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\begin{document}

\begin{titlepage}
\newcommand{\HRule}{\rule{\linewidth}{0.5mm}}
\center
\textsc{\LARGE Universität Tübingen}\\[1.5cm] 
\textsc{\Large F-Praktikum}\\[0.5cm] 
\textsc{\large NAK}\\[0.5cm] 
\HRule \\[0.4cm]
{\huge \bfseries Neutron activation}\\[0.4cm] 
Version 1
\HRule \\[1.0cm]
\begin{minipage}{0.4\textwidth}
    \begin{flushleft} \large
        \emph{Von:}\\
        Group 22\newline
        \underline{Simon Fischer}\newline
        Fabian Kraus
    \end{flushleft}
\end{minipage}
~
\begin{minipage}{0.4\textwidth}
    \begin{flushright} \large
        \emph{Supervisor:}\\
         Shahid Khan
    \end{flushright}
\end{minipage}\\[2cm]
{\large Date: 26.06.2021}\\
{\large Version 1: \today}\\
\end{titlepage}

\newpage

\tableofcontents


\newpage

\section{Introduction}








\newpage

\section{Theoretical background}
In general, nuclei of atoms do not necessarily have to be stable. That is, they can decay under the emission of particles or radiation. Doing so, these nuclei reduce their number of protons and/or neutrons, depending on their initial composition. The different decay channels of radioactive nuclei will be treated in the following.

\subsection{Decay channels}
For our experiment, three different decay channels were relevant, namely the $\alpha$-, $\beta-$ and the $\gamma$-decay, which will be elaborated in detail in the following subsections.

\subsubsection{$\alpha$ decay}
We call the process of a heavy nucleus decaying into a lighter one while emitting an $\alpha$-particle an $\alpha$-decay. An $\alpha$-particle is an ionized helium nucleus, i.e. it consists of two protons and two neutrons. The decay can be written explicitly as
\begin{equation}
    _{Z}^AX \longrightarrow_{Z-2}^{A-4}Y+_{2}^4 He^{2+}.
\end{equation}
An $\alpha$-particle is double positive which results in it being easily screened by a few millimeters or centimeters of matter, for instance paper or air.

\subsubsection{$\beta$ decay}
A $\beta$ decay is either an emission of an electron or of a positron. Thus, the $\beta$ decay can be divided into two subcategories: The $\beta^+$ decay and the $\beta^-$ decay.\\
In the $\beta^-$ decay a neutron decays into a proton and an electron plus one anti electron neutrino, that is
\begin{equation}
    _{Z}^AX \longrightarrow_{Z+1}^{A}Y+e^- + \Bar{\nu_e}.
\end{equation}
On the other hand, the $\beta^+$ decay denotes a proton decaying into a neutron, positron and an electron neutrino:
\begin{equation}
    _{Z}^AX \longrightarrow_{Z-1}^{A}Y+e^+ + \nu_e.
\end{equation}
In both decay channels it can happen that the nucleus is not in the ground state (shell model of the nucleus), leading to an additional emission of $\gamma$ radiation. This is only a secondary decay, which is why we only note it here.\\
Both $\beta^+$ and $\beta^-$ radiation can be shielded with a few centimeter of Pb.


\subsubsection{$\gamma$ decay}
In the case of the $\gamma$ decay, the $\gamma$ radiation, that is a photon, is released due to the de-excitation of a nucleus, which can be described by the shell model for nuclei. We denote this in the following manner:
\begin{equation}
    _{Z}^AX^* \longrightarrow _{Z}^AX + \gamma,
\end{equation}
where the upper $^*$ stands for the excited state of the atomic nucleus.


