1. Aperiodic crystals Incommensurate modulated crystals From G. Pan, Thesis Orsay 1992 Cuprate superconductor Bi 2,2 Sr 1,8 CuO 2 Incommensurate modulated phase ‘‘Satellite’’ reflections around main spots b* c* k 4 indices to index

2. Incommensurate? The NaNO 2 case P Ferroelectric Paraelectric Phase diagram Continuous variation of the satellite position: Incommensurate FerroPara Inc. From Dominique Durand, Thesis, LPS, Orsay BCCD Devil straicase Uhrig (1989)

3. Incommensurate modulation Local property of the crystal has a periodicity irrational with crystal periods Example: displacive modulation NaNO 2 (electric polarisation), alliages (concentration wave), magnetism ADN, Coxeter helixADN

4. RS of modulated structures Calculation of the RS Direct space given by:

5. h=0 a* h=1h=2 k 2k 3k m= 0 1 2 3 -3 -2 Concept of space of higher dimension RS of modulated structures

6. Macroscopic consequence: calaverite G 0012 a* c* +q -q +2q +3q +4q G 2012 G 2014 - - - (201) - (001) Calaverite : Au 1-x Ag x Te, gold ore Facets does not satisfy Haüy’s Law

7. Composites crystals Entanglement of two crystals with mutually incommnsurate lattice constants. a a’ Simple model: RS sum of the two RL a* b*=b’* b=b’ a’* In fact both lattices are intermodulated...

8. Structure of Ba 5.5 GPa12.6 GPa Phase I Body centred Phase II Hexagonal Phase IV Tetragonal inc. 45 GPa Phase V Hexagonal Phase IV : Self-hosting structure Chains of Ba in a matrix of tetragonal Ba 0.341 nm R.J. Nelmes, D.R Allan, M.I McMahon, et S.A. Belmonte, Phys. Rev. Lett., 83 (1999) 4081 C h =0.4696 nm (Centre terre 360=Gpa)

9. Quasi-cristaux Electron diffraction by an Al-Mn alloy D. Shechtman (Chemistry Nobel prize) et al. Phys. Rev. Lett. 53, 1951 (1984)Phys. Rev. Lett. 53, 1951 Quasicrystal discovered by accident (by serendipity) by Schechtman (1982) Who was studying rapidly cooled alloys. Al alloys bad conductivity (I, T) Fragile at 300 K, ductile at HT Diamagnetic Tribologic properties, non stick surfaces AlMn rapidly cooled (imperfect) 1986 : AlLiCu, stable quasicrystals 1988 : Perfect quasicrystals, AlCuFe, AlPdMn, AlPdRe 2000 : Cd 5,7 Yb (Tsai, Nature) Dodecahedral crystal of AlCuFe Photo : Annick Quivy © CNRS - CECM, Vitry-Thiais

10. Are they twinned? X-ray photograph Decagonal microcrystal Al 0.63 Cu 0.175 Co 0.17 Si 0.02 From P. Launois et al., 1991 Linus Pauling (and others) suggested they were microtwins Ex: 5-fold assembly of microcrystals Microscopy and electron diffraction proved the QC existence Quasicrystalline microscopic order: new phase of condensed matter Electron beam (10 nm) X-rays in the 90s (1-100  m) now… also 50 nm From M. Audier (1990) 72° No!

11. Penrose tiling Two types of tiles Two types of tiles Matching rules Matching rules Some quasicrystals can be modelled by penrose tilings Exemple: Al-Fe-Cu 36° 72°

12. Indexation of quasicrystals diffraction diagram FT of the Penrose tiling Indexed by 4 vectors arithmetically independent a1*a1* a4*a4* a3*a3* 4 indices Z-module of rank 4 How to index a non periodical lattice? a2*a2*

13. Indexation of QC X Y a 5 * a 4 * a 1 * a 3 * a 2 * a 6 * Z

14. Definition of a crystal IUCr 1991 ‘‘By ‘crystal’ we mean any solid having an essentially discrete diffraction diagram, and by ‘aperiodic crystal’ we mean any crystal in which three-dimensional lattice periodicity can be considered to be absent.’’ « Par cristal on désigne un solide dont le diagramme de diffraction est essentiellement discret et par cristal apériodique on désigne un cristal dans lequel la périodicité tridimensionnelle peut être considérée absente »

15. Z-module Let’s consider an ‘’object’’ the FT of which is a Z-module of finite rank:

16. Superspace

17. Example in 2D Cut of 2D lattice by a band with irrationnal tangent + Projection of points on the line = Penrose tilings: 2D cuts of 4D superspace Fibonacci sequence

18. Exemples 2D lattice + cut 1D crystal Composite cristal Incommensurate modulation Quasicrystal Quasicrystal: cut and projection Basis are called ‘‘atomic surfaces’’

19. Quasicristal Discontinuous atomic surfaces Physical space Slope  Fibonacci sequence Perp. space Where are the atoms Refinement of electron density in the superespace Decoration of Penrose tilings Approximants Rational slope: approximant

20. Phason: displacement in the perpendicular space Crystal translation… Relative sliding of the two crystals in composites Sliding of incommensurate modulation Atomic rearrangements in quasicrystals Perp. space

21. Edagawa PRL 2000 Phasons in quasicrystals: atomic motions

22. Aperiodic order Indexation of diffraction diagram of a body in dimension D by a finite number N of indices (Case of all known ‘‘crystals’’)

23. Yes, almost periodicity T=76 T=151

24. Esentially discrete… The tiling ‘‘chair’’ is limit-periodic Z-module of infinite rank In nature nothing has been found to be limit-periodic