X RAY DIFFRACTION 101X-RAY DIFFRACTION 101

Wednesdays at OneUNL REU in Nanomaterials andUNL REU in Nanomaterials and

Nanoscience

Jeff ShieldDepartment of Mechanical EngineeringDepartment of Mechanical Engineering

University of Nebraska

What we can discoverWhat we can discover . . .

C t l t t i•Crystal structures: size and shape of unit cells•Atomic positions •Determine phases present p

and their relative amounts•Crystal orientation (“texture”)

•Grain/crystal sizes•Grain/crystal sizes•Residual stress/strain

But first some crystallographyBut first, some crystallography. . .

“Peroidicity”:Peroidicity :equal, predictable spacing between atoms

“C t l” P idi 3D f t“Crystal”: Peroidic 3D array of atoms

More definitionsMore definitions

“Unit Cell”:Unit Cell :Smallest, most symmetric part of crystal

• Requirements: Shape of unit cell must completely fill spacecompletely fill space

Unit CellsUnit CellsSeven “Crystal Systems”

Bravais LatticeBravais LatticeDescribes how to “decorate” the

seven crystal systemsy ySimple (Primitive)—P Body-centered—I Face centered FFace-centered—F

Cubic: P, I, F

T t l P ITetragonal: P, I

Hexagonal: P

Rhombohedral: R

Orthorhombic: P, I, F, C

Monoclinic: P, C

Triclinic: P

Back to X-raysBack to X raysX-radiation: Part of electromagnetic spectrum

Why can x-rays be used to analyze crystals via diffraction?Answer: Their wavelength is of the same order as the distance between atoms

X-ray GenerationX ray GenerationLaboratory-scale:

i

X-ray “Tube”0 V

e-

e-

X

Target

X-rays

-40 kV Target

Characteristic X-raysCharacteristic X raysIncoming electrons ionize atomsRelaxation leads to emission of x-ray e- e-

ΔECharacteristic of atome

e-

eΔE

e-

e-

e-

e-

X-ray

Need to isolate one wavelength:Need to isolate one wavelength:“Monochromation”

WavesWaves

WavesWaves

Bragg’s LawggSo, conditions for constructive interference (“DIFFRACTION”)

Path difference = λ

•Ray A’C’ travels farther than AC by the distance shown in red•Each red segment is dsin θ in length

•So, diffraction occurs when

λ= 2dsin θλ= 2dsin θ

Diffraction Experimentsp“Bragg-Brentano” geometry

Diffraction angle is changed systematically-either sample and detector rotate, or source and point

detector rotate

Source Detector

Diffraction PatternsDiffraction Patterns700080009000

10000

y100020003000400050006000

Inte

nsity

020 40 60 80 100 120

Two Theta

P k t d h diff ti diti ti fi dPeaks are generated when diffraction conditions satisfiedEach peak is from a different set of planes in the crystalIn a powder or polycrystalline material, there will be peaks from all possible planesp p

Alternatively, detection by an “AREA DETECTOR” captures all diffraction peaks at once

Quick but detectors are expensiveQuick, but detectors are expensive

Diffraction Experimentsp“Debye-Scherrer” geometry

Powder sample is hit with x-rays

Sample

AreaD t tDetector

Diffraction PatternsDiffraction Patterns

700080009000

10000

•Crystals with different

200030004000500060007000

Inte

nsity

Bravais lattices will have different diffraction patterns•Two crystals can have

01000

20 40 60 80 100 120

Two Theta

Two crystals can have the same Bravais lattice, but they will always have different lattice

t

8000

10000

12000parameters

Thus, diffraction patterns are UNIQUE to

2000

4000

6000

pa given crystal (aka,

phase)

“PHASE ANALYSIS”0

40 50 60 70 80

PHASE ANALYSIS

Diffraction Peak PositionsDiffraction Peak Positions

700080009000

10000

Peak positions

200030004000500060007000

Inte

nsity

p

•λ=2dsin θ

•For cubic materials:0

1000

20 40 60 80 100 120

Two Theta

•For cubic materials:

d=a/(h2+k2+l2)½

•So, knowing {hkl}, you can find a!

Diffraction Peak IntensitiesDiffraction Peak IntensitiesThe peak intensities tell us two primary things:p p y g

1. Atom positions

10000

2. Crystal orientation

50006000700080009000

ensi

ty

01000200030004000In

te

020 40 60 80 100 120

Two Theta

Diffraction Peak Intensities: Atom Positions

The intensity of a given diffraction peak {hkl}:

80009000

10000

y g p { }

I = KןF 2p ן

30004000500060007000

Inte

nsityK is a constant for a peak

F is the structure factor

010002000

20 40 60 80 100 120

Two Theta

p is the multiplicity

Two Theta

d t itixi, yi and zi are atom positionsf is the atomic scattering factor—depends on atom type and diffraction angle

Diffraction Peak Intensities: Orientations

I = KןF 2p ן

i th lti li it Random orientationp is the multiplicity

“p” is usually assumed

Random orientation

p yfor random orientation. If non-random, p depends on the degree of crystal

Non-random orientation

on the degree of crystal orientation

Diffraction Peak WidthsDiffraction Peak Widths

700080009000

10000

Peak widths tell

200030004000500060007000

Inte

nsityus:

1 Crystallite size0

1000

27 28 29 30

Two Theta

1. Crystallite size

2. Strain

•Smaller grains/crystals → broader peaks•More strain → broader peaks

X-ray DiffractionX ray DiffractionTells us a lot about our material

Crystal/atomic structureOrientationGrain sizeGrain size

Stress/strain

Lots more possible!!!Lots more possible!!!• In situ experiments (temperature, stress, etc.)• High-intensity sources (Synchrotron sources)

Synchrotron I t itiIntensities

The EndThe End