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

2. 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.

3. 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

4. 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

5. 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)

6. 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

7. 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

8. 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

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

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

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

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

13. 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)

14. 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 ) 22 parameter False minimum True minimum – “global” minimum Least-squares cycles  2 surface shape depends on parameter suite

15. 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?)

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

17. 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

18. 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

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

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

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

22. 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

23. 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)

24. 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

25. 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!

26. 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

27. 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 = 

28. 28 CW Axial Broadening Function Finger, Cox & Jephcoat based on van Laar & Yelon 2  Bragg 2i2i 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

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

30. 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

31. 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

32. 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

33. 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)

34. 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

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

36. 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

37. 37 Example - unusual line broadening effects in Na parahydroxybenzoate Sharp lines Broad lines Seeming inconsistency in line broadening - hkl dependent Directional dependence - Lattice defects?

38. 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

39. 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)

40. 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

41. 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!

42. 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

43. Sabine Extinction Coefficient Crystallite grain size = 22 0% 20% 40% 60% 80% 0.025.050.075.0100.0125.0150.0 EhEh Increasing wavelength (1-5 Å)

44. 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

45. 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)

46. 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

47. 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

48. 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!

49. 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)

50. 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)

51. 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 

52. 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 22 fCfC uncorrected Solvent corrected Carbon scattering factor

53. 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

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

55. 55 Neutron TOF - fused silica “quartz”

56. 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

57. 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

58. 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”

59. 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

60. 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.

61. 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

62. 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

63. 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

64. 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

65. 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)

66. 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

67. 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)

68. 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

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

70. 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 }

71. 71 Packing diagram – see fit of C 6 groups

72. 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)

73. 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!!

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

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

76. 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

77. 77 t xyz Q ijk    Residue rigid body model for phenylalanine 3t xyz +3Q ijk +  +  1 +  2 = 9 variables vs 33 unconstrained xyz coordinates

78. 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

79. 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)

80. 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.

81. 81 Thank you - Questions from future Crystallographers?