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Quasi cristalli

Quasi cristalli. Cristalli. 3) Riempimento completo 4) Sharp spots in X diffraction. 1) Invarianza traslazionale 2) Simmetria di rotazione Nel piano:. Reticolo triangolare (esagonale). Reticolo quadrato. Six (three) fold. Four (two) fold. Five fold case (cristallo pentagonale).

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Quasi cristalli

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  1. Quasi cristalli

  2. Cristalli 3) Riempimento completo 4) Sharp spots in X diffraction 1) Invarianza traslazionale 2) Simmetria di rotazione Nel piano: Reticolo triangolare (esagonale) Reticolo quadrato Six (three) fold Four (two) fold

  3. Five fold case (cristallo pentagonale) Simmetria di rotazione No traslazione No riempimento

  4. Esistono simmetrie (di rotazione) che non ammettono simmetrie di traslazione

  5. Pero’ il riempimento del piano puo’ essere fatto con simmetria “fivefold” 4 elementi

  6. Pero’ il riempimento del piano puo’ essere fatto con simmetria “fivefold” 4 elementi

  7. Penrose tiling (1974) 2 elementi Sir Roger Penrose E’ possibile riempire ol piano con simmetria five fold partendo da due figure geometriche e definendo una procedura di suddivisione e iterazione. Questa è legata alla sezione aurea e allasuccessione di Fibonacci Penrose R., “Role of aesthetics in pure and applied research”, Bull. Inst. Maths. Appl. 10 (1974) 266

  8. Penrose tiling fivefold symmetry Bragg diffraction Penrose R., “Role of aesthetics in pure and applied research”, Bull. Inst. Maths. Appl. 10 (1974) 266

  9. Definizione ufficiale In 1992, the International Union for Crystallography’s newly-formed Commission on Aperiodic Crystals decreed a crystal to be “any solid having an essentially discrete diffraction diagram.” In the special case that “three dimensional lattice periodicity can be considered to be absent” the crystal is aperiodic http://www.iucr.org/iucr-top/iucr/cac.html

  10. Proprietà quasi cristallo • Non periodico, ma determina “complete filling” • Ogni regione appare infinite volte • Ordine a lungo raggio • Si costruisce per ricorrenza • Diffrazione X produce Bragg pattern • PhC QC ha band gap anche con basso mismatch dielettrico

  11. Costruzione di un quasi cristallo in 2D Esempio di ricorrenza Due strutture di base Kite Dart

  12. Ricorrenze: Deflation a) b)

  13. Deflation

  14. Tiling: 1 kite 2 kite+1dart Costruiamo il SUN 1 2 10 kites+5 darts 5 kites

  15. Tiling: 1 kite 2 kite+1dart 1 dart 1 kite+1 dart SUN 3 2 10 kites+5 darts

  16. Tiling: 1 kite 2 kite+1dart 1 dart 1 kite+1 dart SUN 3 4

  17. SUN SELF SIMILARITY kites e darts si ripetono con frequenze il cui rapporto è la sezione aurea

  18. Sezione aurea

  19. Sezione aurea

  20. Sezione aurea Triangolo aureo Kites and Darts

  21. Sezione aurea in algebra Frazione continua

  22. Sezione aurea in geometria j-1 j 1 Rettangolo aureo Spirale aurea Rettangolo aureo

  23. Sezione aurea in natura Nautilus pompilius Spirale aurea

  24. Sezione aurea in architettura Piramide di Cheope

  25. Leonardo da Pisa (Fibonacci) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…..

  26. Fibonacci e i frattali http://www.youtube.com/watch?v=4B2DO4I62z8

  27. Frattale 1D Cantor set • Fibonacci spectrum is a self-similar Cantor set remove 1/3 of line, keep end points Total length removed in limit to infinite order? We have removed 1! Infinite number of points, yet length zero. Lebesque measure = 0

  28. Quasi cristalli in arte Darb-i Imam shrine (1453 C.E., Isfahan, Iran)

  29. Kites & Darts

  30. Ricorrenza: Icosaherdal Quasi Crystal in 3D 2 rhombic hexahedrons (romboedri) Rombo aureo a b Oblate RH Prolate RH

  31. Ricorrenza: Icosaherdal Quasi Crystal in 3D b a b a Bilinski's rhombic dodecahedron 2 oblate rhombic hexahedrons + 2 prolate rhombic hexahedrons

  32. Ricorrenza: Icosaherdal Quasi Crystal in 3D 1 Bilinski's rhombic dodecahedron+ 3 oblate rhombic hexahedrons + 3 prolate rhombic hexahedrons rhombic icosahedron

  33. Ricorrenza: Icosaherdal Quasi Crystal in 3D 5 rhombic icosahedron rhombic triacontahedron

  34. Close packing: Icosaherdal Quasi Crystal

  35. Al0.9 Mn0.1after annealing Prima evidenza sperimentale Icosahedral order is inconsistent with traslational symmetry

  36. Primo quasi cristallo in natura Museo di Storia Naturale, Sezione di Mineralogia, Università degli Studi di Firenze, Firenze I-50121, Italy. khatyrkite-bearing sample khatyrkite (CuAl2)

  37. HRTEM Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exterior contains a mixture of spinel, augite, and olivine. The dark material consists predominantly of khatyrkite (CuAl2) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 µm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals. Granulo di Al63Cu24Fe13 QUASI CRISTALLO

  38. HRTEM Fig. 1 (A) The original khatyrkite-bearing sample used in the study. The lighter-colored material on the exterior contains a mixture of spinel, augite, and olivine. The dark material consists predominantly of khatyrkite (CuAl2) and cupalite (CuAl) but also includes granules, like the one in (B), with composition Al63Cu24Fe13. The diffraction patterns in Fig. 4 were obtained from the thin region of this granule indicated by the red dashed circle, an area 0.1 µm across. (C) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodically ordered, fivefold symmetric, real space pattern characteristic of quasicrystals. Granulo di Al63Cu24Fe13 QUASI CRISTALLO

  39. Diffraction Pattern Fig. 4. The fivefold (A), threefold (B), and twofold (C) diffraction patterns obtained from a region (red dashed circle) of the granule in Fig. 1B match those predicted for a FCI quasicrystal, as do the angles that separate the symmetry axes.

  40. Quasi cristalli fotonici

  41. 3D Ph QC (Direct laser writing) Interference pattern of several light beams inside photo resist Photonic QuasiCrystal Group Wegener, Univ Karlsruhe

  42. 3D

  43. 2D Ph QC (lithography)

  44. Quasi cristalli fotonici 1D

  45. Leonardo da Pisa (Fibonacci) 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…..

  46. Fibonacci 1D QuasiCrystal Layer : 157 nm, 69% porosity, n = 1.6 Layer : 105 nm, 47% porosity, n = 2.2 A B A B A B A B A B A A A A B A A B B A B A B A A A A B A A A B B A B 1 2 3 4 5 6 7

  47. Fibonacci band gaps

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