Smolyakov O.V.

Smolyakov O.V.

We propose the method for modelling of quasi-periodic structures based on an algorithm being a geometrical interpretation of the Fibonacci-type numerical sequences. The modelling consists in a recurrent multiplication of basis groups of the sites, which possess the 10-th, 8-th or 12-th order rotatio...

Повний опис

Збережено в:
Бібліографічні деталі
Дата:2019
Автори: Girzhon, V.V., Smolyakov, O.V.
Формат: Стаття
Мова:English
Опубліковано: Інститут металофізики ім. Г.В. Курдюмова НАН України 2019
Назва видання:Успехи физики металлов
Онлайн доступ:http://dspace.nbuv.gov.ua/handle/123456789/167937
Теги: Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:Modelling of lattices of two-dimensional quasi-crystals / V.V. Girzhon, O.V. Smolyakov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 551-583. — Bibliog.: 53 titles. — eng.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine
id irk-123456789-167937
record_format dspace
spelling irk-123456789-1679372020-04-18T01:26:02Z Smolyakov O.V. Girzhon, V.V. Smolyakov, O.V. We propose the method for modelling of quasi-periodic structures based on an algorithm being a geometrical interpretation of the Fibonacci-type numerical sequences. The modelling consists in a recurrent multiplication of basis groups of the sites, which possess the 10-th, 8-th or 12-th order rotational symmetry. The advantage of the proposed method consists in an ability to operate with only two-dimensional space coordinates rather than with hypothetical spaces of dimension more than three. Запропоновано спосіб моделювання квазиперіодичних структур, в основі якого лежить алґоритм, що є геометричною інтерпретацією числових послідовностей типу послідовности Фібоначчі. Моделювання полягає у рекурентному розмноженні базисних груп вузлів, які мають ротаційну симетрію 10, 8 або 12-го порядку. Перевагою запропонованого способу є можливість оперувати координатами лише двовимірного простору, а не гіпотетичних просторів із вимірністю, вищою за три. Предложен способ моделирования квазипериодических структур, в основе которого лежит алгоритм, являющийся геометрической интерпретацией числовых последовательностей типа последовательности Фибоначчи. Моделирование заключается в рекуррентном размножении базисных групп узлов, имеющих ротационную симметрию 10, 8 или 12-го порядка. Преимуществом предлагаемого способа является возможность оперировать координатами только двумерного пространства, а не гипотетических пространств с размерностью, большей трёх. 2019 Article Modelling of lattices of two-dimensional quasi-crystals / V.V. Girzhon, O.V. Smolyakov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 551-583. — Bibliog.: 53 titles. — eng. 1608-1021 DOI: https://doi.org/10.15407/ufm.20.04.551 http://dspace.nbuv.gov.ua/handle/123456789/167937 en Успехи физики металлов Інститут металофізики ім. Г.В. Курдюмова НАН України
institution Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection DSpace DC
language English
description We propose the method for modelling of quasi-periodic structures based on an algorithm being a geometrical interpretation of the Fibonacci-type numerical sequences. The modelling consists in a recurrent multiplication of basis groups of the sites, which possess the 10-th, 8-th or 12-th order rotational symmetry. The advantage of the proposed method consists in an ability to operate with only two-dimensional space coordinates rather than with hypothetical spaces of dimension more than three.
format Article
author Girzhon, V.V.
Smolyakov, O.V.
spellingShingle Girzhon, V.V.
Smolyakov, O.V.
Smolyakov O.V.
Успехи физики металлов
author_facet Girzhon, V.V.
Smolyakov, O.V.
author_sort Girzhon, V.V.
title Smolyakov O.V.
title_short Smolyakov O.V.
title_full Smolyakov O.V.
title_fullStr Smolyakov O.V.
title_full_unstemmed Smolyakov O.V.
title_sort smolyakov o.v.
publisher Інститут металофізики ім. Г.В. Курдюмова НАН України
publishDate 2019
url http://dspace.nbuv.gov.ua/handle/123456789/167937
citation_txt Modelling of lattices of two-dimensional quasi-crystals / V.V. Girzhon, O.V. Smolyakov // Progress in Physics of Metals. — 2019. — Vol. 20, No 4. — P. 551-583. — Bibliog.: 53 titles. — eng.
series Успехи физики металлов
work_keys_str_mv AT girzhonvv smolyakovov
AT smolyakovov smolyakovov
first_indexed 2025-07-15T02:02:12Z
last_indexed 2025-07-15T02:02:12Z
_version_ 1837676587674238976
fulltext ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 551 © V.V. GIrzhoN, o.V. SMolyAKoV, 2019 https://doi.org/10.15407/ufm.20.04.551 V.V. Girzhon and o.V. SmolyakoV zaporizhzhya National University, 66 zhukovsky str., Ua-69600 zaporizhzhya, Ukraine Modelling of lattices of two-diMensional Quasi-crystals We propose the method for modelling of quasi-periodic structures based on an algo- rithm being a geometrical interpretation of the Fibonacci-type numerical sequences. The modelling consists in a recurrent multiplication of basis groups of the sites, which possess the 10-th, 8-th or 12-th order rotational symmetry. The advantage of the proposed method consists in an ability to operate with only two-dimensional space coordinates rather than with hypothetical spaces of dimension more than three. The correspondence between the method of projection of quasi-periodic lat- tices and the method of recurrent multiplication of basis-site groups is shown. As established, the six-dimensional reciprocal lattice for decagonal quasi-crystals can be obtained from orthogonal six-dimensional lattice for icosahedral quasi-crystals by changing the scale along one of the basis vectors and prohibiting the projection of sites, for which the sum of five indices (corresponding to other basis vectors) is not equal to zero. It is shown the sufficiency of using only three indices for de- scribing diffraction patterns from quasi-crystals with 10-th, 8-th and 12-th order symmetry axes. original algorithm enables direct obtaining of information about intensity of diffraction reflexes from the quantity of self-overlaps of sites in course of construction of reciprocal lattices of quasi-crystals. Keywords: quasi-periodic structures, Fibonacci sequence, projection method, basis vectors, rotation symmetry, reciprocal lattice. 1. introduction one of actual problems of modern solid-state physics is the description of quasi-crystalline materials structure. For the establishment and de- scription of crystal structures, the experimental and theoretical basis is well developed. In the same time, formal extrapolation of laws and methods of classical crystallography to quasi-crystalline structures leads to significant difficulties. For instance, the usage of three Miller indi- V.V. Girzhon and O.V. Smolyakov 552 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 ces for denoting of atomic planes (corre- sponding to the reciprocal quasi-lattice sites) leads to the fact that these indices are irra- tional in most cases. In practice, the using of non-integer indices is inconvenient. There- fore, for the indexing of quasi-crystals planes with the symmetry of icosahedron, in paper [1], it was proposed the replacement of three index symbols with six-index integer index as (h/h' k/k' l/l' ), H = h + h'τ, K = k + k'τ, L = l + l'τ, where irrational constant number τ = 2cos (π/5) denotes ‘golden ratio’. Another method of indexing of atomic planes is the result of model- ling method of icosahedral quasi-crystal structures. It consists in pro- jecting the six-dimensional hypercube lattice on the three-dimensional space [2, 3]. In this method, the six-index designation (n1 n2 n3 n4 n5 n6) was proposed for both atomic planes and reciprocal lattice sites, since the symmetry of quasi-crystal lattice is identic to corresponding sym- metry of its reciprocal lattice [4, 5]. In addition, for icosahedral quasi- crystals, authors commonly use the two-index (N, M)-type designation based on the fact that square number of the vector of reciprocal icosa- hedral quasi-lattice can be presented as [1] Q2 = N + Mτ. (1) one of the differences of quasi-crystals, which have 8-th, 10-th or 12-th order symmetry axis, from the quasi-crystals with icosahedral symmetry is the periodicity in direction of higher order axis. The cor- responding index associated with this direction always accepts integer value and there is no need to replace it with the combination of two indices comprising rational and irrational part. The issue is in ambigu- ity of assignment of base vectors for flat quasi-lattice, which is perpen- dicular to symmetry axis of the 8-th, 10-th or 12-th order. In many papers relating to decagonal quasi-crystals [6–10], there are five-index symbols of diffraction reflexes. These symbols include four indices re- ferring to flat quasi-lattice and one index referring to periodicity direc- tion. In papers [11, 12], authors used a six-index notation for such quasi-crystals. In this case, the five-dimensional index