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Quasi cristalli

Quasi cristalli. Dan Shechtman The Nobel Prize in Chemistry 2011. 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.

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Quasi cristalli

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  1. Quasi cristalli Dan Shechtman The Nobel Prize in Chemistry 2011

  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. Simmetrie traslazionali

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

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

  6. Diffrazione Bragg

  7. Diffrazione Bragg

  8. Diffrazione Bragg Materiali amorfi Materiali cristallini

  9. Dan Shechtman The Nobel Prize in Chemistry 2011

  10. Original data

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

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

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

  14. 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

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

  16. Where are the atoms?

  17. 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

  18. 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

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

  20. Ricorrenze: Deflation a) b)

  21. Deflation

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

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

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

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

  26. Sezione aurea

  27. Sezione aurea

  28. Sezione aurea Triangolo aureo Kites and Darts

  29. Sezione aurea in algebra Frazione continua

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

  31. Sezione aurea in natura Nautilus pompilius Spirale aurea

  32. Sezione aurea in architettura Piramide di Cheope

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

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

  35. 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

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

  37. Kites & Darts

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

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

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

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

  42. Close packing: Icosaherdal Quasi Crystal

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

  44. Potential energy surface 8.6x8.6nm2 for Ag on i-Al-Pd-Mn QC. Darker shades indicate strongerinteractions.

  45. 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)

  46. 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

  47. 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

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