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R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory 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) Demonstration – refinement of fluroapatite Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned”

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2 Result from fluoroapatite refinement – powder profile is curve with counting noise & fit is smooth curve NB: big plot is sqrt(I) Rietveld refinement is multiparameter curve fitting I obs + I calc | I o -I c | ) Refl. positions (lab CuK B-B data)

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3 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 2 ) 22 parameter False minimum True minimum – “global” minimum Least-squares cycles 2 surface shape depends on parameter suite

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

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

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6 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 NB 2 – trend with 2 (or TOF) important a – too small LX - too small Ca2(x) – too small too sharp peak shiftwrong intensity

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

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

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9 Result – essentially unchanged Thus, major error in this initial model – peak shapes Ca F PO 4

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10 So how does Rietveld refinement work? Exact overlaps - symmetry Incomplete overlaps Extract structure factors: Apportion I o by ratio of I c to i c & apply corrections IoIo IcIc Residuals: Rietveld Minimize IcIc

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11 Rietveld refinement - Least Squares Theory and a function then the best estimate of the values p i is found by minimizing This is done by setting the derivative to zero Results in n “normal” equations (one for each variable) - solve for p i Given a set of observations G obs

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12 Least Squares Theory - continued Problem - g(p i ) is nonlinear & transcendental (sin, cos, etc.) so can’t solve directly Expand g(p i ) as Taylor series & toss high order terms Substitute above a i - initial values of p i p i = p i - a i (shift) Normal equations - one for each p i ; outer sum over observations Solve for p i - shifts of parameters, NOT values

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

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14 Least Squares Theory - continued Matrix equation Ax=v Solve x = A -1 v = Bv; B = A -1 This gives set of p i to apply to “old” set of a i repeat until all x i ~0 (i.e. no more shifts) Quality of fit – “ 2 ” = M/(N-P) 1 if weights “correct” & model without systematic errors (very rarely achieved) B ii = 2 i – “standard uncertainty” (“variance”) in p i (usually scaled by 2 ) B ij /(B ii *B jj ) – “covariance” between p i & p j Rietveld refinement - this process applied to powder profiles G calc - model function for the powder profile (Y elsewhere)

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15 Rietveld Model: Y c = I o { k h F 2 h m h L h P( h ) + I b } I o - incident intensity - variable for fixed 2 k h - scale factor for particular phase F 2 h - structure factor for particular reflection m h - reflection multiplicity L h - correction factors on intensity - texture, etc. P( h ) - peak shape function - strain & microstrain, etc. I b - background contribution Least-squares: minimize M= w(Y o -Y c ) 2

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Convolution of contributing functions Instrumental effects Source Geometric aberrations Sample effects Particle size - crystallite size Microstrain - nonidentical unit cell sizes Peak shape functions – can get exotic!

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Gaussian – usual instrument contribution is “mostly” Gaussian H - full width at half maximum - expression from soller slit sizes and monochromator angle - displacement from peak position P( k ) = H k 4ln2 e [-4ln2 k 2 / H k 2 ] = G CW Peak Shape Functions – basically 2 parts: Lorentzian – usual sample broadening contribution P( k ) = H k k 2 /H k 2 1 = L Convolution – Voigt; linear combination - pseudoVoigt

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

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19 CW Axial Broadening Function Finger, Cox & Jephcoat based on van Laar & Yelon 2 Bragg 2i2i 2 min Pseudo-Voigt (TCH) = profile function Depend on slit & sample “heights” wrt diffr. radius H/L & S/L - parameters in function (typically ) Debye-Scherrer cone 2 Scan Slit H

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

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

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

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

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Microstrain Broadening a* b* Lorentzian term - usual effect Gaussian term - theory? Remove instrumental part

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

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

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27 Microstrain broadening - continued Broadening – as variance General expression – triclinic – 15 terms Symmetry effects – e.g. monoclinic (b unique) – 9 terms 3 collected terms Cubic – m3m – 2 terms

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

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29 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

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30 Macroscopic Strain Part of peak shape function #5 – TOF & CW d-spacing expression; ij from recip. metric tensor Elastic strain – symmetry restricted lattice distortion TOF: ΔT = ( 11 h 2 + 22 k 2 + 33 l 2 + 12 hk+ 13 hl+ 23 kl)d 3 CW: ΔT = ( 11 h 2 + 22 k 2 + 33 l 2 + 12 hk+ 13 hl+ 23 kl)d 2 tan Why? Multiple data sets under different conditions (T,P, x, etc.)

