Dont be afraid of modulated structures Vclav Petek
Don’t be afraid of modulated structures Václav Petříček and Michal Dušek Institute of Physics Academy of Sciences of the Czech Republic Praha
Outline • Introduction – how to recognize a modulated structure, history remarks, superspace approach • Data collection process • Solution and refinement – programs, special modulation functions • Interpretation of results • Magnetic structures • Jana 2020
Introduction Diffraction pattern Example – Na 2 CO 3
Introduction Additional diffraction spots: Modulated structures: “Gittergeister” U. Dehlinger, Z. Kristallogr. (1927) 65 615– 31. “Satellites” G. D. Preston Proc. R. Soc. (1938) 167 526– 38. M. Korekawa (1967) Theorie der Satellitereflexe Habilitationschrift (München, Germany: Ludwigs-Maximilians-University). Korekawa & Jagodzinski (1967), Schweitz. Miner. Petrogr. Mitt. , 47, 269 -278. The theory of satellite reflections due to various types of modulation waves. Composite crystals: S. van Smallen, (1991), Phys. Rew. B, 43, 11330 -11341. E. Makovicky & B. G. Hide, (1992), Material Science Forum, 100&101, 1 -100.
Introduction
Introduction Superspace theory Ted Janssen Peter M. de Wolff Aloysio Janner Aminoff prize in 1998 Ewald prize in 2014
Introduction e q
Introduction R 3
Introduction Generalization to 3 d Diffraction pattern Fourier transform Charge (nuclear) density 3 d lattice translation symmetry in 3 dimensional space additional satellite spots translation symmetry in (3+d) dimensional space Description in 3+d dimensional superspace
Introduction Fourier expansion of Symmetry in the superspace: basic property unitary operator matrix representation Set of symmetry operations makes superspace group
Introduction General symmetry element: 1. From the basic symmetry, as determined from main reflections 2. Internal space cannot be mixed up with the external one 3. From the metric properties (unitary conditions) →
Introduction Superspace theory in data collection and data reduction procedure: • generation of symmetrically independent reflections in order to plan a single crystal experiment or to model a powder diffraction pattern • merging of symmetry-related reflections • systematic extinctions including satellite reflections Superspace theory in solution, refinement and interpretation of modulated structures: • calculation of structure factors • Fourier synthesis in (3+d) superspace • calculation of geometrical characteristics (distances, angles, BVS) for modulated structures After 1981 when the paper by P. M. de Wolff, T. Janssen and A. Janner [Acta Cryst. (1981). A 37, 625 -636] was published we could start developing programs for modulated structures
Introduction
Introduction Modulated structures were understood for long time as a curiosity not having some practical importance – e. g. Na 2 CO 3 – sodium carbonate. The number of studied modulated crystals grew with improving of experimental facilities – imaging plate, CCD. The real importance of modulations in crystals has been demonstrated by studies of organic conductors and superconductors (e. g. (BEDT-TTF)2 I 3 ) and high temperature Bi superconductors. The modulation can be present even in very simple compounds as oxides – Pb. O, U 4 O 9, Nb 2 Zrx-2 O 2 x+1.
Introduction Composite character of pure metals under high pressure. Nelmes, Allan, Mc Mahon & Belmonte, Phys. Rew. Lett. (1999), 83, 4081 -4084. Barium IV. Schwarz, Grzechnik, Syassen, Loa & Hanfland, Phys. Rew. Lett. (1999), 83, 4085 -4088. Rubidium IV. Modulated protein crystals - profilic: actin C. E. Schutt, U. Lindberg, J. Myslik and N. Strauss, Journal of Molecular Biology , (1989), 209, 735 -746. J. J. Lovelace, K. Narayan, J. K. Chik, H. D. Bellamy, E. H. Snell, U. Lindberg, C. E. Schutt and G. E. O. Borgstahl, J. Appl. Cryst. (2004). 37, 327 -330. Special importance: magnetic materials – helical, cycloidal, skyrmion ordering of magnetic moments
Data collection process
Data collection process
Data collection process
Data collection process
Data collection process History remark Data collection with four-circle diffractometer with a point detector: Average time for one reflection: 120 seconds for 10000 reflections 14 days For modulated structure with only first order satellites (± 1) 42 days Moreover, average time for weak satellites reflections is even longer than for main reflections. With area detectors, more intensive radiation source the measuring time is considerably shorter.
Solution and refinement Average structure: It can be solved by a standard way (direct, heavy atom, charge flipping method) from main reflections in 3 d.
