Determination of crystal structure and
Supervisor: Dr Nikolay. Zotov
• 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
• The background
• Refinement parameter
• Some common problems
• Software's used in Rietveld
• Identify crystal structure & quantify the content of crystalline
• Bragg’s Equation:
1. B.DCULLITY, elements of x-ray diffraction, 2nd edition,Addison-wesley publishing, chapter 3 3
What comes before Rietveld method:
• Debye & Paul Scherer
Large and randomly oriented grains
Difficulty in measuring intensities
2. Fabio pulizzi, powder struggle, nature, (2014),
Figure 1: Debye-Scherrer rings
What is Rietveld Refinement?
• Structure refinement technique
• use of least square method:
𝑀𝑖𝑛 = 𝑖=0
[𝑊𝑖(𝑌𝑜𝑏𝑠 𝑖 − 𝑌𝑐𝑎𝑙𝑐 𝑖)2
𝑌𝑐𝑎𝑙𝑐 𝑖 can be expressed by following formation.
𝑌 𝑐𝑎𝑙𝑐 𝑖= 𝑝ℎ=1
𝑆 𝑝ℎ ℎ𝑘𝑙(𝑝ℎ)(𝐾ℎ𝑘𝑙|𝐹ℎ𝑘𝑙
3. Eric J. Mittemeijer ,U welzel, Modern Diffraction Methods,Chapter-2,WILEYVCH.
Figure 2: minimize differences between calculated and observed pattern by least square method
Why Rietveld Refinements widely used?
• Computational nature
• Fast calculation
• Uses the entire spectrum (as wide as possible)
• Used in various fields
• Provide information about defects & quantitative phase analysis
Stages in Rietveld Refinements.
84. McCusker,Von Dreele, D.E.Cox, D.Louer and P.Scardi. (1999).32, 36-50. J.Appl.Cryst
and closeness of
• 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
Figure 3: Peak fits of three selected reflections of LaB6.
3. Eric J. Mittemeijer ,U welzel, Modern Diffraction Methods, Chapter-2,WILEYVCH.
Peak shape function:
• Pseudo voigt function
• Model over all line broadening
𝜃 + 𝑉𝑡𝑎𝑛𝜃 + 𝑤
Lorentzian Gaussian Pseudo-voigt
Figure 4: Different peak profile functions
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
• Bragg peaks
• Linearizing the background
• Debye formula
𝐼 𝑄 =
, 𝑤𝑖𝑡ℎ 𝑄 = 4𝜋
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.
Software’s used in Rietveld Refinements:
• Full prof
• Many more
• High score
• 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.