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\title{The Franck-Hertz Experiment}
\author{Michael Richmond}
\date{Nov 8, 2010}
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   \begin{abstract} 
       We reproduced the classic Franck-Hertz experiment by
       accelerating electrons across
       varying voltages 
       through gaseous mercury
       and measuring the electrons which have
       enough kinetic energy to pass through a 
       retarding potential.
       Our results show the periodic variations
       one expects, based on the quantum nature
       of the energy required to raise mercury atoms
       out of the ground state.
       However, we measure an potential difference between
       successive peaks of $5.3 \pm 0.1$ Volts,
       which does not correspond within the experimental uncertainty
       with the known $4.9$ eV difference between 
       energy levels in mercury.
       
       
       

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\section*{Introduction}
In the early twentieth century, 
physicists realized that the classical
mechanics did not explain a number of 
phenomena at the atomic level.
Niels Bohr and others postulated that 
some physical quantities, such as the
energy levels of electrons within atoms,
could have only certain specific values;
in other words, that energy levels
were quantized.
In 1914, unaware of Bohr's model,
Gustav Hertz and James Franck 
performed an experiment which provided 
strong evidence in favor of quantized energy
levels within mercury atoms.
We reproduce their work here and find results
which are qualitatively the same as theirs,
matching the predictions of the quantized model.
however, the difference between energy levels
derived from our measurements do not agree 
quantitatively with the known structure of the
mercury atom.




\section*{Theory}
The quantum theory states that electrons within
an atom may have only very specific energies.
In neutral mercury atoms,
the first excited state lies 4.9 eV above the ground
state.
Consider a collision between a mercury atom in its
ground state and a free electron.







Very few people can see an electron, never mind measure it's charge, but we will here.   Here's some equations for your amusement, even with labels.

At the terminal velocity, the acceleration is zero, and Newton's second law gives
\beq
0=6\pi r \eta v_g +\frac{4}{3}\pi r^3 \rho_{air} g - \frac{4}{3}\pi r^3 \rho_{oil} g
\label{eq:freefall}
\eeq
where $v_g$ is the speed of the drop moving downward.

Equation \ref{eq:freefall} can be solved for the radius of the drop;
\beq
r=\sqrt{\frac{9\eta v_g}{2(\rho_{oil}-\rho_{air})g}}
\label {eq:radius}
\eeq

\section*{Experiment}
You can describe the equipment and what you did here.   

For what its worth, I don't care if you rename all these sections something else.

\section*{Analysis}
Here's a way to get a bunch of parameters to print out with the equal signs lined up neatly.
\begin{eqnarray*}
\rho_{oil}&=&872\hspace{.1in} kg/m^3\\
\eta&=&1.85 x 10^{-5}\hspace{.1in} Ns/m^2\\
\rho_{air}&=&1.2\hspace{.1 in} kg/m^3
\end{eqnarray*}

And you can certainly insert a Figure

%\begin{figure}[!h!t]
%	\centering
%		\includegraphics[width=\columnwidth]{boat.jpg}
%	\caption{I'd rather be sailing}
%	\label{fig:sailing}
%\end{figure} 

and then cite it in the text:  See Figure \ref{fig:sailing} for my preferred activity.

\section*{References}

\url{http://www.xkcd.com/242/}
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