貢中元的部落格

Unlimited number of periodic decagonal tiling with a 2(3)-D atom sites quasi-crystal model

Chung Yuan Kung 

Department of Electrical Engineering, National Chung Hsing University, 145 Xingda Road., South District[JC1] , Taichung City 40227, Taiwan. Telephone: 886-4-22850359, E-mail Address: cykung@dragon.nchu.edu.tw.

Malachi Wanake, Department of Chemistry, University of California at Los Angeles (UCLA[JC2] )

Keywords: decagon; coupling; unit cell; atom sites.

Abstract: This study constitutes an attempt to demonstrate the relation between quasi-crystal that is logically associated with Penrose tile by constructing a four-way symmetrical periodical decagonal tiling (i.e., infinite size), and transferring the tiles to a 2-D  atom sites  pattern and making a semi-3-D quasi-crystal model. With this model, a high resistivity periodical metal alloys model is proposed.

Fat and thin rhombus tiles with acute angles of 36 degrees and 72 degrees, respectively[JC3] , can be combined to form six decagonal tiles with different internal structures. Each decagon consists of 10 rhombuses: five thin rhombuses and five thick rhombuses, where a, b, c, d, e, and f are marked for six different decagons (Figure 1). [JC4] 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 symmetric; type-c is neither five-fold symmetric nor mirror symmetric; and the remaining decagons are only mirror symmetric. These six decagons can be regarded as basic units, in which any two of these basic units can be utilized to produce infinite 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 10 different directions, producing 50 coupled pairs. Part of these coupled pairs is shown in Figure 2. Randomly combining (tessellated) all of these coupled pairs forms 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 presented 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 distinct from the original (parent) crystal in Figure 3. This behavior is inconsistent with the understanding of traditional periodic crystals, and constitutes a highly worthwhile direction for future quasi-crystal research.

The[JC5]  five-fold symmetry of Penrose tiles is crucial to the result that the quasi-crystal exhibits a 10-fold symmetry of the TEM diffraction pattern. The quasi-crystal structure must be accompanied by the Penrose tile structure in order to clearly discern [JC6] the existence of the crystal lattice.

In the following, the simplest four-way symmetric Penrose tiling using type-a decagon as a base is demonstrated. Only using a type-a decagon (self-coupling)  in coupling, tessellation can produce periodical four-way symmetrical tiles to infinity, the procedures of which [JC7] are briefly  shown in  Figure 5 (a,b,c).  Figure 5d is the unit cell, composed of 11 large rhombuses (L) and nine small rhombuses (S), and the L/S ratio is also 11/9. In the same way, type-a and type-b (or type-e and type-f) can also be coupled (Figure 5e) and a four-way [JC8] symmetrical tile can be constructed with the same (L/S) ratio in unit cell. The resultant four-way symmetric type-a and type-b [JC9] (unit cell) is presented in Figure 5f.

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This is a better array   for the TEM surface atoms of the  quasicrystals observed L/S =9/18 = 1/2, This L?s values can be varied, if some atom on the intersection of rhombus changed to large atos. 

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Figure[JC10]  6a. Unit cell of a decagonal tile with a unit cell containing 11 thick rhombuses and nine thin rhombuses b) with a circle in each rhombus. A fat rhombus contains a large circle and a thin rhombus contains a small circle, (c) and (d), respectively[JC11] , with the most basic atomic loops (clusters) composition.[JC12] 

The large and small circles, representing two atoms of different sizes, are then embedded into the center of the large rhombus and[u13]   small rhombus, respectively, of Figure 5c, the result of which is shown in Figure 6a. [JC14] After removing  the rhombus lines, the atomic sites structure remains with the large and small circles, as shown in Figure 6b.[JC15]  This is a tile to transfer atomic sites (transformation) in Figure 5c  and Figure 6a. Thereby, Figure  6b shows a reasonable and logical connection between a five-fold symmetric Penrose tiling (or four-way symmetrical tile embedding a five-fold decagonal characteristic) and a 2-D quasi-crystal model exhibiting a ten-fold symmetric TEM diffraction pattern. [JC16] Figure 6b can be thought of as a representation of the atomic sites of a four-way symmetrical tile, which is most logically related to the four-way symmetric decagonal tile, while the inner structure is related to the hidden five-fold symmetrical tiles inside of it[JC17] .

Based on Figure 6b, the unit lengths in both x- and y- directions are identified and marked. Figure 6b presents the unit cell of this four-way symmetrical two-dimensional [JC18] atomic structure. This 2-D atomic structure, intriguingly, has two different four-way symmetrical centers, as marked (0,0,) in Figure 6b[JC19] .  Indeed, this might be the most important characteristic of quasi-crystals, and has also been observed in our other 2-D modes (ref  3). This two-dimensional layer can be arranged in parallel in the x- and y- directions, separated by unit distance, which is the length of the edge of the rectangular unit cell in the x- and y- directions. However, this is a pseudo 3-D crystal structure with a large empty space [JC20] between the unit length.

 The most promising model for a 3-D quasi-crystal model, or a more realistic structure, might be as follows: each symmetrical layer pattern is stacked up along the y-direction at a unit distance “l”, or at a distance of sin 72 degrees of “l” , where “l” is a rhombus side length, and each stacked layer can be rotated clockwise 0, 18,  or 36 degrees, relative to the underlying layers (to the layer beneath).[JC21]  These closely packed semi-three-dimensional structures are very likely to exhibit ten-fold diffraction patterns expected from quasi-crystals.

