\documentclass[11pt]{article}
\usepackage{mhchem}
\usepackage{hyperref}
\usepackage{booktabs}
\usepackage{multirow}
\usepackage{textcomp}
\usepackage{epsfig}
\usepackage[noabbrev, capitalize]{cleveref}
\usepackage[detect-weight=true, per=slash, detect-family=true, separate-uncertainty=true, alsoload=hep]{siunitx}

% \DeclareSIUnit\eVperc{\eV\per\clight}
% \DeclareSIUnit\clight{\text{\ensuremath{c}}}

\begin{document}
\title{Measurement of photons from nuclear muon capture on aluminum}
\author{Nam H. Tran \\ Boston University}
\date{\today}
\maketitle

\begin{abstract}
  The goal is study the emission rate of the \SI{1808.7}{\kilo\eV} gamma rays
  (from the first excited state of \ce{^{26}Mg}) after nuclear muon capture on
  \ce{^{27}Al}. The measured emission rate is \SI{51(5)}{\percent} per muon capture.
\end{abstract}

\section{Experimental set up}
\label{sec:experimental_set_up}
This measurement is part of the AlCap experiment done at PSI, Switzerland.
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:
\begin{itemize}
  \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}
    (half-life: \SI{9.46}{\min})
  \item \SI{1808.7}{\kilo\eV} gamma from the first excited state of
    \ce{^{26}Mg}
\end{itemize}

Low momentum muons (less than \SI[]{40}{\mega\eVperc}) were stopped in
a target after passing a muon counter
(\SI{60}{\mm}$\times$\SI{60}{\mm}$\times$\SI{0.5}{\mm} plastic scintillator). 
% Upstream from the muon counter, a 
% \SI{10}{\cm} $\times$ \SI{10}{\cm}  $\times$ \SI{0.6}{\cm} scintillator with
% a \SI{40}{\mm} diameter hole cut in the center acted as a  beam defining
% veto counter to the incoming muon beam.
There were two 5"$\times$2" liquid scintillator BC501a detectors setup on the
beam right to detect neutrons. For gamma spectrum analysis and normalization
we used an HPGe detector installed on the beam left. In addition, a \ce{LaBr3}
scintillator was also tested if it would be suitable to use in the STM. A 25
LYSO crystal array was placed downstream of the target beam left to observe
high energy photons emitted.

Two identical preamplifier outputs from the HPGe detector were fed into: (a)
a timing filter amplifier for timing information, and (b) a spectroscopy
amplifier for energy information. The timing pulses were read out by a 14-bit
500-MS/s desktop digitizer(CAEN DT5730). In order to accommodate both low
energy X-rays and relatively high energy gamma rays, we used two channels from
the spectroscopy amplifier with different gain settings: (a) a lower gain
channel for photons up to \SI{6.5}{\mega\eV}; and (b) a higher gain channel for
photons up to \SI{2.5}{\mega\eV}. These channels were read out by a 14-bit
100-MS/s VME digitizer (CAEN V1724).

The \ce{LaBr3} crystal is coupled with a photomultiplier, of which output
pulses were large enough so no further amplification was needed. This channel
is read out with the DT5730.

% Detectors' outputs were read out using waveform digitizers. We used a 14-bit
% 100-MS/s VME digitizer (CAEN V1724) to record energy signals from
% HPGe and \ce{LaBr3} detectors. There were two energy outputs from the HPGe
% detector with different gain settings: (a) low gain channel for photons up to
% \SI{6.5}{\mega\eV}; and (b) high gain channel for photons up to
% \SI{2.5}{\mega\eV}. The timing signals from these detectors, and signals from
% plastic and liquid scintillators were fed into a faster digitizer, namely
% a 14-bit 500-MS/s desktop digitizer (CAEN DT5730).
% These fast timing channels
% were also read out using a multihit TDC (CAEN V1290A) as a back up solution.
% All digitizers and TDC were synchronized by an external master clock.

