貢中元的部落格

Unlimited number of periodic decagonal tiling  with a2-D atom sites quasi crystal model貢中元chung yuan kung

Chung Yuan Kung

Department. of Electrical Engineering, National Chung Hsing University. 145 Xingda Road., South Dist. Taichung City 40227. Taiwan. Telephone: 886-4-22850359, E-mail Address: cykung@dragon.nchu.edu.tw.

Key words:  decagon, coupling, unit cell, atom sites

The fat and thin rhombus tiles with acute angles of 36 degrees and 72 degrees can be combined to form six decagonal tiles with different internal structures.  Each decagon consists of ten rhombuses: five thin rhombuses and five thick rhombuses; where a, b, c, d, e, and f are marked for six different decagons.(Figure 1) The thick rhombuses in the decagons of type-a, type-d, type-e and type-f are highlighted in red, magenta, yellow, and light blue, respectively. It should be noted that only type-a decagonal tiles are five-fold symmetrical, type-c is neither five-fold symmetrical nor mirror symmetrical, and the rest of the decagons are only mirror symmetrical. These six decagons can be regarded (considered) as basic units; we can utilize any two two of these basic units to produce (infinity) tiles.

With the coupling and tessellation scheme developed and using decagon type-a as a base, five other different decagons can be coupled with type-a in ten different directions, to produce fifty 50 coupled pairs, part of these coupled pairs, is showing in figure 2.  All these coupled pairs can be randomly combined (tessellated) to form an infinite number of periodic Penrose tiles (crystals), as shown in Figure 3.

The rhombus or rectangular area surrounded by four a-type decagons with the same orientation can be easily recognized from Figure 3, and they are considered to be the potential (elementary) unit cells combined into periodic tiles. Some of these results are shown in Figure 4a.

The tiles in Figure 4a are tessellated to form new translational crystals, as shown below in Figure 4b. This new periodic tiling (crystal) structure is different from the original (parent) crystal in Figure 3 . This behavior is inconsistent with the understanding of traditional periodic crystals and would be one of the most fascinating areas of future quasicrystal research.

If two circles representing two atoms of different sizes are located at the center of two rhombuses, taking the picture on the right side of Figure 4(a) as an example,

We will get periodic decagonal tiles with circles. The large blue circle is located at the center of the thick rhombus, and the small red circle is located at the center of the thin rhombus, as shown in Figure 5(a).

When the rhombus lines appearing in Figure 5a are removed, a periodic arrangement of two atoms of different sizes is obtained as shown in Figure 5b. This is the most indicative and efficient representation of a quasicrystal (exhibiting five-fold symmetry) associated with periodic decagonal tiles, which displays (reveals) a five-fold symmetric diffraction pattern.

This quasicrystal hidden within a decagon is shown in Figure 5b, which is just one example in an infinite unit cell. Therefore, carefully designed quasicrystals, that is, unit cells in the form of atomic sites, can be extensively matched and compared with AFM scanned surfaces of real quasicrystals through image processing to identify differences between different quasicrystals

Figure 1. Six different colorful decagons.

Figure 2. Coupling pairs of the decagons using type-a decagon as base.

Figure 3. Periodic Penrose tiling (crystal)

Figure 4 ( a)unit cell intercepted from tiling in fig 3 (b) periodic tiling  generated by tessellation of unit cell

Figure 5 (a) The unit cell on the right side of Figure 4a with circles representing atoms added, (b) the unit cell in Figure 5a with the rhombus tiles removed

This ic copied from  some where

In addition, due to the different composition and thermodynamic conditions of Al-Cu-Fe ternary alloy, crystalline phase, quasi-crystalline phase and amorphous phase can be obtained respectively. It is worth mentioning that aluminum-based quasicrystals, such as aluminum-copper-iron, contain three elements that are all metals and have good electrical conductivity. However, the conductivity of quasicrystals composed of these three elements is only about 10,000 times that of metals. First, it is even lower than amorphous materials with similar compositions, and the more perfect and defect-free quasicrystals are, the lower their electrical conductivity is, which is exactly the opposite of metals. That is to say, metallic elements exhibit ceramic-like characteristics, which should be very different from ordinary metals and amorphous metals. Figure 3 shows the electron diffraction and SEM images of the aluminum reference crystal material provided by Professor Cai Anbang.

