ELECTRON DIFFRACTION
Electrons : wave or quanta? Measurement of wavelength and momentum
of electrons.
Introduction
Electrons display both wave and particle properties. What is the relationship between the electrons’ wavelength and their momentum?
Theory
Louis de Broglie developed the idea that electrons exhibit both particle and wave properties. He
proposed that, analogous to photons, the wavelength λ of the electron wave is given by
λ=
h
h
=
p
mv
(1)
where h is Planck’s constant and p is the electron momentum which is given by p = mv in the
non-relativistic limit.
In order to observe the wave nature of electrons one directs a beam of electrons through a
thin layer of carbon and analyzes the angles at which electrons emerge. If electrons behave like
particles, the distribution of electrons varies continuously as a function of angle as indicated in
Fig. 1. This distribution will vary only slightly with changes in electron energy.
Theta
carbon target
electron beam
counts
Figure 1: Electrons as Particles
Continuous distribution of electrons as a function of angle.
The picture is different if electrons behave like waves. Analogue to the diffraction of x-rays
(photons with wavelengths of a few Å) by crystals, the crystal may be considered to be a
3 dimensional diffraction grating for electron waves. The diffraction maxima are deduced applying the concept of ’reflecting’ planes in the crystal. The locations of atoms in the two-dimensional
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Incoming Beam
d
d
d
d
d
Hexagonal
Cubic
Figure 2: Atomic Planes of Cubic and Hexagonal Crystals.
Two-dimensional slices of cubic and hexagonal crystals.
Different angles of refraction are indicated in the cubic crystal structure.
slices of cubic and hexagonal crystals are shown in Fig. 2. The entire crystal can be visualized
as a stack of slices, the various lines representing planes intersecting the actual crystal.
Θ
Θ
d sin(Θ )
Θ
d
Θ
Figure 3: Electron Waves refracted from Atomic Planes.
Atomic planes are treated as ’reflecting’ surfaces and allow waves from successive planes to
interfere with one another. If the difference in path length for waves refracted from successive
planes is an integral number of wavelengths (nλ, cf.F ig.3), constructive interference of matter
waves occurs at angles for which
nλ = 2d sin Θ
(2)
holds.
For each plane there will be a series of constructive interference, but the intensity depends on the
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specific geometry of a crystal. In order to measure diffraction angles as a function of wavelength
for a single crystal, one needs to vary not only the wavelength (λ) but also the orientation to find
the diffraction maxima. While q
changing the electron wavelength is easy to obtain by changing
the acceleration potential (v = 2eVa /m and Eq.1), varying the orientation of the crystal inside
an evacuated tube is difficult. In addition single crystals are difficult to obtain.
An easily realizable modification for a diffraction experiment is shining an electron beam of
Theta
carbon target
electron beam
counts
Figure 4: Constructive interference of an electron beam though a polycrystal.
given energy into a polycrystal. A polycrystal is a collection of micro-crystals (very small crystals) where each crystal is randomly oriented.
Most of the micro-crystals will not satisfy Eq. 2 for any of their atomic planes, but some are oriented in a way to allow constructive interference. Since the whole setup is symmetric according
to the beam axis, the diffracted electrons will produce a conical shell of constructive interference
for each atomic plane and each order (n). The most important atomic planes thereby will dominate the diffraction picture, which (using the apparatus described below) will show concentric
rings, one for each atomic plane and order.
In effect, using a polycrystal makes it unnecessary to vary the orientation, as all plane orientations
already exist among the micro-crystals. A typical pattern produced is shown in fig. 4.
Apparatus
An electron diffraction tube consisting of an electron gun, a carbon target, and a luminescent
screen is used for this experiment (see Fig. 5). With this special vacuum tube electrons are
produced and accelerated, and their diffraction on a carbon layer (polycrystal) can be studied.
Electron Gun
The heated cathode and anode make up the electron gun. Electrons are produced by heating a filament that is located inside an oxide-coated metal ”can” called the cathode, electrons are ejected
by thermionic emission from this heated piece of metal (see fig. 5).
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Once emitted, the electrons are accelerated by two pairs of anode rings with adjustable acceleration potential Va (2000 − 5000V ) (kilovolt power supply between the cathode (−) and anode
(+)) and form an electron beam.
Each electron in the beam has a kinetic energy equal to the accelerating electric potential (eVa )
times the electron charge (e). In the non-relativistic limit (eVa << mc2 ), the electrons acquire a
kinetic energy of
1 2
mv = eVa .
(3)
2
Carbon Target
As the electron beam passes through the anode, it meets a very thin mesh containing vaporized
graphite (carbon). The carbon suspension acts as the polycrystal described above. Diffraction
caused by the mesh can be neglected.
Luminescent screen
Leaving the target, diffracted electrons travel a certain distance (L) and strike a phosphor screen.
The beam now appears as concentric rings around a bright center. This pattern can be visualized
as a set of one-dimensional diffraction patterns of bright spots rotated about the axis of the
electron beam (see fig. 4).
As the screen is located inside a sphere, it is necessary to make geometric corrections to data. As
value for the distance L between the carbon target and the screen take L = 148.5 ± 0.5mm and
the radius of the tube R is R = 65.0 ± 0.2mm. A sketch of the tube is shown in fig.6.