\subsection{Activity and activation}
In order to describe the atomic decay quantitatively, we introduce the term \textit{activity} $A$. This quantity equals the number of decays per unit time, that is
\begin{equation}
    A(t):=\frac{dN}{dt}=-\lambda N,
\end{equation}
where $lambda$ is a proportionality constant and is called the \textit{decay constant}. With the boundary condition $N(t=0)=N_0$ we can formally intergate the latter equation to
\begin{equation}
    N(t)=N_0e^{-\lambda t},
\end{equation}
whereas for the activity it holds that
\begin{equation}
    A(t)=A_0e^{-\lambda t},
    \label{nak_activity_1}
\end{equation}
with $A_0 = N_0 \lambda$.\\
The decay constant $\lambda$ turns out to be useful for the definition of the \textit{half-life time}
\begin{equation}
    t_{\frac{1}{2}}=\frac{\ln(2)}{\lambda}.
\end{equation}
This is the time it takes until only half of the initial number of radiative nuclei $N_0$ of a certain type $X$ are left, which corresponds obviously to $N(t_{\frac{1}{2}})=\frac{N_0}{2}$. In case we have an artificial source, we have to take into account the additional flux of of particles. We respect this by adding another term similar to the activity, but with opposite sign
\begin{equation}
    \frac{dN}{dt}=-\lambda N+P.
\end{equation}
Here, we call $P$ the production rate. This is defined by
\begin{equation}
    P=\sigma \Phi n.
\end{equation}
Here, $\sigma$ is the cross-section of interaction and $n$ is the number of particles, $\Phi$ denotes the neutron flow. By that it is possible to determine the activity by integration
\begin{equation}
    A(t)=A_S\left( 1-e^{-\lambda t} \right),
    \label{nak_A_sat}
\end{equation}
where $A_S=P=\sigma \Phi n$.

\subsection{Neutron creation}
As neutrons have only a very short lifetime in free space of $886\,\SI{}{s}$, they do not appear randomly in large enough quantities and have to be created artificially. For that we use an $AmBe$-source. The working principle is that $_{95}^{241}Am$ decays into $_{93}^{237}Np$ and a helium nucleus ($\alpha$-particle), which is then radiated onto a $_{4}^{9}Be$ atom. This then decays via the following chain
\begin{equation}
    \alpha + _{4}^{9}Be \longrightarrow _{6}^{13}C^* + n.
\end{equation}
The excited carbon nucleus deexcites by emitting a neutron.


%Free neutrons have an energy in the magnitude of ~$\SI{}{MeV}$, which are then slowed down by collisions with the hydrogen nuclei in the paraffin (see experimental setup). After the collision, the energy of those neutrons is approximately $25\,\SI{}{MeV}$.

\subsection{Neutron capture} \label{nak_ch_neutron_capture}
In this experiment the targets are Ag and In. By looking at the table of nuclides we find the following possible decays:
\begin{align}
    ^{107}Ag +n \longrightarrow& ^{108}Ag\longrightarrow ^{108}Cd + e^- + \Bar{\nu}_e \\
    ^{109}Ag +n \longrightarrow& ^{110}Ag\longrightarrow ^{110}Cd + e^- + \Bar{\nu}_e.
\end{align}
Here, the first decay channel has a half-life time of $t_{\frac{1}{2}}=142\,\SI{}{s}$ and decays mostly over $\beta^-$ decay. On the other hand, the second decay channel has a half-life time of $t_{\frac{1}{2}}=24\,\SI{}{s}$ and also decays mostly over $\beta^-$ decay.\\
For In we have the following possibilities:
\begin{align}
    ^{113}In +n \longrightarrow& ^{114}In\longrightarrow ^{114}Sn + e^- + \Bar{\nu}_e \\
    ^{115}In +n \longrightarrow& ^{116}In\longrightarrow ^{116}Sn + e^- + \Bar{\nu}_e.
\end{align}
In both cases they mainly decay over $\beta^-$ decay. The first one has a half-life time of $t_{\frac{1}{2}}=71.9\,\SI{}{s}$, the second one has $t_{\frac{1}{2}}=54\,\SI{}{min}$. This vast difference in half-life time is also the reason why in nature $^{113}In$ appears only with $4.3\%$, contrary to $^{115}In$ which appears with $95.7\%$. This is why the neutron capture of $^{113}In$ can be neglected for our purposes.

