1. Rietveld Refinements:
Determination of crystal structure and
crystal imperfections
Speaker: Salman-ul-hassan
Supervisor: Dr Nikolay. Zotov

2. Outline:
• Diffraction methods
• What comes before Rietveld method
• What is Rietveld Refinement?
• Why Rietveld Refinements widely used?
• Stages in Rietveld Refinements
• Le-bail Method
• Fit criteria
• Peak shape function
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• The background
• Refinement parameter
• Some common problems
• Software's used in Rietveld
refinements
• Summary

3. Diffraction Methods:
• Identify crystal structure & quantify the content of crystalline
materials.
• Bragg’s Equation:
nλ=2dsinθ
• Sources:
X-ray
Neutrons
Electrons
1. B.DCULLITY, elements of x-ray diffraction, 2nd edition,Addison-wesley publishing, chapter 3 3

4. What comes before Rietveld method:
• Debye & Paul Scherer
Large and randomly oriented grains
Difficulty in measuring intensities
Increasing complexities
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2. Fabio pulizzi, powder struggle, nature, (2014),
7.http://pd.chem.ucl.ac.uk/pdnn/mod1/ip.htm
Figure 1: Debye-Scherrer rings[7]

5. What is Rietveld Refinement?
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• Structure refinement technique
• use of least square method:
𝑀𝑖𝑛 = 𝑖=0
𝑛=1
[𝑊𝑖(𝑌𝑜𝑏𝑠 𝑖 − 𝑌𝑐𝑎𝑙𝑐 𝑖)2
]
𝑌𝑐𝑎𝑙𝑐 𝑖 can be expressed by following formation.
𝑌 𝑐𝑎𝑙𝑐 𝑖= 𝑝ℎ=1
𝑝ℎ𝑎𝑠𝑒𝑠
𝑆 𝑝ℎ ℎ𝑘𝑙(𝑝ℎ)(𝐾ℎ𝑘𝑙|𝐹ℎ𝑘𝑙
2|⏀ℎ𝑘𝑙(2θ𝑖−2θℎ𝑘𝑙)
3. Eric J. Mittemeijer ,U welzel, Modern Diffraction Methods,Chapter-2,WILEYVCH.

6. 6
Figure 2: minimize differences between calculated and observed pattern by least square method
6. profex.doebelin.org/wp-content/.../Lesson-4-Rietveld-Refinement.pdf

7. Why Rietveld Refinements widely used?
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• Computational nature
• Fast calculation
• Uses the entire spectrum (as wide as possible)
• Used in various fields
• Provide information about defects & quantitative phase analysis

8. Stages in Rietveld Refinements.
84. McCusker,Von Dreele, D.E.Cox, D.Louer and P.Scardi. (1999).32, 36-50. J.Appl.Cryst
Check model
and closeness of
fit
Initial
structure
Background
refinement
Unit cell
dimension
refinement
Refine zero
point
correction
Refine peak
shape
Refine atomic
coordinates
and thermal
parameters

9. Le-bail Method:
• Use for Profile refinement
• Crystal structure determination
How to use?
Modify Rietveld code to set all 𝐹ℎ𝑘𝑙 𝑐𝑎𝑙𝑐 = 1
Extract 𝐹ℎ𝑘𝑙 𝑜𝑏𝑠 using Rietveld algorithm
Set 𝐹ℎ𝑘𝑙 𝑐𝑎𝑙𝑐 from extracted 𝐹ℎ𝑘𝑙 𝑜𝑏𝑠
Repeat 𝐹ℎ𝑘𝑙 𝑜𝑏𝑠 extraction now with 𝐹ℎ𝑘𝑙 𝑐𝑎𝑙𝑐
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Figure 3: Peak fits of three selected reflections of LaB6.[3]
3. Eric J. Mittemeijer ,U welzel, Modern Diffraction Methods, Chapter-2,WILEYVCH.

10. • Residual factors:
𝑅 𝑃 =
𝑖=0
𝑛=1
𝑌𝑜𝑏𝑠 𝑖 − 𝑌𝑐𝑎𝑙𝑐 𝑖
𝑖=0
𝑛=1
𝑌𝑜𝑏𝑠 𝑖
𝑅 𝑤𝑝 =
𝑤 𝑌 𝑜𝑏𝑠 − 𝑌(𝑐𝑎𝑙𝑐)2
𝑤(𝑜𝑏𝑠)2
• Goodness of fit parameter 𝑋2
:
𝑥2 = 𝑖 𝑤𝑖 𝑌𝑜𝑏𝑠 𝑖 − 𝑌𝑐𝑎𝑙𝑐 𝑖
2
𝑀 − 𝑃
=
𝑅 𝑤𝑝
𝑅 𝑒𝑥𝑝
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Fit criteria:

11. Peak shape function:
• Pseudo voigt function
• Model over all line broadening
𝐹𝑊𝐻𝑀2
= 𝑈𝑡𝑎𝑛2
𝜃 + 𝑉𝑡𝑎𝑛𝜃 + 𝑤
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Lorentzian Gaussian Pseudo-voigt
Figure 4: Different peak profile functions
profex.doebelin.org/wp-content/.../Lesson-4-Rietveld-Refinement.pdf

12. 12
Figure 5:The observed (circles), calculated (line) and difference (bottom)(a) a good fit of a peak,
(b) a calculated intensity that is too high and (c) a calculated intensity that is too low
McCusker,Von Dreele, D.E.Cox, D.Louer and P.Scardi. (1999).32, 36-50. J.Appl.Cryst

13. The Background:
• Scattering
• Bragg peaks
• Linearizing the background
• Debye formula
𝐼 𝑄 =
𝑖𝑗=1.𝑁
𝑓𝑖(𝑄)𝑓𝑖(𝑄)
sin(𝑄𝑟𝑖𝑗)
(𝑄𝑟𝑖𝑗)
, 𝑤𝑖𝑡ℎ 𝑄 = 4𝜋
𝑠𝑖𝑛𝜃
𝜆
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14. Refinement parameter:
• Background coefficients, peak shape parameters
• Structural parameters
• Unit cell dimension
• Atomic coordinates
• Thermal parameters
• Occupancy parameters.
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15. 15
Figure 6: PbSO4 X-ray Diffraction, Rietveld Refinement showing structure Refinement
Cu Kα radiation, Pseudo-Voigt profile, h = 0.014*2Q, Rwp = 14% RB = 4.6%

16. Some common problems:
• Background not fitted well.
• Peak shapes shows poor description.
• Peak position mismatch in calculated and observed patterns
• Missed peaks.
• Relative intensities of few reflections are too high.
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17. Software’s used in Rietveld Refinements:
Academic software's
• Full prof
• GSAS
• BGMN
• MAUD
• Brass
• Many more
Commercial software's
• High score
• Topas
• Autoquan
• PDXL
• Jade
• Minx
175. http://www.ccp14.ac.uk/solution/rietveld_software/index.html

18. Summary:
• Modern method for extracting structural details.
• Use directly measured intensity points.
• Use of computer software's.
• Best for peak separation.
• Used in different fields.
• Crystal structure determination.
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