Solution and refinement Modulated structure: • start refinement from small amplitudes of modulation parameters. Recommended strategy - refine first modulations of atoms exhibiting some anomalies - large ADPs, split positions. • start refinement from randomly chosen small amplitudes of modulation parameters. • direct methods Q. Hao, Y. -W. Liu & Fan Hai-Fu, Acta Cryst, A 43, 820 (1987) Fan Hai-Fu, S. van Smaalen, E. J. W. Lam & P. T. Buerskens, Acta Cryst, A 49, 704 (1993) • heavy atom method based on (3+d) Patterson maps W. Steurer, Acta. Cryst. , A 49, 704 (1987). V. Petříček, Aperiodic’ 94, World Scientific, 388, (1995). J. Peterková, M. Dušek V. Petříček & J. Loub, Acta Cryst. B 54, 809 (1998).
Solution and refinement Modulated structure: Charge flipping method – small revolution in solving modulated structures For regular structures: G. Oszlányi and A. Süto, (2004). Acta Cryst. A 60, 134. For modulated structures: L. Palatinus (2004). Acta Cryst. A 60, 604. L. Palatinus and G. Chapuis, (2007). J. Appl. Cryst. 40, 786. It is applicable even for strongly modulated crystals having discontinuously modulated occupancies or positions. Knowledge of the average structure is not explicitly used !!!
Solution and refinement Fourier maps play an important role in finding the best model for modulation functions. For this, the most convenient are x 1 -x 4, x 2 -x 4 and x 3 -x 4 sections (so called de Wolff’s sections): Modulation of atomic position: Produced in Jana From the de Wolff’s paper
Solution and refinement Fourier maps play an important role in finding the best model for modulation functions. For this, the most convenient are x 1 -x 4, x 2 -x 4 and x 3 -x 4 sections (so called de Wolff’s sections): Modulation of atomic occupancy: Produced in Jana From the de Wolff’s paper
Solution and refinement Fourier maps play an important role in finding the best model for modulation functions. For this, the most convenient are x 1 -x 4, x 2 -x 4 and x 3 -x 4 sections (so called de Wolff’s sections): de Wolff’s Patterson map: J. Peterková, M. Dušek V. Petříček & J. Loub, Acta Cryst. B 54, 809 (1998).
Solution and refinement In some cases the modulation functions are not smooth and the number of harmonic waves necessary for the description would be large. In such cases special functions are used to reduce the number of parameters in the refinement. Crenel function V. Petříček, A. van der Lee & M. Evain, (1995). Acta Cryst. , A 51, 529.
Solution and refinement In some cases the modulation functions are not smooth and the number of harmonic waves necessary for the description would be large. In such cases special functions are used to reduce the number of parameters in the refinement. Crenel function V. Petříček, A. van der Lee & M. Evain, (1995). Acta Cryst. , A 51, 529. Another application: Positional modulation - skip between two atomic positions → two crenels
Solution and refinement In some cases the modulation functions are not smooth and the number of harmonic waves necessary for the description would be large. In such cases special functions are used to reduce the number of parameters in the refinement. Saw-tooth function Bi 2 Sr 2 Ca. Cu 2 O 8 - V. Petříček, Y. Gao, P. Lee & P. Coppens, (1990). Phys. Rev. B, 42, 387 -392.
Solution and refinement Refinement programs Akiji Yamamoto, Acta Cryst. A 38, 87 -92, (1982) – REMOS program The first program which could refine modulated structures: • • • Up to three modulation vectors Full use of the superspace symmetry Numerical method for the calculation of structure factors Twining option Later powder version - PREMOS
Solution and refinement Refinement programs MSR – W. Paciorek & D. Kucharczyk, Acta Cryst. (1985). A 41, 462 -466 W. Paciorek & I. Uszynski, J. Appl. Cryst. (1987). 20, 57 -59 • Numerical integration, Bessel functions, generalized Bessel functions • Single crystal refinement XND - J. F. Bérar & G. Baldinozzi, APD 2 nd Conf. , Gaithersburg MD USA (1992) • Numerical integration • Rietveld refinement program GSAS, Fullprof
Solution and refinement Refinement programs Jana- the first version developed during my stay at Philip Coppens lab in 1984, in close collaboration with Pierre Becker - Acta Cryst. (1985). A 41, 478 -483 The latest version • Works up to (3+3)d including composite structures, modulation of occupancies, positions, ADPs up to 6 th order of anharmonicity • Site symmetry restrictions are derived analytically from symmetry operations • (3+3) Fourier maps • It can handle merohedric, reticular and pseudo-merohedric twinning • It can combine data from different sources – x-ray and neutron, powder and single crystal data • Magnetic structures
Interpretation of results Interatomic distances, angles, torsion angles, planes, and BVS should be calculated as a function of the internal coordinate t: Example: Composite structure [M'2 Cu 2 O 3]7[Cu. O 2]10, M' = Bi 0. 04 Sr 0. 96 A. F. Jensen, V. Petříček, F. K. Larsen and E. M. Mc. Carron III, (1997). Acta Cryst. , B 53, 125 -134