\newpage


\section{Experimental setup}

\subsection{Setup of the $AmBe$ source}
We insert the samples from above into the box next to the $AmBe$ source, where the samples themselves are attached to a rod and they themselves face the source as seen in \autoref{nak:fig:AmBe_setup}.
\begin{figure}[htb]
    \centering
    \includegraphics[width=120mm]{08_NAK/NAK_Setup.PNG}
    \centering
    \caption{Setup of the $AmBe$ source. | \textit{Source:} Manual.}
    \label{nak:fig:AmBe_setup}
\end{figure}
The source and the samples are isolated in a box that consists of Pb bricks and paraffine.

\subsection{Setup of the detector}
We use a Geiger-Mueller-tube. It consists of a tube that holds a gas. The molecules/atoms of the gas are splitted by incident particles (or even $\gamma$ radiation). The positively charged part then travels towards the cathode and vice versa.
\begin{figure}[H]
    \centering
    \includegraphics[width=120mm]{08_NAK/NAK_Geiger_Mueller.PNG}
    \centering
    \caption{Setup of the Geiger-Mueller-tube. | \textit{Source:} Manual.}
    \label{nak:fig:AmBe_setup}
\end{figure}
On their way, the particles hit other ones and therefore cause something similar to an avalanche of charged particles traveling towards cathode and anode. When these charged particles hit the wall of the detector, which carries opposite charge, a signal is produced, which manifests in a change in voltage. This change in voltage can then be traced back on the number of incident particles hitting the detector wall. In our experiment we use two of such Geiger-Mueller-tubes that are arranged in the following fashion.
\begin{figure}[H]
    \centering
    \includegraphics[width=120mm]{08_NAK/NAK_setup_detector.PNG}
    \centering
    \caption{Setup of the Geiger-Mueller-tubes used in the experiment. | \textit{Source:} Manual.}
    \label{nak:fig:AmBe_setup_detector}
\end{figure}
\noindent The sample is put in the middle of both tubes in order to gather twice as much data as one would obtain from only one.

\newpage

\section{Evaluation}

\section{Calibration}
The first task was to calibrate the Geiger-Mueller-tubes and to find the plateau region. Each tube collected counts for $60\,\SI{}{s}$ at different voltages. The results can be seen in \autoref{nak:fig:calibration}.
%It should be noted that the continuous lines do \textit{not} imply that there have been continuous measurements or that there as been an interpolation performed. It should rather help understanding the trend of the ten \textit{discrete} measurements taken.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Calibration_2.png}
    \centering
    \caption{Calibration of the Geiger-Mueller-counter. Error bars are calculated by the square root of each value.}
    \label{nak:fig:calibration}
\end{figure}
Unfortunately, no clear trend can be seen for the case of the right tube. The best fit is visible in purple, but shows no clear tendency towards convergence. One would have expected an increase in activity with a certain degree of saturation above that voltage. This can hardly be seen in our data. Still, we can optimistically see a peak at around $370\,\SI{}{V}$, which lead us to choose that value for the following measurements.



\subsection{Background}
We measured the background for about $60\,\SI{}{min}$ in $1\,\SI{}{min}$ step size. The results are shown in the figure below.
\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/background_image.png}
    \centering
    \caption{Background measurement divided by a factor of $60$.}
    \label{nak:fig:AmBe_setup}
\end{figure}
The red line indicates the statistical median, the green ones show the $1\sigma$-standard deviation from it, for which we obtained
\begin{equation}
    A_{background} = \left(0.8833 \pm 0.1172\right)\,\SI{}{Bq}.
\end{equation}

