Explaining Powder XRD Structural Analysis with the Rietveld Method
- 1. "Powder XRD Structural Analysis with the Rietveld Method" Presented by: Anna Zakiyatul Laila (na20101) Yamamoto High-pressure Science Research Group Regional Environment Systems, Shibaura Institute of Technology
- 2. Outline • Powder Diffraction Analysis • Rietveld Method • The Principles of Rietveld Method • Rietveld Refinement Strategies • Rietveld Softwares 1
- 3. Powder Diffraction Pattern Fig 1. Bragg Law Fig 2. XRD Pattern 2
- 4. Fig 3. The Features of XRD Patterns 3
- 5. How to analyze the phase of XRD pattern? Plot the XRD patterns (from experiment) and calculated XRD patterns (from database COD), then compare the peaks between them. For example: Igor Software Use the application to match the the XRD patterns (from experiment) and calculated XRD patterns (from database). For example: Match Software 4
- 6. Fig 4. XRD analysis with Igor Software Fig 5. XRD analysis with Match Software BUT THESE ARE QUALITATIVE ANALYSIS, WE NEED THE QUANTITATIVE ANALYSIS TO MAKE IT CLEAR 5
- 7. Rietveld Method “Rietveld method is a nonlinear fitting method between the curves of the calculated and the measured diffraction patterns based on the crystal structure data by the least squares method” (Hunter BA, 1998) To minimize the residual function using a non-linier least squares algorithm, To refine a crystal structure of a compound (cell parameter, atomic positions, particle size, and etc) GOALS Where wi is the weight factor for i point, and I is intensity for i point
- 8. The Principles of Rietveld Method In the rietveld analysis, the calculated diffraction pattern (model) is fitted to measured diffraction pattern Fig 6. XRD analysis rietveld method
- 9. Fitting is done by changing the parameters in calculated diffraction pattern (model) which is expressed with the diffraction intensity equation: Note: i = index for a point being calculated S = scale factor K = miller index (hkl) LK = Lorentz polarization factor FK = Strutur Factor φ = The peak function 2θi and 2θk = The detector angles are related to point i and the peak of Bragg K PK = Preferred orientation function A = Absorption factor ybi = Background
- 10. Factor structure (FK), a crystal is determined by its crystal structure. This is the important factor, because there are crystallography information, that are: The multiplicity of the k reflection (with h, k, l Miller indices): mk The Atomic scattering factor (fn) The temperature factor: Bn Where the equation is : The peak function φ is Pseudo Voigt, which combines Lorentzian and Gaussian function. Pseudo Voigt is used to estimate the size and strain of measured crystal after after observing the widening of the diffraction peaks, where the size of crystal only contributes on the Lorentzian component and strain of crystal is on Gaussian component. Look at Figure 7.
- 11. Fig 7. The Peak Function of Diffraction Pattern
- 12. The peak asymmetry is expressed with the equation: Where U, V, and W are related with instrument parameters and FWHM of Lorentzian with the equation: The secθ term is related with size of crystal (D), and the tanθ term is related with the strain of crystal (S). Based on the Fig 7, Full Width at Half Maximum (FWHM) of Gaussian varies to diffraction angle with the equation:
- 13. Rietveld Refinement Strategies Example: Al2O3 - Import the phase of crystal from Database COD - Import the measured XRD pattern - Add the instrument parameter
- 14. Refinement Strategy : Mismatches
- 15. For Wrong Peak Positions We need to refine: 1. Unit cell dimensions 2. Sample height displacement 3. Zero-shift (instrument misalignment)
- 17. For Wrong Absolute Intensities We need to refine: Weight fraction (scaling)
- 19. For Wrong Relative Intensities We need to refine: 1. Preferred orientation 2. Graininess 3. Atomic species 4. Atomic coordinates 5. Site occupancies 6. Thermal displacement parameters
- 21. For Wrong Peak Width We need to refine: 1. Crystallite size 2. Micro-strain in crystal structure 3. Surface roughness
- 23. The Result of Refined Crystal Structure
- 24. Summary of Refinement Strategies
- 25. Parameters: Rwp = less than 10% GoF = less than 4
- 27. References: • S. Pratapa, B.H. O'Connor, B. Hunter, Journal Applied Crystallography, vol. 35 (2002) 155-162. • Döbelin, Nicola. Lesson 1 (XRD and Rietveld Refinement). RMS Foundation, Bettlach, Switzerland. • Lutterotti, Luca. Introduction to diffraction and the Rietveld method. Laboratorio Scienza e Tecnologia dei MAteriali