Figure 7. (a ) Rectangular atomic sites of the simplest four-way symmetric. (b)

groups of atoms with “ periodical” parallelogram [JC22] or quadrilateral parallelogram connected diagrams indicated by light blue and green lines, respectively.[JC23] 

In Figure 7, several regular connections with rectangular shapes or parallelogram shapes were made, as indicated by light blue and green lines, respectively[JC24] . All of these diagrams can be extended to different structures periodically (ref 1, 2). As reported (2), when tiles went to infinity, as also presented in Figure 7a,b, the number of these unit cells [JC25] could be infinite. Indeed, this parallelogram or quadrilateral parallelogram connection method may offer in-depth research potential in neural connections and/or cryptographic operations.

In Figure 8a, some random, but ordered, discrete topo-graphs (clusters of atoms) can be identified by a closed series (cycle) of four to seven or eight circles. These clustered (loop) rings could be the basic building truss blocks of the simplest four-way structures of atomic sites. They are both embedded with the structure characteristic of five-fold type-a tile.

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​ ​​​​​ ​​​​​The[JC26]  infinite-size periodic diagrams presented in this paper represent all mature and stable quasi-crystals. They certainly do not represent the observed tiny quasi-crystals on the nanoscale or micron scale. Such tiny crystals observed in quasi-crystalline behavior are most likely in the transition stage from liquid (high temperature) or in the phase coarsening stage of many nucleation stages. In fact, many different nuclei may exist with different phases undergoing a comp eting growth stage, and are subject to the coarsening effect of the processing phase[JC27] . The quasi-crystal discovered was too small; irrespective of what kind of microscope was used, the area illuminated was an extremely small region. This kind of competition result is highly important to explain the phase transformation phenomenon in the growing quasi-crystal. The observed microstructure of quasi-crystals most probably hides nanophase boundaries, which would lead to difficulties in matching between our models and real objects.

It is highly challenging to make a comprehensive comparison of quasi-crystals with the large-size modeling (infinite size) pattern. Therefore, [JC28] we have drawn several special smaller structures that must exist in our quasi-crystal. The symmetrical atomic structure was divided into an even smaller elemental region, which contains only five to seven atoms, as shown in Figure 8. This independent unit serves as the basic building block of the resulting quasi-crystal model (shown as the discrete components in Figure 8), and is also an effective tool for attempting to match the surface of the quasi-crystal observed by atomic force microscopy or HRTEM. The crystal structure in Figure 8 contains a large amount of empty space. This large number of voids in the quasi-crystal is conducive to the storage of hydrogen molecules, marked as green circles.[JC29] 

If this local structure can then be found similarly in the two-dimensional scan of our observed quasi-crystal[JC30] , this would probably constitute a useful starting point for a more in-depth discussion or comparison. From these diagrams, some basic elemental units are generated, in which five or six atoms form (closed series) graphs, as indicated below in Figures 8 and 9[JC31] . These can be considered as the basic graph units from which all different quasi-crystals might contain one or several of these basic graph structures. Consequently, the appearance of these graphs calls for deeper further research.

 From the four-way symmetrical patterns, some basic elemental units are generated, in which four, five, or six atoms form a closed series truss (Figure 9a), as shown below, and these are used to construct a new four-way symmetrical structure. These truss topo-graphs [JC32] can be considered as the basic units from which different quasi-crystals are formed. Such artificial quasi-crystals (as presented in Figure 9b) could increase the likelihood of successful comparative matches with real observed quasi-crystals. The three deformed pentagons are the most important topo-graphs[JC33]  to construct the 10-fold diffraction (TEM) quasi-crystals.[JC34] 

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Figure 9. (a) The created 2-D crystal structure, tessellated by three topo-graphs in (b). The three deformed pentagon is the most important topo-graph to construct the ten-fold diffraction (TEM) quasi-crystals.[JC35] 

  In addition, due to the different composition and thermodynamic conditions of Al-Cu-Fe ternary alloy, the crystalline phase, the quasi-crystalline phase, and the amorphous phase can be obtained, respectively[JC36] . It is worth mentioning that aluminum-based quasi-crystals, such as aluminum-copper-iron, contain three metals and possess good electrical conductivity. However, the conductivity of quasi-crystals composed of these three elements is only ~10,000 times that of metals. It is also even lower than amorphous materials with similar compositions. Moreover, the more perfect and defect-free are the quasi-crystals, the lower is their electrical conductivity, which is exactly the opposite of metals. In other words, metallic elements exhibit ceramic-like characteristics, which should be markedly 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.

Figure 10 presents a bonding model of Al with three (electron) bonds for a small portion of Al clusters (shown in Figure 8) in the two-dimensional unit cell. Each Al can only contribute one electron connection to a nearby atom on the same plane. There are also two other bonds connected to upper and lower layers, respectively.[JC37] 

​     Figure 8a[JC38] , Figure 10 (a).[JC39]  Electron bonding for an atom cluster on the same plane.(b)The model assumes that the large atom Al contributes all three electron bonds in such a way that one is connected to upper layer atoms and one is connected to lower layer atoms.​​[JC40] 

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 copper is (Ar)(3d)10(4s)1, and it has 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.