161 lines
6.5 KiB
TeX
161 lines
6.5 KiB
TeX
\section{Data analysis}
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\subsection{Digital pulse processing}
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\label{sub:digital_pulse_processing}
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Since we recorded all detector outputs using digitizers, offline digital pulse
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processing is needed to extract energy and timing information. Typical output
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pulses from HPGe and \ce{LaBr3} detectors are shown in
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\cref{fig:typical_pulses}.
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\begin{center}
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\begin{figure}[tbp]
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\centering
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\includegraphics[width=1.0\textwidth]{figs/typical_pulses}
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\caption{Typical output pulses of HPGe and \ce{LaBr3} detectors: energy
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output HPGe high gain (top left), energy output HPGe low gain (top
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right), timing output HPGe (bottom left), and \ce{LaBr3} (bottom right).
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Each clock tick corresponds to \SI{10}{\ns} and \SI{2}{\ns} for top and
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bottom plots, respectively.}
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\label{fig:typical_pulses}
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\end{figure}
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\end{center}
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The timing pulses from the HPGe detector were not used in this analysis because
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they are too long and noisy (see bottom left \cref{fig:typical_pulses}).
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Energy of the HPGe detector is taken as amplitude of spectroscopy amplifier
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outputs, its timing is determined by the clock tick where the trace passes
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\SI{30}{\percent} of the amplitude. The timing resolution is \SI{235}{\ns}
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using this method.
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\subsection{Calibrations}
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\label{sub:calibrations}
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The HPGe detector energy scales and acceptance were calibrated
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using \ce{^{152}Eu}, \ce{^{60}Co}, \ce{^{88}Y} sources placed at the target
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position. There was a separate run for background radiation.
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Energy resolutions are better than \SI{3.2}{\keV} for all calibrated peaks.
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\begin{center}
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\begin{figure}[htbp]
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\centering
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\includegraphics[width=1.0\textwidth]{figs/hpge_ecal}
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\caption{Energy calibration spectra for the HPGe detector.}
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\label{fig:hpge_ecal}
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\end{figure}
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\end{center}
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The detector acceptance above \SI{200}{\kilo\eV} were fitted using an empirical
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function:
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\begin{equation}
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A = c_1 \times E ^ {c_2},
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\end{equation}
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where $c_1 = 0.1631$, $c_2 = -0.9257$, and $E$ is photon energy in \si{\keV}.
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Interpolation gives detector acceptance at the peaks of interest as shown in
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\cref{tab:hpge_acceptance}.
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\begin{center}
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\begin{figure}[htbp]
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\centering
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\includegraphics[width=1.0\textwidth]{figs/hpge_higain_acceptance}
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\caption{Acceptance of the HPGe as a function of photon energy.}
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\label{fig:hpge_higain_acceptance}
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\end{figure}
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\end{center}
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\begin{table}[tbp]
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\centering
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\caption{HPGe acceptance for photons of interest}
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\label{tab:hpge_acceptance}
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\begin{tabular}{@{}cccc@{}}
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\toprule
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\multicolumn{2}{c}{\textbf{\begin{tabular}[c]{@{}c@{}}Photon energy\\ {[}keV{]}\end{tabular}}} & \textbf{Acceptance} & \textbf{Error} \\
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\midrule
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$2p-1s$ & 346.8 & \num{8.75E-4} &\num{4.0e-5} \\
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\ce{^{27}Mg} & 843.7 & \num{3.40E-4} &\num{0.9e-5} \\
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% & 1014.4 & \num{2.69e-4} &\num{1.07e-5} \\
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\ce{^{nat}Ti} & 931.96 & \num{3.06E-4} &\num{0.8e-5} \\
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\ce{^{26}Mg}* & 1088.7 & \num{1.51e-4} &\num{0.7e-5} \\
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% 0 346.828 0.000875 0.000040
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% 1 399.268 0.000753 0.000030
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% 2 400.177 0.000751 0.000030
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% 3 476.800 0.000624 0.000022
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% 4 843.740 0.000340 0.000009
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% 5 930.000 0.000306 0.000008
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% 6 931.000 0.000306 0.000008
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% 7 932.000 0.000306 0.000008
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% 8 1014.420 0.000279 0.000008
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% 9 1808.660 0.000151 0.000007
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\bottomrule
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\end{tabular}
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\end{table}
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\subsection{Number of stopped muons}
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% TODO: justification for taking just number from muon counter
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The number of stopped muons are taken as number of muons seen by the muon
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counter, since we used thick targets the muon beam is believed to stop
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completely at the middle of the targets. This assumption is verified for the
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aluminum target where count from muon counter was consistent with number of
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stopped muons calculated from number of $(2p-1s)$ X-rays.
