1. Modulation and its Crystallographic Methodology II Václav Petříček and Michal Dušek Institute of Physics Acadamy of Sciences of the Czech Republic Praha

2. Contents Structure analysis of modulated crystals Solution of the structure Heavy atom method Refinement of modulated crystals Jana2000 Main characteristis Jana2000 for powders

3. Solution of modulated structures The solution of modulated structures is usually divided into two steps. Solution of the average structure Only main reflections are used. Standard techniques can be applied, direct or heavy atom method by using standard programs – SIR, SHELX, … Strongly modulated atoms have usually large atomic displacement parameters or they must be refined as split atoms. Such “wrong” atoms are good candidates to be used as modulated ones when we are start with solution of the modulated structure.

4. Solution of the modulated structure Weak modulations < 0.1 Å The refinement can be started just from small randomly chosen displacements of already known atoms of the basic structure. The similar situation as when we are going from isotropic atomic displacement parameter to anisotropic ones Strong modulations > 0.1 Å A special techniques such as heavy atom or direct methods in superspace are necessary to get proper starting phases for satellite reflection.

5. Direct methods in superspace Q. Hao, Y.-W. Liu & Fan Hai-Fu, Acta Cryst, A43, 820 (1987) Fan Hai-Fu, S. van Smaalen, E.J.W. Lam & P.T. Buerskens, Acta Cryst, A49, 704 (1993) Program DIMS written by Fan Hai-Fu Heavy atom method in superspace W. Steurer, ActaCryst., A49, 704 (1987). V. Petříček, Aperiodic’94, World Scientific, 388, (1995). J. Peterková, M. Dušek V. Petříček & J. Loub Acta Cryst. B54, 809 (1998). Program JANA2000 written by V. Petříček and M.Dušek.

6. Application of heavy atom method to AsKF 4 (OH) 2 superspace group a=4.818, b=16.001, c=6.374 Å,  =99.36 0 Amplitudes of modulations ~ 0.3 Å The anion is modulated in the first approximation as a rigid body 1.Solution and refinement of the average structure Composed from the octahedral anion [AsF 4 (OH) 2 ] - and The cation K + both located at m 2.(3+1) dimensional Patterson map for As atom Symmetrical maximum at between the original position and the position related by the two fold axis or by the inversion center.

7. Modulation function restricted by site symmetry to : no significant modulation visible in x-section

8. Two possibilities : Model map at x 4 =0.25 based on F calc Real map at x 4 =0.25

9. The modulation of the heavy atom was included into the refinement and the subsequent Fourier synthesis allows to find estimation of modulation waves. 3.Fourier synthesis based on known modulation of heavy atom For F atom

10. Refinement of modulated structures kinematical theory of diffraction of modulated crystals  the integrated intensity of diffractions is proportional to the square of the generalized structure factor : Numerical methods Gaussian integrationA.Yamamoto REMOS FFT integration W.Paciorek MSR Analytical Bessel function expansionV.PetříčekJANA Generalized Bessel function W.Paciorek MSR

11. Modulation Functions The periodic modulation function can be expressed as a Fourier expansion: R3R3

12. The necessary number of used terms depends on the complexity of the modulation function. The modulation can generally affect all structural parameter – occupancies, positions and atomic displacement parameters (ADP). The set of harmonic functions used in the expansion fulfils the orthogonality condition, which prevents correlation in the refinement process. In many 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 or set of functions are used to reduce the number of parameters in the refinement.

13. Hexagonal perovskites Sr 1.274 CoO 3 and Sr 1.287 NiO 3 M. Evain, F. Boucher, O. Gourdon, V. Petříček, M. Dušek and P.Bezdíčka, Chem.Matter. 10, 3068, (1998).

14. The strong positional modulation of oxygen atoms can be described as switching between two different positions. This makes octahedral or trigonal coordination of the central Ni/Sr atom and therefore it can have quite different atomic displacement parameters. The regular and difference Fourier through the central atom showed that a modulation of anharmonic displacement parameters of the third order are to be used.

15. Sr at octahedral site Sr at trigonal site

16. Ni at octahedral site Ni at trigonal site

17. Crenel function Fourier transformation  Special modulation function V.Petříček, A.van der Lee & M. Evain, ActaCryst., A51, 529, (1995).

18. Example : TaGe 0.354 Te – F. Boucher, M. Evain & V. Petříček, Acta Cryst.,B52, 100, (1996). The Ge position is either fully occupied or empty: This is typical map for crenel like occupational wave.