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31 Symmetry & macrostrain ij – restricted by symmetry e.g. for cubic T = 11 h 2 d 3 for TOF Result: change in lattice parameters via change in metric coeff. ij ’ = ij -2 ij /C for TOF ij ’ = ij -( /9000) ij for CW Use new ij ’ to get lattice parameters e.g. for cubic

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

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Sabine model - Darwin, Zachariasen & Hamilton Bragg component - reflection Laue component - transmission Extinctio n E h = E b sin 2 + E l cos 2 E b = 1+x 1 Combination of two parts E l = x + 4 x x 3... x < 1 E l = x 2 1 - 8x x x > 1

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Sabine Extinction Coefficient Crystallite grain size = E x 22 0% 20% 40% 60% 80% EhEh Increasing wavelength (1-5 Å)

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35 Random powder - all crystallite orientations equally probable - flat pole figure 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 (100) wire texture (100) random texture What is texture? Nonrandom crystallite grain orientations Pole figure - stereographic projection of a crystal axis down some sample direction Loose powder Metal wire

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36 Texture - measurement by diffraction Debye-Scherrer cones uneven intensity due to texture also different pattern of unevenness for different hkl’s Intensity pattern changes as sample is turned Non-random crystallite orientations in sample Incident beam x-rays or neutrons Sample (111) (200) (220)

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Spherical Distribution Ellipsoidal Distribution - assumed cylindrical Ellipsoidal particles Uniaxial packing Preferred Orientation - March/Dollase Model Integral about distribution - modify multiplicity R o - ratio of ellipsoid axes = 1.0 for no preferred orientation

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Texture - Orientation Distribution Function - GSAS f(g) = l=0 m=-l l n=-l l C l mn T l mn (g) T l mn = Associated Legendre functions or generalized spherical harmonics - Euler angles f(g) = f( ) Probability distribution of crystallite orientations - f(g)

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39 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, C l mn, and 3 orientation angles - sample alignment Texture effect on reflection intensity - Rietveld model

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Absorption X-rays - independent of 2 - flat sample – surface roughness effect - microabsorption effects - but can change peak shape and shift their positions if small (thick sample) Neutrons - depend on 2 and but much smaller effect - includes multiple scattering much bigger effect - assume cylindrical sample Debye-Scherrer geometry

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Model - A.W. Hewat 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

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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 Surface Roughness – Bragg-Brentano only High angle – more penetration (go thru surface roughness) - more dense material; more intensity Low angle – less penetration (scatter in less dense material) - less intensity (a bit more stable)

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Other Geometric Corrections Lorentz correction - both X-rays and neutrons Polarization correction - only X-rays X-rays Neutrons - CW Neutrons - TOF L p = 2sin 2 cos 1 + M cos 2 2 L p = 2sin 2 cos 1 L p = d 4 sin

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44 Solvent scattering – proteins & zeolites? Contrast effect between structure & “disordered” solvent region Babinet’s Principle: Atoms not in vacuum – change form factors f = f o -Aexp(-8 Bsin 2 / 2 ) 2 fCfC uncorrected Solvent corrected Carbon scattering factor

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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) Background scattering

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

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47 Neutron TOF - fused silica “quartz”

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48 Rietveld Refinement with Debye Function 7 terms R i –interatomic distances in SiO 2 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 -quartz 1.60Å Si O 4.13Å 2.63Å 3.12Å 5.11Å 6.1Å -quartz distances

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Summary Non-Structural Features in Powder Patterns 1. Large crystallite size - extinction 2. Preferred orientation 3. Small crystallite size - peak shape 4. Microstrain (defect concentration) 5. Amorphous scattering - background

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50 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”

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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

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52 Constraints vs restraints Constraints – reduce no. of parameters Rigid bodyUserSymmetry Derivative vector Before constraints (longer) Derivative vector After constraints (shorter) Rectangular matrices Restraints – additional information (data) that model must fit Ex. Bond lengths, angles, etc.

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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 U ij ’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 & U 11,U 22,U 33, & U 13 variable on m: y fixed & U 11,U 22, U 33, & U 13 variable Rietveld programs – GSAS automatic, others not

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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) F Mg = 1-F Fe

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55 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 – 2 nd 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 F A +F B +F C +F C’ =1 Then constrain so F A = - F C and F B = - F C’ Multi-atom site fractions - continued

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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 S h & 4 S ph – 6 scales Only 4 refinable – remove 2 by constraints Ex. S 11 = - S 21 & S 12 = - S 22

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57 Rigid body problem – 88 atoms – [FeCl 2 {OP(C 6 H 5 ) 3 } 4 ][FeCl 4 ] 264 parameters – no constraints Just one x-ray pattern – not enough data! Use rigid bodies – reduce parameters P2 1 /c a=14.00Å b=27.71Å c=18.31Å = V=6879Å 3 V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, (2003)

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58 Rigid body description – 3 rigid bodies FeCl 4 – tetrahedron, origin at Fe z x y Fe - origin Cl 1 Cl 2 Cl 3 Cl 4 1 translation, 5 vectors Fe [ 0, 0, 0 ] Cl 1 [ sin(54.75), 0, cos(54.75)] Cl 2 [ -sin(54,75), 0, cos(54.75)] Cl 3 [ 0, sin(54.75), -cos(54.75)] Cl 4 [ 0, -sin(54.75), -cos(54.75)] D=2.1Å; Fe-Cl bond

