The reciprocal space Space of the wave vectors Fourier space Inverse Orthogonal.
Introduction to Reciprocal Space - SSRL · Introduction to Reciprocal Space Apurva Mehta 7th X-ray...
Transcript of Introduction to Reciprocal Space - SSRL · Introduction to Reciprocal Space Apurva Mehta 7th X-ray...
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Introduction to Reciprocal Space
Apurva Mehta
7th X-ray Scattering School
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Scattering Physics
Sample Space
Scattering Space
sample light image
Image Space
lens
Can we create the image without a lens?
Angular Space
Math
Q = 4p sin(q) /l
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Lensless Imaging
Sample Space
Scattering Space
sample light image
lens
Angular Space
Scat
teri
ng
Patt
ern
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Scattering Physics
2q Ki
Ki
Ks
Elastic Scattering |Ki| = |Ks|
DK = Q = momentum transfer
DK = Q = 2*Ki * sin(q)
momentum
Ki = 2p/l
Q = 4psin(q)/l
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Lensless Imaging
Sample Space
Scattering Space
sample light image
lens
Measured in Q = 4p sin(q) /l
Scat
teri
ng
Patt
ern
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Bragg’s Law
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Bragg’s Law
2014 - International year of Crystallography
Proposed in 1912-1913 Nobel Price in Physics - 1915
2dsin(q) = l
Q = 4psin(q)/l Q = 4psin(q)/l =2p/d
Q =2p/d |Q| =2p/d
Q has magnitude and direction
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Bragg’s Law Tells Us
• About the Position of the scattering peaks
• But not the Direction
• And not its Intensity
• Nor its Width
Need to go beyond Bragg’s Law
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Bragg Planes
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Bragg Planes
|Q| =2p/d
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Bragg Planes
|Q| =2p/d
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Reciprocal Lattice
|Q| =2p/d
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Scattering Physics
Sample Space
Scattering Space
sample light image
lens
Measured in Q = 4p sin(q) /l
Scat
teri
ng
Patt
ern
Measured in Q = 4p sin(q) /l
Real Space Lattice Reciprocal (Space) Lattice
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Scattering Physics
sample light image
lens
Measured in Q = 4p sin(q) /l
Real Space Lattice Reciprocal (Space) Lattice
• Reciprical Lattice Points
– Have Position
– Direction
– Intensity
– Width
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Scattering Physics
sample light image
lens
Real Space Lattice Reciprocal (Space) Lattice
Fourier Transform
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Real Space Reciprocal Space
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Recap
• 1: FT (FT (S) ) ~ S
• 2: FT (large) ~ 1/large small – Rec Sp (large) small
• 3: FT (periodic fn) ~ periodic
– Rec Sp (periodic Real Sp) ~ periodic
FT FT
Real Space
Real Space
Reciprocal Space
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Sailing Through Reciprocal Space
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Q1
Q0
QD
19
Ewald’s Sphere
Scattering from a Single Crystal
Reciprocal
Lattice
Elastic
Scattering
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Multi-circle diffractometer •Need at least
•2 angles for the sample •1 for the detector
•But often more for ease, polarization control, environmental chambers •New Diffractometer @7-2
•4 angles for the sample •2 for the detector
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Scattering Pattern and Ewald’s Sphere
Ewald’s Sphere
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Q1
Q0
QD
22
Ewald’s Sphere
2D detectors and Ewald Sphere
Reciprocal
Lattice
Elastic
Scattering
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Q1
Ewald’s
Sphere Reciprocal
Sphere
Scattering from Many Crystallites :polycrystal or powder
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Q0
QD
Q
24
Ewald Sphere
Reciprocal
Sphere
Nested
Powder Diffraction Pattern
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Powder Diffractometer with an Area Detector
X-ray Beam
Detector
Sample
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Diffraction from Polycrystals
111
200
220
311
Nested Reciprocal Spheres
Ewald’s sphere
Diffraction Pattern
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Condition for Polycrystalline/powder Diffraction
• Just 1 angle (detector)
• If large area detector 0 angles
• Nothing moves – Very useful for
fast/time dependent measurements
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Texture
Oriented Polycrystals
Partially filled Reciprocal Sphere
Ewald’s sphere
Diffraction pattern
Partial diffraction ring
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s
s
29 Zurich 2008
Strain Ellipsoid
Deformation of Reciprocal sphere
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30
s
s
•small strain •continuous strain
Strain Ellipsoid
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31
χ
Q0
s s
Q Q
c
χ
Q
Coordinate transformation
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32 Zurich 2008
110 200 211
c c
Q (nm-1) Q (nm-1) Q (nm-1)
c
Measuring Full Strain Tensor
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110 Em = 167 GPa
0.3
400
300
200
100
0
Stre
ss (
MPa
)
0 -.10 .05 .05 .10 .15 .20 .25
% Strain
0 0
Pois
son
’s R
atio
Stress (MPa) 400
eyy ezz
shear
1
200 211 Em = 211 GPa Em = 218 GPa
eyy
ezz Elastic Strain Tensors for Fe
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Resolution Area Detector Point Detector
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Questions?
• Think in Q space
– (yardstick of reciprocal space)
– Q = 4p sin(q) /l
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Effect of Beam Divergence
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Effect of Energy Width