\subsection{Measurements of the samples}
We investigated an Ag-sample and an In-sample. For both we used, in accordance with \eqref{nak_activity_1}, the function
\begin{equation}
    A_{total} = A_1 e^{-\lambda_1(t-t_{offset})} + A_2 e^{-\lambda_2(t-t_{offset})} + A_{background}
\end{equation}
to fit the data, where $A_{background}$ has been determined above. Here, $t_{offset}$ denotes the time it took to transport the sample from the exposure of the source to the Geiger-Mueller-tube, which was located a few meters away from each other. It is clear that during this time, there have already been decays that cannot be neglected. The offset times are listed below

\begin{table}[H]
    \centering
    \begin{tabular}{c|c}
        Activation time $t_a$ [s] & $t_{offset}$ [s] \\ \hline
        
        5 & 13.6\\
        \hline
        10 & 18.5\\
        \hline
        20 & 10.99\\
        \hline
        30 & 12.5\\
        \hline
        45 & 9.97\\
        \hline
        60 & 8.71\\
        \hline
        90 & 9.38\\
        \hline
        120 & 9.17\\
        \hline
        300 & 8.48\\
        \hline
        600 & 10.6\\
        \hline
        900 & 9.09\\
    \end{tabular}
    \caption{Offset times for the different activation times.}
\end{table}


\subsubsection{Ag-sample}
The results for each activation time are listed in the appendix. An example for an activation time of $5$ seconds can be found below.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_5s.png}
    \centering
    \caption{Ag-sample for an activation time of $5$ seconds.}
\end{figure}



The results of the fits are summarized in the table below ordered by the activation time $t_a$


\begin{table}[H]
    \centering
    \begin{tabular}{c|c|c|c|c}
        $t_{a}$ [$\SI{}{s}$] & $A_{108}\pm \Delta A_{108}$ [$\SI{}{Bq}$] & $\lambda_{108}\pm \Delta \lambda_{108}$ [$\SI{}{\frac{1}{s}}$] & $A_{110}\pm \Delta A_{110}$ [$\SI{}{Bq}$] & $\lambda_{110}\pm \Delta \lambda_{110}$ [$\SI{}{\frac{1}{s}}$] \\ \hline
        \hline
        $5$ & $26.97 \pm 3.41$ & $0.03282\pm0.00452$ & $4.558\pm3.274$ & $0.006563
\pm0.00369$\\
        \hline
        $10$ & $37.84 \pm 7.05$ & $0.0396\pm0.00533$ & $14.36\pm6.601$ & $0.01061
\pm0.00313$\\
        \hline
        $20$ & $109.6 \pm 4.2$ & $0.029\pm0.00144$ & $8.105\pm4.224$ & $0.005225
\pm0.002362$\\
        \hline
        $30$ & $109.6 \pm 4.2$ & $0.029\pm0.00144$ & $8.105\pm4.224$ & $0.005225
\pm0.002362$\\
        \hline
        $45$ & $176.2 \pm 10.9$ & $0.03205\pm0.00214$ & $29.28\pm10.92$ & $0.00753
\pm0.001911$\\
        \hline
        $60$ & $232.1 \pm 5.1$ & $0.02807\pm0.00096$ & $16.5\pm5.11$ & $0.00404
\pm0.00131$\\
        \hline
        $90$ & $26.97 \pm 3.41$ & $0.03282\pm0.00452$ & $4.558\pm3.274$ & $0.006563
\pm0.00369$\\
        \hline
        $120$ & $225.4 \pm 38.5$ & $0.02986\pm0.00158$ & $36.37\pm8.48$ & $0.005048
\pm0.00106$\\
        \hline
        $300$ & $253 \pm 9.4$ & $0.02701\pm0.00143$ & $55.32\pm9.69$ & $0.004612
\pm0.000739$\\
        \hline
        $600$ & $232.9 \pm 9.7$ & $0.02592\pm0.00145$ & $60.08\pm9.74$ & $0.004324
\pm0.00067$\\
        \hline
        $900$ & $224.4 \pm 8.9$ & $0.02848\pm0.00161$ & $71.22\pm8.97$ & $0.004776
\pm0.00055$\\
        
