X-ray Diffraction: Principles and Practice

Ashish Garg and Nilesh GuraoDepartment of Materials Science and

EngineeringIndian Institute of Technology Kanpur

Layout of the Lecture

Materials Characterization Importance of X-ray Diffraction

Basics Diffraction X-ray Diffraction Crystal Structure and X-ray Diffraction

Different Methods Phase Analysis Texture Analysis Stress Analysis Particles Size Analysis ………..

Summary

Materials Characterization

Essentially to evaluate the structure and properties

Structural Characterization Diffraction

X-ray and Electron Diffraction Microscopy Spectroscopy

Property Evaluation Mechanical Electrical Anything else

Time Line 1665: Diffraction effects observed by

Italian mathematician Francesco Maria Grimaldi

1868: X-rays Discovered by German Scientist Röntgen

1912: Discovery of X-ray Diffraction by Crystals: von Laue

1912: Bragg’s Discovery

Electromagnetic Spectrum

Generation of X-rays

Commercial X-ray Tube

X-ray Spectrum from an Iron target

Short Wavelength Limit

Continuous spectrum

Characteristic X-ray Moseley’s Law

)( ZCn

kK VVBiI )(

mCS AiZVI

12400(nm)SWL V

λSWL

Use of Filter

Ni filter for Cu Target

Crystal Systems and Bravais Lattices

Structure of Common Materials

Metals Copper: FCC -Iron: BCC Zinc: HCP Silver: FCC Aluminium: FCC

Ceramics SiC: Diamond Cubic Al2O3: Hexagonal MgO: NaCl type

Diffraction A diffracted beam may be defined as a beam

composed of a large number of scattered rays mutually reinforcing each other

ScatteringInteraction with a single particle

DiffractionInteraction with a crystal

Scattering Modes Random arrangement of atoms in space

gives rise to scattering in all directions: weak effect and intensities add

By atoms arranged periodically in space In a few specific directions satisfying Bragg’s law:

strong intensities of the scattered beam :Diffraction

No scattering along directions not satisfying Bragg’s law

d

Diffraction of light through an aperture

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Intensity

an

λsinθ

Minima Maxima

n = 0, 1,.. an2

12sin

n = 1, 2,..

-15 -10 -5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Intensity

d sinθ = mλ, m = 1,2,3…..

d sinθ = (m+½)λ, m = 1,2,3…..

Constructive Interference

Destructive Interference

Young’s Double slit experiment

InterferencePhase Difference = 0˚

Phase Difference = 180˚

Phase Difference = 90˚

Interference and Diffraction

Bragg’s Law

n=2d.sinn: Order of

reflectiond: Plane spacing = : Bragg Angle

Path difference must be integral multiples of the wavelengthin=out

in out

2

2 2 2

a

h k l

Braggs Law

sin2nd

2

1sin d

Geometry of Bragg’s law

The incident beam, the normal to the reflection plane, and the diffracted beam are always co-planar.

The angle between the diffracted beam and the transmitted beam is always 2 (usually measured).

Sin cannot be more than unity; this requires nλ < 2d, for n=1, λ < 2d

λ should be less than twice the d spacing we want to study

Order of reflection

Rewrite Bragg’s law λ=2 sin d/n

A reflection of any order as a first order reflection from planes, real or fictitious, spaced at a distance 1/n of the previous spacing

Set d’ = d/n

An nth order reflection from (hkl) planes of spacing d may be considered as a first order reflection from the (nh nk nl) plane of spacing d’ = d/n

λ=2d’ sin

*The term reflection is only notional due to symmetry between incoming and outgoing beam w.r.t. plane normal, otherwise we are only talking of diffraction.

Reciprocal lattice vectorsUsed to describe Fourier analysis of electron concentration of the diffracted pattern.

Every crystal has associated with it a crystal lattice and a reciprocal lattice.

A diffraction pattern of a crystal is the map of reciprocal lattice of the crystal.

Real space Reciprocal space

Crystal Lattice Reciprocal Lattice

Crystal structure Diffraction pattern

Unit cell content Structure factor

x

y

y’x’

y’x’

Reciprocal space

Reciprocal lattice of FCC is BCC and vice versa

)(*

)(*

)(*

cbabac

cbaacb

cbacba

a

b

c

100

001

010

Ewald sphere

2

hkl2

hkld1

Limiting sphere

Ewald sphere

1

hklk

k'

