Report

Rietveld Refinement with GSAS & GSAS-II Talk will mix both together R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory What does GSAS do in powder pattern analysis? Thanks to Lynn McCusker for maze Includes: - Rietveld refinement - Results - Powder pattern plots -For publication - Bond lengths & angles - Other geometry - CIF (& PDB) files of result - Fourier maps & (some) display - Texture (polefigures) - Utilities Missing: - Indexing - Structure solution Must go elsewhere for these. Form of GSAS PC-GSAS – thin wrapper GUI forplot expedt genles Keyboard interface only powplot fourier .EXP file, etc. disagl GSAS programs – each is a Fortran exe (common library of routines) 3 Form of GSAS & EXPGUI widplt forplot expgui genles fourier disagl expedt GU I powplot Keyboard & mouse liveplot EXPGUI – incomplete GUI access to GSAS but with extras 4 GSAS & EXPGUI interfaces EXPEDT EXPEDT <?> D F K n L P R S X - data setup option (<?>,D,F,K,L,P,R,S,X) > data setup options: Type this help listing Distance/angle calculation set up Fourier calculation set up Delete all but the last n history records Least squares refinement set up Powder data preparation Review data in the experiment file Single crystal data preparation Exit from EXPEDT GSAS – EXPEDT (and everything else) – text based menus with help, macro building, etc. (1980’s user interface!) EXPGUI: access to GSAS Typical GUI – edit boxes, buttons, pull downs etc. Liveplot – powder pattern display (1990’s user interface) 5 GSAS-II: A fresh start Fill in what’s missing from GSAS: - Indexing - Structure solution GSASII – fresh start Base code – python Mixed in old GSAS Fortran Graphics – matplotlib,OpenGL Modern GUI – wxPython Math – numpy,scipy Current: python 2.7 All platforms: Windows, Max OSX & Linux GSAS-II – python code model Slow GUI code – wxPython & common project file name.gpx Fast core processing codes Fast code – numpy array routines (a few fortran routines) Python – ideal for this 7 GSAS-II: A screen shot – 3 frame layout + console Main menu Data tree Submenu Data tabs Data window Graphics window Drawing tabs NB: Dialog box windows will appear wanting a response Rietveld results - visualization Normal Probability 9 Complex peak broadening models m-strain surface NB: mm size & mstrain units 10 Variance-covariance matrix display Useful diagnostic! High V-covV? Forgot a “hold” Highly coupled parms Note “tool tip” 11 Structure drawing Polyhedra Van der Waals atoms Balls & sticks Thermal ellipsoids All selectable by atom Rietveld refinement is multiparameter curve fitting ) (lab CuKa B-B data) Iobs + Icalc | Io-Ic | Refl. positions Result from fluoroapatite refinement – powder profile is curve with counting noise & fit is smooth curve NB: big plot is sqrt(I) Old GSAS example! 13 So how do we get there? Beginning – model errors misfits to pattern Can’t just let go all parameters – too far from best model (minimum c2) False minimum c2 Least-squares cycles True minimum – “global” minimum parameter c2 surface shape depends on parameter suite 14 Fluoroapatite start – add model (1st choose lattice & space group) important – reflection marks match peaks Bad start otherwise – adjust lattice parameters (wrong space group?) 