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Least squares & Rietveld

Least squares & Rietveld. Have n points in powder pattern w/ observed intensity values Y i obs Minimize this function:. Least squares & Rietveld. Minimize this function: Substitute for Y i calc background at point i. Least squares & Rietveld. Minimize this function:

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Least squares & Rietveld

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  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

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

  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. 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. Least squares & Rietveld

  13. Least squares & Rietveld FOMs Profile residual

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

  15. Least squares & Rietveld FOMs Bragg residual

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

  17. Least squares & Rietveld FOMs Goodness of fit

  18. Least squares & Rietveld

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

  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. 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

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