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59 PO – linear, origin at P C 6 – ring, origin at P(!) Rigid body description – continued PO C1C1 C5C5 C3C3 C4C4 C2C2 C6C6 z x P [ 0, 0, 0 ] O [ 0, 0 1 ] D=1.4Å C 1 -C 6 [ 0, 0, -1 ] D 1 =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 ] D 2 =1.38Å; C-C aromatic bond D D1D1 D2D2 (ties them together)

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60 Rigid body description – continued Rigid body rotations – about P atom origin For PO group – R 1 (x) & R 2 (y) – 4 sets For C 6 group – R 1 (x), R 2 (y),R 3 (z),R 4 (x),R 5 (z) 3 for each PO; R 3 (z)=+0, +120, & +240; R 4 (x)=70.55 Transform: X’=R 1 (x)R 2 (y)R 3 (z)R 4 (x)R 5 (z)X 47 structural variables P O C CC CC C z x y R 1 (x) R 2 (y) R 3 (z) R 5 (z) R 4 (x) Fe

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61 Refinement - results R wp =4.49% R p =3.29% R F 2 =9.98% N rb =47 N tot =69

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62 Refinement – RB distances & angles OP(C 6 ) R 1 (x)122.5(13) -76.6(4) 69.3(3) (9) R 2 (y) -71.7(3) -15.4(3) 12.8(3) 69.2(4) R 3 (z) a 27.5(12)51.7(3)-10.4(3)-53.8(9) R 3 (z) b 147.5(12)171.7(3)109.6(3)66.2(9) R 3 (z) c 267.5(12)291.7(3)229.6(3)186.2(9) R 4 (x) 68.7(2)68.7(2)68.7(2)68.7(2) R 5 (z) a 99.8(15) 193.0(14) 139.2(16)64.6(14) R 5 (z) b 81.7(14)88.3(17)135.7(17)-133.3(16) R 5 (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 z x R 1 (x - PO) R 2 (y- PO) R 3 (z) R 5 (z) R 4 (x) Fe } Phenyl twist − C-P-O angle C 3 PO torsion (+0,+120,+240) } PO orientation }

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63 Packing diagram – see fit of C 6 groups

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64 Stereochemical restraints – additional “data” Powder profile (Rietveld)* Bond angles* Bond distances* Torsion angle pseudopotentials Plane RMS displacements* van der Waals distances (if v oi

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65 For [FeCl 2 {OP(C 6 H 5 ) 3 } 4 ][FeCl 4 ] - 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: C 6 to 0.01 – flat phenyl Number = 72 Total = = 290 restraints A lot easier to setup than RB!!

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66 Refinement - results R wp =3.94% R p =2.89% R F 2 =7.70% N tot =277

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

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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

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69 t xyz Q ijk Residue rigid body model for phenylalanine 3t xyz +3Q ijk + + 1 + 2 = 9 variables vs 33 unconstrained xyz coordinates

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70 Q ijk – Quaternion to represent rotations In GSAS defined as: Q ijk = r+ai+bj+ck – 4D complex number – 1 real + 3 imaginary components Normalization: r 2 +a 2 +b 2 +c 2 = 1 Rotation vector: v = a x +b y +c z ; u = (a x +b y +c z )/sin( /2) Rotation angle: r 2 = cos 2 ( /2); a 2 +b 2 +c 2 = sin 2 ( /2) Quaternion product: Q ab = Q a * Q b ≠ Q b * Q a Quaternion vector transformation: v’ = QvQ -1

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71 How effective? PDB 194L – HEWL from a space crystallization; single xtal data,1.40Å resolution X-Plor 3.1 – R F = 25.8% ~4600 variables GSAS RB refinement – R F =20.9% ~2700 variables RMS difference Å main chain & 0.21Å all protein atoms observations; 1148 atoms (1001 HEWL) RB refinement reduces effect of “over refinement”

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72 194L & rigid body model – essentially identical

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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)

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74 GSAS - A bit of history GSAS – conceived in (A.C. Larson & R.B. Von Dreele) 1 st version released in Dec 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

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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

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

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77 GSAS & EXPGUI interfaces 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) GSAS – EXPEDT (and everything else): Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e-5’ all allowed Drag & drop for e.g. file names

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

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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 Gauss FWHM (instrument) Lorentz FWHM (sample) Sum widplt

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80 Powder pattern display - liveplot Zoom (new plot) cum. 2 on updates at end of genles run – check if OK!

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81 Powder pattern display - powplot “publication style” plot – works OK for many journals; save as “emf” can be “dressed up”; also ascii output of x,y table I o -I c Refl. pos. IoIo IcIc

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

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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).

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