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**Rietveld Refinement with GSAS**

Recent Quote seen in Rietveld “Rietveld refinement is one of those few fields of intellectual endeavor wherein the more one does it, the less one understands.” (Sue Kesson) Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned” Demonstration – refinement of fluroapatite R.B. Von Dreele, Advanced Photon Source Argonne National Laboratory

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**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)

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**c2 surface shape depends on parameter suite**

So how do we get there? Beginning – model errors misfits to pattern Can’t just let go all parameters – too far from best model (minimum c2) False minimum Least-squares cycles c2 True minimum – “global” minimum parameter c2 surface shape depends on parameter suite

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**Fluoroapatite start – add model (1st choose lattice/sp. grp.)**

important – reflection marks match peaks Bad start otherwise – adjust lattice parameters (wrong space group?)

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**2nd add atoms & do default initial refinement – scale & background**

Notice shape of difference curve – position/shape/intensity errors

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**NB – get linear combination of all the above **

Errors & parameters? position – lattice parameters, zero point (not common) - other systematic effects – sample shift/offset shape – profile coefficients (GU, GV, GW, LX, LY, etc. in GSAS) intensity – crystal structure (atom positions & thermal parameters) - other systematic effects (absorption/extinction/preferred orientation) NB – get linear combination of all the above NB2 – trend with 2Q (or TOF) important peak shift too sharp wrong intensity a – too small LX - too small Ca2(x) – too small

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**Difference curve – what to do next?**

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

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

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**Result – essentially unchanged**

Ca F PO4 Thus, major error in this initial model – peak shapes

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**So how does Rietveld refinement work?**

Rietveld Minimize Exact overlaps - symmetry Io Residuals: Incomplete overlaps SIc Ic Extract structure factors: Apportion Io by ratio of Ic to Sic & apply corrections

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**Rietveld refinement - Least Squares Theory**

Given a set of observations Gobs and a function then the best estimate of the values pi is found by minimizing This is done by setting the derivative to zero Results in n “normal” equations (one for each variable) - solve for pi

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**Least Squares Theory - continued**

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

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**Least Squares Theory - continued**

Rearrange . Matrix form: Ax=v

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**Least Squares Theory - continued**

Matrix equation Ax=v Solve x = A-1v = Bv; B = A-1 This gives set of Dpi to apply to “old” set of ai repeat until all xi~0 (i.e. no more shifts) Quality of fit – “c2” = M/(N-P) 1 if weights “correct” & model without systematic errors (very rarely achieved) Bii = s2i – “standard uncertainty” (“variance”) in Dpi (usually scaled by c2) Bij/(Bii*Bjj) – “covariance” between Dpi & Dpj Rietveld refinement - this process applied to powder profiles Gcalc - model function for the powder profile (Y elsewhere)

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**Rietveld Model: Yc = Io{SkhF2hmhLhP(Dh) + Ib}**

Least-squares: minimize M=Sw(Yo-Yc)2 Io - incident intensity - variable for fixed 2Q kh - scale factor for particular phase F2h - structure factor for particular reflection mh - reflection multiplicity Lh - correction factors on intensity - texture, etc. P(Dh) - peak shape function - strain & microstrain, etc. Ib - background contribution

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**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

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**Gaussian – usual instrument contribution is “mostly” Gaussian**

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

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**CW Profile Function in GSAS**

Thompson, Cox & Hastings (with modifications) Pseudo-Voigt Mixing coefficient FWHM parameter

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**CW Axial Broadening Function**

Finger, Cox & Jephcoat based on van Laar & Yelon Debye-Scherrer cone 2Q Scan H Slit 2Qmin 2Qi 2QBragg Depend on slit & sample “heights” wrt diffr. radius H/L & S/L - parameters in function (typically ) Ä Pseudo-Voigt (TCH) = profile function

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**How good is this function?**

Protein Rietveld refinement - Very low angle fit ° peaks - strong asymmetry “perfect” fit to shape

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**Bragg-Brentano Diffractometer – “parafocusing”**

Focusing circle Diffractometer circle X-ray source Receiving slit Incident beam slit Sample displaced Sample transparency Beam footprint Divergent beam optics

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**CW Function Coefficients - GSAS**

Shifted difference Sample shift Sample transparency Gaussian profile Lorentzian profile (plus anisotropic broadening terms) Intrepretation?

