Rietveld Refinement with GSAS
& GSAS-II
Talk will mix both together
R.B. Von Dreele, Advanced Photon Source
Argonne National Laboratory
What does GSAS do in powder pattern analysis?
Thanks to Lynn McCusker for maze
Includes:
- Rietveld refinement
- Results
- Powder pattern plots
-For publication
- Bond lengths & angles
- Other geometry
- CIF (& PDB) files of result
- Fourier maps & (some)
display
- Texture (polefigures)
- Utilities
Missing:
- Indexing
- Structure solution
Must go elsewhere for
these.
Form of GSAS
PC-GSAS – thin wrapper GUI
forplot
expedt
genles
Keyboard interface only
powplot
fourier
.EXP file, etc.
disagl
GSAS programs – each is a Fortran exe
(common library of routines)
3
Form of GSAS & EXPGUI
widplt
forplot
expgui
genles
fourier
disagl
expedt
GU
I
powplot
Keyboard & mouse
liveplot
EXPGUI – incomplete GUI
access to GSAS but with
extras
4
GSAS & EXPGUI interfaces
EXPEDT
EXPEDT
<?> D
F
K n L
P
R
S
X
-
data setup option (<?>,D,F,K,L,P,R,S,X) >
data setup options:
Type this help listing
Distance/angle calculation set up
Fourier calculation set up
Delete all but the last n history records
Least squares refinement set up
Powder data preparation
Review data in the experiment file
Single crystal data preparation
Exit from EXPEDT
GSAS – EXPEDT (and everything
else) – text based menus with
help, macro building, etc.
(1980’s user interface!)
EXPGUI: access to GSAS
Typical GUI – edit boxes,
buttons, pull downs etc.
Liveplot – powder pattern
display
(1990’s user interface)
5
GSAS-II: A fresh start
Fill in what’s missing from
GSAS:
- Indexing
- Structure solution
GSASII – fresh start
Base code – python
Mixed in old GSAS Fortran
Graphics –
matplotlib,OpenGL
Modern GUI – wxPython
Math – numpy,scipy
Current: python 2.7
All platforms: Windows,
Max OSX & Linux
GSAS-II – python code model
Slow GUI code –
wxPython
& common project file
name.gpx
Fast core processing codes Fast code – numpy array routines
(a few fortran routines)
Python – ideal for this
7
GSAS-II: A screen shot – 3 frame layout + console
Main menu
Data tree
Submenu
Data tabs
Data window
Graphics window
Drawing tabs
NB: Dialog box windows will appear wanting a response
Rietveld results - visualization
Normal
Probability
9
Complex peak broadening models
m-strain surface
NB: mm size & mstrain units
10
Variance-covariance matrix display
Useful diagnostic!
High V-covV?
Forgot a “hold”
Highly coupled parms
Note “tool tip”
11
Structure drawing
Polyhedra
Van der Waals atoms
Balls & sticks
Thermal ellipsoids
All selectable by atom
Rietveld refinement is multiparameter curve fitting
)
(lab CuKa B-B data)
Iobs +
Icalc |
Io-Ic |
Refl. positions

Result from fluoroapatite refinement – powder profile is curve with counting
noise & fit is smooth curve

NB: big plot is sqrt(I)
Old GSAS example!
13
So how do we get there?
 Beginning – model errors  misfits to pattern
 Can’t just let go all parameters – too far from best model (minimum c2)
False minimum
c2
Least-squares cycles
True minimum – “global” minimum
parameter
c2 surface shape depends on parameter suite
14
Fluoroapatite start – add model
(1st choose lattice & space group)
 important – reflection marks match peaks
 Bad start otherwise – adjust lattice parameters (wrong space group?)
15
2nd add atoms & do default initial refinement –
scale & background
 Notice shape of difference curve – position/shape/intensity errors
16
Errors & parameters?