19. Te atom is also strongly modulated but the modulation is positional

20. Difference Fourier shows that the continuous function does not describe real modulation completely.

21. Splitting of the modulation wave into two parts each circumscribed by crenel function allows to account for discontinuity

22. The superspace approach allows to analyze behaviour of atoms in in the modulated structure. But it is rather cumbersome to present the result in this form to non-specialists. Therefore we should make some 3d pictures showing how the modulation affects arrangement of atoms in the real 3d space. Average structure

23. Only occupational modulation Final result

24. Saw-tooth function Bi 2 Sr 2 CaCu 2 O 8 - V.Petříček, Y.Gao, P.Lee & P.Coppens, Phys.Rew.B, 42, 387-392, (1990) Oxygen atom at Bi layer

25. The displacement u is a linear function of x 4 coordinate:  for not occupied

26. Fourier transform: where The saw-tooth modulation changes the original periodicity and it can indicate some composite character of the compound.

27. Commensurate modulations Modulation vector : The crystal structure may be described as a regular one in the n-fold supercell. The superspace approach can reduce the number of used parameters and moreover it can help to make a systematic study either of different phases of one structure or of different structures of one structural type. The structure factor calculation is reduced to the summation over n different cells having internal coordinates :

29. ``

30. The superspace symmetry operator represents a 3-dimensional operator only if : The supercell 3-dimensional symmetry derived from the same superspace group can be different for different modulation vectors and different values of. Example : Hexagonal perovskites superspace group : Hexagonal perovskites together with analogous sulphides (Sr 1.143 TiS 3,...) belong to the same structure type defined in superspace.

31. t0t0 0 0.25 General

32. Rigid body option The JANA2000 system allows to refine selected groups of atoms as rigid bodies. Each group can be put and refined in several positions in the crystal. This option makes possible : to fix shape of the group to use one rigid body group at different positions to apply TLS approximation to temperature parameters to reduce number of parameters necessary to describe modulation of positional and temperature parameters – rectilinear approximation (OK for rotations < 5deg) to apply local non-crystallographic symmetry ( for C 60 )

33. The actual position of the i-th atom from the rigid body group is calculated from the relevant position of the model atom : where R and t are respectively rotation (proper or improper) and translation of the rigid body and is a point chosen as the rotation center. The rotational matrix can be expressed either with Eulerian or axial angles. The second choice has meaning in cases where the first choice is close to singularity e.g..

34. Modulation of PO 4 tetrahedrons in Na 9 {Fe 2 [PO 4 ](O,F) 2 } B.A.Maximov, R.A.Tamazyan, N.E.Klokova, V.Petříček, A.N.Popov & V.I.Simonov, Kristallografija, 37 (1992) 1152-1163.

35. Restrains of distances and angles Minimized function: stand either for distances or angles the second summation runs over selected distances or angles and for modulated structures also over t for modulated structures the target value need not be specified. Then the restrain will just keep the selected distance or angle constant over t rotational modulations of a rigid group having restrained distances are not limited by 5 deg

36. Conclusions: Superspace description allows to make generalisation of the standard methods of the structure analysis:  collection techniques  data reduction and determination of the crystal symmetry  direct and heavy atom methods for determination of modulation wave  refinement technique  calculation of distances, angles and bond valences

37. Comparing to the standard crystal structure analysis  It is necessary to collect more reflections  Satellite reflections are usually weaker than main reflections  The solution and refinement is more time consuming  Methods for solution are not yet “well established”  The International Tables for Crystallography vol.C contains just basic information on superspace groups

38. Main characteristics of Jana2000 it can be installed on PC under Windows (W98, NT, XP) and on most of UNIX machines it is written mainly in Fortran. C language is used just to make connection to basic graphic functions it uses own graphic objects which makes the program almost independent of the used system applicable for regular, polytype, modulated and composite structures superspace approach for modulated structures even for commensurate cases allows to make data reduction and merging data from different diffractometers (but not different radiation types) Fourier maps (up to 6d), p.d.f., j.p.d.f., deformation maps distance calculation and distance plots up to 6d

39. twinning – up to 18 twin domains, meroedry, pseudo-meroedry, twin index 1 or different from 1, overlap of close satellites Rietveld refinement multiphase up to 6d charge density studies – only 3d symmetry restrictions following from a site symmetry can be applied automatically for most refined parameters restrains of distances and angles rigid-body option to reduce number of regular and/or modulated parameters, TLS tensors, local non-crystallographic symmetry CIF output for regular, modulated structures refined either from single or powder data

40. JANA2000 for powders M. Dušek, V.Petříček, M.Wunschel, R.E.Dinnebier and S. van Smaalen, J. Appl. Cryst. (2001), 34, 398-404. JANA2000 allows to Rietveld refinement against powder diffraction data. All features (modulated structures, rigid body option, ADP,...) of Jana2000 are usable. It provides a state-of-the-art description of the peak profiles.