Dr. T. K. Chaudhuri, Professor and HeadDr. K. C. Patel Research and Development CentreDr. K. C. Patel Research and Development Centre

X-ray Diffraction: Basics and Applications

INTRODUCTION: HISTORY

Soft X-rays: 0.12 to 12 keV (10 to 0.10 nm)

Hard X-rays: 12 to 120 keV (0.10 to 0.01 nm) -------- penetrates solid objects; used for diagnostic radiography and crystallography.

What are X-rays?

1895 X-rays discovered by Roentgen

1914 First diffraction pattern of a crystal made by Knipping and von Laue

1915 Theory to determine crystal structure from diffraction pattern developed by Bragg

1916 First to study polycrystalline powder : Debye and Scherrer

1953 DNA structure solved by Watson and Crick

Now Diffraction improved by computer technology; methods used to determine atomic structures and in medical applications

History of X-ray : Timeline

• Wilhelm Conrad Röntgen discovered X-Rays in 1895

• X-ray : unknown ray• Won 1901 Nobel prize in

PhysicsWilhelm Conrad Röntgen (1845-1923)

A modern radiograph of a hand Bertha Röntgen’s Hand 8 Nov, 1895

The first kind of scatter process to be recognised was discovered by Max von Laue who was awarded the Nobel prize for physics in 1914 "for his discovery of the diffraction of X-rays by crystals". His collaborators Walter Friedrich and Paul Knipping took the picture on the bottom left in 1912. It shows how a beam of X-rays is scattered into a characteristic pattern by a crystal. In this case it is copper sulphate.

Max von Laue (1897-1960)

The X-ray diffraction pattern of a pure substance is like a fingerprint of the substance. The powder diffraction method is thus ideally suited for characterization and identification of polycrystalline phases.

Bragg’s LawThe father and son team of Sir William Henry and William Lawrence Bragg were awarded the Nobel prize for physics "for their services in the analysis of crystal structure by means of Xrays“ in 1915.

Bragg's law was an extremely important discovery and formed the basis for the whole of what is now known as crystallography. This technique is one of the most widely used structural analysis techniques and plays a major role in fields as diverse as structural biology and materials science.

William LawrenceBragg (1890-1971)

Sir William HenryBragg (1862-1942)

BASICS

Crystalline materials are characterized by the orderly periodic arrangements of atoms.

• The unit cell is the basic repeating unit that defines a crystal.

• Parallel planes of atoms intersecting the unit cell are used to define directions and distances in the crystal.

– These crystallographic planes are identified by Miller indices.

The (200) planes of atoms in NaCl

The (220) planes of atoms in NaCl

The atoms in a crystal are a periodic array of coherent scatterers and thus can diffract X-rays

• Diffraction occurs when each object in a periodic array scatters radiation coherently, producing concerted constructive interference at specific angles.

• The electrons in an atom coherently scatter light.

– The electrons interact with the oscillating electric field of the light wave.

• Atoms in a crystal form a periodic array of coherent scatterers.

– The wavelength of X rays are similar to the distance between atoms.

– Diffraction from different planes of atoms produces a diffraction pattern, which contains information about the atomic arrangement within the crystal

• X Rays are also reflected, scattered incoherently, absorbed, refracted, and transmitted when they interact with matter.

Bragg’s law

• Bragg’s law: condition for constructive interference• dhkl determines peak positions. • Type of atoms determines peak intensity

sin2 hkld

dh

kld

hkl

X-RAY DIFFRACTION METHODS

X-Ray Diffraction Method

Laue Rotating Crystal Powder

OrientationSingle Crystal

Polychromatic BeamFixed Angle

Lattice constantSingle Crystal

Monochromatic BeamVariable Angle

Lattice ParametersPolycrystal (powdered)Monochromatic Beam

Variable Angle

Single Crystals

Individual reciprocal lattice points (with specific hkl) projected onto the detector

Used to determine the electron density within a unit cell

Powder XRD

Random crystallites combine to form spheres of reflectionspheres will always intersect the Ewald sphere, at the detector: diffraction rings at angle

Debye Scherrer Camera A very small amount of powdered material is sealed into a fine capillary tube made from glass that does not diffract x-rays.

