<commit_before>\documentclass[herrin-thesis.tex]{subfiles}
\begin{document}

\chapter{Electron Lifetime}
\label{ch:electronlifetime}

\section{Electron Capture on Impurities}
When an electromagnetic process deposits energy in a noble liquid detector, it ionizes the atoms, producing electrons and ions. Some of the electrons will recombine with the ions, which produces scintillation light. However, if the detector has an applied electric field, then the remaining electrons and ions will drift in opposite directions along the field lines. In a detector consisting of perfectly pure noble liquid, the electrons would all reach the anode and would be collected for an energy measurement. In a non-ideal detector, however, electronegative impurities can capture the drifting electrons and form ions. The ions are more massive and drift more slowly, and so they escape inclusion in the signal used for energy measurement.

Electronegative impurities may capture electrons in three ways\cite{Aprile:2006fk}. Denoting the impurities, which may be atoms or molecules, as \(AB\), these are:
\begin{enumerate}
\item Radiative attachment
\begin{equation}
e^{-} + AB \rightarrow AB^{-} + h \nu
\end{equation}
which has a much smaller cross section than the other processes below.
\item Dissociative attachment
\begin{equation}
\begin{split}
e^{-} + AB \rightarrow e^{-} + AB^{*} \rightarrow A^{+} + B^{-} + e^{-} \\
e^{-} + AB \rightarrow AB^{-} \rightarrow A^{+} + B^{-}
\end{split}
\end{equation}
which requires the electron's energy to be much higher than typically found for an electron drifting in a liquid or dense gas.
\item Three-body attachment through the two-stage Bloch-Bradbury reaction
\begin{equation}
\begin{split}
e^{-} + AB \leftrightarrow (AB^{-})^{*} \\
(AB^{-})^{*} + X \rightarrow AB^{-} + X
\end{split}
\label{eq:el_3bodyattachment}
\end{equation}
where X represents the atom or molecule that make up the majority of the liquid.
\end{enumerate}

The three-body reaction shown in \cref{eq:el_3bodyattachment} releases some amount of energy, given by the \emph{electron affinity} of \(AB\). The electron affinity is positive if \(AB\) is electronegative. Noble elements have a negative electron affinity, so the reaction does not take place in a pure detector.

The rate of the reaction shown in \cref{eq:el_3bodyattachment} is given by:
\begin{equation}
\frac{dn_{AB}}{dt} = -k_{3} n_{AB} n_{X} n_{e^{-}}
\label{eq:el_3bodyreactionrate}
\end{equation}
where \(k_3\) is constant for the 3-body reaction, and \(n_{AB}\), \(n_{X}\), and \(n_{e^{-}}\) are the densities of the impurity, the atoms or molecules of the liquid, and the electrons, respectively. \(k_3\) depends on the species of the impurity, the species of the liquid, and the electric field strength.

According to \cref{eq:el_3bodyreactionrate}, electrons will be captured, forming \(AB^{-}\) at a rate proportional to the density of electrons. Thus, the number of free electrons will decay exponentially over time according to:
\begin{equation}
N_{e^{-}}(t) = N_0 \exp (-t/\tau_e)
\label{eq:el_exponentialtaue}
\end{equation}
where \(N_0\) is the original number of electrons, and \(\tau_e\) is the \emph{electron lifetime}.

In general, there can be several different species of electronegative impurity. In that case, they all contribute to the electron lifetime according to:
\begin{equation}
\tau_e^{-1} = \sum_i k_i n_i = \sum_i\tau_i^{-1}
\label{eq:el_tauedefinition}
\end{equation}
where \(n_i\) is the density of an electronegative impurity and \(k_i\) is the cross section for electron capture by that impurity.

\section{Measuring Electron Lifetime}

\subsection{Method}
\Cref{eq:el_exponentialtaue} provides a simple recipe for measuring the electron lifetime: measure the exponential attenuation of a known quantity of ionization as a function of drift time. A source of monoenergetic gamma rays can provide such a signal. The ionization produced by the full absorption of a gamma ray follows a Gaussian distribution, with its width provided by the Fano factor and detector resolution. If some of the ionization is being attenuated, then the peak's mean will shift downward in energy from its true value.