refers to flat quasi-lattice. Quite often, e.g., in refs. [3–15], reflexes are simply de- noted as those related to the quasi-crystalline phase without specifying the corresponding indices. The difference in the number of indices is caused by the fact that for the basis vectors of flat reciprocal quasi- lattice are used five vectors, which directed from the pentagon centre to Fig. 1. basis vectors of planar quasi-lattice with de- cagonal symmetry ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 553 Modelling of Lattices of Two-Dimensional Quasi-Crystals its vertices (±q1, ±q2, ±q3, ±q4, ±q5) (Fig. 1). however, considering equa- tion q1 + q2 + q3 + q4 + q5 = 0, obviously, it can be used only four basic vectors. however, the simplification of indexation, which consists in reducing of number of used indices, leads to the fact that equivalent sites of reciprocal quasi-lattice are differently indexed (Fig. 2). In addition to the problem with indexing, there is also the problem of calculating the diffraction maxima intensity. The main difficulty consists in impossibility of assignment of quasi-crystals elementary cell and, con- sequently, in impossibility of calculating the structural factor. one way of solving this problem is to approximate quasi-crystalline structure with cubic or other lattices with large parameters [16–19]. however, this meth- od is not convenient, since in order to increase the correspondence of the calculated results to the real one, it is necessary to choose the elementary cells of approximant structure with the largest values of lattice parame- ters. In this case, the number of cell basis elements naturally increases. Another method for evaluating the intensity of reflexes is based on the using of periodic lattice in multi-dimensional, in particular, six-di- mensional [20] space. To solve these problems more correctly, the original method of mod- elling the quasi-periodic structures, elucidated in papers [21–25], is proposed. 2. Decagonal Quasi-Periodic lattices Since the concept of the quasi-crystal is closely related to the concepts such as Fibonacci sequence (elements of which are determined by the equation Fn = Fn−1 + Fn−2) and the ‘golden ratio’ (expressed by τ number), then, some kind of geometric interpretation of this sequence is sug- gested for modelling. For two-dimensional decagonal quasi-lattice, the process of model- ling can be demonstrated as follows. The group of sites, set by ten basic vectors (±q1, ±q2, ±q3, ±q4, ±q5), is selected for the first element of the Fig. 2. The ambi- guity of the in- dexing the planar decagonal quasi- lattice sites: (a) five indices (left); (b) four indices (right) 554 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov ‘sequence’. let us call this group as D1 (Fig. 3, a). To simplify the recording, we de- note these ten vectors as qi, where i- index varies from 1 to 10. The group D2 is obtained by placing the centres of additional ten groups D1 in the sites of the initial group (Fig. 3, b, c). Thus, the group D2 is the set of sites given by the set of vectors, {qi}, of the previous group, D1, and the vectors obtained by the addition of vectors, {qi + qj}. Schematically, the procedure for obtaining this group can be written as D2 = D1 + {qi}D1, where the equation {q1}D1 denotes the shifting the centre of the group D1 into corresponding vectors. Then, on ends of the vectors (±τq1, ±τq2, ±τq3, ±τq4, ±τq5) constructed from the cen- tre of the group D2, the centres of the group D1 are placed. As a result, we obtain the group of sites D3 (Fig. 3, d) [21, 22]. For obtaining the group D4 on ends of the vectors (±τ2q1, ±τ2q2, ±τ2q3, ±τ2q4, ±τ2q5) constructed from the cen tre of group D3, ten groups D2 are placed (Fig. 4). Generally, to obtain the group Dn, we have to put the centres of the group Dn−1 at the ends of the vectors (±τn–2q1, ±τn–2q2, ±τn–2q3, ±τn–2q4, ±τn–2q5) constructed from the centre of the group Dn−2. The total algorithm for model- ling the decagonal quasi-crystalline lattice can be written in the form of recursive expression Dn = Dn−1 + + {τn–2qi}Dn−2. Starting with the third group of sites, it is possible to implement two more versions of recursive algorithm: Dn = Dn−2 + + {τn–2qi}Dn−1 and Dn = Dn−1 + {τn–2qi}Dn−1. Therefore, we denote the algorithm Dn = Dn−1 + {τn–2qi}Dn−2 as al- gorithm no. 1, Dn = Dn−2 + {τn–2qi}Dn−1 as algorithm no. 2, and Dn = Dn−1 + + {τn–2qi}Dn−1 as algorithm no. 3. Fig. 4. Group of sites D4 constructed according to algorithm no. 1 Fig. 3. The process of generation of quasi-lattice sites: (a) initial group of sites, (b) displacement of additional initial group of sites along one of the basis vectors, (c) group of sites D2, (d) group of sites D3 ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 555 Modelling of Lattices of Two-Dimensional Quasi-Crystals Table 1. Characteristics of Dn groups constructed according to three algorithms Group Algorithm 1 2 3 Dn = Dn−1 + {τn−2qi}Dn−2 Dn = Dn−2 + {τn−2qi}Dn−1 Dn = Dn−1 + {τn−2qi}Dn−1 Vectors of group Group radius Group vectors Group radius Group vectors Group radius D1 {qi} 1 {qi} 1 {qi} 1 D2 {qi}, {qi + qj} 2 {qi}, {qi + qj} 2 {qi},{qi + qj} 2 D3 {qi}, {qi + qj}, {τqi + qj} τ + 1 {qi}, {τqi + qj + qk}, τ + 2 {qi}, {qi + qj}, {τqi + qj}, {τqi + qj + qk} τ + 2 D4 {qi}, {qi + qj}, {τqi + qj}, {τ2qi + qj}, {τ2qi + qj + qk} τ + 3 {qi}, {qi + qj}, {τ2qi + qj}, {τ2qi + τqj + + qk + ql} 2τ + 3 {qi}, {qi + qj}, {τqi + qj}, {τqi + qj + qk}, {τ2qi + qi}, {τ2qi + qi + qk}, {τ2qi + τqi + qj}, {τ2qi + τqi + qj + qk} 2τ + 3 D5 {qi}, {qi + qj}, {τqi + qj}, {τ2qi + qj}, {τ2qi + qj + qk}, {τ3qi + qj}, {τ3qi + qj + qk}, {τ3qi + τqj + qk} 3τ + 2 {qi}, {τqi + qj + qk}, {τ3qi + qj}, {τ3qi + qj + qk}, {τ3qi + τ2qj + qk}, {τ3qi + τ2qj + + τqk + ql + qm} 4τ + 4 {qi}, {qi + qj}, {τqi + qj}, {τqi + qj + qk}, {τ2qi + qi}, {τ2qi + qj + qk}, {τ2qi + τqj + qk}, {τ2qi + τqj + qk + ql}, {τ3qi + qj}, {τ3qi + qj + qk}, {τ3qi + τqj + qk}, {τ3qi + τqj + qk + ql}, {τ3qi + τ2qj + qk}, {τ3qi + τ2qj + qk + ql}, {τ3qi + τ2qj + τqk + ql}, {τ3qi + τ2qj + τqk + ql + qm} 4τ + 4 It is known [3, 5, 9] that reciprocal for decagonal quasi-crys- talline lattice is also decagonal quasi-periodic lattice. Therefore, obtained models can be compared with electron diffraction pat- terns of real decagonal quasi-crystals having selected certain scale. In fact, these electron diffraction patterns represent the section of three-dimensional recip rocal lattice. The quasi-lattice model constructed according to the first algo- rithm is in a good agreement with the electron diffraction pattern, which was obtained in [26] for the Al–Ni–Co alloy with a decago- nal structure (Fig. 5, a, b). however, the coincidence of model sites is observed only for reflexes with high and medium intensity. Some of the same low-intensity reflexes according to specified al- gorithm are not generated. Using the algorithm no. 2 eliminates this problem (Fig. 5, c). 556 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Using the algorithm no. 3 also leads to similar result. Some characteristics of sites groups constructed by three specified algo- rithms are given in Table 1, from which it is evident that the groups constructed according to the algorithm Dn = Dn−1 + τn–2qi Dn−1 (algorithm no. 3) include also those groups, which are constructed according to the two another algorithms. It should be noted that starting from a group D3 the last sub- sets of vectors in algorithms nos. 2 and 3 (Table 1) contain all other ‘preceding’ subsets of corresponding algorithm. For exam- ple, the subset of {τqi + qj + qk} vectors (D3 group) contains {qi + qj} and {τqi + qj} vectors. It can be easy verified, if to consider some properties of basic vectors, particularly that qi + qj +1 = –τqi + 3. At the same time, the last element in groups constructed according to the first proposed algorithm, in the general case, does not contain all subsets. For example, {τ3qi + qj + qk} subset in D5 group does not contain {qi} vectors. It follows from the fact that |τ3qi – τqi – qi| > > |qi|. Thus, the quasi-lattices constructed according to the second and to the third algorithms are identical with each other and Dn group is reduced to the set of sites given by {qi1 + qi2 + τqi3 + + τ2qi4 +...+ τn –1qin} vectors. 