The specimen is placed in the Debye Scherrer camera and is accurately aligned to be in the centre of the camera. X-rays enter the camera through a collimator.

• Powder Diffraction or polycrystalline diffraction

Samples: powder, sintered pellets, coatings, engine blocks, … • True random sample results into continuous Debye cone.• linear diffraction pattern: detector scans through an arc that

intersects each Debye cone at a single point

MEASUREMENT OF XRD:

DIFFRACTOMETERS

Bragg-Brentano geometry

• Incident angle is always ½ of the detector angle 2 .

:2 instrument, the tube is fixed, the sample rotates at °/min and the detector rotates at 2 °/min.

: instrument, the sample is fixed and the tube rotates at a rate - °/min and the detector rotates at a rate of °/min.

X-ray tube

Detector

1.Quantitative Phase Analysis and Transmission2.Thin film analysis3.Reflectivity4.Small angle Xray scattering

Phillips X’Pert

1. Powder diffraction2. Thin film diffraction3. Small angle

scattering4. In-plane scattering

Rigaku Ultima IV

Bruker D8 Advance

1. powder samples 2. crystallite size determination3. micro strain analysis4. residual stress analysis5. preferred orientation

Shimadzu LabX XRD-6000

1. Powder Diffraction

2. Independent dual axis θ-2θ linkage drive

3. independent 2θ axis and θ axis drives are selectable

Scintag xrd 2000 pole figure

1.Quantitative Phase Analysis and Transmission2.Thin film analysis3.Reflectivity4.Small angle Xray scattering5. Pole Figure (phi-omega scan)

1. Faster desktop unit 2. Powder diffraction3. No external chiller unit4. Accurate and faster solid state detector5. Useful for cement, industrial mineral, chemistry, pharma

applications

Bruker D2 Phaser

Rigaku Miniflex II

1. General XRD

2. Phase composition analysis

X-ray diffractographs: Analysis

Bragg-Brentano diffractometer For single crystals

2

Only those family of planes whose normal bisect the angle made by incident and reflected rays will give peaks

Here only (100) and (200) but not (110)

For polycrystalline / powder sample

2 2 2

• Randomly oriented crystallites throughout: peaks due to all families of planes

• Basic assumption of powder diffraction: equal no. of crystallites for all families of planes oriented to diffract

10 20 30 40

2

PowderCell 2 .0

Atomic distribution in the unit cell

Peak relative intensities

Unit cell Symmetry and size

Peak positions

a

cb

Peak shapes

Particle size and defects

Background

Diffuse scattering, sample holder, matrix, amorphous phases, etc...

X-Ray Diffraction pattern

Powder diffraction data consists of a record of photon intensity versus detector angle 2.

• Diffraction data can be reduced to a list of peak positions and intensities

– Each dhkl corresponds to a family of atomic planes {hkl}

– individual planes cannot be resolved- this is a limitation of powder diffraction versus single crystal diffraction

hkl dhkl (Å) Relative Intensity (%)

{012} 3.4935 49.8

{104} 2.5583 85.8

{110} 2.3852 36.1

{006} 2.1701 1.9

{113} 2.0903 100.0

{202} 1.9680 1.4

Position[°2q]

Intensity [cts]

25.2000 372.0000

25.2400 460.0000

25.2800 576.0000

25.3200 752.0000

25.3600 1088.0000

25.4000 1488.0000

25.4400 1892.0000

25.4800 2104.0000

25.5200 1720.0000

25.5600 1216.0000

25.6000 732.0000

25.6400 456.0000

25.6800 380.0000

25.7200 328.0000

Raw Data Reduced dI list

Diffraction patterns are best reported using dhkl and relative intensity rather than 2 and absolute intensity.

Why?

- 2 depends on instrumental characteristics such as - dhkl is an intrinsic, instrument-independent, material property

- absolute intensity can vary due to instrumental and experimental parameters

- relative intensities should be instrument independent.

