Ciência e arte: da Cristalografia convencional aos quase-cristais

A difração é utilizada no estudo de arranjos de átomos e moléculas. Diffraction is a physical phenomenon, which we will be studying during this course, within the realm of a branch of science called crystallography. In it, we used diffraction for the study of the arrangement of atoms and molecules in crystalline materials, which can be organic or inorganic, metals or proteins and many more. A main difference with other techniques used for the study of materials is that using diffraction we not only say that we have a carbonyl group or a phenyl group or a methyl group, or that this group is attached to this other group. Instead, we can find out how exactly are these groups, molecules, atoms, arranged in space. Which are the distances and angles between these atoms, how far are certain groups from planarity, if they are in cis or trans conformation, and much more. And why do we want to know all this? Well, to start with, most of our planet is made up of crystals. Most rocks of the Earth crust’s consist of silicate crystals, ice and snow are crystalline as well, minerals are crystals, so the study of crystalline materials help us to get a better understanding of the world around us. But crystals are also part of our everyday life, ice crystals cool our drinks, silicon crystals enable computers to work, jewels are very nice looking crystals, we add sugar crystals to our coffee, we add salt to our meals, and many many more examples are all around us.

A difração é utilizada no estudo de arranjos de átomos e moléculas. DNA One very nice example of the use of diffraction for the study of crystalline materials is the discovery of the structure of DNA, which took place 50 years ago. Finding out the structure of proteins help us to understand how living things work, how these large biological molecules work. This in turn gives us a way to influence what they do and how they do it. If a protein is causing trouble and we know the shape of it, a small molecule can be design to bind to this protein and prevent it from causing more harm. In the same way, a better understanding of how drug molecules, medicines, bind to biological molecules can help us to design new drugs and medicines, more effective, needing a lower dose, having less side-effects, etc.

A difração é utilizada no estudo de arranjos de átomos e moléculas YBa2Cu3O7-x (0 < x < 0.5) Superconductor YBa2Cu3O6 Antiferromagnético Another nice example of the use of diffraction for the study of structural problems was the realization of how important oxygen atoms and copper layers are for the existence of high temperature superconductivity. As you may know, superconductors can conduct electricity without any energy losses caused by dissipation of heat. Finding a material with this property for use as a wire to conduct electricity from power stations to our homes could save billions of pounds a year. It has not yet been achieve, but the mechanisms are better understood now, much thanks to the use of diffraction techniques. The famous 1-2-3 superconductor presents superconductivity in directions parallel to the copper planes. If this material is made more deficient in oxygen, as for the 6 case, the material is antiferromagnetic. Superconductivity kicks on after a content of 6.5.

Densidade eletrônica

Resolução

Radiação Eletromagnética

Como então, observar uma molécula?

Interferência construtiva Experimento de Young As fendas agem como fontes secundarias Onda Plana incidente Draw cosine waves on the board to exemplify when they are in phase and when not. Interferência construtiva quando as ondas estão em fase

O 1 2

Para o átomo isolado j Fator de forma Para uma molécula

Porque um cristal?

Convolução Multiplicação

Diffraction Geometry

O que faz um cristalógrafo? h k l Io (Io) -1 -9 -3 557 6 -1 -3 -9 543 8

O problema, tem solução?

Problema das fases

Complexo de Mercúrio

Os cristais

Cristais imperfeitos, condição fundamental para a validade da 1ª aproximação de Born

Cristais ruins  dados ruins Resultados ruins Os cristais devem ser apropriados: Para Raios-x os cristais devem de ter entre ~ 0.1-0.3 mm de lado. Cristais ruins  dados ruins Resultados ruins   Boa qualidade interna: Nem sempre a forma externa é importante

Um pouco da parte experimental

Luz Polarizada

Luz Incidente polarizador analisador Não passa a luz Cristal Isotrópico Não passa a luz Cristal Anisotrópico passa a luz

Como montar um cristal glass fibre crystal Cabeça goniométrica

Simetria cristalina

A simetria se manifesta nos diagramas de difração

von Laue (1912) junto com Friedrich e Knipping descobriram a difração de raios-x por um cristal de sulfeto de cobre pentahidratado. Isso confirmou: A natureza ondulatória dos raios-x A natureza periódica e regular dos cristais.

