ORDENAMIENTO APERIÓDICO

Presentación del tema: "ORDENAMIENTO APERIÓDICO"— Transcripción de la presentación:

1 ORDENAMIENTO APERIÓDICO
Cuasicristales Fernando Hueso González Física del Estado Sólido – 4º de Grado de Física Campus de Burjassot – Facultat de Física - Valencia – UVEG 21 de diciembre de 2010 ferhue#alumni.uv.es Print notas Revisar animaciones Ensayar, controlar tiempos 1

2 INTRODUCCIÓN HISTÓRICA FUNDAMENTOS TEÓRICOS
ÍNDICE INTRODUCCIÓN HISTÓRICA FUNDAMENTOS TEÓRICOS Cristales Figuras de Penrose CUASICRISTALES Descripción Propiedades Aplicaciones BIBLIOGRAFÍA

3 ARQUITECTURA Teselamiento aperiódico  cubrir el espacio sin huecos
INTRODUCCIÓN HISTÓRICA ARQUITECTURA Teselamiento aperiódico  cubrir el espacio sin huecos Cierto orden, no repetitivo Arquitectura medieval islámica  teselas de Girih (año 1200) Templo Darb-i Imam (Isfahan, Iran – 1453) No es posible sólo con pentágonos The five shapes of the tiles are: a regular decagon with ten interior angles of 144°; an elongated (irregular convex) hexagon with interior angles of 72°, 144°, 144°, 72°, 144°, 144°; a bow tie (non-convex hexagon) with interior angles of 72°, 72°, 216°, 72°, 72°, 216°; a rhombus with interior angles of 72°, 108°, 72°, 108°; and a regular pentagon with five interior angles of 108°.

4 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
In 2007, Peter J. Lu of Harvard University and Professor Paul J. Steinhardt of Princeton University published a paper in the journal Science suggesting that girih tilings possessed properties consistent with self-similar fractal quasicrystalline tilings such as Penrose tilings (presentation 1974, predecessor works starting in about 1964) predating them by five centuries.[2][3] "It's only 11 defects out of 3700 Penrose tiles, and each can be corrected by a simple rotation," he says.

5 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
Girih-tile subdivision found in the decagonal girih pattern on a spandrel from the Darb-i Imam shrine, Isfahan, Iran (1453 C.E.). ( ) Photograph of the right half of the spandrel.

6 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
I suspect that once most first-time observers of ancient Islamic architecture get over its astonishing beauty and rigorous interplay of geometry and pattern-work (such as the in-laid tiles above the arch of this 15th-century school in Bukhara, Uzbekistan), they immediately wonder how it could possibly have been built by a bunch of uneducated craftsmen.

7 INTRODUCCIÓN HISTÓRICA
TESELAS DE GIRIH

8 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
iran-isfahan-islamic-or-timurid-tile-mosaic-ii-from-jezand_rani-on-flickr

9 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
Girih strapwork pattern on an interior archway in the Sultan's Lodge in the Green Mosque in Bursa, Turkey (AD 1424) (Image: W B Denny)

10 TESELAS DE GIRIH INTRODUCCIÓN HISTÓRICA
( ) Photograph by A. Sevruguin (~1870s) of the octagonal Gunbad-i Kabud tomb tower in Maragha, Iran (1197 C.E.), with the girih-tile reconstruction of one panel overlaid. ( ) Close-up of the area marked by the dotted yellow rectangle in (B).

11 FIBONACCI Matemático medieval (1170-1250) Liber Abaci
INTRODUCCIÓN HISTÓRICA FIBONACCI Matemático medieval ( ) Liber Abaci Secuencia de Fibonacci 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, Proporción áurea = Número irracional, decimales sin repeticiones Autosimilitud Fractales

12 INTRODUCCIÓN HISTÓRICA
FIBONACCI Secuencia sin patrones repetitivos, sucesión infinita Ordenamiento predecible  Aplicable Figuras de Penrose (1973)

13 KEPLER Kepler 1619 - Harmonice Mundi
INTRODUCCIÓN HISTÓRICA KEPLER Kepler Harmonice Mundi Estudio sobre teselamiento  “Monstruos” Pentágonos, decágonos, estrellas Inspiración para Penrose (1973) In Kepler's 1619 book Harmonice Mundi on tilings, he discussed a tiling built with pentagons, pentagrams, decagons, and "fused decagon pairs." He also called them "monsters." This tiling inspired Penrose tiles in 1973.