Experimental layout  is shown  in \cref{fig:R2015a_setup}.
\begin{center}
  \begin{figure}[!tbp]
    \centering
    \includegraphics[width=0.70\textwidth]{figs/R2015a_setup_2.jpg}
    \caption{Layout of the AlCap 2015 summer run. Muons entered from the top of
      the image. The LYSO detector is not visible in this image, which is
      located further out in the bottom of the image.} 
    \label{fig:R2015a_setup}
  \end{figure}
\end{center}

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}.
% Table~\ref{tab:alcap2015a:datasets} summarizes the data
% sets for the main production run and number of muons
% entering the experiment as counted by the beam scintillator counter TSc. Data
% were also collected on stainless steel, tungsten and mylar targets with
% a substantially reduced amount of data collection time.

\section{Analysis}
\label{sec:analysis}
In this study, the dataset on a \SI{2}{\mm} thick aluminum target is used. It 
was collected in 31 hours of beam time and contains about \num{1.91E9} stopped
muons. Momentum of the muon beam was \SI{36}{\mega\eVperc}.

\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}[htbp]
    \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.

\ce{LaBr3} pulses were passed through a moving average window filter (60
samples wide), then integrated to obtain energy resolution. 

\subsection{Calibrations}
\label{sub:calibrations}
The HPGe and \ce{LaBr3} detectors acceptance and energy scales 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.

\cref{fig:uncalibrated_labr3_spectra} shows \ce{LaBr3}
spectra with calibration sources \ce{^{88}Y}, \ce{^{60}Co}, and background
radiation. It can be seen that below \SI{1.5}{\MeV} region the self activation
from \ce{^{138}La} shows up clearly, and above that products from the chain
decay of \ce{^{227}Ac} dominate the spectrum. The \SI{1173}{\kilo\eV} peak
barely shows up in \ce{^{60}Co} spectrum, while the \SI{1332}{\keV} peak is
buried under the \SI{1436}{\kilo\eV} peak from \ce{^{138}La}. The
\SI{1836}{\kilo\eV} peak of \ce{^{88}Y} and the annihilation peak
\SI{511}{\kilo\eV} can be distinguished, but the \SI{898}{\kilo\eV} has been
distorted by the electrons and \SI{789}{\kilo\eV} gammas from the beta decay of
\ce{^{138}La}. The energy resolution (FWHM) at the \SI{1836}{\kilo\eV} peak was
\SI{5.9}{\percent}.

\begin{center}
  \begin{figure}[htbp]
    \centering
    \includegraphics[width=1.0\textwidth]{figs/labr3_spectra_w_gatedintegration}
    \caption{Calibration of the \ce{LaBr3} detector, top horizontal axis shows
    energy and bottom horizontal axis shows integration of the output pulses.
    The spectra were scaled to make the peak recognition easier.}
    \label{fig:uncalibrated_labr3_spectra}
  \end{figure}
\end{center}

The HPGe spectra are much cleaner as shown in Figure~\ref{fig:hpge_ecal}.
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}
% Energy resolutions were good for all calibration peaks.
The detector acceptance
were fitted as a function of photon energy above \SI{200}{\kilo\eV}:
\begin{equation}
  A = c_1 \times E ^ {c_2},
\end{equation}
where $c_1 = 0.1631$, $c_2 = -0.9257$. 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 (high gain channel) as a function of photon
      energy.}
  \label{fig:hpge_higain_acceptance}
  \end{figure}
\end{center}

\begin{table}[htbp]
\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{7.26e-4}          &\num{4.73e-5}  \\
% 3p-1s   & 399.3      & \num{6.38e-4}          &\num{3.71e-5}  \\
% 4p-1s   & 400.2      & \num{6.36e-4}          &\num{3.70e-5}  \\
% 5p-1s   & 476.8      & \num{5.41e-4}          &\num{2.72e-5}  \\
\ce{^{27}Mg}    & 843.7      & \num{3.19e-4}          &\num{1.20e-5}  \\
        % & 1014.4     & \num{2.69e-4}          &\num{1.07e-5}  \\
\ce{^{26}Mg}*   & 1088.7     & \num{1.57e-4}          &\num{9.80e-6}  \\
\bottomrule
\end{tabular}
\end{table}

\section{Results and discussion}
\label{sec:results_and_discussion}

\subsection{\ce{LaBr3} spectra}
\label{sub:labr3_spectra}
The \ce{LaBr3} energy spectra for the Al dataset are presented in
\cref{fig:labr3_all_al_runs}. The muonic $2p-1s$ peak shows up clearly in
the prompt spectrum as expected, the signal-to-background ratio is
\num{3.13(2)}. The \SI{1809}{\kilo\eV} peak can be
recognized, it has better signal-to-background ratio in the prompt spectrum
than in the delay spectrum (0.88 to 0.33). The background under the
\SI{1809}{\kilo\eV} is dominated by
the $\alpha$ decay of progenies from \ce{^{227}Ac}.

