Programs for computation of powder diffraction patterns • W. Kraus, G. Nolze. Federal Institute for Materials Research and Testing (BAM), Unter den Eichen 87, D-12205 Berlin • http://ccp14.minerals.csiro.au/ccp/webmirrors/powdcell/a_v/v_1/powder/e_cell.html • http://ccp14.minerals.csiro.au/ccp/webmirrors/powdcell/a_v/v_1/powder/details/powcell .htm Structure Determination from Powder Diffraction • Course, experimental point of view • http://www.cristal.org/course/index.html • http://pcb4122.univlemans.fr/iniref/tutorial/indexa.html • http://pcb4122.univ-lemans.fr/iniref.html 1 Number of publications about new structures determined • • • • http://pcb4122.univ-lemans.fr/iniref.htm 1948-1987 28 structures 2000 520 structures 2006 1328 structures Powcel • unit cell parameters, • atomic names, z, coordinates for the asymmetric unit • space group number • information for each space group in file pcwspgr.dat. 2 BCC structured CsCl CsCl input file for PowCell cell 4.123 4.123 4.123 90 90 90 natom 2 Cs 55 0.0 0.0 0.0 Cl 17 0.5 0.5 0.5 rgnr 221 3 Powcell: diffraction pattern of Gold space group 225 F4/m -3 2/m a=b=c=4.078 Diamond 8 atoms in the unit cell: (0,0,0) and (1/4,1/4,1/4) + FCC translations 4 Diamond Diamond 5 Diamond Crystal structure determination • Traditional approach or direct method: using intensities and positions of reflections to solve the structure • Direct space methods: construction of model for the structure (atomic positions) and fitting the model intensity to the diffraction pattern (whole pattern fitting) 6 Structure determination • Determination of lattice parameters • Assignment of crystal symmetry and space group • Structure solution – Initial model directly from experimental diffraction data – Straightforward from single crystal data – Solution from powder diffraction data under quite active research • Structure refinement Crystal structure determination • Problems – Reflections overlap – Background determination – Preferred orientation • Powerful numerical methods needed • Prior information on the structure is useful 7 Neutrons and x-rays • X-rays: heavy elements – Small sample needed when using synchrotron radiation • Neutrons: light elements – Large samples Pattern decomposition • • • • • • Peak search Fitting a model to diffraction lines The diffraction pattern may be split in parts Good profile shape function is important. Taking into account K 1 and K 2 split Needs determination of background 8 Tasks: Indexing • Example: Exhautive search of lattice parameter space by varying lattice parameters in discrete steps and isolating volumes that contain possible solutions (Dichotomy method). • E.g. DICVOL91 (Boultif and Louër 1991) • 84% succesful in triclinic structures How many parameters? • Lattice constants, max. 6 – Cubic: only a – Triclinic: 3 lengths and 3 angles • Experimental: – angular offset – shape of the reflections 3-4 parameters • Number of parameters can be about 10 • 20-30 reflections are needed 9 Equation in matrix form • The equation Xhh h2 + Xkk k2 + Xll l2 + Xhk hk + Xhl hl + Xkl kl = 1/dhkl2 • Eq. in matrix form: HX = D • H nxm matrix • D column (nx1) containinf 1/d2 data • X column (nx1) SVD based algorithm SVD singular value decomposition • The equation Xhh h2 + Xkk k2 + Xll l2 + Xhk hk + Xhl hl + Xkl kl = 1/dhkl2 is solved iteratively using SVD method (A. Coelho. J. Appl. Cryst. 2003, 36, 86-95. • Includes Monte Carlo search 10 Singulaariarvohajotelma • Singulaariarvohajotelma on ominaisarvohajotelman yleistys, jolla on käyttöä numeerisissakin tehtävissä. • Hajotelman avulla voi tarkastella esimerkiksi vektoreiden lineaarisen riippuvuutta. • Hajotelmaa voi käyttää myös ratkaistaessa lineaarisia pienimmän neliösumman tehtäviä. Singular value decomposition • In matrix formulation the singular value decomposition is as follows: for each mxn matrix A (even rectangular matrices, and especially for column and row matrices) there exists unitary mxm matrix U and nxn matrix V, and a mxn matrix S so that A = U S VT. • If A is a square matrix, S is diagonal. The diagonal elements of S are called singular values. 