The WPPD and Rietveld Method
Joseph H. Reibenspies  Ph.D.
Nattamai Bhuvanesh Ph.D.
X-ray Diffraction Laboratory
Texas A & M University
All rights reserved.  Please do not copy, modify or distribute these slideswithout the consent of the authors.
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X-ray Diffraction Laboratory 1.0.2
Pattern fitting
If you know the unit cell, spacegroup and atomiccoordinates in a crystalline material you cangenerate peak positions and predicted intensities.
If you know the peak positions and intensitiesyou can generate a theoretical pattern bycombining a series of peak profiles centered atthose positions.
The whole powder pattern decomposition andRietveld method employ non-linear leastsquares to refine a theoretical pattern until it“fits” the measured pattern.
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Pawley and Rietveld Method
G.S. Pawley and Hugo M. Rietveld 
Minimize 𝐷= 𝑖   𝑤 𝑖     𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐   2   
 𝑦 𝑜𝑏𝑠 
Uncorrected “raw” data
No Smoothing
No Background correction
Best data possible
“garbage in garbage out”
 𝑦 𝑐𝑎𝑙𝑐   :  Scale Factor*intensity*peak_profile + background
Pawley, G.  (1981) Journal of Applied Crystallography  14 , 357-361
H. M. Rietveld (1969)  Journal of Applied Crystallography 2 (2): 65–71. 
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yicalc  (I)
  𝑦 𝑖  𝑐𝑎𝑙𝑐 = 𝑗 𝑁  𝐾 𝑗  𝑝 𝑀  𝐼 𝑝,𝑗 Ω 2 𝜃 𝑖 −2 𝜃 𝑝,𝑗  + 𝑏𝑘𝑔 𝑖   
Kj : Scale factor for phases j to N
 𝐼 𝑝 = integrated intensity (Pawley)
 𝐼 𝑝 = 𝐿𝑝𝑃𝐴 𝐹 2   (Reitveld)
Lp : Lorentzian Polarization
P : Preferred Orientation function
A : Absorption Correction
F  : Structure Factor
Ω 2 𝜃 𝑖 −2 𝜃 𝑝,𝑗   
Peak Shape Function
Gaussian, Cauchy (Lorentzian), Pseudo-Voigt, etc.
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Notes
Ip  Integrated intensity from observed data.
Whole  Powder Pattern Decomposition 
Unit Cell and Spacegroup, no coordinate information is needed.
 𝐼 𝑝 ~ 𝐹 2 
Structure refinement against observed data
Crystal information (unit cell, spacegroup, coordinates etc) is needed
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Intensity
 𝐼 𝑝 ~𝐿𝑝𝑃𝐴
𝐿𝑝= 1+𝑀  cos 2  2𝜃   1+𝑀  sin 2𝜃    
 M =1 (no monochromator)  M=   cos 2  2 𝜃 𝑚  
P : March Dollase Formula
𝑃= 1 𝑚  𝑖 𝑚     𝑃 𝑚𝑑  2   cos 2   𝛼 𝑖 +   sin 2   𝛼 𝑖    𝑃 𝑚𝑑     − 3 2       
Pmd  March-Dollase Parameter
 = angle preferred orientation vector and the plane hkl at p
m sum of all equivalent hkls (at peak position p)
A is the linear absorption coefficient 
Dependent on 2
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I  (scattering factor)
 𝐼 𝑝 ~ 𝐹 2     :  Scattering Factor
  𝐹 ℎ𝑘𝑙  2 =𝑚   𝑛   𝑓 𝑛  𝑒 −𝐵   sin 2   𝜃 ℎ𝑘𝑙    𝜆 2     𝑒 2𝜋𝑖 ℎ 𝑥 𝑛 +𝑘 𝑦 𝑛 +𝑙 𝑧 𝑛       2   
m : multiplicity
fn : atomic scattering factor for the nth atom
B : Temperature Factor
x,y,z : coordinates for the nth atom
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Peak Shape Function
Ω 2 𝜃 𝑖 −2 𝜃 𝑝,𝑗   
Gaussian
Cauchy (Lorentzian)
Pseudo-Voigt (PV)
Split SPV
Pearson VII
Split SPVII
Fundamental Parameters (FP)*
Convolution of the emission profile of the X-rays (L) and the aberration functions (D) of the instrument.
𝜴 𝟐 𝜽 𝒊 −𝟐 𝜽 𝒑,𝒋  =𝐿∗ 𝐷 1𝑝𝑗 ∗ 𝐷 2𝑝𝑗 … 𝐷 𝑛𝑗
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*Cheary, R. and Coelho, A. J. Appl. Cryst. (1992). 25, 109-121.
Peak Shape Parameter
 𝑆 𝑖𝑝𝑗 = 2 𝜃 𝑖 −2 𝜃 𝑝,𝑗  𝐻    
FWHM for crystalline compounds Vary in 2 and follow the Caglioti Formula1
 𝐹𝑊𝐻𝑀 2 =𝑈  tan 2  𝜃 + Vtan 𝜃+𝑊 
Recommended values2
U = 0.003092,  V = -0.00219,  and  W = 0.00476
For a Pearson VII function with the peak shape parameter  would be  ..
