1. Structure determination of incommensurate phases An introduction to structure solution and refinement Lukas Palatinus, EPFL Lausanne, Switzerland

2. Outline This tutorial will cover: introduction to incommensurate structures (very briefly) determination of the symmetry structure solution structure refinement validation of the structure

3. Incommensurate structures Aperiodic structure is a structure that lacks periodicity, but exhibits a long-range order Three main classes: Modulated compositesquasicrystals structures

4. Incommensurate structures Modulated structure Composite Incommensurately modulated structure has a basic 3D periodicity that is perturbed by an incommensurate modulation.

5. Incommensurate structures reciprocal space Reciprocal space is discrete despite of the aperiodicity

6. Incommensurate structures reciprocal space

7. 1-101-10 -120-120 120120

8. 1-100 -1200 1200 1-102 1-10-3 -1301 Incommensurate structures reciprocal space

9. Most current diffractometer softwares allow for indexing of an aperiodic diffraction pattern. However, the q-vector can be only refined, not found automatically. The result is indexing of the pattern by 4 integers: -6 -2 4 2 1970.51001 80.49380 -4 -2 2 0 116733.00000 327.45499 -4 -2 1 -1 280.85901 56.31390 -4 -2 1 -2 156.37300 51.69950 -4 -2 4 -2 135.81400 42.38190 -4 -2 1 0 50292.10156 214.59900 -4 -2 1 -3 21.82130 23.57890 -6 -2 -1 0 1678.30005 69.71670 -4 -2 1 1 372.96399 53.42990 Incommensurate structures reciprocal space

10. Incommensurate structures

11. Superspace

12. Construction of superspace in reciprocal space

13. Superspace Construction of superspace in reciprocal space

14. Superspace Construction of superspace in reciprocal space a* s1 a* s4 q b1b1

15. Superspace Embedding of the structure into superspace R3R3

16. Superspace Embedding of the structure into superspace R3R3

17. Superspace Structure model of a modulated structure consists of: Structure model of basic structure Modulation functions for the parameters of the basic structure: –Modulation of position –Modulation of occupancy –Modulation of displacement parameters Modulation functions are most often modeled by a Fourier series:

18. Superspace

19. Superspace symmetry

20. The symmetry is described by a (3+d)-dimensional space group. A 4D superspace group must be 3+1 reducible = the internal and external dimensions cannot mix together. General form of a symmetry operation: Example of superspace group operations: x 1, -x 2, 1/2+x 3, -x 4 -x 1, -x 2, x 3, 1/2+x 4 Symmetry

21. How can the symmetry be determined? The first three rows are the components of the basic space group. The sign of R I depends on the action of the symmetry operation on the q-vector: 2-fold: -x 1, x 2, -x 3, -x 4 2-fold: -x 1, x 2, -x 3 mirror: x 1, -x 2, x 3, x 4 mirror: x 1, -x 2, x 3

22. Symmetry The translational part is determined from the extinction conditions in complete analogy to the 3D case: in general: hR = h, h.  = integer c-glide: x 1, -x 2, 1/2+x 3 : h0l, l=2n “superspace c-glide” with shift along x 4 : x 1, -x 2, 1/2+x 3, 1/2+x 4 : h0lm, l+m=2n

23. C2/m(  0  )0s Symmetry superspace group symbol

24. C2/m(  0  )0s Herman-Mauguin symbol of the basic space group Symbol of the q-vector Definition of the intrinsic shifts in the fourth dimension s=1/2; t=1/3 q=1/4; h=1/6 Generators:-x 1, x 2, -x 3, (1/2)-x 4 x 1, -x 2, x 3, 1/2+x 4 Centering:1/2 1/2 0 0 Symmetry superspace group symbol

25. Symmetry The search for the superspace group is facilitated by the space group test of Jana2000

26. Symmetry Rational part of the q-vector

27. Symmetry Rational part of the q-vector

28. Symmetry  Rational part of the q-vector Centering vector: 0 1/2 0 1/2

29. Superspace symmetry

30. Structure solution

31. Structure solution means finding a starting model that is good enough to be refined by least-squares. Two cases: 1) small to medium modulations (weak to moderately strong satellites) 2) strong modulations = satellites comparable to or stronger than main reflections

