2. In some cases, for very low energy electrons, k is rather short and a diffraction pattern may contain only Bragg beams corresponding to a particular planes of reciprocal space (e.g. look at the electron diffraction patterns in the lecture notes). An orthorhombic crystal has a = 0.45 nm, b = 0.15 nm and c = 0.20 nm. a) Construct and accurately draw (use graphing software if it helps) a section of the plane of reciprocal space containing a* and b* (i.e. the a*b* plane). (Your section should be large enough to allow you to complete parts c) and d)). (4) b) Indicate the relative orientation of the real space lattice vectors a, b and c relative to the unit vectors of the reciprocal lattice. (1) c) Suppose the crystal is now illuminated with a beam of electrons with wavevector k = -2a*-b*. Use the Ewald sphere construction to determine the allowed Bragg reflections in this plane of reciprocal space. Sketch the cross section of the Ewald sphere on your diagram and give the Miller indices for allowed reflections. (For TEM experts: For this analysis, assume the relrods are very small and ignore directional variations in the electron scattering due to the atomic scattering factor, f.). (6) d) On your diagram, draw and list in vector notation (i.e. g in terms of a*, b* and c*) the reciprocal lattice vectors, g, corresponding to the Bragg reflections. (3)

2. In some cases, for very low energy electrons, k is rather short and a diffraction pattern may contain only Bragg beams corresponding to a particular planes of reciprocal space (e.g. look at the electron diffraction patterns in the lecture notes). An orthorhombic crystal has a = 0.45 nm, b = 0.15 nm and c = 0.20 nm. a) Construct and accurately draw (use graphing software if it helps) a section of the plane of reciprocal space containing a* and b* (i.e. the ab plane). (Your section should be large enough to allow you to complete parts c) and d)). (4) b) Indicate the relative orientation of the real space lattice vectors a, b and c relative to the unit vectors of the reciprocal lattice. (1) c) Suppose the crystal is now illuminated with a beam of electrons with wavevector k = -2a-b. Use the Ewald sphere construction to determine the allowed Bragg reflections in this plane of reciprocal space. Sketch the cross section of the Ewald sphere on your diagram and give the Miller indices for allowed reflections. (For TEM experts: For this analysis, assume the relrods are very small and ignore directional variations in the electron scattering due to the atomic scattering factor, f.). (6) d) On your diagram, draw and list in vector notation (i.e. g in terms of a, b and c*) the reciprocal lattice vectors, g, corresponding to the Bragg reflections. (3)

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2) A thin crystal of the metal oxide with a tetragonal crystal structure and lattice parameters a = 4.05 Å and c = 2.517 Å is tilted so that a 200 kV electron beam travels along the [001] zone axis. Create a schematic diagram of zero order Laue zone (ZOLZ) diffraction pattern for a crystals of thickness: a) 75 Å b) 400 Å You should assume there are no forbidden reflections and that the diffraction intensity drops to zero when the excitation error, s, exceeds the first minimum of the relrod. You should show the approximate number of beams in the ZOLZ circle. (10)
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