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**Crystallite Size Broadening**

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

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**Microstrain Broadening**

Lorentzian term - usual effect Gaussian term - theory? Remove instrumental part

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**Microstrain broadening – physical model**

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

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**Microstrain broadening - continued**

Terms in variance Substitute – note similar terms in matrix – collect terms

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**Microstrain broadening - continued**

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

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**Example - unusual line broadening effects**

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

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**H-atom location in Na parahydroxybenzoate**

Good DF map allowed by better fit to pattern DF contour map H-atom location from x-ray powder data

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Macroscopic Strain Part of peak shape function #5 – TOF & CW d-spacing expression; aij from recip. metric tensor Elastic strain – symmetry restricted lattice distortion TOF: ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d3 CW: ΔT = (d11h2+d22k2+d33l2+d12hk+d13hl+d23kl)d2tanQ Why? Multiple data sets under different conditions (T,P, x, etc.)

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**Symmetry & macrostrain**

dij – restricted by symmetry e.g. for cubic DT = d11h2d3 for TOF Result: change in lattice parameters via change in metric coeff. aij’ = aij-2dij/C for TOF aij’ = aij-(p/9000)dij for CW Use new aij’ to get lattice parameters e.g. for cubic

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**Nonstructural Features**

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

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**Sabine model - Darwin, Zachariasen & Hamilton **

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

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**Sabine Extinction Coefficient**

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

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**What is texture? Nonrandom crystallite grain orientations**

Random powder - all crystallite orientations equally probable - flat pole figure Pole figure - stereographic projection of a crystal axis down some sample direction Loose powder (100) random texture (100) wire texture Crystallites oriented along wire axis - pole figure peaked in center and at the rim (100’s are 90º apart) Orientation Distribution Function - probability function for texture Metal wire

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**Texture - measurement by diffraction**

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

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**Preferred Orientation - March/Dollase Model**

Uniaxial packing Ellipsoidal Distribution - assumed cylindrical Ro - ratio of ellipsoid axes = 1.0 for no preferred orientation Ellipsoidal particles Spherical Distribution Integral about distribution - modify multiplicity

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**å Texture - Orientation Distribution Function - GSAS**

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

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**Texture effect on reflection intensity - Rietveld model**

Projection of orientation distribution function for chosen reflection (h) and sample direction (y) K - symmetrized spherical harmonics - account for sample & crystal symmetry “Pole figure” - variation of single reflection intensity as fxn. of sample orientation - fixed h “Inverse pole figure” - modification of all reflection intensities by sample texture - fixed y - Ideally suited for neutron TOF diffraction Rietveld refinement of coefficients, Clmn, and 3 orientation angles - sample alignment

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

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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 mR>1 GSAS – New more elaborate model by Lobanov & alte de Viega – works to mR>10 Other corrections - simple transmission & flat plate

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**Surface Roughness – Bragg-Brentano only**

Low angle – less penetration (scatter in less dense material) - less intensity High angle – more penetration (go thru surface roughness) - more dense material; more intensity Nonuniform sample density with depth from surface Most prevalent with strong sample absorption If uncorrected - atom temperature factors too small Suortti model Pitschke, et al. model (a bit more stable)

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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 = 2 s i n Q c o 1 + M L p = 2 s i n Q c o 1 L p = d 4 s i n Q

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**Solvent scattering – proteins & zeolites?**

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

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**Background scattering**

Manual subtraction – not recommended - distorts the weighting scheme for the observations & puts a bias in the observations Fit to a function - many possibilities: Fourier series - empirical Chebyschev power series - ditto Exponential expansions - air scatter & TDS Fixed interval points - brute force Debye equation - amorphous background (separate diffuse scattering in GSAS)

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**real space correlation function especially good for TOF terms with**

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

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**Neutron TOF - fused silica “quartz”**

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**Rietveld Refinement with Debye Function**

1.60Å 4.13Å Si 3.12Å 2.63Å 5.11Å 6.1Å a-quartz distances 7 terms Ri –interatomic distances in SiO2 glass 1.587(1), 2.648(1), 4.133(3), 4.998(2), 6.201(7), 7.411(7) & 8.837(21) Same as found in a-quartz

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**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

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**“A Rietveld refinement is never perfected, merely abandoned” **

Time to quit? Stephens’ Law – “A Rietveld refinement is never perfected, merely abandoned” Also – “stop when you’ve run out of things to vary” What if problem is more complex? Apply constraints & restraints “What to do when you have too many parameters & not enough data”

