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EXPERIMENT 19 Electron Diffraction the Purpose of This Experiment Is To EXPERIMENT 19 Electron Diffraction The purpose of this experiment is to show that electrons can be diffracted by a crystal and hence exhibit wave behavior. Theory In l924, de Broglie proposed that particles could exhibit wave behavior in the same fashion that light was described as having both particle and wave behavior. He hypothesized that the wavelength associated with a moving particle of velocity v is given by l = h/p = h/mv (1) where p = mv is the momentum of the particle and h is Planck's constant, namely 6.626 x 10-34 J-s. If an electron of mass m = 9.11 x 10-31 kg is accelerated through a potential difference V, then its kinetic energy is given by 1 mv2 = eV (2) 2 where e = 1.602 x 10-19 C is the charge of the electron. Combining equations (1) and (2), the de Broglie wavelength for an electron can be written as l = h/ 2meV (3) Note: It has been assumed that the electron velocity v is small compared to the speed of light so that a relativistic correction is not necessary. In order to test de Broglie's hypothesis for the wave behavior of electrons, a diffraction-type experiment is conducted where a beam of high-speed electrons is allowed to strike a crystalline material. The observed diffraction pattern serves as evidence for the wave nature of the electrons, while measurements of the pattern can be used to verify the wavelength . 19-2 Electron Diffraction by Graphite A calculation using equation (3) predicts that electrons accelerated through a potential difference of 4000 volts would have a wavelength of about 0.2 Angstroms. If an ordinary ruled grating is used for diffracting these electrons, the condition for 1st order diffraction would be l = d sinq (4) where d is the grating spacing and is the angle of diffraction. However, using the best man-made grating (typically 40,000 lines/inch), the angle would too small (~0.002 degrees) for diffraction to be observable. d11 d10 x (21) (12) b b (11) (10) plane Figure 1 A model showing the hexagonal crystalline structure of graphite. It was Max von Laue, in connection with X-ray studies, who suggested that crystals could serve as a "grating" where d is the spacing between parallel rows of atoms. In this experiment, graphite is used as a suitable crystalline grating for the diffraction of electrons. The structure of graphite consists of layers of carbon atoms which are linked into arrays of hexagons as shown in figure (1). Note the various sets of parallel lines that can be drawn in different directions through the carbon atoms in figure (1). The lines denoted by (10), (01), (11) have a spacing equivalent to d10. The lines denoted by (11), (12), (21) have a spacing r r equivalent to d11. These spacings have been defined in terms of the unit vectors a and b where a = b in the case of the hexagonal structure. The indices (10), (11), etc. are known as Miller indices and are explained in the Bragg Diffraction experiment. 19-3 In this experiment a graphite sample is mounted inside an evacuated Electron Diffraction tube as shown in figure (2). A filament heater "boils" electrons off the cathode, and then the electrons are accelerated by a high voltage towards the anode. The electrons pass through the graphite and are diffracted into a circular diffraction pattern which is visible on the luminescent screen of the tube. There are two diffraction patterns consisting of concentric rings visible on the screen. The inner and outer rings corresponds to diffraction patterns from the d10 and d11 row of atoms. Figure 2 Schematic showing the wiring of the equipment. The pattern consists of a ring instead of diffraction "spots" which normally occur for a single plane grating. This is due to the graphite sample consisting of many randomly oriented layers of atoms, so that the spots form into a ring. If the diameter D of a diffraction ring is measured, the diffraction angle can be found using = D/2L (5) where L is the distance from the graphite sample to the screen. The wavelength of the electrons can be determined using the 1st order diffraction equation given by æ D dD l = d sinq = d sin ö » (6) è 2Lø 2L 19-4 where sin( ) for small angles. Equating from equation (6) with the de Broglie relation from equation (3), we have for small angles 2Lh D = (7) d 2meV Procedure 1. Read the instruction sheet at the end of the lab to understand how to use the FLUKE high voltage probe. Note that a FLUKE 8010A multimeter has an input of 10 MW when used as a voltmeter. The voltage output from the probe is divided by 1000. 