1. Rietveld Refinement with GSAS
Recent Quote seen in Rietveld
“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

2. 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)

3. c2 surface shape depends on parameter suite
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
Least-squares cycles c2
True minimum – “global” minimum
parameter
c2 surface shape depends on parameter suite

4. Fluoroapatite start – add model (1st choose lattice/sp. grp.)
important – reflection marks match peaks
Bad start otherwise – adjust lattice parameters (wrong space group?)

5. 2nd add atoms & do default initial refinement – scale & background
Notice shape of difference curve – position/shape/intensity errors

6. NB – get linear combination of all the above
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
too sharp
wrong intensity
a – too small
LX - too small
Ca2(x) – too small

7. Difference curve – what to do next?
Characteristic “up-down-up”
profile error
NB – can be “down-up-down” for too “fat” profile
Dominant error – peak shapes? Too sharp?
Refine profile parameters next (maybe include lattice parameters)
NB - EACH CASE IS DIFFERENT

8. Result – much improved!
maybe intensity differences left – refine coordinates & thermal parms.

9. Result – essentially unchangedCa F
PO4
Thus, major error in this initial model – peak shapes

10. So how does Rietveld refinement work?
Rietveld Minimize
Exact overlaps
- symmetry Io
Residuals:
Incomplete overlaps
SIc Ic
Extract structure factors:
Apportion Io by ratio of Ic to Sic & apply corrections

11. 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

12. 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

13. Least Squares Theory - continued
Rearrange .
Matrix form: Ax=v

14. Least Squares Theory - continued
Matrix equation Ax=v
Solve x = A-1v = Bv; B = A-1
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/(N-P)  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)

15. Rietveld Model: Yc = Io{SkhF2hmhLhP(Dh) + 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(Dh) - peak shape function - strain & microstrain, etc.
Ib - background contribution

16. 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

17. Gaussian – usual instrument contribution is “mostly” Gaussian
CW Peak Shape Functions – basically 2 parts:
Gaussian – usual instrument contribution is “mostly” Gaussian P ( D k ) = H p 4 l n 2 e [ - / ] G
Lorentzian – usual sample broadening contribution P ( D k ) = p H 2 1 + 4 / L
H - full width at half maximum - expression
from soller slit sizes and monochromator
angle
- displacement from peak position
Convolution – Voigt; linear combination - pseudoVoigt

18. CW Profile Function in GSAS
Thompson, Cox & Hastings (with modifications)
Pseudo-Voigt
Mixing coefficient
FWHM parameter

19. 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 )
Ä Pseudo-Voigt (TCH)
= profile function

20. How good is this function?
Protein Rietveld refinement - Very low angle fit
° peaks - strong asymmetry “perfect” fit to shape

21. Bragg-Brentano Diffractometer – “parafocusing”
Focusing circle
Diffractometer
circle
X-ray source
Receiving slit
Incident beam
slit
Sample
displaced
Sample
transparency
Beam footprint
Divergent beam optics

22. CW Function Coefficients - GSAS
Shifted difference
Sample shift
Sample transparency
Gaussian profile
Lorentzian profile
(plus anisotropic broadening terms) Intrepretation?

23. Crystallite Size Broadening
Dd*=constant a* b*
Lorentzian term - usual
K - Scherrer const.
Gaussian term - rare
particles same size?

24. Microstrain Broadening
Lorentzian term - usual effect
Gaussian term - theory?
Remove instrumental part

25. Microstrain broadening – physical model
Model – elastic deformation of crystallites
Stephens, P.W. (1999). J. Appl. Cryst. 32,
Also see Popa, N. (1998). J. Appl. Cryst. 31,
d-spacing expression
Broadening – variance in Mhkl

26. Microstrain broadening - continued
Terms in variance
Substitute – note similar terms in matrix – collect terms

27. 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

28. Example - unusual line broadening effects
in Na parahydroxybenzoate
Sharp lines
Broad lines
Directional dependence -
Lattice defects?
Seeming inconsistency in line broadening
- hkl dependent

29. H-atom location in Na parahydroxybenzoate
Good DF map allowed by better fit to pattern
DF contour map
H-atom location
from x-ray powder data

30. Macroscopic Strain
Part of peak shape function #5 – TOF & CW
d-spacing 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.)

31. Symmetry & macrostrain
dij – restricted by symmetry
e.g. for cubic
DT = 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

32. Nonstructural Features
Affect the integrated peak intensity and not peak shape
Bragg Intensity Corrections:
Extinction
Preferred Orientation
Absorption & Surface Roughness
Other Geometric Factors L h

33. Sabine model - Darwin, Zachariasen & Hamilton
Extinction
Sabine model - Darwin, Zachariasen & Hamilton
Bragg component - reflection
Laue component - transmission E b = 1 + x E l = 1 - 2 x + 4 8 5 3 .
< E l = p x 2 é ê ë 1 - 8 3 . ù ú û
> E h = b s i n 2 Q + l c o
Combination of two parts

34. Sabine Extinction Coefficient
Crystallite grain size = 0%
20%
40%
60%
80%
0.0
25.0
50.0
75.0
100.0
125.0
150.0
Increasing
wavelength
(1-5 Å) Eh 2Q

35. 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

36. Texture - measurement by diffraction
(220)
Non-random crystallite orientations in sample
(200)
Incident beam
x-rays or neutrons
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

37. 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

38. å Texture - Orientation Distribution Function - GSAS
Probability distribution of crystallite orientations - f(g)
f(g) = f(F1,Y,F2) F1 F2 Y f ( g ) = å
l=0
m=-l l
n=-l C m n T
Tlmn = Associated Legendre functions or generalized spherical harmonics
F1,Y,F2 - Euler angles