14 CRISTALES Redes de Bravais Simetría de traslación
FUNDAMENTOS TEÓRICOS CRISTALES Redes de Bravais Celda unidad primitiva Vectores de la red Sin huecos Simetría de traslación Teorema de restricción cristalográfica Simetrías de rotación de orden 2, 3, 4, 6 (únicas permitidas) Estructura repetitiva periódica No es posible teselar sólo con pentágonos

15 FIGURAS DE PENROSE Roger Penrose 1973
FUNDAMENTOS TEÓRICOS FIGURAS DE PENROSE Roger Penrose 1973 Es posible teselar el espacio de manera no repetitiva, sin huecos Rombos, polígonos (distintos grupos) Ordenamiento aperiódico, sin patrones repetidos Método predecible (jerarquía de inflación – crecimiento)

16 FIBONACCI Y PENROSE Proporción áurea entre rombos
FUNDAMENTOS TEÓRICOS FIBONACCI Y PENROSE Proporción áurea entre rombos Secuencia de Fibonacci (teselación aperiódica de 1D)  Proyección plano 2D periódico sobre una recta con pendiente irracional Construct the perpendicular to our line. Project the unit square around the origin onto the perpendicular. The tilings of the line that result from the above procedure are known as Fibonacci Tilings. They have many beautiful properties. For instance the ratio of the lengths of the tiles is the inverse of the golden mean. In addition, they are aperiodic: meaning they have no translational symmetry. For if they did then the line they are on would pass through some integer lattice point. This cannot happen since the slope is irrational.

17 FUNDAMENTOS TEÓRICOS FIBONACCI Y PENROSE Red de Bravais en 2D  Proyección sobre recta (1D) con pendiente irracional  Teselamiento aperiódico de esta recta

18 FUNDAMENTOS TEÓRICOS PENROSE Y AMMANN Red Bravais 2D  Proyección sobre recta (1D) con pendiente irracional  Teselamiento aperiódico 1D Generalizar a más dimensiones Espacio de 5 dimensiones  Proyección sobre 2D Se obtienen las figuras de Penrose, etc. Hiperplanos, Teoría de grupos, ... Robert Ammann  Figuras 3D (Penrose + Romboedros áureos) ¿Existen físicamente? ¿Artificial o natural?

19 DESCRIPCIÓN Dan Shechtman 1984  aleación Al-Mn (METALES)
CUASICRISTALES DESCRIPCIÓN Dan Shechtman 1984  aleación Al-Mn (METALES) Patrón de difracción rayos X (2D)  revela simetrías prohibidas Existen artificialmente  SÍ – ¿Y de manera natural? AlAg icosahedral ZnMgHo quasicrystals – 10 fold symmetry

20 DESCRIPCIÓN CUASICRISTALES
New Ames Laboratory Logo, showing two decagonal and one icosahedral single-grain quasicrystals

21 DESCRIPCIÓN CUASICRISTALES
Single-grain sample of a quasicrystaline compound AlPdRe

22 DESCRIPCIÓN CUASICRISTALES
Figure 4: Quasicrystal of an alloy of aluminium, copper and iron, displaying an external form consistent with their icosahedral symmetry [12].

23 DESCRIPCIÓN Existen artificialmente  SÍ – ¿Y de manera natural?
CUASICRISTALES DESCRIPCIÓN Existen artificialmente  SÍ – ¿Y de manera natural? Koryak – Rusia, 2009

24 DESCRIPCIÓN CUASICRISTALES
( ) 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 ( ), 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 mm across. ( ) The inverted Fourier transform of the HRTEM image taken from a subregion about 15 nm across displays a homogeneous, quasiperiodi- cally ordered, fivefold symmetric, real space pattern characteristic of quasicrystals.

25 DESCRIPCIÓN CUASICRISTALES
The fivefold ( ), threefold ( ), and twofold ( ) 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.

