Rietveld Refinement with GSAS. Recent Quote seen in Rietveld email:. “Rietveld refinement is one of those few fields of intellectual endeavor wherein the more one does it, the less one understands.” (Sue Kesson). Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned”.
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Recent Quote seen in Rietveld email:
“Rietveld refinement is one of those few fields of intellectual endeavor wherein the more one does it, the less one understands.” (Sue Kesson)
Stephens’ Law –
“A Rietveld refinement is never perfected,
merely abandoned”
Demonstration – refinement of fluroapatite
R.B. Von Dreele, Advanced Photon Source
Argonne National Laboratory
(lab CuKa BB data)
)
Iobs+
Icalc
IoIc
Refl. positions
False minimum
Leastsquares cycles
c2
True minimum – “global” minimum
parameter
c2 surface shape depends on parameter suite
 other systematic effects – sample shift/offset
 other systematic effects (absorption/extinction/preferred orientation)
NB – get linear combination of all the above
NB2 – trend with 2Q (or TOF) important
peak shift
wrong intensity
too sharp
LX  too small
a – too small
Ca2(x) – too small
NB – can be “downupdown” for too “fat” profile
So how does Rietveld refinement work?
Rietveld Minimize
Exact overlaps
 symmetry
Residuals:
Io
Incomplete overlaps
SIc
Ic
Extract structure factors:
Apportion Io by ratio of Ic to Sic & apply corrections
Rietveld refinement  Least Squares Theory
Given a set of observations Gobs
and a function
then the best estimate of the values pi is found by
minimizing
This is done by setting the derivative to zero
Results in n “normal” equations (one for each variable)  solve for pi
Least Squares Theory  continued
Problem  g(pi) is nonlinear & transcendental (sin, cos, etc.)
so can’t solve directly
Expand g(pi) as Taylor series & toss high order terms
ai  initial values of pi
Dpi = pi  ai (shift)
Substitute above
Normal equations  one for each Dpi; outer sum over observations
Solve for Dpi  shifts of parameters, NOT values
Least Squares Theory  continued
Matrix equation Ax=v
Solve x = A1v = Bv; B = A1
This gives set of Dpi to apply to “old” set of ai
repeat until all xi~0 (i.e. no more shifts)
Quality of fit – “c2” = M/(NP) 1 if weights “correct” & model without systematic errors (very rarely achieved)
Bii = s2i – “standard uncertainty” (“variance”) in Dpi
(usually scaled by c2)
Bij/(Bii*Bjj) – “covariance” between Dpi & Dpj
Rietveld refinement  this process applied to powder profiles
Gcalc  model function for the powder profile (Y elsewhere)
Rietveld Model: Yc = Io{SkhF2hmhLhP(Dh) + Ib}
Leastsquares: minimize M=Sw(YoYc)2
Io  incident intensity  variable for fixed 2Q
kh  scale factor for particular phase
F2h  structure factor for particular reflection
mh  reflection multiplicity
Lh  correction factors on intensity  texture, etc.
P(Dh)  peak shape function  strain & microstrain, etc.
Ib  background contribution
Peak shape functions – can get exotic!
Convolution of contributing functions
Instrumental effects
Source
Geometric aberrations
Sample effects
Particle size  crystallite size
Microstrain  nonidentical unit cell
sizes
1
2
2
)
=
=
L
P
(
D
[

4
l
n
2
D
/
H
]
4
l
n
2
2
p
H
2
k
P
(
D
)
=
e
=
G
k
k
/
H
1
+
4
D
k
H
p
k
k
k
k
CW Peak Shape Functions – basically 2 parts:
Gaussian – usual instrument contribution is “mostly” GaussianLorentzian – usual sample broadening contribution
Convolution – Voigt; linear combination  pseudoVoigt
Thompson, Cox & Hastings (with modifications)
PseudoVoigt
Mixing coefficient
FWHM parameter
Finger, Cox & Jephcoat based on van Laar & Yelon
DebyeScherrer
cone
2Q Scan
H
Slit
2QBragg
2Qmin
2Qi
Depend on slit & sample “heights” wrt diffr. radius
H/L & S/L  parameters in function
(typically 0.002  0.020)
Ä PseudoVoigt (TCH)
= profile function
Protein Rietveld refinement  Very low angle fit
1.04.0° peaks  strong asymmetry “perfect” fit to shape
BraggBrentano Diffractometer – “parafocusing”
Focusing circle
Xray source
Diffractometer
circle
Receiving slit
Incident beam
slit
Sample
displaced
Sample
transparency
Beam footprint
Divergent beam optics
CW Function Coefficients  GSAS
Shifted difference
Sample shift
Sample transparency
Gaussian profile
Lorentzian profile
(plus anisotropic broadening terms) Intrepretation?
a*
Crystallite Size Broadening
Dd*=constant
Lorentzian term  usual
K  Scherrer const.