A single calibration source at the cathode of the detector illuminates both TPCs, and the gamma rays interact throughout the entire drift region. After a sufficiently long calibration run, the TPC is divided into drift time bins. The size of the bins is a compromise: a larger bin in drift time will have more events, and thus the measurement of the full absorption peak energy will be better. A smaller bin will have fewer events, but will suffer less energy smearing because events will have similar drift times. In practice, 16 bins per TPC seems a good compromise.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4034_dt_bin_fit.pdf}
\caption[An example fit in a drift time bin]{A fit of the simple Gaussian + complementary error function model to one single drift time bin. In this example, the full absorption peak is the 2615 keV gamma line from \thorium{228}.}
\label{fig:el_dtbinfit}
\end{figure}

In addition to the full-absorption peak, which is Gaussian, the energy spectrum from a gamma ray source will contain a Compton shoulder. Some gamma rays will interact without depositing their full energy, and then scatter out of the detector. A simple model for this shoulder is a step function convolved with a Gaussian smearing representing the effects of energy resolution, producing a complementary error function. For each drift time bin, this simple Gaussian + complementary error function model is fit to the energy spectrum of that bin using an unbinned maximum likelihood fit. This fit is performed twice, first over a broad energy range to find the full-absorption peak, then in the range \((-3.0\sigma, +2.5\sigma)\) of the found peak to precisely determine the peak energy. \Cref{fig:el_dtbinfit} shows an example fit.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4228_fit.pdf}
\caption[An example fit to exponential attenuation]{Measuring the electron lifetime by fitting a decaying exponential to the full-absorption peak energies binned by drift time. TPC 2 is assigned a negative drift time for convenience in visualization. Both fits to the individual TPCs and to both TPCs together are shown. In this case, there is no large difference between the combined fit and the individual fits.}
\label{fig:el_elfit}
\end{figure}

Plotting the full absorption peak energy from each drift time bin as a function of drift time reveals the exponential decay described in \cref{eq:el_exponentialtaue}. Fitting an exponential to each TPC yields a measurement of the electron lifetime for each. Alternatively, a fit with a single electron lifetime to the entire detector uses information from both TPCs. In all cases, the amplitude of the exponential is allowed to float in the fit, since only the relative decay matters when measuring the electron lifetime. Presently, the separate TPC lifetimes are used when correcting for electron lifetime in EXO-200, while the single measurement is used when monitoring the detector and data quality. \Cref{fig:el_elfit} shows an example.

\begin{figure}[htb]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4252_profile.pdf}
\caption[A profile scan around the best-fit electron lifetime]{Confidence intervals around the best fit electron lifetime come from a profile scan, shown here. As this figure shows, this is superior to simply estimating the 1\(\sigma\) errors from the second derivative at the best fit value, since the profile is asymmetric around the best fit (indicated by the red dot). The blue line indicates \(\Delta\chi^2 = 1\), corresponding to a 68\% confidence interval.}
\label{fig:el_profileel}
\end{figure}

The electron lifetime measurement comes from minimizing the \(\chi^2\) statistic. Confidence intervals for the measurement come from doing a profile scan. Short electron lifetimes are easily distinguished, while longer electron lifetimes are not, and so the intervals will be asymmetric about the minimum. For a profile scan the electron lifetime is set to some fixed value away from the best fit value, and the amplitude is allowed to vary to minimize \(\chi^2\). All electron lifetime values for which this reminimized \(\chi^2\) is less than 1 above the minimum value define the 1\(\sigma\) (68\%) confidence band. \Cref{fig:el_profileel} shows an example.