Ewald sphere

J. Krawit, Introduction to Diffraction in Materials Science and Engineering, Wiley New York 2001

Two Circle Diffractometer For polycrystalline Materials

Four Circle Diffractometer

For single crystals

2 Circle diffratometer 2 and 3 and 4 circle diffractometer 2θ, ω, φ, χ 6 circle diffractometer θ, φ, χ and δ, γ, µ

www.serc.carleton.edu/Hong et al., Nuclear Instruments and Methods in Physics Research A 572 (2007) 942

NaCl crystals in a tube facing X-ray beam

Powder Diffractometer

Calculated Patterns for a Cubic Crystal

(100

)

(110

)

(200

)(1

11)

(210

)(2

11)

(220

)(3

30)(2

21)

(310

)(3

11)

(222

)(3

20)

(321

)

(400

)(4

10)

Structure Factor

2 ( )

1

n n n

Ni hu kv lw

hkl nF f e − h,k,l : indices of the diffraction plane under consideration − u,v,w : co-ordinates of the atoms in the lattice− N : number of atoms − fn : scattering factor of a particular type of atom

Bravais Lattice

Reflections possibly present

Reflections necessarily absent

Simple All NoneBody Centered

(h+k+l): Even (h+k+l): Odd

Face Centered

h, k, and l unmixed i.e. all odd or all even

h, k, and l: mixed

Intensity of the diffracted beam |F|2

Systematic Absences

Simple Cubic

(100), (110), (111), (200), (210), (211), (220), (300), (221) ………

BCC (110), (200), (211), (220), (310), (222)….

FCC (111), (200), (220), (311)…..

Permitted Reflections

Diffraction Methods

Method Wavelength Angle Specimen

Laue Variable Fixed Single Crystal

Rotating Crystal

Fixed Variable (in part)

Single Crystal

Powder Fixed Variable Powder

Laue Method

• Uses Single crystal• Uses White Radiation• Used for determining crystal orientation and quality

Transmission

Zone axis

crystal

Incident beamFilm

Reflection Zone axis

crystal

Incident beam Film

Rotating Crystal Method

Determination of unknown crystal structures

Powder Method

• Useful for determining lattice parameters with high precision and for identification of phases

Incident Beam Sample

Film

Indexing a powder pattern

1

2

Sθ (for front reflections)or2W

Sθ 1 (for back reflections)2 W

Bragg’s Lawn = 2d sin

For cubic crystals

hkl 2 2 2

2 2

2 2 2 2

adh k l

which gives rise to

sin θh k l 4awhich is a constant

IndexingBCC

S1 (mm) () sin2 h2+k2+l2 sin2/ h2+k2+l2

Not BCC

38 19.0 0.11 2 0.055

45 22.5 0.15 4 0.038

66 33.0 0.30 6 0.050

78 39.0 0.40 8 0.050

83 41.5 0.45 10 0.045

97 49.5 0.58 12 0.048

113 56.5 0.70 14 0.050

118 59.0 0.73 16 0.046

139 69.5 0.88 18 0.049

168 84.9 0.99 20 0.050

Not Constant

Simple CubicS1 (mm) () sin2 h2+k2+l2 sin2/ h2+k2+l2

Not Simple Cubic

38 19.0 0.11 1 0.1145 22.5 0.15 2 0.7566 33.0 0.30 3 0.1078 39.0 0.40 4 0.1083 41.5 0.45 5 0.0997 49.5 0.58 6 0.097113 56.5 0.70 8 0.0925118 59.0 0.73 9 0.081139 69.5 0.88 10 0.088168 84.9 0.99 11 0.09

Not Constant

FCC; wavelength=1.54056ÅS1 (mm)

() sin2 h2+k2+l2

sin2/ h2+k2+l2

Lattice Parameter, a (Å)

38 19.0 0.11 3 0.037 4.02345 22.5 0.15 4 0.038 3.97866 33.0 0.30 8 0.038 3.97878 39.0 0.40 11 0.036 4.03983 41.5 0.45 12 0.038 3.97897 49.5 0.58 16 0.036 4.046113 56.5 0.70 19 0.037 4.023118 59.0 0.73 20 0.037 4.023139 69.5 0.88 24 0.037 4.023168 84.9 0.99 27 0.037 4.023

Constant; so it is FCC

But what is the lattice parameter?

