Introduction to Reciprocal Space
Apurva Mehta
7th X-ray Scattering School
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
Lensless Imaging
Sample Space
Scattering Space
sample light image
lens
Angular Space
Scat
teri
ng
Patt
ern
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
Lensless Imaging
Sample Space
Scattering Space
sample light image
lens
Measured in Q = 4p sin(q) /l
Scat
teri
ng
Patt
ern
Bragg’s Law
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
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
Bragg Planes
Bragg Planes
|Q| =2p/d
Bragg Planes
|Q| =2p/d
Reciprocal Lattice
|Q| =2p/d
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
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
Scattering Physics
sample light image
lens
Real Space Lattice Reciprocal (Space) Lattice
Fourier Transform
Real Space Reciprocal Space
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
Sailing Through Reciprocal Space
Q1
Q0
QD
19
Ewald’s Sphere
Scattering from a Single Crystal
Reciprocal
Lattice
Elastic
Scattering
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
Scattering Pattern and Ewald’s Sphere
Ewald’s Sphere
Q1
Q0
QD
22
Ewald’s Sphere
2D detectors and Ewald Sphere
Reciprocal
Lattice
Elastic
Scattering
Q1
Ewald’s
Sphere Reciprocal
Sphere
Scattering from Many Crystallites :polycrystal or powder
Q0
QD
Q
24
Ewald Sphere
Reciprocal
Sphere
Nested
Powder Diffraction Pattern
Powder Diffractometer with an Area Detector
X-ray Beam
Detector
Sample
Diffraction from Polycrystals
111
200
220
311
Nested Reciprocal Spheres
Ewald’s sphere
Diffraction Pattern
Condition for Polycrystalline/powder Diffraction
• Just 1 angle (detector)
• If large area detector 0 angles
• Nothing moves – Very useful for
fast/time dependent measurements
Texture
Oriented Polycrystals
Partially filled Reciprocal Sphere
Ewald’s sphere
Diffraction pattern
Partial diffraction ring
s
s
29 Zurich 2008
Strain Ellipsoid
Deformation of Reciprocal sphere
30
s
s
•small strain •continuous strain
Strain Ellipsoid
31
χ
Q0
s s
Q Q
c
χ
Q
Coordinate transformation
32 Zurich 2008
110 200 211
c c
Q (nm-1) Q (nm-1) Q (nm-1)
c
Measuring Full Strain Tensor
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
Resolution Area Detector Point Detector
Questions?
• Think in Q space
– (yardstick of reciprocal space)
– Q = 4p sin(q) /l
Effect of Beam Divergence
Effect of Energy Width