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**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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**Constraints vs restraints**

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

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**Space group symmetry constraints**

Special positions – on symmetry elements Axes, mirrors & inversion centers (not glides & screws) Restrictions on refineable parameters Simple example: atom on inversion center – fixed x,y,z What about Uij’s? – no restriction – ellipsoid has inversion center Mirrors & axes ? – depends on orientation Example: P 2/m – 2 || b-axis, m ^ 2-fold on 2-fold: x,z – fixed & U11,U22,U33, & U13 variable on m: y fixed & U11,U22, U33, & U13 variable Rietveld programs – GSAS automatic, others not

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**Multi-atom site fractions**

“site fraction” – fraction of site occupied by atom “site multiplicity”- no. times site occurs in cell “occupancy” – site fraction * site multiplicity may be normalized by max multiplicity GSAS uses fraction & multiplicity derived from sp. gp. Others use occupancy If two atoms in site – Ex. Fe/Mg in olivine Then (if site full) FMg = 1-FFe

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**Multi-atom site fractions - continued**

If 3 atoms A,B,C on site – problem Diffraction experiment – relative scattering power of site “1-equation & 2-unknowns” unsolvable problem Need extra information to solve problem – 2nd diffraction experiment – different scattering power “2-equations & 2-unknowns” problem Constraint: solution of J.-M. Joubert Add an atom – site has 4 atoms A, B, C, C’ so that FA+FB+FC+FC’=1 Then constrain so DFA = -DFC and DFB = -D FC’

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**Multi-phase mixtures & multiple data sets**

Neutron TOF – multiple detectors Multi- wavelength synchrotron X-ray/neutron experiments How constrain scales, etc.? Histogram scale Phase scale Ex. 2 phases & 2 histograms – 2 Sh & 4 Sph – 6 scales Only 4 refinable – remove 2 by constraints Ex. DS11 = -DS21 & DS12 = -DS22

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**Rigid body problem – 88 atoms – [FeCl2{OP(C6H5)3}4][FeCl4]**

V=6879Å3 264 parameters – no constraints Just one x-ray pattern – not enough data! Use rigid bodies – reduce parameters V. Jorik, I. Ondrejkovicova, R.B. Von Dreele & H. Eherenberg, Cryst. Res. Technol., 38, (2003)

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**Rigid body description – 3 rigid bodies**

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

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**Rigid body description – continued**

PO – linear, origin at P C6 – ring, origin at P(!) C4 C2 x (ties them together) D2 C1 D1 D z C6 P O C1-C6 [ 0, 0, ] D1=1.6Å; P-C bond C1 [ 0, , ] C2 [ sin(60), 0, -1/2 ] C3 [-sin(60), 0, -1/2 ] C4 [ sin(60), 0, -3/2 ] C5 [-sin(60), 0, -3/2 ] C6 [ 0, , ] D2=1.38Å; C-C aromatic bond C5 C3 P [ 0, 0, 0 ] O [ 0, ] D=1.4Å

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**Rigid body description – continued**

Rigid body rotations – about P atom origin For PO group – R1(x) & R2(y) – 4 sets For C6 group – R1(x), R2(y),R3(z),R4(x),R5(z) 3 for each PO; R3(z)=+0, +120, & +240; R4(x)=70.55 Transform: X’=R1(x)R2(y)R3(z)R4(x)R5(z)X P O C z x y R1(x) R2(y) R3(z) R5(z) R4(x) Fe 47 structural variables

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

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**} } } Refinement – RB distances & angles OP(C6)3 1 2 3 4**

R1(x) 122.5(13) (4) 69.3(3) (9) R2(y) (3) (3) 12.8(3) 69.2(4) R3(z)a 27.5(12) 51.7(3) -10.4(3) -53.8(9) R3(z)b (12) 171.7(3) 109.6(3) 66.2(9) R3(z)c 267.5(12) 291.7(3) 229.6(3) 186.2(9) R4(x) 68.7(2) 68.7(2) 68.7(2) 68.7(2) R5(z)a 99.8(15) (14) (16) 64.6(14) R5(z)b 81.7(14) 88.3(17) 135.7(17) (16) R5(z)c 155.3(16) 63.8(16) 156.2(15) 224.0(16) P-O = 1.482(19)Å, P-C = 1.747(7)Å, C-C = 1.357(4)Å, Fe-Cl = 2.209(9)Å } PO orientation } C3PO torsion (+0,+120,+240) p − C-P-O angle } Phenyl twist x R5(z) R4(x) R1(x - PO) R3(z) R2(y- PO) Fe z

