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Electron Diffraction
1 Objectives
- Electrons as waves.
- Study and verification of the de Broglie hypothesis λ = h/p.
- Measurement of the spacing of diffracting planes in graphite.
2 Theory
In a bold and daring hypothesis in his 1924 doctoral dissertation Louis de Broglie reasoned that if electromagnetic radiation can be interpreted as both particles (Photoelectric Effect, Compton Scattering) and waves (diffraction), then perhaps the electron, which had tradi- tionally been interpreted as a particle, could also have a wave interpretation. De Broglie hypothesized that all particles have a wave behavior with a universal relationship between the wavelength and momentum given by λ = h/p. This expression is called the de Broglie relationship and the wavelength is called the de Broglie wavelength. The momentum in this relationship is the momentum that is conserved in collisions, i.e. the relativistic momentum. The de Broglie relationship holds for all particles. Note that it is identical to the one for photons (E = hν).
Diffraction phenomena represent clear evidence for wave properties. How can an electron be both wave and particle?
In this experiment you will investigate the diffraction of electrons passing through a thin layer of graphite (carbon), which acts as a diffraction grating. It was Max von Laue, who in 1912 suggested (in connection with x-ray studies) that the basic granularity of matter at the atomic level might provide a suitable grating. Lawrence Bragg, using the crystalline structure of NaCl, with a cubic unit cell, first calculated the inter-atomic spacings and showed them to be of the right order for x-rays.
Figure 1: Structure of Graphite
Figure 1 shows the hexagonal structure of graphite used in this experiment with the two characteristic spacings of 0.123 and 0.213 nm. These two spacings will produce two diffraction patterns.
3 Apparatus
Our electron diffraction tube, see Fig. 2, comprises an electron gun which emits a narrow, converging beam of electrons within an evacuated clear glass bulb on the front surface of which is deposited a luminescent screen. Across the exit aperture of the gun lies a micro- mesh nickel grid, onto which a very thin layer (only a few molecular layers!) of graphite has been deposited.
The electron beam penetrates through this graphite target to become diffracted into two rings corresponding to the separation of the carbon atoms of 0.123 and 0.213 nm. The diffraction pattern appears as rings due to the polycrystalline nature of graphite. The source of the electron beam is an indirectly-heated oxide-coated cathode.
Figure 3: Circuit Diagram
applying the HV (anode voltage VA).
6 Data Taking and Analysis
Diffraction patters will show as two concentric rings, related to the two characteristic spacings of Graphite. The rings are faint. Be sure the room is very dark. The rings are quite distinct at the full 5 kV, but get very hard to see as you go down in voltage. Try to go as low as 2. kV. For each experimental run, take at least 10 different data points measuring the diameter of the two rings with the calipers. And perform at least 4 experimental runs. You can adjust the position of the central spot in the fluorescent screen by using the little magnet on the neck of the tube. Make sure that the rings are centered on the fluorescent screen.
For your data analysis you will need to analyze the geometry of the light bulb. Obtain the relationship between the measured diameter with the extrapolated diameter D. This is necessary for correcting for the curvature (and thickness?) of the glass of the tube-face (see Figure 4). The length of the tube L in Figure 4 is taken as 13.0 ± 0.2 cm.
The kinetic energy and momentum of the electrons are related to the accelerating poten- tial. From Fig. 4 we also know that D = 2L tan θ, where D is the extrapolated diameter, which you need to obtain from your measured diameters. By using the de Broglie relation and the diffraction law for a crystalline lattice, show that (derive the equation) the lattice spacing d in graphite is related to the accelerating voltage VA approximately by
d =
4 πLℏc D
2 eVAmc^2
Figure 4: Curvature and glass thickness of the electron diffraction tube. The length L =
- 0 ± 0 .2 cm.
Plot D as a function of V (^) A− 1 /^2 for all your data. Include in your plot the estimated error in D as vertical error bars. The straight lines you should get verify the theory and substantiate de Broglie’s hypothesis.
- Calculate the two values for d (d 1 and d 2 ) from your data (no need to make the above approximation!) and their uncertainties using Eq. (1) and the relation between your measured diameters and D.
- As a second method obtain the spacings d 1 and d 2 from linear fits to Eq. (1).
Compare your two determinations of each characteristic spacing d 1 and d 2 using the methods above with the accepted values of 0.123 and 0.213 nm.
7 References
[1] Melissinos and Napolitano.
Any good Modern Physics textbook.