Diffraction from a variety of materials

(From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley)

Reality

0.9cos

B

tB

Crystallite size can be calculated using Scherrer Formula

Instrumental broadening must be subtracted(From “Elements of X-ray Diffraction”, B.D. Cullity, Addison Wesley)

• polarization factor• structure factor (F2)• multiplicity factor• Lorentz factor• absorption factor• temperature factor

For most materials the peaks and their intensity are documented

JCPDS

ICDD

Intensity of diffracted beam

Name and formulaReference code: 00-001-1260

PDF index name: Nickel Empirical formula: NiChemical formula: NiCrystallographic parameters

Crystal system: Cubic Space group: Fm-3m

Space group number: 225a (Å): 3.5175 b (Å): 3.5175 c (Å): 3.5175 Alpha (°): 90.0000 Beta (°): 90.0000 Gamma (°): 90.0000

Measured density (g/cm^3): 8.90 Volume of cell (10^6 pm^3): 43.52

Z: 4.00 RIR: -Status, subfiles and quality

Status: Marked as deleted by ICDDSubfiles: InorganicQuality: Blank (B)

ReferencesPrimary reference: Hanawalt et al., Anal. Chem., 10, 475, (1938)

Optical data: Data on Chem. for Cer. Use, Natl. Res. Council Bull. 107 Unit cell: The Structure of Crystals, 1st Ed.

http://ww1.iucr.org/cww-top/crystal.index.html

Stick pattern from JCPDS

Bulk electrodeposited nanocrystalline nickel

Lattice parameter, phase diagrams

Texture, Strain (micro and residual)

Size, microstructure (twins and dislocations)

Actual Pattern

Powder X-ray diffraction

is essentially a misnomer and should be replaced by

Polycrystalline X-ray diffraction

Information in a Diffraction Pattern

Phase Identification Crystal Size Crystal Quality Texture (to some extent) Crystal Structure

Analysis of Single PhaseIn

tens

ity (a

.u.)

2(˚) d (Å) (I/I1)*100

27.42 3.25 1031.70 2.82 10045.54 1.99 6053.55 1.71 556.40 1.63 3065.70 1.42 2076.08 1.25 3084.11 1.15 3089.94 1.09 5I1: Intensity of the strongest peak

Procedure Note first three strongest peaks at d1, d2, and d3 In the present case: d1: 2.82; d2: 1.99 and d3: 1.63 Å Search JCPDS manual to find the d group belonging to the strongest

line: between 2.84-2.80 Å There are 17 substances with approximately similar d2 but only 4

have d1: 2.82 Å Out of these, only NaCl has d3: 1.63 Å It is NaCl……………Hurrah

Specimen and Intensities Substance File Number

2.829 1.999 2.26x 1.619 1.519 1.499 3.578 2.668

(ErSe)2Q 19-443

2.82x 1.996 1.632 3.261 1.261 1.151 1.411 0.891

NaCl 5-628

2.824 1.994 1.54x 1.204 1.194 2.443 5.622 4.892

(NH4)2WO2Cl4 22-65

2.82x 1.998 1.263 1.632 1.152 0.941 0.891 1.411

(BePd)2C 18-225

Caution: It could be much more tricky if the sample is oriented or textured or your goniometer is not calibrated

Presence of Multiple phases More Complex Several permutations combinations possible e.g. d1; d2; and d3, the first three strongest

lines show several alternatives Then take any of the two lines together and

match It turns out that 1st and 3rd strongest lies

belong to Cu and then all other peaks for Cu can be separated out

Now separate the remaining lines and normalize the intensities

Look for first three lines and it turns out that the phase is Cu2O

If more phases, more pain to solve

d (Å) I/I1

3.01 52.47 722.13 282.09 1001.80 521.50 201.29 91.28 181.22 41.08 201.04 30.98 50.91 40.83 80.81 10

*

*

*

*

**

*

Pattern for Cud (Å) I/I1

2.088 1001.808 461.278 201.09 171.0436

5

0.9038

3

0.8293

9

0.8083

8

Remaining Linesd

(Å)I/I1

Observed

Normalized

3.01

5 7

2.47

72 100

2.13

28 39

1.50

20 28

1.29

9 13

1.22

4 6

0.98

5 7

Pattern of Cu2O

d (Å) I/I1

3.020 9

2.465 100

2.135 37

1.743 1

1.510 27

1.287 17

1.233 4

1.0674

2

0.9795

4

Broadeing 2 2 tan

dbd

Lattice Strain

Non-uniform Strain

Uniform Strain

No Strain

do

2

2

2

d strain

Texture in Materials Grains with in a polycrystalline are not

completely randomly distributed Clustering of grains about some particular

orientation(s) to a certain degree Examples:

Present in cold-rolled brass or steel sheets Cold worked materials tend to exhibit some

texture after recrystallization Affects the properties due to anisotropic

nature

Texture Fiber Texture

A particular direction [uvw] for all grains is more or less parallel to the wire or fiber axis

e.g. [111] fiber texture in Al cold drawn wire Double axis is also possible

Example: [111] and [100] fiber textures in Cu wire Sheet Texture

Most of the grains are oriented with a certain crystallographic plane (hkl) roughly parallel to the sheet surface and certain direction [uvw] parallel to the rolling direction