32. Structure solution Case 1 - small modulations: a) Solve the average structure from main reflections b) Refine the modulations from small starting values

33. Structure solution Case 2 - large modulations: no reasonable average structure exists The structure can be solved by two methods: superstructure approximation: the components of a q-vector are approximated by commensurate values and the structure is solved as superstructure: q=(0.345, 0, 0.478) ==> q  (1/3, 0, 1/2) => 6-fold supercell directly in superspace by charge flipping (lecture tomorrow, 13:30). Both the average structure and modulation functions can be obtained at the same time.

34. Structure solution In Jana2000 you can: Directly call Sir97/Sir2004. The data are prepared, sent to Sir2004, and the model is imported back. Manually export data into SHELX format, solve the average structure by SHELX and import the structure back to Jana2000. Prepare input files for the charge flipping calculation with Superflip and EDMA. Superflip returns the density map and a list of structure factors in Jana2000 format, EDMA can provides a structure model of the average structure.

35. Structure solution

36. Structure refinement

37. Two step procedure: Refine the average structure against the main reflections using standard crystallographic methods. Refine the modulation parameters of the atoms, namely: –Occupational modulation (1 function) –Positional modulation (1 function for the x, y and z components) –Modulation of ADP’s (1 function per parameter = up to 6 functions)

38. Structure refinement Initial modulation refinement cookbook Recommended: Start with the heaviest atoms or with atoms with largest modulation If you suspect strong occupational modulation of some atoms, start with occupational modulation, otherwise refine positional modulation first.

39. Structure refinement Initial modulation refinement cookbook Recommended II: Watch the R-values of the satellites AND the Fourier maps of the modulation functions

40. Structure refinement Initial modulation refinement cookbook Discouraged: Don’t use more modulation waves than you have satellite orders Reason: The contribution of the higher harmonics to low- order satellites is negligible. If it were there, high-order satellites would be observed.

41. Structure refinement Initial modulation refinement cookbook Discouraged II: Don’t switch off automatic refinement keys and automatic symmetry restrictions of Jana2000 unless you are sure it is necessary. For temporary fixing of some parameters use Refine commands/Fixed commands

42. Structure refinement Initial modulation refinement cookbook Discouraged III: Don’t refine the ADPs in the initial stages of the refinement unless you see the evidence in the difference Fourier map

43. Structure refinement Special functions

44. Crenel function (block wave) Sawtooth function

45. Structure refinement Special functions + harmonic modulation Harmonic functions are mutually orthogonal on the interval. Shorter interval leads to severe correlation between the parameters.

46. Structure refinement

47. Evaluation of the structure

48. Evaluation of the structure Fourier maps Fourier maps are indispensable: Check, if the modulation functions match the shape of the electron density:

49. Evaluation of the structure t-plots R3R3

51. Conclusions Structure solution and refinement of an incommensurately modulated structure can be a relatively straightforward undertaking if: The symmetry is determined correctly The modulation is not too strong The modulation is refined step by step from the most significant to the least significant waves If becomes less straightforward if: The modulation is very strong Special functions are needed for description of the modulation Acknowledgement: Special thanks to Michal Dusek for providing me his set of lectures on modulated structures

52. Incommensurate structures How many q-vectors? Each rationally independent q-vector counts as one q- vector = one additional dimension in superspace b* a* q2q2 q1q1 (3+2)D -q 1 -q 2

53. Structure refinement Special functions Crenel function (step function, block wave)  Saw-tooth function

54. Structure refinement Setting of special functions Find the parameters in the Fourier map.

55. Structure refinement Setting of special functions Check the function in the Fourier map after setting.

56. Structure refinement Special functions Special functions allow to describe discontinuous modulation functions with few parameters 3 harmonic waves = 6 parameters; crenel function = 2 parameters

57. Structure solution Case 1 - small modulations: a) Solve the average structure from main reflections b) Refine the modulations from small starting values The basic structure often gives a hint on the nature of the modulation.