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\author{Anthony Tricarichi:106718074}
\title{Lab 09: Radioactive Decay}
\begin{document}
\maketitle
\section{Introduction}
The purpose of this lab is to determine the decay constant of a sample of $^{137}$Ba.
\section{Procedure}
\subsection{Setup}
The setup for this lab consisted of a Geiger-M\"uller tube, a power supply, and a frequency counter. 
\subsection{Lab}
There are two main parts of this lab, in the first part we found the background noise of the room to normalize our data. We set the counter to count for 10 minutes then divided that result by 20 to see how much background noise we got per 30 seconds. The second part of the lab we would take 30 second counts with 10 second intervals in between. This data is what we use to determine the decay constant of the sample.

\section{Data}
Table 1: Lab Data
\\
\begin{tabular}{|l|l|l|l|}
\hline
Time Elapsed (s) & Normalized data\footnote{Data is normalized to 171 counts per 10 mins which is 9 taken away from each result} & N/Second & Ln(N/s)\\
\hline
0-30 & 69 & 2.3 & .832 \\
\hline
40-70 & 52 & 1.7 & .55 \\
\hline
80-110 & 38 & 1.26 & .236 \\
\hline
120-150 & 25 & .833 & -.1823  \\
\hline
160-190 & 27 & .9 & -.105 \\
\hline 
200-230 & 22 & .733 & -.31 \\
\hline
240-270 & 25 & .83 & -.19 \\
\hline
280-310 & 16 & .53 & -.63 \\
\hline
320-350 & 18 & .60 & -.51\\
\hline
360-390 & 8  & .27 & -1.31\\
\hline 
400-430 & 10 & .3 & -1.2 \\
\hline 
440-470 & -2 & .07 & -2.66 \\
\hline 
\end{tabular}
\\
With $t^{1/2}$ being 156 second we can find the activity to be .0044 using formula 3
\\
Using this value we can compute this to our data 
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\includegraphics[scale=.4, angle=-90]{Plot1.eps}\\
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In our graph we have a slope of -.00595 which is fairly close to our expected value of -.0044
\section{Formulas}
\begin{eqnarray}
N(t) &=& N_oe^{\lambda t} \\
ln[N(t)] = -\lambda t + ln[N_o] \\
\lambda = \frac{ln[.5]}{t^{1/2}}
\end{eqnarray}
\subsubsection{Sample Lab Calculations}
Here we have a sample calculation for finding the expect lambda
\begin{eqnarray}
\nonumber
\lambda &=& \frac{ln[.5]}{t^{1/2}} \\ \nonumber
\lambda &=& \frac{ln[.5]}{156} \\ \nonumber
\lambda &=& -.0044 \nonumber
\end{eqnarray}

\section{Error}
Error in the lab can be defined by deviation in the expected value of the natural log of N. 
\begin{eqnarray}
\nonumber
\Delta ln(N) = \frac{\Delta N}{N}  \nonumber
\end{eqnarray}
Each value is calculated and included as the y error bars in the graph.

\section{Analysis \& Conclusion}
The lab was overall successful we got a value which was relatively close to the expected, given the spontaneity of the lab, our error wasn't off the charts. The biggest problem with the lab was assuring good timing and not hindering results by being too late or early. The biggest change to this lab would be making it more automated, but seeing this is a 200 level lab, i don't think that is a major concern. Overall the lab was successful and we yielded some good results

\end{document}