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electron_diffraction2

electron_diffraction2 - University of Michigan Physics...

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University of Michigan Physics 441-442 2/9/06 Advanced Physics Laboratory Electron Diffraction and Crystal Structure 1. Introduction In classical mechanics we describe motion by assigning momenta to point particles. In quantum mechanics we learn that the motion of particles is also described by waves, with the crucial parameters of the two viewpoints related through the de Broglie relation: ! = h p [1] where p is the momentum, λ is the wavelength, and h is Planck’s constant h = 6.626 ! 10 " 34 J # s = 4.136 ! 10 " 15 eV # s . To observe wave-like behavior, we require some kind of grating where the “distance between slits” is of order the wavelength. At typical laboratory energies, the electron’s de Broglie wavelength is of order one Angstrom (10 –8 cm), about the same size as the interatomic spacings in common crystals. The regular atomic arrays in crystals are thus perfectly scaled gratings for creating a “matter wave” diffraction pattern, measuring their wavelength, and verifying Eq. 1. As an added bonus, with the principle verified, the diffraction patterns then become powerful tools for the study of crystal structure. In this experiment, you will use a cathode ray tube with a graphite crystal target that shows the diffraction pattern on the screen. You will verify the de Broglie relation, and analyze crystal structures, including measurement of the inter-atomic distance in the crystal. 2. Basic Principles a. The de Broglie Wavelength vs. Voltage In the cathode ray tube the electron is accelerated through high voltage V. Its energy and momentum are then given by E = p 2 2 m = eV [2] Solving for the momentum, and substituting into Eq. 1 gives: ! = h 2 eVm [3] You should verify for yourself that this can be re-written in the practical form ! ( Angstroms ) = 151.3 V ( volts ) [4] Thus, a 150 V electron has a de Broglie wavelength of 1 Angstrom, and the wavelength should vary in inverse proportional to the accelerating voltage. b. Crystal Lattice Spacing A crystal is a very regular array of atoms. The regularity can be quantified in terms of certain small patterns of atoms, called unit cells , which are repeated over and over again. Since the vertices of
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2/9/06 2 Electron Diffraction the unit cell are atoms, the size of the unit cell is related to the inter-atomic spacing, or lattice constant, which is usually called a . This experiment will be done with a graphite (carbon) crystal that has a hexagonal structure. For a simple hexagonal crystal such as graphite, the lattice is as shown below. The (100) and (110) planes, which respectively give rise to the inner and outer rings in the electron diffraction tube, are shown at right; the ratio of the d-spacings d 100 /d 110 = 3 :1. These spacings have been defined in terms of the unit vectors a and b where a = b in the case of the hexagonal structure.
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