    \end{tabular}
    \caption{Results of the fits for Ag.}
\end{table}
With that we can calculate the half-life time by
\begin{equation}
    t_{\frac{1}{2}} = \frac{\ln(2)}{\lambda}.
\end{equation}
We thus obtain for $Ag_{108}$ the time $t_{\frac{1}{2},108}$ and for $Ag_{110}$ the time $t_{\frac{1}{2},110}$ as follows:

\begin{table}[H]
    \centering
    \begin{tabular}{c|c|c}
        Activation time $t_a$ [s] & $\left(t_{\frac{1}{2},108}\pm\Delta t_{\frac{1}{2},108}\right)$ [s] & $\left(t_{\frac{1}{2},110}\pm \Delta t_{\frac{1}{2},110}\right)$ [s] \\ \hline
        
        5 & $105.614\pm 59.381$ & $21.120 \pm 2.909$\\
        \hline
        10 & $65.330\pm 19.273$ & $17.504 \pm 2.356$\\
        \hline
        20 & $132.660\pm 59.970$ & $23.902\pm 1.187$\\
        \hline
        30 & $151.210\pm 72.703$ & $23.902 \pm 1.138$\\
        \hline
        45 & $92.051\pm 23.363$ & $21.627 \pm 1.444$\\
        \hline
        60 & $171.571\pm 55.634$ & $24.694 \pm 0.845$\\
        \hline
        90 & $168.526\pm 44.744$ & $25.437 \pm 0.915$\\
        \hline
        120 & $137.311\pm 28.834$ & $23.213 \pm 1.229$\\
        \hline
        300 & $150.292\pm 24.082$ & $25.663 \pm 1.359$\\
        \hline
        600 & $160.302\pm24.839$ & $26.742 \pm 1.496$\\
        \hline
        900 & $145.131\pm 16.714$ & $24.338 \pm 1.376$\\
    \end{tabular}
    \caption{Offset times for the different activation times.}
    \label{nak:tab:halflife_times_results}
\end{table}
\noindent It is clearly visible that the data set of $t_s = 45 \, \SI{}{s}$ seems to be corrupted. Aside from that every set displays a good trend as we will see later. Therefore, we will disregard this one activation time. We assume that in this particular case we we recorded the data in the wrong manner. But since every other activation time yields good results, we do not consider this mistake as drastic.\\
\\[0.5pt]
\noindent We will now calculate the median for the half-life time and then compare it to the literature. Afterwards we will briefly discuss the trend of the activation itself.\\
\\[0.5pt]
\noindent Regarding the half-life time for $Ag_{108}$ we get the following graph by plotting the previously obtained values.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_halflife_108.png}
    \centering
    \caption{Half-life time for $Ag_{108}$.}
\end{figure}
We calculated the mean value to be
\begin{equation}
    t_{\frac{1}{2},108} = \left( 145.131 \pm 28.834 \right)\,\SI{}{s},
\end{equation}
where we determined the error by Gaussian error propagation in \autoref{nak:tab:halflife_times_results} and subsequently taking the mean value of the error due to the vast difference in the error intervals. It is evident that half-life times with bigger error bars do not contribute significantly to a shift of the mean value caused by values with smaller error bars. Qualitatively, one can think of larger errors as "smeared out" mean values; hence, the smaller contribution arises compared to values with smaller errors. The literature [\ref{NAK_Lit2}] suggests a value of $t_{\frac{1}{2},108}=142.2\,\SI{}{s}$, which comes very close to the value we found.\\
\\[0.5pt]
\noindent On the other hand, for the $Ag_{110}$-sample, the same procedure yields the following graph.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_halflife_110.png}
    \centering
    \caption{Half-life time for $Ag_{110}$.}
\end{figure}
\noindent We obtained the mean value
\begin{equation}
    t_{\frac{1}{2},110} = \left( 23.902 \pm 2.910 \right)\,\SI{}{s},
\end{equation}
where we chose conservatively the largest error of \autoref{nak:tab:halflife_times_results}. Compared to the literature [\ref{NAK_Lit1}], where a value of $t_{\frac{1}{2},110}=24.6\,\SI{}{s}$ is given, we conclude that also here the value from the literature lies well within the uncertainties of our result.\\
\\[0.5pt]
\noindent Finally, let us take a brief look at the activation for each isotope of Ag.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_A_110_2.png}
    \centering
    \caption{Activity $Ag_{110}$ w.r.t. activation time.}
\end{figure}