position – lattice parameters, zero point (not common)
- other systematic effects – sample shift/offset
shape – profile coefficients (GU, GV, GW, LX, LY, etc. in GSAS)
intensity – crystal structure (atom positions & thermal parameters)
- other systematic effects (absorption/extinction/preferred orientation)
NB – get linear combination of all the above
NB2 – trend with 2Q (or TOF) important
peak shift
a – too small
too sharp
LX - too small
wrong intensity
Ca2(x) – too small
17
Difference curve – what to do next?
Characteristic “up-down-up”
profile error
NB – can be “down-updown” for too “fat” profile
 Dominant error – peak shapes? Too sharp?
 Refine profile parameters next (maybe include lattice parameters)
 NB - EACH CASE IS DIFFERENT
18
Result – much improved!
 maybe intensity differences remain
– – refine coordinates & thermal parms.
19
Result – essentially unchanged
Ca
F
PO4
Thus, major error in this initial model – peak shapes
20
Pawley/Rietveld refinement
Io
SIc
Ic
Exact overlaps
- symmetry
Incomplete overlaps
Residual:

R wp 
w(I
o
 Ic)
 wI
2
2
o
Processing:
GSAS – point by point
GSAS-II – reflection by reflection
Minimize
MR 
 w (I
o
 Ic )
2
21
Least Squares Theory
MR 
Minimize
 w (I
o
 Ic )
2
This is done by setting the derivative of MR to zero

w I 

 I c
 I  I o  I c (a i )

 p  p i  p  0
i
i
j

I c
ai - initial values of pi
I c (p i )  I c (a i )  
pi
i p i
p = p - a (shift)
a i, j 

w
I c
I c I c
p i p j
i
x j  p j
vi 

i
i
w ( I)
I c
p i
Normal equations - one for each pi; outer sum over observations
Solve for pi - shifts of parameters, NOT values
Matrix form: Ax=v & B = A-1 so x = Bv = p
Least Squares Theory - continued
Matrix equation Ax=v
Solve
x = A-1v = Bv; B = A-1
This gives set of pi to apply to “old” set of ai
repeat until all xi~0 (i.e. no more shifts)
Quality of fit – “c2” = M/(N-P)  1 if weights “correct” & model without
systematic errors (very rarely achieved)
Bii = s2i – “standard uncertainty” (“variance”) in pi
(usually scaled by c2)
Bij/(Bii*Bjj) – “covariance” between pi & pj
Rietveld refinement - this process applied to powder profiles
Gcalc - model function for the powder profile (Y elsewhere)
23
Rietveld Model: Yc = Io{SkhF2hmhLhP(h) + Ib}
Least-squares: minimize M=Sw(Yo-Yc)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(h) - peak shape function - strain & microstrain, etc.
Ib - background contribution
24
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
CW Peak Shape Functions – basically 2 parts:
Gaussian – usual instrument contribution is “mostly” Gaussian
𝟒𝒍𝒏𝟐
𝜟𝟐
𝑮 𝜟, 𝜞 =
𝒆𝒙𝒑 −𝟒𝒍𝒏𝟐 𝟐
𝜞
𝜞 𝝅
Lorentzian – usual sample broadening contribution
𝟐
𝟏
𝑳 𝜟, 𝜞 =
𝝅𝜞 𝟏 + 𝟒𝜟𝟐
𝜞𝟐
G - full width at half maximum – expression from soller slit
sizes and monochromator angle & sample broadening
- displacement from peak position
Convolution – Voigt; linear combination - pseudoVoigt
CW Profile Function in GSAS & GSAS-II
Thompson, Cox & Hastings (with modifications)
Pseudo-Voigt
𝑷 𝜟, 𝜸, 𝚪 = 𝜼𝑳 𝜟, 𝑯 + 𝟏 − 𝜼 𝑮 𝜟, 𝑯
𝟑
Mixing coefficient
𝜼=
𝒌𝒋
𝒋=𝟏
𝟓
FWHM parameter
𝜸
𝟐
𝜞
𝟓
𝒄𝒊 𝜞𝟓−𝒊 𝜸𝒊
𝑯=
𝒊=𝟏
Where Lorentzian FWHM = g and Gaussian FWHM = G
27
CW Axial Broadening Function
Finger, Cox & Jephcoat based on van Laar & Yelon
Debye-Scherrer
cone
2Q Scan
H
Slit
2Qmin
2Qi
2QBragg
Depend on slit & sample “heights” wrt diffr. radius
H/L & S/L - parameters in function (combined as S/L+H/L; S = H)
(typically 0.002 - 0.020)
 Pseudo-Voigt (TCH)
= profile function
28
How good is this function?
Protein Rietveld refinement - Very low angle fit
1.0-4.0° peaks - strong asymmetry “perfect” fit to shape
29
Bragg-Brentano Diffractometer – “parafocusing”
Focusing circle
Diffractometer
circle
X-ray source
Receiving slit
Incident beam
slit
Sample
displaced
Beam footprint
Sample
transparency
Divergent beam optics
30
CW Function Coefficients – GSAS & GSAS-II
Shifted difference
Sample shift
𝜟′ = 𝜟 + 𝑺𝒔 𝒄𝒐𝒔 𝜣 + 𝑻𝒔 𝒔𝒊𝒏 𝟐𝜣
−𝝅𝑹𝑺𝒔
𝒔=
𝟑𝟔𝟎𝟎𝟎
−𝟗𝟎𝟎𝟎
Sample transparency 𝝁
𝒆𝒇𝒇 =
𝝅𝑹𝑻𝒔
𝟐
𝟐
𝐏
Gaussian profile 𝚪 = 𝐔 𝐭𝐚𝐧 𝚯 + 𝐕 𝐭𝐚𝐧 𝚯 + 𝐖 + 𝐜𝐨𝐬 𝟐𝚯
𝐗
+ 𝐘 𝐭𝐚𝐧 𝚯
Lorentzian profile 𝛄 =
𝐜𝐨𝐬 𝚯
(plus anisotropic broadening terms) Intrepretation?
NB: P term not in GSAS-II;
sample shift, meff refined directly as parameters
31
Crystallite Size Broadening
d*=constant
d* 
d
d

