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S. PraveenPhD Scholar MME, IITM
Main Graphics
Additional Graphics
List Pane
Basic steps for any analysis
• Determine Back ground • Search Peaks• Fit Profile
Back ground determination
Bending Factor – 0 to 2Granularity – 15 to 30
Search Peaks
Garbage in Garbage out !
Profile Fitting
Peak matching ‐ ICSD
Peak matching ‐ ICSD
Peak Details
Anatomy of XRD pattern
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Dr. Sharon Mitchell, Prof. Javier Pérez‐RamírezAdvanced Catalysis Engineering, Institute for Chemical and BioengineeringETH Zürich, Switzerland
Peak shape
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AssumptionGaussian – Strain Lorentzian ‐ Size
Profile Functions
• Gaussian– I(2θ) = Imax exp [ − π{ (2θ − 2θ0)2 / β2 } ]
• Lorentzian– I(2θ) =Imax / {(β2 / π)+ (2θ − 2θ0)2 }
• From convolution integral (O(x) = I(x)*S(x) + background)– Gaussian ‐ β2
OG = β2iG + β2
SG
– Lorentzian ‐ βOL = βIL + βSG
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x-ray diffraction line profiles cannot be well represented with a simple Cauchy or Gauss function
Profile Functions
• Voigt (Convolution of G and L)
–
• Pearson VII (Exponential mixing of G and L)
– I(2θ) = Imaxβ2m / [ β 2 + (21/m − 1) (2θ − 2θ0)2 ] m
• Pseudo‐Voigt (Linear addition of G and L) – I(2θ) = Ihkl [η L (2θ − 2θ0) + (1 − η) G (2θ − 2θ0) ]
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Profile Functions
• Voigt (Convolution of G and L)– S = G * L
• Pearson VII (Exponential mixing of G and L)
– S = (x )m
• Pseudo‐Voigt (Linear addition of G and L) – S = (n) L + (1‐n) G
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Profile Functions
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Ceria – 111Powder Diffr., Vol. 23, No. 1, March 2008 Comparison methods of variance and line profile ...
Gauss
Lorentz
Pseudo‐voigt
voigt
Pearson VII
Coherent domain size and strain
• Coherent domain size – Stacking (deformation) or twin (growth) fault, small‐angle boundaries (dislocation), grains
• Strain – disruption of a regular lattice
• dislocations and different point defects
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H.P. Klug & L.E. Alexander, X‐Ray Diffraction Procedures
Diffraction line broadening ‐ Instrument
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21/05/11
0
100000
200000
Counts
Position [°2Theta] (Copper (Cu))46.50 47 47.50 48
Si std 0.02, 0.05 23 5 2011 Si std 0.05, 0.05 23 5 2011 Si std 0.1, 0.05 23 5 2011
Instrumental Broadening
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Diffraction line broadening ‐ Sample
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• O(x) = I(x)*S(x) + background– I(x) = w(x)*g(x)– S(x) = G(x)*L(x)
Size and Strain ‐methods
• Sherrer– <Dv> = Kλ/{β cos θ}
• Stokes and Wilson– ε = β/{4 tan θ}
• Williamson and Hall– {βobs − βinst}cos θ = λ/Dv + 4 εstr{sin θ} (Linear)– {β2
obs − β 2inst} cos θ2 = λ/Dv + { 4 εstr sin θ}2 (quadratic)
• Langford method4/22/2013 21
Gaussian - β2OG = β2
iG + β2 sG
Lorentzian - βOL = βiL + β sL
Nanoscale, 2011, 3, 792–810
Double Voigt method• Double Voigt method (Langford, Balzar)
– Voigt/ pseudo‐voigt/ Pearson VII
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Langford method (average S‐S) plot
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Langford, J. I. (1980). In Accuracy in Powder Diffraction
Powder Diffr., Vol. 23, No. 1, March 2008 Comparison methods of variance and line profile
Methods
• Analytical – Profile functions – Biased – Single Peak
• Integral Breadth
– Multiple Peak• William Son Hall
– Linear, Quadratic
• Langford Method – Average
• Double – Voigt method
• Fourier– Fourier Transform– Un biased– Warren – Averbach
• Fourier series• Stokes deconvolution
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30 40 50 60 70 80 90 100 110
Counts
0
20000
40000
60000 Si std 15 06 2011
0.00
0.03
0.06
0.09
0.12
0.15FWHM^2 = 0.010(2) + -0.010(4) * Tan(Th) + 0.010(2) * Tan(Th)^2, Chi sq.: 4.66601830391445E-6
FWHM Left [°2Th.]
Instrumental Broadening FWHM
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Nanoscale, 2011, 3, 792–810
0.208 0.206
0.569
0.5050.47 0.46
0.423
0.172 0.165
0.3650.322 0.301 0.297 0.292
0
0.1
0.2
0.3
0.4
0.5
0.6
Integral Breadth FWHM
2429
32 3440
43
55
63 6568
0
10
20
30
40
50
60
70
80
SPS HT + 6 h HT + 12 h HT + 18 h HT + 24h
Crystallite Size (nm)
Integral Breadth FWHM
FWHM/Integral Breadth ?
Displacement Error
0
5000
10000
15000
Counts
Position [°2Theta] (Copper (Cu))40 45
NM sample height 1mm above NM sample height 0mm above NM sample height2mm above
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Nanoscale, 2011, 3, 792–810
• Simplified Integral Breadth method– Integral Breadth
• Single peak (Sherrer, Stokes) • Multiple peak (W‐H plot)
• Voigt method – De‐convolution of Gaussian and lorentzian
• Single peak (Sherrer, Stokes) • Multiple peak (Langford method)
• Double‐Voigt method– single peak– Fourier size coefficients
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Summary
THANK YOU
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• Scattering factor , f– Ratio of Amplitude of wave scatterd by an atom to amplitude of wave scattered by an electron
– At higher angle, wave scattered by an electron –out of phase , f decreases
• Structure factor, F– Amplitude of wave scattered by all the atoms to amplitude of wave scattered by one electron
– BCC – (h+k+l)‐ even– FCC – unmixed, either even or odd
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