1. Interaction of X-Rays with Materials
Concepts and Vocabulary
P Kidd

2. Agenda X-ray and Atom Theories Diffraction from Crystals Bragg's Law
The Reciprocal Lattice in XRD
The Reciprocal Lattice and Bragg's Law
The Ewald Sphere
Reciprocal Lattice of a Single Crystal in 3D
Reciprocal Lattice of Powder or Polycrystalline Solid
2Theta/Omega Powder Scans in Reciprocal Space
Polycrystalline Materials with Preferred Orientation and Texture
Pole Figure Measurement
Stress Measurement
SAXS and Reflectivity
Single Crystal Diffraction to Solve for Molecular Structure
Single Crystal Substrates and Thin Films
Thin Layers and Multilayers
Resolution
Summary

3. X-ray and Atom
An X-ray photon interacts with electrons

4. Fluorescence and Scattering
Emission at a different wavelength (non-elastic)

5. Fluorescence and Scattering
Emission at the same wavelength (elastic)
Transmitted +
scattered
scattered
Dipole oscillations of electrons

6. Diffraction = Sum of Scattering
Scattered X-rays adding together in phase to give a diffracted beam (Interference)
Diffracted Beam
Angle of diffraction
Transmitted beam
Incident beam

7. PANalytical Products An instrument Analysis Software Provides X-rays
Aligns a sample
Detects diffraction pattern
Analysis Software
Performs calculations with peak positions, widths, intensities
Simulates and fits diffraction patterns 
2  S
Detector
source
sample

8. Scattering Centres in Materials
Any material is a mass of scattering centres (electric field distribution)
Where any length scale is repeated sufficiently often there will be enhanced scattering intensity in some direction
If the scattering centres are very ordered and periodic the peaks in scattered intensity form a diffraction pattern
Inhomogeneous electric field distribution
Grain boundary
homogeneous electric field distribution
Random distribution of bond lengths
Amorphous solid or Liquid
homogeneous electric field distribution
Periodic array
Crystal

9. Theories: Common Names
Maxwell
Equations describing interactions of electromagnetic waves (X-rays)
Laue
Interference from 3D array of scattering centres in crystals
Ewald
Worked with on general solution for 3D array
Bragg
Simplified equation based on planes of scattering centres
Fresnel
Optical model based on uniform refractive index of a material (reflectivity)
Dynamical Diffraction (Darwin, Prins, James)
Theory for scattering from perfect single crystals including multiple scattering events within the crystal
Kinematical Theory
Simplified theory for “small crystals” not including multiple scattering

10. Diffraction From Crystals
How to simplify diffraction from 3D arrays of atoms
Simplification (Bragg)
Describe a crystal as sets of planes
Principle (Laue/Ewald)
Add scattering from every individual scattering centre

11. Bragg's Law n = 2dsin  Plane normal Incident beam Diffracted beam 
This is called a ‘reflection’
Plane normal
Incident beam
Diffracted beam    
Crystal planes
dhkl 2
Path difference = n
Incident beam is inclined by  with respect to crystal planes
Scattered beam is at 2 with respect to incident beam
Incident beam, plane normal, diffracted beam are coplanar

12. Bragg's Law in Diffraction Patterns
Bragg’s law is used to identify the scattering angle
Work out the intensity in different directions:
“Form Factor”
Consider the crystal structure (the density of scattering centres in a plane)
“Structure factor”
Consider the microstructure of the material
Polycrystalline powder
Polycrystalline solid
Single crystal
Consider the Experimental set up
Analysis of diffraction pattern:
Different theoretical approaches are used depending upon these properties

13. Crystals: Unit Cells (vectors)
Simplify repeat groups (molecules) to a Lattice (simple scattering point) .
Define the crystal in terms of unit cells with vectors
-|a|, |b| and |c| are unit cell dimensions given in Å c b a
Give each scattering point a vector coordinate r
r = ua + vb + wc

14. Unit Cells: Planes and Miller Indeces (hkl)
Classify the scattering points into planes
Designate the planes (hkl) by Miller indices a b c n
c/l
b/k
a/h
Plane (hkl)
Plane hkl cuts the unit cell
a vector at a/h
b vector at b/k
c vector at c/l

15. Formulae for Calculating Plane Spacings and Angles
Available in crystallography books, databases etc

16. The Reciprocal Lattice in XRD
Why do we need this?
Bragg’s law concept is a simplification that is useful in a limited number of situations
XRD methods are advancing, we need a clear way of understanding them all.
The competition are becoming educated in these areas. We need to stay ahead of them.

