04/20/23

Interaction of X-Rays with Materials

Concepts and Vocabulary P Kidd

04/20/23

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

04/20/23

X-ray and Atom

• An X-ray photon interacts with electrons

04/20/23

Fluorescence and Scattering• Fluorescence

– Emission at a different wavelength (non-elastic)

04/20/23

Fluorescence and Scattering• Scattering

– Emission at the same wavelength (elastic)

scattered

Transmitted + scattered

Dipole oscillations of electrons

04/20/23

Diffraction = Sum of Scattering

• Scattered X-rays adding together in phase to give a diffracted beam (Interference)

Incident beamTransmitted beam

Diffracted Beam

Angle of diffraction

PANalytical Products

• An instrument– Provides X-rays– Aligns a sample– Detects diffraction pattern

• Analysis Software– Performs calculations with peak positions,

widths, intensities– Simulates and fits diffraction patterns

2

SDetector source

sample

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 distributionGrain boundary

homogeneous electric field distributionRandom distribution of bond lengths

Amorphous solid or Liquid

homogeneous electric field distribution

Periodic array Crystal

04/20/23

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

04/20/23

Diffraction From Crystals

• How to simplify diffraction from 3D arrays of atoms

• Principle (Laue/Ewald)– Add scattering from every

individual scattering centre

• Simplification (Bragg)– Describe a crystal as sets

of planes

04/20/23

Bragg's Law

dhkl

2

Incident beam Diffracted beam

Crystal planes

n = 2dsin

Incident beam is inclined by with respect to crystal planesScattered beam is at 2 with respect to incident beamIncident beam, plane normal, diffracted beam are coplanar

Plane normal

Path difference = n

This is called a ‘reflection’

04/20/23

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

04/20/23

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

ab

c

•Give each scattering point a vector coordinate r

r = ua + vb + wc

04/20/23

Unit Cells: Planes and Miller Indeces (hkl)

Classify the scattering points into planes

Designate the planes (hkl) by Miller indices

a

bc

nc/l

b/k

a/h

Plane (hkl)

Plane hkl cuts the unit cell a vector at a/hb vector at b/kc vector at c/l

Formulae for Calculating Plane Spacings and Angles

• Available in crystallography books, databases etc

C u b i c p l a n a r s p a c i n g s : 222

2

lkh

ad

H e x a g o n a l p l a n a r s p a c i n g s :

2

2

222

3

4)(

1

c

l

ahkkh

d

C u b i c i n t e r p l a n a r a n g l e s : 222222 '''

'''cos

lkhlkh

llkkhh

H e x a g o n a l i n t e r p l a n a r a n g l e s :

2221

'''

'

3

4''''cos

c

ll

akhhkkkhhdd lkhhkl

04/20/23

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.

04/20/23

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.

000

001

002

110

111

112

d*(112

)

1/d112

001

002112

111110

04/20/23

The Reciprocal “Lattice” of a 3D Array of Scattering Centres

Scattering centres in a real space crystal lattice

Reciprocal Lattice

Fourier transform

Scattering centres in a random group (e.g. amorphous material)

Fourier transform

The reciprocal of any repeated length scales give reciprocal “lattice” features

1/L

L

04/20/23

Reciprocal Lattice and Scattering Vectors

000

d*hkl

Reciprocal lattice vector d*hkl

Length 1/dDirection, normal to hkl planes

Incident beam vector, k0,Length n/Direction, with respect to sample surface

Scattered beam vector, kH,Length n/ (user defined)Direction, 2 with respect to k0

2

Diffraction vector, S,S = kH – k0

d*hkl

k0

kH

S

k0kH

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

S

04/20/23

Scattering Vectors Related to a Real Experiment

PhiPsi

2

S

Detector source

sample

04/20/23

Reciprocal Lattice and Bragg’s Law

000

d*hkl

= 2k0

kH

By rotating kH and ko and/or the sample we can achieve S = d*hkl

then Bragg diffraction occurs

S

|ko| = 1/

|d*hkl | = 1/d

sin = |d*hkl | /2 |ko|

= 2d sin

Trigonometry:

04/20/23

The Ewald Sphere• This is a popular way of showing the reciprocal lattice and

scattering using vector algebra– Follows the same principle as previously

Ko

KH

S2d*hkl

Ko incident beam vectorKH diffracted beam vectorS scattering vectord*hkl reciprocal lattice vector

000

hkl

|KH| = |Ko| = radius of Ewald sphere = n/

KH -Ko = S

At maximum intensity:

