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\rightline{(revised 1/23/13)}
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\bf{RAMSAUER - TOWNSEND EFFECT}
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{Advanced Laboratory, Physics 407,}\\
{University of Wisconsin}\\
{Madison, Wisconsin 53706}
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\begin{abstract}
The scattering cross section of electrons on noble gas atoms exhibits a
very small value at electron energies near 1 eV. This is the Ramsauer-Townsend
effect and provides an example of a phenomenon which requires a quantum mechanical
description of the interaction of particles.
\end{abstract}
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\section*{Theory}
 The Ramsauer-Townsend effect can be observed as long as the scattering
does not become inelastic by excitation of the first excited state of the atom. This condition
is best fulfilled by the closed shell noble gas atoms. Physically, the Ramsauer-Townsend effect
may be thought of as a diffraction of the electron around the rare-gas atom, in which the wave
function inside the atom is distorted in just such a way that it fits on smoothly to an undistorted
wave function outside. The effect is analagous to the perfect transmission found at particular
energies in one-dimensional scattering from a square well.
Appendix A (from Ref.[2]) contains a one-dimensional treatment of scattering from a square well.
This is the first model which you will use to analyze the data. A three-dimensional treatment using
partial waves is given in Ref. [4], pp 396-402.

\section*{Apparatus}

\subsection*{Thyratron - (RCA 2D21)}
The tube contains Xenon gas. The assembly is
mounted on a stand so that the filament of the tube is uppermost and so that
the tube may be dipped into a liquid nitrogen dewar.
(Note that the voltages being used here are NOT the voltages which are normally
used in thyratron circuits).
\subsection*{Regulated DC Power Supply}
The supply provides the voltage
to accelerate the electrons. The supply provides 0 to 30 volts but is
difficult to adjust for very low voltages. For this reason a control box containing a
potentiometer is used to accurately set the lower voltages.
\subsection*{4-Volt Transformer}
The transformer provides the power for the thyratron filament.
The tube normally uses 6.3 volts AC but by running the cathode at a lower
temperature the spread in electron energies is reduced. The transformer is contained
in the control box.
\subsection*{Dewar Flask}
The dewar will hold the liquid nitrogen necessary for freezing
out the Xenon in the thyratron tube. The cold data is used to correct for thyratron geometry effects.
\subsection*{Digital Multimeters - (6 1/2 digit Keithley 2100)}
These are high impedance meters used to measure the plate voltage, $V_p$;
the shield voltage, $V_s$; and the cathode to shield voltage, $(V - V_s)$.
\subsection*{Ramsauer.exe Program}
The ramsauer.exe program will allow you to record all the DVM readings in a .txt file
which you can name. The program will also plot the $V - V_s$ readings and both $V_p$ and $V_p$
as a function of $V - V_s$

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\includegraphics[width=5.0in]{ramsauer_circuit.pdf}
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Circuit Diagram
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{\ub{Thratron Socket Wiring Color Code}}
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\begin{tabular}{|c|c|c|}
\hline
Pin & Internal Connection & Color of Wire \\
\hline
 1 & grid \#1 & green* \\
 2 & cathode & black \\
 3 & heater & red \\
 4 & heater & red \\
 5 & shield (grid \#2) & no connection \\
 6 & anode & yellow \\
 7 & shield (grid \#2) & green* \\
\hline
\end{tabular}
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{* grid \#1 and shield (grid \#2) are joined externally}
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\includegraphics[width=\textwidth]{ramsauer_insert1.pdf}
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\includegraphics[width=\textwidth]{ramsauer_insert2.pdf}
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\includegraphics[width=\textwidth]{ramsauer_insert3.pdf}
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\includegraphics[width=\textwidth]{ramsauer_insert4.pdf}
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\includegraphics[width=\textwidth]{ramsauer_insert5.pdf}
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\section*{Procedure}
\begin{enumerate}
\item
Read the article by S.G. Kukolich in the Am. Jour. Phys. \ub{36}, 1968 and
understand the one-dimensional scattering from a square well.
\item
Set up the circuit as in the diagram on page 4.
\item
Allow 5 minutes for the tube filament, cathode and multimeters
to heat up and become stable.
\item
In order for the ramsauer.exe program to display correctly the DVMs have to be
connected to the control box in a specific way. The bottom DVM (V1) should connect
to $V - V_s$ as a negative voltage. The middle DVM (V2) connects to the grid ($V_s$)
and the top DVM (V3) connects to the plate ($V_p$). The program will enter the DVM
readings into a file (which you will name) by clicking on the READ button. The Append/Overwrite
button should be set to Append for data taking. The file may be viewed by clicking on the
ViewLogFile button. The file can be saved as a.txt file and viewed externally with WordPad.
\item
Measure the voltages $V_s$ and $V_p$ as a function of
the cathode to shield voltage $(V-V_s)$ with the thyratron at room
temperature. Use values of $(V-V_s)$ as follows:
\vskip .5cm
\begin{tabular}{lr@{\quad to\quad }r@{\quad volts in steps of\quad
}l@{\quad volts}}
 from & 0.25  & 1.00  & 0.1  \\
%  & 0.40  & 1.00  & 0.05   \\
  & 1.00  & 2.00  & 0.1    \\
  & 2.00  & 3.00  & 0.2    \\
  & 3.00  & 5.00  & 0.5    \\
  & 5.00  & 10.00  & 1.0
  \end{tabular}