With \eqref{nak_A_sat} we can determine the saturation activity, which results in
\begin{equation}
    A_S = \left( 240.6 \pm 23.9 \right)\,\SI{}{Bq}.
\end{equation}
Due to the fit, we also received a result for the half-life time, which is
\begin{equation}
    t_{\frac{1}{2},110} = \left( 23.105 \pm 5.969 \right)\,\SI{}{s},
\end{equation}
which almost perfectly coincides with the result found abobe.\\
\\[0.5pt]
\noindent Repeating the same procedure for $Ag_{108}$ yields equally matching results, as seen below.

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_A_108_2.png}
    \centering
    \caption{Activity $Ag_{108}$ w.r.t. activation time.}
\end{figure}

With \eqref{nak_A_sat} we can determine the saturation activity, which results in
\begin{equation}
    A_S = \left( 66.46 \pm 11.35 \right)\,\SI{}{Bq}.
\end{equation}
Again, we also obtained a result for the half-life time, which reads
\begin{equation}
    t_{\frac{1}{2},108} = \left( 113.00 \pm 50.81 \right)\,\SI{}{s},
\end{equation}
which lies within the error range as the value found previously for $t_{\frac{1}{2},108}$.


\newpage


\section{In-sample}

The In-sample had been activated only once but for around three hours. To be exact, the exposure time was $3$ hrs and $8$ min, i.e. $188$ min. The offset time, that is the time between removing the sample from the tank and transporting it to the Geiger-Mueller counter was $t_{offset}=11.63\,\SI{}{s}$.\\
As discussed in chapter \ref{nak_ch_neutron_capture}, $^{116m}In$ can be neglected due to its low abundance. Still, we considered it for the sake of completeness. Thus, we again fit \eqref{nak_A_sat} to the data and find the following graph.
\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Indium_decay.png}
    \centering
    \caption{Decay of activated Indium in its natural mixture (see chapter \ref{nak_ch_neutron_capture}).}
\end{figure}
The results from the fit are summarized below.
\begin{table}[H]
    \centering
    \begin{tabular}{c|c|c|c|c}
        $t_{a}$ [$\SI{}{s}$] & $A_{116m}\pm \Delta A_{116m}$ [$\SI{}{Bq}$] & $\lambda_{116m}\pm \Delta \lambda_{116m}$ [$\SI{}{\frac{1}{s}}$] & $A_{116}\pm \Delta A_{116}$ [$\SI{}{Bq}$] & $\lambda_{116}\pm \Delta \lambda_{116}$ [$\SI{}{\frac{1}{s}}$] \\ \hline
        \hline
        $11280$ & $182.8 \pm 3.1$ & $0.00031\pm0.0000728$ & $40.73\pm5.84$ & $0.0641
\pm0.017$\\
        \hline
        
    \end{tabular}
    \caption{Results of the fit for In.}
\end{table}
This yields half-life times of
\begin{align}
    t_{\frac{1}{2},116m} =& \left( 2193.50 \pm 505.33 \right)\,\SI{}{s}\\
    t_{\frac{1}{2},116} =& \left( 10.82 \pm 2.85 \right)\,\SI{}{s},
\end{align}
for $^{116m}In$ and $^{116}In$. Compared to the literature [\ref{NAK_Lit2}], where the values
\begin{align}
    t_{\frac{1}{2},116m} =& 3257\,\SI{}{s}\\
    t_{\frac{1}{2},116} =& 14.1\,\SI{}{s},
\end{align}
are given, we see that our result for $t_{\frac{1}{2},116}$ lies close, but not quite inside the error range. Furthermore, $t_{\frac{1}{2},116m}$ also comes fairly close to the one given by the literature, whereas it also does not lie withing its error boundaries.