2. Wire the apparatus as shown in figure (2). The filament is heated using a 6.3 volt transformer. The electrons are accelerated through a potential difference applied between the anode and cathode. 3. Starting with an accelerating voltage of 2250 volts, measure the curved diameter of the inner and outer diffraction rings. Repeat the measurements increasing the accelerating voltage in 250 volt steps to a maximum of 5000 volts. The anode current must be kept below .4 mA. The distance from the graphite sample to the screen L is 13.5 ± .1 cm. Analysis 1. For both diffraction rings, plot the ring diameter D vs the inverse square root of the voltage 1 on the same graph. V 2. Find the atomic spacing d and the uncertainty d for both the (10) and (11) family of planes from the slopes of your linear graphs. What is your experimental value for the ratio d10/d11? 3. Using trigonometry, calculate spacing d between the rows of atoms for the (10) and (11) family of planes in a hexagonal structure as shown in figure (1). Express the spacing in terms of the distance x between the nearest neighbor atoms as shown in figure (1). 4. Calculate the distance, x, between the nearest neighbor atoms using the graphical results for d for both family of planes. Questions (1) Calculate the distance between the rows of atoms for the (10) and for the (11) family of planes in a simple square lattice. (2) Calculate the theoretical ratio d10/d11 for hexagonal and square lattice. Compare these ratios to your experimental ratio. What conclusions can you make regarding the structure of graphite. 19-5 (3) Is equation (3) valid for a 10 MeV electron beam. If not, why not? (4) Graphite consists of layers of carbon atoms with an average separation between layers of 3.40 Å. Assume that each layer has the hexagonal structure shown in figure (1). Use the following information to compute the nearest neighbor distance x between atoms within each carbon layer. A convenient choice of unit cell is the equilateral triangle shown in figure (3) below. N = Avogadro's Number = 6.02 x 1023 atoms/mole W = Atomic weight of carbon = 12 g/mole r = Density of carbon = 2.25 g/cm3 Figure 3 Hexagonal structure. 19-6 Rm = Voltmeter input impedance in MW FLUKE INSTRUCTION SHEET (>10 MW) MODEL 80K-40 HIGH VOLTAGE PROBE Example: If Rm = 100 MW. INTRODUCTION 100 x 10 1000 Rs = = = 11.1 MW The Model 80K40 is a high voltage 100 - 10 90 accessory probe designed to extend the voltage measuring capability of an ac/dc voltmeter up to b. Use the following formula to calculate a 40,000 volts. In essence the probe is a precision correction factor: 1000:1 voltage divider formed by two matched metal- film resistors. The unusually high input impedance 1.11 + Rm offered by these resistors minimizes circuit loading Cf = and, thereby, optimizes measurement accuracy. A 1.11 x Rm special plastic body houses the divider and provides where: Cf = Correction factor (multiplier the user with isolation and protection from the voltage for meter reading) being measured. Rm = Voltmeter input impedance in MW SPECIFICATIONS Example: If Rm = 1MW, The 80K40 will achieve rated accuracy when used with a 0.25% voltmeter (ac or dc) having an Cf = 1.11 + 1 = 2.11 = 1.901 input impedance of 10 MW±10%. Specifications for 1.11 x 1 1.11 the probe are as follows: Voltage Range: 1 kV to 40 kV dc or peak ac, 28 Therefore: A meter reading of 0.526 volts represents kV rms ac an input of: 0.526 x 1.901 = 1 or 1 kV Circuit Loading Input Resistance: 1000 MW Division Ratio: 1000:1 (1000X attenuator) Accuracy DC: The 80K40 represents a 1000 MW load to Overall Accuracy: 20 kV to 30 kV ± 2% the circuit being measured, or 1 mA per 1 kV. Table (calibrated 1% at 25 kV) 1 shows the circuit loading and input/output Upper Limit: Changes linearly from 2% at characteristics of the probe over its measurement 30 kV to 4% at 40 kV range. Lower Limit: Changes linearly from 2% at 20 kV to 4% at 1 kV Table 1 Accuracy AC: ± 5% at 60 Hz 80K40 Circuit Loading and Input/Output Characteristics MEASUREMENT CONSIDER ATIONS Input Voltage Loading Current Output Voltage Before attempting to use the 80K40, the ________________________________________ following paragraphs should be read and understood. 10V 10 nA 10mV Particular attention should be given to Operator 100V 100 nA 100mV Safety. 1 kV 1mA 1V Voltmeter Compatibility 10 kV 20 mA 10V The 80K40 is compatible with any ac or dc 20 kV 20 mA 20V voltmeter that has an input impedance of 10 MW ± 30 kV 30 mA 30V 10%.
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