Notation: (hkl)[uvw]

Texture in materials

[uvw] i.e. perpendicular to the surface of all grains is parallel to a direction [uvw]

Also, if the direction [u1v1w1] is parallel for all regions, the structure is like a single crystalHowever, the direction [u1v1w1] is not aligned for all regions, the structure is like a mosaic structure, also called as Mosaic Texture

Pole Figures

(100) pole figures for a sheet material(a) Random orientation (b) Preferred orientation

Thin Film Specimen

Smaller volume i.e. less intensity of the scattered beam from the film

Grazing angle Useful only for polycrystalline specimens

Substrate

Film or CoatingB BGrazing angle (very small, ~1-5)

Thin Film XRD Precise lattice constants measurements derived from 2-

scans, which provide information about lattice mismatch between the film and the substrate and therefore is indicative of strain & stress

Rocking curve measurements made by doing a q scan at a fixed 2 angle, the width of which is inversely proportionally to the dislocation density in the film and is therefore used as a gauge of the quality of the film.

Superlattice measurements in multilayered heteroepitaxial structures, which manifest as satellite peaks surrounding the main diffraction peak from the film. Film thickness and quality can be deduced from the data.

Glancing incidence x-ray reflectivity measurements, which can determine the thickness, roughness, and density of the film. This technique does not require crystalline film and works even with amorphous materials.

Thin Films Specimens

If the sample and substrate is polycrystalline, then problems are less

But if even if one of them is oriented, problems arise In such situations substrate alignment is necessary

(hkl) plane of the substrate

B B

Single Crystal SubstrateSingle Crystal Substrate

B1 B2

B1B2

i.e. No Diffraction from hkl plane

Diffraction from hkl plane occurs

Oriented thin films Bismuth Titanate

thin films on oriented SrTiO3

substrates Only one type of

peaks It apparent that

films are highly oriented

004 00

8

0010

0012 0014

0016

0018

0020 00

2200

24

0026

0028

△△

△

*

2216

△ △

014

4016

/04

16

△00

6 SrTiO3 (100)

SrTiO3 (110)

SrTiO3 (111)

Degree of orientation

Substrate

Film

Side view

[uvw] corresponding to planes parallel to the surface

But what if the planes when looked from top have random orientation?

Top view

Pole Figure

4 Peaks at ~50 Excellent in-

plane orientation

1

1

1

1

2

22

2

1

1

12

2

2

3

3

3

1

2

3

2 sets of peaks at ~ 5, 65 and 85°

Indicating a doublet or opposite twin growth

3 sets of peaks at ~ 35 and 85°

indicating a triplet or triple twin growth

(117) Pole Figures for Bismuth Titanate Films

SrTiO3 (100) SrTiO3 (110) SrTiO3 (111)

Texture Evolution

BNdT/SrTiO3 (111)

3 (100) planes inclined at 54.7° to (110) plane, separated by 120°STO(111)

54.7°

STO(111)

BNdT/SrTiO3 (110)

STO(110)

45

Two (100) planes inclined at 45° to (110) plane in opposite directions

BNdT/SrTiO3 (100)

BNdT [100

]

STO [100]

STO(100)

BNdT(001)

SrTiO3 (100)Film

Rocking Curve An useful method for evaluating the quality of oriented samples such

as epitaxial films is changed by rocking the sample but B is held constant Width of Rocking curve is a direct measure of the range of orientation

present in the irradiated area of the crystal

Inte

nsity

(a

.u.)

()17.5 17.6 17.7 17.8

FWHM = 0.07°

(0010) Rocking curve of (001)-oriented SrBi2Ta2O9 thin film

Inte

nsity

(a

.u.)

()32.4 32.6 32.8 33.0 33.2

(2212) Rocking curve of (116)-oriented SrBi2Ta2O9 thin film

FWHM = 0.171°

B

Normal

Order Disorder Transformation Structure factor is dependent on the

presence of order or disorder within a material

Present in systems such as Cu-Au, Ti-Al, Ni-Fe

Order-disorder transformation at specific compositions upon heating/cooling across a critical temperature

Examples: Cu3Au, Ni3Fe

Order Disorder Transformation Structure factor is dependent on the presence

of order or disorder within a material.

