this is for holding javascript data
William P. Gammel added subsubsection_Charge_Density_Since_we__.tex
over 8 years ago
Commit id: bad5c19e9278de0fa8e7dce10d0a1d0df2e2ad08
deletions | additions
diff --git a/subsubsection_Charge_Density_Since_we__.tex b/subsubsection_Charge_Density_Since_we__.tex
new file mode 100644
index 0000000..841baf3
--- /dev/null
+++ b/subsubsection_Charge_Density_Since_we__.tex
...
\subsubsection{Charge Density}
Since we know that both have positive and negative muons contribute to the mean muon lifetime, it is also important to have some sort of quantitative understanding of the ratio, $\rho$ of muons to antimuons is. We can calculate $\rho$ via Equation ()
\begin{equation}
\rho=-\frac{\tau^{+}}{\tau^{-}}(\frac{\tau^{-}-\tau_{obs}}{\tau^{+}-\tau_{obs}})=\frac{N^{+}}{N^{-}}
\end{equation}
Where $\tau^{+}$ is the known value of the mean lifetime for an antimuon,$2.19 \mu\textrm{seconds}$, $\tau^{-}$ is the known value of the mean lifetime for a muon, $2.03 \mu\textrm{seconds}$, and $\tau_{obs}$ is the value $2.04\pm0.04 \mu\textrm{seconds}$, taken from the best fit of our data.
Thus the calculated ratio is approximately $0.072\pm.001$. This value is quite small, which confirms our previous hypothesis that the number of muon decays detected was significantly greater than the number of antimuon decays detected.
\subsubsection{Fermi Coupling Constant}
The Fermi Coupling Constant is the strength of the weak force, and can be calculated from the observed muon lifetime, $\tau_{obs}$. The relationship between the muon lifetime and the Fermi Coupling Constant is shown below,
\begin{equation}
\tau=\frac{192\pi^{3}\hslash^{7}}{G_{f}^{2}m^{5}c^{4}}
\end{equation}
In order to find the Fermi Coupling Constant we can rework the equation such that,
\begin{equation}
G_{f}=\sqrt[]{\frac{192\pi^{3}\hslash^{7}}{\tau m^{5}c^{4}}}
\end{equation}
\subsection{Gamma Ray Spectroscopy}
\subsubsection{Determining the Unknown Sample}
In order to determine what element the unknown sample was, we kept all the same settings from the calibration trial. Since the software had been calibrated, the resulting spectrum that was displayed was plotted as the number of gamma rays observed at each integer value, versus Energy, in MeV. The sample was in a rather large container, so it was placed as close as possible to 2 cm, which was the distance away from the scintillator of the sources used in the calibration. The sample ran for $3670$ seconds Live Time, which is the actual time the data is being recorded. The program also lists a quantity called Real Time, which is () since it includes the time it takes the () to process the data. The spectral distribution obtained from the multi-channel analyzer for the mystery source is shown in Figure ().
Determining what the unknown element is is a fairly straightforward process, since our plot of spectral distribution has already been calibrated. From Figure () we find that the height of the photopeak appears to be (). We then referred to the Smith College Physics radiation safety protocols, which contained information on what radioactive materials the school possessed. There were five () that the sample could have been, namely, Cesium-137, Cobalt 60, Sodium 22, and Strontium 90. Out of the five potential sources, the spectral distribution obtained from the multi-channel analyzer seemed to match most closey to the spectral distribution of Sodium-22, both qualitatively and quantitatively. Figure () shows a comparison between the observed spectral distribution of the mystery source and the known spectral distribution for Sodium-22, for qualitative comparison. Both the observed distribution and the known distribution appear to have two photopeaks. Guassians were fit to the plot of observed spectral distribution, to obtained a photopeak height of $0.53655 \pm 7.11 \cdot 10^{-5} \textrm{ MeV}$ for the first peak, and a height of $1.2791 \pm 0.000406 \textrm{ MeV}$. for the second peak. It is important to note that a Gaussian curve was fit to the observed data because there is fluctuation in the height of the voltage pulse produced by a gamma ray, which results in a broadening of the photopeak. Thus, a Guassian is believed to be the best fit for our data. Furthermore, the full width half maximum can be found from the Gaussian fits, and is a good measure of the resolution of the instrument. The expected value for Sodium-22's two peaks is $.5110034 \textrm{ KeV}$ and $1.2745 \textrm{ KeV}$ respectively. Our observed values are not consistent within uncertainty of the expected values of Sodium-22, however, the percent error between the expected and observed values is very slight ($5\%$ for the first peak and $0.36\%$ for the second peak) , thus with uphold that we have correctly determined the mystery sample. Furthermore, when compared to the expected photopeak values of the other possible elements the discrepancy between expected and observed values was far larger.