S = d*hkl

Vector algebra:

KH

04/20/23

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)

04/20/23

Reciprocal Lattice of a Single Crystal in 3D

004

113

224

115

440

-440

d*| d*| = 1/dhkl

Just a few points are shown for clarity

•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)

-2-24

04/20/23

Why Sample Alignment is Important for Single Crystals

2

S 1/

1/

1/

• 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*

04/20/23

Psi and Phi Alignment

PhiPsi

2

S

Detector source

sample

Psi =

Phi =

Omega =

Theta =

04/20/23

Reciprocal Lattice of Powder or Polycrystalline Solid

• Simultaneous illumination of many small crystals

– Random orientations

04/20/23

Reciprocal Lattice of Powder

004

113

115

400

d*

• Add the reciprocal lattices of all the crystals

Single crystal lots of single crystals

Concentric Spherical Shells

• A sufficient number of randomly oriented crystals forms a reciprocal “lattice” of spherical shells

000

113

0 0 4

hkl

Just a part of the shells are shown for clarity

04/20/23

Alignment of Powders or Polycrystalline Solids?!

• Bragg’s law can be satisfied for any and – Providing 2 is correct

2

S = 1/dhkl

S

Spherical shell radius 1/dhkl

1/dhkl

One hkl reflection

04/20/23

2Theta/Omega “Powder” Scans in Reciprocal Space

2Theta/Omega scan

scattering vector S

Reciprocal lattice points

04/20/23

2Theta/Omega “powder” scans in reciprocal space

111

2

2Theta/Omega scan

04/20/23

2Theta/Omega “Powder” Scans in Reciprocal Space

111

220

311

2

2Theta/Omega scan

04/20/23

2Theta/Omega “Powder” Scans in Reciprocal Space

111

220

311

004 331

2

2Theta/Omega scan

04/20/23

2Theta/Omega “Powder” Scans in Reciprocal Space

111

220

311

004 331422

511

2

2Theta/Omega scan

04/20/23

Preferred Orientation and Texture

• A polycrystalline solid may not have a truly random orientation of crystallitesA powder sample may have preferred orientation of not properly prepared.

• What happens to the reciprocal lattice?

Reciprocal Lattice of Non-Random Polycrystalline Material

• Non uniform reciprocal lattice– Different intensities at different directions

2

2

S = 1/dhkl

S

Spherical shell radius 1/dhkl

1/dhkl

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

2

2S

One hkl reflection

Pole Figure Displayed

• Intensity displayed as a contour map, hemisphere is “flattened out”

Al 111 2 = 38o

04/20/23

Reciprocal Lattice of Fibre Textured Material• Something between single

crystal and random polycrystalline

Sharp Fibre texture:Spots and ringsRandom in but not in

Weak Fibre texture:Arcs and ringsRandom in spread in

Nb 110Al111

38o

Two hkl reflections

Pole Figure

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!)

2

2

S = 1/dhkl

S

1/dhkl not constant

One hkl reflection

“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 requiredSpherical shell distorted

2

2S

1/dhkl varies with position

One hkl reflection

04/20/23

Amorphous Material

Fourier transform

1/LL

2

2Theta/Omega scan

Amorphous Halo

04/20/23

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

SAXS - particles Reflectivity – thin films

04/20/23

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

Ln

2

2Theta/Omega scan

Range of lengths

0o 3o

04/20/23

Reflectivity• Reciprocal lattice is very small• Scattering vector must be small

Fourier transform

2

2Theta/Omega scan

0o 3o

04/20/23

Back to Single Crystals

004

113

224

115

440

-440

d*| d*| = 1/dhkl

Just a few points are shown for clarity

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

Single Crystal Substrates and Thin Films

• We investigate the fine structure of individual reciprocal lattice spots

004

113

224

115

440

-440

This requires high resolution instrumentation

“Reciprocal space map” “Scan”

04/20/23

Thin Layers and Multi-layers

• The reciprocal lattices of the crystals and the multilayer combine

Fourier transform

004

113

224

115

-440

004

113

224

115

-440

Reflectivity is known as the 000 reflection

04/20/23

Resolution

Single crystal silicon – measured in 0.001 to 0.01 degree steps 2

Textured Nb/Al multilayer peakmeasured in 0.01 to 1 degree steps in 2

Normal Resolution

High Resolution

Summary• An instrument

– 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

SDetector source

sample

When MRD?• When high resolution is

necessary– 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