ELECTRON DIFFRACTION Electrons : wave or quanta? Measurement of wavelength an momentum of electrons. Introuction Electrons isplay both wave an particle properties. What is the relationship between the electrons wavelength an their momentum? Theory Louis e Broglie evelope the iea that electrons exhibit both particle an wave properties. He propose that, analogous to photons, the wavelength λ of the electron wave is given by λ = h p = h mv (1) where h is Planck s constant an p is the electron momentum which is given by p = mv in the non-relativistic limit. In orer to observe the wave nature of electrons one irects a beam of electrons through a thin layer of carbon an analyzes the angles at which electrons emerge. If electrons behave like particles, the istribution of electrons varies continuously as a function of angle as inicate in Fig. 1. This istribution will vary only slightly with changes in electron energy. Theta carbon target electron beam counts Figure 1: Electrons as Particles Continuous istribution of electrons as a function of angle. The picture is ifferent if electrons behave like waves. Analogue to the iffraction of x-rays (photons with wavelengths of a few Å) by crystals, the crystal may be consiere to be a 3 imensional iffraction grating for electron waves. The iffraction maxima are euce applying the concept of reflecting planes in the crystal. The locations of atoms in the two-imensional 1
Incoming Beam Cubic Hexagonal Figure 2: Atomic Planes of Cubic an Hexagonal Crystals. Two-imensional slices of cubic an hexagonal crystals. Different angles of refraction are inicate in the cubic crystal structure. slices of cubic an hexagonal crystals are shown in Fig. 2. The entire crystal can be visualize as a stack of slices, the various lines representing planes intersecting the actual crystal. Θ sin( Θ) Θ Θ Θ Figure 3: Electron Waves refracte from Atomic Planes. Atomic planes are treate as reflecting surfaces an allow waves from successive planes to interfere with one another. If the ifference in path length for waves refracte 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 = 2 sin Θ (2) hols. For each plane there will be a series of constructive interference, but the intensity epens on the 2
specific geometry of a crystal. In orer to measure iffraction angles as a function of wavelength for a single crystal, one nees to vary not only the wavelength (λ) but also the orientation to fin the iffraction maxima. While changing the electron wavelength is easy to obtain by changing the acceleration potential (v = 2eV a /m an Eq.1), varying the orientation of the crystal insie an evacuate tube is ifficult. In aition single crystals are ifficult to obtain. An easily realizable moification for a iffraction 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 ranomly oriente. Most of the micro-crystals will not satisfy Eq. 2 for any of their atomic planes, but some are oriente in a way to allow constructive interference. Since the whole setup is symmetric accoring to the beam axis, the iffracte electrons will prouce a conical shell of constructive interference for each atomic plane an each orer (n). The most important atomic planes thereby will ominate the iffraction picture, which (using the apparatus escribe below) will show concentric rings, one for each atomic plane an orer. In effect, using a polycrystal makes it unnecessary to vary the orientation, as all plane orientations alreay exist among the micro-crystals. A typical pattern prouce is shown in fig. 4. Apparatus An electron iffraction tube consisting of an electron gun, a carbon target, an a luminescent screen is use for this experiment (see Fig. 5). With this special vacuum tube electrons are prouce an accelerate, an their iffraction on a carbon layer (polycrystal) can be stuie. Electron Gun The heate cathoe an anoe make up the electron gun. Electrons are prouce by heating a filament that is locate insie an oxie-coate metal can calle the cathoe, electrons are ejecte by thermionic emission from this heate piece of metal (see fig. 5). 3
Once emitte, the electrons are accelerate by two pairs of anoe rings with ajustable acceleration potential V a (2000 5000V ) (kilovolt power supply between the cathoe ( ) an anoe (+)) an form an electron beam. Each electron in the beam has a kinetic energy equal to the accelerating electric potential (ev a ) times the electron charge (e). In the non-relativistic limit (ev a << mc 2 ), the electrons acquire a kinetic energy of 1 2 mv2 = ev a. (3) Carbon Target As the electron beam passes through the anoe, it meets a very thin mesh containing vaporize graphite (carbon). The carbon suspension acts as the polycrystal escribe above. Diffraction cause by the mesh can be neglecte. Luminescent screen Leaving the target, iffracte electrons travel a certain istance (L) an strike a phosphor screen. The beam now appears as concentric rings aroun a bright center. This pattern can be visualize as a set of one-imensional iffraction patterns of bright spots rotate about the axis of the electron beam (see fig. 4). As the screen is locate insie a sphere, it is necessary to make geometric corrections to ata. As value for the istance L between the carbon target an the screen take L = 148.5 ± 0.5mm an the raius of the tube R is R = 65.0 ± 0.2mm. A sketch of the tube is shown in fig.6. Proceure Equipment Neee electron iffraction tube HV-power supply flexible ruler electrometer cables Equipment Notes A arkene room will help you see the iffraction patterns. Use high voltage cable between the anoe (sie plug on iffraction tube) an the electrometer. 4