15 2nd add atoms & do default initial refinement – scale & background Notice shape of difference curve – position/shape/intensity errors 16 Errors & parameters? position – lattice parameters, zero point (not common) - other systematic effects – sample shift/offset shape – profile coefficients (GU, GV, GW, LX, LY, etc. in GSAS) intensity – crystal structure (atom positions & thermal parameters) - other systematic effects (absorption/extinction/preferred orientation) NB – get linear combination of all the above NB2 – trend with 2Q (or TOF) important peak shift a – too small too sharp LX - too small wrong intensity Ca2(x) – too small 17 Difference curve – what to do next? Characteristic “up-down-up” profile error NB – can be “down-updown” for too “fat” profile Dominant error – peak shapes? Too sharp? Refine profile parameters next (maybe include lattice parameters) NB - EACH CASE IS DIFFERENT 18 Result – much improved! maybe intensity differences remain – – refine coordinates & thermal parms. 19 Result – essentially unchanged Ca F PO4 Thus, major error in this initial model – peak shapes 20 Pawley/Rietveld refinement Io SIc Ic Exact overlaps - symmetry Incomplete overlaps Residual: R wp w(I o Ic) wI 2 2 o Processing: GSAS – point by point GSAS-II – reflection by reflection Minimize MR w (I o Ic ) 2 21 Least Squares Theory MR Minimize w (I o Ic ) 2 This is done by setting the derivative of MR to zero w I I c I I o I c (a i ) p p i p 0 i i j I c ai - initial values of pi I c (p i ) I c (a i ) pi i p i p = p - a (shift) a i, j w I c I c I c p i p j i x j p j vi i i w ( I) I c p i Normal equations - one for each pi; outer sum over observations Solve for pi - shifts of parameters, NOT values Matrix form: Ax=v & B = A-1 so x = Bv = p Least Squares Theory - continued Matrix equation Ax=v Solve x = A-1v = Bv; B = A-1 This gives set of pi to apply to “old” set of ai repeat until all xi~0 (i.e. no more shifts) Quality of fit – “c2” = M/(N-P) 1 if weights “correct” & model without systematic errors (very rarely achieved) Bii = s2i – “standard uncertainty” (“variance”) in pi (usually scaled by c2) Bij/(Bii*Bjj) – “covariance” between pi & pj Rietveld refinement - this process applied to powder profiles Gcalc - model function for the powder profile (Y elsewhere) 23 Rietveld Model: Yc = Io{SkhF2hmhLhP(h) + Ib} Least-squares: minimize M=Sw(Yo-Yc)2 Io - incident intensity - variable for fixed 2Q kh - scale factor for particular phase F2h - structure factor for particular reflection mh - reflection multiplicity Lh - correction factors on intensity - texture, etc. P(h) - peak shape function - strain & microstrain, etc. Ib - background contribution 24 Peak shape functions – can get exotic! Convolution of contributing functions Instrumental effects Source Geometric aberrations Sample effects Particle size - crystallite size Microstrain - nonidentical unit cell sizes CW Peak Shape Functions – basically 2 parts: Gaussian – usual instrument contribution is “mostly” Gaussian , = − Lorentzian – usual sample broadening contribution , = + G - full width at half maximum – expression from soller slit sizes and monochromator angle & sample broadening - displacement from peak position Convolution – Voigt; linear combination - pseudoVoigt CW Profile Function in GSAS & GSAS-II Thompson, Cox & Hastings (with modifications) Pseudo-Voigt , , = , + − , Mixing coefficient = = FWHM parameter − = = Where Lorentzian FWHM = g and Gaussian FWHM = G 