Gaussian term  rare
particles same size?
a*
Microstrain Broadening
Lorentzian term  usual effect
Gaussian term  theory?
Remove instrumental part
Microstrain broadening – physical model
Model – elastic deformation of crystallites
Stephens, P.W. (1999). J. Appl. Cryst. 32, 281289.
Also see Popa, N. (1998). J. Appl. Cryst. 31, 176180.
dspacing expression
Broadening – variance in Mhkl
Microstrain broadening  continued
Terms in variance
Substitute – note similar terms in matrix – collect terms
Microstrain broadening  continued
Broadening – as variance
3 collected terms
General expression – triclinic – 15 terms
Symmetry effects – e.g. monoclinic (b unique) – 9 terms
Cubic – m3m – 2 terms
Example  unusual line broadening effects
in Na parahydroxybenzoate
Sharp lines
Broad lines
Directional dependence 
Lattice defects?
Seeming inconsistency in line broadening
 hkl dependent
Hatom location in Na parahydroxybenzoate
Good DF map allowed by better fit to pattern
DF contour map
Hatom location
from xray powder data
Part of peak shape function #5 – TOF & CW
dspacing expression; aij from recip. metric tensor
Elastic strain – symmetry restricted lattice distortion
TOF:
ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d3
CW:
ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d2tanQ
Why? Multiple data sets under different conditions (T,P, x, etc.)
dij – restricted by symmetry
e.g. for cubic
DT = d11h2d3 for TOF
Result: change in lattice parameters via change in metric coeff.
aij’ = aij2dij/C for TOF
aij’ = aij(p/9000)dij for CW
Use new aij’ to get lattice parameters
e.g. for cubic
h
Nonstructural Features
Affect the integrated peak intensity and not peak shape
Bragg Intensity Corrections:
Extinction
Preferred Orientation
Absorption & Surface Roughness
Other Geometric Factors
E
=
1
+
x
b
2
3
x
x
5
x
E
=
1

+

.
.
.
x
<
1
2
4
4
8
l
2
1
3
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ù
E
=
1


.
.
.
x
>
1
p
x
ê
ú
8
x
2
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1
2
8
x
ë
û
2
2
E
=
E
s
i
n
Q
+
E
c
o
s
Q
h
b
l
Extinction
Sabine model  Darwin, Zachariasen & Hamilton
Bragg component  reflection
Laue component  transmission
Combination of two parts
E
x
60%
40%
20%
0%
0.0
25.0
50.0
75.0
100.0
125.0
150.0
Sabine Extinction Coefficient
Crystallite grain size =
Increasing
wavelength
(15 Å)
Eh
2Q
What is texture? Nonrandom crystallite grain orientations
Random powder  all crystallite orientations equally probable  flat pole figure
Pole figure  stereographic projection of a crystal axis down some sample direction
Loose powder
(100) random texture
(100) wire texture
Crystallites oriented along wire axis  pole figure peaked in center and at the rim (100’s are 90º apart)
Orientation Distribution Function  probability function for texture
Metal wire
Texture  measurement by diffraction
(220)
Nonrandom crystallite orientations in sample
(200)
Incident beam
xrays or neutrons
Sample
(111)
Preferred Orientation  March/Dollase Model
Uniaxial packing
Ellipsoidal Distribution 
assumed cylindrical
Ro  ratio of ellipsoid
axes = 1.0 for no
preferred orientation
Ellipsoidal particles
Spherical Distribution
Integral about distribution
 modify multiplicity
l
¥
m
n
m
n
å
å
å
f
(
g
)
=
C
T
(
g
)
l
l
l=0
m=l
n=l
F1
F2
Y
Texture  Orientation Distribution Function  GSAS
Probability distribution of crystallite orientations  f(g)
f(g) = f(F1,Y,F2)
Tlmn = Associated Legendre functions or generalized spherical harmonics
F1,Y,F2  Euler angles
Texture effect on reflection intensity  Rietveld model
Xrays  independent of 2Q
 flat sample – surface roughness effect
 microabsorption effects
 but can change peak shape and shift
their positions if small (thick sample)
Neutrons  depend on 2Q and l but much
smaller effect
 includes multiple scattering
