Rietveld Refinement with GSAS
Recent Quote seen in Rietveld e-mail:
“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
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)
2
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
3
Fluoroapatite start – add model (1st choose lattice/sp. grp.)
 important – reflection marks match peaks
 Bad start otherwise – adjust lattice parameters (wrong space group?)
4
2nd add atoms & do default initial refinement –
scale & background
 Notice shape of difference curve – position/shape/intensity errors
5
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
6
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
7
Result – much improved!
 maybe intensity differences left – refine coordinates & thermal parms.
8
Result – essentially unchanged
Ca
F
PO4
Thus, major error in this initial model – peak shapes
9
So how does Rietveld refinement work?
Rietveld Minimize
Exact overlaps
M R   w( I o  I c )2
- symmetry
Io
Incomplete overlaps
SIc
Ic
Residuals:
Rwp 
2
w(I

I
)
 o c
 wI
2
o
Extract structure factors:
Apportion Io by ratio of Ic
to Sic & apply corrections
 Ic
1
F 
I o 

Lp
  Ic
2
o




10
Rietveld refinement - Least Squares Theory
Given a set of observations Gobs
and a function Gcalc  g(p1, p 2 , p 3 ..., pn )
then the best estimate of the values pi is found by
minimizing
M   w(G  G )2
o
c
This is done by setting the derivative to zero
 w (Go  Gc )
 Gc
0
p j
Results in n “normal” equations (one for each
variable) - solve for pi
11
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
Gc (pi )  Gc (ai )  
i
Gc
pi
pi
ai - initial values of pi
pi = pi - ai (shift)
Substitute above

 Gc
Gc
pi 
0
 w  G  
i pi

 p j
G  Go  Gc (ai )
Normal equations - one for each pi; outer sum
over observations
Solve for pi - shifts of parameters, NOT values
12
Least Squares Theory - continued
Rearrange

Gc  n Gc
Gc
w

p

w

G



i
p1  i1 pi
p1

.
.
.

Gc  n Gc
Gc
pi    wG

w
pn  i1 pi
pn

Matrix form: Ax=v
a i, j   w
Gc Gc
p i  p j
x j  p j
vi   w (G)
Gc
p i
13
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)
14
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
15
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
2
2
[-4ln2 k/ Hk]
P(k) = 4ln2 e
=G
Hk p
Lorentzian – usual sample broadening contribution
1
=L
P( k ) = 2
p Hk 1 + 4 2/H 2
k
k
H - full width at half maximum - expression
from soller slit sizes and monochromator
angle
- displacement from peak position
Convolution – Voigt; linear combination - pseudoVoigt
CW Profile Function in GSAS
Thompson, Cox & Hastings (with modifications)
Pseudo-Voigt
P(T)  L(T, )  (1 )G(T, )
  k j (  )j

j1
3
Mixing coefficient
FWHM parameter
5
5
5 i i
c

 ig 
i1
18
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
(typically 0.002 - 0.020)
 Pseudo-Voigt (TCH)
= profile function
19
How good is this function?
Protein Rietveld refinement - Very low angle fit
1.0-4.0° peaks - strong asymmetry “perfect” fit to shape
20
Bragg-Brentano Diffractometer – “parafocusing”
Focusing circle
Diffractometer
circle
X-ray source
Receiving slit
Incident beam
slit
Sample
displaced
Divergent beam optics
Sample
Beam footprint transparency
21
CW Function Coefficients - GSAS
Shifted difference
T'  T  Ss cosQ  Ts sin2Q
 pRS s
36000
 9000
 eff 
Sample transparency
pRTs
Sample shift
s
Gaussian profile g2  U tan2 Q  V tan Q  W  P
Lorentzian profile
cos2 Q
X

 Y tan Q
cos Q
(plus anisotropic broadening terms) Intrepretation?
22
Crystallite Size Broadening
d*=constant
d Q cot Q

