first draft r15a xray

r15a_xray/tex/analysis.tex

@@ -1,2 +1,146 @@
 \section{Data analysis}
+\subsection{Digital pulse processing}
+\label{sub:digital_pulse_processing}
+Since we recorded all detector outputs using digitizers, offline digital pulse
+processing is needed to extract energy and timing information. Typical output
+pulses from HPGe and \ce{LaBr3} detectors are shown in
+\cref{fig:typical_pulses}. 
 
+\begin{center}
+  \begin{figure}[tbp]
+    \centering
+    \includegraphics[width=1.0\textwidth]{figs/typical_pulses}
+    \caption{Typical output pulses of HPGe and \ce{LaBr3} detectors: energy
+      output HPGe high gain (top left), energy output HPGe low gain (top
+      right), timing output HPGe (bottom left), and \ce{LaBr3} (bottom right).
+      Each clock tick corresponds to \SI{10}{\ns} and \SI{2}{\ns} for top and
+      bottom plots, respectively.}
+    \label{fig:typical_pulses}
+  \end{figure}
+\end{center}
+
+The timing pulses from the HPGe detector were not used in this analysis because
+they are too long and noisy (see bottom left \cref{fig:typical_pulses}).
+Energy of the HPGe detector is taken as amplitude of spectroscopy amplifier
+outputs, its timing is determined by the clock tick where the trace passes
+\SI{30}{\percent} of the amplitude. The timing resolution is \SI{235}{\ns}
+using this method.
+
+\subsection{Calibrations}
+\label{sub:calibrations}
+The HPGe detector energy scales and acceptance were calibrated
+using \ce{^{152}Eu}, \ce{^{60}Co}, \ce{^{88}Y} sources placed at the target
+position. There was a separate run for background radiation.
+
+Energy resolutions are better than \SI{3.2}{\keV} for all calibrated peaks.
+\begin{center}
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[width=1.0\textwidth]{figs/hpge_ecal}
+    \caption{Energy calibration spectra for the HPGe detector.}
+  \label{fig:hpge_ecal}
+  \end{figure}
+\end{center}
+
+The detector acceptance above \SI{200}{\kilo\eV} were fitted using an empirical
+function:
+\begin{equation}
+  A = c_1 \times E ^ {c_2},
+\end{equation}
+where $c_1 = 0.1631$, $c_2 = -0.9257$, and $E$ is photon energy in \si{\keV}.
+Interpolation gives detector acceptance at the peaks of interest as shown in
+\cref{tab:hpge_acceptance}.
+
+\begin{center}
+  \begin{figure}[htbp]
+    \centering
+    \includegraphics[width=1.0\textwidth]{figs/hpge_higain_acceptance}
+    \caption{Acceptance of the HPGe as a function of photon energy.}
+  \label{fig:hpge_higain_acceptance}
+  \end{figure}
+\end{center}
+
+\begin{table}[tbp]
+\centering
+\caption{HPGe acceptance for photons of interest}
+\label{tab:hpge_acceptance}
+\begin{tabular}{@{}cccc@{}}
+\toprule
+\multicolumn{2}{c}{\textbf{\begin{tabular}[c]{@{}c@{}}Photon energy\\ {[}keV{]}\end{tabular}}} & \textbf{Acceptance} & \textbf{Error} \\
+\midrule
+$2p-1s$   & 346.8      & \num{8.75E-4}          &\num{4.0e-5}  \\
+\ce{^{27}Mg}    & 843.7      & \num{3.40E-4}          &\num{0.9e-5}  \\
+        % & 1014.4     & \num{2.69e-4}          &\num{1.07e-5}  \\
+\ce{^{nat}Ti}   & 931.96     & \num{3.06E-4}          &\num{0.8e-5}  \\
+\ce{^{26}Mg}*   & 1088.7     & \num{1.51e-4}          &\num{0.7e-5}  \\
+% 0 	346.828 	0.000875 	0.000040
