Least squares & Rietveld

1. Least squares & Rietveld Have n points in powder pattern w/ observed intensity values Yiobs Minimize this function:

2. Least squares & Rietveld Minimize this function: Substitute for Yicalc background at point i

3. Least squares & Rietveld Minimize this function: Substitute for Yicalc scale factor

4. Least squares & Rietveld Minimize this function: Substitute for Yicalc no. of Bragg reflections contributing intensity to point i

5. Least squares & Rietveld Minimize this function: Substitute for Yicalc integrated intensity of j th Bragg reflection (area under peak)

6. Least squares & Rietveld Minimize this function: Substitute for Yicalc peak shape function

7. Least squares & Rietveld Minimize this function: Substitute for Yicalc xj= 2qjcalc – 2qi

8. Least squares & Rietveld FOMs Profile residual

9. Least squares & Rietveld FOMs Profile residual Weighted profile residual

10. Least squares & Rietveld FOMs Bragg residual

11. Least squares & Rietveld FOMs Bragg residual Expected profile residual

12. Least squares & Rietveld FOMs Goodness of fit

13. Least squares & Rietveld

14. Least squares & Rietveld Best data possible Best models possible Vary appropriate parameters singly or in groups

15. Least squares & Rietveld Best data possible Best models possible Vary appropriate parameters singly or in groups Watch correlation matrix – adjust as necessary Watch parameter shifts – getting smaller? Watch parameter standard deviations – compare to shifts

16. Least squares & Rietveld Best data possible Best models possible Vary appropriate parameters singly or in groups Watch correlation matrix – adjust as necessary Watch parameter shifts – getting smaller? Watch parameter standard deviations – compare to shifts Check FOMs - Converging? Always inspect plot of obs and calc data, and differences

17. Rietveld- background Common background function - polynomial bi = S Bm (2qi)m determine Bs to get backgrd intensity bi at ith point N m=0

18. Rietveld- background Common background function - polynomial bi = S Bm (2qi)m determine Bs to get backgrd intensity bi at ith point Many other functions bi = B1 + S Bm cos(2qm-1) Amorphous contribution bi = B1 + B2 Qi + S (B2m+1 sin(QiB2m+2))/ QiB2m+2 Qi = 2π/di N m=0 N m=2 N-2 m=1

19. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6

20. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 axial divergence

21. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 axial divergence p1 = –h2 K1/3R R = diffractometer radius p2 = –h2 K2/3R K1,K2 = constants for collimator h = specimen width

22. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 flat sample

23. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 flat sample p3 = – a2/K3a = beam divergence K3 = constant

24. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 specimen transparency

25. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 specimen transparency p4 = 1/2meffR meff = effective linear absorption coefficient

26. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 specimen displacement p5 = –2s/R s = displacement

27. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 zero error

28. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 p4, p5, &p6 strongly correlated when refined together

29. Rietveld-peak shift 2qobs = 2qcalc + D2q where D2q= p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q + p5 cos q + p6 p4, p5, &p6 strongly correlated when refined together When instrument correctly aligned, generally need get only p5

30. Preferred orientation In powder diffractometry, usually assume random orientation For this, need >106 randomly oriented particles

31. Preferred orientation In powder diffractometry, usually assume random orientation For this, need >106 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry

32. Preferred orientation S = s - so so s In powder diffractometry, usually assume random orientation For this, need >106 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry

33. Preferred orientation March-Dollase function (a la GSAS) plates needles

34. Preferred orientation March-Dollase function (a la GSAS) plates needles multiplier in intensity equation # symmetrically equivalent reflections

35. Preferred orientation March-Dollase function (a la GSAS) plates needles multiplier in intensity equation # symmetrically equivalent reflections preferred orientation parameter (refined)

36. Preferred orientation March-Dollase function (a la GSAS) plates needles multiplier in intensity equation # symmetrically equivalent reflections preferred orientation parameter (refined) angle betwn orientation axis & diffraction vector for hkl

37. Preferred orientation March-Dollase function - needles probability of reciprocal lattice point to be in reflecting position

38. Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation

39. Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation harmonic coefficients harmonic functions

40. Preferred orientation Preferred orientation model using 2nd & 4th order spherical harmonics for (100) in orthorhombic