11 Space group determination • • • • Forbidden reflections Problem: overlapping reflections Trial and error Structure: comparing model and experimental intensity • Iterative improvement Structure analysis: R-factor • A measure of agreement of the structural model with the experiment • R = ( ||Fhkl|obs - |Fhkl|calc| )/ |Fhkl|obs • For some purposes also • R = whkl (|Fhkl|obs - |Fhkl|calc )2 where whkl are weighting factors. 12 Patterson function from intensities of the reflections hkl Generally I(q) = P(r) exp(i q·r) d3r P(r) ~ I(q) exp(-i q·r) d3q If the electron density is periodic, P is called the Patterson function. It is then P(r) ~1/V hkl | F(hkl) |2 exp(-iqhkl·r) ~ 1/V hkl | F(hkl) |2 cos (hx+ky+lz) Phase problem: special cases • The structure contains heavy atoms, whose positions are known. The phase of these heavy atoms determine phases of other structure factors. • General constraints as >0. • Known sructures 13 Minimization methods • Plenty of fitting parameters • Powerful fitting method needed – Monte Carlo – Simulated annealing – Genetic algorithms • Reliable intensity information essential Crystal structure solution • Patterson method: only structure factor amplitudes are used. • P(r) = 1/V |F(h)|2 exp(-2 h·r). – Each peak in P corresponds to an interatomic distance within the unit cell. – Heights of the peak are proportional to the scattering power of the atoms 14 Crystal structure solution • Direct methods. Attempts to solve phases directly from the diffraction data. Constraint: electron density is positive in the unit cell. • Example. P-methoxy benzoic acid. Monoclinic structure from 169 nonoverapping and 203 overlapping reflections. Rietvelt refinement programs • FullProf: Rodríguez-Carvajal. Laboratoire Léon Brillouin (CEA-CNRS) CEA/Saclay http://www.ccp14.ac.uk/ http://www.ccp14.ac.uk/ccp/ccp14/ftpmirror/fullprof/pub/divers/fullprof.2k/Windo ws/ 15 Rietvelt refinement programs • GSAS: Larson and von Dreele. Los Alamos National Laboratory • Scheme in GSAS Indexation – – – – Structure factor amplitude extraction Method for structure factors selection Conventional Patterson or direct methods Model building, molecule location, unconventional methods... – Completion of the structure – Final Rietveld refinement http://www.ccp14.ac.uk/about.htm Example. Kaolinite Al2Si2O5(OH)4 b c 16 Kaolinite (triclinic C1) • Platelike orientation (110) 8215 -1-11 kaolinite1993 (pref.Or) 25 30 003 -201 -1-31 -131 200 022 0-22 -112 1-11 -1-12 002 021 0-21 20 -111 020 15 110 001 0 10 111 1-10 4107 35 Example. Kaolinite • Neutron powder diffraction 1.5 K • Wavelength 1.9102 Å • Most thermal contraction occur in 001 direction due to decrease in interlayer distance. • H atom positions? •DL Bish. Clays and Clay Minerals. 41, 6, 738-744, 1993 17 Refinement • Missing reflections h+k = odd • Young and Hewat kaolinite structure model 1988 • Program GSAS • Anisotropic displacement parameters for H • H1 random positional disorder • OH bond lengths between 0.976 and 0.982 Å •DL Bish. Clays and Clay Minerals. 41, 6, 738-744, 1993 Example. New study on kaolinite • Single crystal refinement • Synchrotron radiation, microfocus beam line at ESRF • Wavelength 0.6883 Å • Room temperature • Space group C1 • Positions of intralayer H atoms could not be refined. • The diffraction patterns showed diffuse scattering in streaks parallel to [001] direction caused by stacking faults. Neder et al. Clays and clay minerals 1999, 47, 4, 487-494. 18 Example. K3Ba3C60 • The crystal structure of bodycentered cubic K3Ba3C60. The Ba2+ and K+ ions are shifted from the tetrahedral sites, (0, 1/2, 1/4). • Fullerenes in the corners and in the middle of the cell. Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000. Example. K3Ba3C60 • • • High-resolution synchrotron X-ray diffraction measurements were performed on the K3Ba3C60 sample sealed in a 0.5 mm diameter glass capillary. Data were collected in continuous scanning mode using nine Ge(111) analyzer crystals on the BM16 beamline at ESRF at 10 and 295 K (wavelength 0.83502 Å). Analysis of the diffraction data at both temperatures was performed with the GSAS suite of Rietveld analysis programs. • • • Temperature-dependent synchrotron X-ray diffraction measurements were also performed on the SwissNorwegian beamline (BM1A) at the ESRF with a 300 mm diameter Mar Research circular image plate. The sample was cooled from 320 to 105 K at a rate of 25 K/h. Diffraction patterns were measured every 5 min with a sample to detector distance of 200 mm and with an exposure time of 40 s. Data analysis was performed with the Fullprof suite of Rietveld analysis programs. Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000. 19 Example. K3Ba3C60 • The neutron diffraction experiment was undertaken with the high-resolution diffractometer D2b (1.5944 Å) at the Institute Laue Langevin, Grenoble, France. • The sample (0.58 g) was loaded in a cylindrical vanadium can (diameter = 5 mm) sealed with indium wire and then placed in a standard ILL liquid helium cryostat. • The data were collected in the scattering angle range 0164.5 in steps of 0.05 deg. • A full diffraction profile was measured with counting time of 10 h at 10 K. • The data analysis was performed with the GSAS software. Margadonna et al. Chem. Mater., 12 (9), 2736 -2740, 2000. Example. Structures of the Polymer Electrolyte Complexes • Polymer electrolytes consist of salts, e.g., LiCF3SO3, dissolved in solid highmolecular-weight polymers, such as PEO (CH2CH2O)n. • Neutron diffraction experiment • Refinement was carried out using the GSAS program package. Rietveld refinement • The refinements involved 1734 data points, 50 atoms in the asymmetric unit, 162 variables, and 135 soft constraints. • For both PEO6:LiPF6 and PEO6:LiSbF6 the powder data could be indexed on monoclinic cells and with systematic absences that unambiguously determined the space group as P21/a. Gadjourova et al. Structures of the Polymer Electrolyte Complexes PEO6:LiXF6 (X = P, Sb), Determined from Neutron Powder Diffraction Data. Chem. Mater., 13 (4), 1282 -1285, 2001. 20 Rietvelt refinement programs • A. A. Coelho Whole-profile structure solution from powder diffraction data using simulated annealing. J. Appl. Cryst. (2000). 33, 899-908 • Starting point: space group and lattice parameters • Solution of structures by simulated annealing • The simulated annealing control parameters have been systematically investigated. – Most significant: electrostatic-potential penalty functions. Rietvelt refinement programs • EXPO (Altomare et al. J. Appl. Cryst. 1999, 32, 338-340, and J. Appl. Cryst. 35, 182-184. • Electron density modification procedure • Diagonal least squares refinement • Ratio of the number of observations to the number of parameters should be greater than 3. 21 References • JI Langford and D Louëb. Powder diffraction. Rep.Prog. Phys. 59, 1996, 131-234 • K Harris and M Tremayne. Crystal structure determination from powder diffraction data. Chem. Mater. 1996, 8, 2554-2570. • BM Kariuki et al. The application of genetic algorithm for solving crystal structures from powder diffraction data. Chem. Phys. Lett. 280, 1997, 189-195. • H Putz, JC Schön, M Jansen. Combined method for ab initio structure solution from powder diffraction data. J. Appl. Cryst. 32, 1999, 864-870 – cost function E + R, simulated annealing – E: Lennard-Jones –type potential Model independent solution • MEM • A model-independent maximum-entropy method is presented which will produce a structural model from small-angle X-ray diffraction data of disordered systems using no other prior information. In this respect, it differs from conventional maximum-entropy methods which assume the form of scattering entities a priori. The method is demonstrated using a number of different simulated diffraction patterns, and applied to real data obtained from perfluorinated ionomer membranes, in particular Nafion(TM), and a liquid crystalline copolymer of 1,4-oxybenzoate and 2,6-oxynaphthoate (B-N). • Elliot and Hanna. J. Appl. Cryst. 1999, 32(6), 1069-1083. 22

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