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123B_1_EE123B W 11 Lecture3

123B_1_EE123B W 11 Lecture3 - Lecture 3 Kittel Chapter 2...

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Lecture 3 Kittel Chapter 2 January 11, 2010 Hand in HW#1.
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Homework #2 due 1/18 Tuesday Kittel Chapter 2 2.1 – interplanar distance 2.2 – primitive cell calculations volume, vectors, Brillouin zone sketch for HCP 2.4 – examine scattering linewidth F, K, G 2.5 – structure factor
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Last lecture Chapter 1 cont’d More BCC, FCC lattice characteristics More diamond, zincblende lattices Chapter 2 Diffraction in a crystal Bragg’s law Fourier Analysis Reciprocal lattice
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Constructive Interference Constructive interference occurs when reflected beam path lengths differ by nλ 2dsinθ=nλ Bragg Law dsinθ θ θ θ d
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Today’s lecture Chapter 2 Reciprocal Space Brillouin Zone Scattering amplitude, F Structure factor Atomic form factor
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Diffraction Conditions – map real to reciprocal Reciprocal lattice vectors G determines possible reflection We want an expression to relate Bragg Condition to G. Next define G using reciprocal lattice vectors 3 3 2 2 1 1 b v b v b v G + + = 3 2 1 3 2 1 2 a a a a a b × × = π Translates reciprocal to real 3 2 1 1 3 2 2 a a a a a b × × = π 3 2 1 2 1 3 2 a a a a a b × × = π
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Laue Equations – also derive E.S. Laue equations relate to real lattice to reciprocal lattice 3 2 1 , a a b ij j i a b πδ 2 = 1 3 2 , a a b 2 1 3 , a a b 3 3 2 2 1 1 b v b v b v G + + = 3 2 1 3 2 1 2 a a a a a b × × = π 3 2 1 1 3 2 2 a a a a a b × × = π 3 2 1 2 1 3 2 a a a a a b × × = π Axis vectors of the reciprocal lattice: Each vector is orthogonal to two axis vectors of the crystal lattice: Thus: j i ij = = if 1 δ j i ij = if 0 δ Points in the reciprocal lattice are: are integers, is a reciprocal lattice vector 3 2 1 , , v v v G
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Laue condition contd. Thus can be rewritten in terms of k G , G a k a = 1 1 1 3 3 2 2 1 1 1 2 ) ( v b v b v b v a π = + + 2 3 3 2 2 1 1 2 2 ) ( v b v b v b v a π = + + 3 3 3 2 2 1 1 3 2 ) ( v b v b v b v a π = + + Relates back to Ewald Sphere 3 2 1 , , a a a k G ' k θ θ 2 1 2 But why a.G?
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Brillouin Zone Provide geometrical, visual interpretation of Use Wigner-Seitz primitive cell in reciprocal space.
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