39. Texture effect on reflection intensity - Rietveld model
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

40. 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 l but much
smaller effect
- includes multiple scattering
much bigger effect
- assume cylindrical sample
Debye-Scherrer geometry

41. Model - A.W. Hewat
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

42. 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
(a bit more stable)

43. Other Geometric Corrections
Lorentz correction - both X-rays and neutrons
Polarization correction - only X-rays
X-rays
Neutrons - CW
Neutrons - TOF L p = 2 s i n Q c o 1 + M L p = 2 s i n Q c o 1 L p = d 4 s i n Q

44. Solvent scattering – proteins & zeolites?
Contrast effect between structure & “disordered” solvent region
f = fo-Aexp(-8pBsin2Q/l2) 2 4 6 2Q fC
uncorrected
Solvent
corrected
Carbon scattering factor
Babinet’s Principle:
Atoms not in vacuum –
change form factors

45. 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)

46. real space correlation function especially good for TOF terms with
Debye Equation - Amorphous Scattering
real space correlation function especially good for TOF terms with
vibration
amplitude
distance

47. Neutron TOF - fused silica “quartz”

48. Rietveld Refinement with Debye Function
1.60Å
4.13Å Si
3.12Å
2.63Å
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

49. 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

50. “A Rietveld refinement is never perfected, merely abandoned”
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”

51. 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

52. Constraints vs restraints
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.

53. 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

54. 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

55. 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 DFA = -DFC and DFB = -D FC’

56. Multi-phase mixtures & multiple data sets
Neutron TOF – multiple detectors
Multi- wavelength synchrotron
X-ray/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

57. Rigid body problem – 88 atoms – [FeCl2{OP(C6H5)3}4][FeCl4]
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, (2003)

58. Rigid body description – 3 rigid bodies
FeCl4 – tetrahedron, origin at Fe
1 translation, 5 vectors
Fe [ , , ]
Cl1 [ sin(54.75), 0, cos(54.75)]
Cl2 [ -sin(54,75), 0, cos(54.75)]
Cl3 [ , sin(54.75), -cos(54.75)]
Cl4 [ , -sin(54.75), -cos(54.75)]
D=2.1Å; Fe-Cl bond z
Fe - origin
Cl2
Cl1 y
Cl4 x
Cl3

59. Rigid body description – continued
PO – linear, origin at P
C6 – ring, origin at P(!) C4 C2 x
(ties them together) D2 C1 D1 D z C6 P O
C1-C6 [ 0, 0, ]
D1=1.6Å; P-C bond
C1 [ 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, , ]
D2=1.38Å; C-C aromatic bond C5 C3
P [ 0, 0, 0 ]
O [ 0, ]
D=1.4Å

60. 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 P O C z x y
R1(x)
R2(y)
R3(z)
R5(z)
R4(x) Fe
47 structural variables

61. Refinement - results
Rwp=4.49%
Rp =3.29%
RF2 =9.98%
Nrb =47
Ntot =69

62. } } } Refinement – RB distances & angles OP(C6)3 1 2 3 4
R1(x) 122.5(13) (4) 69.3(3) (9)
R2(y) (3) (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 (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) (14) (16) 64.6(14)
R5(z)b 81.7(14) 88.3(17) 135.7(17) (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)
R4(x)
R1(x - PO)
R3(z)
R2(y- PO) Fe z

63. Packing diagram – see fit of C6 groups

64. 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 0.1-3)

65. A lot easier to setup than RB!!
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 = = 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 = = 126
Planes: C6 to 0.01 – flat phenyl
Number = 72
Total = = 290 restraints
A lot easier to setup than RB!!

66. Refinement - results
Rwp=3.94%
Rp =2.89%
RF2 =7.70%
Ntot =277

67. Stereochemical restraints – superimpose on RB results
Nearly identical with RB refinement
Different assumptions – different results

68. 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

69. Residue rigid body model for phenylalanine
Qijk c2
txyz c1 y
3txyz+3Qijk+y+c1+c2 = 9 variables
vs 33 unconstrained xyz coordinates

70. 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

71. 21542 observations; 1148 atoms (1001 HEWL)
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”

72. 194L & rigid body model – essentially identical

73. 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)

74. GSAS - A bit of history
GSAS – conceived in (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

75. 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

76. PC-GSAS – GUI only for access to GSAS programs
pull down menus for GSAS programs
(not linux)

77. GSAS & EXPGUI interfaces
GSAS – EXPEDT (and everything else):
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
On console screen
Keyboard input – text & numbers
1 letter commands – menu help
Layers of menus – tree structure
Type ahead thru layers of menus
commands)
Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e-5’ all allowed
Drag & drop for e.g. file names

78. GSAS & EXPGUI interfaces
Access to GSAS
Typical GUI – edit boxes,
buttons, pull downs etc.
Liveplot – powder pattern

79. 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)

80. Powder pattern display - liveplot
Zoom
(new plot)
updates at end of genles run – check if OK!
cum. c2 on

81. Powder pattern display - powplotIo 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

82. Powplot options – x & y axes – “improved” plot?
Sqrt(I)
Refl. pos.
rescale y by 4x
Q-scale (from Q=pl/sinq)

83. Citations:
GSAS:
A.C. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los Alamos National Laboratory Report LAUR (2004).
EXPGUI:
B. H. Toby, EXPGUI, a graphical user interface for GSAS, J. Appl. Cryst. 34, (2001).