The purpose of the uneven steps is to give the best detail
between 0.3 and 1.0 on the plot of $\sqrt{V-V_s}$.
Above 10 V the Xenon begins to ionize and the tube currents become large.
The true accelerating potential will be different from $(V-V_s)$ due to
the contact potential difference cathode and shield. The contact potential
has to be measured separately to make this correction.

\item
Turn off the filament and gently immerse only the lower blackened
part of the thyratron in liquid nitrogen. Allow it to cool for 15 minutes then
turn on the filament again and allow a further 5 minutes for temperatures to
stabilize. The Xenon will have condensed and frozen at the cold end of the
tube.
\item
Repeat measurements of Step 4 above at the same values of $(V-V_s)$ to
obtain $V^*_s$ and $V^*_p$. Adjust the tube from time to time to keep the
lower end in the liquid nitrogen.
\item
Plot $I_p$ and $I^*_p$ against $\sqrt{V-V_s}$.
\item
Calculate the probability of transmission (no scattering):
$$T = { I_p I^*_s \over I_s I^*_p}.$$
Since $V_p = I_p R_p$
$$V^*_p = I^*_p R_p$$
$$V_s = I_s R_s$$
$$V^*_s = I^*_s R_s,$$
it is easier to calculate:
$$T = { V_p V^*_s \over V_s V^*_p}.$$
Plot $T$ against $\sqrt{V - V_s}$ (which is proportional to the electron
momentum).

Plot $T$ against $V- V_s$ (which is proportional to the electron energy).

Note the value of $(V-V_s)$ corresponding to maximum $T$. Correct your
result for the contact potential difference.

The contact potential is
best determined  by measuring the the value of $V-V_s$ which makes the
grid current $I_s$ equal to zero. If there were no contact potential, $I_s
= 0$ would correspond to $V-V_s = 0$. You will find that the required
value of $V-V_s$ to make $I_s = 0$ is a reverse polarity. Reverse the
power supply voltage and take data with increasing $(V-V_s)$ so that the
the value of $(V-V_s)$ which makes $I_s = 0$ can be determined.
The value of this offset voltage will be the contact potential.

\item
Assume that the diameter of a Xenon atom is about 2.8 \AA
(Xenon is smaller than Cesium (5.5 \AA) because Xenon has closed shells).
From your data and using one-dimensional Quantum Mechanics estimate the average
depth of the square well seen by the electrons.
\item
A somewhat more realistic result for the depth of the square well
seen by the electrons can be made by using the three-dimensional square well
as a model. Theory predicts that the scattering will be a minimum when the
phase shift $\delta_0$ of the $\ell =0$ partial wave is $n \pi$ provided that
all other partial wave contributions are negligible. The condition that the
wave function and its derivative must be continuous at the boundary $r=a$ then
becomes
$$k_2 a \tan k_1 a = k_1 a \tan k_2 a$$
where $k = { 2 \pi \over \lambda}$, $\lambda_1 =$ wave length of the electron
inside the square well, and $\lambda_2=$ wave length of the free electron. Use
this relation to make another estimate of the depth of the square well.
\end{enumerate}

\begin{thebibliography}{99}

\bibitem{ref1}
``Demonstration of the Ramsauer - Townsend Effect in a Xenon
Thyratron", S.G. Kukolich, Am. J. Phys.\ {\bf 36}, 1968, pages 701 - 701,
included in this description.
\bibitem{ref2}
{\it An Introduction to Quantum Physics}\ (Norton, 1978), A.P. French and E.F. Taylor,
\bibitem{ref3}
{\it Quantum Mechanics, 3rd Ed.}\ (Wiley, 1998), E. Merzbacher.
\bibitem{ref4}
{\it Modern Physics and Quantum Mechanics}\ (Saunders, 1971), E.E. Anderson.

\end{thebibliography}
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