GEOMETRICAL MODEL OF ICOSAHEDRAL QUASICRYSTAL:
CONSTRUCTION OF QUASIPERIODIC LATTICE BY MEANS
OF SYMMETRY OPERATIONS
A.A. Polyakov
South Ural State University, Chelyabinsk, Russia
A method for construction in three dimensions of quasiperiodic structure with
icosahedral symmetry of icosahedra of two types is offered. Main feature of the
method is an application of symmetry operations (action of planes and axes of
symmetry). Also deflation rules for the structure has been formulated. The icosahedra
linear size of one type are in  times more the size of the others (= 2 cos 36º = 1.618
— golden mean). All the icosahedra have the same orientation, neighbouring
icosahedra of an identical type share an edge, and the icosahedra of a different type
can touch each other by vertices. A notion of a compound star – convexo-concave
dodecahedron is used for the not conflicting description of the structure. The nucleus
of this star is a compound icosahedron, constructed as 13-atomic Mackey's
icosahedron. The structure consists of the compound stars, and the petals of these stars
can be crossed. Each icosahedron in the field of crossing belongs simultaneously to
the several stars. The small to large amount ratio of icosahedra in this structure is
close to 3 : 1 that corresponds to 81 % of small icosahedra.
INTRODUCTION
Authors of discovery of quasicrystals (1) have suggested structural model of
icosahedral quasicrystals in which one size regular icosahedra are connected. All the
icosahedra are oriented uniformly and can share the edges with neighbours. In eighties
(2) similar models named "icosahedral glass" were used in which icosahedra are
connected by vertices, by sides also. Nowadays popular geometrical models are based
on Penrose tilings in a plane (3), and in three dimensions (4).
Diffraction investigations of icosahedral quasicrystals structure (5) had shown
existence of icosahedral atomic clusters. The structures was geometrically described
in two ways: 1) clusters was built by means of atomic decoration of three dimensional
Penrose tiles; 2) rules of cluster deflation was formulated on basis of the cluster
strucure. The structure obtained by cluster deflation has pores and additional rules for
filling the pores are necessary.
So there are two main geometrical approaches to building regular
quasiperiodic structures: tiles — closely adjoin structural elements — and clusters
with elements, having space between each other. Recently a new approach (6) to
constructing quasiperiodic lattice has been offered. It was offered to exploit one sort
of units (decagons) in a plane, which can both touch and overlap each other. After the
deflation the overlapped decagons does not conflict with each other. The property of
building units coincidence in overlapped area after deflation we shall name as "mutual
transparency".
We have offered a method of building quasiperiodic lattice in a plane by
means of symmetry operation (7). The construction process is accompanied by
forming of mutually overlapped clusters, e.i. clusters having property of mutual
transparency.
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PROPERTIES OF THE MODEL
The goal of this paper is to present new approach to geometric constructing of
quasiperiodic icosahedral structure. The main construction elements are regular
icosahedra of two types - i and i, linear size of i in times greater than i size(2
cos 36° = 1.618 — golden mean). All the icosahedra are oriented equally, the same
Figure 1. Composite icosahedron i(I) and stellated icosahedron Z.
icosahedra can share an edge and the ones of different type can touch each other by
vertices. Composite 13-element icosahedron i(I) is a basis cluster which consists of
icosahedron i located at the cluster's centre (further — b0 structure element) and of
twelve i (а0 elements), which are touch all the vertices of the b0 element. Each
icosahedron а0 has five common edges with neighbour icosahedra — а0 elements.
The deflation of such clusters will bring to pores formation. How it can be
avoided? Let us look at cavity in i(I), it has a form of stellated icosahedron (convexoconcave dodecahedron) (Fig.1). If in deflation process one should changed the
icosahedra not by composite icosahedra, but by composite stellated icosahedra,
having nucleus in form of i(I), then crossing petals of these stars would densely fill
up all the space and pores would not appear.
The composite stellated icosahedron Z(I) consists of the following parts
(Fig.2):
a) nucleus — i(I), consisting of b0 and 12 а0 elements;
b) 30 b1 elements of structure — iicosahedra, edges of which touches the edges
of i(I), i.e. vertices of b1 touches vertices of two а0;
c) 20 a1 elements — i, its vertices are coincide with vertices of three b1
elements;
d) 30 b2 elements — i, touching vertices of two a1 elements.