1Caglioti, G., Paoletti, A., Ricci, F.P.  Nucl. Instr. and Meth., 3 (1958), p. 223-228.
Lowe-Ma, C.K., Cline, J.P., Crowder, C.E., Kaduk, J.A., Robie, S.B., Smith, D.K., Young,R.A. Adv. Xray Anal. (1997) 40, 390.
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𝜴 𝟐 𝜽 𝒊 −𝟐 𝜽 𝒑,𝒋   =𝑪  𝟏+𝟒( 𝟐  𝟏 𝒎  −𝟏)   𝑺 𝒊𝒑𝒋   𝟐    −𝒎  𝒎=𝟏,𝟐…∞
Background (bkg)
Background 
Reciprocal
𝑏𝑘𝑔 2  𝑖  = 𝑛    𝑎 𝑛  2 𝜃 𝑖   
Polynomial
𝑏𝑘𝑔 2  𝑖  =  𝑛   𝑎 𝑛   2  𝑖   𝑛  
Chebyshev polynomial
𝑏𝑘𝑔 2  𝑖  =  𝑛=0 𝑚  𝑎 𝑛  𝑡 𝑛 ( 2 𝑖 ) 
1st  kind
2nd kind
n
tn(x)  1st Kind
0
1
1
x
2
2x2-1
3
4x3-3x
4
8x4-8x2+1
n>2
tn(x) = 2x.tn-1(x)-tn-2(x)
n
tn(x) 2nd kind
0
1
1
2x
2
4x2-1
3
8x3-4x
4
16x4-12x2+1
n>2
tn(x) = 2x.tn-1(x)-tn-2(x)
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Refinement
Non-Linear Least Squares
The goal is to minimize  𝑖   𝑤 𝑖     𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐   2   
Non-linear least squares “fit” will involve many cyclic determinations.
Each determination provides parameter shifts which when applied to the yield better values
The refinement converges when the maximum shift is much less than the error for any given parameter.
Weights are applied to the refinement.
Errors in very large values of data will increase the sum more than errors in weak values.
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Non-linear Least Squares
𝐷= 𝑖   𝑤 𝑖     𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐   2
 𝜕𝐷 𝜕 𝑝 𝑗  =0= 𝑖   𝑤 𝑖    𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐 ( 𝑝 1 , 𝑝 2 , 𝑝 3 ,…)  × 𝜕  𝑦 𝑖  𝑐𝑎𝑙𝑐   𝑝 1 , 𝑝 2 , 𝑝 3 ,…  𝜕 𝑝 𝑗
Estimate:     𝑦 𝑐 =  𝑦 0  𝑐 +∆ 𝑦 𝑐 =  𝑦 0  𝑐 + 𝜕 𝑦 𝑐  𝜕 𝑝 1  ∆ 𝑝 1 + 𝜕 𝑦 𝑐  𝜕 𝑝 2  ∆ 𝑝 2 +… 𝜕 𝑦 𝑐  𝜕 𝑝 𝑛  ∆ 𝑝 𝑛
0= 𝑖   𝑤 𝑖    𝑦 𝑖  𝑜 −  𝑦 0  𝑐 − 𝜕 𝑦 𝑐  𝜕 𝑝 1  ∆ 𝑝 1 −… 𝜕 𝑦 𝑐  𝜕 𝑝 𝑛  ∆ 𝑝 𝑛   × 𝜕  𝑦 𝑖  𝑐  𝜕 𝑝 𝑗
   𝑤   𝑦 𝑜 − 𝑦 𝑐   =    𝑤   𝜕 𝑦 𝑐  𝜕 𝑝 1    2 ∆ 𝑝 1 +   𝑤 𝜕 𝑦 𝑐 𝜕 𝑦 𝑐  𝜕 𝑝 1 𝜕 𝑝 2  ∆ 𝑝 2 +…     𝑤 𝜕 𝑦 𝑐 𝜕 𝑦 𝑐  𝜕 𝑝 1 𝜕 𝑝 𝑛  ∆ 𝑝 𝑛
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Non-Linear Least Squares
 𝑣 1 = 𝑎 11 ∆ 𝑝 1 + 𝑎 12 ∆ 𝑝 2 +… 𝑎 1𝑛 ∆ 𝑝 𝑛
 𝑣 2 = 𝑎 21 ∆ 𝑝 1 + 𝑎 22 ∆ 𝑝 2 +… 𝑎 2𝑛 ∆ 𝑝 𝑛
 𝑣 𝑛 = 𝑎 𝑛1 ∆ 𝑝 1 + 𝑎 𝑛2 ∆ 𝑝 2 +… 𝑎 𝑛𝑛 ∆ 𝑝 𝑛
e.g.    𝑎 12 =   𝑤 𝜕 𝑦 𝑐 𝜕 𝑦 𝑐  𝜕 𝑝 1 𝜕 𝑝 2
𝜈=   𝑤   𝑦 𝑜 − 𝑦 𝑐
   𝑣 1  .  𝑣 𝑛   =   𝑎 11  ⋯  𝑎 1𝑛  ⋮ ⋱ ⋮  𝑎 𝑛1  ⋯  𝑎 𝑛𝑛     ∆ 𝑝 1  . ∆ 𝑝 𝑛
           𝒗=𝑨∆𝒑
 𝑨 −𝟏 𝒗=∆𝒑
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Residuals
  𝑅 𝑤𝑝  2 =  𝑖   𝑤 𝑖     𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐   2    𝑖  𝑤    𝑦 𝑖  𝑜𝑏𝑠   2     
  𝑅 𝑒𝑥𝑝  2 =   𝑁 𝑜𝑏𝑠 − 𝑁 𝑣𝑎𝑟    𝑖  𝑤    𝑦 𝑖  𝑜𝑏𝑠   2   
 𝜒 2 =  𝑖   𝑤 𝑖     𝑦 𝑖  𝑜𝑏𝑠 −  𝑦 𝑖  𝑐𝑎𝑙𝑐   2     𝑁 𝑜𝑏𝑠 − 𝑁 𝑣𝑎𝑟
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Standard Uncertainty
𝜎  𝑝 𝑖  =    𝑏 𝑖𝑖     𝑣 2   𝑁−𝑛  
𝜎  𝑝 𝑖  =   𝑏 𝑖𝑖  𝜒 2  
𝜎  𝑝 𝑖  =   𝑏 𝑖𝑖     𝑅 𝑤𝑝   𝑅 𝑒𝑥𝑝
