Transcript Rietveld method

Rietveld method
Uses every datum (yobs) collected, individually
Each yobs compared with a corresponding calculated value (ycalc)
Must be able to calculate ycalc
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
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 =
S wi (yobs
–
i
i
ycalc)2
i
Rietveld method
Change parameters according to the least squares criterion
Minimize
R =
Si wi (yobs
i –
ycalc
)2
i
Simple example – straight line fit
What is best straight line to represent these data?
Rietveld method
Change parameters according to the least squares criterion
Minimize
R =
Si wi (yobs
i –
ycalc
)2
i
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
Rietveld method
Change parameters according to the least squares criterion
Minimize
R =
Si wi (yobs
i –
ycalc
)2
i
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)
Rietveld method
Change parameters according to the least squares criterion
Minimize
R =
Si wi (yobs
i –
ycalc
)2
i
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 =
S (yobs – (mx + b))2
Minimize R
∂R/∂m = ∂R/∂b = 0
–2S (yobs – (mx + b))x = 0
–2S (yobs – (mx + b)) = 0
Rietveld method
Change parameters according to the least squares criterion
Minimize
R =
Si wi (yobs
i –
ycalc
)2
i
Simple example – straight line fit
–2S (yobs – (mx + b))x = 0
S
–2S (yobs – (mx + b)) = 0
Si yobsi =
i
S
obs = m
x y
i i
x2 + b S x
i
i
i
m S x +i b S 1
i
i
i
Least squares
In general:
Least squares
In general:
(AT A) x = (AT y )
x = (AT A)-1 (AT y )
Rietveld method
Rietveld method
What parameters should be determined?
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Least squares
ƒs are not linear in xi
Least squares
ƒs are not linear in xi
Expand ƒs in Taylor series
Least squares
ƒs are not linear in xi
Expand ƒs in Taylor series
Least squares
Solve:
Least squares
Solve:
Least squares
Solve:
Least squares
Weighting factors matrix:
Least squares
So:
Need set of initial parameters xjo
Problem solution gives shifts ∆xj, not xj
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
Least squares
How good are final parameters?
Use usual procedure to calculate standard deviations, (xj)
no. observations
no. parameters
Least squares
Warning: Frequently, all parameters cannot be
“let go” at the same time
How to tell which parameters can be refined simultaneously?
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
Rietveld method
Rietveld method