Practical X-Ray Diffraction

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Practical X-Ray Diffraction

• Prof. Thomas Key
• School of Materials Engineering

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Instrument Settings

• Source
• Cu Ka
• Slits
• Less than 3.0
• Type of measurement
• Coupled 2?
• Detector scan
• Etc.
• Angle Range
• Increment
• Rate (deg/min)
• Detector
• LynxEye (1D)

Bruker D8 Focus
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Coupled 2? Measurements
Detector
Motorized Source Slits
X-ray tube
F
q
w
2q

• In Coupled 2? 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 (?) 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 (F)

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Braggs law and Peak Positions.

• For parallel planes of atoms, with a space dhkl
  between the planes, constructive interference
  only occurs when Braggs 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 Ka11.540598 Å and Cu Ka21.544426 Å
• Or Cu Ka(avg)1.54278 Å
• dhkl is dependent on the lattice parameter
  (atomic/ionic radii) and the crystal structure
• IhklIopCLPFhkl2 determines the intensity of
  the peak

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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

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Z-Displacements
Detector

• Tetragonal PZT
• a4.0215Å
• b4.1100Å

R
?
2?
It is important that your sample be at the
correct height
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Z-Displacements vs. Change in Lattice Parameter
Change In Lattice Parameter Strain/Composition?

• Lattice Parameters
• a4.0215 Å
• c4.1100 Å

a4.07A c4.16A
101/110
Tetragonal PZT
002/200
111
Z-Displaced Fit Disp.1.5mm
Shifts due to z-displacements are systematically
different and differentiable from changes in
lattice parameter
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Sample Preparation

• Crystal Orientation Matters

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Orientations Matter in Single Crystals(a big
piece of rock salt)
2q
The (200) planes would diffract at 31.82 2q
however, they are not properly aligned to produce
a diffraction peak
The (222) planes are parallel to the (111)
planes.
At 27.42 2q, Braggs law fulfilled for the (111)
planes, producing a diffraction peak.
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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

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Sample Preparation

• Inducing Texture In A Powder Sample

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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 lt10mm 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

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An Examination of Table Salt

• Salt Sprinkled on double stick tape
• What has Changed?

NaCl
With Randomly Oriented Crystals
Hint Typical Shape Of Crystals
Its the same sample sprinkled on double stick
tape but after sliding a glass slide across the
sample
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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 cant or dont want to do this
  to many samples.

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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 2? 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
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Phase Identification

• One of the most important uses of XRD

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For cubic structures it is often possible to
distinguish crystal structures by considering
the periodicity of the observed reflections.
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Identifying Non-Cubic Phases
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ICCD JCPDS Files
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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

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Measuring Changes In A Single Phases Composition

• by X-Ray Diffraction

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Vegards 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

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Measuring Changes In Phase Fraction

• Using I/Icor
• and
• Direct Comparison Method

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Phase Fractions

• Using I/Icorr
• Where
• ? weight fraction
• I(hkl)References relative intensity
• Iexp(hkl)Experimental integrated intensity

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Phase Fractions

• Direct Comparison Method
• Where
• vVolume fraction
• VVolume of the unit cell

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Strain Effects

• Peak Shifts
• and
• Peak Broadening

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Other Factors contributing to contribute tothe
observed peak profile
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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 contributions

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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) p102-113.
• Remember
• Instrument contributions must be subtracted

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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
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Williamson-Hull Plot
y-intercept
slope
Grain size and strain broadening
Grain size broadening
K0.94
Gausian Peak Shape Assumed
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Which of these diffraction patterns comes from a
nanocrystalline material?
Hint Why are the intensities different?

• 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

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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

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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.

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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.

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Reporting Data
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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.
• Braggs 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.

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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
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
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
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Extra Examples

• Crystal Structure
• vs.
• Chemistry

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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
200
210
211
2? (Deg.)
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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
R(Y3) 0.104Å R(Zr4) 0.079Å
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Questions
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Supplimental Information
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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

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Dealing With Different Integral Breadth/FWHM
Contributions Contributions

• Lorentzian and Gaussian Peak shapes are treated
  differently
• BFWHM or ß 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)
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