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nam
2014-10-09 18:13:21 +09:00
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thesis/chapters/chap6_analysis.tex
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@@ -91,9 +91,9 @@ the beam rate was generally less than \SI{8}{\kilo\hertz}.
To make sure that we will analyse good data, a low level data quality checking
was done on the whole data sets. The idea is plotting the variations of basic
parameters, such as noise level, length of \tpulseisland{}, \tpulseisland{}
rate, time correlation to hits on $\mu$Sc, \ldots on each channel during the
data collecting period. Runs with significant difference from the nominal
parameters, such as noise level, length of raw waveforms, pulse rate, time
correlation to hits on the muon counter on each channel during the data
collecting period. Runs with significant difference from the averaging
values were further checked for possible causes, and would be discarded if such
discrepancy was too large or unaccounted for. Examples of such trend plots are
shown in \cref{fig:lldq}.
@@ -120,9 +120,10 @@ the data quality checking shown in \cref{fig:lldq}. The data set contains
\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
of nuclear muon capture to obtain an emission rate. Finally, the rate is
compared with that from the literature.
the active target is calculated. The charged particles yield then is normalised
to the number of nuclear muon capture to obtain an emission rate.
%Finally, the
%rate is compared with that from the literature.
\subsection{Event selection}
\label{sub:event_selection}
@@ -196,22 +197,21 @@ starting from at least 1200~ns, therefore another cut is introduced:
\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
\num{9.32E+6}.
~\eqref{eqn:sir2_2ndhit_cut} reduce the sample to the size of \num{9.82E+5}.
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 \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:
detector hits:
\begin{equation}
\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 \SI{2}{\MeV}}
\subsection{Number of charged particles with energy above \SI{3}{\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
@@ -235,11 +235,11 @@ the energy spectrum recorded by the active target:
\centering
\includegraphics[width=0.45\textwidth]{figs/sir2_meas_spec}
\includegraphics[width=0.45\textwidth]{figs/sir2_mc_pdfs}
\caption{The observed spectrum in the timing window 1500 -- 9500~ns (left)
\caption{The observed spectrum in the timing window 1300 -- 10000~ns (left)
and its components from MC simulation (right). The charged particles
spectrum is obtained assuming the spectrum shape and emission rate from
Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scale between
components is arbitrarily chosen for the purpose of illustration.}
Sobottka and Wills~\cite{SobottkaWills.1968}. The relative scales between
components are arbitrarily chosen for the purpose of illustration.}
\label{fig:sir2_mc_pdfs}
\end{figure}
@@ -270,7 +270,8 @@ in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
\begin{align}
n_{\textrm{beam e}} &= 23756 \pm 581\\
n_{\textrm{dio}} &= 111340 \pm 1245\\
n_{\textrm{sig}} &= 207201 \pm 856
n_{\textrm{sig}} &= 2.57 \pm 856
\label{eqn:sir2_n_chargedparticles}
\end{align}
\begin{figure}[htb]
\centering
@@ -282,12 +283,6 @@ in~\ref{fig:sir2_mll_fit}, the yields of backgrounds and signal are:
\label{fig:sir2_mll_fit}
\end{figure}
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}
\end{equation}
% subsection number_of_charged_particles_with_energy_from_8_10_mev (end)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Number of nuclear muon captures}
@@ -336,92 +331,34 @@ or correction was made.
%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}.
%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}.
Using the same procedure of fitting and correcting described in
\cref{sub:germanium_detector}, the number of X-rays is calculated to be 370.
Details of the correction factors are given in \cref{tab:sir2_xray_corr}.
\begin{table}[htb]
\begin{center}
\begin{tabular}{l}
\toprule
\textbf{Col1}\\
\midrule
item1\\
\bottomrule
\end{tabular}
\end{center}
\caption{Corrections for the number of X-rays from the active target.}
\label{tab:sir2_xray_corr}
\end{table}
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{\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}
k_{\textrm{self absorption}} &= \dfrac{\mu t}{1 - e^{-\mu t}} \nonumber\\
&= \dfrac {9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}}
{1 - e^{-9.614\times 10^{-2} \times 2.33 \times 0.75 \times 10^{-1}}}\nonumber \\
%&= \dfrac{1}{0.992} \nonumber\\
&= 1.008
\end{align}
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
$\rho = \SI{2.33}{\g\per\cm^3}$ 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 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
\cref{fig:sir2_ge_deadtime}.
