Lecture18-XRD-LorentzPolarizationFactor

Lecture18-XRD-LorentzPolarizationFactor - EMSE 312 –...

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Unformatted text preview: EMSE 312 – Diffraction Principles Lecture 18 Powder X-ray Diffraction (XRD) Techniques Intensity of Diffracted Beams: Polarization Factor [P( θ )] Lorentz Factor [L( θ )] ⇒ Lorentz Polarization Factor [LP( θ )] EMSE 312 DIFFRACTION PRINCIPLES EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • Consider ZnS – Intensity • Even: (h+k+l) = 4n = 2(2n) • Odd: (h+k+l) = 4n + 1 • Even: (h+k+l) = 4n + 2 =2(2n+1) • Odd: (h+k+l) = 4n + 3 ZnS Basis Lattice g g g F F F = × ( ) ( ) ( ) { } 2 2 ZnS X X Lattice g S Zn g i I F f f e x p h k l F 2 π θ θ ⎡ ⎤ ∝ = + + + × ⎢ ⎥ ⎣ ⎦ ( ) ( ) Basis X X g S Zn F f f θ θ ⎡ ⎤ = + ⎣ ⎦ ( ) ( ) ( ) 2 X X S Zn I 1 6 f f θ θ = ⋅ + ( ) ( ) Basis X X g S Zn F f i f θ θ ⎡ ⎤ = + ⋅ ⎣ ⎦ ( ) ( ) ( ) 2 2 X X S Zn I 1 6 f f θ θ ⎡ ⎤ ⎡ ⎤ = ⋅ + ⎣ ⎦ ⎣ ⎦ ( ) ( ) Basis X X g S Zn F f i f θ θ ⎡ ⎤ = − ⋅ ⎣ ⎦ ( ) ( ) ( ) 2 2 X X S Zn I 1 6 f f θ θ ⎡ ⎤ ⎡ ⎤ = ⋅ + ⎣ ⎦ ⎣ ⎦ ( ) ( ) Basis X X g S Zn F f f θ θ ⎡ ⎤ = − ⎣ ⎦ ( ) ( ) ( ) 2 X X S Zn I 1 6 f f θ θ = ⋅ − High Med Med Low EMSE 312 – Diffraction Principles Diffraction Techniques - XRD ZnS Basis Lattice g g g F F F = × Prediction: (100) – mixed - zero intensity (110) – mixed - zero intensity (111) – 4n+3 - med intensity (200) – 4n+2 - low intensity (210) – mixed - zero intensity (211) – mixed - zero intensity (220) – 4n - high intensity (221) – mixed - zero intensity (300) – mixed - zero intensity (310) – mixed - zero intensity (311) – 4n+1 - med intensity (222) – 4n+2 - low intensity (321) – mixed - zero intensity etc. • Consider ZnS Prediction: (111) – 4n+3 - med intensity (200) – 4n+2 - low intensity (220) – 4n - high intensity (311) – 4n+1 - med intensity (222) – 4n+2 - low intensity (400) – 4n - high intensity Something else must also be important EMSE 312 – Diffraction Principles • Consider the scattering by an electron – The intensity of the scattered beam depends on the scattering angle – J.J. Thompson (1897) • Scattering of an x-ray • intensity of the incident beam • charge of an electron • mass of an electron [rest mass] • permeability of vacuum Diffraction Techniques - XRD acceleration α 2 θ 2 4 2 2 2 2 2 e K I I sin I sin 4 m r r μ α α π ⎛ ⎞ ⎛ ⎞ = = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ O P ____ r O P = ( ) 7 4.7 10 Vs Am μ − = × ( ) 19 e 1 . 6 1 C − = × ( ) 31 m 9 . 1 1 k g − = × ( ) I 2 2 e K 4 m μ π ⎛ ⎞ = ⎜ ⎟ ⎝ ⎠ Intensity at point P e − EMSE 312 – Diffraction Principles Diffraction Techniques - XRD • X-rays are un-polarized EM waves – Consider an X-ray traveling along the x-axis – The electric polarization vector is in a random direction in the y-z plane – The electric polarization vector can be resolved into two plane polarized components and ; i.e., and – On average – The intensity is proportional to and E y E 2 2 2 2 y z E E...
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Lecture18-XRD-LorentzPolarizationFactor - EMSE 312 –...

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