reciprocalLattice+BrillouinZone - Brillouin Zones Physics...

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Brillouin Zones Physics 3P41 Chris Wiebe
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Direct space to reciprocal space ij j i a a πδ 2 * = Reciprocal space Real (direct) space Note: The real space and reciprocal space vectors are not necessarily in the same direction
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Laue Equations Another way of expression the diffraction condition k = G is by the Laue equations, which can be derived by taking the scalar product of k and G with a 1 , a 2 , and a 3 These equations have a simple interpretation: for a Bragg reflection, k must lie on a certain cone about the direction of a 1 (for example). Likewise, it must be on a cone for a 2 and a 3 . Thus, a Bragg reflection which satisfies all three conditions must lie at the intersection of these three cones, which is at a point in reciprocal space. a 1 k = 2 πν 1 a 2 k = 2 πν 2 a 3 k = 2 πν 3 (Laue Equations)
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Single crystal diffraction Ewald sphere k Each time k = G, a reciprocal lattice vector, you get a Bragg reflection. This is a point of intensity at some 2 θ angle, and some Φ angle in space Bragg reflections
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Actual single crystal diffraction patterns
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Experimental setup
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So, Bragg peaks are usually at a single spot in real space for single crystals. For powder samples, however, which are composed of many tiny single crystals randomly oriented, the G vectors exist on a cone of scattering (think of taking the Ewald circle, and rotating about k). k
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This note was uploaded on 04/21/2010 for the course MS&E 2060 taught by Professor Robinson during the Spring '10 term at Cornell University (Engineering School).

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reciprocalLattice+BrillouinZone - Brillouin Zones Physics...

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