2
Q cot Q

d
 2 Q cot Q sin Q

b*
 2Q 
a*
Lorentzian term - usual
K - Scherrer const.
Gaussian term - rare
particles same size?
p 
p 
180 K 
 " LX "
180 K 
 " GP "
NB: In GSAS-II size is refined directly in mm
d
2
d cos Q
Microstrain Broadening
d
d
d
d
b*

 cons tan t
d *
d*
 2Q 
 Q cot Q
2d
d
a*
S  100 %
tan Q

180
" LY "
Lorentzian term - usual effect

Gaussian term - theory?
S  100 %
 " GU "
180
(No, only a misreading)
Remove instrumental part
NB: In GSAS-II mstrain refined directly; no conversion needed)
Microstrain broadening – physical model
Model – elastic deformation of crystallites
Stephens, P.W. (1999). J. Appl. Cryst. 32, 281-289.
Also see Popa, N. (1998). J. Appl. Cryst. 31, 176-180.
d-spacing expression
1
2
d hkl
 M hkl  a 1 h  a 2 k  a 3 l  a 4 kl  a 5 hl  a 6 hk
2
2
2
Broadening – variance in Mhkl
s
2
 M hkl    S ij
i,j
M M
a i a
j
34
Microstrain broadening - continued
Terms in variance
M
a 1
h ,
2
M
a 2
k ,
2
M
a 3
M
l ,
2
a 4
 kl ,
M
 hl ,
a 5
M
a 6
 hk
Substitute – note similar terms in matrix – collect terms
 h4
 2 2
h k
 h2l 2
M M
  2
a i a j
 h kl
 h3l
 3
 h k
2
h k
2
2 2
h kl
h l
2 2
k l
3
hk l
h l
2
4
k l
2 2
l
k l
3
kl
3
k l
2
hl
3
hkl
3
hkl
k
k l
hk l
hk
4
2
3
hl
2 2
hkl
2
h l
kl
2
3
2
hk l
3
2
2 2
2
h kl
3
h k 
3 
hk 
2
hkl 

2
hk l 
2
h kl 

2 2
h k 
35
Microstrain broadening - continued
Broadening – as variance
s
 M hkl    S HKL h H k K l L
2
,H K L 4
HKL
3 collected terms
General expression – triclinic – 15 terms
s
2
 M hkl  

S 400 h  S 040 k  S 004 l  3 S 220 h k  S 202 h l  S 022 k l
4

4 S
4
4
2 S 310 h k  S 103 hl
3
2
 S 031 k l  S 130 hk
3
3
h kl  S 121 hk l  S 112 hkl
2
211
2
2
2
2 2
3

2 2
 S 301 h l  S 013 kl
3
3


Symmetry effects – e.g. monoclinic (b unique) – 9 terms
s
2
 M hkl  
S 400 h  S 040 k
4
4
 S 004 l  3 S 202 h l  3 ( S 220 h k
2 S 301 h l  S 103 hk
3
4
3
 4S
2
2
2
2
 S 022 k l ) 
2
2
121
hk l
Cubic – m3m – 2 terms
s
2
 M hkl  
S 400 h
4
k
4
l
4
  3 S h
220
2
k
2
h l
2
2
k l
2
2

