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Eq. (1) |
In this part of the lab you will demonstrate that electrons diffract from an ordered crystal as if they were waves. You will also verify deBroglie's relations for the energy dependence of the electron's wavelength and measure the spacing between carbon atoms in graphite.
Figure 1 shows the electrical connections and a schematic view of the electron gun and tube. Electrons are emitted from the cathode when it is heated by the 0-7V heater supply. The electrons are focused by the focus element and accelerated to the graphite target by the high voltage supply. After passing through the target they have sufficient energy to light up the phosphor screen when they hit it. Since the electrons are waves, they have a wavelength λ given by the deBroglie relation: λ = h/p, where h is Planck's constant and p is the electron momentum. For electrons accelerated through a potential eV = p2/2m their wavelength is given by:
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Eq. (1) |
where, m and e are the mass and charge of the electron, respectively. Roughly, for a 150eV energy, the electron's deBroglie wavelength is ~1 Å. The graphite sample is composed of graphite crystals that have a regular or periodic arrangement of carbon atoms. Planes of these atoms, separated by a distance d, act like a diffraction grating (see Fig. 2). If the electrons behave like waves they will diffract only into certain angles given by Bragg's Law.
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Eq. (2) |
Figure 1: Schematic drawing of the electron diffraction system showing electrical connections.
Figure 2: Bragg scattering geometry
Graphite has a hexagonal structure with neighboring atoms bonded to each other by a spacing ao as shown in Fig. 3a. The two smallest d spacings are shown. Because of the 6-fold symmetry of the hexagonal structure, we would expect to see 6 spots on the screen for d1 and 6 for d2. Each of the six spots for each d spacing would have the same θ but would be azimuthally rotated about the direct beam, each separated by 60°. However, the hexagonal structure (see model in the lab) is made of planes of hexagonal rings that are only loosely bonded to each other. When graphite is formed, the crystal structure is disordered because of the weak inter-plane bonding. Each plane is randomly oriented to the other. This type of material is called a "powdered crystal" containing small crystallites randomly oriented. The resulting diffraction pattern from this powdered is made up of a large set of 6 diffraction beams for each d spacing; with each set randomly oriented azimuthally around the direct beam. Instead of seeing 6 spots per d spacing you will see a single ring, or for two d spacings, you will see two rings.
Figure 3: Hexagonal (a) and Cubic (b) lattices. a0 is the bond distance between atoms.
Personnel
The high voltage supply is potentially lethal. The mechanical and electrical integrity of this part of the circuit should be checked prior to use and care should be exercised to avoid inadvertently partially disconnecting leads while making adjustments or readings in the darkened room. It is intended that all dangerous leads are secured with tape - check this. An earthed ground should run to the grounding plug of the high voltage supply.
The glass diffraction tube will implode if unduly stressed - so be careful not to hit it.
Apparatus
Excessive anode current, Ie on the diagram, will ruin the tube either by limiting the lifetime of the cathode or by burning a hole in the target. The maximum value is 0.2 mA, but there should be no reason to go above 0.1 mA. Monitor this current whenever making changes in various voltages. As the experiment progresses you will acquire a "feeling" for the appropriate brightness of the screen; an increase in brightness signals an increase in anode current.
In most of the experiments in this course, you are encouraged to disconnect old wiring you may find left from a previous use and rewire yourself. In this experiment, a mix up in the wiring can lead to conditions dangerous either to personnel or apparatus. Hence, check the wiring before use, even though the apparatus is prewired.
Part A: Operational procedures
Before turning on any power supplies or plugging in any equipment, read the following instruction for Part IA completely through. This will prevent you from damaging the equipment and also give you hints on how to achieve the best diffraction patterns.
Start-Up procedure
The Anode current, Ie, must never exceed 0.2 mA; monitor this carefully whenever you are changing something. The anode current will tend to rise with increasing high voltage. To control the anode current, adjust the focus voltage; also, you can reduce the heater voltage. For high voltages in excess of 4000V you probably should reduce the heater voltage to 4 v while running the focus voltage up to its maximum value of 50 V. You will probably find the diffraction pattern most distinct with anode current much less than the maximum allowed, perhaps 0.01 to 0.04 mA (observe with room lights out). Maximum high voltage is 5000 V.