\subsection{Comparison to Simulation}

\begin{figure}[htb]
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=1.0\columnwidth]{./plots/el_sim_error_both.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=1.0\columnwidth]{./plots/el_sim_error_indiv.pdf}
\end{subfigure}
\caption[Error on reconstructed electron lifetime from simulation]{The absolute error on the electron lifetime measurement for the electron lifetime measured in both TPCs simultaneously (left) and individually (right). The points show the mean error of 20 simulations. The lines show the size of the 68\% confidence bands returned by the measurement for those simulations. The measurement method systematically underestimates the purity, and the error grows as the electron lifetime gets large. Typical calibration runs include 2.5k events in the full absorption peak (black), but using more events would improve the error.}
\label{fig:el_sim_err}
\end{figure}

\Cref{fig:el_sim_err} shows a comparison between a known simulated electron lifetime and the measurement of that lifetime using the method described above. The error is small for electron lifetimes below \SI{1}{\ms}. For large electron lifetimes, however, the method consistently underestimates the electron lifetime, with the effect getting worse as the electron lifetime improves. 

This effect seems to be due to a some \(z\)-dependence introduced in processing the data. In simulations of a \thorium{228} source at the cathode with infinite electron lifetime, the method reports electron lifetimes of \SI{3.0\pm0.9d4}{\micro\second} measured in both TPCs simultaneously, and \SI{4.3\pm0.9d4}{\micro\second} measured in a single TPC, for runs with 2.5k events in the full absorption peak. It reports electron lifetimes of \SI{5.0\pm0.7d4}{\micro\second} measured in both TPCs simultaneously and \SI{4.9\pm0.8d4}{\micro\second} measured in a single TPC, for runs with 5k events in the full absorption peak.

The error due to this effect is small, however. A measured electron lifetime of \SI{3000}{\micro\second} corresponds to a true electron lifetime of roughly \((1/3000-1/40000)^{-1} =\) \SI{3250}{\micro\second}. This will only move the corrected ionization signal 0.3\% higher than its true value, if the ionization drifts over the full \SI{120}{\micro\second} drift time. The effect on the energy resolution in the ionization channel will be half of this this, and even smaller in the rotated spectrum. It is not understood what in the processing chain causes the effect, or if it is also present in real data. Therefore, this effect is accepted as a slight worsening in energy resolution, rather than corrected for.<commit_msg>Suggested sentence addition from Marty.<commit_after>\documentclass[herrin-thesis.tex]{subfiles}
\begin{document}

\chapter{Electron Lifetime}
\label{ch:electronlifetime}

\section{Electron Capture on Impurities}
When an electromagnetic process deposits energy in a noble liquid detector, it ionizes the atoms, producing electrons and ions. Some of the electrons will recombine with the ions, which produces scintillation light. However, if the detector has an applied electric field, then the remaining electrons and ions will drift in opposite directions along the field lines. In a detector consisting of perfectly pure noble liquid, the electrons would all reach the anode and would be collected for an energy measurement. In a non-ideal detector, however, electronegative impurities can capture the drifting electrons and form ions. The ions are more massive and drift more slowly, and so they escape inclusion in the signal used for energy measurement.

Electronegative impurities may capture electrons in three ways\cite{Aprile:2006fk}. Denoting the impurities, which may be atoms or molecules, as \(AB\), these are:
\begin{enumerate}
\item Radiative attachment
\begin{equation}
e^{-} + AB \rightarrow AB^{-} + h \nu
\end{equation}
which has a much smaller cross section than the other processes below.
\item Dissociative attachment
\begin{equation}
\begin{split}
e^{-} + AB \rightarrow e^{-} + AB^{*} \rightarrow A^{+} + B^{-} + e^{-} \\
e^{-} + AB \rightarrow AB^{-} \rightarrow A^{+} + B^{-}
\end{split}
\end{equation}
which requires the electron's energy to be much higher than typically found for an electron drifting in a liquid or dense gas.
\item Three-body attachment through the two-stage Bloch-Bradbury reaction
\begin{equation}
\begin{split}
e^{-} + AB \leftrightarrow (AB^{-})^{*} \\
(AB^{-})^{*} + X \rightarrow AB^{-} + X
\end{split}
\label{eq:el_3bodyattachment}
\end{equation}
where X represents the atom or molecule that make up the majority of the liquid.
\end{enumerate}

The three-body reaction shown in \cref{eq:el_3bodyattachment} releases some amount of energy, given by the \emph{electron affinity} of \(AB\). The electron affinity is positive if \(AB\) is electronegative. Noble elements have a negative electron affinity, so the reaction does not take place in a pure detector.