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\subsection{Muonic X-ray spectra}
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We use the HPGe spectra to look for characteristic muonic X-rays from elements
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of interest. Energies of these muonic X-rays are listed
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in~\cref{tab:hpge_acceptance}.
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\subsubsection{Titanium}
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We are looking at X-rays from $(2p-1s)$ transitions in titanium. Natural
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titanium has 5 stable isotopes: \ce{^{46}Ti}, \ce{^{47}Ti}, \ce{^{48}Ti},
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\ce{^{49}Ti}, and \ce{^{50}Ti}, with the \ce{^{48}Ti} being the
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most abundant at 73.72\%. The fine splitting between muonic $2p_{3/2}
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$ and $2p_{1/2}$ levels in these stable isotopes are about
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\SI{2}{keV}~\cite{Wohlfahrt1981}, comparable to the resolution of our HPGe
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detector. The $(2p-1s)$ X-rays therefore show up as a broad, asymmetric peak
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with a longer tail on the low energy side. The peak is fitted as two
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Gaussian peaks on top of a first-order polynomial.
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\subsection{Fraction of muon captured by a nucleus}
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An atomic captured muon at the 1S state has only two choices, either to decay
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in orbit or to be captured on the nucleus. The total disappearance rate for
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negative muon, $\Lambda_{tot}$, is given by:
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\begin{equation}
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\Lambda_{tot} = \Lambda_{cap} + Q \Lambda_{free},
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\label{eq:mu_total_capture_rate}
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\end{equation}
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where $\Lambda_{cap}$ and $\Lambda_{free}$ are nuclear capture rate and free
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decay rate, respectively, and $Q$ is the Huff factor, which is corrects for the
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fact that muon decay rate in a bound state is reduced because of the binding
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energy reduces the available energy.
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Using mean lifetime measured by Suzuki et.al.~\cite{SuzukiMeasday.etal.1987}
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and fractions of muons captured by element of interest are calculated and
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listed in~\cref{tab:capture_frac}.
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\begin{table}[tbp]
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\centering
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\caption{Nuclear capture probabilities calculated from mean lifetimes taken
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from measurements of Suzuki et.al.~\cite{SuzukiMeasday.etal.1987}}
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\label{tab:capture_frac}
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\begin{tabular}{cccc}
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\toprule
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Element & Mean lifetime & Huff factor & Nuclear capture\\
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& [\si{ns}] & & probability [\%]\\
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\midrule
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\ce{^{nat}Al} & \num{864.0 \pm 1.0} & \num{0.993} &\num{60.95(5)} \\
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\ce{^{nat}Ti} & \num{329.3 \pm 1.3} & \num{0.981} &\num{85.29(6)} \\
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\ce{^{nat}W} & \num{78.4 \pm 1.5} & \num{0.860} &\num{96.93(6)} \\
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\bottomrule
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\end{tabular}
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\end{table}
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Number of stopped and captured muons in our targets are:
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\begin{table}[tbp]
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\centering
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\caption{Number of muons stopped and captured}
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\label{}
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\begin{tabular}{ccc}
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Target & Number of muons stopped & Number of muons captured \\
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\midrule
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\ce{^{nat}Al} & $(2.96002\pm 0.00017) \times 10^8$ & $(1.8041\pm 0.0015) \times 10^8$\\
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\ce{^{nat}Ti} & $(2.17237\pm 0.00015) \times 10^8$ & $(1.8530\pm 0.0013) \times 10^8$\\
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\ce{^{nat}W} & & \\
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\end{tabular}
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\end{table}
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