36
2
Example - unusual line broadening effects
in Na parahydroxybenzoate
Sharp lines
Broad lines
Directional
dependence Lattice defects?
Seeming inconsistency in line broadening
- hkl dependent
37
H-atom location in Na parahydroxybenzoate
Good F map allowed by better fit to pattern
F contour map
H-atom location
from x-ray powder data
38
Macroscopic Strain
Part of peak shape function #5 – TOF & CW
d-spacing expression; aij from recip. metric tensor
1
2
d hkl
 M hkl  a 1 h  a 2 k  a 3 l  a 4 kl  a 5 hl  a 6 hk
2
2
2
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.)
NB: In GSAS-II generally available (CW only at present)
39
Symmetry & macrostrain
dij – restricted by symmetry
e.g. for cubic
T = d11h2d3 for TOF (in GSAS)
Result: change in lattice parameters via
change in metric coeff.
aij’ = aij-2dij/C for TOF
aij’ = aij-(/9000)dij for CW
Use new aij’ to get lattice parameters
e.g. for cubic
a 
1
a ij
'
40
Nonstructural Features
Affect the integrated peak intensity and not peak shape
Bragg Intensity Corrections:
Lh
Extinction
Absorption & Surface Roughness
Preferred Orientation/Texture
Other Geometric Factors
}
diagnostic:
Uiso too small!
Extinction – only GSAS for now
Sabine model - Darwin, Zachariasen & Hamilton
Bragg component - reflection
E
=
b
1
1+x
Laue component - transmission
x
x2 5x3
E = 1 +
...x < 1
2
4
4
8
l
E =
l
2 
1
3
x > 1
1
.
.
.
x 

8x 128x2


Combination of two parts
E
= E
h
s in 2Q + E c o s 2Q
b
l
Sabine Extinction Coefficient
  Fh 
x  Ex

 V 
2
Crystallite grain size =
80%
Increasing
wavelength
(1-5 Å)
60%
Eh
40%
20%
0%
0.0
25.0
50.0
75.0
2Q
100.0
125.0
150.0
𝐸𝑥
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 90
apart)
Metal wire
Orientation Distribution Function - probability
function for texture
44
Texture - measurement by diffraction
(220)
Non-random crystallite
orientations in sample
Incident beam
x-rays or neutrons
(200)
Sample
(111)
Debye-Scherrer cones
•uneven intensity due to texture
•also different pattern of unevenness for different hkl’s
•Intensity pattern changes as sample is turned
45
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
Ah 
1
M
n

j 1
 2
sin 
 R o cos 2  

Ro

2





3
2
Integral about distribution
- modify multiplicity
Texture effect on reflection intensity – Sph. Harm. model

l
l
4
mn
m
n
A(h, y ) 

l0
C


2l  1
l
K l (h) K l ( y )
ml nl
• Projection of orientation distribution function for
chosen reflection (h) and sample direction (y)
• K - symmetrized spherical harmonics - account for
sample & crystal symmetry
• “Pole figure” - variation of single reflection intensity as
fxn. of sample orientation - fixed h
• “Inverse pole figure” - modification of all reflection
intensities by sample texture - fixed y - Ideally suited
for neutron TOF diffraction
• Rietveld refinement of coefficients, Clmn, and 3
orientation angles - sample alignment
NB: In GSAS-II as correction & texture analysis
47
Absorption
X-rays - 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  but much
smaller effect
- includes multiple scattering
much bigger effect
- assume cylindrical sample
Debye-Scherrer geometry
Diagnostic: thermal parms. too small!
Model - A.W. Hewat
A h  exp(  T1 A B   T2 A B  )
2
2
For cylinders and weak absorption only
i.e. neutrons - most needed for TOF data
not for CW data – fails for mR>1
GSAS & GSAS-II – New more elaborate model by
Lobanov & alte de Viega – works to mR>10
Other corrections - simple transmission & flat plate
(GSAS only for now)
Surface Roughness – Bragg-Brentano & GSAS 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
SR
q