Interpretation of results Interatomic distances, angles, torsion angles, planes, and BVS should be calculated as a function of the internal coordinate t: Example: Composite structure [M'2 Cu 2 O 3]7[Cu. O 2]10, M' = Bi 0. 04 Sr 0. 96 A. F. Jensen, V. Petříček, F. K. Larsen and E. M. Mc. Carron III, (1997). Acta Cryst. , B 53, 125 -134
Interpretation of results Drawing programs such as Mercury, Diamond or VESTA cannot handle superspace symmetry but Jana 2006 program can generate all atomic parameters within a selected region and create a CIF file. There already programs which can directly draw modulated structures: Jmol for modulated structures made by Bob Hanson Mole. Cool. Qt written by Christian B. Hübschle Jana 2020
Interpretation of results The modulation curves and corresponding density maps play a crucial role during the solution and refinement process. But a fine presentation should be made in the 3 d real space. Several cells and more different sections are to be presented to see various configurations in the modulated crystal: Ta. Ge 0. 354 Te – F. Boucher, M. Evain & V. Petříček, (1996). Acta Cryst. , B 52, 100. Average structure
Interpretation of results The modulation curves and corresponding density maps play a crucial role during the solution and refinement process. But a fine presentation should be made in the 3 d real space. Several cells and more different sections are to be presented to see various configurations in the modulated crystal: Ta. Ge 0. 354 Te – F. Boucher, M. Evain & V. Petříček, (1996). Acta Cryst. , B 52, 100. Occupation modulation
Interpretation of results The modulation curves and corresponding density maps play a crucial role during the solution and refinement process. But a fine presentation should be made in the 3 d real space. Several cells and more different sections are to be presented to see various configurations in the modulated crystal: Ta. Ge 0. 354 Te – F. Boucher, M. Evain & V. Petříček, (1996). Acta Cryst. , B 52, 100. Final result
Interpretation of results L. Bindi, P. Bonazzi, M. Dušek, V. Petříček and G. Chapuis, Acta Cryst. (2001). B 57, 739 -746.
Magnetic structures Juan Manuel Pérez-Mato Harold T. Stokes Branton Campbell Juan Rodrigues-Carvajal
Magnetic structures Aleksey Vasilyevich Shubnikov Magnetic space groups • Magnetic moment – axial vector • Any symmetry operation can be combined with “time inversion” - , which inverts the magnetic moment
Magnetic structures Magnetic superspace groups
Magnetic structures Magnetic superspace groups
Magnetic structures Representational analysis
Magnetic structures List of kernels and epikernels (isotropic subgroups):
Magnetic structures ISODISTORT – H. T. Stokes, B. J. Campbell and D. M. Hatch
Magnetic structures Bilbao Crystallographic Server – M. Aroyo, J. M. Pérez-Mato
Magnetic structures Visualization of magnetic structure with VESTA program (Koichi Momma and Fujio Izumi)
Jana 2020
Jana 2020
Jana 2020
Jana 2020
Conlusions • Data collection and data reduction with more than 3 diffraction indices – included in integration programs • Superspace group can be determined by a space group test analogical to regular structures • Structure solution – Superflip program • Structure refinement – different modulation functions can be applied to occupancies, atomic coordinates and ADPs • Interpretation of results – calculation of “modulated” distances, angles, BVS, . . . • Visualization - Jmol, Mole. Cool. Qt, Jana 2020
Monographies Sander van Smaalen: Incommensurate Crystallography. Oxford University Press, Oxford (2007). Ted Janssen, Gervais Chapuis & Marc de Boissieu. Aperiodic Crystals. Oxford University Press, Oxford (2007).
Acknowledgements My first steps in field of modulated structures and the first version of Jana program were made during my stay in Philip Coppens’s lab in 1984. Philip recognized, long time before discovering of high Tc superconductors, how important role plays modulation in crystals for property of materials. My stay in his lab changed my scientific life.
Acknowledgements Sander van Smaalen– University of Bayreuth – composite structures, powder refinement Juan Manuel Perez-Mato – Bilbao University – magnetic structures Nadja Bolotina – Institute of Crystallography Moscow Andrei Mironov – State University in Moscow Sven Lidin – Stockholm University Gervais Chapuis, Alla Arakcheeva - University of Lausanne Michel Evain, Philippe Deniard – University of Nantes, CNRS Nantes Guido Baldinozzi – Ecole Central Paris Theo Woike, Dominik Schaniel – University of Köln Jürg Schefer, Peter Fischer – PSI Villigen Pascal Roussel, Olivier Mentré – CNRS Lille Dominique Grebille, Olivier Perez, Philippe Boullay – CRISMAT Caen Olivier Gourdon, ICDD, USA to all Jana 98/Jana 2000/Jana 2006/Jana 2020 users for their suggestions
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