123B_1_EE123B W 11 Lecture3

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

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Unformatted text preview: Lecture 3 Kittel Chapter 2 January 11, 2010 Hand in HW#1. 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 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 Constructive Interference • Constructive interference occurs when reflected beam path lengths differ by nλ 2dsinθ=nλ Bragg Law dsinθ θ θ θ d Today’s lecture • Chapter 2 • Reciprocal Space • Brillouin Zone • Scattering amplitude, F • Structure factor • Atomic form factor 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 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 δ Points in the reciprocal lattice are: are integers, is a reciprocal lattice vector 3 2 1 , , v v v G 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? Brillouin Zone...
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This note was uploaded on 02/20/2012 for the course EE 123B taught by Professor Dianahuffaker during the Winter '11 term at UCLA.

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

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