p  1  q  exp 

sin
Q



p  1  q  exp   q 
(a bit more stable)
SR
q

1  p  1


2
sin
Q
sin
Q



1  p  pq
Other Geometric Corrections
Lorentz correction - both X-rays and neutrons
Polarization correction - only X-rays
X-rays
1 + M c o s 22 Q
L =
p
2 s in 2Q c o s Q
Neutrons - CW
L
Neutrons - TOF
L
p
p
=
1
2 s in 2Q c o s Q
4
= d s in Q
Solvent scattering – proteins & zeolites?
Contrast effect between structure & “disordered” solvent region
f = fo-Aexp(-8Bsin2Q/2)
Carbon scattering factor
uncorrected
6
4
fC
Babinet’s Principle:
Atoms not in vacuum –
change form factors
Solvent
corrected
2
0
0
5
10
15
20
2Q
(GSAS only)
52
Background scattering
Manual subtraction –
not recommended - distorts the weighting scheme for the
observations
& puts a bias in the observations
Fit to a function - many possibilities:
Fourier series - empirical
Chebyschev power series - ditto
Exponential expansions - air scatter & TDS (only GSAS)
Fixed interval points - brute force
Debye equation - amorphous background
(separate diffuse scattering in GSAS; part of bkg. in GSAS-II)
Debye Equation - Amorphous Scattering
real space correlation function
especially good for TOF
terms with
Ai
amplitude
sin( QR i )
QR
exp( 
i
1
2
2
B iQ )
vibration
distance
Neutron TOF - fused silica “quartz”
55
Rietveld Refinement with Debye Function
O
1.60Å
4.13Å
Si
2.63Å
3.12Å
5.11Å
6.1Å
a-quartz 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 a-quartz
56
Non-Structural 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
When to quit?
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”
58
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: (NB: not GSAS-II yet!)
rigid body description – fixed geometry/fewer parameters
stereochemical restraints – more data
59
Constraints vs restraints
Constraints – reduce no. of parameters
Derivative vector
After constraints
(shorter)
F
v i
 R il U lk S kj
F
p j
Derivative vector
Before constraints
(longer)
Rigid body User Symmetry
Rectangular matrices
Restraints – additional information (data) that model must fit
Ex. Bond lengths, angles, etc.
60
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 || b-axis, m ^ 2-fold
on 2-fold: x,z – fixed & U11,U22,U33, & U13 variable
on m: y fixed & U11,U22, U33, & U13 variable
Rietveld programs – GSAS, GSAS-II automatic, others not
61
Multi-atom site fractions
“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 & GSAS-II 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 = 1-FFe
62
Multi-atom site fractions - continued
If 3 atoms A,B,C on site – problem
Diffraction experiment – relative scattering power of site
“1-equation & 2-unknowns” unsolvable problem
Need extra information to solve problem –
2nd diffraction experiment – different scattering power
“2-equations & 2-unknowns” 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 FA = -FC and FB = - FC’
NB: More direct in GSAS-II as constraints are on values!
63
Multi-phase mixtures & multiple data sets
Neutron TOF – multiple detectors
Multi- wavelength synchrotron
X-ray/neutron experiments
How constrain scales, etc.?
I c  I b  I d  S h  S ph Y ph
p
Histogram scale
Phase scale
Ex. 2 phases & 2 histograms – 2 Sh & 4 Sph – 6 scales
Only 4 refinable – remove 2 by constraints
Ex. S11 = -S21 & S12 = -S22
64
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 x-ray pattern – not enough data!
Use rigid bodies – reduce parameters
V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, 174-181 (2003)
65
Rigid body description – 3 rigid bodies
FeCl4 – tetrahedron, origin at Fe
z
Fe - origin
Cl2
Cl1
Cl4
y
x
Cl3