Peak areas are much more reliable than peak heights as a measure of intensity

APPLICATIONS OF X-ray Diffraction

Application of XRDDifferentiation between crystalline and amorphous materials

Diffraction angle 2

Inte

nsit

y

Gas

LiquidAmorphous

Crystal

Diffraction angle 2

Inte

nsit

y

Diffraction angle 2

Inte

nsit

y

Applications of Powder XRD

• Phase Composition of a Sample– Quantitative Phase Analysis: relative amount of each phase

• Unit cell lattice parameters and Bravais lattice symmetry– Index peak positions– Lattice parameters also effect of stress and doping– Residual Strain (macrostrain)

• Crystal Structure– By full profile Rietveld refinement

• Epitaxy/Texture/Orientation

• Crystallite Size and Microstrain– by peak broadening– Other defects (stacking faults, etc.) from peak shapes and width

Peak position-Unit cell parameter refinement -Pattern indexing-Space group determination-Anisotropic thermal exapansion-Microstress analysis (sin2method)-Phase identification

Intensity -Phase abundance-Reaction kinetics-crystal structure analysis (full profile)-phase identification-texture analysis

Width and shape-Instrumental resolution function-Microstructure-Crystallite size-Size distribution-Lattice distortion-Structure mistakes-Dislocations-Composition gradient

Non ambient and dynamic diffraction-Reaction kinetics

Diffraction peak parameters and their applications

Phase Identification

• Diffraction pattern of each phase is its fingerprint

Phases with the same chemical composition can have drastically different diffraction patterns.

peak position and intensity used to detect particular phase

Powder Diffraction File (PDF) or JCPDS File

• Maintained by Joint Committee on Powder Diffraction Standards • Standard for thousands of crystalline phases.• Over 200,000 diffraction patterns.• Quicker phase search by computer assistance• Other useful information like reference also

Quantitative Phase Analysis

• To Know amount of each phase

• Relative peak intensity proportional to relative phase amount

• Thus quantitative phase analysis: to find const. of proportionality

• RIR method: fast and semi-quantitative results

• Whole pattern fitting/Rietveld refinement: accurate but complicated

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8 1

X(phase a)/X(phase b)

I(pha

se a

)/I(p

hase

b)

..

Unit Cell Lattice Parameter Refinement

• By accurate measurement of peak positions over long range• measure peak positions with a peak search algorithm or profile

fitting• profile fitting is more accurate but more time consuming• Can be use to quantify the effect due to alloying, substitutional

doping, temperature and pressure, etc • identify and correct for systematic errors such as specimen

displacement and zero shift

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41

2 (deg.)

Inte

ns

ity

(a

.u.)

00-043-1002> Cerianite- - CeO

2

Crystallite Size and Microstrain

• Both small crystallite (<120nm) & microstrain cause peak broadening

• Crystallite size from peak broadening using Scherrer relation • must know contribution of peak width from the instrument • Williamson-Hull plot can let you separate microstrain and

crystallite size

cos

2L

KB

Crystallite size

β ｓ Particle size

What is Crystallite size?

Preferred Orientation (texture)

• Preferred orientation of crystallites can create a systematic variation in diffraction peak intensities– can qualitatively analyze using a 1D diffraction pattern– a pole figure maps the intensity of a single peak as a

function of tilt and rotation of the sample• this can be used to quantify the texture

(111)

(311)(200)

(220)

(222)(400)

40 50 60 70 80 90 100

Two-Theta (deg)

x103

2.0

4.0

6.0

8.0

10.0

Inte

nsity

(Cou

nts)

00-004-0784> Gold - Au

COMPONENTS OF X-RAY DIFFRACTOMETER

Essential Parts of the Diffractometer• X-ray Tube: the source of X Rays• Incident-beam optics: condition the X-ray beam before it hits

the sample• The goniometer: the platform that holds and moves the sample,

optics, detector, and/or tube• The sample & sample holder• Receiving-side optics: condition the X-ray beam after it has

encountered the sample• Detector: count the number of X Rays scattered by the sample

Parafocusing geometry of Bragg-Brentano Diffractometer

• Detector moves maintaining 2 angle wrt incident ray

• Sample maintains angle wrt incident ray

• Parafocussing: reflected rays from again collimated onto detector

F: X-ray sourceDS: incident-beam divergence-limiting slitSS: Soller slit assemblyS: sampleRS: diffracted-beam receiving slitC: monochromator crystalAS: the anti-scatter slit

X-radiation sources

• Sealed X-ray tubes operate at 1.8 to 3 kW.