Because of the long time constant in the high voltage supply, it is best to take data by starting at low voltages. The time constant for increasing the voltage is much smaller than when you decrease it. Starting at lower voltages and then increasing will make it less likely that the voltage will change during your measurements.
Once you turn on the system, you should see an undeflected beam and two diffraction rings on the fluorescent screen. The observed pattern is characteristic of a "powder crystal" sample as described in the next section.
SPECIAL NOTE: The electron beam can have enough energy to heat the graphite target. Monitor the appearance of the carbon target area (with the lights out). It should not glow! If a faint red glow should be detected, reduce high voltage and be sure the anode current is well below 0.2 mA. Consult with the instructor to make sure you understand if you are operating the tube within the correct limits.
Turn-Off Procedure
Part B: Experimental procedures
1. You will measure the diameter of the diffraction rings as a function of accelerating voltage. The ring diameter is related to the Bragg scattering angle as described below. You should be able to say whether the outer ring corresponds to the larger or smaller d spacing (hint: use Bragg's Law). Cover the range from 1500 V to ≈ 5000 V, in about 10-20 voltage steps. It is probably best to start with 2000 V and obtain the lower voltage data after you gain some familiarity with the apparatus; the lower voltage patterns are more difficult to see. At each voltage you will observe two concentric circles, in addition to a central bright region. Measure the arc lengths, 2s, of each ring (see Fig. 4) as a function of accelerating voltage. Also record heater voltage, focus voltage, and anode current. Read the procedure sheet, and observe precautions to avoid excess anode current. The patterns are easier to see if they are not too bright so observations need to be made in a darkened room. BE CAREFUL OF THE HIGH VOLTAGE.
Figure 4: Scattering geometry. Note that the dome radius R is not equal to half the distance from the target to the screen l.
In order to determine the scattering angle 2θ, from the diffraction ring arc lengths, s, you must first measure the target-to-screen distance, l, and the dome radius, R. Figure 4 defines these parameters. Use the calipers to measure l and R being careful not to hit and break the glass. Also note the error in making these measurements since it effects overall errors in the rest of the experiment.
The angle α subtended by the diffraction rings is given by:
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Eq. (3) |
From the geometry of Fig. 3 the scattering angle can be related to α, l and R.
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Eq. (4) |
Using the trig identity sin(α - 2θ) = sin α cos 2θ - cos α sin 2θ, the scattering angle is given by:
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Eq. (5) |
2. Prepare a table which lists: accelerating voltage, electron velocity (v), s1, α1, θ1 for the smaller diameter ring; s2, α2, θ2 for the larger diameter ring. Do not put this table in the main body of the lab report. Instead put it in an appendix. To find v use mv2/2 = eV.
3. From the deBroglie relation Eq. (1), and Bragg’s law Eq. (2), we can derive a relationship between the scattering angle, d and V. We observe rings for two values of the spacing between atomic planes, d1 and d2, with n = 1 for both cases. θ is eliminated between the two equations to give:
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Eq. (6) |
For both ring, plot sin θ versus 1/√V . If the electron behaves according to deBroglies's relation, these plots should be a straight line. Draw the best straight lines through each set of points. If you prefer you can make a least squares fit to the data. In either case you may choose to de-emphasize data at the lower voltages if you find that the plot is not linear there. (In the case of a least squares fit, that implies assignment of varying weights to different data points.)
4. The slope of the straight lines drawn in part (3) are inversely related to the lattice spacings from Eq. (5). Use this result to determine d1, d2, and a0, for the carbon target. From Fig. 3, d1 = 3a0 / 2 and d2 = a0√3/2 (The result for d2 should be, very roughly, 1 Å = 10-10 m.). Compare your answer for the atomic separation with the published value of graphite. (Do not confuse graphite with diamond even though they are both made of carbon. They have different values for the atomic separation.)
5. The ratio of the d spacings allows you to judge whether the graphite lattice is hexagonal or cubic since d1/d2 is √3 or √2 for hexagonal or cubic lattices, respectively. For each accelerating voltage determine the ratio of the two d’s and state your judgement of the average value for this ratio. Interpret the result in terms of consistency of a cubic vis-a-vis an hexagonal arrangement of atoms in the target.
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