quasi cristalli
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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Quasi cristalli
Dan ShechtmanThe Nobel Prize in Chemistry 2011
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Cristalli
1) 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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Simmetrie traslazionali
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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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Diffrazione Bragg
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Diffrazione Bragg
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Materiali amorfi
Materiali cristallini
Diffrazione Bragg
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Dan ShechtmanThe Nobel Prize in Chemistry 2011
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Original data
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Five fold case (cristallo pentagonale)
Simmetria di rotazione
No traslazioneNo riempimento
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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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Where are the atoms?
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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
KiteDartDart2
1
2
1
2
1
KiteDartKite 12
1
2
1a)
b)
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Deflation
KiteDartDart2
1
2
1
2
1
KiteDartKite 12
1
2
1
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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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32
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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SUN
SELF SIMILARITY
kites e darts si ripetono con frequenze il cui rapporto è
la sezione aurea 618.12
51
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2
51
BD
AB
AB
BC
Sezione aurea
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2
51
Triangolo aureo
Sezione aurea
Kites and Darts
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Sezione aurea in algebra
11
1
11
11
1
Frazione continua
1
1
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Rettangolo aureo
Sezione aurea in geometria
er Spirale aurea
1
1
1
Rettangolo aureo
2
51
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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
FFF
FF
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*3/23/1
0
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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Potential energy surface 8.6x8.6nm2 for Ag on i-Al-Pd-Mn QC. Darker shades indicate strongerinteractions.
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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
FFF
FF
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
Tra
nsm
issi
on
Wavelength (nm)1400 1600 1800 2000 2200 2400
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Tra
nsm
issi
on
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 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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