2
d
d
2Q cot Q sin Q


 d
 2Q  2
d cos Q
d* 
b*
a*
Lorentzian term - usual
K - Scherrer const.
p
180K
p" LX"
Gaussian term - rare
particles same size?
p
180K
p " GP"
Microstrain Broadening
d
 cons tan t
d
d d *

 Q cot Q
d
d*
b*
a*
Lorentzian term - usual effect
Gaussian term - theory?
Remove instrumental part
2 d
 2Q 
tan Q
d
S  100%
p
" LY"
180
p
S  100%
" GU"
180
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
2
2

M

a
h

a
k

a
l
 a 4 kl  a 5 hl  a 6 hk
hkl
1
2
3
2
d hkl
Broadening – variance in Mhkl
s2 M hkl    Sij
i,j
M M
ai a j
25
Microstrain broadening - continued
Terms in variance
M
M
M
M
M
2
2 M
2
h ,
k ,
l ,
 kl ,
 hl ,
 hk
a1
a 2
a 3
a 4
a 5
a 6
Substitute – note similar terms in matrix – collect terms
 h4
 2 2
h k
M M  h2 l 2
 2
a i a j  h kl
 h3 l
 3
 h k
h2 k 2
h2 l 2
h2 kl
k4
k 2l 2
k 3l
k 2l 2
k 3l
l4
kl 3
kl 3
k 2l 2
hk 2 l
hl3
hkl2
hk 3
hkl2
hk 2 l
h3 k 

hk 2 l hk 3 
hl3 hkl2 
2
2 
hkl hk l 
h2 l 2 h2 kl 

2
2 2
h kl h k 
h3 l
26
Microstrain broadening - continued
Broadening – as variance
s 2 M hkl    SHKL h H k K l L , H  K  L  4
HKL
3 collected terms
General expression – triclinic – 15 terms
s 2 M hkl   S400 h4  S040 k 4  S004 l 4  3 S220 h2 k 2  S202 h2 l 2  S022 k 2 l 2  

4 S

2 S310 h3 k  S103 hl3  S031 k 3 l  S130 hk3  S301 h3 l  S013 kl 3 
2
2
2
h
kl

S
hk
l

S
hkl
211
121
112

Symmetry effects – e.g. monoclinic (b unique) – 9 terms
s 2 M hkl   S 400 h 4  S 040 k 4  S 004 l 4  3S 202 h 2 l 2  3( S 220 h 2 k 2  S 022 k 2 l 2 ) 


2 S 301 h 3l  S103 hk 3  4S121 hk 2 l
Cubic – m3m – 2 terms
s 2 M hkl   S400 h4  k 4  l 4   3 S220 h2 k 2  h2 l 2  k 2 l 2 
27
Example - unusual line broadening effects
in Na parahydroxybenzoate
Sharp lines
Broad lines
Directional
dependence Lattice defects?
Seeming inconsistency in line broadening
- hkl dependent
28
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
29
Macroscopic Strain
Part of peak shape function #5 – TOF & CW
d-spacing expression; aij from recip. metric tensor
1
2
2
2

M

a
h

a
k

a
l
 a 4 kl  a 5 hl  a 6 hk
hkl
1
2
3
2
d hkl
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.)
30
Symmetry & macrostrain
dij – restricted by symmetry
e.g. for cubic
T = d11h2d3 for TOF
Result: change in lattice parameters via
change in metric coeff.
aij’ = aij-2dij/C for TOF
aij’ = aij-(p/9000)dij for CW
Use new aij’ to get lattice parameters
e.g. for cubic
1
a
a ij'
31
Nonstructural Features
Affect the integrated peak intensity and not peak shape
Bragg Intensity Corrections:
Lh
Extinction
Preferred Orientation
Absorption & Surface Roughness
Other Geometric Factors
Extinctio
n
Sabine model - Darwin, Zachariasen & Hamilton
Eb =
Bragg component - reflection
1
1+x
Laue component - transmission
2
3
El = 1 - x + x - 5x . . . x < 1
2 4 48
El =
2 1 - 1 - 3 . . .  x > 1

px  8x 128x2


Combination of two parts
2
2
Eh = Eb sin Q + El cos Q
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 90º
apart)
Orientation Distribution Function - probability
function for texture
Metal wire
35
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
36
Preferred Orientation - March/Dollase Model
Uniaxial packing
Ellipsoidal Distribution assumed cylindrical
Ellipsoidal particles
Spherical Distribution
Ro - ratio of ellipsoid
axes = 1.0 for no
preferred orientation
1 n  2
sin2  
2

