $\pi$E1
%beam line with the same factor, the muon beam momentum would be scaled with the
%same factor.
%The first study was on the range of muons in an active silicon target. The SiL2
%detector was placed perpendicular to the nominal beam path, after an oval
%collimator. The beam momentum scaling factor was scanned from 1.10 to 1.60,
%muon momenta and energies in the measured points are listed in
%\cref{tab:mu_scales}.
%\begin{table}[htbp]
%\begin{center}
%\begin{tabular}{c c c c}
%\toprule
%\textbf{Scaling} & \textbf{Momentum} & \textbf{Kinetic energy}
%& \textbf{Momentum spread}\\
%\textbf{factor} & \textbf{(MeV\per\cc)} & \textbf{(MeV)}
%& \textbf{(MeV\per\cc, 3\% FWHM)}\\
%\midrule
%1.03 & 28.84 & 3.87& 0.87\\
%1.05 & 29.40 & 4.01& 0.88\\
%1.06 & 29.68 & 4.09& 0.89\\
%1.07 & 29.96 & 4.17& 0.90\\
%1.10 & 30.80 & 4.40& 0.92\\
%1.15 & 32.20 & 4.80& 0.97\\
%1.20 & 33.60 & 5.21& 1.01\\
%1.30 & 36.40 & 6.09& 1.09\\
%1.40 & 39.20 & 7.04& 1.18\\
%1.43 & 40.04 & 7.33& 1.20\\
%1.45 & 40.60 & 7.53& 1.22\\
%1.47 & 41.16 & 7.73& 1.23\\
%1.50 & 42.00 & 8.04& 1.26\\
%\bottomrule
%\end{tabular}
%\end{center}
%\caption{Muon beam scaling factors, energies and momenta.}
%\label{tab:mu_scales}
%\end{table}
% subsection muon_momentum_scanning (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% section detector_calibration (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charged particles following muon capture on a thick silicon target}
\label{sec:charged_particles_from_muon_capture_on_silicon_thick_silicon}
This analysis was done on a subset of the active target runs 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 \sn{6.43}{7}
muon events.
%64293720
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
%\num[fixed-exponent=2, scientific-notation = fixed]{6.4293720E7} muon events.
\num{6.43E7} muon events.
Firstly, the number of charged particles emitted after nuclear muon capture on
the active target is calculated. This number then is normalised to the number
@@ -309,15 +128,15 @@ compared with that from the literature.
\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 a narrow momentum spread beam was used. The
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
and about 5 MeV energy corresponds to stopped muons in the thick target. The
band below 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.
two pulses. It is the mentioned limitation of the current pulse parameter
extraction method where no pile up or double pulses is accounted for.
\begin{figure}[htb]
\centering
@@ -352,12 +171,12 @@ 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<50\textrm{ 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:
\begin{equation}
3.4 \textrm{ MeV}<E_{\textrm{target}} < 5.6 \textrm{ MeV}
\SI{3.4}{\MeV} \le E_{\textrm{target}} \le \SI{5.6}{\MeV}
\label{eqn:sir2_muE_cut}
\end{equation}
\end{enumerate}
@@ -371,45 +190,44 @@ starting from at least 1200~ns, therefore another cut is introduced:
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 1300\textrm{
ns}
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 muon events sample to the size of
\sn{9.32}{6}.
\num{9.32E+6}.
The number of stopped muons can also be calculated from the number of muonic
X-rays recorded by the germanium detector. The X-rays are emitted during the
cascading of the muon to the muonic 1S state in the time scale of \sn{}{-9}~s,
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
hits:
\begin{equation}
\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < 500\textrm{ ns}
\lvert t_{\textrm{Ge}} - t_{\mu\textrm{ counter}} \rvert < \SI{500}{\ns}
\label{eqn:sir2_ge_cut}
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of charged particles with energy above 2~MeV}
\subsection{Number of charged particles with energy above \SI{2}{\MeV}}
\label{sub:number_of_charged_particles_with_energy_from_8_10_mev}
As shown in \cref{fig:sir2_1us_slices} and illustrated by MC simulation
in \cref{fig:sir2_mc_pdfs}, there are several components in
the energy spectrum recorded by the active target:
\begin{enumerate}
\item charged particles from nuclear muon capture, this is the signal we are
interested in;
\item beam electrons with a characteristic Landau peak around 800~keV,
dominating at large timing (from 6500 ns), causing background at energy
lower than 1~MeV which drops sharply at energy larger than 3~MeV;
\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei from
neutron emitting muon captures, peak at
around 300~keV, dominate the region where energy smaller than 1~MeV and
timing less than 3500~ns. This component is intrinsic background, having
the same time structure as that of the signal;
\item charged particles from nuclear muon capture, this is the signal of
interest;
\item beam electrons with a characteristic Landau peak around \SI{800}{\keV},
dominating at large delay (from \SI{6500}{\ns}), causing background at
energy lower than \SI{1}{\MeV} which drops sharply at energy larger than
\SI{3}{\MeV};
\item electrons from muon decay-in-orbit (DIO) and recoiled nuclei
from neutron emitting muon captures, peak at
around \SI{300}{\keV}, dominate the region where energy smaller than
\SI{1}{\MeV} and delay less than \SI{3500}{\ns}. This component is
intrinsic background, having the same time structure as that of the signal;
\item stray muons scattered into the target, this component is small compares
to the charged particles yield so it is not considered further.
