Electron Diffraction
====================
Background
----------
In 1923, in his doctoral dissertation, Louis de Broglie proposed that
all forms of matter have wave as well as particle properties, just
like light. The wavelength, :math:`\lambda`, of a particle, such as
an electron, is related to its momentum, :math:`p`, by the same
relationship as for a photon:
.. math::
:label: wl
\lambda = h/p
where :math:`h` is Planck's constant. The wave properties of
electrons are illustrated in this experiment by the interference,
which results when they are scattered from successive planes of atoms
in a target composed of graphite micro crystals. The spacing between
successive planes is obtainable from the interference pattern.
.. _ed_fig1:
.. figure:: pictures/bragg_reflection_simple.png
:align: center
:scale: 30 %
Figure 1: Reflection of electron waves from atomic planes. Ray 1
and 2 have no path length difference, :math:`2 \Delta` is the
path length difference between ray 1 from the top plane and ray 3
from the one below. The short-dashed lines show another possible
set of atomic planes.
For a single crystal, strong reflection of waves occurs when the Bragg
condition is met:
.. math::
:label: br
2d sin(\theta) = n\lambda\ \ (n = 1.,2, ...)
In the graphite target, there are very many
perfect micro crystals randomly oriented to one another.
.. _ed_fig2:
.. figure:: pictures/electron_diffraction_schematic.png
:align: center
:scale: 40 %
Figure 2: Geometry for diffraction from a single graphite crystal
Therefore the strongly emerging beam will be of a conical shape of
half-angle :math:`2\theta` as shown in :ref:`Fig. 3`. If this
beam falls on a phosphor-coated screen, rings of light will be formed.
.. _ed_fig3:
.. figure:: pictures/electron_diffraction_cone.png
:align: center
:scale: 40 %
Figure 3: Diffraction from a large number of micro crystals
Experimental Setup
------------------
The apparatus is shown in :ref:`Fig. 4 `. Electrons emitted by thermionic
emission from a heated filament (4) inside the cathode are accelerated towards
the graphite target (9) of the anode by a potential difference,
:math:`V_a` between the cathode and anode. A focusing electrode (8) is
located in front of the target to focus the electron beam
in order to provide a sharp interference pattern on the screen (11).
.. _ed_fig4:
.. figure:: pictures/electron-diffraction_overview.png
:align: center
:scale: 40 %
Figure 4: Overview of the electron diffraction tube.(1)4-mm socket for
filament heating supply, (2) 2-mm socket for cathode
connection, (3) internal resistor, (4) filament. (5) cathode, (6)
anode, (7) 4-mm plug for anode connection (HV), (8) focusing
electrode, (9) polycrystalline graphite grating, (10) Boss, (11)
fluorescent screen.
Their kinetic energy, :math:`K`, on reaching the target is equal to
their loss of potential energy:
.. math::
:label: kin
K = \frac{p^2}{2m} = eV_a
from equations :eq:`wl` and :eq:`kin` one obtains:
.. math::
:label: kin1
\lambda = \frac{h}{\sqrt{2 m V_a e}}
\lambda(nm) = 1.228/\sqrt(V_a)
Experimental Procedure
----------------------
Connect the power supplies and micro-ammeter to the vacuum tube as
shown in :ref:`Fig. 5`
.. _ed_fig5:
.. figure:: pictures/electron_diffraction_wiring.png
:align: center
:scale: 40 %
Figure 5: Connecting the electron diffraction tube
**HAVE THE CIRCUIT CHECKED BY YOUR INSTRUCTOR BEFORE
TURNING ANYTHING ON.**
Adjust the voltage controls on the power supplies to zero and then
turn them on. Wait a few minutes for the filament to warm up. Set
the focusing/intensity voltage to around 30 V and then slowly increase the
accelerating voltage to 3 kV. Monitor the electron beam current to
ensure that it **never** exceeds 0.2 mA. A bright central spot and two
rings should be observable on the screen. The rings are due to first
order (n = 1) diffraction from two different sets of atomic planes
having different spacings. Adjust the focusing/intensity voltage
until the rings are as sharply defined as possible and then measure
their diameters. When adjusting the focus check the current, it should
not exceed 0.2 mA. Try to find a setting that produces the best images for
the lowest accelerating voltage, where the rings are barely visible,
as well as the highest accelerating voltage :math:`V_a` (about 5kV) Obtain sets of values of D for different
accelerating voltages, :math:`V_a`, over as wide a range as possible, but not
exceeding 5 kV.
You can use a camera to take pictures of the rings and analyze them
using the
`ImageAnalyzer `_.
For help ask an instructor.
Analysis
--------
The geometry for the vacuum tube is shown in :ref:`Fig. 6`.
.. _ed_fig6:
.. figure:: pictures/electron_diffraction_geometry.png
:align: center
:scale: 40 %
Figure 6: Vacuum tube geometry
The value of :math:`\theta` can be obtained from
.. math::
:label: g1
tan(2\theta) = \frac{D/2}{l_1 + l_2}
l_1 = L - R
l_2 = \sqrt{R^2 - (D/2)^2}
here :math:`R = 6.25\ cm` and :math:`L = 13.7\ cm`. The structure of
graphite is shown in :ref:`Fig. 7`. The two possible layers are
indicated by dashed lines.
#. Using these parameters together with your
measured voltages and the associated diameters, determine the two
different spacings between the planes.
#. Make a plot of your result as a
function of the voltages. Include all error bars. Determine the
weighted average values and their uncertainties.
#. From your measured data
determine the distance :math:`s` between the carbon atoms and compare
your result to the
value provided in :ref:`Fig. 7`. Does it agree within error bars ?
.. _ed_fig7:
.. figure:: pictures/electron_diffraction_graphite.png
:align: center
:scale: 40 %
Figure 7: The structure of graphite with Bragg planes.