The object of this experiment is to test the de Broglie relationship for matter waves,

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1 Experiment #58 Electron Diffraction References Most first year texts discuss optical diffraction from gratings, Bragg s law for x-rays and electrons and the de Broglie relation. There are many appropriate reference books in the Library. Introduction The object of this experiment is to test the de Broglie relationship for matter waves, λ = h p (1) where λ is the wavelength, h is Planck s constant and p is the particle momentum. The first experimental evidence that a wavelength can be associated with a particle was obtained by Davisson and Germer in 1926 when they studied the scattering of electrons from the surface of a nickel crystal. Neglecting relativistic effects it can be readily shown that the wavelength (λ) of an electron when accelerated by a potential V is given by λ = where C = nm V 1 2. (Verify this equation.) C V (2) A crystal or crystallite is made up of an ordered array atoms. When an electron beam passes through a crystal the electrons are scattered by the atoms, giving rise to strong diffracted beams in certain directions. There is a strong analogy between electron diffraction and the diffraction of light (principal maxima) from a grating; but with electron diffraction the grating is 3-dimensional. Bragg showed that both x-ray and electron diffraction can be considered in terms of reflection from atomic planes and that, for an incident beam of wavelength λ, a diffraction beam occurs if 2d sin θ = nλ (3) 1

2 where, as illustrated in Fig. 1, d is the separation between a set of atomic planes in the solid, θ is the angle between the incident beam direction and the atomic planes and n is an integer called the diffraction order. Figure 1: Edge-on view of lattice planes, illustrating Bragg s law Equation 3 is called Bragg s law. Different sets of atomic planes with different d- spacings in a crystal (d, d 2 d 3, d 4 in Fig. 1) can give rise to different diffraction angles, but only if equation 3 is satisfied. If d is known for a crystal, say from x-ray work, then the diffraction of electrons can be used to determine the electron wavelength. This is what is to be done in the first part of this experiment. If a single crystal of something is placed in an electron beam in an arbitrary orientation then it is highly unlikely that equation 3 will be satisfied because λ is fixed and it is improbable that any set of planes have the correct θ. If, instead, one uses polycrystalline material (an assembly of small crystallites such as common metal wire, sheets, doorknobs etc.) then some of these will have sets of planes at the correct θ by chance. Different crystallites all with the same θ will exist, but oriented differently by rotation about an axis parallel to the electron beam. These will give rise to a diffraction ring. (See Fig. 2). Different sets of planes with different d spacings will give rise to different rings. Note that electrons with energies of approximately 10 kv will only penetrate through very thin foils 10 5 cm thick or less; x-rays used for diffraction can typically penetrate through a thickness of approximately 0.1 mm. In this experiment an electron beam passes through thin foils made of either aluminum or graphite. The aluminum foil consists of many very small and randomly oriented crystallites. Aluminum crystals have a face centred cubic (fcc) structure as shown in Fig. 3. Because of phase cancellations it turns out for fcc structures that diffraction is only observed for 2

3 Figure 2: Formation of diffraction rings for powder or polycrystalline samples. Only one crystallite is shown. 2a sin θ = λ (4) h2 + k 2 + l2 where h, k and l, called Miller indices, are either all odd or all even integers. (This holds for fcc only). In Eq. 4, the diffraction order n is contained in the Miller indices. The graphite foil also consists of small crystallites. However, graphite has a pseudo-two-dimensional, layered structure and the crystallites in the graphite sample are oriented such that the atomic layers are all parallel and are aligned perpendicular to the direction of the electron beam (the c axis parallel to the beam see Fig. 4). There are a few areas of graphite foil where only one or two graphite crystallites are located. Graphite crystals have hexagonal symmetry in the layers. (See Fig. 4.) In this experiment, diffraction from the d 100 and d 110 lattice spacings are observed. For graphite the a spacing is nm as measured from x-ray work. Because the graphite crystals used in this experiment are extremely thin (approximately 100 atomic layers) it turns out for this special geometry the Bragg equation(eq. 3) does not need to be satisfied. However, the electron diffraction is similar to the diffraction of light from an optical grating, and the diffraction maxima are given by the grating equation d sin φ = nλ (5) where d is the spacing between lattice planes as shown in Fig. 4 and φ is the angle between the incident beam and the diffraction beam. In Fig. 4 each spot represents two carbon atoms. The intensities of the diffracted beams are affected by the positions of the carbon atoms, but it turns out that the 3