27 CW Axial Broadening Function Finger, Cox & Jephcoat based on van Laar & Yelon Debye-Scherrer cone 2Q Scan H Slit 2Qmin 2Qi 2QBragg Depend on slit & sample “heights” wrt diffr. radius H/L & S/L - parameters in function (combined as S/L+H/L; S = H) (typically 0.002 - 0.020) Pseudo-Voigt (TCH) = profile function 28 How good is this function? Protein Rietveld refinement - Very low angle fit 1.0-4.0° peaks - strong asymmetry “perfect” fit to shape 29 Bragg-Brentano Diffractometer – “parafocusing” Focusing circle Diffractometer circle X-ray source Receiving slit Incident beam slit Sample displaced Beam footprint Sample transparency Divergent beam optics 30 CW Function Coefficients – GSAS & GSAS-II Shifted difference Sample shift ′ = + + − = − Sample transparency = Gaussian profile = + + + + Lorentzian profile = (plus anisotropic broadening terms) Intrepretation? NB: P term not in GSAS-II; sample shift, meff refined directly as parameters 31 Crystallite Size Broadening d*=constant d* d d 2 Q cot Q d 2 Q cot Q sin Q b* 2Q a* Lorentzian term - usual K - Scherrer const. Gaussian term - rare particles same size? p p 180 K " LX " 180 K " GP " NB: In GSAS-II size is refined directly in mm d 2 d cos Q Microstrain Broadening d d d d b* cons tan t d * d* 2Q Q cot Q 2d d a* S 100 % tan Q 180 " LY " Lorentzian term - usual effect Gaussian term - theory? S 100 % " GU " 180 (No, only a misreading) Remove instrumental part NB: In GSAS-II mstrain refined directly; no conversion needed) Microstrain broadening – physical model Model – elastic deformation of crystallites Stephens, P.W. (1999). J. Appl. Cryst. 32, 281-289. Also see Popa, N. (1998). J. Appl. Cryst. 31, 176-180. d-spacing expression 1 2 d hkl M hkl a 1 h a 2 k a 3 l a 4 kl a 5 hl a 6 hk 2 2 2 Broadening – variance in Mhkl s 2 M hkl S ij i,j M M a i a j 34 Microstrain broadening - continued Terms in variance M a 1 h , 2 M a 2 k , 2 M a 3 M l , 2 a 4 kl , M hl , a 5 M a 6 hk Substitute – note similar terms in matrix – collect terms h4 2 2 h k h2l 2 M M 2 a i a j h kl h3l 3 h k 2 h k 2 2 2 h kl h l 2 2 k l 3 hk l h l 2 4 k l 2 2 l k l 3 kl 3 k l 2 hl 3 hkl 3 hkl k k l hk l hk 4 2 3 hl 2 2 hkl 2 h l kl 2 3 2 hk l 3 2 2 2 2 h kl 3 h k 3 hk 2 hkl 2 hk l 2 h kl 2 2 h k 35 Microstrain broadening - continued Broadening – as variance s M hkl S HKL h H k K l L 2 ,H K L 4 HKL 3 collected terms General expression – triclinic – 15 terms s 2 M hkl S 400 h S 040 k S 004 l 3 S 220 h k S 202 h l S 022 k l 4 4 S 4 4 2 S 310 h k S 103 hl 3 2 S 031 k l S 130 hk 3 3 h kl S 121 hk l S 112 hkl 2 211 2 2 2 2 2 3 2 2 S 301 h l S 013 kl 3 3 Symmetry effects – e.g. monoclinic (b unique) – 9 terms s 2 M hkl S 400 h S 040 k 4 4 S 004 l 3 S 202 h l 3 ( S 220 h k 2 S 301 h l S 103 hk 3 4 3 4S 2 2 2 2 S 022 k l ) 2 2 121 hk l Cubic – m3m – 2 terms s 2 M hkl S 400 h 4 k 4 l 4 3 S h 220 2 k 2 h l 2 2 k l 2 2 36 2 Example - unusual line broadening effects in Na parahydroxybenzoate Sharp lines Broad lines Directional dependence Lattice defects? Seeming inconsistency in line broadening - hkl dependent 37 H-atom location in Na parahydroxybenzoate Good F map allowed by better fit to pattern F contour map H-atom location from x-ray powder data 38 Macroscopic Strain Part of peak shape function #5 – TOF & CW d-spacing expression; aij from recip. metric tensor 1 2 d hkl M hkl a 1 h a 2 k a 