Practical X-Ray Diffraction

Prof. Thomas Key

School of Materials Engineering

Instrument Settings

• Source– Cu Kα

• Slits– Less than 3.0

• Type of measurement– Coupled 2θ – Detector scan– Etc.

• Angle Range– Increment– Rate (deg/min)

• Detector– LynxEye (1D) Bruker D8 Focus

Coupled 2θ Measurements

X-ray tube

Detector

Φ

• In “Coupled 2θ” Measurements:– The incident angle is always ½ of the detector angle 2 . – The x-ray source is fixed, the sample rotates at °/min and the detector

rotates at 2 °/min.• Angles

– The incident angle (ω) is between the X-ray source and the sample.– The diffracted angle (2) is between the incident beam and the detector. – In plane rotation angle (Φ)

Motorized Source Slits

Bragg’s law and Peak Positions.

• For parallel planes of atoms, with a space dhkl between the planes, constructive interference only occurs when Bragg’s law is satisfied.

– First, the plane normal must be parallel to the diffraction vector• Plane normal: the direction perpendicular to a plane of atoms• Diffraction vector: the vector that bisects the angle between the incident and diffracted

beam – X-ray wavelengths are:

• Cu Kα1=1.540598 Å and Cu Kα2=1.544426 Å• Or Cu Kα(avg)=1.54278 Å

– dhkl is dependent on the lattice parameter (atomic/ionic radii) and the crystal structure

– Ihkl=IopCLP[Fhkl]2 determines the intensity of the peak

sin2 hkld

Sample Preparation(Common Mistakes and Their Problems)

• Z-Displacements– Sample height matters– Causes peaks to shift

• Sample orientation of single crystals – Affects which peaks are observed

• Inducing texture in powder samples– Causes peak integrated intensities to vary

Z-Displacements

011

110

111

002200

sin

cos2

DetectorActual R

Disp

d

d

R• Tetragonal PZT

– a=4.0215Å– b=4.1100Å

sincos21

DetectorRDisp

MeasuredActual

dd

Disp 2θ

θ

It is important that your sample be at the correct height

Detector

Z-Displacements vs. Change in Lattice Parameter

• Lattice Parameters– a=4.0215 Å

– c=4.1100 Å

Z-Displaced Fit

Disp.=1.5mm

Change In Lattice Parameter Strain/Composition?

101/110

111 002/200

Disp

a=4.07A c=4.16A

Tetragonal PZT

Shifts due to z-displacements are systematically different and differentiable from changes in lattice parameter

Sample Preparation

Crystal Orientation Matters

Orientations Matter in Single Crystals(a big piece of rock salt)

2

At 27.42 °2, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

The (200) planes would diffract at 31.82 °2; however, they are not properly aligned to produce a diffraction peak

The (222) planes are parallel to the (111) planes.

111

200

220

311

222

For phase identification you want a random powder (polycrystalline) sample

2 2 2

• When thousands of crystallites are sampled, for every set of planes, there will be a small percentage of crystallites that are properly oriented to diffract

• All possible diffraction peaks should be exhibited • Their intensities should match the powder diffraction file

111

200

220

311

222

Sample Preparation

Inducing Texture In A Powder Sample

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

• Salt Sprinkled on double stick tape

• What has Changed?

NaCl

An Examination of Table Salt

It’s the same sample sprinkled on double stick tape but after sliding a glass slide across the sample

<100>Hint

Typical Shape Of Crystals

200

111220 311 222

With Randomly Oriented Crystals

Texture in Samples

• Common Occurrences– Plastically deformed metals

(cold rolled)– Powders with particle

shapes related to their crystal structure

• Particular planes form the faces• Elongated in particular

directions (Plates, needles, acicular,cubes, etc.)

• How to Prevent– Grind samples into fine

powders– Unfortunately you can’t or

don’t want to do this to many samples.

A Simple Means of Quantifying Texture

• Lotgering degree of orientation (ƒ)– A comparison of the relative intensities of a

particular family of (hkl) reflections to all observed reflections in a coupled 2θ powder x-ray diffraction (XRD) Spectrum

– ƒ is specifically considered a measure of the “degree of orientation” and ranges from 0% to 100%

– po is p of a sample with a random crystallographic orientation.

