Quasi cristalli

58
Quasi cristalli

description

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). - PowerPoint PPT Presentation

Transcript of Quasi cristalli

Page 1: Quasi cristalli

Quasi cristalli

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Cristalli1) Invarianza traslazionale 2) Simmetria di rotazione

Nel piano:

Four (two) fold Six (three) fold

3) Riempimento completo 4) Sharp spots in X diffraction

Reticolo quadrato Reticolo triangolare (esagonale)

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Five fold case (cristallo pentagonale)

Simmetria di rotazione

No traslazioneNo riempimento

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

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Pero’ il riempimento del piano puo’ essere fatto con simmetria “fivefold”

4 elementi

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Pero’ il riempimento del piano puo’ essere fatto con simmetria “fivefold”

4 elementi

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Penrose tiling (1974)

Penrose R., “Role of aesthetics in pure and applied research ”, Bull. Inst. Maths. Appl. 10 (1974) 266

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 alla successione di Fibonacci

2 elementi

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Penrose tiling

Penrose R., “Role of aesthetics in pure and applied research ”, Bull. Inst. Maths. Appl. 10 (1974) 266

fivefold symmetry Bragg diffraction

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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

Definizione ufficiale

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1. Non periodico, ma determina “complete filling”2. Ogni regione appare infinite volte3. Ordine a lungo raggio

4. Si costruisce per ricorrenza5. Diffrazione X produce Bragg pattern6. PhC QC ha band gap anche con basso mismatch dielettrico

Proprietà quasi cristallo

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Kite Dart

Costruzione di un quasi cristallo in 2DEsempio di ricorrenza

Due strutture di base

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Ricorrenze: Deflation

KiteDartDart21

21

21

KiteDartKite 121

21

a)

b)

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Deflation

KiteDartDart21

21

21

KiteDartKite 121

21

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1 2

5 kites 10 kites+5 darts

Tiling: 1 kite 2 kite+1dart

Costruiamo il SUN

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10 kites+5 darts

Tiling: 1 kite 2 kite+1dart 1 dart 1 kite+1 dart

SUN

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3 4

Tiling: 1 kite 2 kite+1dart 1 dart 1 kite+1 dart

SUN

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SUNSELF SIMILARITY

kites e darts si ripetono con frequenze il cui rapporto è

la sezione aurea 618.12

51

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251

BDAB

ABBC

Sezione aurea

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251

Triangolo aureo

Sezione aurea

Kites and Darts

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Sezione aurea in algebra

111

11

11

1

Frazione continua

11

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Rettangolo aureo

Sezione aurea in geometria

er Spirale aurea

1

11

Rettangolo aureo

251

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Sezione aurea in natura

Nautilus pompilius

er Spirale aurea

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Piramide di Cheope

Sezione aurea in architettura

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aureaSezioneFF

FFFFF

nnn

nnn

618.1/

11

1

11

10

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…..

Leonardo da Pisa (Fibonacci)

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http://www.youtube.com/watch?v=4B2DO4I62z8

Fibonacci e i frattali

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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

13*3/13/21

13/1)3/2(3/13/1*9/43/1*3/23/10

n

n

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

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Kites &Darts

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2 rhombic hexahedrons (romboedri)

Oblate RH Prolate RH

a

b

Ricorrenza: Icosaherdal Quasi Crystal in 3D

aureaSezione

618.12

51

Rombo aureo

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a

b

b

a

Bilinski's rhombic dodecahedron

2 oblate rhombic hexahedrons +2 prolate rhombic hexahedrons

Ricorrenza: Icosaherdal Quasi Crystal in 3D

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rhombic icosahedron 1 Bilinski's rhombic dodecahedron+ 3 oblate rhombic hexahedrons +3 prolate rhombic hexahedrons

Ricorrenza: Icosaherdal Quasi Crystal in 3D

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rhombic triacontahedron 5 rhombic icosahedron

Ricorrenza: Icosaherdal Quasi Crystal in 3D

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Close packing: Icosaherdal Quasi Crystal

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Al0.9 Mn0.1 after annealing

Icosahedral order is inconsistent with traslational symmetry

Prima evidenza sperimentale

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Museo di Storia Naturale, Sezione di Mineralogia, Università degli Studi di Firenze, Firenze I-50121, Italy.

khatyrkite-bearing sample khatyrkite (CuAl2)

Primo quasi cristallo in natura

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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.

HRTEM

Granulo di

Al63Cu24Fe13

QUASI CRISTALLO

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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.

HRTEM

Granulo di

Al63Cu24Fe13

QUASI CRISTALLO

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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.

Diffraction Pattern

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Quasi cristalli fotonici

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3D Ph QC (Direct laser writing)Interference pattern of several light beams inside photo resist

Group Wegener, Univ Karlsruhe

Photonic QuasiCrystal

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3D

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2D Ph QC (lithography)

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Quasi cristalli fotonici1D

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aureaSezioneFF

FFFFF

nnn

nnn

618.1/

11

1

11

10

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,…..

Leonardo da Pisa (Fibonacci)

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AB AB A AB AB A AB A AB AB A AB AB A AB A AB AB A AB

1 2 3 4 5 6 7

nnn FFF

AFBF

11

10

Fibonacci 1D QuasiCrystal

Layer : 157 nm, 69% porosity, n = 1.6

Layer : 105 nm, 47% porosity, n = 2.2

A

B

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1400 1600 1800 2000 2200 24000.00

0.05

0.10

0.15

0.20

0.25

0.30

Tran

smis

sion

Wavelength (nm)1400 1600 1800 2000 2200 2400

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Tran

smis

sion

Wavelength (nm)

Fibonacci band gaps

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Self-similarity in spectra

12th order 9th order

Effetto della finitezza della successione

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Self similar mode structureWavelet analysis on 15th order Fibonacci

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Fibonacci states map (12th order)

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Fibonacci band gap

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Fibonacci band gap

Propagazione sugli stati di band edge

Significant delay and stretching close to pseudo bandgap

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DFB Lasers

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