3 l a 4 kl a 5 hl a 6 hk 2 2 2 Elastic strain – symmetry restricted lattice distortion TOF: ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d3 CW: ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d2tanQ Why? Multiple data sets under different conditions (T,P, x, etc.) NB: In GSAS-II generally available (CW only at present) 39 Symmetry & macrostrain dij – restricted by symmetry e.g. for cubic T = d11h2d3 for TOF (in GSAS) Result: change in lattice parameters via change in metric coeff. aij’ = aij-2dij/C for TOF aij’ = aij-(/9000)dij for CW Use new aij’ to get lattice parameters e.g. for cubic a 1 a ij ' 40 Nonstructural Features Affect the integrated peak intensity and not peak shape Bragg Intensity Corrections: Lh Extinction Absorption & Surface Roughness Preferred Orientation/Texture Other Geometric Factors } diagnostic: Uiso too small! Extinction – only GSAS for now Sabine model - Darwin, Zachariasen & Hamilton Bragg component - reflection E = b 1 1+x Laue component - transmission x x2 5x3 E = 1 + ...x < 1 2 4 4 8 l E = l 2 1 3 x > 1 1 . . . x 8x 128x2 Combination of two parts E = E h s in 2Q + E c o s 2Q b l Sabine Extinction Coefficient Fh x Ex V 2 Crystallite grain size = 80% Increasing wavelength (1-5 Å) 60% Eh 40% 20% 0% 0.0 25.0 50.0 75.0 2Q 100.0 125.0 150.0 What is texture? Nonrandom crystallite grain orientations Random powder - all crystallite orientations equally probable - flat pole figure Pole figure - stereographic projection of a crystal axis down some sample direction Loose powder (100) random texture (100) wire texture Crystallites oriented along wire axis - pole figure peaked in center and at the rim (100’s are 90 apart) Metal wire Orientation Distribution Function - probability function for texture 44 Texture - measurement by diffraction (220) Non-random crystallite orientations in sample Incident beam x-rays or neutrons (200) Sample (111) Debye-Scherrer cones •uneven intensity due to texture •also different pattern of unevenness for different hkl’s •Intensity pattern changes as sample is turned 45 Preferred Orientation - March/Dollase Model Uniaxial packing Ellipsoidal Distribution assumed cylindrical Ro - ratio of ellipsoid axes = 1.0 for no preferred orientation Ellipsoidal particles Spherical Distribution Ah 1 M n j 1 2 sin R o cos 2 Ro 2 3 2 Integral about distribution - modify multiplicity Texture effect on reflection intensity – Sph. Harm. model l l 4 mn m n A(h, y ) l0 C 2l 1 l K l (h) K l ( y ) ml nl • Projection of orientation distribution function for chosen reflection (h) and sample direction (y) • K - symmetrized spherical harmonics - account for sample & crystal symmetry • “Pole figure” - variation of single reflection intensity as fxn. of sample orientation - fixed h • “Inverse pole figure” - modification of all reflection intensities by sample texture - fixed y - Ideally suited for neutron TOF diffraction • Rietveld refinement of coefficients, Clmn, and 3 orientation angles - sample alignment NB: In GSAS-II as correction & texture analysis 47 Absorption X-rays - independent of 2Q - flat sample – surface roughness effect - microabsorption effects - but can change peak shape and shift their positions if small (thick sample) Neutrons - depend on 2Q and but much smaller effect - includes multiple scattering much bigger effect - assume cylindrical