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\begin{document}
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\title{Study of the self-activity of a LaBr$_{3}$:Ce scintillation detector}
%\subtitle{Do you have a subtitle?\\ If so, write it here}
\author{Raphaël-David Lasseri \& Andrea Mogini}
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\date{10/10/2014}

\maketitle
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\begin{abstract}
\paragraph{}
In this study the properties of a Lathanium Bromide LaBr$_{3}$:Ce scintillator were investigated, with a particular attention for the study of the internal radiocativity of this crystal. Using coincidence measurements and fine spectral analysis of the crystal with an HpGe detector we were able to get a full reconstruction of the radiocative isotopes contained in the crystal.
The difference of reconstruction of an $\alpha$ $\gamma$ and $e^-$ by the LaBr$_{3}$:Ce were also discussed. Finally this study tries to higlight some of the major advantages and inconvenients of LaBr$_{3}$ used as a particle detector.
\end{abstract}

\section{Introduction}
\label{intro}
\paragraph{}
LaBr$_{3}$:Ce scintillation detectors currently present an interesting set of characteristics for nuclear physics and $\gamma$-spectroscopy in general. Their time resolution is excellent \cite{nicolini}, making them a suitable candidate for triggering purposes and for neutron-$\gamma$ discrimination by time of flight. Moreover, the high density of the crystal lattice and the high Z of lanthanum ensure both a small Compton cross section at low energies and a good stopping power for very energetic $\gamma$-rays. Since LaBr$_{3}$:Ce detectors are also relatively stable in efficiency over a broad range of energies, as shown in Table \ref{tab:eff}, they are good candidates for high-energy applications.

\begin{table}[h]
\begin{center}
\begin{tabular}{l|ll}
\hline
\noalign{\smallskip}
relative efficiency & 1064 keV & 1771 keV\\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
\ hpGe & x1 & y1 \\
\ LaBr$_{3}$:Ce & x2 & y2 \\ 
\noalign{\smallskip}
\hline
\end{tabular}
\caption{\textit{Comparison of the relative efficiencies of a germanium detector and a lanthanum(III) bromide scintillator. Results are obtained using a \textsuperscript{207}Bi source and under the hypothesis that all the activity at 569 keV is seen by both detectors.}}\label{tab:eff}   % Give a unique label
\end{center}
\end{table}

Finally, LaBr$_{3}$:Ce detectors exhibit the best energy resolution among scintillators, with a theoretical value of 2,6$\%$ and operational values of $\sim$3$\%$ at 662 keV \cite{resol}\cite{knoll}. For our device, a 6$\times$6 cm\textsuperscript{2} crystal connected to a cylindrical photomultiplier tube of the same diameter, we measured an optimal resolution of 4,3$\%$ at this same energy. The discrepancy between these values is mainly due to the geometry of the scintillator-to-photomultiplier juncture. In fact, when a fraction of the effective area of the scintillator is lost the resolution increases proportionally. This equates to a $4/\pi$ factor in our configuration, which, when taken into account, brings the intrinsic value for the crystal's energy resolution down to $\sim$3,4$\%$. However, that is probably still an overestimation since it neglects the contribution of the electronics to the final resolution.

The many qualities of these scintillation detectors make of paramount importance a precise knowledge of their self-activity, especially if one wishes to investigate phenomena that occur at very low rates. This internal activity is the consequence of the presence in the crystal of ${}^{138}La$ and ${}^{227}Ac$, respectively a rare isotope and a chemical homologue of \textsuperscript{140}La. Fig. \ref{fig:selfact} %change to real one
summarizes all the different contributions to the self-activity.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.47\textwidth]{figures/selfact4.png}
\caption{\textit{LaBr$_{3}$:Ce internal activity \cite{book}. Diagonal : decay chain of ${}^{227}Ac$. Top-left corner : ${}^{138}La$ decay modes.}}\label{fig:selfact}       % Give a unique label
\end{center}
\end{figure}

\section{Experimental Setup}
\paragraph{}

Two main approaches were taken in order to investigate the internal activity. The first one simply consisted in shielding the LaBr$_{3}$:Ce scintillator using lead layers and in using it to register every single event. The resulting signal was then amplified and fed directly to the acquisition unit. 

This method allows for a complete reconstruction of every contribution to the self-activity but doesn't produce a satisfactory signal over background ratio, especially at low energies. For this reason, a threshold at $\sim$150 keV is used with this configuration. In addition, it is very difficult with this setup to isolate modes that overlap in energy or to study the ${}^{138}La$ $\beta$\textsuperscript{-} electron. Lastly, cutting at $\sim$3 MeV is found to be reasonable because of the quenching effects for $\alpha$ particles, which are due to their high ionization power.

