Least squares & Rietveld
Have n points in powder pattern w/ observed intensity values Yiobs
Minimize this function:
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
background at point i
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
scale factor
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
no. of Bragg reflections contributing intensity to point i
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
integrated intensity of j th Bragg reflection
(area under peak)
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
peak shape function
Least squares & Rietveld
Minimize this function:
Substitute for Yicalc
xj = 2qjcalc – 2qi
Least squares & Rietveld
FOMs
Profile residual
Least squares & Rietveld
FOMs
Profile residual
Weighted profile residual
Least squares & Rietveld
FOMs
Bragg residual
Least squares & Rietveld
FOMs
Bragg residual
Expected profile residual
Least squares & Rietveld
FOMs
Goodness of fit
Least squares & Rietveld
Least squares & Rietveld
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
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
Rietveld - background
Common background function - polynomial
N
bi = S Bm (2qi)m
m=0
determine Bs to get backgrd intensity bi at ith point
Rietveld - background
Common background function - polynomial
N
bi = S Bm (2qi)m
m=0
determine Bs to get backgrd intensity bi at ith point
Many other functions
N
bi = B1 + S Bm cos(2qm-1)
m=2
Amorphous contribution
N-2
bi = B1 + B2 Qi + S (B2m+1 sin(QiB2m+2))/ QiB2m+2
m=1
Qi = 2π/di
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
axial divergence
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + 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
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
flat sample
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
flat sample
p3 = – a2/K3 a = beam divergence
K3 = constant
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
specimen transparency
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
specimen transparency
p4 = 1/2meffR
meff = effective linear absorption coefficient
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
specimen displacement
p5 = –2s/R
s = displacement
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
zero error
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
p4, p5, & p6 strongly correlated when refined together
Rietveld - peak shift
2qobs = 2qcalc + D2q
where
D2q = p1/tan 2q + p2/sin 2q + p3/tan q + p4 sin 2q +
p5 cos q + p6
p4, p5, & p6 strongly correlated when refined together
When instrument correctly
aligned, generally need get
only p5
Preferred orientation
In powder diffractometry, usually assume random orientation
For this, need >106 randomly oriented particles
Preferred 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
Preferred orientation
In powder diffractometry, usually assume random orientation
For this, need >106 randomly oriented particles
Extremes:
diffraction vector
plates
S = s - so
so
needles
s
diffraction vector normal
cylindrical symmetry
Preferred orientation
March-Dollase function (a la GSAS)
plates
needles
Preferred orientation
March-Dollase function (a la GSAS)
plates
multiplier
in intensity
equation
needles
# symmetrically equivalent reflections
Preferred orientation
March-Dollase function (a la GSAS)
plates
multiplier
in intensity
equation
# symmetrically equivalent reflections
needles
preferred
orientation
parameter
(refined)
Preferred orientation
March-Dollase function (a la GSAS)
plates
multiplier
in intensity
equation
# symmetrically equivalent reflections
needles
angle betwn
orientation
axis &
diffraction
vector for hkl
preferred
orientation
parameter
(refined)
Preferred orientation
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 2nd & 4th order spherical harmonics
for (100) in orthorhombic