Rietveld refinement is a widely used technique for determining crystal structures and quantifying crystalline materials from powder diffraction data. It works by minimizing the difference between observed and calculated diffraction patterns using least squares refinement. Key aspects include modeling the background, peak shape, unit cell parameters, atomic positions, and other structural details. Common software packages are used to perform the iterative refinement calculations.

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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