\newpage


\section{Conclusion}
Using two Geiger-Mueller counters and a neutron source, we were able to experimentally verify results for the decay rates, lifetimes and initial activities of In- and Ag-samples. The Ag-samples had been activated for $5$, $10$, $30$, $30$, $45$, $60$, $90$s and for $2$, $5$, $10$ and $15$ minutes. The Indium sample had been activated for $188$ min.\\
All in all our results match well with the literature. Especially those for the Ag-sample yield very good results (see [\ref{NAK_Lit1}]) and importantly, consistent results with different methods. The results for Ag are
\begin{equation}
    t_{\frac{1}{2},110} = \left( 23.902 \pm 2.910 \right)\,\SI{}{s},
\end{equation}
which matches with $t_{\frac{1}{2},110}=24.6\,\SI{}{s}$ from [\ref{NAK_Lit1}]. Further,
\begin{equation}
t_{\frac{1}{2},110} = \left( 23.902 \pm 2.910 \right)\,\SI{}{s},
\end{equation}
also lies within the uncertainty of $t_{\frac{1}{2},110}=24.6\,\SI{}{s}$ from the literature.\\
\\[0.5pt]
The results for the In-sample, while the literature values are not lying inside the error intervals, come close to the values given in [\ref{NAK_Lit2}]. The obtained results are
\begin{align}
    t_{\frac{1}{2},116m} =& \left( 2193.50 \pm 505.33 \right)\,\SI{}{s}\\
    t_{\frac{1}{2},116} =& \left( 10.82 \pm 2.85 \right)\,\SI{}{s},
\end{align}
for $^{116m}In$ and $^{116}In$. Compared to the literature [\ref{NAK_Lit2}], where the values
\begin{align}
    t_{\frac{1}{2},116m} =& 3257\,\SI{}{s}\\
    t_{\frac{1}{2},116} =& 14.1\,\SI{}{s},
\end{align}
they come close, but do not match within the error intervals.

\newpage


\section{Literature}

\begin{enumerate}
    \item \label{NAK_Lit1} ”Aktivierung von Silber und Indium durch langsame Neutronen (NAK)”, Physikalisches Institut
der Universität Tübingen, 28.03.2011
    
    \item \label{NAK_Lit2} http://www.periodensystem-online.de (entsprechend die Isotope aufrufen), 07.03.2021 16:11
    
    
\end{enumerate}

\newpage

\section{Appendix}


\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_10s.png}
    \centering
    \caption{Ag-sample for an activation time of $10$ seconds.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_20s.png}
    \centering
    \caption{Ag-sample for an activation time of $20$ seconds.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_30s.png}
    \centering
    \caption{Ag-sample for an activation time of $30$ seconds.}
\end{figure}


\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_45s.png}
    \centering
    \caption{Ag-sample for an activation time of $45$ seconds.}
\end{figure}




\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_60s.png}
    \centering
    \caption{Ag-sample for an activation time of $60$ seconds.}
\end{figure}




\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_90s.png}
    \centering
    \caption{Ag-sample for an activation time of $90$ seconds.}
\end{figure}


\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_120s.png}
    \centering
    \caption{Ag-sample for an activation time of $120$ seconds.}
\end{figure}



\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_5min.png}
    \centering
    \caption{Ag-sample for an activation time of $5$ minutes.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_10min.png}
    \centering
    \caption{Ag-sample for an activation time of $10$ minutes.}
\end{figure}

\begin{figure}[H]
    \centering
    \includegraphics[width=160mm]{08_NAK/Ag_15min.png}
    \centering
    \caption{Ag-sample for an activation time of $15$ minutes.}
\end{figure}



\end{document}