2.1. Relation between Decagonal and Icosahedral Quasi-Lattices; Indexing of Diffraction Reflexes Writing five unit basis vectors (Fig. 1) in a form 1 1 2  τ = + τ γ  q i j , ( )2 1 0 2 2 = + τ τ q i j , 3 1 2  τ = +  τ γ  q i j , 2 4 1 1 2   = τ  τ γ  +i jq , 2 5 1 1 2   = + τ  τ γ  iq j , (2) Fig. 5. overlaying the model groups of lattice sites on electron diffraction pattern of real decagonal Al–Ni–Co quasicrystal [26] (a), where the sites are constructed via the algorithms nos. 1 (b) and 2 (c) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 557 Modelling of Lattices of Two-Dimensional Quasi-Crystals where 1 2γ = τ + , then, we can show that, in a case of plain decagonal quasi-lattice, the equation for square distance between the random site and origin of coordinates (Q = n1q1 + n2q2 + n3q3 + n4q4 + n5q5 ) can be reduced to the form: |Q|2 = (n1 2 + n2 2 + n3 2 + n4 2 + n5 2 − n1n2 − n2n3 − n3n4 − n4n5 − n5n1) + + (n1n2 + n2n3 + n3n4 + n4n5 + n5n1 − n1n3 − n2n4 − n3n5 − n4n1 − n5n2) τ. (3) Using denotations N* = n1 2 + n2 2 + n3 2 + n4 2 + n5 2 − n1n2 − n2n3 − n3n4 − n4n5 − n5n1, M* = n1n2 + n2n3 + n3n4 + n4n5 + n5n1 − n1n3 − n2n4 − n3n5 − n4n1 − n5n2, (4) it can be derived the equation, which is similar by the form to obtained in ref. [1] for icosahedral quasi-crystals: |Q|2 = N* + M*τ. (5) Identical form of eqs. (1) and (5) is due to the relation between icosahedral and decagonal lattices. To prove this statement, let us use the method of projection and select six orthogonal vectors in the recip- rocal six-dimensional space, which the general view was reported in ref. [1]: u1 = [τ 1 0 1 τ− 0], u2 = [0 τ 1 0 1 τ−], u3 = [1 − 0 τ τ 0 1], u4 = [0 τ− 1 0 1 − τ−], (6) u5 = [τ 1− 0 1 τ 0], u6 = [1 0 τ τ− 0 1], let us consider the first triple and the second one of components for each vector as the Cartesian coordinates of reciprocal spaces: physical (XYZ) and ‘perpendicular’ (X′Y′Z′) ones. The vectors (6) determine six vertices of icosahedron both in physical and ‘perpendicular’ spaces. Thus, the projection of six-dimensional periodic structure constructed on the set of vectors (6) specifies the reciprocal icosahedral lattice. Us- ing rotation matrix of the form 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 γτ γ       γ γτ   γ γτ      γτ γ  , (7) 558 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov the system (10) can be converted in a manner that vector u6 is projected only onto Z and Z′ axis, while projections of the rest of five vectors on XOY and X′OY′ planes specifies the vertices of regular pentagon: 2 1 2 2 3 2 4 2 5 6 [ 1 ], [ 1 ], [2 0 2 0 ], [ 1 ], [ 1 ], [0 0 1/ 0 0 1/ ]. = γτ γτ γ τ γτ = γ τ γτ γτ γτ = γτ γτ γτ γτ = γ τ γτ γτ γτ = γτ γτ γ τ γτ = γ γ u u u u u u (8) The set of vectors (8) remains orthogonal, and next statement is correct for both the (8) and (6) vectors: 1 2 3 4 5 6| | | | | | | | | | | | 2( 2)= = = = = = τ +u u u u u u . (9) linear combination of the first five vectors (8) q1 * = (u1 − u3), q2 * = (u2 − u4), q3 * = (u3 − u5), q4 * = (u4 − u1), q5 * = (u5 − u2) (10) represents five vectors in reciprocal six-dimensional space, which pro- jections onto physical and ‘perpendicular’ spaces are coplanar between each other: * 1 * 2 * 3 * 2 4 * 2 5 [ / 1 0 1/ 0], [ 0 2 0 0 2 0], [ / 1 0 1/ 0], [1/ 0 1/ 1/ 0], [1/ 0 1/ 1 / 0]. = τ γ γτ τ = τ = τ γ γτ τ = γ τ γ τ = γ τ γ τ q q q q q (11) Comparing eqs. (2) and (11), we can write = τ * 1 1 1 2 q q , = τ * 2 2 1 2 q q , = τ * 3 3 1 2 q q , = τ * 4 4 1 2 q q , = τ q * 5 5 1 2 q ; (12) here, qi *|| are projections of qi * vectors onto reciprocal space. Thus, the basis vectors {qi} of reciprocal decagonal quasi-lattice in physical space are expressed through the similar basis vectors of re- ciprocal icosahedral lattice. Using equations (12), it is possible to ob - tain the relations between (N*, M*) and (N, M), which appear in eqs. (1) and (5): N* + M*τ = 1/(2τ)2 (N + Mτ), N = 4/(N* + M*), M = 4/(N* + 2M*). (13) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 559 Modelling of Lattices of Two-Dimensional Quasi-Crystals let complement the system (11) with sixth vector and divide all vectors by 2τ: 6 2 1 6 2 6 2 3 6 2 4 6 2 5 6 6 [1/2 1/2 0 1/2 1/2 0], [0 1 0 0 1/ 0], [1/2 1/2 0 1/2 1/2 0], [1/2 /2 0 1/2 1/2 0], [1/2 /2 0 1/2 1/2 0], [0 0 / 2 0 0 / 2 ]. = γ τ γτ = τ = γ τ γτ = γτ τ γτ τ = γτ τ γτ τ = λ γτ λ γτ q q q q q q (14) establishment of dimensionless coefficient λ for vector u6 is equiva- lent to the substitution of the six-dimensional cubic lattice by the or- thogonal non-cubic one. It is necessary to note that 6 6 6 6 6 1 2 3 4 5 3= = = = = − τq q q q q , * 6 3= λ − τq . (15) According to (10) and (14), the indices of random site of reciprocal decagonal lattice (n1 n2 n3 n4 n5 n6) can be expressed through the indices of reciprocal icosahedral (non-cubic) lattice (k1 k2 k3 k4 k5 k6) by next equation: 1 1 2 2 3 3 4 4 5 5 6 6 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 1 0 1 0 0 0 0 0 0 1 n k n k n k n k n k n k                         =                                 . (16) It can be easily shown that the sum of the first five indices ki derived from eq. (16) is equal to zero. Thus, the reciprocal decagonal lattice can be constructed by the pro- jection of six-dimensional orthogonal non-cubic lattice (which corre- sponds to distorted icosahedral lattice) onto the physical space. Addi- tionally, it is necessary to prohibit the projection of sites, in which the sum of the first five indices is not equal to zero. Considering (14), eqs. (3) and (5) can be written as |Q|2 = (n1 2 + n2 2 + n3 2 + n4 2 + n5 2 − n1n2 − n2n3 − n3n4 − n4n5 − n5n1) + + (n1n2 + n2n3 + n3n4 + n4n5 + n5n1 − n1n3 − n2n4 − − n3n5 − n4n1 − n5n2) τ + λ2n, (17) |Q2| = N* + M*τ + λ2L2, (18) where N = n6. 560 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Table 2. Site indices of the flat decagonal quasi-lattice based on five base vectors and corresponding indices N* and M*. The relation between the value of the parameter τN* − M* and the self-overlapping quantity in the model group D6 No. (n1 n2 n3 n4 n5) N* M* |Q| τN* − M* Quantity of self- overlaps of sites 1 (1 1 − 1 0 0) 5 −3 2 − τ 11.09 131 2 (1 0 1 0 0) 2 −1 τ − 1 4.24 538 3 (1 1 − 0 1 1 − ) 7 −4 7 4− τ 15.33 24 4 (1 0 0 0 0) 1 0 1 1.62 850 5 (1 1 − 0 0 0) 3 −1 3 − τ 5.85 424 6 (2 0 2 0 0) 8 −4 2τ − 2 16.94 5 7 (2 0 1 0 0) 5 −2 5 2− τ 10.09 167 8 (2 0 1 0 1) 4 −1 4 − τ 7.47 287 9 (1 1 0 0 0) 1 1 τ 0.62 1033 10 (2 1 − 1 0 1) 8 −3 8 3− τ 15.94 14 11 (1 0 1 − 0 0) 2 1 2 + τ 2.24 764 12 (2 1 − 0 0 0) 7 −2 7 2− τ 13.33 57 13 (2 0 0 0 0) 4 0 2 6.47 347 14 (1 1 1 − 0 0) 3 1 3 + τ 3.85 514 15 (1 1 1 − 0 1 − ) 5 0 5 8.09 259 16 (2 0 1 − 1 0) 7 −1 7 − τ 12.33 76 17 (2 1 0 0 0) 3 2 3 2+ τ 2.85 615 18 (2 0 1 − 0 1 − ) 8 −1 8 − τ 13.94 41 19 (1 1 0 1 − 0) 2 3 τ + 1 0.24 1018 20 (2 0 1 1 − 1 − ) 9 −1 9 − τ 15.56 15 21 (2 1 0 0 1 − ) 6 1 6 + τ 8.71 216 22 (2 0 0 1 − 0) 5 2 5 2+ τ 6.09 338 23 (2 1 − 1 − 0 0 ) 7 1 7 + τ 10.33 150 24 (3 0 0 0 0) 9 0 3 14.56 27 25 (1 1 0 1 − 1 − ) 3 4 3 4+ τ 0.85 944 26 (2 1 − 0 1 − 1) 8 1 8 + τ 11.94 80 27 (2 1 1 − 0 0) 5 3 5 3+ τ 5.09 405 28 (2 2 0 0 0) 4 4 2τ 2.47 623 ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 561 Modelling of Lattices of Two-Dimensional Quasi-Crystals Proceeding from the above, it can be proposed quadratic form for decagonal lattice: * * 2 2 * * 2 2 2 2 2 1 1 , . N M L N M L c ad a a c + τ + λ + τ = = + = λ (19) Using equations (4) and (18), it is possible to proceed from the six- indexes’ notation (n1 n2 n3 n4 n5 n6) to the three-indexes’ one (NML), which is more convenient in the case of indexing the XrD-patterns of polycrystalline samples. The values of N and M for the plain quasi-lat- tice are presented in Table 2 in ascending order of |Q|. equation (19) is formally identical to quadratic form for tetragonal lattice. 2.2. Intensity of Diffraction Reflexes In paper [1], it has been shown that, for icosahedral quasi-crystals, the intensity of diffraction reflexes is determined by the value of τ (τN − M) that is the distance between the site of hyper-lattice and its correspond- ing projection onto physical space. Moreover, the intensity increases with decreasing of this distance. In our case of two-dimensional decago- nal quasi-lattice, the distance from the site of six-dimensional lattice to physical space is determined by modulus of vector: Q⊥ = n1q1 ⊥ + n2q2 ⊥ + n3q3 ⊥ + n4q4 ⊥ + n5q5 ⊥, (20) where qi ⊥ are projections of six-dimensional vectors (14) onto ‘perpen- dicular’ space. Using the set of eqs. (14), it can be shown that the square value of Q⊥ modulus can be expressed through the same param- eters N* and M*, which determine the square value of vector modulus in physical space (5): |Q⊥| 2 = τ–3(N*τ + M*). (21) Therewith, the square value of six-dimensional vector modulus is equal to1.5 |Q6|2 = |Q|2 + |Q⊥| 2 (3 − τ) (N* + M*). (22) It occurs multiple overlapping of sites during modelling the two- dimensional decagonal lattice ac- cording to definite algorithm since various combinations of basis vec- tors can lead to the same result. Fig. 6. Correlation between the intensity of reflexes on electron diffraction pattern from decagonal Al–Ni–Co quasicrystal [26] and the quantity of self-overlaps of sites at the construction of the quasi-lat- tice according to algorithm no. 3 (group of sites D6) 562 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Computer analysis of constructed lattices allowed indicating distinct correlation between the value of (N*τ − M*) parameter, experimental intensity of reflexes and the number of overlaps (Fig. 6, Table 2). The revealed correlation evidences for correctness of the selection of six-di- mensional lattice basis vectors (14) that is in agreement with the data of paper [1] for icosahedral lattice. The selection of alternative basis, which projection onto the physical space also determines the vectors (6), may interrupt this correlation. For example, in paper [27], it has been proposed orthogonal basis in five-dimensional space: 5 1 0 0 0 0 5 2 1 1 3 3 5 3 2 2 1 1 5 4 3 3 4 4 5 5 4 4 2 2 [ 1/ 2], [ 1/ 2], [ 1/ 2], [ 1/ 2], [ 1/ 2], c s c s c s c s c s c s c s c s c s c s = = = = = q q q q q (23) where cr = cos (2r π/5), sr = sin (2r π/5), and the first two vectors compo- nents are referred to the physical space, while the rest ones are referred to ‘perpendicular’ space. In this case, correlation between the distance from the site of five-dimensional lattice to physical space and the inten- sity has not been observed. Comparing systems (14) and (23), we can propose the criterion for selection of decagonal-lattice basis vectors in the space with dimension- ality, which is higher than 3: the sum of five basis vectors has to be equal to zero. Correlation between the intensity of reflexes and the number of overlapping could be interpreted in the following way. basis sites of quasi-crystalline lattice are obtained from the projection of hyper-lat- tice sites ‘closely’ located to physical space. Moreover, according to eq. (26) and Table 2, (10100)- and (10000)-type sites are located at the same minimal distance from coordinate start in six-dimensional space. how- ever, the (10000)-type sites are ‘closer’ located to physical space and this determines their selection as the basis ones. The only one site of six-dimensional hyper-lattice, which located in the real (physical) space, is the origin of coordinate. This is necessary condition of aperiodicity of this hyper-lattice projection in any direction. In fact, the overlapping of geometric group shifted by certain vec- tor means ‘parallel transfer’ of physical space so that another site of hyper-lattice closely located to the physical space is turned out in the real space (this site corresponds to τn–2qi vector). Intensity of diffraction reflexes is determined by the distance from the hyper-lattice site to physical space [20]. Since that, indicated correlation of intensity can be ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 563 Modelling of Lattices of Two-Dimensional Quasi-Crystals interpreted in terms of probability for hyper-lattice site to be in projec- tion region at the ‘shifting’ of the physical space during the generation of sites groups. In this manner, within this algorithm, the sites located closer to the initial physical space are generated more frequently as compared to those located at higher distances. Therefore, multiple gen- eration of the same sites enables to get information on the intensity of appropriate diffraction reflexes. Figure 7 illustrates mutual orientation of the basis vectors projec- tions onto the physical qi and ‘perpendicular’ qi ⊥ spaces according to the set (14). Value of q1 + q2 = − τq4 type defines one of the shifting the group of sites during modelling process. It corresponds to ‘perpendicular’ shift- ing to q1 ⊥ + q2 ⊥ = − q4 ⊥/τ vector. It follows that the radii of sites groups in the model (algorithms nos. 2 and 3) in the physical and ‘perpendicular’ spaces are defined by equations: rn = 1 + 1 + τ +...+ τn–2, rn ⊥ = 1 + 1 + 1/τ +...+ 1/τn–2, (24) respectfully. The second equation in (24) shows that the radius of sites groups in ‘perpendicular’ space is limited: r ⊥n→∞ = 1 + (1 − 1/τ)–1 = 1 + τ2 = τ + 2. (25) It follows that only those sites of six-dimensional lattice, which are located at the distance not higher than τ + 2 from the physical space, are projected within discussed model. Then, during the construction sites’ groups of high orders, the density of its location will be limited due to finite size of projection region [21]. 3. Quasi-Periodic lattices with octagonal Symmetry let show that algorithm Dn = Dn –1 + {τn–2qi}Dn –1 proposed for decagonal quasi-crystals is appropriate for using to quasi-crystalline lattices with octagonal symmetry. Fig. 7. Mutual ori- entation of basis vectors in the physical (a) and ‘perpendicular’ (b) spaces 564 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov 3.1. Real Space The set of basis vectors has to be selected with two options, which are different in mutual orientation (Fig. 8): ( )1 1 0= +q i j , ( )2 2 2   = +     q i j , ( )3 0 1= +q i j , ( )4 2 2   = − −     q i j (26) or ( )1 i j= +q 1 0 , ( )2 2 2   = +     q i j , ( )3 2 2   = − −     q i j , ( )4 0 1= −q i j . (27) As we can see, there is some ambiguity in selection of basis vectors. Then, if we consider qi as reciprocal lattice vectors, the ambiguity in indexing of diffraction reflexes of octagonal quasi-crystals will exist. For example, let consider set (26) as the basis. Initial sites group O1 is constructed with (±q1, ±q2, ±q3, ±q4) set of qi vectors. Algorithm for modelling the lattice can be expressed in the form On = On –1 + {δs n–2qi} On –1, (28) where we use ‘silver ratio’ ( 1 2)sδ = + as parameter by analogy with ‘golden ratio’ τ [28]. one of the properties of number δs is that exponent va- lues for it can be expressed as δs n = Knδs + Kn –1; (29) here, Kn are Pell’s numbers (0; 1; 2; 5; 12; 29; 70; 169; Fig. 9. Model for construction of the octagonal quasi-lattice Fig. 8. options for selection of the ba - sis vectors for quasi- lattice, which pos- sesses the octago- nal sym metry ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 565 Modelling of Lattices of Two-Dimensional Quasi-Crystals 408; …), which satisfy to Kn = 2Kn –1 + Kn –2 condition [29]. It should be noted that there is a relation between the basis vectors: q1 + q2 + q3 = δsq2. (30) Then, using equations (29) and (30), we can write as following: 2 2 1 2 3 2 1 3 2 3 2 1 2 3 2 1 3 2 4 2 1 2 3 2 1 3 2 2 1 2 3 1 2 1 3 1 2 2 ( ) 2 ( ) 3 ; 5 ( ) 2 5 ( ) 7 ; 12 ( ) 5 12 ( ) 17 ; ... ... ... ( ) ( ) ( ) . s s s n s n n n n nK K K K K− − δ = + + + = + + δ = + + + = + + δ = + + + = + + δ = + + + = + + + q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (31) Thus, it is evidently that any site of On = On –1 + {δs n–2 qi} On –1 group can be expressed as linear combination of basis vectors in the form Q = n1q1 + n2q2 + n3q3 + n4q4. Figure 9 illustrates the example of applica- tion of specified algorithm for O4 sites group. It is important that algorithm (28) can be modified by substitution of one or few numeral coefficients (Fig. 10): (a) O2 = O1 + {qi} O1, …, On = On –1 + {δs n–2qi} On –1; (b) O2 = O1 + {qi} O1, O3 = O2 + {2qi} O2, …, On = On –1 + {δs n–3qi} On –1; (c) O2 = O1 + {qi} O1, 3 2 2{ 2 }iO O O= + q , …, On = On –1 + {δs n–3qi} On –1; (d) 2 1 1{ 2 }O O O= + qi , …, On = On –1 + {δs n–2qi} On –1. Fig. 10. Fragments of octagonal lattices for different algorithms, where O1 group is marked 566 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Also momentous is condition that specified coefficients are ex- pressed through relations between basis vectors similar to eq. (30). In contradistinction to known modelling methods [30–34], this proposed method for multiplying of sites groups allows to classify qua- si-crystalline structures. For exam- ple, Fig. 11 illustrates two-di men- sional quasi-periodic structure [35]. It is evident that this structure is in agreement with the mo -del shown in Fig. 10, d. Such structure ac- cording to numeric coefficients in algorithm can be expressed as the structure of O (δs − 1, δs n–2) type. The structures obtained with other structure algorithms (Fig. 10, a, b, c) can be denoted as O (1, δs n–2), O (1, 2, δs n–3) and O (1, δs − 1, δs n–2), respectively. It is easy to show that variation of algorithm consists in rearrangement of coefficients at qi. For instance, O2 = O1 + {δsqi} O1, O3 = O2 + {qi} O2, O4 = O3 + {2qi} O3, …, On = On –1 + {δs n–3qi} On –1 leads to construction the structure, which is the same as obtained with O2 = O1 + {qi} O1, O3 = O2 + {2qi} O2, …, On = On –1 + {δs n–3qi} On –1. That is why it is advi - sable to note the coefficients in notation of structural class in ascen - ding order. It is known [36, 37] that quasi-crystalline lattice can be represented as projection of periodic lattice in the space with dimensionality R onto space with dimensionality d. In the case of octagonal plain quasi-lattice, it can be proposed the projection of four-dimensional hyper-cubic lattice onto the plain. If the basis of four-dimensional lattice are represented as orthogonal vectors, [ ] [ ] 1 2 3 4 1 0 1 0 , 2 2 2 2 2 2 2 2 , 0 1 0 1 , 2 2 2 2 2 2 2 2 ,  = = −   = − = −  u u u u (32) then, the first two coordinates of each vector correspond to basis vec- tors. Two of rest coordinates correspond to the vectors 1 2 3 4 (1 0 ), ( ( 2 2) ( 2 2) ), (0 1 ), (( 2 