17. The Reciprocal Lattice from Planes
Create reciprocal lattice (RL), where each point represents a set of planes (hkl)
-The points are generated from the RL origin where the vector, d*(hkl), from the origin to the RLP has the direction of the plane normal and length given by the reciprocal of the plane spacing.
002
112
1/d112
001
d*(112)
111
112
002
000
110
111
001
110

18. The Reciprocal “Lattice” of a 3D Array of Scattering Centres
Scattering centres in a real space crystal lattice
Reciprocal Lattice
Fourier transform
The reciprocal of any repeated length scales give reciprocal “lattice” features
Scattering centres in a random group (e.g. amorphous material)
Fourier transform
1/L L

19. Reciprocal Lattice and Scattering Vectors
Reciprocal lattice vector d*hkl
Length 1/d
Direction, normal to hkl planes
d*hkl S
d*hkl kH k0
Incident beam vector, k0,
Length n/
Direction,  with respect to sample surface k0  2
000 kH
Scattered beam vector, kH,
Length n/ (user defined)
Direction, 2 with respect to k0
By rotating kH and ko the diffraction vector S can be made to scan through reciprocal space.
When S = d*hkl then Bragg diffraction occurs
Diffraction vector, S,
S = kH – k0 S

20. Scattering Vectors Related to a Real Experiment
Psi
Phi
source
Detector S  2
sample

21. Reciprocal Lattice and Bragg’s Law
Trigonometry: S
d*hkl kH  k0
sin  = |d*hkl | /2 |ko|
 =  2
000
|ko| = 1/
|d*hkl | = 1/d
By rotating kH and ko and/or the sample we can achieve S = d*hkl then Bragg diffraction occurs
 = 2d sin 

22. The Ewald Sphere
This is a popular way of showing the reciprocal lattice and scattering using vector algebra
Follows the same principle as previously
Vector algebra:
KH -Ko = S
hkl KH
At maximum intensity:
d*hkl 2 S
S = d*hkl Ko KH
Ko incident beam vector
KH diffracted beam vector
S scattering vector
d*hkl reciprocal lattice vector
000
|KH| = |Ko| = radius of Ewald sphere = n/

23. Notation Different people use different notation e.g.
Ko KI incident beam vector
KH Kd diffracted beam vector
S Q scattering vector
d*hkl r*hkl reciprocal lattice vector
|Ko| = |KH| = n/ Where n = 1 or 2 (for example)

24. Reciprocal Lattice of a Single Crystal in 3D
115
-2-24
There are families of planes
All planes in the same family have the same length |d*|, but different directions
The family members have the same 3 indices (in different orders e.g. 400,040,004 etc)
004
224
113 d*
| d*| = 1/dhkl
-440
440
Just a few points are shown for clarity

25. Why Sample Alignment is Important for Single Crystals
For n = 2dsin
Use  and  to bring a rlp into the diffraction plane
Use the right combination of  and 2 so that S coincides with d*
1/
1/ S
1/ 2   

26. Psi and Phi Alignment Psi =  Phi =  Omega =  Theta =  Psi Phi
2  S
Detector
source
sample

27. Reciprocal Lattice of Powder or Polycrystalline Solid
Simultaneous illumination of many small crystals
Random orientations

28. Reciprocal Lattice of Powder
Add the reciprocal lattices of all the crystals
115
004
113 d*
400
Single crystal lots of single crystals

29. Concentric Spherical Shells
A sufficient number of randomly oriented crystals forms a reciprocal “lattice” of spherical shells
000
113
hkl
0 0 4
Just a part of the shells are shown for clarity

30. Alignment of Powders or Polycrystalline Solids?!
Bragg’s law can be satisfied for any  and 
Providing 2 is correct
One hkl reflection
Spherical shell radius 1/dhkl S 2 
1/dhkl 
S = 1/dhkl

31. 2Theta/Omega “Powder” Scans in Reciprocal Space
2Theta/Omega scan
Reciprocal lattice points
scattering vector S

32. 2Theta/Omega “powder” scans in reciprocal space
2Theta/Omega scan
111 2

33. 2Theta/Omega “Powder” Scans in Reciprocal Space
2Theta/Omega scan
111
220
311 2

34. 2Theta/Omega “Powder” Scans in Reciprocal Space
2Theta/Omega scan
111
220
311
004
331 2

35. 2Theta/Omega “Powder” Scans in Reciprocal Space
2Theta/Omega scan
111
220
311
422
004
331
511 2

36. Preferred Orientation and Texture
A polycrystalline solid may not have a truly random orientation of crystallites
A powder sample may have preferred orientation of not properly prepared.
What happens to the reciprocal lattice?

