least squares & rietveld have n points in powder pattern w/ observed intensity values y i obs...
TRANSCRIPT
![Page 1: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/1.jpg)
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:
![Page 2: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/2.jpg)
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
![Page 3: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/3.jpg)
Least squares & Rietveld Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
scale factor
Minimize this function:
Substitute for Yicalc
scale factor
![Page 4: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/4.jpg)
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
![Page 5: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/5.jpg)
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)
![Page 6: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/6.jpg)
Least squares & Rietveld Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
peak shape function
Minimize this function:
Substitute for Yicalc
peak shape function
![Page 7: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/7.jpg)
Least squares & Rietveld Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
xj = 2jcalc – 2i
Minimize this function:
Substitute for Yicalc
xj = 2jcalc – 2i
![Page 8: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/8.jpg)
Least squares & Rietveld Least squares & Rietveld
FOMs
Profile residual
FOMs
Profile residual
![Page 9: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/9.jpg)
Least squares & Rietveld Least squares & Rietveld
FOMs
Profile residual
Weighted profile residual
FOMs
Profile residual
Weighted profile residual
![Page 10: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/10.jpg)
Least squares & Rietveld Least squares & Rietveld
FOMs
Bragg residual
FOMs
Bragg residual
![Page 11: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/11.jpg)
Least squares & Rietveld Least squares & Rietveld
FOMs
Bragg residual
Expected profile residual
FOMs
Bragg residual
Expected profile residual
![Page 12: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/12.jpg)
Least squares & Rietveld Least squares & Rietveld
FOMs
Goodness of fit
FOMs
Goodness of fit
![Page 13: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/13.jpg)
Least squares & Rietveld
![Page 14: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/14.jpg)
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
![Page 15: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/15.jpg)
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
![Page 16: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/16.jpg)
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
![Page 17: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/17.jpg)
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
![Page 18: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/18.jpg)
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
![Page 19: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/19.jpg)
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
![Page 20: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/20.jpg)
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
![Page 21: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/21.jpg)
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
![Page 22: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/22.jpg)
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
![Page 23: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/23.jpg)
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
![Page 24: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/24.jpg)
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
![Page 25: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/25.jpg)
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
![Page 26: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/26.jpg)
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
![Page 27: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/27.jpg)
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
![Page 28: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/28.jpg)
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
![Page 29: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/29.jpg)
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
![Page 30: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/30.jpg)
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
![Page 31: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/31.jpg)
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
![Page 32: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/32.jpg)
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
![Page 33: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/33.jpg)
Preferred orientation Preferred orientation
March-Dollase function (a la GSAS)
plates
needles
March-Dollase function (a la GSAS)
plates
needles
![Page 34: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/34.jpg)
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
![Page 35: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/35.jpg)
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)
![Page 36: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/36.jpg)
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
![Page 37: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/37.jpg)
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
![Page 38: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/38.jpg)
Preferred orientation Preferred orientation
Spherical harmonics (a la GSAS)Spherical harmonics (a la GSAS)
hklhkl sample orientation
sample orientation
![Page 39: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/39.jpg)
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
![Page 40: 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](https://reader035.vdocuments.site/reader035/viewer/2022062804/5697bf761a28abf838c80b4d/html5/thumbnails/40.jpg)
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