Let us consider location of the structure elements. The centres of а0 are
situated at the points Ri = ( + 1) ei, here ei — 12 vectors, pointed from centre to all
vertices of i icosahedron — b0 element. Location of b1 is correspondent to vectors
rb1 = Ri + Rj, where Ri, Rj are neighbour vectors similar to shown above. Situation of
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Figure 2. Composite stellated icosahedron Z(I): 1 — а0 structural units, 2 — а0 and
b1, 3 — а0, b1 and a1, 4 — а0, b1, a1 and b2 structural units, 5 — part of Z(I).
a1 are described by vectors ra1 = Ri + Rj + Rk, where Ri, Rj, Rk are neighbours and
vector rb2 = Ri + Rj + Rk + Rl corresponds to b2 elements, where vectors Ri, Rj, Rk and
Rj, Rk, Rl are neighbours.
In a composite star Z(I), the а elements are the i icosahedra and the elements
of a b type are absent. In a cluster Z(II), the a elements are the stars Z(I), the
elements of a b type correspond to the iicosahedra. The stars nuclei sizes increase in
times in a sequence i, i, Z(I), Z(I), Z(II), Z(II), …
CONSTRUCTING BY MEANS OF SYMMETRY OPERATIONS
The approach to constructing quasiperiodic lattice in a plane by means of symmetry
operations (7) may be used for creation of quasiperiodic structure in three dimensions.
Let us build the Z(I):
1. Twenty mirror planes do simultaneously on central star — Z, all the planes touch
the outer edges of the star (Fig.3). As a result we shall get 20 Z star — b1 elements.
2. The b0, b1 petals surround cavities having form of i icosahedra — а0 nuclei.
Symmetry axes of the b0 and b1 cross in the а0, a1 centres. We shall guess that in these
points Z — а0 and a1 elements will be created.
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Figure 3. Action of mirror planes on a b0 structure element.
3. Symmetry axes and mirror planes of a1 pairs cross in points, which correspond to
b2 elements. Also we shall guess, that in these points Z stars — b2 elements are
created.
So there is complete composite Z(I) cluster, consisting of 32 i and 61 i. If i, i to
replace by Z, Z stars then Z(I) would have form of stellated icosahedron plus some
extra volume, which occupy petals of Z, Z.
The building of Z(I) cluster by means of symmetry operations from Z we
shall call as "opening of a star" on the analogy with two dimension variant (7). The
further growth (Z(I) -> Z(III)) is like previous one but the operation "closing of the
star" will appear:
1. By the action of mirror planes to b0 element (Z(I)) the b1 will appear.
2. The b0, b1 petals surround cavities in form of nuclei of Z(II) — а0 element.
For the fill of the cavities the "closing of the star" is used, which operates in
opposite direction with "opening of a star". The Z is formed at а0, a1
centres by the operation "closing of the star".
3. The b2 element will be formed in the way analogues to the formation of b2
in Z(I) star.
It should be noted that the а0 forming is limited by the mirror planes which
divide pairs of touching а0 icosahedra nuclei, i.e. the growth of each а0 star petals will
be limited by the planes.
Deflation rules for discussing structure have been formulated. The Z(N) star
consists of Z(N – 1), Z(N – 1) stars and the Z(N) star consists of Z(N – 1), Z(N –
2) stars. The deflation is simply increasing of number N by one. There no problem in
replacement Z(N) by Z(N+1) and Z (N) by Z (N+1) for all elements of structure but
the а0. The deflation of composite star, which touch neighbour by an edge of its
nucleus, have been changed: mirror plane, dividing neighbours, cuts off all the
element's icosahedra, lying on opposite side from the nucleus of the element. In all the
other cases intersection of composite star petals does not bring to inconsistency: in the
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region of intersection the icosahedra from different stars coincide (property of mutual
transparency).
In order to estimate composition, the recursion formulae for icosahedra
counting in section of the structure has been derived. It is found that amount ratio of i
to i in section N1:N = 2:1. The section of i corresponds to the layer, in which the i
icosahedra centres are situated, its thickness in  times greater than the i section layer
thickness. Because of it, amount of i icosahedra in volume is increasing in  times,
i.e. N1:N = 3:1 or 81% of i icosahedra.
Our approach is related to "icosahedra glass" (2), but the first is completely
regular and is cluster one. The constructing by symmetry operations is a novel
method, it allow to confirm consistency of obtained quasiperiodic structures. The two
types icosahedra, which sizes are different in  times, has not been used for
constructing of quasiperiodic lattices earlier, at the same time such icosahedra
arrangement of atoms were observed in experimental investigations of approximants
(8). The building with help of two types icosahedra is very flexible, so modelling of
periodic lattice with twinnings, having a global icosahedral symmetry, can be
performed also.
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