A simetria se manifesta na morfologia externa dos cristais

Hextetrahedral Forms (4i3m) Tetrahedron

Diploidal Forms (2/m3i) Pyritohedron

Pyritohedron                                   A pyritohedron has 30 edges, divided into two lengths: 24 and 6 in each group. Face polygon irregular pentagon Faces 12 Edges 30 (6+24) Vertices 20 (8+12) Symmetry group Th, [4,3+], (3*2) Dual polyhedron Pseudoicosahedron Properties convex

This crystal is a 'pyritohedron' This crystal is a 'pyritohedron'. It looks similar to regular dodecahedron—but it's not! At the molecular level, iron pyrite has little cubic crystal cells. But these cubes can form a pyritohedron

dodecahedron

icosahedron

Dodecahedron e Icosahedron possuindo simetria pentagonal são incompatíveis com a simetria traslacional

Porém, em 1984... Daniel Shechtman

As velocidades de esfriamento alcançáveis pelo método de “melt-spinning” são da ordem de 104–107 graus kelvin por segundo (K/s).

Electron diffraction pattern of an icosahedral Ho-Mg-Zn quasicrystal

Padrão de Difração

O primeiro “tiling” de Penrose's usa pentágonos e outras três formas: uma “estrela” de cinco pontas,um “barco“ e um “diamante”

Será uma pseudo-periodicidade?

Teoria de twinning de Linus Pauling Nature,317, 512-514, 1985

Schectman's is an interesting story, involving a fierce battle against established science, ridicule and mockery from colleagues and a boss who found the finding so controversial, he asked him to leave the lab.

Penrose “tilings” com só dois motivos

Roger Penrose in the foyer of the Mitchell Institute for Fundamental Physics and Astronomy, Texas A&M University, standing on a floor with a Penrose tiling

A Ho-Mg-Zn icosahedral quasicrystal formed as a dodecahedron

Pipas e Dardos (Kites and Darts) φ 1

Losangos

The darts and kites can be obtained from a rhombus with degree measures of 72° and 108° by dividing the long diagonal into two segments in the golden ratio f = (1+√5)/2 = 1.618… then joining the dividing point to the obtuse corners as shown in Figure 2a. Both rhombuses are composed of two golden triangles, the first having sides with ratios of 1:f:f, and the other having sides with ratios of 1:1:f. The ratio of the areas of the kite and dart is the golden ratio as well

The other common polygons used in Penrose tilings are Penrose rhombs, which are also composed of golden triangles. One rhomb has degree measures of 72 and 108, while the other has degree measures of 36 and 144. Since both sets can be derived from golden triangles, it makes sense that a kite and dart tiling can be translated into a rhomb tiling, and vice versa

Uma variante de “tiling” que não é quseperiódica Uma variante de “tiling” que não é quseperiódica. Não é um Penrose tiling porque a sistemática de empacotamento não foi seguida

In geometry, a pyritohedron is an irregular dodecahedron with pyritohedral (Th) symmetry. Like the regular dodecahedron, it has twelve identical pentagonal faces, with three meeting in each of the 20 vertices. However, the pentagons are not regular, and the structure has no fivefold symmetry axes. Its 30 edges are divided into two sets - containing 24 and 6 edges of the same length. Although regular dodecahedra do not exist in crystals, the distorted pyritohedron form occurs in the crystal pyrite, and it may be an inspiration for the discovery of the regular Platonic solid form.

A “Proporção Divina” ou Razão Áurea (Golden Rate)

Razão áurea e números de Fibonacci 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765

Números de Fibonacci Fn =Fn-1+Fn-2

Retângulos com a “razão áurea”

Comprimento das arestas seguem a sequência de Fibonacci

Uma curva conectando números de Fibonacci consecutivos da origem a uma forma muito especial chamada a “Espiral Áurea”