\end{enumerate}
@@ -425,10 +243,10 @@ the energy spectrum recorded by the active target:
\label{fig:sir2_mc_pdfs}
\end{figure}
An energy cut at 2~MeV is introduced to reduce the domination of the beam
electrons. In order to obtain the yields of backgrounds above 2~MeV, a binned
maximum likelihood fit was done. The likelihood of getting the measured
spectrum is defined as:
An energy cut at \SI{2}{\MeV} is introduced to avoid the domination of the
beam electrons at low energy. In order to obtain the yields of backgrounds
above \SI{2}{\MeV}, a binned maximum likelihood fit was done. The likelihood of
getting the measured spectrum is defined as:
\begin{equation}
L = \frac{e^{-\mu}\mu^n}{n!}\prod_i \frac{\mu_i^{n_i} e^{-\mu_i}}{n_i!}
\label{eqn:llh_def}
@@ -466,11 +284,10 @@ in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
The total number of charged particles from time zero is then calculated to be:
\begin{equation}
N_{\textrm{charged particles}} =(149.9\pm 0.6)\times 10^4
\label{eqn:sir2_Nchargedparticle}
N_{\textrm{charged particles}} =(149.9\pm 0.6)\times 10^4
\label{eqn:sir2_Nchargedparticle}
\end{equation}
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of nuclear muon captures}
@@ -478,8 +295,7 @@ N_{\textrm{charged particles}} =(149.9\pm 0.6)\times 10^4
The number of nuclear captures can be inferred from the number of recorded
muonic X-rays. The reference values of the parameters needed for the
calculation taken from Suzuki et al.~\cite{SuzukiMeasday.etal.1987} and Measday
et al.~\cite{MeasdayStocki.etal.2007} are
listed in \cref{tab:mucap_pars}.
et al.~\cite{MeasdayStocki.etal.2007} are listed in \cref{tab:mucap_pars}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l l l}
@@ -501,31 +317,35 @@ listed in \cref{tab:mucap_pars}.
The muonic X-ray spectrum emitted from the active target is shown in
\cref{fig:sir2_xray}. The $(2p-1s)$ line is seen at
399.5~\si{\kilo\electronvolt}, 0.7~\si{\kilo\electronvolt}\ off from the
reference value of 400.177~\si{\kilo\electronvolt}. 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.
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.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/sir2_xray}
\caption{Muonic X-rays spectrum from the active silicon target, the two major
lines $(2p-1s)$ and $(3p-1s)$ are clearly distinguishable at 400 and 476
keV, respectively. The $(5p-1s)$ line at 504 keV and $(6p-1s)$ line at 516
keV can also be seen.
\caption{Prompt muonic X-rays spectrum from the active silicon target, the
two major lines $(2p-1s)$ and $(3p-1s)$ are clearly distinguishable at 400
and 476 keV, respectively. The $(5p-1s)$ line at 504 keV and $(6p-1s)$ line
at 516 keV can also be seen.
}
\label{fig:sir2_xray}
\end{figure}
The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5 \pm 65.6$,
obtained by subtracting the background of 101.5 from the spectral integral of
2083 in the region from 396 to 402 keV. This number of X-rays needs to be
corrected for several effects:
%The area of the $(2p-1s)$ peak is $N_{(2p-1s)} = 2981.5 \pm 65.6$,
%obtained by subtracting the background of 101.5 from the spectral integral of
%2083 in the region from 396 to 402 keV.
The area of the $(2p-1s)$ peak is $2929.7 \pm 56.4$ obtained by fitting
a Gaussian peak on top of a first-order polynomial background to the spectrum
in \cref{fgi:sir2_xray} in the region from \SIrange{395}{405}{\keV}.
