1. Practical X-Ray Diffraction
Prof. Thomas Key
School of Materials Engineering

2. Instrument Settings Bruker D8 Focus Source Slits Type of measurement
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

3. Coupled 2θ Measurements
Detector
Motorized Source Slits
X-ray tube Φ w q 2q
In “Coupled 2θ” Measurements:
The incident angle w is always ½ of the detector angle 2q .
The x-ray source is fixed, the sample rotates at q °/min and the detector rotates at 2q °/min.
Angles
The incident angle (ω) is between the X-ray source and the sample.
The diffracted angle (2q) is between the incident beam and the detector.
In plane rotation angle (Φ)

4. 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 l are:
Cu Kα1= Å and Cu Kα2= Å
Or Cu Kα(avg)= 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
draw the diffraction vector on this slide, or make a second slide explicitly illustrating the diffraction vector

5. 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

6. It is important that your sample be at the correct height
Z-Displacements
Detector
Tetragonal PZT
a=4.0215Å
b=4.1100Å R
011
110
111
002
200 θ
Disp 2θ
It is important that your sample be at the correct height

7. Z-Displacements vs. Change in Lattice Parameter
Change In Lattice Parameter Strain/Composition?
Lattice Parameters
a= Å
c= Å
a=4.07A c=4.16A
101/110
Tetragonal PZT
002/200
111
Z-Displaced Fit
Disp.=1.5mm
D= 1.5 mm
a [A] 4.07
c [A] 4.16
Disp
Shifts due to z-displacements are systematically different and differentiable from changes in lattice parameter 7

8. Crystal Orientation Matters
Sample Preparation
Crystal Orientation Matters

9. Orientations Matter in Single Crystals (a big piece of rock salt)
111
200
220
311
222 2q
The (200) planes would diffract at °2q; however, they are not properly aligned to produce a diffraction peak
The (222) planes are parallel to the (111) planes.
At °2q, Bragg’s law fulfilled for the (111) planes, producing a diffraction peak.

10. For phase identification you want a random powder (polycrystalline) sample
200
220
111
222
311 2q 2q 2q
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

11. Inducing Texture In A Powder Sample
Sample Preparation
Inducing Texture In A Powder Sample

12. 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 <10mm 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

13. An Examination of Table Salt
200
111
220
311
222
Salt Sprinkled on double stick tape
What has Changed?
NaCl
<100>
With Randomly Oriented Crystals
Hint Typical Shape Of Crystals
It’s the same sample sprinkled on double stick tape but after sliding a glass slide across the sample

14. Texture in Samples Common Occurrences How to Prevent
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.

15. 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.
Where for (00l)
Ihkl is the integrated intensity of the (hkl) reflection
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

16. One of the most important uses of XRD
Phase Identification
One of the most important uses of XRD

17. For cubic structures it is often possible to distinguish crystal structures by considering the periodicity of the observed reflections.

18. Identifying Non-Cubic Phases

20. ICCD: JCPDS Files

21. 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

23. Measuring Changes In A Single Phase’s Composition
by X-Ray Diffraction

24. Good for alloys with continuous solid solutions
Vegard’s Law
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

25. Measuring Changes In Phase Fraction
Using I/Icor
and
Direct Comparison Method

26. Phase Fractions Using I/Icorr Where ω= weight fraction
I(hkl)=Reference’s relative intensity
Iexp(hkl)=Experimental integrated intensity 1

27. Phase Fractions Direct Comparison Method Where
v=Volume fraction
V=Volume of the unit cell
Because this is already a complicated method, many choose to go ahead and use Rietveld Refinement

28. Peak Shifts and Peak Broadening
Strain Effects
Peak Shifts
and
Peak Broadening

30. Other Factors contributing to contribute to the observed peak profile

31. Many factors may contribute to the 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 contributions

32. Crystallite Size Broadening
Peak Width B(2q) 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) p
Remember:
Instrument contributions must be subtracted

33. Methods used to Define Peak Width
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 q
(deg.)
Intensity (a.u.)
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
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 q
(deg.)
Intensity (a.u.)

34. Williamson-Hull Plot y-intercept slope K≈0.94
Grain size and strain broadening
Grain size broadening
K≈0.94
Gausian Peak Shape Assumed

35. Hint: Why are the intensities different?
Which of these diffraction patterns comes from a nanocrystalline material? 66 67 68 69 70 71 72 73 74 2 q
(deg.)
Intensity (a.u.)
Hint: Why are the intensities different?
These two diffraction patterns come from the exact same sample (silicon).
The apparent difference in peak broadening is due to the instrument optics, not due to specimen broadening
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 35

36. 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
TEM images collected by Jane Howe at Oak Ridge National Laboratory.

37. 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

38. 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.

39. Reporting Data

40. Diffraction patterns are best reported using dhkl and relative intensity rather than 2q and absolute intensity.
The peak position as 2q 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 2q 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.

41. Powder diffraction data consists of a record of photon intensity versus detector angle 2q.
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
Raw Data
Reduced dI list
Position
[°2q]
Intensity
[cts]
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

42. Extra Examples
Crystal Structure
vs.
Chemistry

43. 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.
Assume that they are both random powder samples
SrTiO3 and
CaTiO3
Ca Z=20; Sr Z=38
Zr Z=40; Y Z=39
Plays into why having an un-textured sample is important
200
210
211
2θ (Deg.) 43

44. Two samples of Yttria stabilized Zirconia
Why might the two patterns differ?
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 45 50 55 60 65
2θ (Deg)
Intensity(Counts)
Ca Z=20; Sr Z=38
Zr Z=40; Y Z=39
Plays into why having an un-textured sample is important
R(Y3+) = 0.104Å
R(Zr4+) = 0.079Å 44

45. Questions

46. Supplimental Information

47. Free Software Empirical Peak Fitting Whole Pattern 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

48. 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.
Lorentzian (Cauchy)
Gaussian
Integral Breadth (PV)