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

X-ray lens with resolution better than ~10nm don’t exist

image light sample lens

Sample Scattering Image Space Space Space

X-ray Scattering/diffraction is about probing the structure without a lens Sample to Scattering Space

Diffraction Pattern:

Contains all the l contrast relevant information at the 4q resolution of l/2sin(q)

Transformation of distance into angle

Fourier Transform: sample space  scattering space Scattering Physics: Bragg’s Law

Real Space 2dsin(q) = l

d 1/2d =sin(q)/l 1. Distance  Angle “Reciprocal” relation

2. Fundamental unit is not q but sin(q)/l

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

Elastic Scattering Momentum change

DK = 2sin(q) * 2p/l

Momentum 2q DK = Q = 4p sin(q) /l K0 = 2p/l K0 = 2p/l Reciprocal Space Ordered Lattice can only provide

discrete momentum kicks: Bloch X-ray Diffraction Elastic Momentum Transfer Space Scattering

Real Space Lattice Reciprocal Lattice Graphical Solution: Single crystal

Q QD 1

Ewald’s Sphere

10 Conditions for Single Crystal Diffraction Bragg’s law provides condition Bragg’s law for only the detector = scalar 2dsin(q) = l Need a vector law = Laue’s Eq Necessary but not Q QD 1 sufficient

Ewald’s Sphere

Detector AND the crystal at desired location/angles. 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 Diffraction from Polycrystals

Ewald Reciprocal Sphere Sphere

13 Diffraction from Polycrystals

Ewald Sphere Q

QD Reciprocal Sphere

Diffraction Cone

14 Diffraction from Polycrystals

311

220

200

111

Ewald’s sphere

Diffraction Pattern Nested Reciprocal Spheres 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

Detector

X-ray Beam

Sample Powder Average and Rocking

Oscillate while collecting data

Ewald’s sphere Texture

Oriented Diffraction pattern Polycrystals

Ewald’s sphere Partially filled Reciprocal Sphere Partial diffraction ring Strain Global Scale

s s d0 df

d  d df  = f 0 d0 Local Scale d0

d s f s df d0

Zurich 2008 20 Strain in Polycrystals

Reciprocal Sphere s

Strain Ellipsoid

TMS 2008 21 Coordinate Transformation

Q Q χ c Q0 χ s s

TMS 2008 22 Strain Relaxation

110 200 211

c c c

Q (nm-1) Q (nm-1) Q (nm-1)

TMS 2008 23 Elastic Strain Tensor: bcc Fe z z

yy

110 Em = 211 200 Em = 167 211 Em = 218 GPa GPa GPa

400 1

shear 300

200

0.3

Stress (MPa) Stress   100 zz yy Ratio Poisson’s

0 0 -.10 .05 0 .05 .10 .15 .20 .25 0 Stress (MPa) 400 % Strain

Zurich 2008 24 Scattering Physics

X-ray lens with resolution better than ~10nm don’t exist

image light sample lens

Sample ReciprocalScattering Image Space Space Space

X-ray Scattering/diffraction is about probing the structure without a lens Thanks

 Reciprocal Space is the map of diffraction pattern

 Think in Q space  (yardstick of reciprocal space)  Q = 4p sin(q) /l