It is clear that this
\ce{LaBr3} detector in the current set up is not good enough to measure the
\SI{1809}{\keV} line. The situation of the $2p-1s$ line is a little better, but
more studies is needed to understand the background and possible interferences
around the peak. On another note, there have been steady progress in
manufacturing \ce{LaBr3} detectors, and better performance has been observed.

\begin{center}
  \begin{figure}[htbp]
    \centering
    \includegraphics[width=1.0\textwidth]{figs/labr3_all_al_runs}
    \caption{\ce{LaBr3} spectra: prompt (less than \SI{100}{\ns} from muon
      hit), delay ($>$ \SI{100}{\ns} from muon hit), and all hits.}
    \label{fig:labr3_all_al_runs}
  \end{figure}
\end{center}

\subsection{HPGe spectrum}
\label{sec:hpge_spectrum}
The HPGe photon spectrum for the aluminum dataset is shown in
\cref{fig:GeCHH_all_al_runs}. Both the \SI{347}{\kilo\eV}
and \SI{1809}{\keV} peaks are clearly visible with the X-ray peak dominates in
the prompt spectrum. The apperance of the \SI{347}{\keV} (and other X-ray
peaks) in the delay spectrum can be explained by a second muon stopped in the
aluminum target shortly after the trigger muon.
\begin{center}
  \begin{figure}[htbp]
    \centering
    \includegraphics[width=1.0\textwidth]{figs/GeCHH_all_al_runs}
    \caption{HPGe high gain spectra: prompt (less than \SI{500}{\ns} from muon
      hit), delay ($>$ \SI{500}{\ns} from muon hit), and all hits.}
    \label{fig:GeCHH_all_al_runs}
  \end{figure}
\end{center}

\subsection{Number of stopped muons}
\label{sub:number_of_stopped_muons}

The number of stopped muons is calculated by two methods:
\begin{itemize}
  \item infering from the number of $2p-1s$ X-rays,
  \item counting the muon hits on the muon counter.
\end{itemize}
The latter gives the number of stopped muons as:
\begin{equation}
  N_{\mu} = 3.03 \times 10^8 \pm 1.7 \times 10^4.
  \label{eqn:n_mu_TSc}
\end{equation}

The number of $2p-1s$ X-rays is calculated by fitting a Gaussian peak with
a linear background to the region \SIrange{340}{350}{\keV} around the peak in
the prompt HPGe spectrum: 
\begin{equation}
  N_{346.8} = (191.27 \pm 0.42) \times 10^3.
\end{equation}
Using the acceptance of the $2p-1s$ photons in \cref{tab:hpge_acceptance},
number of stopped muons is:
\begin{equation}
N_{\mu} = \frac{N_{346.8}}{A_{346.8}} = (3.30 \pm 0.22) \times 10^8,
\end{equation}
which is consistent with that in \cref{eqn:n_mu_TSc}.

\subsection{Emission rate of \SI{1809}{\keV} photons}
\label{sub:emission_rate_of_1809_kev_photons}
Number of the \SI{1809}{\keV} photons is calculated using the same method for
the \SI{347}{\keV} photons:
\begin{equation}
  N_{1808.7} = 16032.54 \pm 166.19.
\end{equation}
Therefore the emission rate per nuclear capture is:
\begin{equation}
  R_{1808.7} = \frac{N_{1808.7}}{A_{1808.7} \times N_{\mu} \times 0.609} = 0.51 \pm 0.05,
\end{equation}
, where the factor 0.609 comes from the fact that only \SI{60.9}{\percent} of
stopped muons are captured. This result is consistent with the rate
\num{0.51(5)} reported by Measday et al.

\end{document}