2) ( 2 2) ), ⊥ ⊥ ⊥ ⊥ = + = − + = − = + q i j q i j q i j q i j (33) which are projection of set (32) onto ‘perpendicular’ space. Mutual ori- entation of basis vectors in ‘perpendicular’ space with preset basis in Fig. 11. Two-dimensional colloidal quasi- crystal [35] ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 567 Modelling of Lattices of Two-Dimensional Quasi-Crystals physical space is presented in Fig. 12. evidently, the vector of physical space q1 + q2 + q3 corresponds to the vector of ‘perpendicular’ space q1 ⊥ + q2 ⊥+ q3 ⊥, whose modulus has a minimal value for all combinations of three basis vectors. We show that algorithm (28) corresponds to the sites of four-dimen- sional hyper-cubic lattice, which are closely located to physical space. by this way, it will prove that proposed method and projected method are equivalent between each other. For this, it is enough to show that ‘the radius’ of sites group in ‘perpendicular’ space (maximal distance from sites of four-dimensional space to physical space) is finite. As seen from Fig. 12, the next equation is valid during execution of eq. (30): 1 2 3 2 2( 2 1) (1 )s ⊥ ⊥ ⊥ ⊥ ⊥+ + = − − = − δq q q q q . (34) We can show from eqs. (31) and (34) that boundary radii of sites’ groups rn→∞ and r ⊥n→∞ are equal to 2 2 1 n n s n r ∞ − →∞ = = + δ = ∞∑ , 2 1 2 11 2 1 1 1 2 2 21 n s n s n s r ∞ ⊥ − →∞ − = δ + = + δ = + = + = +∑ − δ . Thus, the distance from projected four-dimensional lattice to physical space does not exceed 2 2 2+ . hence, proposed method is quite correct. 3.2. Reciprocal Octagonal Lattice let us analyse the possibility of using the proposed model for reciprocal lattice of the octagonal quasi-crystals. We can reduce the square values of modules of vectors Q|| = n1q1 + + n2q2 + n3q3 + n4q4, Q⊥ = n1q1 ⊥ + n2q2 ⊥ + n3q3 ⊥ + n4q4 ⊥, and Q = n1u1 + n2u2 + + n3u3 + n4u4 (in physical, ‘perpendicular’, and four-dimensional spaces, respectively) to the form: ⊥ ⊥ = + + + + + + − = + + + − + + − = + = + + +   2 2 2 2 2 1 2 3 4 1 2 2 3 3 4 1 4 2 2 2 2 2 1 2 3 4 1 2 2 3 3 4 1 4 22 2 2 2 2 1 2 3 4 ( ) ( ) 2, ( ) ( ) 2, | | 2( ). n n n n n n n n n n n n n n n n n n n n n n n n n n n n Q Q Q Q Q Fig. 12. Mutual orientation of ba- sis vectors (32) at their projection to physical and ‘per- pendicular’ spa ces 568 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov 2 2 2 2 2 1 2 3 4 1 2 2 3 3 4 1 4 2 2 2 2 2 1 2 3 4 1 2 2 3 3 4 1 4 22 2 2 2 2 1 2 3 4 ( ) ( ) 2, ( ) ( ) 2, | | 2 ( ). n n n n n n n n n n n n n n n n n n n n n n n n n n n n ⊥ ⊥ = + + + + + + − = + + + − + + − = + = + + + Q Q Q Q Q   (35) Using denotations 2 2 2 2 1 2 3 4 1 2 2 3 3 4 1 4 1 2 2 3 3 4 1 4 ( ) ( ), ( ), N n n n n n n n n n n n n M n n n n n n n n = + + + − + + − = + + − (36) we can deduce |Q||| 2 = N + Mδs (37) that has similar form to equations for icosahedral (1) quasi-crystals [1] as well as for plain lattice of decagonal (5) quasi-crystals [21]. At the same time, the squared distance from site of four-dimensional lattice to its corresponding projection in physical space is defined by Nδs − M value: |Q⊥| 2 = (Nδs – M)/δs. (38) According to refs. [18, 20, 34, 36], the value of |Q⊥| 2 defines the intensity of diffraction reflexes. It is important that |Q⊥| 2 ∝ (Nτ – M) for the icosahedral and decagonal lattices. The translation of On−1 sites’ groups on δs n–2qi value corresponds to shifting its centres to positions of (n1n2n3n4) sites of the (1110)-, (2320)-, (5750)-, (1217120)-, …, (Kn, Kn + Kn−1, Kn, 0)-type according to eqs. (29) and (30). The substitution of these indices in eq. (36) gives the pairs of values N = K2 n + K2 n –1 and M = 2 (K2 n + Kn Kn –1): (1, 2); (5, 12); (29, 70); (169, 408); … . Thus, squared modulus values for shifting vectors of sites groups can be expressed throw the pairs of N and M numbers, which are neighbouring elements in Pell’s sequence. The corresponding pairs of numbers satisfy to the condition M/N < δs, which is necessary Fig. 13. The overlap of the groups of O4 sites on electron diffraction pattern from octagonal quasi-crystal of the Mn4(Al,Si) system oriented by its symmetry axis of the 8-th order along the electron beam (diffraction pattern adopted from paper [38]) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 569 Modelling of Lattices of Two-Dimensional Quasi-Crystals condition according to eq. (38). It is can be verified that the value of |Q⊥| 2 defined from eq. (38) is small for these numbers’ pairs as compared to any other numbers’ pairs. Table 3. Characteristics of some sites of O7 groups constructed according to algorithms (28) and (39) No. (n1 n2 n3 n4) N M |Q⊥| 2 Quantity of overlaps (28) Quantity of overlaps (39) 1 (1 −1 1 0) 5 −2 5.828 5 78 2 (−1 2 −1 0) 10 −4 11.657 11 3 (0 0 1 −1) 3 −1 3.414 12 108 4 (1 0 0 0) 1 0 1 43 223 5 (1 1 −1 1) 6 −2 6.828 48 6 (−1 1 1 −2) 11 −4 12.657 12 7 (2 −1 1 0) 9 −3 10.243 22 8 (0 2 −1 0) 7 −2 7.828 46 9 (0 0 2 −2) 12 −4 13.657 6 10 (1 0 1 0) 2 0 2 26 170 11 (1 0 1 −1) 3 0 3 25 150 12 (1 1 0 0) 1 1 0.586 48 224 13 (2 0 0 0) 4 0 4 14 119 14 (0 1 1 −2) 7 −1 7.414 54 15 (2 0 1 0) 5 0 5 10 96 16 (1 1 1 0) 1 2 0.172 73 257 17 (2 1 0 1) 6 0 6 4 78 18 (1 1 1 −1) 2 2 1.172 52 236 19 (0 2 1 −1) 5 1 4.586 16 112 20 (2 1 0 0) 3 2 2.172 28 162 21 (1 1 2 −1) 5 2 4.172 15 110 22 (2 1 1 0) 3 3 1.756 42 222 23 (1 2 1 0) 2 4 0.343 78 286 24 (2 2 −1 0) 7 2 6.172 61 25 (1 2 1 −1) 3 4 1.343 44 196 26 (2 2 0 0) 4 4 2.343 32 202 27 (1 3 0 0) 7 3 5.757 8 96 28 (2 1 2 0) 5 4 3.343 17 131 29 (2 1 2 −1) 6 4 4.343 20 131 30 (3 1 1 0) 7 4 5.343 5 81 31 (1 2 2 −1) 5 5 2.929 32 192 32 (2 2 1 0) 3 6 0.515 48 213 33 (1 3 1 0) 5 6 2.515 27 151 34 (2 2 1 −1) 3 7 0.101 104 332 35 (1 3 1 −1) 6 6 3.515 32 172 36 (2 3 0 0) 7 6 4.515 7 89 37 (2 2 2 0) 4 8 0.686 54 276 38 (3 1 2 −1) 9 6 6.515 3 63 570 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Figure 13, a illustrates the over- lapping of O4 sites group on the elec- tron diffraction pattern for octago- nal quasi-crystal of Mn4(Al,Si) system. evidently, the sites of con- structed lattice totally coincide with reflexes from diffraction pattern. Therewith, there are reflexes with low intensities, which have no corresponding site on the model (some of them are marked with the point in Fig. 13). Changing algorithm for construction of O2 = O1 + {qi} O1 into 2 1 1{ 2 }O O O= + qi (the next steps of algorithm remain unchanged) causes the appearance of additional sites, which coincide with marked reflexes (Fig. 13, b). Thus, diffrac- tion pattern for octagonal quasi-crystal Mn4(Al,Si) is related to O (δs − − 1, δs n–2), class by geometry. Such algorithm change corresponds to ex- tending of projection region in four-dimensional space, because 2 2 2 ( 2 1)nr ⊥ →∞ = + + − in this case. Table 3 presents the characteristics of some reciprocal lattice sites, which are the most closely located to coordinate start. These sites have been generated according to algorithms (28) and the following relation- ship: On = On –1 + {δs n–3qi} On –1. (39) As a result of construction of the octagonal quasi-lattices, using described algorithms (as well as in the case of the construction of de- cagonal quasi-lattices), there is a multiple mutual overlapping of the sites. The quantity of this overlaps for various algorithms is presented in the last two columns of Table 3. As shown, the correlation between overlapping quantity and |Q⊥| 2 value is observed for all proposed algo- rithms as well as for decagonal quasi-lattice (Table 2). Figure 14 shows indices and the quantity of overlaps (algorithm (28)) for appropriate reflexes on electron diffraction pattern for octago- nal quasi-crystal of Mn4(Al,Si) system. As seen, the quantity of overlaps is in a distinct agreement with intensity of diffraction reflexes. reflexes with the next values of indices (N, M) should have suffi- ciently high intensity according to obtained results: (1,0); (2,0); (1,1); (1,2); (2,4); (3,4); (3,6); (3,7); (4,8); (5,1); …. (40) reasoning from the three-dimensionality of octagonal quasi-crystals and its periodicity along 8th-order symmetry axis, inter-planar distanc- Fig. 14. Indices (N, M) and the quantity of site overlaps (algorithm (28), O7 group) for the corresponding reflex on electron diffraction pattern adopted from ref. [38] ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 571 Modelling of Lattices of Two-Dimensional Quasi-Crystals es can be calculated by the equation, which is similar to obtained one for decagonal quasi-crystals (19): 1/d2 = (N + Mδs)/a 2 + L2/c2; (41) here, a is spacing parameter of plain quasi-lattice, с is spacing param- eter along 8th-order symmetry axis. In practice, value of L index does not exceed 2 during indexing of XrD (x-ray diffraction) patterns. That is why the number of possible combinations of three indices (N, M, L) is rather small. It should be noted that reflexes of (0, 0, L)-type can also be observed on diffraction patterns in addition to reflexes of (N, M, L)-type (with N and M indices, which correspond to values of eq. (40)). Therefore, the indexing of XrD-patterns for octagonal quasi-crystals should be considered as simi- lar to indexing of crystalline materials, which belong to middle crystals’ systems. 