This number of X-rays needs to be corrected for following effects:
\begin{itemize}
\item Self-absorption effect: the X-rays emitted could be absorbed by the
target itself, the probability of self-absorption becomes larger in case of
thick sample and low energy photons.
For this silicon target of 1500~\si{\micro\meter}\ thick and the photon energy of
For this silicon target of 1500~\si{\um}\ thick and the photon energy of
400~keV, and assuming a narrow muon stopping distribution at the centre of
the target, the self-absorption correction is estimated to be:
\begin{align}
@@ -535,15 +355,15 @@ corrected for several effects:
%&= \dfrac{1}{0.992} \nonumber\\
&= 1.008
\end{align}
where $t = 0.075\textrm{ cm}$ is the thickness of the target, and $\mu$ is the
linear attenuation coefficient of silicon for a photon of 400~keV. The
where $t = \SI{0.075}{\cm}$ is the thickness of the target, and $\mu$ is
the linear attenuation coefficient of silicon for a photon of 400~keV. The
value of $\mu$ is calculated as product of the density of silicon
$\rho = 2.33 \textrm{ g/cm}^3$ and its mass attenuation coefficient
$\mu/\rho = 9.614\times 10^{-2} \textrm{ cm}^2/\textrm{g}$ taken
$\rho = \SI{2.33}{\g\per\cm^3}$ and its mass attenuation coefficient
$\mu/\rho = \SI{9.614E-2}{\cm^2\per\g}$ taken
from the NIST's X-ray Mass Attenuation Coefficients
table~\footnote{\url{http://www.nist.gov/pml/data/xraycoef}}.
\item Dead time of the germanium detector system: there are two types of dead
time in our germanium detector, (a) the insensitive period due to long
\item Dead time of the germanium detector system: there are two causes of
dead time in our germanium detector, (a) the insensitive period due to long
pulse time, and (b) the reset pulses of the transistor reset preamplifier.
The effects of the two dead time could be calculated by examining the
interval between two consecutive pulses on the germanium detector in
@@ -552,8 +372,8 @@ corrected for several effects:
\centering
\includegraphics[width=0.85\textwidth]{figs/sir2_ges_self_tdiff}
\caption{Interval between to consecutive pulses on the germanium
detector. The peak at 57~\si{\micro\second}\ indicates the pulse length, and
the bump at about 2000~\si{\micro\second}\ shows the width of the reset
detector. The peak at 57~\si{\us}\ indicates the pulse length, and
the bump at about 2000~\si{\us}\ shows the width of the reset
pulses. The average count rate of this detector is extracted from the
decay constant of the time spectrum to be
$5.146 \times 10^{-7}\textrm{ ns}^{-1} = 514.6 \textrm{ s}^{-1}$}
@@ -585,6 +405,13 @@ corrected for several effects:
origin of the X-rays have a finite spatial distribution. The correction
factor is estimated to be \ldots
\end{itemize}
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/ge_eff_mc_finitesize_vs_pointlike}
\caption{Ratio between geometrical acceptance of the germanium detector in
two cases: point-like source and finite-size source.}
\label{fig:ge_eff_mc_finitesize_vs_pointlike}
\end{figure}
The number of X-rays after applying all above corrections is 3293.5. The X-ray
intensity in \cref{tab:mucap_pars} was normalised to the number of stopped
@@ -686,13 +513,13 @@ So, the emission rate is:
%\end{figure}
%The spectrum measured by Sobottka and Wills~\cite{SobottkaWills.1968} is
%reproduced in \cref{fig:sobottka_spec}, the spectral integral in the
%energy region from 8 to 10~\si{\mega\electronvolt}\ is $2086.8 \pm 45.7$.
%The authors obtained the spectrum in a 4~\si{\micro\second}\ gate period which began
%1~\si{\micro\second}\ after a muon stopped, which would take 26.59\% of the emitted
%energy region from 8 to 10~\si{\MeV}\ is $2086.8 \pm 45.7$.
%The authors obtained the spectrum in a 4~\si{\us}\ gate period which began
%1~\si{\us}\ after a muon stopped, which would take 26.59\% of the emitted
%particles into account. The number of stopped muons was not explicitly stated,
%but can be inferred to be $55715/0.06 = 92858.3$.
