Least squares & RietveldLeast squares & Rietveld

Have n points in powder pattern w/ observed intensity values Yi

obs

Minimize this function:

Have n points in powder pattern w/ observed intensity values Yi

obs

Minimize this function:

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

background at point i

Minimize this function:

Substitute for Yicalc

background at point i

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

scale factor

Minimize this function:

Substitute for Yicalc

scale factor

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

no. of Bragg reflections contributing intensity to point i

Minimize this function:

Substitute for Yicalc

no. of Bragg reflections contributing intensity to point i

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

integrated intensity of j th Bragg reflection

(area under peak)

Minimize this function:

Substitute for Yicalc

integrated intensity of j th Bragg reflection

(area under peak)

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

peak shape function

Minimize this function:

Substitute for Yicalc

peak shape function

Least squares & Rietveld Least squares & Rietveld

Minimize this function:

Substitute for Yicalc

xj = 2jcalc – 2i

Minimize this function:

Substitute for Yicalc

xj = 2jcalc – 2i

Least squares & Rietveld Least squares & Rietveld

FOMs

Profile residual

FOMs

Profile residual

Least squares & Rietveld Least squares & Rietveld

FOMs

Profile residual

Weighted profile residual

FOMs

Profile residual

Weighted profile residual

Least squares & Rietveld Least squares & Rietveld

FOMs

Bragg residual

FOMs

Bragg residual

Least squares & Rietveld Least squares & Rietveld

FOMs

Bragg residual

Expected profile residual

FOMs

Bragg residual

Expected profile residual

Least squares & Rietveld Least squares & Rietveld

FOMs

Goodness of fit

FOMs

Goodness of fit

Least squares & Rietveld

Least squares & Rietveld 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 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 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 - backgroundRietveld - backgroundCommon background function - polynomial

bi = Bm (2i)m

determine Bs to get backgrd intensity bi at ith point

Common background function - polynomial

bi = Bm (2i)m

determine Bs to get backgrd intensity bi at ith point

m=0m=0

NN

Common background function - polynomial

bi = Bm (2i)m

determine Bs to get backgrd intensity bi at ith point

Many other functions

bi = B1 + Bm cos(2m-1)

Amorphous contribution

bi = B1 + B2 Qi + (B2m+1 sin(QiB2m+2))/ QiB2m+2

Qi = 2π/di

Common background function - polynomial

bi = Bm (2i)m

determine Bs to get backgrd intensity bi at ith point

Many other functions

bi = B1 + Bm cos(2m-1)

Amorphous contribution

bi = B1 + B2 Qi + (B2m+1 sin(QiB2m+2))/ QiB2m+2

Qi = 2π/di

m=0m=0

NN

NN

m=2m=2

m=1m=1

N-2N-2

Rietveld - backgroundRietveld - background

Rietveld - peak shiftRietveld - peak shift2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

axial divergence

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

axial divergence

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

axial divergence

p1 = –h2 K1/3R R = diffractometer radius

p2 = –h2 K2/3R K1, K2 = constants for collimator

h = specimen width

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

axial divergence

p1 = –h2 K1/3R R = diffractometer radius

p2 = –h2 K2/3R K1, K2 = constants for collimator

h = specimen width

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

flat sample

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin

2

p5 cos p6

flat sample

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

flat sample

p3 = – 2/K3 = beam divergence

K3 = constant

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

flat sample

p3 = – 2/K3 = beam divergence

K3 = constant

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen transparency

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen transparency

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen transparency

p4 = 1/2effR

eff = effective linear absorption coefficient

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen transparency

p4 = 1/2effR

eff = effective linear absorption coefficient

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen displacement

p5 = –2s/R

s = displacement

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

specimen displacement

p5 = –2s/R

s = displacement

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

zero error

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

zero error

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

p4, p5, & p6 strongly correlated when refined together

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

p4, p5, & p6 strongly correlated when refined together

Rietveld - peak shift Rietveld - peak shift 2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

p4, p5, & p6 strongly correlated when refined together

When instrument correctly

aligned, generally need get

only p5

2obs = 2calc + 2

where

2= p1/tan 2p2/sin 2p3/tan p4 sin 2

p5 cos p6

p4, p5, & p6 strongly correlated when refined together

When instrument correctly

aligned, generally need get

only p5

Preferred orientationPreferred orientation

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

Preferred orientationPreferred orientation

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

Extremes: diffraction vector

plates

needles

diffraction vector normal

cylindrical symmetry

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

Extremes: diffraction vector

plates

needles

diffraction vector normal

cylindrical symmetry

Preferred orientationPreferred orientation

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

Extremes: diffraction vector

plates

needles

diffraction vector normal

cylindrical symmetry

In powder diffractometry, usually assume random orientation

For this, need >106 randomly oriented particles

Extremes: diffraction vector

plates

needles

diffraction vector normal

cylindrical symmetry

sos

S = s - so

Preferred orientation Preferred orientation

March-Dollase function (a la GSAS)

plates

needles

March-Dollase function (a la GSAS)

plates

needles

Preferred orientation Preferred orientation

March-Dollase function (a la GSAS)

plates

needles

March-Dollase function (a la GSAS)

plates

needles

# symmetrically equivalent reflections# symmetrically equivalent reflectionsmultiplier in intensity equation

multiplier in intensity equation

Preferred orientation Preferred orientation

March-Dollase function (a la GSAS)

plates

needles

March-Dollase function (a la GSAS)

plates

needles

# symmetrically equivalent reflections# symmetrically equivalent reflectionsmultiplier in intensity equation

multiplier in intensity equation

preferred orientation parameter(refined)

preferred orientation parameter(refined)

Preferred orientation Preferred orientation

March-Dollase function (a la GSAS)

plates

needles

March-Dollase function (a la GSAS)

plates

needles

# symmetrically equivalent reflections# symmetrically equivalent reflectionsmultiplier in intensity equation

multiplier in intensity equation

preferred orientation parameter(refined)

preferred orientation parameter(refined)

angle betwn orientation axis & diffraction vector for hkl

angle betwn orientation axis & diffraction vector for hkl

Preferred orientation 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 Preferred orientation

Spherical harmonics (a la GSAS)Spherical harmonics (a la GSAS)

hklhkl sample orientation

sample orientation

Preferred orientation Preferred orientation

Spherical harmonics (a la GSAS)Spherical harmonics (a la GSAS)

hklhkl sample orientation

sample orientation

harmonic coefficients

harmonic coefficients

harmonic functionsharmonic functions

Preferred orientation Preferred orientation

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