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thesis/chapters/chap6_analysis.tex

@@ -382,7 +382,8 @@ the cut on protons is estimated to be small compared to the statistical ones.
 \subsection{Corrections for the number of protons}
 \label{sub:corrections_for_the_number_of_protons}
 The protons spectra observed by the silicon detectors have been modified by
-the energy loss inside the target so correction (or unfolding) is necessary.
+the energy loss inside the target so correction (also called unfolding, or
+reconstruction) 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
@@ -413,4 +414,57 @@ method is implemented.
   \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 
+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. 
+
+Comparing the reconstructed spectra from \SIrange{5}{8}{\MeV}, the protons
+yields are consistent with each other:
+\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
+\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
+\end{equation}
+
+\begin{figure}[htb]
+  \centering
+    \includegraphics[width=0.85\textwidth]{figs/al100_unfold}
+  \caption{Unfolded proton spectra from the 100-\si{\um} aluminium target.}
+  \label{fig:al100_unfold}
+\end{figure}
+
+\subsection{Number of nuclear captures}
+\label{sub:number_of_nuclear_captures}
+\begin{figure}[htb]
+  \centering
+  \includegraphics[width=0.85\textwidth]{figs/al100_ge_spec}
+  \caption{X-ray spectrum from the aluminium target, the characteristic
+  $(2p-1s)$ line shows up at 346.67~keV}
+\label{fig:al100_ge_spec}
+\end{figure}
+ 
+The X-ray spectrum on the germanium detector is shown on
+\cref{fig:al100_ge_spec}. Fitting the double peaks on top of a first-order
+polynomial gives the X-ray peak area of $5903.5 \pm 109.2$. With the same
+procedure as in the case of the active target, the number stopped muons and
+the number of nuclear captures are:
+\begin{align}
+  N_{\mu \textrm{ stopped}} &= (1.57 \pm 0.05)\times 10^7\\
+  N_{\mu \textrm{ nucl. cap.}} &= (9.57\pm 0.31)\times 10^6
+\end{align}
+The proton emission rate in the range from \SIrange{5}{8}{\MeV} is therefore:
+\begin{equation}
+  R_{\textrm{p}} = \frac{110.6\times 10^3}{9.57\times 10^6} = 1.16\times
+  10^{-2}
+  \label{eq:proton_rate_al}
+\end{equation}
+
+\subsection{Uncertainties of the emission rate}
+\label{sub:uncertainties_of_the_emission_rate}
+