26 PROPIEDADES Ordenamiento aperiódico en espacio posiciones
CUASICRISTALES PROPIEDADES Ordenamiento aperiódico en espacio posiciones Transformada de Fourier  Picos dispuestos simétricamente Patrón de difracción de rayos X  simetrías prohibidas (pero cubre todo) Artificialmente  subenfriamiento rápido Natural  ¿? Proceso geológico Ausencia de simetría en los QC  efectos relevantes en la conductividad eléctrica y térmica Malos conductores calor/electricidad pese a estar constituidos por metales Sin propiedades magnéticas acusadas (diamagnetismo) Fricción superficial muy pequeña No hay alineamiento entre superficies (aperiódicas) Muy duros y elásticos (altas T), resistentes corrosión Semiconductores, bandas complejas, conductividad óptica (FIR) No pueden acceder a posiciones de equilibrio, transición de fase brusca

27 PROPIEDADES Orden Cristal Desorden Orden aperiódico
CUASICRISTALES PROPIEDADES Orden Cristal Periodicidad Desorden Cristal amorfo Aleatorio Orden aperiódico Cuasicristal Ordenamiento predecible No repetitivo - Término intermedio Ecuaciones desarrolladas no válidas  contradicción experimento Herramientas matemáticas POR DESARROLLAR Ecuación de schrödinger conduce a resultados contradictorios con el experimento Funciones de Bloch (periódicas - espacio)

28 CONDUCTIVIDAD TÉRMICA
CUASICRISTALES CONDUCTIVIDAD TÉRMICA Figure 9.1: The temperature dependence of the experimentally determined thermal conductivity of AlFeCu icosahedral quasicrystals compared with that of zircon [14]. Such a reduced range of real extended phonons should result in very poor thermal conductivity, reaching a maximum value around 70K when all possible phonons are fully excited (The Dulong-Petite regime) [15]. Most of these predictions are observed experimentally. The thermal conductivity values K(T) are indeed very small - much smaller than those expected for purely metallic compounds. For instance, at room temperature, K(T) for quasicrystalline AlFeCu and AlPdMn is more than two orders of magnitude smaller than for aluminium, more than one order of magnitude smaller than for steel, and about one-half that for zircon which is currently considered to be one of the best thermal insulators [20] (indeed, it is widely used because of its insulator properties). The phonon saturation effect is also observed as plateau in the K(T) curves covering a temperature range from about 25K to 100K [21, 22]. The unexpected feature is that at higher temperature, K(T) resumes an increasing trend (Fig. 9.1). This may be understood in terms of nonlinearity effects [15] which allow vibration modes to interact. The temperature dependence is quite different from classical metal samples for example Lithium (Fig. 9.2).

29 CONDUCTIVIDAD TÉRMICA
CUASICRISTALES CONDUCTIVIDAD TÉRMICA Cond. Térm. QC menor que en metales  1% Alum, 10% Acero, 50% Zircon (aislante térmico) Pocos fonones (saturación a baja T) Plateau Mayoría portadores libres: electrones Efectos no lineales a altas T Figure 9.2: The temperature dependence of the experimentally determined thermal conductivity of Li metal sample [22] For similar reasons, most of the electronic transport is expected to come from phonon-assisted collective electron hopping between equivalent sites in the structure, and QC must behave in almost the same way as insulators at low temperature, with recovery of some conductivity as the temperature increases. Such behavior, based on a recurring hierarchical localization of the bonding electrons, is in fact in good agreement with what has been observed in resistivity data for the highest quality AlPdRe QC alloys [22]. At around 0.5K, the electrical resistivity of this QC can exceed 30Ωcm, a value which is 103 to 104 times larger than that of pure aluminium (Fig. 10.1). The conductivity

30 CONDUCTIVIDAD ELÉCTRICA
CUASICRISTALES CONDUCTIVIDAD ELÉCTRICA increases almost linearly upon heating6, reaching a value at room temperature that is a factor of some 200-times larger than its value at the temperature of liquid helium. Chemical composition and quasicrystaline perfection are critical: defects and slight departures from ideal stoichiometry give a fairly dramatic recovery of the conductivity. Figure 10.1: The electrical conductivity σ(T) at low temperatures of AlPdRe quasicrystals showing the 5 ~ 0 behavior for a prefect icosahedral phase (lower curves) as well as the 5 ~ 0/ law for a slightly less good sample (two uppermost curves). A log-log plot is shown in the insert for the uppermost and lower curves [23].

