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PC1144 Physics IV
Electron Diffraction
1
Purpose
• Demonstrate the wave behaviour of electrons.
• Determine experimental values for the interatomic spacing of graphite via electron
diffraction patterns.
2
Equipment
• Electron diffraction tube with universal stand
• Power supplies
• Ammeter
• Vernier caliper
3
Theory
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, λ, of a particle, such as
an electron, is related to its momentum, p, by the same relationship as for a photon:
λ=
h
p
(1)
where h is Planck’s constant. The first experimental evidence of the existence of matter waves
was obtained by Davisson and Germer in 1927.
The wave properties of electrons are illustrated in this experiment by the interference
when they are scattered from successive planes of atoms in a target composed of graphite
microcrystals. The spacing between successive planes can be deduced from the interference
pattern.
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Electron Diffraction
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Figure 1: Diffraction from a single crystal.
When the beam of electrons strikes a family of parallel crystal atomic planes, each plane
will reflect part of the waves. If the reflected waves from O and Q, as indicated in Figure 1,
are to be in phase (interfere constructively), the path difference
P Q + QR = 2d sin θ
(2)
must equal an integral number of wavelengths, or
2d sin θ = nλ
(n = 1, 2, 3, . . .)
(3)
where d is the separation between two adjacent planes. Equation (3) is known as Bragg’s law
for constructive interference. For any incident angle other than those satisfying equation (3),
there is no reflected beam because of destructive interference.
Figure 2: Diffraction from a large number of microcrystals.
In the graphite target, there are very many perfect microcrystals randomly oriented to one
another. Therefore the strongly emerging beam will be of a conical shape of half-angle 2θ as
shown in Figure 2. If this beam falls on a phosphor-coated screen, rings of light will then be
formed. Now, the condition for constructive interference becomes
ψ
2d sin
2
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!
= nλ
(4)
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Electron Diffraction
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which for small angles and first order diffraction (n = 1) becomes
d
D
=λ
2L
(5)
where D is the diameter of the diffraction ring and L is the distance from the graphite target
to the luminescent screen.
Figure 3: Schematic diagram of the electron diffraction tube.
This experiment uses Thomson’s method for sending electrons through a thin film of
graphite target to investigate the resulting ring diffraction pattern with the aid of Bragg’s law.
A schematic diagram for the apparatus is shown in Figure 3. Electrons emitted by thermionic
emission from a heated filament (the cathode) are accelerated towards the graphite target
(the anode) by a potential difference, V . Their kinetic energy, K, on reaching the target is
equal to their loss of potential energy:
K=
p2
= eV
2m
(6)
where p, m and e are, respectively, the momentum, mass and charge of the electron. Combining equations (1) and (6), the wavelength of the electrons is given by
λ= √
h
2meV
(7)
With equation (7), equation (5) becomes
D
d
h
1
√
=√
2L
2me V
(8)
For the apparatus used in this experiment, the distance from the graphite target to the
luminescent screen L equals to 0.140 m .
Graphite possesses a hexagonal sheet structure as shown in Figure 4. Each layer has
strong internal bonds but weak bonds between layers which stack into crystals. The properties
parallel and perpendicular to the sheets are markedly different. The two sets of parallel atomic
planes responsible for our observation are all perpendicular to the sheets as shown in Figure 5.
The interatomic distance is 0.142 nm and all bond angles are 120◦ .
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Electron Diffraction
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Figure 5: Parallel atomic planes
responsible for the observations.
Figure 4: Atomic structure of graphite.
4
Experimental Procedure
Warning: High voltage is dangerous. The diffraction tube is delicate and expensive. You
must take all the necessary precautions and follow the instructions closely. If for any reason
the tube must be moved or the wires disconnected, call your demonstrator.
P1. 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.
P2. Switch on the filament voltage for at least one minute to allow the cathode temperature
to stabilize before applying anode voltage.
P3. Increase the anode voltage VA gradually. Monitor the anode current to ensure that it
NEVER exceeds 0.2 mA. A bright central spot and two diffraction rings should appear
on the screen at VA ≈ 2.5 kV. You may apply the external bias (0–50 V d. c.) to regulate
the anode current. The external bias also provides a way to improve the focusing which
sharpens the diffraction pattern for better observation.
Note: The anode current should never be allowed to exceed 0.2 mA throughout the
whole experiment. The graphitized carbon target bombarded by the electron beam is
only a few molecular layers in thickness which can be punctured by current overload.
P4. Measure the diameter of the inner diffraction ring at THREE different positions along
the ring. Record the results as dinner in Data Table 1.
P5. Measure the diameter of the outer diffraction ring at THREE different positions along
the ring. Record the results as douter in Data Table 1.
P6. With the diffraction rings sharply defined and the anode current under control by carefully adjusting the external bias voltage, repeat steps P4–P5 SIX times for different
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Electron Diffraction
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anode voltage VA . You may need to re-adjust the external dias as necessary. Record
your values in Data Table 1.
Note: The anode voltage should not exceed 4.5 kV.
P7. At the end of the experiment, reduce the anode voltage to zero first before switching off
the filament voltage and other voltages.
5
Data Processing
D1. Perform a suitable linear least squares fit to the data in Data Table 1 and 2 respectively.
Determine the slopes and intercepts with the corresponding uncertainties of the least
squares fit to the data.
D2. Plot suitable graphs as well as the best fitted lines obtained above.
D3. Determine your best experimental values for dinner and douter respectively with the corresponding uncertainties.
6
Questions
Q1. Calculate the spacing between adjacent planes responsible for our observation dinner and
douter as indicated in Figure 5. Use percentage discrepancy to compare your experimental
values for dinner and douter with their theoretical values.
Hint: The percentage discrepancy is defined as
Percentage discrepancy =
|Experimental value − Theoretical value|
× 100%
Theoretical value
Q2. Justify with clear explanation that special relativistic effect can be ignored in this experiment.
Q3. Why do we use atomic layers as our diffraction grating for electron diffraction and not
the usual diffraction grating to diffract our electron beam?
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Department of Physics
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