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; Fe-Cl bond
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Rigid body description – continued
C4
C6
x
C2
D2
C1
D1
D
P
C5
P [ 0, 0, 0 ]
O [ 0, 0 1 ]
D=1.4Å
C3
PO – linear, origin at P
C6 – ring, origin at P(!)
(ties them together)
O
z
C1-C6 [ 0, 0, -1 ]
D1=1.6Å; P-C 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; C-C aromatic bond
67
Rigid body description – continued
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
C
C
C
x
R5(z)
C
C
C
47 structural variables
P
R4(x)
R1(x)
y R2(y)
R3(z)
z
O
Fe
68
Refinement - results
Rwp=4.49%
Rp =3.29%
RF2 =9.98%
Nrb =47
Ntot =69
69
Refinement – RB distances & angles
OP(C6)3
1
2
3
R1(x)
122.5(13)
-76.6(4)
69.3(3)
R2(y)
-71.7(3)
-15.4(3)
12.8(3)
R3(z)a
27.5(12)
51.7(3)
-10.4(3)
R3(z)b
147.5(12)
171.7(3)
109.6(3)
R3(z)c
267.5(12)
291.7(3)
229.6(3)
R4(x)
68.7(2)
68.7(2)
68.7(2)
R5(z)a
99.8(15)
193.0(14) 139.2(16)
R5(z)b
81.7(14)
88.3(17)
135.7(17)
R5(z)c
155.3(16)
63.8(16)
156.2(15)
P-O = 1.482(19)Å, P-C = 1.747(7)Å, C-C = 1.357(4)Å,
Fe-Cl = 2.209(9)Å
R5(z)
R2(y- PO)
4
-158.8(9)
69.2(4)
-53.8(9)
66.2(9)
186.2(9)
68.7(2)
64.6(14)
-133.3(16)
224.0(16)
}
PO orientation
}
C3PO torsion
(+0,+120,+240)
 − C-P-O angle
} Phenyl twist
x
R4(x)
R1(x - PO)
R3(z)
z
Fe
70
Packing diagram – see fit of C6 groups
71
Stereochemical restraints – additional “data”
M  f Y  w i Y oi  Y ci 
 f a  w  a oi  a ci 
2
2
i
 f d  w i  d oi  d ci 
 f t  w i   t ci 
2
Bond angles*
Bond distances*
Torsion angle pseudopotentials
4
 f p  w i   p ci 
Plane RMS displacements*
2
 f v  w i v oi  v ci 
4
 f h  w i  h oi  h ci 
2
 f x  w i  x oi  x ci 
 f R  w i (  R ci )
Powder profile (Rietveld)*
2
van der Waals distances (if voi<vci)
Hydrogen bonds
Chiral volumes**
4
“/y” pseudopotential
wi = 1/s2 weighting factor
fx - weight multipliers (typically 0.1-3)
72
For [FeCl2{OP(C6H5)3}4][FeCl4] - restraints
Bond distances:
Fe-Cl = 2.21(1)Å, P-O = 1.48(2)Å, P-C = 1.75(1)Å, C-C = 1.36(1)Å
Number = 4 + 4 + 12 + 72 = 92
Bond angles:
O-P-C, C-P-C & Cl-Fe-Cl = 109.5(10) – assume tetrahedral
C-C-C & P-C-C = 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!!
73
Refinement - results
Rwp=3.94%
Rp =2.89%
RF2 =7.70%
Ntot =277
74
Stereochemical restraints – superimpose on RB results
Nearly identical with RB refinement
Different assumptions – different results
75
New rigid bodies for proteins (actually more general)
 Proteins have too many parameters
 Poor data/parameter ratio - especially for powder data
 Very well known amino acid bonding –
e.g. Engh & Huber
 Reduce “free” variables – fixed bond lengths & angles
 Define new objects for protein structure –
flexible rigid bodies for amino acid residues
 Focus on the “real” variables –
location/orientation & torsion angles of each residue
 Parameter reduction ~1/3 of original protein xyz set
76
Residue rigid body model for phenylalanine
Qijk
c2
txyz
c1
y
3txyz+3Qijk+y+c1+c2 = 9 variables
vs 33 unconstrained xyz coordinates
77
Qijk – Quaternion to represent rotations
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’ = QvQ-1
78
Conclusions – constraints vs. restraints
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)
79
Citations:
 GSAS:
A.C. Larson and R.B. Von Dreele, General Structure Analysis
System (GSAS), Los Alamos National Laboratory Report LAUR
86-748 (2004).
 EXPGUI:
B. H. Toby, EXPGUI, a graphical user interface for GSAS, J.
Appl. Cryst. 34, 210-213 (2001).
 GSAS-II:
None yet except the web site
https://subversion.xor.aps.anl.gov/pyGSAS
We’ll have a paper soon.
80
Thank you Questions from future Crystallographers?
81