• Rotating anode X-ray tubes have efficient heat distribution: operate at 9-18 kW >produce much more flux .

– X-ray generation: rapid decceleration of e’s onto W target

– The target must be water cooled.

– The target and filament must be contained in a vacuum.

Cu

H2O In H2O Out

e-

Be

XRAYS

windowBe

XRAYS

FILAMENT

ANODE

(cathode)

AC CURRENT

window

metal

glass

(vacuum) (vacuum)

X rays production

• Electrons from the filament strike the target anode, producing characteristic radiations of anode via the photoelectric effect.

• Monochromator for filtering particular wavelength

KL

M

Spectral Contamination in Diffraction Patterns

K1

K2

K W L1

K1

K2K1

K2

• The K1 & K2 doublet will almost always be present– Very expensive optics can remove the K2 line– K1 & K2 overlap heavily at low angles and are more

separated at high angles• W lines form as the tube ages: the W filament contaminates the target anode and becomes a new X-ray source

• W and K lines can be removed with optics

Wavelengths for X-Radiation are Sometimes Updated

Copper

Anodes

Bearden

(1967)

Holzer et al.

(1997)

Cobalt

Anodes

Bearden

(1967)

Holzer et al.

(1997)

Cu K1 1.54056Å 1.540598 Å Co K1 1.788965Å 1.789010 Å

Cu K2 1.54439Å 1.544426 Å Co K2 1.792850Å 1.792900 Å

Cu K 1.39220Å 1.392250 Å Co K 1.62079Å 1.620830 Å

Molybdenum

Anodes

Chromium

Anodes

Mo K1 0.709300Å 0.709319 Å Cr K1 2.28970Å 2.289760 Å

Mo K2 0.713590Å 0.713609 Å Cr K2 2.293606Å 2.293663 Å

Mo K 0.632288Å 0.632305 Å Cr K 2.08487Å 2.084920 Å

X-ray Shutter

• X-rays exit the tube through X-ray transparent Be windows.

• X-Ray safety shutters contain the beam so that you may work in the diffractometer without being exposed to the X-rays.

• Being aware of the status of the shutters is the most important factor in working safely with X rays.

Requirement of X-ray optics

• X Rays from an X-ray tube are: – divergent– contain multiple characteristic wavelengths as well as

Bremmsstrahlung radiation• Optics are used to:

– limit divergence of the X-ray beam– refocus X rays into parallel paths– remove unwanted wavelengths

sin2 hkld

Divergence slits

• The slits block X-rays that have too great a divergence.

• The size of the divergence slit influences peak intensity and peak shapes.

• Narrow divergence slits:

– reduce the intensity of the X-ray beam

– reduce the length of the X-ray beam hitting the sample

– produce sharper peaks

• the instrumental resolution is improved so that closely spaced peaks can be resolved.

Monochromators

• crystal monochromator: to select one wavelength of radiation and provide energy discrimination.

• An incident-beam monochromator might be used to select only K1 radiation for the tube source.

• A diffracted-beam monochromator, may be used to remove fluoresced photons, K, or W-contimination photons from reaching the detector.

Other optics:

• limit divergence of the X-ray beam– Divergence limiting slits– Parallel plate collimators– Soller slits

• refocus X rays into parallel paths– “parallel-beam optics”– parabolic mirrors and capillary

lenses– focusing mirrors and lenses

• remove unwanted wavelengths– monochromators– K filters

Parallel Plate Collimator & Soller Slits block divergent X-rays, but do not restrict beam size like a divergent slit

Göbel Mirrors and capillary lenses collect a large portion of the divergent beam and refocus it into a nearly parallel beam

Detectors

• point detectors– observe one point of space at a time

• slow, but compatible with most/all optics– scintillation and gas proportional detectors count all photons,

within an energy window, that hit them– Si(Li) detectors can electronically analyze or filter wavelengths