4 spots shown in the figure are sufficient to enable the positions of diffracted beams to be calculated from the d values shown. Note that in the plane of the graphite layers the spots are periodically repeated with a period a in the horizontal direction in Figure 4b, and with the same period a at 120 to the horizontal direction. Prelab Questions 1. Verify Eq Draw a hexagonal array of points like that in Fig. 4b. Three of the points are shown in the diagram, and the vectors a 1 and a 2 are defined as shown. The c axis is perpendicular to the paper. Do not join the points with lines. Now from each point, at a displacement of a a 2 3 place another point. (These points should be at the centres of half of the triangles formed by the points shown in Fig. 4b). Now draw a line from each of these new points to its three nearest neighbours; you should end up with a lattice that looks like a honeycomb. Draw a carbon atom at each point. The diagram you obtain is a view in the c direction of one layer of carbon atoms in a piece of graphite. Apparatus Electron Diffraction Apparatus Camera Ruler (6) 4

5 Caution NEVER EXCEED 10 kv AND ALWAYS KEEP THE TARGET CURRENT BELOW 10 µa. Experiment The electron diffraction tube used in this experiment is a cathode ray tube in which thin foils of graphite and aluminum are supported on a grid between the electron gun and the screen. The distance between the foils and the screen is cm. Proper operation will increase the useful life of the tube: Make sure that the voltage and intensity controls are turned fully counterclockwise. Turn on the power and allow the filament to warm up for approximately thirty seconds. Increase the voltage slowly and look for a spot on the screen. Play with the intensity, focus and horizontal and vertical controls to see what effect they have. Move the spot around to locate the two samples. Defocussing the beam sometimes makes it easier to find the samples. When you are not actually taking data turn the intensity down, leaving the power on. When you are finished turn the voltage and intensity controls fully counterclockwise and turn off the power. NEVER EXCEED 10kV AND ALWAYS KEEP THE TARGET CURRENT BELOW 10 µa. 1. Using accelerating voltages of 8 to 10 kv, and a target current of 2 to 5 µa, observe the circular diffraction rings of polycrystalline Al, and the spot patterns for graphite. A nice single crystal pattern for graphite can be found near (0.2 cm, 2.1 cm) and ( 1.4 cm, 1 cm). 2. Carefully measure the diffraction ring diameters for Al for a series of accelerating voltages from about 6 or 7 kv up to a maximum of 10 kv. (Keep the target current less than 10 microamps.) Use Eq. 4 and the lattice constant for Al (a 5

6 = nm) determined from x-ray measurements to compute the de Broglie wavelengths for electrons for the various accelerating voltages used. Compare the results with theory (Eq. 2) by plotting λ 2 vs. 1/V for your results and the theoretical curve on the same graph. 3. Use the camera to photograph the diffraction pattern for graphite for one accelerating voltage. Cover the bright central spot with black tape. Using the theoretical value for the electron wavelength and Eq. 4, determine the lattice constant, a, for graphite. (See Fig. 4.) Compare it with the accepted value. Do the results agree within your experimental uncertainty? DJH,AEC/1996, NA/2010 6

7 Figure 3: The fcc structure of Al. The circles represent Al atoms. Figure 4: (a) The electron beam is incident along the c-axis, which is perpendicular to the hexagonal layers. (b) View along the c-axis. The spots represent pairs of carbon atoms. The lattice is hexagonal and the lattice constant is a. 7