*Posted December 14, 2021 by beauty33*

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Quasicrystals, also known as "quasicrystals" or "pseudocrystals", are a solid structure between crystalline and amorphous. In the atomic arrangement of quasicrystals, its structure is long-range order, which is similar to crystals; but quasicrystals do not have translational symmetry, which is different from crystals. Ordinary crystals have second-order, third-order, fourth-order, or sixth-order rotational symmetry, but the Bragg diffraction pattern of quasicrystals has other symmetry, such as fifth-order symmetry or more than six-order symmetry.

The composition of matter is determined by the characteristics of its atomic arrangement. A solid substance with periodic arrangement of atoms is called a crystal, a disordered arrangement of atoms is called an amorphous substance, and the one in between is called a quasicrystal. The discovery of quasicrystals was a breakthrough in crystallographic research in the 1980s.

On April 8, 1982, Shechtman first observed an "abnormal" phenomenon under an electron microscope: the atoms of the aluminum-manganese alloy were arranged in a non-repetitive, non-periodical but symmetrical and orderly manner. At that time, it was generally believed that the atoms in the crystal were arranged in a symmetrical pattern that was repeated periodically. This repetitive structure was necessary for the formation of a crystal. It is impossible in nature to have the kind of arrangement of atoms discovered by Shechtman. Crystal. Subsequently, scientists have produced more and more various quasi-crystals in the laboratory, and discovered pure natural quasi-crystals for the first time in 2009.

This quasi-crystal is also related to the Fibonacci sequence. In the Fibonacci sequence, each number is the sum of the previous two numbers. In 1753, Robert Simson, a mathematician at the University of Glasgow, discovered that as the number increases, the ratio between the two numbers is getting closer and closer to the golden ratio (an infinite non-recurring decimal similar to the pi, its value is about 0.618). Scientists later proved that the distance between atoms in the quasi-crystal is also in full compliance with the golden ratio. In 1982, when Shechtman conducted a "diffraction grating" experiment, he let electrons diffract through an aluminum-manganese alloy, and found that countless concentric circles were each surrounded by 10 light spots, which was exactly a 10-fold symmetry. Shechtman thought "this is impossible" at the time, and wrote in his notebook: "10 times?" However, in 1987, French and Japanese scientists successfully produced a quasi-crystal structure in the laboratory; 2009 Scientists have discovered natural quasi-crystals in mineral samples obtained from Lake Khatelka in eastern Russia. This new mineral called icosahedrite (taken from the icosahedron) is composed of aluminum, copper and iron. Composition; A Swedish company also found quasicrystals in one of the most durable steels used in razor blades and surgical needles for eye surgery.

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Quintic rotational symmetry

In classical crystallography, whether it is 14 kinds of Bravais lattices or 230 kinds of space groups, it is not allowed to have quintic symmetry, because quintic symmetry will destroy the translational symmetry of the spatial lattice, that is, it is impossible to use regular five sides. The shape fills the two-dimensional plane, and it is impossible to fill the three-dimensional space with an icosahedron. The discovery of quasicrystals overturned this concept. One of the characteristics of quasicrystals is the five-fold symmetry. In fact, the opal in the mineral world, the boron ring compound in organic chemistry, and the virus in biology all show five-fold symmetry, and mathematicians have already prepared the theoretical foundation for quasicrystals. In 1974, the British Penrose (Roger Penrose) proposed a solution to cover the plane with two types of quadrilateral puzzles based on previous work. For Shechtman's quasi-crystal diffraction pattern and Penrose's puzzle, there is a fascinating property, that is, hidden in their shape is the wonderful mathematical constant τ, which is the golden ratio of 0.618... The Penrose puzzle is made up of two types of quadrilaterals, one fat and one thin (internal angles are 72 degrees, 108 degrees and 36 degrees, 144 degrees). The ratio of the number of the two quadrilaterals is exactly τ; the same, in quasicrystals , The ratio of the distance between atoms tends to approach this value. Then, from 1981 to 1982, Mackay extended the concept of Penrose to three-dimensional space. The symmetry of the icosahedron was obtained by interspersing the two types of hexahedrons, and used an optical converter to obtain a five-fold symmetric optical diffraction pattern.

Quasi-cyclical

As we all know, quintic symmetry and periodicity cannot coexist. If you insist on five-fold symmetry, you must consider quasi-cyclicality. , Looking along an axis orthogonal to the 5th axis, the length of the line segment is not arbitrary, but there are only one length and one short. Their ratio is exactly the golden ratio 0.618..., and all the included angles in the figure are Integer multiples of /5. In other words, although there is no periodicity in this two-dimensional structure, it is not completely chaotic and disorderly. Both the length and the included angle have a fixed value.

Quasi-periodic is characterized by irrational numbers. In a one-dimensional quasi-periodic lattice, in addition to the translation unit 1, it can also be translated. A two-dimensional square lattice, select the strip with the slope, and project the points on it into the one-dimensional space E// (horizontal space) to form a one-dimensional quasi-periodic lattice with lengths L and S, LSLLSLSL... …. The characteristic of this one-dimensional quasi-periodic lattice is that there is no neighbor of S on both sides of S, and there is at most one neighbor of L on both sides of L. Since the slope of the strip is an irrational number, its edge can only pass through one lattice point. If its slope is changed to a rational number of 2/1, the projection of the strip in the parallel space becomes a periodic LLSLLS... It can be seen that both a one-dimensional periodic lattice and a one-dimensional quasi-periodic lattice can be obtained by a two-dimensional periodic lattice projection. The only difference is the slope of the selected projection zone. The former is a rational number and the latter is an irrational number.

Quasicrystals and Art

What is interesting is that the combination of the rich colors of light and the unique geometric structure of quasi-crystals will show extraordinary artistic quality. The quasicrystal patterns presented in the research of many fields, such as the quasicrystalline polymer structure, the diffraction pattern of the quasicrystal, and the distribution intensity of the resonance state in the photonic quasicrystal, have high artistic appreciation value.

The non-periodic mosaic patterns in the Alhambra Palace in Spain and the Darb-iImam Mosque in Iran reflect the perfect combination of quasicrystalline patterns and architectural art.

Japanese artist Akio Hizume took inspiration from quasicrystals and used 510 small bamboo poles to create three-dimensional quasicrystals.

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Last Updated | December 14, 2021 |

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