sample Debye-Scherrer geometry Diagnostic: thermal parms. too small! Model - A.W. Hewat A h exp( T1 A B T2 A B ) 2 2 For cylinders and weak absorption only i.e. neutrons - most needed for TOF data not for CW data – fails for mR>1 GSAS & GSAS-II – New more elaborate model by Lobanov & alte de Viega – works to mR>10 Other corrections - simple transmission & flat plate (GSAS only for now) Surface Roughness – Bragg-Brentano & GSAS only Low angle – less penetration (scatter in less dense material) - less intensity High angle – more penetration (go thru surface roughness) - more dense material; more intensity Nonuniform sample density with depth from surface Most prevalent with strong sample absorption If uncorrected - atom temperature factors too small Suortti model Pitschke, et al. model SR q p 1 q exp sin Q p 1 q exp q (a bit more stable) SR q 1 p 1 2 sin Q sin Q 1 p pq Other Geometric Corrections Lorentz correction - both X-rays and neutrons Polarization correction - only X-rays X-rays 1 + M c o s 22 Q L = p 2 s in 2Q c o s Q Neutrons - CW L Neutrons - TOF L p p = 1 2 s in 2Q c o s Q 4 = d s in Q Solvent scattering – proteins & zeolites? Contrast effect between structure & “disordered” solvent region f = fo-Aexp(-8Bsin2Q/2) Carbon scattering factor uncorrected 6 4 fC Babinet’s Principle: Atoms not in vacuum – change form factors Solvent corrected 2 0 0 5 10 15 20 2Q (GSAS only) 52 Background scattering Manual subtraction – not recommended - distorts the weighting scheme for the observations & puts a bias in the observations Fit to a function - many possibilities: Fourier series - empirical Chebyschev power series - ditto Exponential expansions - air scatter & TDS (only GSAS) Fixed interval points - brute force Debye equation - amorphous background (separate diffuse scattering in GSAS; part of bkg. in GSAS-II) Debye Equation - Amorphous Scattering real space correlation function especially good for TOF terms with Ai amplitude sin( QR i ) QR exp( i 1 2 2 B iQ ) vibration distance Neutron TOF - fused silica “quartz” 55 Rietveld Refinement with Debye Function O 1.60Å 4.13Å Si 2.63Å 3.12Å 5.11Å 6.1Å a-quartz distances 7 terms Ri –interatomic distances in SiO2 glass 1.587(1), 2.648(1), 4.133(3), 4.998(2), 6.201(7), 7.411(7) & 8.837(21) Same as found in a-quartz 56 Non-Structural Features in Powder Patterns Summary 1. Large crystallite size - extinction 2. Preferred orientation 3. Small crystallite size - peak shape 4. Microstrain (defect concentration) 5. Amorphous scattering - background When to quit? Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned” Also – “stop when you’ve run out of things to vary” What if problem is more complex? Apply constraints & restraints “What to do when you have too many parameters & not enough data” 58 Complex structures (even proteins) Too many parameters – “free” refinement fails Known stereochemistry: Bond distances Bond angles Torsion angles (less definite) Group planarity (e.g. phenyl groups) Chiral centers – handedness Etc. Choice: (NB: not GSAS-II yet!) rigid body description – fixed geometry/fewer parameters stereochemical restraints – more data 59 Constraints vs restraints Constraints – reduce no. of parameters Derivative vector After constraints (shorter) F v i R il U lk S kj F p j Derivative vector Before constraints (longer) Rigid body User Symmetry