The rate of the reaction shown in \cref{eq:el_3bodyattachment} is given by:
\begin{equation}
\frac{dn_{AB}}{dt} = -k_{3} n_{AB} n_{X} n_{e^{-}}
\label{eq:el_3bodyreactionrate}
\end{equation}
where \(k_3\) is constant for the 3-body reaction, and \(n_{AB}\), \(n_{X}\), and \(n_{e^{-}}\) are the densities of the impurity, the atoms or molecules of the liquid, and the electrons, respectively. \(k_3\) depends on the species of the impurity, the species of the liquid, and the electric field strength.

According to \cref{eq:el_3bodyreactionrate}, electrons will be captured, forming \(AB^{-}\) at a rate proportional to the density of electrons. Thus, the number of free electrons will decay exponentially over time according to:
\begin{equation}
N_{e^{-}}(t) = N_0 \exp (-t/\tau_e)
\label{eq:el_exponentialtaue}
\end{equation}
where \(N_0\) is the original number of electrons, and \(\tau_e\) is the \emph{electron lifetime}.

In general, there can be several different species of electronegative impurity. In that case, they all contribute to the electron lifetime according to:
\begin{equation}
\tau_e^{-1} = \sum_i k_i n_i = \sum_i\tau_i^{-1}
\label{eq:el_tauedefinition}
\end{equation}
where \(n_i\) is the density of an electronegative impurity and \(k_i\) is the cross section for electron capture by that impurity.

\section{Measuring Electron Lifetime}

\subsection{Method}
\Cref{eq:el_exponentialtaue} provides a simple recipe for measuring the electron lifetime: measure the exponential attenuation of a known quantity of ionization as a function of drift time. A source of monoenergetic gamma rays can provide such a signal. The ionization produced by the full absorption of a gamma ray follows a Gaussian distribution, with its width provided by the Fano factor and detector resolution. If some of the ionization is being attenuated, then the peak's mean will shift downward in energy from its true value.

A single calibration source at the cathode of the detector illuminates both TPCs, and the gamma rays interact throughout the entire drift region. After a sufficiently long calibration run, the TPC is divided into drift time bins. The size of the bins is a compromise: a larger bin in drift time will have more events, and thus the measurement of the full absorption peak energy will be better. A smaller bin will have fewer events, but will suffer less energy smearing because events will have similar drift times. In practice, 16 bins per TPC seems a good compromise.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4034_dt_bin_fit.pdf}
\caption[An example fit in a drift time bin]{A fit of the simple Gaussian + complementary error function model to one single drift time bin. In this example, the full absorption peak is the 2615 keV gamma line from \thorium{228}.}
\label{fig:el_dtbinfit}
\end{figure}

In addition to the full-absorption peak, which is Gaussian, the energy spectrum from a gamma ray source will contain a Compton shoulder. Some gamma rays will interact without depositing their full energy, and then scatter out of the detector. A simple model for this shoulder is a step function convolved with a Gaussian smearing representing the effects of energy resolution, producing a complementary error function. For each drift time bin, this simple Gaussian + complementary error function model is fit to the energy spectrum of that bin using an unbinned maximum likelihood fit. This fit is performed twice, first over a broad energy range to find the full-absorption peak, then in the range \((-3.0\sigma, +2.5\sigma)\) of the found peak to precisely determine the peak energy. \Cref{fig:el_dtbinfit} shows an example fit.

\begin{figure}[htbp]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4228_fit.pdf}
\caption[An example fit to exponential attenuation]{Measuring the electron lifetime by fitting a decaying exponential to the full-absorption peak energies binned by drift time. TPC 2 is assigned a negative drift time for convenience in visualization. Both fits to the individual TPCs and to both TPCs together are shown. In this case, there is no large difference between the combined fit and the individual fits.}
\label{fig:el_elfit}
\end{figure}