Complete Disorder Example: AB with A and B atoms

randomly distributed in the lattice Lattice positions: (000) and (½ ½ ½)Atomic scattering factor

favj= ½ (fA+fB)Structure Factor, F, is given by

F = Σf exp[2i (hu+kv+lw)] = favj [1+e( i (h+k+l))] = 2. favj when h+k+l is even = 0 when h+k+l is odd

The expected pattern is like a BCC crystal

AB

Order Disorder Transformation Complete Order Example: AB with A at (000) and B at (½

½ ½)

Structure Factor, F, is given by

F = fA e[2i (h.0+k.0+l.0)]+ fA e[2i (h. ½+k. ½+l. ½)]

= fA+fB when h+k+l is even = fA-fB when h+k+l is odd

The expected pattern is not like a BCC crystal, rather like a simple cubic crystal where all the reflections are present.

Extra reflections present are called as superlattice reflections

AB

Order-Disorder Transformation

Disordered Cu3Au

Ordered Cu3Au

Instrumentation

Diffractometer

Source

Optics

Detector

Source Incident Beam Optics

Sample Diffracted Beam Optics

Detector

Incident Beam Part Diffracted Beam Part

Geometry and Configuration

Source Incident Beam Optics

Sample Diffracted Beam Optics

Detector

Incident Beam Part Diffracted Beam Part

Theta-Theta Source and detector move θ, sample fixed

Theta-2Theta Sample moves θ and detector 2θ , source fixed

Vertical configuration Horizontal sample

Horizontal configuration Vertical sample

XYZ translation

Z translation sample alignment

Sample exactly on the diffractometer circle

Knife edge or laser

Video microscope with laser

XY movement to choose area of interest

Sample translation

X-ray generation X-ray tube (λ = 0.8-2.3 Ǻ)

Rotating anode (λ = 0.8-2.3 Ǻ)

Liquid metal

Synchrotron (λ ranging from infrared to X-ray)

Electrons

X-raysBe window

Metal anodeW cathode

X-ray tube

Rotating anode

Small angle anode Large angle anode

Small focal spot Large focal spot

Rotating anode of W or Mo for high flux

Microfocus rotating anode 10 times brighter

Liquid anode for high flux 100 times brighter and small beam size

Gallium and Gallium, indium, tin alloys

Synchrotron provides intense beam but access is limited

Brighter than a thousand suns

High brilliance and coherence

X-ray bulb emitting all radiations from IR to X-rays

http://www.coe.berkeley.edu/AST/srms

Synchrotron

Moving charge emits radiation

Electrons at v~c

Bending magnet, wiggler and undulator

Straight section wiggler and undulator

Curved sections Bending magnet

Filter to remove Kβ For eg. Ni filter for Cu Kβ

Reduction in intensity of Kα

Choice of proper thickness

Slits To limit the size of beam (Divergence slits)

To alter beam profile (Soller slit angular divergence ) Narrow slits Lower intensity + Narrow peak

Mirror focusing and remove Kα2

Mono-chromator remove Kα2

Si Graphite

Beam Profile

Detector

Sample

Mirror

Source

Soller slit

Mirror

Detector

Sample

Source

Parallel beam

Para-focusing

Detector

Sample

SourcePoint focus

ComparisonParallel beam Para-focusingX-rays are aligned X-rays are divergingLower intensity for bulk samples

Higher intensity

Higher intensity for small samples

Lower intensity

Instrumental broadening independent of orientation of diffraction vector with specimen normal

Instrumental broadening dependent of orientation of diffraction vector with specimen normal

Suitable for GI-XRD Suitable for Bragg-Brentano

Texture, stress Powder diffraction

Detectors

Single photon detector (Point or 0D) scintillation detector NaI proportional counter, Xenon gas semiconductor

Position sensitive detector (Linear or 1D) gas filled wire detectors, Xenon gas charge coupled devices (CCD)

Area detectors (2D) wire CCD

3D detector

X-ray photon

Photoelectron orElectron-hole pair

Photomultiplier tube oramplifier Electrical signal

Resolution: ability to distinguish between energies

Energy proportionality: ability to produce signal proportioanl to energy of x-ray photon detected

Sensitivity: ability to detect low intensity levels

Speed: to capture dynamic phenomenon

Range: better view of the reciprocal space

Data collection and analysis

Choose 2θ range

Step size and time per step

Hardware: slit size, filter, sample alignment

Fast scan followed with a slower scan

Look for fluorescence

Collected data: Background subtraction, Kα2 stripping

Normalize data for comparison I/Imax

Summary X-ray Diffraction is a very useful to characterize

materials for following information Phase analysis Lattice parameter determination Strain determination Texture and orientation analysis Order-disorder transformation and many more things

Choice of correct type of method is critical for the kind of work one intends to do.

Powerful technique for thin film characterization