Procedure
Equipment Needed
• electron diffraction tube
• HV-power supply
• flexible ruler
• electrometer
• cables
Equipment Notes
• A darkened room will help you see the diffraction patterns.
• Use high voltage cable between the anode (side plug on diffraction tube) and the electrometer.
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• Make sure power is turned off before handling connections.
anode
heater
6.3 V ac
carbontarget
cathode
A
screen
-
+
0 .. 5000 V dc
Figure 5: Electron Diffraction Experimental Diagram.
Connections
1. Connect the electron diffraction tube to the power supply as shown in fig. 5.
2. Connect the electro-meter with the cathode and the negative port of the power supply.
Power Up
• Set the electro-meter at the sensitivity of 600µA.
• Make sure that the high voltage supply is set to minimum. Switch on the power and
wait one minute for the cathode to reach a stable temperature.
Controlling the Current
The normal operation current should be kept below 150µA during measurements at voltages
below 4.0kV . When it is necessary to go above this range, measurements should be made quickly
and high voltage should be decreased as soon as results are obtained.
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—————————————————————————————–
Caution: DO NOT EXCEED 200µA of current between cathode and anode! If you do, turn
off high voltage immediately!
Watch the carbon target! If it starts to glow turn of high voltage immediately!
—————————————————————————————–
Slowly increase the high voltage until you observe a pattern (≈ 2.2kV ). Monitor the current and
take precautions when exceeding the normal (150µA) range.
Measurements
Take readings of accelerating voltage vs. ring diameter for each observed ring. Measure the
diameter arc length (diameter bent around the screen) using a flexible, transparent ruler (provided
by lab).
Accelerating Voltage vs. Ring Diameter
• Start at minimum voltage and record diameters and voltages at 0.1kV increments for each
ring.
At voltages above 4kV , take each reading quickly and decrease the voltage to zero after the
measurements.
Geometric Analysis
• The distance L from the carbon target to the screen is 148.5 ± 0.5mm. The screen has a
radius of curvature of 65mm.
Remember: For small angles one often can assume sin(θ) ≈ θ ( θ in radian!).
Analysis
1. Substitute Eq. 3 into the de Broglie relation Eq. 1 and solve for the de Broglie wavelength λ.
Show that the result gives m (meter) as unit.
What is the wavelength of an electron accelerated with 2(3, 4)kV ?
2. Use the diffraction Eq. 2, the result obtained in the previous question and geometry (see
fig.6) to find a relation between plane separation d, ring diameter D and accelerating voltage Va (assume L = l).
√
3. Show that if you assume sin Φ = Φ, the ring diameter D = 2r is proportional to 1/ Va .
Find the constant of proportionality. What error do you expect do to this assumption.
√
4. Make a plot 1/ Va versus D = 2r for each ring. Does a straight line fit your data? If so,
use the values of the slope to calculate the plane separation d that corresponds to each ring.
Discuss the errors of your raw data. Sketch them in your graph (y- and x-axis).
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L
l
r
screen
l´
carbon target
r
a
as ur ed di ameter
me
R
F
curvature R=65 mm
1.5 mm
Figure 6: Geometric Layout of the Electron Diffraction Tube
5. For each ring, calculate the plane separation d for each Va . Use a equation similar to
Analysis 3. Find the mean and standard deviation. In addition discuss the systematic error.
How does the result compare to the previous deduced value (Analysis 4)? Do both values
agree within error limits?
6. Using the result of Analysis 3 and using as plane separation d1 = 2.13 · 10−10 m and
d2 = 1.23 · 10−10 m, calculate the value of the plank constant h for all measurements.
Combine all data to a final value including statistic and systematic error.
Questions
1. What is the total direction change Φ in terms of Θ? Which order of diffraction (i.e., value
of n) should be visible at smallest Φ. Show using a numeric example that for ’small’ angles
one can assume sin Φ ∼ Φ.
2. If the crystal structure of carbon is cubic, what would be the spacing of the atoms? (Assume
a density of 2.0g/cm3 and note that 12 grams of carbon contain Avogadro’s number of
atoms). What acceleration potential Va (approximately) should be applied to the electrons
to produce a first-order (n = 1) ring at 200 .
3. In principle an analysis of the ring diameters can tell what the crystal structure is like hexagonal or cubic, and what the atomic spacing is in the carbon. Fig.2 shows the complexity of this analysis. One must be satisfied with obtaining the correct order of magnitude
for the spacing and showing that the electrons have a wavelength λ = constant/p; where
the constant is something close to h, Planck’s constant. Explain how the data shows this to
be true. If the energy of a photon is E = hf , why is its momentum p = h/λ?
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4. If the kinetic energy of a particle exceeds 0.1mc2 , its behavior must be described by relativistic equations. What acceleration potential Va would be needed to produce relativistic
electrons? How does this potential compare to the maximum potential (≈ 5kV )?
5. In a diffraction experiment with ordinary light (wavelengths of order 5000Å or 5×10−7 m),
a grating is made by scratching thin lines on glass. Why doesn’t this work well for electrons
or x-rays? Why is a typical crystal the best choice (see Question 2)?
6. Consider the mesh in the experiment. Suggest several methods that the tube designers
could have used to ensure that diffraction in the mesh is negligible (i.e. electrons it diffracts
produce a faint or invisible pattern).
7. Do you have the impression to have performed a ’high-precision-measurement’? How
does your answer to this question match to the overall results.
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