Make sure power is turne off before hanling connections. heater anoe 6.3 V ac carbontarget cathoe A screen - + 0.. 5000 V c Figure 5: Electron Diffraction Experimental Diagram. Connections 1. Connect the electron iffraction tube to the power supply as shown in fig. 5. 2. Connect the electro-meter with the cathoe an 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 an wait one minute for the cathoe to reach a stable temperature. Controlling the Current The normal operation current shoul be kept below 150µA uring measurements at voltages below 4.0kV. When it is necessary to go above this range, measurements shoul be mae quickly an high voltage shoul be ecrease as soon as results are obtaine. 5
Caution: DO NOT EXCEED 200µA of current between cathoe an anoe! If you o, turn off high voltage immeiately! Watch the carbon target! If it starts to glow turn of high voltage immeiately! Slowly increase the high voltage until you observe a pattern ( 2.2kV ). Monitor the current an take precautions when exceeing the normal (150µA) range. Measurements Take reaings of accelerating voltage vs. ring iameter for each observe ring. Measure the iameter arc length (iameter bent aroun the screen) using a flexible, transparent ruler (provie by lab). Accelerating Voltage vs. Ring Diameter Start at minimum voltage an recor iameters an voltages at 0.1kV increments for each ring. At voltages above 4kV, take each reaing quickly an ecrease the voltage to zero after the measurements. Geometric Analysis The istance L from the carbon target to the screen is 148.5 ± 0.5mm. The screen has a raius of curvature of 65mm. Remember: For small angles one often can assume sin(θ) θ ( θ in raian!). Analysis 1. Substitute Eq. 3 into the e Broglie relation Eq. 1 an solve for the e Broglie wavelength λ. Show that the result gives m (meter) as unit. What is the wavelength of an electron accelerate with 2(3, 4)kV? 2. Use the iffraction Eq. 2, the result obtaine in the previous question an geometry (see fig.6) to fin a relation between plane separation, ring iameter D an accelerating voltage V a (assume L = l). 3. Show that if you assume sin Φ = Φ, the ring iameter D = 2r is proportional to 1/ V a. Fin the constant of proportionality. What error o you expect o to this assumption. 4. Make a plot 1/ V a versus D = 2r for each ring. Does a straight line fit your ata? If so, use the values of the slope to calculate the plane separation that correspons to each ring. Discuss the errors of your raw ata. Sketch them in your graph (y- an x-axis). 6
r r L l carbon target R l curvature R=65 mm screen measure iameter 1.5 mm Figure 6: Geometric Layout of the Electron Diffraction Tube 5. For each ring, calculate the plane separation for each V a. Use a equation similar to Analysis 3. Fin the mean an stanar eviation. In aition iscuss the systematic error. How oes the result compare to the previous euce value (Analysis 4)? Do both values agree within error limits? 6. Using the result of Analysis 3 an using as plane separation 1 = 2.13 10 10 m an 2 = 1.23 10 10 m, calculate the value of the plank constant h for all measurements. Combine all ata to a final value incluing statistic an systematic error. Questions 1. What is the total irection change Φ in terms of Θ? Which orer of iffraction (i.e., value of n) shoul 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 woul be the spacing of the atoms? (Assume a ensity of 2.0g/cm 3 an note that 12 grams of carbon contain Avogaro s number of atoms). What acceleration potential V a (approximately) shoul be applie to the electrons to prouce a first-orer (n = 1) ring at 20 0. 3. In principle an analysis of the ring iameters can tell what the crystal structure is like - hexagonal or cubic, an what the atomic spacing is in the carbon. Fig.2 shows the complexity of this analysis. One must be satisfie with obtaining the correct orer of magnitue for the spacing an showing that the electrons have a wavelength λ = constant/p; where the constant is something close to h, Planck s constant. Explain how the ata shows this to be true. If the energy of a photon is E = hf, why is its momentum p = h/λ? 7
4. If the kinetic energy of a particle excees 0.1mc 2, its behavior must be escribe by relativistic equations. What acceleration potential V a woul be neee to prouce relativistic electrons? How oes this potential compare to the maximum potential ( 5kV )? 5. In a iffraction experiment with orinary light (wavelengths of orer 5000Å or 5 10 7 m), a grating is mae by scratching thin lines on glass. Why oesn t this work well for electrons or x-rays? Why is a typical crystal the best choice (see Question 2)? 6. Consier the mesh in the experiment. Suggest several methos that the tube esigners coul have use to ensure that iffraction in the mesh is negligible (i.e. electrons it iffracts prouce a faint or invisible pattern). 7. Do you have the impression to have performe a high-precision-measurement? How oes your answer to this question match to the overall results. 8