37. Reciprocal Lattice of Non-Random Polycrystalline Material
Non uniform reciprocal lattice
Different intensities at different directions
Spherical shell radius 1/dhkl S 2 2
1/dhkl
S = 1/dhkl

38. Pole Figure Measurement
A Pole figure maps out the intensity over part of the spherical shell
2 stays fixed, the sample is scanned over all  at different  positions 
One hkl reflection  S 2 2

39. Pole Figure Displayed
Intensity displayed as a contour map, hemisphere is “flattened out”
Al 111
2 = 38o  

40. Reciprocal Lattice of Fibre Textured Material
Something between single crystal and random polycrystalline
Nb 110
Al111
38o
Two hkl reflections  
Pole Figure 
Sharp Fibre texture:
Spots and rings
Random in  but not in 
Weak Fibre texture:
Arcs and rings
Random in  spread in 

41. Residual Stress Analysis in Polycrystalline Materials
Non uniform reciprocal lattice
Different d-spacings at different directions
Polycrystalline components subjected to external mechanical stresses
Spherical shell distorted
(not to scale!)
One hkl reflection S 2 2
1/dhkl not constant
S = 1/dhkl

42. “Stress” Measurement
A stress measurement determines dhkl at a series of Psi positions
The sample is stepped to different  positions, 2 scan at each position to obtain peak position
Repeated for different  positions as required
Spherical shell distorted
One hkl reflection  S 2 2
1/dhkl varies with position

43. Amorphous Material 2Theta/Omega scan 2 Amorphous Halo
Fourier transform
1/L L
Amorphous Halo 2

44. Length Scales Other Than d(hkl)
SAXS and Reflectivity
We have discussed scattering from atoms as scattering centres
Bundles of atoms, namely large molecules or particles can also form interference patterns
Reflectivity – thin films
SAXS - particles

45. Nano- Length Scales in Reciprocal Space: SAXS
Reciprocal lattice is very small
Scattering vector must be small
SAXS is for random array of particles
Fourier transform
1/L
Range of lengths Ln
2Theta/Omega scan 0o 3o 2

46. Reflectivity Reciprocal lattice is very small
Scattering vector must be small
Fourier transform
2Theta/Omega scan 0o 3o 2

47. Back to Single Crystals
004
113
224
115
440
-440 d*
| d*| = 1/dhkl
Just a few points are shown for clarity

48. Single Crystal Diffraction to Solve Molecular Structure
Collect all Bragg reflections and analyse position and intensity
We don’t do this!
We do solve for polycrystalline and powders

49. Single Crystal Substrates and Thin Films
We investigate the fine structure of individual reciprocal lattice spots
115
004
224
113
“Reciprocal space map”
“Scan”
-440
440
This requires high resolution instrumentation

50. Thin Layers and Multi-layers
The reciprocal lattices of the crystals and the multilayer combine
115
004
113
224
115
-440
004
224
113
-440 
Fourier transform
Reflectivity is known as the 000 reflection

51. Resolution Textured Nb/Al multilayer peak
Normal Resolution
Textured Nb/Al multilayer peak
measured in 0.01 to 1 degree steps in 2
High Resolution
Single crystal silicon – measured in to 0.01 degree steps 2

52. Summary An instrument A Material An Experiment Provides X-rays
Aligns a sample
Detects diffraction pattern
A Material
Reciprocal “Lattice” Structure
An Experiment
Designed to suit the material
Designed to answer the question 
2  S
Detector
source
sample

53. When MRD? When high resolution is necessary When alignment is critical
Investigate fine features in reciprocal space
When alignment is critical
Single crystals
For reciprocal space mapping
Any Material
Measurements using Psi and Phi
Texture, Stress
For X-Y Sample mapping
Versatility
Many different materials types in one lab