Where bii = diagonalelement of A-1  for pi and N= number of reflections andn = number of parameters
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Residual Problems
Problem with residuals
  𝑦 𝑖  𝑐𝑎𝑙𝑐 = 𝑗 𝑁  𝐾 𝑗  𝑝 𝑀  𝐼 𝑝,𝑗 Ω 2 𝜃 𝑖 −2 𝜃 𝑝,𝑗  + 𝑏𝑘𝑔 𝑖   
  𝑦 𝑖  𝑐𝑎𝑙𝑐 =𝐴+𝐵
B >> A then       𝑦 𝑖  𝑐𝑎𝑙𝑐 ≅𝐵=𝑏𝑘𝑔
  𝑅 𝑤𝑝  2 =  𝑖      𝑦 𝑖  𝑜𝑏𝑠 − 𝑏𝑘𝑔 𝑖   2    𝑖  𝑤    𝑦 𝑖  𝑜𝑏𝑠   2     and  𝜒 2 =  𝑖      𝑦 𝑖  𝑜𝑏𝑠 − 𝑏𝑘𝑔 𝑖   2     𝑁 𝑜𝑏𝑠 − 𝑁 𝑣𝑎𝑟   
Both residual values measure how well the background is modeled!
Background plot will tell you if the background has been over modeled (to high ordered)
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Background Plots
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Background is
Over corrected
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BackgroundPlot
Residual Plots
Residual plot is the difference plot betweenyobs  and ycalc.
In TOPAS  the residual plot is always shownbelow the pattern.
Residual plots are important to judge thequality of the powder pattern fit.
All reports should contain the residual plot
Published papers MUST show a residual plot!
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TOPAS- Residual Plot
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Position
Fitted Peakless the bkg
ResidualPlot
Raw Data
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TOPAS-BAD peak shape
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Peak too wide
Peak too narrow
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Up curvedpeak
NegativePeak
TOPAS More
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Derivative-like shape
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Missingpeakshape
Peak Position Error
Missing Peak
Uses
Quantitative Analysis
Percent Crystallinity
Residual Stress
Crystallite size
Structural Investigations
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Quantitative Analysis
MAUD, TOPAS etc
Free
JAVA based
Windows, MAC, Linux
Example  Al2O3 and TPZ mixture (from MAUDtutorial)
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MAUD
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Mixture of AL2O3 and T-PSZ
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Contains a small database of
Common minerals/compounds
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Choose Quantitative Analysis and GO
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Percent Crystallinity
SiO2 (crystalline) / SiO2 (amorphous) mixture
SiO2 (amorphous) 			area = 13474
SiO2 (crystalline) 			area = 22834
% crystallinity  =   𝐶𝑟𝑦𝑠𝑡(𝑎𝑟𝑒𝑎) 𝐶𝑟𝑦𝑠𝑡 𝑎𝑟𝑒𝑎 +𝐴𝑚𝑜𝑟𝑝(𝑎𝑟𝑒𝑎) ×100=  63%
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Amorphous
Crystalline
Crystalline (modeled PVII)
Amorphous
(modeled SPV)
Compton and Air Scatter:
High angle/Low angle
Background Correction
End of Lecture 7
Lecture 8   The Results