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