much bigger effect
 assume cylindrical sample
DebyeScherrer geometry
For cylinders and weak absorption only
i.e. neutrons  most needed for TOF data
not for CW data – fails for mR>1
GSAS – New more elaborate model by
Lobanov & alte de Viega – works to mR>10
Other corrections  simple transmission & flat plate
Surface Roughness – BraggBrentano only
Low angle – less penetration (scatter in less dense material)  less intensity
High angle – more penetration (go thru surface roughness)  more dense material; more intensity
Nonuniform sample density with depth from surface
Most prevalent with strong sample absorption
If uncorrected  atom temperature factors too small
Suortti model Pitschke, et al. model
(a bit more stable)
1
+
M
c
o
s
2
Q
L
=
2
p
2
s
i
n
Q
c
o
s
Q
1
L
=
2
p
2
s
i
n
Q
c
o
s
Q
4
L
=
d
s
i
n
Q
p
Other Geometric Corrections
Lorentz correction  both Xrays and neutrons
Polarization correction  only Xrays
Xrays
Neutrons  CW
Neutrons  TOF
uncorrected
6
4
fC
Solvent
corrected
2
0
0 5 10 15 20
2Q
Solvent scattering – proteins & zeolites?
Contrast effect between structure & “disordered” solvent region
f = foAexp(8pBsin2Q/l2)
Babinet’s Principle:
Atoms not in vacuum –
change form factors
Manual subtraction – not recommended  distorts the weighting scheme for the observations& puts a bias in the observationsFit to a function  many possibilities:
Fourier series  empirical
Chebyschev power series  ditto
Exponential expansions  air scatter & TDS
Fixed interval points  brute force
Debye equation  amorphous background
(separate diffuse scattering in GSAS)
Debye Equation  Amorphous Scattering
real space correlation functionespecially good for TOFterms withvibration
amplitude
distance
Rietveld Refinement with Debye Function
O
1.60Å
4.13Å
Si
2.63Å
3.12Å
5.11Å
6.1Å
aquartz distances
7 terms Ri –interatomic distances in SiO2 glass
1.587(1), 2.648(1), 4.133(3), 4.998(2), 6.201(7), 7.411(7) & 8.837(21)
Same as found in aquartz
NonStructural Features in Powder Patterns
Summary
1. Large crystallite size  extinction
2. Preferred orientation
3. Small crystallite size  peak shape
4. Microstrain (defect concentration)
5. Amorphous scattering  background
Stephens’ Law –
“A Rietveld refinement is never perfected,
merely abandoned”
Also – “stop when you’ve run out of things to vary”
What if problem is more complex?
Apply constraints & restraints
“What to do when you have
too many parameters
& not enough data”
Complex structures (even proteins)
Too many parameters – “free” refinement fails
Known stereochemistry:
Bond distances
Bond angles
Torsion angles (less definite)
Group planarity (e.g. phenyl groups)
Chiral centers – handedness
Etc.
Choice:
rigid body description – fixed geometry/fewer parameters
stereochemical restraints – more data
Constraints – reduce no. of parameters
Derivative vector
Before constraints
(longer)
Derivative vector
After constraints
(shorter)
Rigid body
User
Symmetry
Rectangular matrices
Restraints – additional information (data) that model must fit
Ex. Bond lengths, angles, etc.
Space group symmetry constraints
Special positions – on symmetry elements
Axes, mirrors & inversion centers (not glides & screws)
Restrictions on refineable parameters
Simple example: atom on inversion center – fixed x,y,z
What about Uij’s?
– no restriction
– ellipsoid has inversion center
Mirrors & axes ? – depends on orientation
Example: P 2/m – 2  baxis, m ^ 2fold
on 2fold: x,z – fixed & U11,U22,U33, & U13 variable
on m: y fixed & U11,U22, U33, & U13 variable
Rietveld programs – GSAS automatic, others not
“site fraction” – fraction of site occupied by atom
“site multiplicity” no. times site occurs in cell
“occupancy” – site fraction * site multiplicity
may be normalized by max multiplicity
GSAS uses fraction & multiplicity derived from sp. gp.
