Rietveld method Uses every datum (y obs ) collected, individually

1. Rietveld method Uses every datum (yobs) collected, individually Each yobs compared with a corresponding calculated value (ycalc) Must be able to calculate ycalc

2. Rietveld method Uses every datum (yobs) collected, individually Each yobs compared with a corresponding calculated value (ycalc) Must be able to calculate ycalc Need models for all scattering effects - both Bragg peaks & backgrd Models all involve parameters

3. i i i Rietveld method Uses every datum (yobs) collected, individually Each yobs compared with a corresponding calculated value (ycalc) Must be able to calculate ycalc Need models for all scattering effects - both Bragg peaks & backgrd Models all involve parameters Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2

4. i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit What is best straight line to represent these data?

5. i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit What is best straight line to represent these data? Minimize sum of squares of these distances R = S(yobs– ycalc)2

6. i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit What is best straight line to represent these data? Minimize sum of squares of these distances R = S(yobs– ycalc)2 ycalc values unknown except y = mx + b (straight line)

7. i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit What is best straight line to represent these data? Minimize sum of squares of these distances R = S(yobs– ycalc)2 ycalc values unknown except y = mx + b (straight line) Then R =  (yobs– (mx + b))2 Minimize R ∂R/∂m = ∂R/∂b = 0 –2S (yobs– (mx + b))x = 0 –2S (yobs– (mx + b)) = 0

8. i i i i i i i i i i i i i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit –2S (yobs– (mx + b))x = 0 S x yobs= m Sx2 + bS x –2S (yobs– (mx + b)) = 0 S yobs= mS x + bS1

9. Least squares In general:

10. Least squares In general: (AT A) x = (AT y) x = (AT A)-1 (AT y)

11. Rietveld method

12. Rietveld method What parameters should be determined? 1 2 3 4 5 6 7 8 9 10 11 12 13 14

13. Least squares ƒs are not linear in xi

14. Least squares ƒs are not linear in xi Expand ƒs in Taylor series

15. Least squares ƒs are not linear in xi Expand ƒs in Taylor series

16. Least squares Solve:

17. Least squares Solve:

18. Least squares Solve:

19. Least squares Weighting factors matrix:

20. Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj

21. Least squares So: Need set of initial parameters xjo Problem solution gives shifts ∆xj, not xj Eqns not exact, so refinement process requires no. of cycles to complete the refinement Add shifts ∆xj to xjo for each new refinement cycle

22. Least squares How good are final parameters? Use usual procedure to calculate standard deviations, (xj) no. observations no. parameters

23. Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously?

24. Least squares Warning: Frequently, all parameters cannot be “let go” at the same time How to tell which parameters can be refined simultaneously? Use correlation matrix: Calc correlation matrix for each refinement cycle Look for strong interactions (rij > + 0.5 or < – 0.5, roughly) If 2 parameters interact, hold one constant

25. Rietveld method

26. Rietveld method