The internal activity was also measured in coincidence with a high purity germanium detector. The setup for this acquisition mode is shown in Fig. \ref{fig:setup}. A fast amplification is performed on the signals of both detectors. These are then used to determine the occurrence of a coincidence, which serves as a triggering condition. Upon the fulfillment of said condition, decoupled signals from LaBr$_{3}$:Ce and hpGe are registered by the acquisition card. In order to improve the performance, the delay between the two bare signals was measured with a time-to-amplitude converter and found to be of $\sim$23 ns. Satisfactory results were obtained when applying gating of 25 ns to the fast LaBr$_{3}$:Ce signal and of 20 ns to the hpGe one. 

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.43\textwidth]{figures/Scheme_Coincidence.png}
\caption{\textit{Schematics for the coincidence setup : an opportune electronic treatment and delaying are performed on the signals to produce a coincidence trigger which is then used by the acquisition card.}}\label{fig:setup}       % Give a unique label
\end{center}
\end{figure}

The signal to noise ratio is far greater in this configuration. Moreover, the ${}^{138}La$ $\beta$\textsuperscript{-} electron and several $\alpha$ decays from the ${}^{227}Ac$ chain are easily identifiable, which provides a good test for the hypothesis made for explaining the internal activity. The limit of this approach is of course the absence, in the resulting matrix, of all the contributions incapable of producing coincidences, for instance the $\alpha$ particles not associated with $\gamma$-emission. The joint use of informations from these two methods allows to reconstruct fully the self-activity spectrum.

\section{Results}
\subsection{Coincidence Matrix Analysis}
With the second setup proposed above, one can get access to the events detected in coincidence between the detectors.
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.5\textwidth]{figures/coincidence_matrix.png}
\caption{\textit{Representation of a coicidence matrix between HpGe and $LaBr_3$:Ce detectors.Events labeled 1-4 corresponds to $\alpha$ decays when 5 is a $\beta$ decay }}\label{fig:Coincidence matrix}       % Give a unique label
\end{center}
\end{figure}
We then shall study each of this events to determine wich are the main elements contributing to the internal activity of the $LaBr_3$.
\subsection{$\alpha $ decay chain}
\subsubsection{Identification}
We are expecting the presence of Actinium impurity in the detector, indeed ${}^{138}La$ and ${}^{227}Ac$ are elements of the same column in the Mendeleiv Classification as a consequence it is extremly difficult to separate this elements when synthetising a Lanthanium crystal.
We will search for a signature of the decay chain \ref{fig:selfact} of Actinium, wich are mailny composed of $\alpha$ decay.
In fact we are looking for events involving an $\alpha$ particle produced in coincidence with a $\gamma$ in the Lanthanium-Bromide Crystal, the $\alpha$ will be detected by the $LaBr_3:Ce$, when sometimes the associated $\gamma$ will be for instance detected in the HpGe. This decay will be visible in a coincidence study and will allow the charectarisation of the $\alpha$ emmiters.
We had to select the best candidates decays that may be observed in a coincidence experiment (data from \cite{nicolini}), for instance we are selecting the decays for the Actinium, wich produce an $\alpha$ particle in coincidence with a $\gamma$ with a branching ratio above 10 \%,The $\alpha$ decay region is zoomed in on the insert on \ref{fig:Coincidence matrix}. 
With the full set of events we are able to identify each of the candidates proposed,:
\begin{table}[h]
\begin{center}
\begin{tabular}{l|l|l|l|l|}
\hline
\noalign{\smallskip}
Isotope & {$E_\alpha$} (keV) & {$E_\gamma$} (keV) & Branch & Id  \\
\noalign{\smallskip}
\hline
\noalign{\smallskip}
${}^{227}Th$ & 5755           & 256 & 12.3 \% & 4 \\
${}^{219}Rn$ & 6553     & 272 & 10.8 \% & 1 \\ 
${}^{223}Ra$ & 5610     & 270 & 13.7  \% & 3 \\
${}^{211}Bi$ & 6278     & 350 & 13.1  \% & 2 \\
\noalign{\smallskip}
\hline
\end{tabular}
\caption{$\alpha$-$\gamma$ emitters in the Actinium decay chain where Id is the label of the isotope in \ref{fig:Coincidence matrix}}\label{tab:candidates}   % Give a unique label
\end{center}
\end{table}
With the simultaneous presence of this isotopes we can conclude that there is definitly a contamination of the crystal by ${}^{227}Ac$. Moreover one must note that the calibration for $\alpha$ and $\gamma$ particle is not the same, indeed instead of the $E_\alpha$ energy values expected in \ref{tab:candidates} we are measuring lower energy. This difference may be explained by a difference on the deposit mechanism for light or massless particles as the $\gamma$ and heavy particle as the $\alpha$. We are able to find the callibration for the $\alpha$ particle, withe the measured energies $E_\alpha = 2.58 \cdot x+1.31 (MeV)$ where $x$ is the energy measured   with the initial $\gamma$ calibration. This simple results show than the light yield for an $\alpha$ particle is $\simeq 2.58$ lower than for a $      \gamma$
\subsubsection{Shape of the $\alpha$ deposits}
Projecting the signal corresponding to an alpha decay give us some information on the energy deposit mechanism:
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.35\textwidth]{figures/alpha-final.png}
\caption{\textit{Distribution of the energy deposit of an $\alpha$ particle from ${}^{211}Bi$ on the LaBr$_{3}$ at a real energy of $E_{\alpha}=6278 keV$}}\label{fig:Coincidence matrix}       % Give a unique label
\end{center}
\end{figure}