Others use occupancy
If two atoms in site – Ex. Fe/Mg in olivine
Then (if site full) FMg = 1FFe
Multiatom site fractions  continued
If 3 atoms A,B,C on site – problem
Diffraction experiment – relative scattering power of site
“1equation & 2unknowns” unsolvable problem
Need extra information to solve problem –
2nd diffraction experiment – different scattering power
“2equations & 2unknowns” problem
Constraint: solution of J.M. Joubert
Add an atom – site has 4 atoms A, B, C, C’
so that FA+FB+FC+FC’=1
Then constrain so DFA = DFC and DFB = D FC’
Multiphase mixtures & multiple data sets
Neutron TOF – multiple detectors
Multi wavelength synchrotron
Xray/neutron experiments
How constrain scales, etc.?
Histogram scale
Phase scale
Ex. 2 phases & 2 histograms – 2 Sh & 4 Sph – 6 scales
Only 4 refinable – remove 2 by constraints
Ex. DS11 = DS21 & DS12 = DS22
Rigid body problem – 88 atoms – [FeCl2{OP(C6H5)3}4][FeCl4]
P21/c
a=14.00Å
b=27.71Å
c=18.31Å
b=104.53
V=6879Å3
264 parameters – no constraints
Just one xray pattern – not enough data!
Use rigid bodies – reduce parameters
V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, 174181 (2003)
Rigid body description – 3 rigid bodies
FeCl4 – tetrahedron, origin at Fe
1 translation, 5 vectors
Fe [ 0, 0, 0 ]
Cl1 [ sin(54.75), 0, cos(54.75)]
Cl2 [ sin(54,75), 0, cos(54.75)]
Cl3 [ 0, sin(54.75), cos(54.75)]
Cl4 [ 0, sin(54.75), cos(54.75)]
D=2.1Å; FeCl bond
z
Fe  origin
Cl2
Cl1
y
Cl4
x
Cl3
Rigid body description – continued
PO – linear, origin at P
C6 – ring, origin at P(!)
x
C4
C2
(ties them together)
D2
C1
D1
D
z
C6
P
O
C1C6 [ 0, 0, 1 ]
D1=1.6Å; PC bond
C1 [ 0, 0, 0 ]
C2 [ sin(60), 0, 1/2 ]
C3 [sin(60), 0, 1/2 ]
C4 [ sin(60), 0, 3/2 ]
C5 [sin(60), 0, 3/2 ]
C6 [ 0, 0, 2 ]
D2=1.38Å; CC aromatic bond
C5
C3
P [ 0, 0, 0 ]
O [ 0, 0 1 ]
D=1.4Å
Rigid body description – continued
C
C
x
R5(z)
R4(x)
P
C
R1(x)
C
R3(z)
y
R2(y)
z
Fe
C
C
O
Rigid body rotations – about P atom origin
For PO group – R1(x) & R2(y) – 4 sets
For C6 group – R1(x), R2(y),R3(z),R4(x),R5(z)
3 for each PO; R3(z)=+0, +120, & +240; R4(x)=70.55
Transform: X’=R1(x)R2(y)R3(z)R4(x)R5(z)X
47 structural variables
Refinement – RB distances & angles
OP(C6)3 1 2 3 4
R1(x) 122.5(13) 76.6(4) 69.3(3) 158.8(9)
R2(y) 71.7(3) 15.4(3) 12.8(3) 69.2(4)
R3(z)a 27.5(12) 51.7(3) 10.4(3) 53.8(9)
R3(z)b 147.5(12) 171.7(3) 109.6(3) 66.2(9)
R3(z)c 267.5(12) 291.7(3) 229.6(3) 186.2(9)
R4(x) 68.7(2) 68.7(2) 68.7(2) 68.7(2)
R5(z)a 99.8(15) 193.0(14) 139.2(16) 64.6(14)
R5(z)b 81.7(14) 88.3(17) 135.7(17) 133.3(16)
R5(z)c 155.3(16) 63.8(16) 156.2(15) 224.0(16)
PO = 1.482(19)Å, PC = 1.747(7)Å, CC = 1.357(4)Å,
FeCl = 2.209(9)Å
}
PO orientation
}
C3PO torsion
(+0,+120,+240)
p − CPO angle
}
Phenyl twist
x
R5(z)
R4(x)
R1(x  PO)
R3(z)
R2(y PO)
Fe
z
Packing diagram – see fit of C6 groups
Stereochemical restraints – additional “data”
Powder profile (Rietveld)*
Bond angles*
Bond distances*
Torsion angle pseudopotentials
Plane RMS displacements*
van der Waals distances (if voi<vci)Hydrogen bonds
Chiral volumes**
“f/y” pseudopotential
wi = 1/s2 weighting factor
fx  weight multipliers (typically 0.13)
For [FeCl2{OP(C6H5)3}4][FeCl4]  restraints
Bond distances:
FeCl = 2.21(1)Å, PO = 1.48(2)Å, PC = 1.75(1)Å, CC = 1.36(1)Å
Number = 4 + 4 + 12 + 72 = 92
Bond angles:
OPC, CPC & ClFeCl = 109.5(10) – assume tetrahedral
CCC & PCC = 120(1) – assume hexagon
Number = 12 + 12 + 6 + 72 + 24 = 126
Planes: C6 to 0.01 – flat phenyl
Number = 72
Total = 92 + 126 + 72 = 290 restraints
A lot easier to setup than RB!!