%The partial rate of charged particle from 8 to 10~\si{\mega\electronvolt}\ is then
%The partial rate of charged particle from 8 to 10~\si{\MeV}\ is then
%calculated to be:
%\begin{equation}
%R_{\textrm{8-10 MeV}}^{lit.} =
@@ -700,7 +527,7 @@ So, the emission rate is:
%= 1.28 \times 10^{-2}
%\end{equation}
%The authors did not mentioned how the uncertainties of their measurement was
%derived, but quoted the emission rate below 26~\si{\mega\electronvolt}\ to be $0.15
%derived, but quoted the emission rate below 26~\si{\MeV}\ to be $0.15
%\pm 0.02$, which translates to a relative uncertainty of 0.133. The statistical
%uncertainty from the spectral integral and the number of stopped muons is:
%\begin{equation*}
@@ -708,14 +535,14 @@ So, the emission rate is:
%\end{equation*}
%Then their systematic uncertainty would be: $0.133 - 0.009 = 0.124$.
%For the partial spectrum from 8 to 10~\si{\mega\electronvolt}, the statistical
%For the partial spectrum from 8 to 10~\si{\MeV}, the statistical
%contribution to the uncertainty is:
%\begin{equation*}
%\dfrac{1}{\sqrt{2086.8}} + \dfrac{1}{\sqrt{92858.3}} = 2.5 \times 10^{-2}
%\end{equation*}
%So, the combined uncertainty of this partial rate calculation is: $0.124
%+ 0.025 = 0.150$. The partial rate of charged particles from 8 to
%10~\si{\mega\electronvolt} per muon capture is:
%10~\si{\MeV} per muon capture is:
%\begin{equation}
%R_{\textrm{8-10 MeV}}^{lit.} = (1.28 \pm 0.19) \times 10^{-2}
%\end{equation}
@@ -725,7 +552,7 @@ So, the emission rate is:
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Charged particles following muon capture on a thin silicon target}
\label{sec:charged_particles_following_muon_capture_on_a_thin_silicon_target}
In this measurement, a passive, 62-\si{\micro\meter}-thick silicon target was used as the
In this measurement, a passive, 62-\si{\um}-thick silicon target was used as the
target. The silicon target is $5\times5$~\si{\centi\meter}$^2$ in area. The muon
momentum was chosen to be 1.06 after a scanning to maximise the stopping ratio.
The charged particles were measured by two arms of silicon detectors. The
@@ -750,17 +577,17 @@ tree contains total $1.452 \times 10^8$ muon events. %145212698
\subsection{Particle identification by dE/dx and proton selection}
\label{sub:particle_identification_by_de_dx}
%All silicon hits with energy deposition larger than
%200~\si{\kilo\electronvolt}\ that happened within $\pm 10$~\si{\micro\second}\ of the
%200~\si{\keV}\ that happened within $\pm 10$~\si{\us}\ of the
%muon hit are then
%associated to the muon and stored in the muon event tree. The
%200~\si{\kilo\electronvolt}\ cut effectively rejects all MIPs hits on thin silicon
%detectors of which the most probable value is about 40~\si{\kilo\electronvolt}.
%200~\si{\keV}\ cut effectively rejects all MIPs hits on thin silicon
%detectors of which the most probable value is about 40~\si{\keV}.
%In order to use dE/dx for particle identification, $\Delta$E and total E are
%needed.
The charged particle selection 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{\micro\second}\ around the thick
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 with
$\Delta$E being the energy of the thin hit, and total E being the sum energy of
the two hits. Particle identification is done using correlation between
@@ -782,8 +609,8 @@ $\Delta$E-E plots can be identified as follows:
\end{itemize}
%The electrons either from Michel decay or from the beam are MIPs particles,
%which would deposit about 466~keV on the 1500-\si{\micro\meter}-thick silicon detector,
%and about 20~keV on the 65-\si{\micro\meter}-thick silicon detector. Therefore our thin
%which would deposit about 466~keV on the 1500-\si{\um}-thick silicon detector,
%and about 20~keV on the 65-\si{\um}-thick silicon detector. Therefore our thin
%silicon counters could not distinguish electrons from electronic
%noise. The brightest spots on the $\Delta$E-E plots are identified as electrons
%due to
@@ -860,7 +687,7 @@ The double peaks of muonic X-rays from the lead shield at 431 and 438~keV are
very intense, reflects the fact that the low momentum muon beam of
29.68~MeV\cc\ (scaling factor 1.06) was strongly scattered by the upstream
counters. After a prompt cut that requires the photon
hit occured in $\pm 1$~\si{\micro\second}\ around the muon hit, the peaks from lead
hit occured in $\pm 1$~\si{\us}\ around the muon hit, the peaks from lead
remain prominent which is an expected result because of all the lead shield
inside the chamber was to capture stray muons. The cut shows its effect on
reducing the background level under the 400.177 keV peak by about one third.
