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R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory Rietveld Refinement with GSAS & GSAS-II Talk will mix both together

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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.

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3 Form of GSAS genles disagl fourier expedt forplot powplot PC-GSAS – thin wrapper GUI GSAS programs – each is a Fortran exe (common library of routines).EXP file, etc. Keyboard interface only

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4 genles disagl fourier forplot powplot Form of GSAS & EXPGUI widplt liveplot GU I expedt expgui EXPGUI – incomplete GUI access to GSAS but with extras Keyboard & mouse

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GSAS & EXPGUI interfaces 5 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 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)

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GSAS-II: A fresh start GSASII – fresh start Fill in what’s missing from GSAS: - Indexing - Structure solution 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

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7 GSAS-II – python code model Fast core processing codes (a few fortran routines) Slow GUI code – wxPython & common project file name.gpx Fast code – numpy array routines Python – ideal for this

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GSAS-II: A screen shot – 3 frame layout + console Data tree Data window Graphics window Data tabs Main menu Submenu Drawing tabs NB: Dialog box windows will appear wanting a response

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9 Rietveld results - visualization Easy zoom I/ (I) Normal Probability

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10 Complex peak broadening models -strain surface NB: m size & strain units

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11 Variance-covariance matrix display Useful diagnostic! High V-covV? Forgot a “hold” Highly coupled parms Note “tool tip”

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Structure drawing Polyhedra Van der Waals atoms Balls & sticks Thermal ellipsoids All selectable by atom

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13 Result from fluoroapatite refinement – powder profile is curve with counting noise & fit is smooth curve NB: big plot is sqrt(I) Old GSAS example! 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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14 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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15 Fluoroapatite start – add model (1 st choose lattice & space group) important – reflection marks match peaks Bad start otherwise – adjust lattice parameters (wrong space group?)

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16 2 nd add atoms & do default initial refinement – scale & background Notice shape of difference curve – position/shape/intensity errors

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17 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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18 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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19 Result – much improved! maybe intensity differences remain –– refine coordinates & thermal parms.

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

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Pawley/Rietveld refinement 21 Exact overlaps - symmetry Incomplete overlaps IoIo IcIc Residual: IcIc Minimize Processing: GSAS – point by point GSAS-II – reflection by reflection

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Least Squares Theory This is done by setting the derivative of M R to zero 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 Matrix form: Ax=v & B = A -1 so x = Bv = p Minimize

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23 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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24 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 - full width at half maximum – expression from soller slit sizes and monochromator angle & sample broadening - displacement from peak position CW Peak Shape Functions – basically 2 parts: Lorentzian – usual sample broadening contribution Convolution – Voigt; linear combination - pseudoVoigt

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27 CW Profile Function in GSAS & GSAS-II Thompson, Cox & Hastings (with modifications) Pseudo-Voigt Mixing coefficient FWHM parameter Where Lorentzian FWHM = and Gaussian FWHM =

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28 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 (combined as S/L+H/L; S = H) (typically 0.002 - 0.020) Debye-Scherrer cone 2 Scan Slit H

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29 How good is this function? Protein Rietveld refinement - Very low angle fit 1.0-4.0° peaks - strong asymmetry “perfect” fit to shape

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30 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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31 CW Function Coefficients – GSAS & GSAS-II Sample shift Sample transparency Gaussian profile Lorentzian profile (plus anisotropic broadening terms) Intrepretation? Shifted difference NB: P term not in GSAS-II; sample shift, eff refined directly as parameters

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Crystallite Size Broadening a* b* d*=constant Lorentzian term - usual K - Scherrer const. Gaussian term - rare particles same size? NB: In GSAS-II size is refined directly in m

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Microstrain Broadening a* b* Lorentzian term - usual effect Gaussian term - theory? (No, only a misreading) Remove instrumental part NB: In GSAS-II strain refined directly; no conversion needed)

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34 Microstrain broadening – physical model Stephens, P.W. (1999). J. Appl. Cryst. 32, 281-289. Also see Popa, N. (1998). J. Appl. Cryst. 31, 176-180. Model – elastic deformation of crystallites d-spacing expression Broadening – variance in M hkl

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

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36 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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37 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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38 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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39 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.) NB: In GSAS-II generally available (CW only at present)

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40 Symmetry & macrostrain ij – restricted by symmetry e.g. for cubic T = 11 h 2 d 3 for TOF (in GSAS) 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: L h Extinction Absorption & Surface Roughness Preferred Orientation/Texture Other Geometric Factors Affect the integrated peak intensity and not peak shape Nonstructural Features } diagnostic: U iso too small!

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Sabine model - Darwin, Zachariasen & Hamilton Bragg component - reflection Laue component - transmission Extinction – only GSAS for now E h = E b sin 2 + E l cos 2 E b = 1+x 1 Combination of two parts E l = 1 - 2 x + 4 x 2 - 48 5x 3... x < 1 E l = x 2 1 - 8x 1 - 128x 2 3... x > 1

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Sabine Extinction Coefficient Crystallite grain size = 22 0% 20% 40% 60% 80% 0.025.050.075.0100.0125.0150.0 EhEh Increasing wavelength (1-5 Å)

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44 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 90 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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45 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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47 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 NB: In GSAS-II as correction & texture analysis Texture effect on reflection intensity – Sph. Harm. 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 Diagnostic: thermal parms. too small!

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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 & GSAS-II – New more elaborate model by Lobanov & alte de Viega – works to R>10 Other corrections - simple transmission & flat plate (GSAS only for now)

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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 & GSAS 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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52 Solvent scattering – proteins & zeolites? Contrast effect between structure & “disordered” solvent region Babinet’s Principle: Atoms not in vacuum – change form factors (GSAS only) f = f o -Aexp(-8 Bsin 2 / 2 ) 0 2 4 6 0 5 10 15 20 22 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 (only GSAS) Fixed interval points - brute force Debye equation - amorphous background (separate diffuse scattering in GSAS; part of bkg. in GSAS-II) 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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55 Neutron TOF - fused silica “quartz”

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56 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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58 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”

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59 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

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60 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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61 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, GSAS-II automatic, others not

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62 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) F Mg = 1-F Fe

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63 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’ NB: More direct in GSAS-II as constraints are on values! Multi-atom site fractions - continued

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64 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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65 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 =104.53 V=6879Å 3 V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, 174-181 (2003)

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66 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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67 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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68 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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69 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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70 Refinement – RB distances & angles OP(C 6 ) 3 1234 R 1 (x)122.5(13) -76.6(4) 69.3(3) -158.8(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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71 Packing diagram – see fit of C 6 groups

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72 Stereochemical restraints – additional “data” Powder profile (Rietveld)* Bond angles* Bond distances* Torsion angle pseudopotentials Plane RMS displacements* van der Waals distances (if v oi <v ci ) Hydrogen bonds Chiral volumes** “ ” pseudopotential w i = 1/ 2 weighting factor f x - weight multipliers (typically 0.1-3)

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73 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 = 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: C 6 to 0.01 – flat phenyl Number = 72 Total = 92 + 126 + 72 = 290 restraints A lot easier to setup than RB!!

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

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

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76 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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77 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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78 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(a/2) Rotation angle: r 2 = cos 2 (a/2); a 2 +b 2 +c 2 = sin 2 (a/2) Quaternion product: Q ab = Q a * Q b ≠ Q b * Q a Quaternion vector transformation: v’ = QvQ -1

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79 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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80 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 https://subversion.xor.aps.anl.gov/pyGSAS We’ll have a paper soon.

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81 Thank you - Questions from future Crystallographers?

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