Plotting the full absorption peak energy from each drift time bin as a function of drift time reveals the exponential decay described in \cref{eq:el_exponentialtaue}. Fitting an exponential to each TPC yields a measurement of the electron lifetime for each. Alternatively, a fit with a single electron lifetime to the entire detector uses information from both TPCs. In all cases, the amplitude of the exponential is allowed to float in the fit, since only the relative decay matters when measuring the electron lifetime. Presently, the separate TPC lifetimes are used when correcting for electron lifetime in EXO-200, while the single measurement is used when monitoring the detector and data quality. \Cref{fig:el_elfit} shows an example. In EXO-200, the electron lifetime is typically \(>\)~\SI{1}{\ms} over a \SI{100}{\micro\s} drift time, and so the attenuation begins to look approximately linear.

\begin{figure}[htb]
\centering
\includegraphics[width=0.6\columnwidth]{./plots/el_run4252_profile.pdf}
\caption[A profile scan around the best-fit electron lifetime]{Confidence intervals around the best fit electron lifetime come from a profile scan, shown here. As this figure shows, this is superior to simply estimating the 1\(\sigma\) errors from the second derivative at the best fit value, since the profile is asymmetric around the best fit (indicated by the red dot). The blue line indicates \(\Delta\chi^2 = 1\), corresponding to a 68\% confidence interval.}
\label{fig:el_profileel}
\end{figure}

The electron lifetime measurement comes from minimizing the \(\chi^2\) statistic. Confidence intervals for the measurement come from doing a profile scan. Short electron lifetimes are easily distinguished, while longer electron lifetimes are not, and so the intervals will be asymmetric about the minimum. For a profile scan the electron lifetime is set to some fixed value away from the best fit value, and the amplitude is allowed to vary to minimize \(\chi^2\). All electron lifetime values for which this reminimized \(\chi^2\) is less than 1 above the minimum value define the 1\(\sigma\) (68\%) confidence band. \Cref{fig:el_profileel} shows an example.

\subsection{Comparison to Simulation}

\begin{figure}[htb]
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=1.0\columnwidth]{./plots/el_sim_error_both.pdf}
\end{subfigure}%
\begin{subfigure}[b]{0.5\linewidth}
\centering
\includegraphics[width=1.0\columnwidth]{./plots/el_sim_error_indiv.pdf}
\end{subfigure}
\caption[Error on reconstructed electron lifetime from simulation]{The absolute error on the electron lifetime measurement for the electron lifetime measured in both TPCs simultaneously (left) and individually (right). The points show the mean error of 20 simulations. The lines show the size of the 68\% confidence bands returned by the measurement for those simulations. The measurement method systematically underestimates the purity, and the error grows as the electron lifetime gets large. Typical calibration runs include 2.5k events in the full absorption peak (black), but using more events would improve the error.}
\label{fig:el_sim_err}
\end{figure}

\Cref{fig:el_sim_err} shows a comparison between a known simulated electron lifetime and the measurement of that lifetime using the method described above. The error is small for electron lifetimes below \SI{1}{\ms}. For large electron lifetimes, however, the method consistently underestimates the electron lifetime, with the effect getting worse as the electron lifetime improves. 

This effect seems to be due to a some \(z\)-dependence introduced in processing the data. In simulations of a \thorium{228} source at the cathode with infinite electron lifetime, the method reports electron lifetimes of \SI{3.0\pm0.9d4}{\micro\second} measured in both TPCs simultaneously, and \SI{4.3\pm0.9d4}{\micro\second} measured in a single TPC, for runs with 2.5k events in the full absorption peak. It reports electron lifetimes of \SI{5.0\pm0.7d4}{\micro\second} measured in both TPCs simultaneously and \SI{4.9\pm0.8d4}{\micro\second} measured in a single TPC, for runs with 5k events in the full absorption peak.

The error due to this effect is small, however. A measured electron lifetime of \SI{3000}{\micro\second} corresponds to a true electron lifetime of roughly \((1/3000-1/40000)^{-1} =\) \SI{3250}{\micro\second}. This will only move the corrected ionization signal 0.3\% higher than its true value, if the ionization drifts over the full \SI{120}{\micro\second} drift time. The effect on the energy resolution in the ionization channel will be half of this this, and even smaller in the rotated spectrum. It is not understood what in the processing chain causes the effect, or if it is also present in real data. Therefore, this effect is accepted as a slight worsening in energy resolution, rather than corrected for.
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