4. Dodecagonal Quasi-Periodic lattices The formation of condensed matter with quasi-periodic long-range order and with the 12th-order symmetry axis has been established not only for metal systems (as like Ni–V [39], Cr–Ni [40], bi–Mn [41], Ta–Te [42], and Mn–Si–V [43]), but also for liquid crystals [44], colloidal solutions [45], and polymer systems [46]. Interpretation of the electron and x-ray diffraction patterns for dodecagonal quasi-crystals, as well as for all others, is also ambiguous because of indetermination of indexing of diffraction reflections. Such ambiguity is caused by inflation–deflation symmetry, which is native for quasi-crystals. As a result, the ratio of the absolute values of the reciprocal lattice vectors is expressed in terms of so-called scaling factor [20–23]. In electron diffraction studies of quasi-crystals, basis vectors are commonly match with reflections closest to the trace of the primary beam, which have a very low intensity, as a rule. For this reason, the minimal (basis) reciprocal lattice vectors (determined in diffraction ex- periments) are dependent on the experimental conditions. For construction of two-dimensional reciprocal quasi-lattice, let use algorithm in the form of recurrent equation: Dn = Dn –1 + {kn–2qi} Dn –1. (42) In this case, the k parameter (for dodecagonal lattice, let us denote it as t) was chosen from geometric interpretation of τ and δs numbers and from the condition that this numbers belong to Pisot numbers [5, 31, 34, 47] τ = 2 cos (2 π/10) and δs = 1 + 2 cos (2 π/8): k = t = 1 + 2 cos (2 π/12), (43) k = t1 = 2 + 2 cos (2 π/12). (44) 572 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov Parameters (43) and (44) have been used as scaling factors for a dode- cagonal lattice in papers [5, 31, 48]. As shown earlier, the application of algorithm (42) for the octagonal and decagonal quasi-crystals results to complete agreement between ob- tained quasi-lattices and experimental electron diffraction patterns. The implementation of algorithm (43) and (44) is illustrated in Fig. 15. The comparison of this lattice with the electron diffraction pattern of a do- decagonal quasi-crystal (Fig. 16) [26] shows the qualitative conformity between them. Fig. 16. Comparison of fragment of group D5 sites (a) (algorithm (42) and parameter (44)) with electron diffraction pattern from dodecagonal quasi-crystal (b) of Ta–Te system obtained in ref. [49] Fig. 15. Groups of sites obtained according to algorithm (42) and parameter (43) (D1 is an initial group of sites) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 573 Modelling of Lattices of Two-Dimensional Quasi-Crystals The usage of parameter (44) for the implementation of algorithm (42) leads to discontinuities of the lattice. The conformity of the mo- del quasi-lattice with above-speci- fied electron diffraction is obser- ved after the replacement of algo- rithm (1) with the algorithm pro- posed earlier in Ref. [27], which can be written in the form of the following recurrent relations (Fig. 17): D2  D1 {qi} D1, D3  D2 {2qi} D2, D4  D3 {t1qi} D3, D5  D4 {2t1qi} D4, D6  D5 {t2 1 qi} D5, D7  D6 {2t2 1 qi} D6. (45) The numbers t and t1 are the solutions for quadratic equations x2  2x  2 and x2  4x  1, respectively. It follows, hence, that any power of t and t1 can be expressed in terms of these numbers proper (e.g., t3  6t  4, t4  16t  12, … ; t3 1  15t  4, t4 1  56t  15, …). We must take into ac- count that basis vectors qi of a dodecagonal lattice relate as q1  q2  q3   tq2 and q1  2q2  q3  t1q2. Therefore, one can easy see that the posi- tions of all sites appearing in the realization of the above algorithms can be expressed in terms of a linear combination of qi vectors. Thus, each site of model quasi-lattices can be indexed. Let us compare the proposed method of recurrent multiplication of site groups with the projecting method. Since six basis vectors are used for a 2D-dodecagonal lattice, it is logically to use a six-dimensional hyper-cubic lattice. We require that the first two components of the coordinates of six-dimensional basis vectors represent the basis coordi- nates of a 2D-dodecagonal quasi-lattice. Then, we can use the unit or- thogonal basis vectors proposed in paper [31], Fig. 17. The overlap of group D5 (algo- rithm (45)) on the electron diffraction pattern [49] from quasi-crystal of Ta–Te system 1 2 3 4 5 6 (1, 0, 1, 0, 1 2 , 1 2) 3 , ( 3 2, 1 2, 3 2, 1 2, 1 2 , 1 2) 3 , ( 1 2, 3 2, 1 2, 3 2, 1 2 , 1 2) 3 , (0, 1, 0, 1, 1 2 , 1 2) 3 , ( 1 2, 3 2, 1 2, 3 2, 1 2 , 1 2) 3 , ( 3 2, 1 2, 3 2, 1 2, 1 2 , 1 2) 3 ,                    u u u u u u (46) 574 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov or suggest another set of vectors: 1 2 3 4 5 6 (1, 0, 1, 0, 1, 0) 3 , ( 3 2, 1 2, 3 2, 1 2, 0, 1) 3 , (1 2, 3 2, 1 2, 3 2, 1, 0) 3 , (0, 1, 0, 1, 0, 1) 3 , ( 1 2, 3 2, 1 2, 3 2, 1, 0) 3 , ( 3 2, 1 2, 3 2, 1 2, 0, 1) 3 . = = − − = − = − − = − − = − − u u u u u u (47) each vector in eq. (46) or (47) has two components corresponding to the two-dimensional physical (i.e., real) space and two components corre- sponding to the ‘perpendicular’ space. Therefore, we can write these vectors as ui = (qi ||; qi ⊥). For each site in the physical space, to correspond uniquely to a vector in the ‘perpendicular’ space, it is necessary that, for the linear combination of vectors that gives a zero vector (e.g., q1 || − q3 || + + q5 || = 0 and q2 || − q4 || + q6 || = 0 for vectors (48) given below), the correspond- ing combination of vectors qi ⊥ can be also equal to zero. As revealed, vectors (46) and (47) do not satisfy this requirement. Then, as the basis, we can choose vectors obtained from set (47) in the following manner: * 1 2 6 * 2 1 3 * 3 2 4 * 4 3 5 * 5 4 6 * 6 5 1 ( ) (1, 0, 1, 0, 0, 0), ( ) ( 3 2, 1 2, 3 2, 1 2, 0, 0), ( ) (1 2, 3 2, 1 2, 3 2, 0, 0), ( ) (0, 1, 0, 1, 0, 0), ( ) ( 1 2, 3 2, 1 2, 3 2, 0, 0), ( ) ( 3 2, 1 2, 3 2, 1 2 = − = − = + = = + = − − = + = = + = − − = − = − − u u u u u u u u u u u u u u u u u u , 0, 0). (48) omitting in these expressions (48) the last two coordinates, we can obtain the four-dimensional non-orthogonal basis of the lattice, which is Fig. 18. reflexes of electron diffraction pattern [49] corresponding to basis vec- tors according to the algorithm (45) (num- bered reflexes are described in Table 4) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 575 Modelling of Lattices of Two-Dimensional Quasi-Crystals analogous to the basis proposed in ref. [31]. It can be verified that the re ciprocal angles between the triples of four-dimensional vectors (u* 1, u* 3, u * 5) and (u* 2, u * 4, u * 6) are equal to 60° and 120°. At the same time, each vector from a triple is orthogonal to vectors from another set. It fol- lows, hence, that we can consider the given four-dimensional lattice as a combination of two 2D hexagonal sublattices, the spaces of which are mutually orthogonal. According to ref. [50], such a lattice belongs to a bi-isohexagonal orthogonal system. In the given basis, only four vectors are linearly independent. Therefore, two vectors (e.g., u* 5 and u* 6) can be omitted, writing the basis of the 4D lattice in the form: * * 1 2 * * 3 4 (1,0, 1,0); ( 3 2,1 2, 3 2,1 2); (1 2, 3 2, 1 2, 3 2); (0,1,0,1). = − = = − − = q q q q (49) evidently, if we put vectors (49) in correspondence to the basis group of sites in the proposed model, the sites generated in accordance with algorithm (45) will be projections of certain sites in the indicated four-dimensional lattice. It is easy to see that, in both cases of the octagonal and dodecagonal lattices during the multiplication of sites of a dodecagonal lattice in ac- cordance with algorithm (45), a correlation between the number of self- overlaps of sites and the intensity of the corresponding diffraction re- flections is also observed (Fig. 18, Table 4). Table 4. Indices and characteristics of the sites indicated in Fig. 18 No. (n1 n2 n3 n4) N*; M* N; M N1; M1 |Q|||2 |Q⊥|2 Quantity of site self-overlaps 1 (2 −2 0 1) 7; −4 11; −4 15; −4 0.072 13.928 37 2 (1 0 −1 1) 2; −1 3; −1 4; −1 0.268 3.732 182 3 (1 −1 1 0) 4; −2 6; −2 8; −2 0.536 7.464 118 4 (−1 1 2 −2) 6; −3 9; −3 12; −3 0.804 11.196 76 5 (1 0 0 0) 1; 0 1; 0 1; 0 1 1 245 6 (2 −1 0 1) 5; −2 7; −2 9; −2 1.536 8.464 144 7 (0 1 1 −1) 2; 0 2; 0 2; 0 2 2 266 8 (1 1 −1 1) 4; −1 5; −1 6; −1 2.268 5.732 194 9 (1 0 1 0) 3; 0 3; 0 3; 0 3 3 350 10 (1 1 0 0) 2; 1 1; 1 0; 1 3.732 0.268 326 11 (0 2 0 0) 4; 0 4; 0 4; 0 4 4 335 12 (2 0 0 1) 5; 0 5; 0 5; 0 5 5 328 13 (1 1 0 1) 4; 1 3; 1 2; 1 5.732 2.268 346 14 (−1 2 2 −1) 6; 0 6; 0 6; 0 6 6 292 15 (1 1 1 0) 4; 2 3; 2 1; 2 7.464 0.536 387 16 (1 1 1 1) 6; 3 