Rectangular matrices Restraints – additional information (data) that model must fit Ex. Bond lengths, angles, etc. 60 Space group symmetry constraints Special positions – on symmetry elements Axes, mirrors & inversion centers (not glides & screws) Restrictions on refineable parameters Simple example: atom on inversion center – fixed x,y,z What about Uij’s? – no restriction – ellipsoid has inversion center Mirrors & axes ? – depends on orientation Example: P 2/m – 2 || b-axis, m ^ 2-fold on 2-fold: x,z – fixed & U11,U22,U33, & U13 variable on m: y fixed & U11,U22, U33, & U13 variable Rietveld programs – GSAS, GSAS-II automatic, others not 61 Multi-atom site fractions “site fraction” – fraction of site occupied by atom “site multiplicity”- no. times site occurs in cell “occupancy” – site fraction * site multiplicity may be normalized by max multiplicity GSAS & GSAS-II uses fraction & multiplicity derived from sp. gp. Others use occupancy If two atoms in site – Ex. Fe/Mg in olivine Then (if site full) FMg = 1-FFe 62 Multi-atom site fractions - continued If 3 atoms A,B,C on site – problem Diffraction experiment – relative scattering power of site “1-equation & 2-unknowns” unsolvable problem Need extra information to solve problem – 2nd diffraction experiment – different scattering power “2-equations & 2-unknowns” problem Constraint: solution of J.-M. Joubert Add an atom – site has 4 atoms A, B, C, C’ so that FA+FB+FC+FC’=1 Then constrain so FA = -FC and FB = - FC’ NB: More direct in GSAS-II as constraints are on values! 63 Multi-phase mixtures & multiple data sets Neutron TOF – multiple detectors Multi- wavelength synchrotron X-ray/neutron experiments How constrain scales, etc.? I c I b I d S h S ph Y ph p Histogram scale Phase scale Ex. 2 phases & 2 histograms – 2 Sh & 4 Sph – 6 scales Only 4 refinable – remove 2 by constraints Ex. S11 = -S21 & S12 = -S22 64 Rigid body problem – 88 atoms – [FeCl2{OP(C6H5)3}4][FeCl4] P21/c a=14.00Å b=27.71Å c=18.31Å b=104.53 V=6879Å3 264 parameters – no constraints Just one x-ray pattern – not enough data! Use rigid bodies – reduce parameters V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, 174-181 (2003) 65 Rigid body description – 3 rigid bodies FeCl4 – tetrahedron, origin at Fe z Fe - origin Cl2 Cl1 Cl4 y x Cl3 1 translation, 5 vectors Fe [ 0, 0, 0 ] Cl1 [ sin(54.75), 0, cos(54.75)] Cl2 [ -sin(54,75), 0, cos(54.75)] Cl3 [ 0, sin(54.75), -cos(54.75)] Cl4 [ 0, -sin(54.75), -cos(54.75)] D=2.1Å; Fe-Cl bond 66 Rigid body description – continued C4 C6 x C2 D2 C1 D1 D P C5 P [ 0, 0, 0 ] O [ 0, 0 1 ] D=1.4Å C3 PO – linear, origin at P C6 – ring, origin at P(!) (ties them together) O z C1-C6 [ 0, 0, -1 ] D1=1.6Å; P-C bond C1 [ 0, 0, 0 ] C2 [ sin(60), 0, -1/2 ] C3 [-sin(60), 0, -1/2 ] C4 [ sin(60), 0, -3/2 ] C5 [-sin(60), 0, -3/2 ] C6 [ 0, 0, -2 ] D2=1.38Å; C-C aromatic bond 67 Rigid body description – continued Rigid body rotations – about P atom origin For PO group – R1(x) & R2(y) – 4 sets For C6 group – R1(x), R2(y),R3(z),R4(x),R5(z) 3 for each PO; R3(z)=+0, +120, & +240; R4(x)=70.55 Transform: X’=R1(x)R2(y)R3(z)R4(x)R5(z)X C C C x R5(z) C C C 47 structural variables P R4(x) R1(x) y R2(y) R3(z) z O Fe 68 Refinement - results Rwp=4.49% Rp =3.29% RF2 =9.98% Nrb =47 Ntot =69 69 Refinement – RB distances & angles OP(C6)3 1 2 3 R1(x) 122.5(13) -76.6(4) 