• position sensitive detectors– linear PSDs observe all photons scattered along a line from 2 to

10° long– 2D area detectors observe all photons scattered along a conic

section– gas proportional (gas on wire; microgap anodes)

• limited resolution, issues with deadtime and saturation– CCD

• limited in size, expensive – solid state real-time multiple semiconductor strips

• high speed with high resolution, robust

SAMPLE / SPECIMEN PREPARATION

Preparing a powder specimen

• An ideal powder sample should have many crystallites in random orientations

– the distribution of orientations should be smooth and equally distributed amongst all orientations

• If the crystallites in a sample are very large, there will not be a smooth distribution of crystal orientations. You will not get a powder average diffraction pattern.

– crystallites should be <10m in size to get good powder statistics

• Large crystallite sizes and non-random crystallite orientations both lead to peak intensity variation

– the measured diffraction pattern will not agree with that expected from an ideal powder

– the measured diffraction pattern will not agree with reference patterns in the Powder Diffraction File (PDF) database

Important characteristics of samples for XRPD

• a flat plate sample for XRPD should have a smooth flat surface– if the surface is not smooth and flat, X-ray absorption may

reduce the intensity of low angle peaks– parallel-beam optics can be used to analyze samples with

odd shapes or rought surfaces• Densely packed• Randomly oriented grains/crystallites• Grain size less than 10 microns• ‘Infinitely’ thick

XRD EXAMPLES

ZnO flowers 3-5 m

XRD of ZnO powder : polycrystalline ZnO powder JCPDF:36-1451

2(degree)

I/Io % d, nm

Miller indices (hkl)

2(degree)

I/Io % d (nm)

31.569 56.91 0.2831 (100) 31.769 57 0.2814 34.228 61.39 0.2617 (002) 34.421 44 0.2603 36.054 100.00 0.2489 (101) 36.252 100 0.2475 47.281 22.60 0.1920 (102) 47.538 23 0.1911 56.396 38.92 0.1630 (110) 56.602 32 0.1624

62.607 34.37 0.1482 (103) 62.862 29 0.1477 66.124 5.52 0.1411 (200) 66.378 4 0.1407 67.704 28.75 0.1382 (112) 67.961 23 0.1378 68.997 12.50 0.1359 (201) 69.098 11 0.1358 72.295 2.91 0.1305 (004) 72.560 2 0.1301 76.828 3.84 0.1239 (202) 76.953 4 0.1238 81.207 2.17 0.1183 (104) 81.368 1 0.1181 89.303 7.95 0.1096 (203) 89.605 7 0.1093 92.487 2.73 0.1066 (210) 92.781 3 0.1063 95.050 8.83 0.1044 (211) 95.301 6 0.1042

98.802 2.97 0.1014 (114) 98.609 4 0.1015 102.742 3.55 0.0986 (212) 102.942 2 0.0984 103.981 6.22 0.0977 (105) 104.130 5 0.0976 107.313 0.65 0.0956 (204) 107.426 1 0.0955 110.033 3.82 0.0940 (300) 110.388 3 0.0938 116.078 7.98 0.0907 (213) 116.274 8 0.0906

(302) 121.567 4 0.0882 (006) 125.183 1 0.0867 (205) 133.925 3 0.0837 (106) 136.513 1 0.0829 (214) 138.505 2 0.0823

(220) 142.910 3 0.0812

200 nm

Aligned ZnO nanorod-array

Texture Coefficient, TC

TC(002) = I(002)/Io(002) ____________________________________________________________________________________________________________

1/3 {I(100)/Io(100)+I(002)/Io(002) + I(101)/Io(101)}

where

I = the measured intensity;

Io = the standard intensity (JCPDS: 36-1451)

ZnO nanorod-array films deposited on glass substrate: preferred orientation

ZnO rod-array film JCPDF:36-1451

2(degree)

I/Io %

d, nm

Miller indices (hkl)

2(degree)

I/Io %

d (nm)

