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

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

3. 3 Least squares & Rietveld Minimize this function: Substitute for Y i calc scale factor

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

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

6. 6 Least squares & Rietveld Minimize this function: Substitute for Y i calc peak shape function

7. 7 Least squares & Rietveld Minimize this function: Substitute for Y i calc x j = 2  j calc – 2  i

8. 8 Least squares & Rietveld

9. 9 So what are the parameters to be determined? background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters

10. 10 Least squares & Rietveld So what are the parameters to be determined? background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters unit cell dimensions preferred orientation, absorption, porosity, extinction parameters scale factor(s)

11. 11 Least squares & Rietveld So what are the parameters to be determined? background function parameters sample displacement, sample transparency, zero-shift errors peak shape parameters unit cell dimensions preferred orientation, absorption, porosity, extinction parameters scale factor(s) atom positional parameters atom site occupancy parameters atom thermal motion parameters

12. 12 Least squares & Rietveld

13. 13 Least squares & Rietveld FOMs Profile residual

14. 14 Least squares & Rietveld FOMs Profile residual Weighted profile residual

15. 15 Least squares & Rietveld FOMs Bragg residual

16. 16 Least squares & Rietveld FOMs Bragg residual Expected profile residual

17. 17 Least squares & Rietveld FOMs Goodness of fit

18. 18 Least squares & Rietveld

19. 19 Rietveld Best data possible Best models possible Vary appropriate parameters singly or in groups

20. 20 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

21. 21 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