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**Packing diagram – see fit of C6 groups**

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**Stereochemical restraints – additional “data”**

Powder profile (Rietveld)* Bond angles* Bond distances* Torsion angle pseudopotentials Plane RMS displacements* van der Waals distances (if voi<vci) Hydrogen bonds Chiral volumes** “f/y” pseudopotential wi = 1/s2 weighting factor fx - weight multipliers (typically 0.1-3)

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**A lot easier to setup than RB!!**

For [FeCl2{OP(C6H5)3}4][FeCl4] - restraints Bond distances: Fe-Cl = 2.21(1)Å, P-O = 1.48(2)Å, P-C = 1.75(1)Å, C-C = 1.36(1)Å Number = = 92 Bond angles: O-P-C, C-P-C & Cl-Fe-Cl = 109.5(10) – assume tetrahedral C-C-C & P-C-C = 120(1) – assume hexagon Number = = 126 Planes: C6 to 0.01 – flat phenyl Number = 72 Total = = 290 restraints A lot easier to setup than RB!!

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

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**Stereochemical restraints – superimpose on RB results**

Nearly identical with RB refinement Different assumptions – different results

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**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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**Residue rigid body model for phenylalanine**

Qijk c2 txyz c1 y 3txyz+3Qijk+y+c1+c2 = 9 variables vs 33 unconstrained xyz coordinates

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**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

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**21542 observations; 1148 atoms (1001 HEWL)**

How effective? PDB 194L – HEWL from a space crystallization; single xtal data,1.40Å resolution 21542 observations; 1148 atoms (1001 HEWL) X-Plor 3.1 – RF = 25.8% ~4600 variables GSAS RB refinement – RF=20.9% ~2700 variables RMS difference - 0.10Å main chain & 0.21Å all protein atoms RB refinement reduces effect of “over refinement”

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**194L & rigid body model – essentially identical**

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**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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GSAS - A bit of history GSAS – conceived in (A.C. Larson & R.B. Von Dreele) 1st version released in Dec. 1985 Only TOF neutrons (& buggy) Only for VAX Designed for multiple data (histograms) & multiple phases from the start Did single crystal from start Later – add CW neutrons & CW x-rays (powder data) SGI unix version & then PC (MS-DOS) version also Linux version (briefly HP unix version) 2001 – EXPGUI developed by B.H. Toby Recent – spherical harmonics texture & proteins New Windows, MacOSX, Fedora & RedHat linux versions All identical code – g77 Fortran; 50 pgms. & 800 subroutines GrWin & X graphics via pgplot EXPGUI – all Tcl/Tk – user additions welcome Basic structure is essentially unchanged

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**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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**PC-GSAS – GUI only for access to GSAS programs**

pull down menus for GSAS programs (not linux)

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**GSAS & EXPGUI interfaces**

GSAS – EXPEDT (and everything else): EXPEDT data setup option (<?>,D,F,K,L,P,R,S,X) > EXPEDT data setup options: <?> - Type this help listing D - Distance/angle calculation set up F - Fourier calculation set up K n - Delete all but the last n history records L - Least squares refinement set up P - Powder data preparation R - Review data in the experiment file S - Single crystal data preparation X - Exit from EXPEDT On console screen Keyboard input – text & numbers 1 letter commands – menu help Layers of menus – tree structure Type ahead thru layers of menus commands) Numbers – real: ‘0.25’, or ‘1/3’, or ‘2.5e-5’ all allowed Drag & drop for e.g. file names

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**GSAS & EXPGUI interfaces**

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

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**Unique EXPGUI features (not in GSAS)**

CIF input – read CIF files (not mmCIF) widplt/absplt coordinate export – various formats instrument parameter file creation/edit widplt Sum Lorentz FWHM (sample) Gauss FWHM (instrument)

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**Powder pattern display - liveplot**

Zoom (new plot) updates at end of genles run – check if OK! cum. c2 on

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**Powder pattern display - powplot**

Io Ic Refl. pos. Io-Ic “publication style” plot – works OK for many journals; save as “emf” can be “dressed up”; also ascii output of x,y table

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**Powplot options – x & y axes – “improved” plot?**

Sqrt(I) Refl. pos. rescale y by 4x Q-scale (from Q=pl/sinq)

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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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