31.80 2.31 0.2852 (100) 31.769 57 0.2814

35.10 100 0.2595 (002) 34.421 44 0.2603

36.8 2.21 0.2482 (101) 36.252 100 0.2475

48.1 2.11 0.1944 (102) 47.538 23 0.1911

63.45 6.0 0.1535 (103) 62.862 29 0.1477

73.1 3.5 0.1374 (004) 72.560 2 0.1301

Biaxially textured YBa2Cu3O7 (YBCO) film on STO substrate

Spray pyrolysis, Thickness: 2 μm, JC ~ 1 MA/sq. cm, Orthorhombic

80x103

70

60

50

40

30

20

10

0

Inte

nsit

y (c

ps)

60°5040302010

2 Theta (degrees)

(00

1)

(00

3)+

STO

(00

4)

(00

5)

(00

6)

+ S

TO

(00

7)

<--

(00

8)

(00

2)

50x103

40

30

20

10

0

Inte

nsity

(cps

)

22°2120191817

Omega (degrees)

YBCO, (005) reflection

FWHM = 1°

Out of plane texturing / orientation

YBCO, (103) reflection

50x103

40

30

20

10

0

Inte

nsi

ty (

cps)

350°300250200150100500

Phi (degrees)

YBCO (103) reflection

FWHM = 1.2°In plane texturing / orientation

FWHM = 36.5 -35.93 = 0.57o= ~0.01 rad.Peak Position 2 = 36.218o

= 18.019o= ~ 0.316 rad.; Cos = 0.999

cosKd

Scherrer relation (<100 nm)

K = 0.89, λ =1.5406 ×10-10 m

Crystallite size = 15 nm

XRD of ZnO nanopowder : use of Scherrer equation

Varying Irradiated area of the sample

• the area of your sample that is illuminated by the X-ray beam varies as a function of:– incident angle of X rays– divergence angle of the X rays

• at low angles, the beam might be wider than your sample– “beam spill-off”

The constant volume assumption

• In a polycrystalline sample of ‘infinite’ thickness, the change in the irradiated area as the incident angle varies is compensated for by the change in the penetration depth

• These two factors result in a constant irradiated volume– (as area decreases, depth increase; and vice versa)

• This assumption is important for many aspects of XRPD– Matching intensities to those in the PDF reference database– Crystal structure refinements– Quantitative phase analysis

• This assumption is not necessarily valid for thin films or small quantities of sample on a ZBH

Sources of Error in XRD Data

• Sample Displacement– occurs when the sample is not on the focusing circle

(or in the center of the goniometer circle)– Can be minimized by using a zero background sample

holder– Can be corrected by using an internal calibration

standard – Can be analyzed and compensated for with many data

analysis algorithms– can be eliminated by using parallel-beam optics

Texture analysis Using Pole figures

Totally different setup: 4 circle goniometer

Scan Axes

Detector

Sample

ψ

2

Source

Pole Figures

z-axis(sample normal)

South pole

-axis

(hkl) plane

Equatorial plane Projected position of the (hkl) plane

Schematic diagram showing the construction of a pole figure.

Projected distance from origin is given by:

r tan (psi/2)

(where r is the radius )

Choosing which planes• Often necessary to use a number of different crystal planes to determine a complicated texture.

• The planes should be chosen from the theta 2 theta pattern. We need:

• strongly diffracting planes (high intensity in theta 2 theta)

• discrete peaks preferably at 2 theta values between 20 degrees and 50 degrees for optimum results.

x x

o

o

oo o

o

(11

1)

(20

0)

(10

0)

(11

0)

(11

1)

(20

0)

(21

0) (2

11

)

Ho

lde

r

Ho

lde

rx x

o

o

oo o

o

(11

1)

(20

0)

(10

0)

(11

0)

(11

1)

(20

0)

(21

0) (2

11

)

Ho

lde

r

Ho

lde

r

x Ag o LaNiO3

Interpretation

•No general methods

• sometimes by visual comparisons with known sample

•an unknown texture or combination of textures can be identified by carrying out a series of pole figures and calculating an Orientation Distribution Function (ODF).

Example: c-axis aligned superconducting thin films.

(b)

(a)

Biaxial Texture (105 planes) Random in-plane alignment