+% 1 	399.268 	0.000753 	0.000030
+% 2 	400.177 	0.000751 	0.000030
+% 3 	476.800 	0.000624 	0.000022
+% 4 	843.740 	0.000340 	0.000009
+% 5 	930.000 	0.000306 	0.000008
+% 6 	931.000 	0.000306 	0.000008
+% 7 	932.000 	0.000306 	0.000008
+% 8 	1014.420 	0.000279 	0.000008
+% 9 	1808.660 	0.000151 	0.000007
+\bottomrule
+\end{tabular}
+\end{table}
+
+\subsection{Number of stopped muons}
+% TODO: justification for taking just number from muon counter
+
+The number of stopped muons are taken as number of muons seen by the muon
+counter, since we used thick targets the muon beam is believed to stop
+completely at the middle of the targets. This assumption is verified for the
+aluminum target where count from muon counter was consistent with number of
+stopped muons calculated from number of $(2p-1s)$ X-rays.
+
+\subsection{Muonic X-ray spectra}
+We use the HPGe spectra to look for characteristic muonic X-rays from elements
+of interest. Energies of these muonic X-rays are listed
+in~\cref{tab:hpge_acceptance}. 
+
+\subsubsection{Titanium}
+We are looking at X-rays from $(2p-1s)$ transitions in titanium. Natural
+titanium has 5 stable isotopes: \ce{^{46}Ti}, \ce{^{47}Ti}, \ce{^{48}Ti},
+\ce{^{49}Ti}, and \ce{^{50}Ti}, with the \ce{^{48}Ti} being the
+most abundant at 73.72\%. The fine splitting between muonic $2p_{3/2}
+$ and $2p_{1/2}$ levels in these stable isotopes are about
+\SI{2}{keV}~\cite{Wohlfahrt1981}, comparable to the resolution of our HPGe
+detector. The $(2p-1s)$ X-rays therefore show up as a broad, asymmetric peak
+with a longer tail on the low energy side. The peak is fitted as two
+Gaussian peaks on top of a first-order polynomial. 
+
+\subsection{Fraction of muon captured by a nucleus}
+An atomic captured muon at the 1S state has only two choices, either to decay
+in orbit or to be captured on the nucleus. The total disappearance rate for
+negative muon, $\Lambda_{tot}$, is given by: 
+\begin{equation}
+  \Lambda_{tot} = \Lambda_{cap} + Q \Lambda_{free},
+  \label{eq:mu_total_capture_rate}
+\end{equation}
+where $\Lambda_{cap}$ and $\Lambda_{free}$ are nuclear capture rate and free
+decay rate, respectively, and $Q$ is the Huff factor, which is corrects for the
+fact that muon decay rate in a bound state is reduced because of the binding
+energy reduces the available energy.
+
+Using mean lifetime measured by Suzuki et.al.~\cite{SuzukiMeasday.etal.1987}
+and fractions of muons captured by element of interest are calculated and
+listed in~\cref{tab:capture_frac}.
+\begin{table}[tbp]
+  \centering
+  \caption{Nuclear capture probability calculated from mean lifetimes taken
+    from measurements of Suzuki et.al.~\cite{SuzukiMeasday.etal.1987}}
+  \label{tab:capture_frac}
+  \begin{tabular}{cccc}
+    \toprule
+    Element & Mean lifetime & Huff factor & Nuclear capture\\ 
+            & [\si{ns}]     &             & probability [\%]\\ 
+    \midrule
+    \ce{^{nat}Al}   & \num{864.0 \pm 1.0}   & \num{0.993} &\num{60.95(5)}  \\
+    \ce{^{nat}Ti}   & \num{329.3 \pm 1.3}   & \num{0.981} &\num{85.29(6)}  \\
+    \ce{^{nat}W}    & \num{78.4 \pm 1.5}    & \num{0.860} &\num{96.93(6)}  \\
+    \bottomrule
+  \end{tabular}
+\end{table}

