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The Rietveld method uses collected data (yobs) to calculate corresponding values (ycalc) and determine scattering effects. Parameters are adjusted using the least squares criterion to minimize the sum of squares of the differences between yobs and ycalc. Straight line fitting is used as a simple example for data representation. The method involves solving equations and requires initial parameter values. Standard deviations can be calculated for the final parameters. The refinement process may require multiple cycles. The correlation matrix can be used to identify parameters that can be refined simultaneously.
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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
i i i 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 = Swi (yobs– ycalc)2
i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit What is best straight line to represent these data?
i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 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
i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 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)
i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 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 = (yobs– (mx + b))2 Minimize R ∂R/∂m = ∂R/∂b = 0 –2S (yobs– (mx + b))x = 0 –2S (yobs– (mx + b)) = 0
i i i i i i i i i i i i i i i Rietveld method Change parameters according to the least squares criterion Minimize R = Swi (yobs– ycalc)2 Simple example – straight line fit –2S (yobs– (mx + b))x = 0 S x yobs= m Sx2 + bS x –2S (yobs– (mx + b)) = 0 S yobs= mS x + bS1
Least squares In general:
Least squares In general: (AT A) x = (AT y) x = (AT A)-1 (AT y)
Rietveld method What parameters should be determined? 1 2 3 4 5 6 7 8 9 10 11 12 13 14
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