31 CONDUCTIVIDAD ELÉCTRICA
CUASICRISTALES CONDUCTIVIDAD ELÉCTRICA Menor conductividad eléctrica que en metales Proporcional a T (a pocos K) Resonancia en infrarrojo (1mm) Figure 10.2: Temperature dependence of the electrical resistivity ρ (which equals 89 ) showing the ρ ~ 0 behavior for various pure metal samples (Au, Na, Cu, Al, Ni), where ΘD is the Debye temperature of the metal. This is known as the Bloch-Grϋneisen law. This behaviour is quite unexpected since the behaviour of all the components is quite different as can be seen in Fig The same law applies to elements such as Pd, Re, which are present in the previously mentioned quasicrystal. electronic and computing devices.

32 CONDUCTIVIDAD ÓPTICA Sandwich  QC entre óxidos SiO2 ó Al2O3
CUASICRISTALES CONDUCTIVIDAD ÓPTICA Sandwich  QC entre óxidos SiO2 ó Al2O3 -Gap en reflectividad en el visible APLICACIONES Ventanas Placas solares The frequency dependence σ(ω) of the electrical (or so-called optical) conductivity also deviates strongly from that for metallic behavior [27, 28]. In agreement with direct-current measurements, σ(ω) remains small for most of the frequency range, especially towards low-energy values. However, a rather strong resonance shows up (Fig. 12) at around or 290THz (corresponding to infrared radiation with a wavelength of about 1 mm). Assuming this resonance comes from electron-plasma oscillation a mean-free amplitude of about 22 Å for majority carrier can be deduced. Such a value is very close to the distance between two elementary clusters in the structure of AlPdMn QC (Fig. 6), something that would be expected if a hopping mechanism is relevant for conductivity in QC. Practical consequences should be very interesting, one of which is illustrated in Fig. 13 [29] showing that sandwiches of QC between oxide layers such as SiO2 or Al2O3 have reflectivity (R) gap centered on the visible wavelength range10 A. straightforward application of such a property involves energy coating for solar cells, insulator screens or window glass. Figure 13: The reflectivity of quasicrystal layer sandwiched between two layers of three different types of oxide insulators [29].

33 APLICACIONES Dureza, elasticidad, fricción, resistencia a corrosión
CUASICRISTALES APLICACIONES Dureza, elasticidad, fricción, resistencia a corrosión Protectores antiadherentes  sartenes comercializadas Resistente temperatura, arañazos, ... Termometría (Conductividad térmica proporcional a T) EN DESARROLLO: Piezas de maquinaria de muy baja fricción Rodamientos Partes deslizantes Pistones de motores Implantes quirúrgicos, prótesis, articulaciones Complicado evitar contaminación por gases atmosféricos  se adhieren provocando pérdida de lubricidad Pantallas térmicas en cohetes y aviones (alta plasticidad a alta T) Revestimiento de placas solares, aislantes ventanas No pueden acceder a posiciones de equilibrio, transición de fase brusca

34 CONCLUSIONES Teselamiento aperiódico es posible y existe
RESULTADOS CONCLUSIONES Teselamiento aperiódico es posible y existe Comprensión requiere matemáticas muy complejas Campo de estudio muy reciente, totalmente distinto No aplican ecuaciones usuales (premisas inválidas) Dificultad de explicar con teorías las observaciones experimentales Muchas incógnitas Camino por recorrer Propiedades interesantes Múltiples aplicaciones, todavía por plantear y desarrollar

35 Dirección de contacto:
BIBLIOGRAFÍA Peter J. Lu and Paul J. Steinhardt (2007). "Decagonal and Quasi-crystalline Tilings in Medieval Islamic Architecture". Science 315 (5815): 1106–1110 M. Senechal. Quasicrystals and geometry. Cambridge University Press, Cambridge, 1995. Penrose, Roger (1974), "Role of aesthetics in pure and applied research", Bulletin of the Institute of Mathematics and its Applications 10: 266ff Bindi, L.; Steinhardt, P. J.; Yao, N.; Lu, P. J. (2009). "Natural Quasicrystals". Science 324 (5932): 1306 The Properties and Applications of Quasicrystals - Seminar II, Simon Jazbec Dirección de contacto: ferhue#alumni.uv.es Página Web: 35