Least squares

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