Razão Áurea, Números de Fibonacci e o mundo que habitamos

Since the original discovery of Dan Shechtman, hundreds of quasicrystals have been reported and confirmed. Undoubtedly, the quasicrystals are no longer a unique form of solid; they exist universally in many metallic alloys and some polymers. Quasicrystals are found most often in aluminium alloys (Al-Li-Cu, Al-Mn-Si, Al-Ni-Co, Al-Pd-Mn, Al-Cu-Fe, Al-Cu-V, etc.), but numerous other compositions are also known (Cd-Yb, Ti-Zr-Ni, Zn-Mg-Ho, Zn-Mg-Sc, In-Ag-Yb, Pd-U-Si, etc.). In theory, there are two types of quasicrystals. The first type, polygonal (dihedral) quasicrystals, have an axis of eight, ten, or 12-fold local symmetry (octagonal, decagonal, or dodecagonal quasicrystals, respectively). They are periodic along this axis and quasiperiodic in planes normal to it. The second type, icosahedral quasicrystals, are aperiodic in all directions. Regarding thermal stability, three types of quasicrystals are distinguished: Stable quasicrystals grown by slow cooling or casting with subsequent annealing,Metastable quasicrystals prepared by melt spinning, and Metastable quasicrystals formed by the crystallization of the amorphous phase. Except for the Al–Li–Cu system, all the stable quasicrystals are almost free of defects and disorder, as evidenced by x-ray and electron diffraction revealing peak widths as sharp as those of perfect crystals such as Si. Diffraction patterns exhibit fivefold, threefold, and twofold symmetries, and reflections are arranged quasiperiodically in three dimensions. The origin of the stabilization mechanism is different for the stable and metastable quasicrystals. Nevertheless, there is a common feature observed in most quasicrystal-forming liquid alloys or their undercooled liquids: a local icosahedral order. The icosahedral order is in equilibrium in the liquid state for the stable quasicrystals, whereas the icosahedral order prevails in the undercooled liquid state for the metastable quasicrystals.

Galáxias

Orelha humana

Over the centuries, it has been designed by trial and error, without the aid of computers. What is interesting is that it was not made to the Golden Ratio intentionally. What the designers found was that the closer the design was to the Golden Ratio, the better the quality of sound. It appears that even sound waves and harmonics tend to the Golden Ratio.

Os anéis de Saturno estão na razão áurea

Here is a full segment of a DNA Here is a full segment of a DNA. It is roughly 21 angstroms wide & 34 angstroms long for each full cycle of its double helix spiral. 21 & 34 are consecutive Fibonacci Numbers….. But wait….there is more….if you look at the two grooves created by the twisting of the double helix strand. It creates a major groove and minor groove that is to the Golden Proportion. By now you probably won’t be surprised, but the major grooves and minor groves that created form twisting the DNA strand are consecutive Fibonacci numbers 21 & 13.

Número de espirais na alcachofra

Obtenção de uma sequência quase periódica com a repetição periódica de uma dada operação.

1 2 3 5 8 13 21

Sequência de Fibonacci e quasicristais de “quasesimetria” pentagonal

If the slope of EII in 2-D is rational with respect to rows of the square lattice, the projected 1-D structure is a discrete periodic set of sites. On the contrary, if the slope of EII is irrational, the projected I-D structure is a very dense, nonperiodic set of points. To restore physical distances in I-D, the only projected sites of 2-D are those within a strip S2 extending infinitely parallel to E II but having a finite cross section A1 11 in the subspace E1 that is orthogonal to E II in2-D. Thus the resulting I-D projected structure is a nonperiodic, perfectly ordered at long distance, discrete set of sites. The 1-D subspace E II is now tiled with two segments whose lengths are in the (irrational) ratio of the slope of Ell in 2-D, If the whole 2-D square lattice as pictured in Fig. 13 is Fourier transformed into its reciprocal counterpart, the result is another 2-D square distribution of Dirac functions and "diffraction spots" will be found at scattering vectors Qz having two components (n"n z) on the orthogonal axes of the square distribution

. Let Et and Et be the I-D reciprocal subspaces associated with E II and E1 , respectively, and QII and Ql the incommensurate projections of Q2 onto E t and E t. The QII subset is the only component of Q2 that is actually scanned in real l-D experiments. If the scheme is extended to the generation of 3-D structures, the Q6 vectors have six orthogonal components in 6-D (nl,n2,n3,n4,nS,n6) which project onto the 6 fivefold symmetry directions ej in 3-D to combine into QII = .L. nj Qoej •i=1 The nj have been proposed as the indexing set for the diffraction peaks by Bancel et a/. 2l and Elser.

Se a inclinação da reta é irracional, por exemplo na razão áurea, a estrutura projetada unidimensional é um conjunto denso, não periódico, de sitios discretos. Limitando a projecão a uma faixa de largura finita, o subespaço pode ser “tiled” com dois segmentos cujos comprimentos estão, precisamente, na razão áurea.