Start from here  Fjg 6

準晶體結構必須伴隨著彭羅斯tile結構，才能清楚感受到晶格的存在

The quasicrystal structure must be accompanied by the Penrose tile structure in order to clearly feel the existence of the crystal lattice.

準晶體表現出十重對稱 TEM 衍射圖案的結果，這對於五重對稱彭羅斯瓦片至關重要

Fig 6 a and 6 ( b)  show  a reasonable and logic connection between a quasicrystal exhibiting a ten-fold symmetric TEM diffraction pattern and a five-fold symmetric Penrose t tiling.

, Figure 6 (a) Unit cell of a decagonal tile with 18 thick rhombuses and 11 thin rhombuses with a circle in each rhombus. A fat rhombus contains a large circle and a thin rhombus contains a small circle (b) Remove the rhombus lines, leaving only 18 large circles and 11 small circles.

( this is for Present)

Fig xx  the swirl style Penrose tile with L/S ratio is 1. A. five fold and mirror symmetrical  b> neuther

Prof. Tsai found Al65 Cu23 Fe 12 is a stable I phase QC, In this three dimensional I phase QC, composed of three different metal elements Al, Cu, and Fe. The ratio of contented metal is Al 65, Cu 23, Fe 12 .  Large metallic Al atom has an atomic radius of 0.143 nm, small atom Cu and Fe, has atomic radius 0.128 nm and 0.126 nm. respectively. The large and small atom ratio ( is about 1.8.

The ratio of large atoms (Al) and small atoms (Cu + Fe) is about 1.8, which is the three-dimensional volume ratio. For a two-dimensional plane, the ratio will be 1.2,

We have made all kind of Penrose tiling, with large rhombus over small rhombus (L/S) of 1 to 1.9. The one with plane ratio (L/S) closed to 1.2 is the most simple four way symmetrical tile, as show  fig 7  , L/S is 1.2222. Large atom is 55% and small atom 1s 45 %.

Figure 7a is the simplest four-way symmetric Penrose tiling. B) Penrose tile with a circle inside. C) The atomic sites of the simplest four-way symmetric Penrose tiling after removing the tiling lines. D) Label groups of atoms with periodic

The reason we take the unit cell with the ratio of large rhombus and small rhombus 18 over 11, as show in fig 6a as an example. Because of following reasons: This unit cell is intercepted from another four way symmetrical tile.  From two-dimensional observation, the ratio of large atoms to small atoms on the plane is (18/11)1.636. Observed from the three-dimensional volume, this ratio is around 2, which is the square root of 1.636*3. The alloy contains a large amount of different

Elements, large atom is 50% , small atom is also 50%.

Base on the electronic configuration

The electronic configuration of Aluminum is (Ne)(3s)2(3p)1, and it has an atomic radius of 0.143 nm. The electronic configuration of coper is (Ar)(3d)10(4s)1, and an atomic radius of 0.128 nm.

The electronic configuration of the Fe atom is (Ar)(3d)6(4s)2, and it has an atomic radius of 0.126 nm.

Figure 6 shows a bonding model of Al with three (electron) bonds for a small portion of Al clusters in the two dimensional unit cell. Each Al can only contribute on contribute  one electron connection to the near by atoms on the same plane,

Fig  8a, the electron bonding for an atom cluster on the same plane

Fig 8b

The model assumes that the large atom Al contributes all 3 electron bonds in such a way, one to any atom in the upper layer and one to any atom in the lower layer. The last and only one is connected to the closest (or available) atom on the same 2D plane

The electronic bonding of this two-dimensional atomic site model must be explained..

One must explain why a periodic quasicrystal with all metallic elements (Al(x)-Cu(y)-Fe(z) ) does not behave like a metal

Why resistivity is so low? 

Because, although the atoms in the lattice atomic sites are periodic, the electron bonds (on different unit cells) may not always be the same ( as shown in fig 8a, 8b), so we cannot use traditional free electron-based models to explain decagonal quasicrystals electronic behavior . Fig 8a, the electron bonding for an atom cluster on the same plane, fig b Fig 8b, the electron bonding for an atom cluster on the same plane, but also connect to up layer and low layer.

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