Thinking in Reciprocal Space - SSRL · 2012. 8. 28. · Momentum Transfer Space Reciprocal Space...
Transcript of Thinking in Reciprocal Space - SSRL · 2012. 8. 28. · Momentum Transfer Space Reciprocal Space...
Thinking in Reciprocal Space
Apurva Mehta
Outline
Introduce Reciprocal Space
Reciprocal Space maybe at first appear strange
Examples thinking in reciprocal space
makes intepretting scattering experiments easier.
Why X-ray Scattering?
Structure of Materials
10s of nm to A
Scattering Physics
Sample Space
Scattering Space
sample light image
Image Space
lens
X-ray lens with resolution better than ~10nm don’t exist
X-ray Scattering/diffraction is about probing the structure without a lens
Sample to Scattering Space
Diffraction Pattern: Contains all the contrast relevant information at the resolution of l/2sin(q)
4q
l
Transformation of distance into angle
Fourier Transform: sample space scattering space
Real Space
Scattering Physics: Bragg’s Law
2q
2dsin(q) = l
1/2d =sin(q)/l
1. Distance Angle
“Reciprocal” relation
2. Fundamental unit is not q but sin(q)/l
d
Fundamental Units for scattering
s = sin(q)/l
Q = 4p sin(q)/l
Think in Q
Or provide both q and l
l= hc/E – synchrotron units are E
Scattering Physics
Momentum K0 = 2p/l
2q
K0 = 2p/l
= 2sin(q) * 2p/l
DK = Q = 4p sin(q) /l
DK
Elastic Scattering
Momentum change
Elastic Scattering
X-ray Diffraction Momentum Transfer Space
Reciprocal Space
Real Space Lattice Reciprocal Lattice
Ordered Lattice can only provide discrete momentum kicks: Bloch
Q1
Q0
QD
10
Ewald’s Sphere
Graphical Solution: Single crystal
Conditions for Single Crystal Diffraction
Q1
Q0
QD
Ewald’s Sphere
Detector AND the crystal at desired location/angles.
2dsin(q) = l
Bragg’s law provides condition for only the detector
Necessary but not sufficient
Bragg’s law = scalar Need a vector law = Laue’s Eq
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
Q1
13
Ewald Sphere
Reciprocal Sphere
Diffraction from Polycrystals
Q0
QD
Q
14
Ewald Sphere
Reciprocal Sphere
Diffraction from Polycrystals
Diffraction Cone
Diffraction from Polycrystals
111
200
220
311
Nested Reciprocal Spheres
Ewald’s sphere
Diffraction Pattern
Condition for Polycrystalline/powder Diffraction
Just 1 angle (detector)
Bragg’s law sufficient
If large area detector 0 angles
Very useful for fast/time dependent measurements
Powder Diffractometer with an Area Detector
X-ray Beam
Detector
Sample
Powder Average and Rocking
Ewald’s sphere
Oscillate while collecting data
Texture
Oriented Polycrystals
Partially filled Reciprocal Sphere
Ewald’s sphere
Diffraction pattern
Partial diffraction ring
s s df
df
20 Zurich 2008
Local Scale
d0
d0 s s
df
df
Global Scale
d0
d0
0
0
f
d
d d
=
Strain
s
s
21 TMS 2008
Strain Ellipsoid
Strain in Polycrystals
Reciprocal Sphere
χ
Q0
s s
Q
22 TMS 2008
Q
c
χ
Q
Coordinate Transformation
23 TMS 2008
110 200 211
c c
Q (nm-1) Q (nm-1) Q (nm-1)
c
Strain Relaxation
110 Em = 167 GPa
0.3
400
300
200
100
0
Str
ess (
MPa)
0 -.10 .05 .05 .10 .15 .20 .25
% Strain
0 0
Pois
son’s
Ratio
Stress (MPa) 400
yy zz
shear
1
200 211 Em = 211 GPa
Em = 218 GPa
24 Zurich 2008
yy
z
z
Elastic Strain Tensor: bcc Fe
Scattering Physics
Sample Space
Scattering Space
sample light image
Image Space
lens
X-ray lens with resolution better than ~10nm don’t exist
X-ray Scattering/diffraction is about probing the structure without a lens
Reciprocal Space
Thanks
Reciprocal Space is the map of diffraction pattern
Think in Q space
(yardstick of reciprocal space)
Q = 4p sin(q) /l