\begin{figure}[htb]
\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{\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
$5.146 \times 10^{-7}\textrm{ ns}^{-1} = 514.6 \textrm{ s}^{-1}$}
\label{fig:sir2_ge_deadtime}
\end{figure}
The correction factor for the pulse length is calculated by the formula:
\begin{align}
k_{\textrm{pulse length}} &= e^{2\times \textrm{(pulse length)}
\times \textrm{(count rate)}} \nonumber\\
&= e^{2\times 57\times10^{-6} \times 514.6} \nonumber\\
&= 1.06
\end{align}
The 2-ms-long reset pulses effectively reduce the actual measurement time
compared to other channels, so the correction factor for the effect is:
\begin{align}
k_{\textrm{reset pulse}} &= \frac{\textrm{(measurement time)}}
{\textrm{(measurement time)}
- \textrm{(number of reset)}\times
\textrm{(reset pulse length)}}\nonumber\\
&= 1.033
\end{align}
\item The true coincidence summing is negligibly small due to the far
geometry as mentioned in the calibration process, so no correction is made.
%%TODO
\item The geometrical acceptance of the detector: the absolute efficiency
calibration was done with a point-like source, but the actual points of
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
muons, so the number of stopped muons is:
The X-ray intensity in \cref{tab:mucap_pars} was normalised to the number of
stopped muons, so the number of stopped muons is:
\begin{align}
N_{\mu\textrm{ stopped}} &=
\dfrac{N_{(2p-1s)}}{\epsilon_{2p-1s}\times I_{(2p-1s)}}\nonumber\\
&= \dfrac{3293.5}{4.54\times10^{-4} \times 0.803} \\
&= 9.03\times10^6 \nonumber
&= \dfrac{370}{4.38\times10^{-4} \times 0.803} \\
&= 1.05\times10^6 \nonumber
\end{align}
where $\epsilon_{(2p-1s)}$ is the calibrated absolute efficiency of the
detector for the 400.177~keV line in \cref{tab:xray_eff}, and
@@ -431,15 +368,15 @@ $I_{(2p-1s)}$ is the probability of emitting this X-ray per stopped muon
Taking the statistical uncertainty of the peak area, and systematic
uncertainties from parameters of muon capture, the number of stopped muons
calculated from the X-ray measurement is
$(9.03 \pm 0.31)\times 10^6$. This figure is consistent with the number of
stopped muons of $9.32\times10^6$ after the cuts described in the event
$(10.50 \pm 0.65)\times 10^5$. This figure is consistent with the number of
stopped muons of $9.82\times10^5$ after the cuts described in the event
selection process.
The number of nuclear captured muons is:
\begin{equation}
N_{\mu\textrm{ nucl.capture}} =
N_{\mu\textrm{ stopped}}\times f_{\textrm{cap.Si}}
= 9.03 \times 10^6 \times 0.658 = 7.25 \times 10^6
= 10.05 \times 10^5 \times 0.658 = 6.91 \times 10^5
\label{eqn:sir2_Ncapture}
\end{equation}
where the $f_{\textrm{cap.Si}}$ is the probability of nuclear capture per
@@ -738,7 +675,7 @@ X-rays recorded and the number of captures are shown in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Lifetime measurement}
\label{sub:lifetime_measurement}
\label{sub:lifetime_measurement}w
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{\us}\ time steps on the proton
@@ -762,7 +699,7 @@ before that, the time constants are shorter ($655.9\pm 9.9$ and $731.1\pm8.9$)
indicates the contamination from muon captured on material with higher $Z$.
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
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{\um}.
\begin{figure}[htb]

3
thesis/mythesis.sty
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@@ -58,7 +58,8 @@ bookmarks
\RequirePackage[final]{listings}
\RequirePackage{xfrac}
%% Units
\RequirePackage[]{siunitx}
%\RequirePackage[]{siunitx}
\RequirePackage[detect-weight=true, detect-family=true]{siunitx}
\RequirePackage{hepnames}
%% Various fonts ...
%\RequirePackage[T1]{fontenc}