3; 3 0; 3 11.196 0.804 440 576 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov let us put in correspondence the intensities of reflections to the distance from the sites of a 4D lattice to the physical space. each site of this lattice can be represented as Q = (Q||;Q⊥), where = ∑  4 1 i inQ q and 4 1 i in⊥ ⊥= ∑Q q . (50) Then, squared values of vectors Q|| and Q⊥ are as follow: 4 2 2 * * 1 3 2 4 1 2 2 3 3 41 | | ( ) 3 3in n n n n n n n n n n N M = Σ + + + + + = +    Q , (51) 4 2 2 * * 1 3 2 4 1 2 2 3 3 41 | | ( ) 3 3in n n n n n n n n n n N M⊥  = Σ + + − + + = −    Q . (52) The calculation of |Q⊥| value is based on eq. (52) for reflections in Fig. 18 shows that a correlation is observed between |Q⊥|, the number of self-overlaps of sites in the modelling, and the intensity of reflections (Table 1). The equation for calculation of |Q||| in both cases of using t and t1 parameters can be reduced to the form similar to eqs. (5) and (37): |Q|||2 = N + Mt, |Q|||2 = N1 + M1t1; (53) here, N = N* − M*, M = M*, N1 = N* − 2M*, M1 = M*. Within the value of |Q⊥| (in contrast of to those of icosahedral, octagonal, and dodecagonal Table 5. Indices of the sites (Fig. 18) with basis vectors corresponding to the reflexes located near the central spot No. (n1 n2 n3 n4)ch Nch; Mch |Q|| ch| 2 |Q⊥ ch| 2 1 (1 0 0 0) 1; 0 1 1 2 (1 1 0 0) 2; 1 3.732 0.268 3 (1 1 1 0) 4; 2 7.464 0.536 4 (1 2 1 −1) 6; 3 11.196 0.804 5 (2 2 0 −1) 7; 4 13.928 0.072 6 (2 2 1 0) 11; 6 21.392 0.608 7 (2 3 1 −1) 14; 8 27.856 0.144 8 (2 3 1 0) 16; 9 31.588 0.412 9 (2 3 2 0) 21; 12 41.785 0.215 10 (3 4 1 −1) 26; 15 51.981 0.019 11 (2 4 2 0) 28; 16 55.713 0.287 12 (3 4 2 0) 35; 20 69.641 0.359 13 (2 4 3 1) 40; 23 79.837 0.163 14 (1 4 4 1) 42; 24 83.569 0.431 15 (3 5 3 0) 52; 30 103.96 0.038 16 (2 5 5 2) 78; 45 155.94 0.058 ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 577 Modelling of Lattices of Two-Dimensional Quasi-Crystals quasi-lattices) cannot be reduced to the form |Q|2  (Nk M): |Q|  N 2M/t, |Q|  N1 M1/t1. (54) Basis vectors of reciprocal lattice are ascribed in [48] to low-inten- sity reflections that are closest to the trace of primary beam. In this case, the indices of the reflections and the magnitudes of corresponding vectors are recalculated by the following formulas: Nch  7N* 12M*, Mch  4N* 7M*, 2 2 2 ch | | | | | | (7 4 3) 7 4 3     Q Q Q    , 2 2 2 ch | | | | | | (7 4 3) 7 4 3       Q Q Q . The results of calculations for characteristics of reflexes (Fig. 18) obtained with Eq. (55) are presented in Table 5. Note that, with such indexing of intense reflections, the rounding of ch 3M value to the larger integer yields the value of Nch. The values of |Q ch| 2 and |Q ch| 2 are also determined only by Mch value, i.e.,  ch ch[ 3 1]N M ,    2 ch ch ch| [ 3 1] 3M MQ , 2 ch ch ch| | [ 3 1] 3M M   Q . According to [51, 52], low-intensive reflections in the vicinity of central spot on electron diffraction patterns are the results of multiple diffraction typical of quasi-crystals. At the same time, many authors take these reflections as those corresponding to basis vectors [48, 49]. In our model, the basis vectors of the reciprocal lattice correspond to reflections of type 5 (Fig. 18), which is in agreement with the results obtained in [51, 52]. Therefore, the proposed model of recurrent multi- plication of site groups takes into account the effect of multiple diffrac- tion and, at the same time, correctly maps the basis vectors on the dif- fraction pattern. The existence of correlation between the quantity of self-overlaps of sites and the intensity of diffraction reflections indi- cates that the procedure of recurrent construction of site groups is a certain analog of multiple diffraction processes. To pass from the reciprocal space to the real one, we write vectors (49) as * * * 4 2i D iaa q where a* 4D is a space parameter of four-dimen- sional reciprocal lattice. Using the condition a* i aj  δij, we can define the basis vectors of the direct lattice as follows 4 4 1 2 4 4 3 4 ( 3 2, 1 2, 3 2,1 2), (1,0,1,0), 2 2 (0,1,0, 1), ( 1 2, 3 2, 1 2, 3 2); 2 2 D D D D a a a a          a a a a (57) (56) (55) 578 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov here, * 4 42/( 3 )D Da a= is a lattice parameter. Denoting the interplanar distance corresponding the basis vector of reciprocal lattice as dq and considering that * 4 / 2 1/D qa d= , we obtain the equation for the param- eter of four-dimensional lattice and quasi-parameter a of four-dimen- sional quasi-lattice: 4 2/3D qa d= , 4 / 2 / 3D qa a d= = . (58) Then, to calculate the interplanar distances, we can use expression * * * * ( , ) 3 3 N M d a N M= + . (59) If there were detected reflexes corresponding to basis vectors, which are closely located to primary electron beam, then, equation remains similar to eq. (59): ch ch( , ) ch ch ch3 3N Md a N M= + , (60) where ch 17 4 3a a a t= + = . From the physical point of view, (N*, M*) indices are more correct, since they relate to the fundamental vectors of the reciprocal quasi- crystal lattice. however, indices (Nch, Mch) are more convenient, be- cause, if we know only one index from this pair, we can easily determine the second index and estimate the intensity of the corresponding reflec- tions (see eq. (56)). Thus, the dodecagonal system ‘falls out’ of the general relation |Q⊥|2 ∝ (Nk − M); this is observed for other existing types of quasi-crys- tals. however, it is still possible to indexing diffraction reflections us- Fig. 19. Comparison of atomic structure of baTio3 thin layer on platinum substrate (a) [53] and fragment of group D5 (b) (algorithm (61)) ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 579 Modelling of Lattices of Two-Dimensional Quasi-Crystals ing integers. At the same time, taking into account the periodicity of dodecagonal quasi-crystals along the 12th-order symmetry axis, all dif- fraction peaks on the powder diffraction patterns can be indexed with three indices, as for the octagonal and decagonal quasi-crystals. except the analysis of the diffraction pattern from quasi-crystalline materials, the description and classification of quasi-crystalline struc- ture is a complicated problem. We proposed above the method for de- scription of the variety of octagonal quasi-lattices. Such description is possible because we can change the coefficients of vectors in initial al- gorithm (42). For example, a change of even one coefficient changes the quasi-lattice without affecting its symmetry. For instance, the image of the atomic structure of a thin baTio3 layer on a platinum substrate was obtained in ref. [53]. We obtained almost the same geometry of the ar- rangement of sites (Fig. 19) using the following algorithm: 1 2 1 1 3 2 2 4 3 3 5 4 4 {2 } , {2 } , { } , {2 } . i i i i D D t D D D D D D t D D D t D −= + = + = + = + q q q q (61) Taking into account earlier proposed denotation of quasi-crystalline structures classes, the structure illustrated in Fig. 18 can be denoted as (2/ , 2, , 2 )D t t t . 5. Conclusions The method of modelling the quasi-periodic structures, which act as a geometric interpretation of Fibonacci-type sequences, is proposed. The correspondence between projection method for periodic lattices and the method of recurrent multiplication of basis sites’ group is ob- tained. The possibility of using only three indices (NML) for describing dif- fraction patterns for quasi-crystals with 10th-order, 8th-order, and 12th-order symmetry axis is proved. Using constructed algorithm for quasi-crystalline structures, we can directly obtain information about the intensity of diffraction reflexes. Described method of modelling is simpler as compared with projec- tion method. It enables to operate the coordinates of two-dimensional space unlike to coordinates with dimensionality, greater than three. reFereNCeS J.W1. . Cahn, D. Shechtman, and D. Gratias, J. Mater. Res., 1, No. 1: 30 (1986). https://doi.org/10.1557/JMr.1986.0013 А. Katz and м. Duneau, 2. Scr. Metall., 20, No. 9: 1211 (1986). https://doi.org/10.1016/0036-9748(86)90033-5 D. levine and P.J. Steinhardt, 3. Phys. Rev. B, 34, No. 2: 596 (1986). https://doi.org/10.1103/Physrevb.34.596 580 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov S4. . ebalard and F. Spaepen, J. Mater. Res., 4, No. 1: 39 (1989). https://doi.org/10.1557/JMr.1989.0039 J. Socolar, 5. Phys. Rev. B, 39, No. 15: 10519 (1989). https://doi.org/10.1103/Physrevb.39.10519 W. bogdanowicz, 6. Cryst. Res. Technol., 38, Nos. 3–5: 307 (2003). https://doi.org/10.1002/crat.200310036 W. bogdanowicz, 7. Cryst. Res. Technol., 40, Nos. 4–5: 488 (2005). https://doi.org/10.1002/crat.200410372 D.A. Shulyatev, 8. Crystallogr. Rep., 52, No. 6: 938 (2007). https://doi.org/10.1134/S106377450706003X W. Steurer, T. haibach, and b. zhang, 9. Acta Cryst. B, 49: 661 (1993). https://doi.org/10.1107/S0108767393001758 A. yamamoto and K. N. Ishihara, 10. Acta Cryst. A, 44: 707 (1988). https://doi.org/10.1107/S0108767388002958 P. Schall, M. Feuerbacher, and K. Urban, 11. Phys. Rev. B, 69, No. 13: 134105 (2004). https://doi.org/10.1103/Physrevb.69.134105 y. yan, r. Wang, J. Gui, and M. Dai, 12. Acta Cryst., 49: 435 (1993). https://doi.org/10.1107/S0108768192010127 X.b. liu, G. C. yang, J.F. Fan, and G.S. Song, 13. J. Mater. Sci. Lett., 22, No. 2: 103 (2003). https://doi.org/10.1023/A:1021850417176 X.b. liu, 14. J. Mater. Sci., 38, No. 5: 885 (2003). https://doi.org/10.1023/A:1022300503414 G. rosas, C. Angeles-Chavez, and r. Perez, 15. J. New Mater. Electrochem. Syst., 8: 149 (2005). M. bostr16. öm and S. hovmöller, Solid State Chem., 153, No. 2: 398 (2000). https://doi.org/10.1006/jssc.2000.8790 z.M. Mo and K.h. Kuo, 17. Mater. Sci. Eng., 294–296: 242 (2000). https://doi.org/10.1016/S0921-5093(00)01333-2 y. Vekilov and M. Chernikov, 18. Phys.