@@ -868,7 +695,7 @@ reducing the background level under the 400.177 keV peak by about one third.
\begin{figure}[htb]
\centering
\includegraphics[width=0.98\textwidth]{figs/si16p_xray}
\caption{X-ray spectrum from the passive 62-\si{\micro\meter}-thick silicon target with
\caption{X-ray spectrum from the passive 62-\si{\um}-thick silicon target with
and with out timing cut.}
\label{fig:si16_xray}
\end{figure}
@@ -914,10 +741,10 @@ X-rays recorded and the number of captures are shown in
\label{sub:lifetime_measurement}
To check the origin of the protons recorded, lifetime measurements were made by
cutting on time difference between a hit on one thick silicon and the muon
hit. Applying the time cut in 0.5~\si{\micro\second}\ time steps on the proton
hit. Applying the time cut in 0.5~\si{\us}\ time steps on the proton
events in \cref{fig:si16p_proton_after_ecut}, the number of surviving
protons on each arm are plotted on \cref{fig:si16p_proton_lifetime}. The
curves show decay constants of $762.9 \pm 13.7$~\si{\nano\second}\ and $754.6 \pm
curves show decay constants of $762.9 \pm 13.7$~\si{\ns}\ and $754.6 \pm
11.9$,
which are consistent with the each other, and with mean life time of muons in
silicon in the literatures of $758 \pm 2$~\cite{}. This is the confirmation
@@ -937,7 +764,7 @@ Therefore a timing cut from 500~ns is used to select good silicon events, the
remaining protons are shown in \cref{fig:si16p_proton_ecut_500nstcut}.
The spectra have a low energy cut off at 2.5~MeV because protons with energy
lower than that could not pass through the thin silicon to make the cuts as the
range of 2.5~MeV protons in silicon is about 68~\si{\micro\meter}.
range of 2.5~MeV protons in silicon is about 68~\si{\um}.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/si16p_proton_ecut_500nstcut}
@@ -1054,7 +881,7 @@ The ratio between the number of protons and other particles at 500~ns is $(1927
%\centering
%\includegraphics[width=0.85\textwidth]{figs/si16p_toyMC}
%\caption{An example of response function between the observed energy and
%initial energy of protons in a 62-\si{\micro\meter}-target.}
%initial energy of protons in a 62-\si{\um}-target.}
%\label{fig:si16p_toyMC}
%\end{figure}
@@ -1092,7 +919,7 @@ The ratio between the number of protons and other particles at 500~ns is $(1927
%\subsection{Proton emission rate and uncertainties estimation}
%\label{sub:proton_emission_rate_and_uncertainties_estimation}
%The rate of proton emission from 2.5--10~\si{\mega\electronvolt} is:
%The rate of proton emission from 2.5--10~\si{\MeV} is:
%\begin{equation}
%R =
%\end{equation}
@@ -1115,7 +942,7 @@ The ratio between the number of protons and other particles at 500~ns is $(1927
\section{Proton emission following muon capture on an aluminium target}
\label{sec:proton_emission_following_muon_capture_on_an_aluminium_target}
The aluminium is the main object of the AlCap experiment, in this preliminary
analysis I chose one target, Al100 the 100-\si{\micro\meter}-thick target, on
analysis I chose one target, Al100 the 100-\si{\um}-thick target, on
a sub-range of the data set runs 2808--2873, as a demonstration.
Because this is a passive target, the same procedure and cuts used in the
passive silicon runs were applied.
@@ -1159,8 +986,8 @@ proton energy spectrum is shown in \cref{fig:al100_proton_spec}.
The lifetime of these protons are shown in
\cref{fig:al100_proton_lifetime}, the fitted decay constant on the right
arm is consistent with the reference value of $864 \pm 2$~\si{\nano\second}~\cite{}.
But the left arm gives $918 \pm 16.1$~\si{\nano\second}, significantly larger than
arm is consistent with the reference value of $864 \pm 2$~\si{\ns}~\cite{}.
But the left arm gives $918 \pm 16.1$~\si{\ns}, significantly larger than
the reference value.
%The longer lifetime suggested some contributions from
%other lighter materials, one possible source is from muons captured on the back
@@ -1176,7 +1003,7 @@ the reference value.
Further investigation of the problem of longer lifetime was made and the first
channel on the thin silicon detector on that channel was the offender. The
lifetime measurement with out that SiL1-1 channel gives a reasonable result,
and the decay constant on the SiL1-1 alone was nearly about 1000~\si{\micro\second}.
and the decay constant on the SiL1-1 alone was nearly about 1000~\si{\us}.
The reason for this behaviour is not known yet. For this emission rate
calculation, this channel is discarded and the rate on the left arm is scaled
with a factor of 4/3. The proton spectrum from the aluminium target is plotted