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

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

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

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

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

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

Least squares & Rietveld FOMs Profile residual FOMs Profile residual

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

Least squares & Rietveld FOMs Bragg residual FOMs Bragg residual

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

Least squares & Rietveld FOMs Goodness of fit FOMs Goodness of fit

Least squares & Rietveld

Best data possible Best models possible Vary appropriate parameters singly or in groups Best data possible Best models possible Vary appropriate parameters singly or in groups

Least squares & 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 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

Least squares & 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 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

Rietveld - background Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point m=0 N N

Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point Many other functions b i = B 1 +  B m cos(2  m-1 ) Amorphous contribution b i = B 1 + B 2 Q i +  (B 2m+1 sin(Q i B 2m+2 ))/ Q i B 2m+2 Q i = 2π/d i Common background function - polynomial b i =  B m (2  i ) m determine Bs to get backgrd intensity b i at i th point Many other functions b i = B 1 +  B m cos(2  m-1 ) Amorphous contribution b i = B 1 + B 2 Q i +  (B 2m+1 sin(Q i B 2m+2 ))/ Q i B 2m+2 Q i = 2π/d i m=0 N N N N m=2 m=1 N-2 Rietveld - background

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence p 1 = –h 2 K 1 /3RR = diffractometer radius p 2 = –h 2 K 2 /3RK 1, K 2 = constants for collimator h = specimen width 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  axial divergence p 1 = –h 2 K 1 /3RR = diffractometer radius p 2 = –h 2 K 2 /3RK 1, K 2 = constants for collimator h = specimen width

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample p 3 = –  2 /K 3  = beam divergence K 3 = constant 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  flat sample p 3 = –  2 /K 3  = beam divergence K 3 = constant

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency p 4 = 1/2  eff R  eff = effective linear absorption coefficient 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6  specimen transparency p 4 = 1/2  eff R  eff = effective linear absorption coefficient

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 specimen displacement p 5 = –2s/R s = displacement 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 specimen displacement p 5 = –2s/R s = displacement

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 zero error 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 zero error

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together

Rietveld - peak shift 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together When instrument correctly aligned, generally need get only p 5 2  obs = 2  calc +  2  where  2   = p 1 /tan 2   p 2 /sin 2   p 3 /tan   p 4 sin 2   p 5 cos   p 6 p 4, p 5, & p 6 strongly correlated when refined together When instrument correctly aligned, generally need get only p 5

Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles

Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry

Preferred orientation In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry In powder diffractometry, usually assume random orientation For this, need >10 6 randomly oriented particles Extremes: diffraction vector plates needles diffraction vector normal cylindrical symmetry soso s S = s - s o

Preferred orientation March-Dollase function (a la GSAS) plates needles March-Dollase function (a la GSAS) plates needles

Preferred orientation March-Dollase function (a la GSAS) plates needles March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation

Preferred orientation March-Dollase function (a la GSAS) plates needles March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation preferred orientation parameter (refined) preferred orientation parameter (refined)

Preferred orientation March-Dollase function (a la GSAS) plates needles March-Dollase function (a la GSAS) plates needles # symmetrically equivalent reflections multiplier in intensity equation preferred orientation parameter (refined) preferred orientation parameter (refined) angle betwn orientation axis & diffraction vector for hkl

Preferred orientation March-Dollase function - needles probability of reciprocal lattice point to be in reflecting position March-Dollase function - needles probability of reciprocal lattice point to be in reflecting position

Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation

Preferred orientation Spherical harmonics (a la GSAS) hkl sample orientation harmonic coefficients harmonic functions

Preferred orientation Preferred orientation model using 2 nd & 4 th order spherical harmonics for (100) in orthorhombic