Oh    Ro cos  
M j1 
Ro 

3
2
Integral about distribution
- modify multiplicity
Texture - Orientation Distribution Function - GSAS
Probability distribution of crystallite orientations - f(g)
f(g) = f(F1,Y,F2)

f(g) = 
l
l

l=0 m=-l n=-l
F1
Y
F2
F1,Y,F2 - Euler angles
C
mn
l
T
mn
l
Tlmn = Associated
Legendre functions or
generalized spherical
harmonics
(g)
Texture effect on reflection intensity - Rietveld model
4p l l mn m
n
A(h, y)  
C
K
(
h
)
K


l
l
l ( y)
l 0 2l  1 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
39
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
Model - A.W. Hewat
Ah  exp(T1AB  T2 AB2 2 )
For cylinders and weak absorption only
i.e. neutrons - most needed for TOF data
not for CW data – fails for R>1
GSAS – New more elaborate model by
Lobanov & alte de Viega – works to R>10
Other corrections - simple transmission & flat plate
Surface Roughness – Bragg-Brentano 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
q

 1

p  1  qexp  q
1

p

2
sinQ 
sinQ
sin
Q


SR 
SR 
p  1  qexp q
1 p  pq
(a bit more stable)
Other Geometric Corrections
Lorentz correction - both X-rays and neutrons
Polarization correction - only X-rays
2
X-rays
Lp = 1 + M2 cos 2Q
2sin Q cosQ
Neutrons - CW
Lp =
Neutrons - TOF
1
2
2sin Q cosQ
4
Lp = d sinQ
Solvent scattering – proteins & zeolites?
Contrast effect between structure & “disordered” solvent region
f = fo-Aexp(-8pBsin2Q/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
44
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
Fixed interval points - brute force
Debye equation - amorphous background
(separate diffuse scattering in GSAS)
Debye Equation - Amorphous Scattering
real space correlation function
especially good for TOF
terms with
sin(QR i )
1
Ai
exp( BiQ2 )
QR i
2
vibration
amplitude
distance
Neutron TOF - fused silica “quartz”
47
Rietveld Refinement with Debye Function
O
1.60Å
4.13Å
Si
3.12Å
5.11Å
2.63Å
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
48
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
Time 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”
50
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
51
Constraints vs restraints
Constraints – reduce no. of parameters
Derivative vector
After constraints
(shorter)
F
F
 R il U lk Skj
v i
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.
52
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 automatic, others not
53
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 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
54
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’
55
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  Sh  Sph Yph
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
56
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)
57
Rigid body description – 3 rigid bodies
FeCl4 – tetrahedron, origin at Fe
z
Fe - origin
Cl1
Cl4
x
Cl2
y
Cl3
1 translation, 5 vectors
Fe [
0,
0,
Cl1 [ sin(54.75), 0,
Cl2 [ -sin(54,75), 0,
Cl3 [ 0, sin(54.75),
Cl4 [ 0, -sin(54.75),
D=2.1Å; Fe-Cl bond
0
]
cos(54.75)]
cos(54.75)]
-cos(54.75)]
-cos(54.75)]
58
Rigid body description – continued
C4
PO – linear, origin at P
C6 – ring, origin at P(!)
x
C2
(ties them together)
D2
C6
C1 D1
D
P
C5
P [ 0, 0, 0 ]
O [ 0, 0 1 ]
D=1.4Å
C3
z
O
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
59
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
60
Refinement - results
Rwp=4.49%
Rp =3.29%
RF2 =9.98%
Nrb =47
Ntot =69
61
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)
P-O = 1.482(19)Å, P-C = 1.747(7)Å, C-C = 1.357(4)Å,