69.3(3) R2(y) -71.7(3) -15.4(3) 12.8(3) R3(z)a 27.5(12) 51.7(3) -10.4(3) R3(z)b 147.5(12) 171.7(3) 109.6(3) R3(z)c 267.5(12) 291.7(3) 229.6(3) R4(x) 68.7(2) 68.7(2) 68.7(2) R5(z)a 99.8(15) 193.0(14) 139.2(16) R5(z)b 81.7(14) 88.3(17) 135.7(17) R5(z)c 155.3(16) 63.8(16) 156.2(15) P-O = 1.482(19)Å, P-C = 1.747(7)Å, C-C = 1.357(4)Å, Fe-Cl = 2.209(9)Å R5(z) R2(y- PO) 4 -158.8(9) 69.2(4) -53.8(9) 66.2(9) 186.2(9) 68.7(2) 64.6(14) -133.3(16) 224.0(16) } PO orientation } C3PO torsion (+0,+120,+240) − C-P-O angle } Phenyl twist x R4(x) R1(x - PO) R3(z) z Fe 70 Packing diagram – see fit of C6 groups 71 Stereochemical restraints – additional “data” M f Y w i Y oi Y ci f a w a oi a ci 2 2 i f d w i d oi d ci f t w i t ci 2 Bond angles* Bond distances* Torsion angle pseudopotentials 4 f p w i p ci Plane RMS displacements* 2 f v w i v oi v ci 4 f h w i h oi h ci 2 f x w i x oi x ci f R w i ( R ci ) Powder profile (Rietveld)* 2 van der Waals distances (if voi<vci) Hydrogen bonds Chiral volumes** 4 “/y” pseudopotential wi = 1/s2 weighting factor fx - weight multipliers (typically 0.1-3) 72 For [FeCl2{OP(C6H5)3}4][FeCl4] - restraints Bond distances: Fe-Cl = 2.21(1)Å, P-O = 1.48(2)Å, P-C = 1.75(1)Å, C-C = 1.36(1)Å Number = 4 + 4 + 12 + 72 = 92 Bond angles: O-P-C, C-P-C & Cl-Fe-Cl = 109.5(10) – assume tetrahedral C-C-C & P-C-C = 120(1) – assume hexagon Number = 12 + 12 + 6 + 72 + 24 = 126 Planes: C6 to 0.01 – flat phenyl Number = 72 Total = 92 + 126 + 72 = 290 restraints A lot easier to setup than RB!! 73 Refinement - results Rwp=3.94% Rp =2.89% RF2 =7.70% Ntot =277 74 Stereochemical restraints – superimpose on RB results Nearly identical with RB refinement Different assumptions – different results 75 New rigid bodies for proteins (actually more general) Proteins have too many parameters Poor data/parameter ratio - especially for powder data Very well known amino acid bonding – e.g. Engh & Huber Reduce “free” variables – fixed bond lengths & angles Define new objects for protein structure – flexible rigid bodies for amino acid residues Focus on the “real” variables – location/orientation & torsion angles of each residue Parameter reduction ~1/3 of original protein xyz set 76 Residue rigid body model for phenylalanine Qijk c2 txyz c1 y 3txyz+3Qijk+y+c1+c2 = 9 variables vs 33 unconstrained xyz coordinates 77 Qijk – Quaternion to represent rotations In GSAS defined as: Qijk = r+ai+bj+ck – 4D complex number – 1 real + 3 imaginary components Normalization: r2+a2+b2+c2 = 1 Rotation vector: v = ax+by+cz; u = (ax+by+cz)/sin(a/2) Rotation angle: r2 = cos2(a/2); a2+b2+c2 = sin2(a/2) Quaternion product: Qab = Qa * Qb ≠ Qb * Qa Quaternion vector transformation: v’ = QvQ-1 78 Conclusions – constraints vs. restraints Constraints required space group restrictions multiatom site occupancy Rigid body constraints reduce number of parameters molecular geometry assumptions Restraints add data molecular geometry assumptions (again) 79 Citations: GSAS: A.C. Larson and R.B. Von Dreele, General Structure Analysis System (GSAS), Los Alamos National Laboratory Report LAUR 86-748 (2004). EXPGUI: B. H. Toby, EXPGUI, a graphical user interface for GSAS, J. Appl. Cryst. 34, 210-213 (2001). GSAS-II: None yet except the web site https://subversion.xor.aps.anl.gov/pyGSAS We’ll have a paper soon. 80 Thank you Questions from future Crystallographers? 81