r15a_xray/tex/intro.tex

@@ -1,2 +1,11 @@
 \section{Introduction}
-Why are we even doing this measurement? Here is a very thorough study~\cite{Zinatulina2019}
+Why are we even doing this measurement?
+\begin{itemize}
+  \item targets for mu-e conversion experiments
+  \item why did we measure \ce{W}, \ce{H_2O}, \ldots: background for Xrays of
+    interest in Mu2e
+  \item existing data? focused on nuclear charge radii, did not report muonic
+    X-ray yields. This is true for \ce{^{nat}Ti}~\cite{Wohlfahrt1981}
+
+\end{itemize}
+

r15a_xray/tex/results.tex

@@ -1,2 +1,28 @@
 \section{Results and discussions}
+\subsection{Titanium}
+Number of stopped muons in the natural titanium target was:
+\begin{equation}
+  N_{\mu} = (88296 \pm 9) \times 10^3 \,.
+  \label{eqn:Nmu_Ti_Tsc}
+\end{equation}
 
+Fitting the peak around \SI{931}{keV} in the photon spectrum gives the 
+center of gravity at \SI{931.6 \pm 0.7}{keV} (see~\cref{fig:ti_931keV_fit}),
+consistent with previously reported value~\cite{Wohlfahrt1981}.
+Number of $(2p-1s)$ X-rays in the \SI{931.6}{keV} peak is:
+\begin{equation}
+  N_{931.6} = (20750 \pm 764) \,.
+\end{equation}
+
+\begin{figure}[tbp]
+  \centering
+  \includegraphics[width=0.8\textwidth]{figs/ti_931keV_fit}
+  \caption{Fitting $(2p-1s)$ peaks}
+  \label{fig:ti_931keV_fit}
+\end{figure}
+
+The emission rate of the $(2p-1s)$ muonic X-rays is calculated as:
+\begin{equation}
+  R_{Ti} = \frac{N_{931.6}}{A_{931.6} \times N_{\mu} \times f_{capTi}} = 0.90
+  \pm 0.04 \,.
+\end{equation}

r15a_xray/tex/setup.tex

@@ -5,14 +5,14 @@ The 2015 summer run focused on the detection of neutral particles: low energy
 X-ray, gamma ray and neutron emission after the muon is captured by the
 nucleus. 
 
-The X-rays and gamma rays of interest are:
+% The X-rays and gamma rays of interest are:
-\begin{itemize}
+% \begin{itemize}
-  \item muonic $2p-1s$ transition in aluminum: \SI{346.8}{\kilo\eV}
+  % \item muonic $2p-1s$ transition in aluminum: \SI{346.8}{\kilo\eV}
-  \item \SI{843.7}{\kilo\eV} gamma from the $\beta^-$ decay of \ce{^{27}Mg}
+  % \item \SI{843.7}{\kilo\eV} gamma from the $\beta^-$ decay of \ce{^{27}Mg}
-    (half-life: \SI{9.46}{\min})
+    % (half-life: \SI{9.46}{\min})
-  \item \SI{1808.7}{\kilo\eV} gamma from the first excited state of
+  % \item \SI{1808.7}{\kilo\eV} gamma from the first excited state of
-    \ce{^{26}Mg}
+    % \ce{^{26}Mg}
-\end{itemize}
+% \end{itemize}
 
 Low momentum muons (less than \SI[]{40}{\mega\eVperc}) were stopped in
 a target after passing a muon counter
@@ -58,11 +58,16 @@ Experimental layout  is shown  in \cref{fig:R2015a_setup}.
 \begin{center}
   \begin{figure}[tbp]
     \centering
-    \includegraphics[width=0.70\textwidth]{figs/r15a_setup_photo}
+    \begin{minipage}{0.45\textwidth}
-    \caption{Layout of the AlCap 2015 summer run. Muons entered from the top of
+      \includegraphics[width=1.0\textwidth]{figs/r15a_setup_photo}
-      the image. The LYSO detector is not visible in this image, which is
+    \end{minipage}
-      located further out in the bottom of the image.} 
+    \begin{minipage}{0.45\textwidth}
-    \label{fig:R2015a_setup}
+      \includegraphics[width=1.0\textwidth]{figs/alcap_r15a_setup}
+    \end{minipage}
+    \caption{Layout of the AlCap experiment in the summer 2015 run. Negative
+      muons entered from the top of the photo. The LYSO detector is not visible
+      in this image, which is located further out in the bottom of the image.} 
+    \label{fig:r15a_setup}
   \end{figure}
 \end{center}
 
@@ -70,4 +75,6 @@ There were several runs with different targets made of aluminum, titanium,
 lead, water. All targets were sufficiently thick to stop the muon beam with
 momenta up to \SI{40}{\mega\eVperc}.
 
+TODO: a table of targets and details
+
 

r15a_xray/xray.bib

@@ -30,6 +30,32 @@
   url       = {https://link.aps.org/doi/10.1103/PhysRevC.23.533},
 }
 
+@Article{SuzukiMeasday.etal.1987,
+  author    = {Suzuki, T and Measday, David F and Roalsvig, JP},
+  title     = {Total nuclear capture rates for negative muons},
+  journal   = {Physical Review C},
+  year      = {1987},
+  volume    = {35},
+  number    = {6},
+  pages     = {2212},
+  file      = {Published version:SuzukiMeasday.etal.1987.pdf:PDF},
+  owner     = {NT},
+  publisher = {APS},
+  timestamp = {2014-07-13},
+  url       = {http://journals.aps.org/prc/abstract/10.1103/PhysRevC.35.2212},
+}
+
+@Article{refId0,
+  author  = {{Zinatulina, Daniya} and {Brian\c{c}on, Chantal} and {Brudanin, Victor} and {Egorov, Viacheslav} and {Perevoshchikov, Lev} and {Shirchenko, Mark} and {Yutlandov, Igor} and {Petitjean, Claude}},
+  title   = {Electronic catalogue of muonic X-rays},
+  journal = {EPJ Web Conf.},
+  year    = {2018},
+  volume  = {177},
+  pages   = {03006},
+  doi     = {10.1051/epjconf/201817703006},
+  url     = {https://doi.org/10.1051/epjconf/201817703006},
+}
+
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