-Usp., 53: 537 (2010). https://doi.org/10.3367/UFNr.0180.201006a.0561 J.S. Wu and K.h. Kuo, 19. Metall. Mater. Trans. A, 28, No. 3: 729 (1997). https://doi.org/10.1007/s11661-997-0059-9 V. elser, 20. Phys. Rev. B, 32, No. 8: 4892 (1985). https://doi.org/10.1103/Physrevb.32.4892 V.V. Girzhon, V.M. Kovalyova, o.V. Smolyakov, and M.I. zacharenko, 21. J. Non-Cryst. Solids, 358: 137 (2012). https://doi.org/10.1016/j.jnoncrysol.2011.09.017 V.V. Girzhon, o.V. Smolyakov, and I.V. Gayvoronsky, 22. Sposib Modelyuvannya Struktury Dodekagonalnykh Kvazykrystaliv: Patent No. 80699 MPK G09b 23/26 G09b 23/06 (Patent na korysnu model No. 11) (2012) (in Ukrainian). V.V. Girzhon, o.V. Smolyakov, and M.I. zakharenko, 23. ZhETF, 5, No. 11: 973 (2014). https://doi.org/10.7868/S0044451014110091 o.V. Smolyakov and V.V. Girzhon, 24. ZhETF, 3, No. 9: 521 (2017). https://doi.org/10.7868/S0044451017090097 V.V. Girzhon, o.V. Smolyakov, and I.V. Gayvoronsky, 25. Visnyk Lvivskogo Univer- sytetu. Seriia Fizychna, 54: 13 (2017). S. ritsch, 26. Philos. Mag. Lett., 74, No. 2: 99 (1996). https://doi.org/10.1080/095008396180452 S.A. ranganathan, e.A. lord, N.K. Mukhopadhyay, and A. Singh, 27. Acta Cryst. A, 63, No. 1: 1 (2007). https://doi.org/10.1107/S0108767306041298 ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 581 Modelling of Lattices of Two-Dimensional Quasi-Crystals J. Dubois, 28. Useful Quasicrystals (Singapore–london: World Scientific: 2005). M.A. bicknell, 29. The Fibonacci Quart, 13, No. 4: 345 (1975). S. burkov, 30. Phys. Rev. B, 47: 12325 (1993). https://doi.org/10.1103/Physrevb.47.12325 W. Steurer and S. Deloudi 31. Crystallography of Quasicrystals: Concepts, Methods and Structures (london–New york: Springer: 2009). r. Amman, b. Gr32. ünbaum, and G. Shephard, Discrete & Computational Geometry, 8, No. 1: 1 (1992). F.P.N. beenker, 33. Algebraic Theory of Non-Periodic Tilings of the Plane by Two Simple Building Blocks: a Square and a Rhombus (eindhoven: eindhoven Uni- versity of Technology: 1982). C. Janot, 34. Quasicrystals (oxford: oxford Clarendon Press: 1994). y. roichman and D. Grier, 35. Holographic Assembly of Quasicrystalline Photonic Heterostructures (New york: Department of Physics and Center for Soft Matter research: 2005). W. Steurer and S. Deloudi, 36. Crystallography of Quasicrystals: Concepts, Methods and Structures (london–New york: Springer: 2009). D. levine, P. Steinhardt, 37. Phys. Rev. Lett., 53: 2477 (1984). https://doi.org/10.1103/Physrevlett.53.2477 K. Kuo, 38. J. Non-Cryst. Solids, 117–118: 756 (1990). https://doi.org/10.1016/0022-3093(90)90639-4 h. Chen, D.X. li, and K.h. Kuo, 39. Phys. Rev. Lett., 60, No. 16: 1645 (1988). https://doi.org/10.1103/Physrevlett.60.1645 T. Ishimasa, h.-U. Nissen, and y. Fukano, 40. Phys. Rev. Lett., 55, No. 5: 511 (1985). https://doi.org/10.1103/Physrevlett.55.511 K. yoshida, T. yamada, and y. Taniguchi, 41. Acta Cryst. B, 45: 40 (1989). https://doi.org/10.1107/S0108768188011024 M. Uchida and S. horiuchi, 42. J. Appl. Cryst., 31: 634 (1998). https://doi.org/10.1107/S0021889898003008 h. Iga, M. Mihalkovic, and T. Ishimasa, 43. Philos. Mag., 91, Nos. 19–21: 2624 (2011). https://doi.org/10.1080/14786435.2010.508448 X. zeng, G. Ungar, y. liu, V. Percec, A.e. Dulcey, and J.K. hobbs, 44. Nature, 428: 157 (2004). https://doi.org/10.1038/nature02368 S. Fischer, A. exner, K. zielske, J. Perlich, S. Deloudi, W. Steurer, P. lindner, 45. and S. Förster, Proc. Natl. Acad. Sci. U.S.A., 108, No. 5: 1810 (2011). https://doi.org/10.1073/pnas.1008695108 K. hayashida, T. Dotera, and A. Takano, 46. Phys. Rev. Lett., 98, No. 19: 195502-1 (2007). https://doi.org/10.1103/Physrevlett.98.195502 A.e. Madison, 47. Struct. Chem., 26, No. 4: 923 (2015). https://doi.org/10.1007/s11224-014-0559-3 S. Iwami and T. Ishimasa, 48. Philos. Mag. Lett., 95, No. 4: 229 (2015). https://doi.org/10.1080/09500839.2015.1038332 M. Conrad, F. Krumeich, and b. harbrecht, 49. Angew. Chem. Int. Ed., 37, No. 10: 1383 (1998). https://doi.org/10.1002/(SICI)1521-3773(19980605)37:10<1383::AID- ANIe1383>3.0.Co;2-r b. Souvignier, 50. Acta Cryst. A, 59: 210 (2003). https://doi.org/10.1107/s0108767303004161 P.A. bancel, 51. Phys. Rev. Lett., 54, No. 22: 2422 (1985). https://doi.org/10.1103/Physrevlett.54.2422 582 ISSN 1608-1021. Prog. Phys. Met., 2019, Vol. 20, No. 4 V.V. Girzhon and O.V. Smolyakov F. G52. ähler, Quasicrystalline Materials. Proc. I.L.L. / Codest Workshop, Grenoble, 21–25 March 1988 (Singapore: World Scientific: 1988), p. 272. S. F53. örster, K. Meinel, r. hammer, M. Trautmann, and W. Widdra, Nature, 502: 215 (2013). https://doi.org/10.1038/nature12514 received June 14, 2019; in final version, october 10, 2019 В.В. Гіржон, О.В. Смоляков Запорізький національний університет, вул. жуковського, 66, 69600 Запоріжжя, Україна моДелюВАННя ґрАТНиць ДВоВимІрНих КВАЗиКриСТАлІВ Запропоновано спосіб моделювання квазиперіодичних структур, в основі якого лежить алґоритм, що є геометричною інтерпретацією числових послідовностей типу послідовности Фібоначчі. моделювання полягає у рекурентному розмно жен- ні базисних груп вузлів, які мають ротаційну симетрію 10, 8 або 12-го порядку. перевагою запропонованого способу є можливість оперувати координатами лише двовимірного простору, а не гіпотетичних просторів із вимірністю, вищою за три. показано відповідність між методою проєціювання періодичних ґратниць і методою рекурентного розмноження груп базисних вузлів. Встановлено, що шестивимірну обернену ґратницю для декагонального квазикристалу можна одержати з ортогональної шестивимірної ґратниці для ікосаедричного квазикри- сталу за допомогою зміни масштабу вздовж одного з базисних векторів і заборо- ни на проєціювання вузлів, для яких сума п’ятьох індексів (відповідних іншим базисним векторам) не дорівнює нулю. показано достатність використання лише трьох індексів для опису дифрактограм від квазикристалів з осями симетрії 10, 8 та 12-го порядків. ориґінальний алґоритм уможливлює безпосереднє одержан- ня інформації про інтенсивність дифракційних рефлексів за кількістю самона- кладань вузлів у процесі побудови обернених ґратниць квазикристалів. Ключові слова: квазіперіодичні структури, послідовність Фібоначчі, метод проє- ціювання, базисні вектори, ротаційна симетрія, обернена ґратниця. В.В. Гиржон, А.В. Смоляков Запорожский национальный университет, ул. жуковского, 66, 69600 Запорожье, Украина моДелироВАНие решёТоК ДВУмерНых КВАЗиКриСТАллоВ предложен способ моделирования квазипериодических структур, в основе кото- рого лежит алгоритм, являющийся геометрической интерпретацией числовых последовательностей типа последовательности Фибоначчи. моделирование за- ключается в рекуррентном размножении базисных групп узлов, имеющих ро- тационную симметрию 10, 8 или 12-го порядка. преимуществом предлагаемого способа является возможность оперировать координатами только двумерного пространства, а не гипотетических пространств с размерностью, большей трёх. показано соответствие между методом проецирования периодических решёток и методом рекуррентного размножения групп базисных узлов. Установлено, что шестимерную обратную решётку для декагональных квазикристаллов можно ISSN 1608-1021. Usp. Fiz. Met., 2019, Vol. 20, No. 4 583 Modelling of Lattices of Two-Dimensional Quasi-Crystals получить из ортогональной шестимерной решётки для икосаэдрических квази- кристаллов с помощью изменения масштаба вдоль одного из базисных векторов и запрета на проецирование узлов, для которых сумма пяти индексов (соответ- ствующих другим базисных векторам) не равна нулю. показана достаточность использования только трёх индексов для описания дифрактограм от квазикри- сталлов с осями симметрии 10, 8 и 12-го порядков. оригинальный алгоритм даёт возможность непосредственного получения информации об интенсивности дифракционных рефлексов по количеству самоналожений узлов в процессе по- строения обратных решёток квазикристаллов. Ключевые слова: квазипериодические структуры, последовательность Фибонач- чи, метод проецирования, базисные векторы, ротационная симметрия, обратная решётка.

Схожі ресурси