Post on 26-Jan-2016
description
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