Neste exemplo a rede recíproca coincide com a rede “direta” (quadrada de lado unitário). A transformada de Fourier antes da projeção na rede direta é a convolução da rede recíproca com a transformada da função escalão.

Projeção num espaço corresponde a nível zero no espaçp conjugado Mas,

Fator de Estrutura

Al-Pd-Mn by Yamamoto (1993)

octagonal QC:  V-Ni-Si  Cr-Ni-Si  Mn-Si  Mn-Si-Al  Mn-Fe-Si  decagonal QC:   Al-TM (TM=Ir,Pd,Pt,Os,Ru,Rh,Mn,Fe,Co,Ni,Cr)   Al-Ni-Co *   Al-Cu-Mn  Al-Cu-Fe  Al-Cu-Ni  Al-Cu-Co *   Al-Cu-Co-Si *  Al-Mn-Pd *   V-Ni-Si  Cr-Ni   

dodecagonal QC:   Cr-Ni  V-Ni  V-Ni-Si  icosahedral QC:   Al-Mn  Al-Mn-Si  Al-Li-Cu *   Al-Pd-Mn *   Al-Cu-Fe  Al-Mg-Zn Zn-Mg-RE * (RE=La,Ce,Nd,Sm,Gd,Dy,Y)   Ti-TM (TM=Fe, Mn, Co, Ni)   Nb-Fe  V-Ni-Si  Pd-U-Si   

Então, essa nova forma da matéria condensada existe no Universo só como criação tecnológica?

Proceedings of the National Academy of Sciences Evidence for the extraterrestrial origin of a natural quasicrystal. PNAS January 31, 2012 vol. 109 no. 5 1396-1401 Luca Bindi John M. Eiler Yunbin Guan Lincoln S. Hollister Glenn MacPherson Paul J. Steinhardt Nan Yao

Schechtman produced quasicrystals in the laboratory in 1982, but until 2008 nobody had found a naturally occurring quasicrystal. Now researchers in Italy and the United States have examined the rock that contained these natural quasicrystals and determined it may actually be part of a meteorite. In 2008, Luca Bindi of the Museo di Storia Naturale in Firenze, Italy approached Paul Steinhardt at Princeton University to investigate a curious rock collected in eastern Russia during the late 1970s. The researchers (including Bindi, Steinhardt, Nan Yao, and Peter Lu) found it contained naturally occurring quasicrystal grains—the first ever identified.

The rock sample consists of grains of more ordinary metallic and silicon compounds interspersed with the quasicrystal grains, so it's not wholly a quasicrystal. Some structures in the rock are only formed under high shocks (unlike sedimentary or volcanic rocks), and one particular silicate structure, known as stishovite, is most strongly associated with meteorites. To confirm this suspicion, the researchers investigated the ratios of various oxygen isotopes, 18O/16O and 17O/16O, and compared them to the ratios found on Earth in analogous minerals. Because of the differences in formation and environment, meteorites have a distinctive isotope signature compared to their chemically similar terrestrial cousins. The scientific team found the sample containing the quasicrystals looked like it had an extraterrestrial source.

The quasicrystal within the rock is a known type, Al63Cu24Fe13, first synthesized in a lab in 1987. However, if the Russian rock is as similar to a chondrite meteorite as its composition suggests, that places it around 4.5 billion years old, meaning quasicrystals evidently were present (at least in small amounts) at the start of our Solar System's history. The great age also speaks to the long-term stability of quasicrystals, at least under some conditions. How this particular meteorite formed is still a mystery, though. The metallic aluminum present in the rock usually requires a very different set of processes to form, and it has not been found in any other meteorites. In other words, while the isotope ratios indicate an extraterrestrial origin for the rock, its composition marks it as a new type of meteorite, one with uncertain origins. The authors suggest a high-velocity impact may have broken this rock sample off a larger parent body, and some combination of the conditions before impact and the collision itself may have made the strange combination of metals seen. However, they admit that's speculation; one thing that is certain is this rock has a lot to tell us beyond containing the first quasicrystals observed in nature.

Extraterrestial quasicrystal of Al63Cu24Fe13, first synthesized in a lab in 1987

Fim de esta estória, mais este século será verá o desenvolvimento de uma fascinante área na pesquisa de novos materiais.