Stereochemical restraints – superimpose on RB results
Nearly identical with RB refinement
Different assumptions – different results
e.g. Engh & Huber
flexible rigid bodies for amino acid residues
location/orientation & torsion angles of each residue
Residue rigid body model for phenylalanine
Qijk
c2
txyz
c1
y
3txyz+3Qijk+y+c1+c2 = 9 variables
vs 33 unconstrained xyz coordinates
In GSAS defined as: Qijk = r+ai+bj+ck – 4D complex number – 1 real + 3 imaginary components
Normalization: r2+a2+b2+c2 = 1
Rotation vector: v = ax+by+cz; u = (ax+by+cz)/sin(a/2)
Rotation angle: r2 = cos2(a/2); a2+b2+c2 = sin2(a/2)
Quaternion product: Qab = Qa * Qb≠ Qb * Qa
Quaternion vector transformation: v’ = QvQ1
21542 observations; 1148 atoms (1001 HEWL)
XPlor 3.1 – RF = 25.8% ~4600 variables
GSAS RB refinement – RF=20.9% ~2700 variables
RMS difference 
0.10Å main chain & 0.21Å all protein atoms
RB refinement reduces effect of “over refinement”
Conclusions – constraints vs. restraints crystallization; single xtal data,1.40
Constraints required
space group restrictions
multiatom site occupancy
Rigid body constraints
reduce number of parameters
molecular geometry assumptions
Restraints
add data
molecular geometry assumptions (again)
GSAS  A bit of history crystallization; single xtal data,1.40
Structure of GSAS crystallization; single xtal data,1.40
1. Multiple programs  each with specific purpose
editing, powder preparation, least squares, etc.
2. User interface  EXPEDT
edit control data & problem parameters for
calculations  multilevel menus & help listings
text interface (no mouse!)
visualize “tree” structure for menus
3. Common file structure – all named as “experiment.ext”
experiment name used throughout, extension
differs by type of file
4. Graphics  both screen & hardcopy
5. EXPGUI – graphical interface (windows, buttons, edit boxes, etc.); incomplete overlap with EXPEDT but with useful extra features – by B. H. Toby
pull down menus for GSAS programs
(not linux)
GSAS – EXPEDT (and everything else):
On console screen
Keyboard input – text & numbers
1 letter commands – menu help
Layers of menus – tree structure
Type ahead thru layers of menus
Macros (@M, @R & @X commands)
EXPEDT data setup option (<?>,D,F,K,L,P,R,S,X) >EXPEDT data setup options: <?>  Type this help listing D  Distance/angle calculation set up F  Fourier calculation set up K n  Delete all but the last n history records L  Least squares refinement set up P  Powder data preparation R  Review data in the experiment file S  Single crystal data preparation X  Exit from EXPEDT
Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e5’ all allowed
Drag & drop for e.g. file names
EXPGUI:
Access to GSAS
Typical GUI – edit boxes,
buttons, pull downs etc.
Liveplot – powder pattern
widplt
Sum
Lorentz FWHM
(sample)
Gauss FWHM
(instrument)
Zoom
(new plot)
cum. c2 on
updates at end of genles run – check if OK!
Io
Ic
Refl. pos.
IoIc
“publication style” plot – works OK for many journals; save as “emf”
can be “dressed up”; also ascii output of x,y table
Sqrt(I)
Refl. pos.
rescale y by 4x
Qscale (from Q=pl/sinq)
A.C. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los Alamos National Laboratory Report LAUR 86748 (2004).
B. H. Toby, EXPGUI, a graphical user interface for GSAS, J. Appl. Cryst. 34, 210213 (2001).