Lecture17-XRD-UnitCellStructureFactor

Lecture17-XRD-UnitCellStructureFactor - EMSE 312 –...

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Unformatted text preview: EMSE 312 – Diffraction Principles Lecture 17 Powder X-ray Diffraction (XRD) Techniques Unit Cell Structure Factors Diffracted Intensities EMSE 312 DIFFRACTION PRINCIPLES EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • Scattering by a Unit Cell – first order reflections • Plane (h00): • Angular phase shift: hkl d B θ B θ 2 2’ 1 1’ x 3 3’ ( ) B 1'3' 2xsin δ θ λ = < ( ) B 1'2' hkl 2d sin δ θ λ = = h00 a d h = ( ) ( ) B B 1'2' h00 h00 2d sin sin 2d λ δ θ λ θ = = → = ( ) ( ) 1'3' h00 x 2x h x / a 2d a / h λ λ δ λ ⋅ = = = ⋅ ⋅ ( ) ( ) 1'3' 2 x 2 h x / a a / h π λ φ π λ ⋅ = ⋅ = 2 δ φ π λ = EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • 1 D → 3D • Angular Phase Shift: Atomic positions: – X-axis: – Y-axis: – Z-axis: • Total Angular Phase Shift: hkl d B θ B θ 2 2’ 1 1’ x 3 3’ ( ) x 2 h x / a φ π = ( ) y 2 k y / b φ π = ( ) z 2 l z / c φ π = ( ) x / a u = ( ) y / b v = ( ) z / c w = ( ) Total x y z 2 h u k v l w φ φ φ φ π = + + = ⋅ + ⋅ + ⋅ EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • Unit Cell Structure Factor: – Atomic positions (relative positions with in unit cell) – Scattering Amplitude = The Atomic Scattering factor – Sum over All Atoms “j” in the Unit Cell (the BASIS) • The Structure Factor depends on the plane (hkl) – The plane (hkl) defines the scattering angle through the interplanar spacing ( ) ( ) { } X g h k l j j j j j F F f e x p 2 i h u k v l w θ π = = ⋅ ⋅ + ⋅ + ⋅ ∑ ( ) B hkl sin 2d λ θ = hkl 2 2 2 a d h k l = + + e.g., Cubic crystals ( ) X j f θ ( ) j j j u ,v ,w EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • Lattice + Basis Example: Consider NaCl – Cl Ions at 0,0,0; ½,½,0; ½,0,½; and 0,½,½ – Na Ions at ½,½,½; 0,0,½; 0,½,0; and ½,0,0 ( ) ( ) { } NaCl NaCl X g h k l j j j j j F F f e x p 2 i h u k v l w θ π = = ⋅ ⋅ + ⋅ + ⋅ ∑ ( ) ( ) { } ( ) ( ) ( ) X X Cl Cl X X Cl Cl 1 1 f e x p 2 i h k l f e x p 2 i h k l 2 2 1 1 1 1 f e x p 2 i h k l f e x p 2 i h k l 2 2 2 2 θ π θ π θ π θ π ⎧ ⎫ ⎛ ⎞ = ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⎨ ⎬ ⎜ ⎟ ⎝ ⎠ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭ ( ) ( ) ( ) ( ) X X Na Na X X Na Na 1 1 1 1 f e x p 2 i h k l f e x p 2 i h 1 k 1 l 2 2 2 2 1 1 f exp 2 i h 1 k l 1 f exp 2 i h k 1 l 1 2 2 θ π θ π θ π θ π ⎧ ⎫ ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭ ⎧ ⎫ ⎧ ⎫ ⎛ ⎞ ⎛ ⎞ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + ⋅ ⎨ ⎬ ⎨ ⎬ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎩ ⎭ ⎩ ⎭ EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • Consider NaCl – Factor out ( ) ( ) { } ( ) { } ( ) { } { } ( ) ( ) { } ( ) { } ( ) { } ( ) { } { } NaCl X g C l X Na F f 1 exp i h...
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This note was uploaded on 12/01/2011 for the course EMSE 312 taught by Professor Lagerlof,p during the Spring '08 term at Case Western.

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Lecture17-XRD-UnitCellStructureFactor - EMSE 312 –...

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