Fe-Cl = 2.209(9)Å
} PO orientation
}
C3PO torsion
(+0,+120,+240)
p − C-P-O angle
}Phenyl twist
x
R5(z)
R2(y- PO)
R4(x)
R1(x - PO)
R3(z)
z
Fe
62
Packing diagram – see fit of C6 groups
63
Stereochemical restraints – additional “data”
M  fY  wi Yoi  Yci 
2
Powder profile (Rietveld)*
2
Bond angles*
 f a  wi aoi  aci 
 f d  wi d oi  d ci 
2
 f t  wi  tci 
4
Bond distances*
Torsion angle pseudopotentials
 f p  wi  pci 
2
Plane RMS displacements*
 f v  wi voi  vci 
4
 f h  wi hoi  hci 
2
 f x  wi  xoi  xci 
2
van der Waals distances (if voi<vci)
Hydrogen bonds
Chiral volumes**
 f R  wi ( Rci ) 4
“/y” pseudopotential
wi = 1/s2 weighting factor
fx - weight multipliers (typically 0.1-3)
64
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!!
65
Refinement - results
Rwp=3.94%
Rp =2.89%
RF2 =7.70%
Ntot =277
66
Stereochemical restraints – superimpose on RB results
Nearly identical with RB refinement
Different assumptions – different results
67
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
68
Residue rigid body model for phenylalanine
Qijk
c2
txyz
c1
y
3txyz+3Qijk+y+c1+c2 = 9 variables
vs 33 unconstrained xyz coordinates
69
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
70
How effective? PDB 194L – HEWL from a space
crystallization; single xtal data,1.40Å resolution
21542 observations; 1148 atoms (1001 HEWL)
X-Plor 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”
71
194L & rigid body model – essentially identical
72
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)
73
GSAS - A bit of history
GSAS – conceived in 1982-1983 (A.C. Larson & R.B. Von Dreele)
1st version released in Dec. 1985
•Only TOF neutrons (& buggy)
•Only for VAX
•Designed for multiple data (histograms) & multiple phases from
the start
•Did single crystal from start
Later – add CW neutrons & CW x-rays (powder data)
SGI unix version & then PC (MS-DOS) version
also Linux version (briefly HP unix version)
2001 – EXPGUI developed by B.H. Toby
Recent – spherical harmonics texture & proteins
New Windows, MacOSX, Fedora & RedHat linux versions
All identical code – g77 Fortran; 50 pgms. & 800 subroutines
GrWin & X graphics via pgplot
EXPGUI – all Tcl/Tk – user additions welcome
Basic structure is essentially unchanged
74
Structure of GSAS
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
75
PC-GSAS – GUI only for access to GSAS programs
pull down menus for GSAS programs
(not linux)
76
GSAS & EXPGUI interfaces
GSAS – EXPEDT (and everything else):
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
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)
Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e-5’ all allowed
Drag & drop for e.g. file names
77
GSAS & EXPGUI interfaces
EXPGUI:
Access to GSAS
Typical GUI – edit boxes,
buttons, pull downs etc.
Liveplot – powder pattern
78
Unique EXPGUI features (not in GSAS)




CIF input – read CIF files (not mmCIF)
widplt/absplt
coordinate export – various formats
instrument parameter file creation/edit
widplt
Sum
Lorentz FWHM
(sample)
Gauss FWHM
(instrument)
79
Powder pattern display - liveplot
Zoom
(new plot)
updates at end of genles run – check if OK!
cum. c2 on
80
Powder pattern display - powplot
Io
Ic
Refl. pos.
Io-Ic
“publication style” plot – works OK for many journals; save as “emf”
can be “dressed up”; also ascii output of x,y table
81
Powplot options – x & y axes – “improved” plot?
Sqrt(I)
Refl. pos.
Io  Ic
rescale y by 4x
Q-scale (from Q=p/sinq)
82
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).
83