nam/writeup
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

prog saved

Browse Source
This commit is contained in:
nam
2014-10-28 22:30:26 +09:00
parent 788e96cc54
commit 078802df95
4 changed files with 169 additions and 55 deletions
Download Patch File Download Diff File Expand all files Collapse all files

193
thesis/chapters/chap6_analysis.tex
View File

@@ -337,13 +337,15 @@ The band of protons is then extracted by cut on likelihood probability
calculated as:
\begin{equation}
p_{i} = \dfrac{1}{\sqrt{2\pi}\sigma_{\Delta E}}
e^{\frac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta E}}}
\exp{\left[\dfrac{(\Delta E_{meas.} - \Delta E_i)^2} {2\sigma^2_{\Delta
E}}\right]}
\end{equation}
where $\Delta E_{\textrm{meas.}}$ is measured energy deposition in the thin
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. A cut value of $3\sigma_{\Delta E}$, or
$p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}).
caused by the proton calculated by MC study. A threshold is set to extract
protons at 0.011 (equivalent to $3\sigma_{\Delta E}$), the band of protons is
shown in (\cref{fig:al100_protons}).
\begin{figure}[htb]
\centering
\includegraphics[width=0.47\textwidth]{figs/al100_protons}
@@ -354,6 +356,12 @@ $p_i \ge 0.011$, was used to extract protons (\cref{fig:al100_protons}).
\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.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proton emission rate from aluminium}
\label{sec:proton_emission_rate_from_aluminium}
@@ -365,25 +373,20 @@ of protons is normalised to the number of nuclear muon captures.
\subsection{Number of protons emitted}
\label{sub:number_of_protons_emitted}
From the particle identification above, number of protons having energy in the
range from \SIrange{2.2}{8.5}{\MeV} hitting the two arms are:
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_} it is shown that
muons stopped off centre to the right arm.
%%TODO
The uncertainties are statistical only. The systematic uncertainties due to
the cut on protons is estimated to be small compared to the statistical ones.
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 (also called unfolding, or
reconstruction) is necessary.
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
@@ -413,28 +416,65 @@ method is implemented.
\caption{Response functions for the two silicon 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.
%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. 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.
Comparing the reconstructed spectra from \SIrange{5}{8}{\MeV}, the protons
yields are consistent with each other:
%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 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]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_cutoff_study}
\caption{Unfolded spectra with different 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.
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_integral_comparison}
\caption{Proton yields calculated from two arms.}
\label{fig:al100_integral_comparison}
\end{figure}
The yields of protons from \SIrange{4}{8}{\MeV} are:
\begin{align}
N_{\textrm{p reco. left}} &= (110.9 \pm 2.0)\times 10^3\\
N_{\textrm{p reco. right}} &= (110.2 \pm 2.3)\times 10^3
N_{\textrm{p unfold left}} &= (165.4 \pm 2.7)\times 10^3\\
N_{\textrm{p unfold right}} &= (173.1 \pm 2.9)\times 10^3
\end{align}
Therefore, the number of emitted protons is taken as average value:
\begin{equation}
N_{\textrm{p reco.}} = (110.6 \pm 2.2) \times 10^3
N_{\textrm{p unfold}} = (169.3 \pm 2.9) \times 10^3
\end{equation}
\begin{figure}[htb]
\centering
\includegraphics[width=0.85\textwidth]{figs/al100_unfold}
\includegraphics[width=0.85\textwidth]{figs/al100_unfolded_lr}
\caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
\label{fig:al100_unfold}
\end{figure}
@@ -458,29 +498,84 @@ the number of nuclear captures are:
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}
The proton emission rate in the range from \SIrange{5}{8}{\MeV} is therefore:
\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{110.6\times 10^3}{9.57\times 10^6} = 1.16\times
R_{\textrm{p}} = \frac{169.3\times 10^3}{9.57\times 10^6} = 1.74\times
10^{-2}
\label{eq:proton_rate_al}
\end{equation}
%\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 protons:
%\begin{itemize}
%\item statistical uncertainty of the measured spectra;
%\item systematic uncertainty due to misidentification;
%\item systematic uncertainty from the unfolding
%\end{itemize}
%\item uncertainties in the number of nuclear captures:
%\begin{itemize}
%\item statistical uncertainty of the number of X-rays;
%\item uncertainty of the detector acceptance;
%\item uncertainty from the corrections: random summing and transistor
%reset amplifier
%\end{itemize}
%\end{itemize}
The total proton emission rate can be estimated by assuming a spectrum shape
with the same parameterisation as in \eqref{eqn:EH_pdf}. The fit parameters
are shown in . With such parameterisation, the integration in
range from \SIrange{4}{8}{\MeV} is 51\% of the total number of protons. The
total proton emission rate is therefore $3.5\times 10^{-2}$.
\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: spectra
(left), and yields (right). The ratio is defined as $\textrm{(Unfold - MC
truth)/(MC truth)}$}
\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 & 1.6\% \\
Systematic from unfolding & 5.0\% \\
Systematic from PID & \textless0.5\% \\
\midrule
Total & 6.1\%\\
\bottomrule
\end{tabular}
\end{center}
\caption{Uncertainties of the proton emission rate.}
\label{tab:al100_uncertainties_all}
\end{table}
The proton emission rate is then $(3.5 \pm 0.2)$\%.