Transcript Practical X-Ray Diffraction

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
Detector
X-ray
tube
Motorized
Source Slits
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)
Bragg’s law and Peak Positions.
l  2d hkl sin q
•
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=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
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
Detector
• Tetragonal PZT
– a=4.0215Å
– b=4.1100Å
R
θ
011
Disp
2θ
110
111
200
002
d
d Actual
Disp  cos2 q

RDetector  sin q
d Actual 
d Measured
1
Disp cos2 q
RDetector sin q
It is important that your sample
be at the correct height
Z-Displacements vs. Change in
Lattice Parameter
Change In Lattice Parameter
Strain/Composition?
a=4.07A
c=4.16A
• Lattice Parameters
–
–
a=4.0215 Å
c=4.1100 Å
Tetragonal PZT
101/110
111
002/200
Z-Displaced Fit
Disp.=1.5mm
Disp
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)
200
220
111
222
311
2q
At 27.42 °2q, Bragg’s law
fulfilled for the (111) planes,
producing a diffraction peak.
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.
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
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 <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
An Examination of Table Salt
200
NaCl
With Randomly Oriented
Crystals
Hint
• Salt Sprinkled on
<100>
double stick tape
Typical Shape
Of Crystals
• What has Changed?
111
220
311 222
It’s the same sample sprinkled on double
stick tape but after sliding a glass slide
across the sample
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.
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
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.
d hkl
a2

h2  k 2  l 2
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
Good for alloys with
continuous solid solutions
d hkl
a2

h2  k 2  l 2
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
I exp hkl
I hkl I Icor  hkl w

I  exp HKL I Icor  hkl I  hkl w
– Where
•
1
I
I cor

Inte nsityof samples' 100%pe ak
Inte nsityof C orundum'
s 100%pe ak
• ω= weight fraction
• I(hkl)=Reference’s relative
intensity
• Iexp(hkl)=Experimental
integrated intensity
Phase Fractions
• Direct Comparison Method
I exp hkl
R  v

I  exp HKL R  v 
– Where

1
2
R  pC LP Fhkl 
V
Ihkl  I0pCLP Fhkl 

2
• 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
Strain Effects
Peak Shifts
and
Peak Broadening
Other Factors contributing to
contribute to
the observed peak profile
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
Crystallite Size Broadening
0.94l
B2q  
Sizecosq
• 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
Methods used to Define Peak Width
• Full Width at Half Maximum
(FWHM)
– 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
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
2q (deg.)
Intensity (a.u.)
• Integral Breadth
Intensity (a.u.)
– the width of the diffraction
peak, in radians, at a height
half-way between background
and the peak maximum
FWHM
46.7
46.8
46.9
47.0
47.1
47.2
47.3
47.4
2q (deg.)
47.5
47.6
47.7
47.8
47.9
Williamson-Hull Plot
FWHM  cosq  
K l
 Strain  4  sinq 
Size
(FWHMobs-FWHMinst)
cos(q )
y-intercept
Gausian Peak Shape Assumed
slope
Grain size broadening
4 x sin(q)
K≈0.94
Which of these diffraction patterns comes
from a nanocrystalline material?
Intensity (a.u.)
Hint: Why are the
intensities different?
66
67
68
69
70
71
72
73
74
2q (deg.)
• 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
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
To determine aspect
ratios
Use 200 and 400
peaks
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 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.
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
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
Reduced dI list
hkl
dhkl (Å)
Relative
Intensit
y (%)
{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
Extra Examples
Crystal Structure
vs.
Chemistry
Two Perovskite Samples
• What are the differences?
•
Assume that they are both
random powder samples
– 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
200
210
2θ (Deg.)
211
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
Intensity(Counts)
•
R(Y3+) = 0.104Å
R(Zr4+) = 0.079Å
45
50
55
2θ (Deg)
60
65
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.
Lorentzian (Cauchy)
BExp  BSize  BStrain  BInst
B
Exp

 BInst  BSize  BStrain
Gaussian
2
2
2
2
BExp
 BSize
 BStrain
 BInst
2
2
2
2
BExp
 BInst
 BSize
 BStrain


Integral Breadth (PV)
2
Exp
 Lorentzian Exp  Gaussian