1. Contents Introduction Superspace approach Metric consideration Symmetry in the superspace Basic modulation types Occupational (substitutional) modulation Positional modulation Composite structure Examples

2. Introduction Experiment Additional diffraction spots – satellites are present in the diffraction pattern of modulated crystals. Satellites are regularly spaced but they cannot be indexed with three reciprocal vectors. One or more additional, modulation vectors have to be added to index all diffraction spots. Consequence Diffraction pattern has no more 3d lattice character  the basic property of crystal 3d translation symmetry is violated but in a specific regular way.

3. Additional satellite diffractions are often as sharp as main spots and can be integrated and used to describe modulation of aperiodic crystals. The term “aperiodic crystal” covers modulated, composite crystals and quasicrystals. The effect of positional modulation was described first by Dehling, (1927) Z.Kristallogr., 65, 615-631. Occupational modulation was described later by Korekawa & Jagodzinski (1967), Schweitz.Miner.Petrogr.Mitt., 47, 269-278. Composite crystals approach was introduced by Makovicky & Hide, Material Science Forum, 100&101, 1-100.

4. Structural analysis of modulated crystals is based on theoretical works of P.M. de Wolff, A.Janner and T.Janssen. 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. Moreover 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 – PbO, U 4 O 9, Nb 2 Zr x-2 O 2x+1.

5. In 1999 two papers reported incommensurate composite character of pure metals under high pressure : Nelmes, Allan, McMahon & Belmonte, Phys.Rew.Lett. (1999), 83, 4081-4084. Barium IV.

6. Schwarz, Grzechnik, Syassen, Loa & Hanfland, Phys.Rew.Lett. (1999), 83, 4085-4088. Rubidium IV. Theory of aperiodic crystals and computer program – Jana2000 – has been later applied to solve and refine several analogical structures.

7. Superspace approach - Metric considerations Additional diffraction spots : modulation vector q can be expressed as a linear combination of ..... all rational  commensurate structure..... at least one irrational  incommensurate structure It is rather difficult to prove irrationality only from measured values. The higher the denominators the smaller difference between commensurate and incommensurate approach. But there are clearly distinguished cases 1/2,1/3 where commensurability plays an important role.

8. Modulated and composite crystals can be described in a (3+d) dimensional superspace. The theory has developed by P.M.DeWolff, A.Janner and T.Janssen (Aminoff prize 1998). This theory allows to generalize concept of symmetry and also to modify all method used for structure determination and refinement of aperiodic crystals. The basic idea is that a real diffraction pattern can be realized by a projection from the (3+d) dimensional superspace: Superspace

9. e q

10. 1.The important assumption is that all satellites are clearly separated. This is true for the commensurate case or for the incommensurate case when the intensities diminish for large satellite index. 2. The additional vector e is perpendicular to the real space and plays only an ancillary role. All diffraction spots form a lattice in the four-dimensional superspace  there is a periodic generalized (electron) density in the four-dimensional superspace. Reciprocal base : Direct base :

11. Then the generalized density fulfill the periodic condition : and therefore it can be expressed as a 4-dimensional Fourier series : where: From the definition of the direct and reciprocal base it follows:

12. But the real diffraction pattern is a projection of the 4d pattern into R 3 which means that the term has to be constant. Conclusion : A real 3d density can be found as a section through the generalized density.

13. Example : positionally modulated structure R3R3

14. Symmetry in the superspace Basic property of 3+d dimensional crystal - generalized translation symmetry :    basic property unitary operator matrix representation Trivial symmetry operator - translation symmetry :

15. The rotational part of a general symmetry element 1. The right upper part of the matrix is a column of three zeros. It is consequence of the fact that the additional ancillary vector e cannot be transformed into the real space. 2.From the condition that symmetry operator has to conserve scalar product it follows 3. These conditions show that any superspace group is a four-dimensional space group but on the other hand not every four ‑ dimensional space group is a superspace group. The superspace groups are 3+1 reducible. This allows to derive possible rotations and translation is same way as for 3d case.

16. Examples 1. Inversion centre There is no modulation for the second case and therefore the inversion centre has to have

17. 2. Two-fold axis along z direction monoclinic axial case monoclinic planar case

18. 3. Mirror with normal parallel to z direction monoclinic axial case monoclinic planar case

19. Translation part Symbol Reflection condition for

20. Super-space group symbols There are three different notation The rational part of the modulation vector represents an additional centring. It is much more convenient to use the centred cell instead of the explicit use of the rational part of the modulation vector. 

21. Basic modulation types Occupational (substitutional) modulation One harmonic wave

22. Form factor of changes from to Only main reflections and first order satellites will appear:

23. Occupational modulation – only 1 st harmonic Fourier map

24. Occupational modulation – only 1 st harmonic Diffraction pattern

25. Occupational modulation – 1 st and 2 nd harmonic Fourier map

26. Occupational modulation – 1 st and 2 nd harmonic Diffraction pattern

27. Crenel line modulation

28. Occupational modulation – crenel function Fourier map

29. Occupational modulation – crenel function Diffraction pattern

30. Positional modulation - longitudinal Modulation vector : Modulation wave :

31. Positional modulation longitudinal 1 st harmonic 0.1Å Fourier map

32. Positional modulation longitudinal 1 st harmonic 0.1Å Diffraction pattern

33. Positional modulation longitudinal 1 st harmonic 0.5Å Fourier map

34. Positional modulation longitudinal 1 st harmonic 0.5Å Diffraction pattern

35. Positional modulation - transversal Modulation vector : Modulation wave :

36. Positional modulation transversal – 1 st harmonic 0.5Å Fourier map

37. Positional modulation transversal – 1 st harmonic 0.5Å Diffraction pattern l=0

38. Positional modulation transversal – 1 st harmonic 0.5Å Diffraction pattern l=1

39. Composite structure

40. Composite structure – without modulation Fourier map

41. Composite structure – without modulation Diffraction pattern

42. Composite structure – with modulation Fourier map

43. Composite structure – with modulation Diffraction pattern

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

45. 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.

46. 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).

47. 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.

48. Sr at octahedral site Sr at trigonal site

49. Ni at octahedral site Ni at trigonal site

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

51. 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.

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

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

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

55. 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

56. Only occupational modulation Final result

57. 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

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

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

60. 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

61. 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

62. 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.