Jacob L. Jones, Elliott B. Slamovich, and Keith J. Bowman, “Critical evaluation of the Lotgering degree of orientation texture indicator,” J. Mater. Res., Vol. 19, No. 11, Nov 2004

Where for (00l)

– Ihkl is the integrated intensity of the (hkl) reflection

Phase Identification

One of the most important uses of XRD

For cubic structures it is often possible to distinguish crystal structures by considering

the periodicity of the observed reflections.

222

2

lkh

adhkl

Identifying Non-Cubic Phases

ICCD: JCPDS Files

Phase Identification

• One of the most important uses of XRD• Typical Steps

– Obtain XRD pattern– Measure d-spacings– Obtain integrated intensities– Compare data with known standards in the – JCPDS file, which are for random orientations

• There are more than 50,000 JCPDS cards of inorganic materials

Measuring Changes In A Single Phase’s Composition

by X-Ray Diffraction

Vegard’s Law

222

2

lkh

adhkl

Good for alloys with continuous solid solutions

Ex) Au-Pd • To create the plot on the right

Using the crystal structure of the alloy calculate “a” for each metal

Draw a straight line between them as shown on the chart to the left.

• To calculate the composition

• Calculate “a” from d-spacings

• “a” will be an atomic weighted fraction of “a” of the two metal

Measuring Changes In Phase Fraction

Using I/Icor

and

Direct Comparison Method

Phase Fractions• Using I/Icorr

–Where

•

• ω= weight fraction• I(hkl)=Reference’s relative intensity

• Iexp(hkl)=Experimental integrated intensity

whklIhkl

whklhklI

HKLI

hklI

IcorI

IcorI

exp

exp

1

peak 100% sCorundum' of Intensity

peak 100% ssample' of Intensity

I

I

cor

Phase Fractions• Direct Comparison Method

– Where

• v=Volume fraction• V=Volume of the unit cell

vR

vR

HKLI

hklI

exp

exp

2 hklP FpCL

1R

V

2 hklP0hkl FpCLII

Because this is already a

complicated method, many choose to go

ahead and use Rietveld Refinement

Strain Effects

Peak Shifts and

Peak Broadening

Other Factors contributing to contribute to

the observed peak profile

Many factors may contribute tothe observed peak profile

• Instrumental Peak Profile– Slits– Detector arm length

• Crystallite Size• Microstrain

– Non-uniform Lattice Distortions (aka non-uniform strain)– Faulting– Dislocations– Antiphase Domain Boundaries– Grain Surface Relaxation

• Solid Solution Inhomogeneity• Temperature Factors

• The peak profile is a convolution of the profiles from all of these The peak profile is a convolution of the profiles from all of these contributionscontributions

Crystallite Size Broadening

• Peak Width B(2) varies inversely with crystallite size• The constant of proportionality, K (the Scherrer constant) depends

on the how the width is determined, the shape of the crystal, and the size distribution– The most common values for K are 0.94 (for FWHM of spherical

crystals with cubic symmetry), 0.89 (for integral breadth of spherical crystals with cubic symmetry, and 1 (because 0.94 and 0.89 both round up to 1).

– K actually varies from 0.62 to 2.08– For an excellent discussion of K,

JI Langford and AJC Wilson, “Scherrer after sixty years: A survey and some new results in the determination of crystallite size,” J. Appl. Cryst. 11 (1978) p102-113.

• Remember: – Instrument contributions must be subtracted

cos

94.02

SizeB

46.746.846.947.047.147.247.347.447.547.647.747.847.9

2 (deg.)

Inte

nsity

(a.

u.)

46.7 46.8 46.9 47.0 47.1 47.2 47.3 47.4 47.5 47.6 47.7 47.8 47.9

2 (deg.)

Inte

nsity

(a.

u.)

Methods used to Define Peak Width• Full Width at Half Maximum

(FWHM)– the width of the diffraction

peak, in radians, at a height half-way between background and the peak maximum

• Integral Breadth– the total area under the peak

divided by the peak height– the width of a rectangle having

the same area and the same height as the peak

– requires very careful evaluation of the tails of the peak and the background

FWHM

Williamson-Hull Plot

4 x sin()

(FW

HM

ob

s-F

WH

Min

st)

c

os

()

sin4cos

StrainSize

KFWHM

y-intercept slope

K≈0.94

Grain size broadeningGrain size and stra

in broadening

Gausian Peak Shape Assumed

66 67 68 69 70 71 72 73 74

2 (deg.)