Despite the low statistic we are able to see an assymetry of the peak, with the presence of a low energy tail. This could be explained by a recoil phenomenum in the crystal, when hit by an $\alpha$
\subsection{$\beta^-$ decay}
We are also expecting evidence for a $\beta$ decay on the crystal that will simultenaously produce an electron in coincidence with a $\gamma$ according to this mechanism:
${}^{138}La\xrightarrow{\beta^-}{}^{138}Ce^*+e^-+\bar{\nu_e}\rightarrow{}^{138}Ce+e^-+\bar{\nu_e}+\gamma$
We are then expecting to detect in the LaBr$_{3}$ the electron and in coincidence the $\gamma$ in the HpGe.
\subsubsection{Identification and shape of the $\beta^-$ decay}
In the coincidence matrix \ref{fig:Coincidence matrix} this decay was labeled 5, the associated $\gamma$ match the expected energy of 789 keV.
To be sure that it is indeed the correct $\beta^-$ decay we may look at the kinetic distribution of energy of this decay.

\begin{figure}[h]
\begin{center}
\includegraphics[width=0.32\textwidth]{figures/beta+alpha_final2.png}
\caption{\textit{Distribution of the kinetic energy of a $\beta^-$ decay in the LaBr$_{3}$}}\label{fig:Coincidence matrix}       % Give a unique label
\end{center}
\end{figure}
We identify the expected shape corresponding to the prediction for the Fermy theory for $\beta$ decay.
We also get the energy threshold of the reaction Q from this plot, wich we evalute $Q\simeq 262$ keV wich again match exactly with the data from \cite{knoll}
We can also note an important result, the calibration for the electron and a $\gamma$ does not differ significantly, it is an important point that confirm our hypothesis on the difference between the kind of energetival deposit of heavy ($\alpha$) and light massless particle ($e^{-}$ and $\gamma$).

\subsection{Relative activity of ${}^{138}La$ and ${}^{227}Ac$ }
Another important parameter of this study will be the relative activity of the Lanthanium and the Actinium, such a study will give us information about the relative quantity of the isotopes in the crystal and by this way we shall get an access to the mass fraction of the polluting Actinium.
To perform this analysis we did a long acquisition of the spectrum of the LaBr$_{3}$ with the HpGe:
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.53\textwidth]{figures/self_final.png}
\caption{\textit{Full self activity spectrum of the LaBr$_{3}$ with the Actinium $\alpha$ decays contribution zoomed in and the peak of ${}^{138}La$ at $\simeq$ 1.46 MeV }}\label{fig:LaBr spectrum}       % Give a unique label
\end{center}
\end{figure}
We can numerically compute the number of events detected for both Lanthanium and Actinium, but we need to take into account the fact that only $frac{2}{3}$ of the ${}^{138}La$ is detected (insert reason here) and we need also to weight the calculus with the efficiency of the  LaBr$_{3}$ wich is as shown below a function of the energy, then with $\eta$ the efficiency
\begin{center}
R=$\frac{\frac{4}{3}Activity of {}^{138}La}{Activity of  {}^{227}Ac}\,\frac{\eta(1460 keV)}{\eta(2.2 MeV)}=4.02$
\end{center}
Meaning that the activity of ${}^{138}La$ is in average 4 time higher than the activity of the Actinium. 



\section{Discussion}


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\bibitem{nicolini}R. Nicolini et al., Nucl. Instr. and Meth. A 582 (2007) 554-561.
\bibitem{resol}F. Quarati et al., Nucl. Instr. and Meth. A 574 (2007) 115-120.
\bibitem{knoll}G.F. Knoll, Radiation Detection and Measurement 3\textsuperscript{rd} ed., John Wiley and Sons, Inc., New York (2000).
\bibitem{book}E. Brown, J.M. Dairiki, R.E. Doebler et al., Table of Isotopes, John Wiley and Sons, Inc., New York (1978).
%\bibitem{Boh}A. Bohr and B.R. Mottelson, Nuclear Structure, vol. 2, Benjamin,
%New York, 1975.

%\bibitem{ipn-web} \url{http://ipnweb.in2p3.fr}

%\bibitem{trip} Nick Park, \textsl{A Grand Day Out}, 1989, \url{http://en.wikipedia.org/wiki/A_Grand_Day_Out}

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\end{document}