Inte

ns

ity

(a

.u.)

Which of these diffraction patterns comes from a nanocrystalline material?

• These diffraction patterns were produced from the exact same sample

• The apparent peak broadening is due solely to the instrumentation– 0.0015° slits vs. 1° slits

Hint: Why are the intensities different?

Remember, Crystallite Size is Different than Particle Size

• A particle may be made up of several different crystallites

• Crystallite size often matches grain size, but there are exceptions

Anistropic Size Broadening• The broadening of a single diffraction peak is the product of the

crystallite dimensions in the direction perpendicular to the planes that produced the diffraction peak.

Use 111 and 222 peaks

Use 200 and 400 peaks

To determine aspect ratios

Crystallite Shape

• Though the shape of crystallites is usually irregular, we can often approximate them as:– sphere, cube, tetrahedra, or octahedra– parallelepipeds such as needles or plates– prisms or cylinders

• Most applications of Scherrer analysis assume spherical crystallite shapes

• If we know the average crystallite shape from another analysis, we can select the proper value for the Scherrer constant K

• Anistropic peak shapes can be identified by anistropic peak broadening– if the dimensions of a crystallite are 2x * 2y * 200z, then (h00) and (0k0)

peaks will be more broadened then (00l) peaks.

Reporting Data

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

intensity.• The peak position as 2 depends on instrumental characteristics

such as wavelength.– The peak position as dhkl is an intrinsic, instrument-independent,

material property.• Bragg’s Law is used to convert observed 2 positions to dhkl.

• The absolute intensity, i.e. the number of X rays observed in a given peak, can vary due to instrumental and experimental parameters. – The relative intensities of the diffraction peaks should be instrument

independent.• To calculate relative intensity, divide the absolute intensity of every peak by

the absolute intensity of the most intense peak, and then convert to a percentage. The most intense peak of a phase is therefore always called the “100% peak”.

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

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[°2]

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

Extra Examples

Crystal Structure

vs.

Chemistry

Two Perovskite Samples• What are the differences?

– Peak intensity

– d-spacing

• Peak intensities can be strongly affected by changes in electron density due to the substitution of atoms with large differences in Z, like Ca for Sr.

SrTiO3 and CaTiO3

2θ (Deg.)

Assume that they are both random powder samples

200 210 211

45 50 55 60 65

2θ (Deg)

Inte

nsi

ty(C

ou

nts

)

Two samples of Yttria stabilized Zirconia

• Substitutional Doping can change bond distances, reflected by a change in unit cell lattice parameters

• The change in peak intensity due to substitution of atoms with similar Z is much more subtle and may be insignificant

10% Y in ZrO2

50% Y in ZrO2

Why might the two patterns differ?

R(Y3+) = 0.104ÅR(Zr4+) = 0.079Å

Questions

Supplimental Information

Free Software

• Empirical Peak Fitting– XFit– WinFit

• couples with Fourya for Line Profile Fourier Analysis

– Shadow• couples with Breadth for Integral Breadth Analysis

– PowderX– FIT

• succeeded by PROFILE

• Whole Pattern Fitting– GSAS– Fullprof– Reitan

• All of these are available to download from http://www.ccp14.ac.uk

Dealing With Different Integral Breadth/FWHM Contributions Contributions

• Lorentzian and Gaussian Peak shapes are treated differently

• B=FWHM or β in these equations

• Williamson-Hall plots are constructed from for both the Lorentzian and Gaussian peak widths.

• The crystallite size is extracted from the Lorentzian W-H plot and the strain is taken to be a combination of the Lorentzian and Gaussian strain terms.

2222InstStrainSizeExp BBBB

InstStrainSizeExp BBBB

Gaussian

Lorentzian (Cauchy)

Integral Breadth (PV)

StrainSizeInstExp BBBB

2222StrainSizeInstExp BBBB

GaussianExpLorentzianExp 2