MODELING QUASI-LATTICE WITH OCTAGONAL SYMMETRY
V. V. Girzhon", O. V. Smolyakov"*,
Zaporizhzhya National University 69063, Zaporizhzhya, Ukraine
b Tains Shevchenko Kiev National University 01680, Kiev, Ukraine
M. I. Zakharenko
Received January 28, 2014
We prove the possibility to use the method of modeling of a quasi-lattice with octagonal symmetry similar to that proposed earlier for the decagonal quasicrystal. The method is based on the multiplication of the groups of basis sites according to specified rules. This model is shown to be equivalent to the method of the periodic lattice projection, but is simpler because it considers merely two-dimensional site groups. The application of the proposed modeling procedure to the reciprocal lattice of octagonal quasicrystals shows a fairly good matching with the electron diffraction pattern. Similarly to the decagonal quasicrystals, the possibility of three-index labeling of the diffraction reflections is exhibited in this case. Moreover, the ascertained ratio of indices provides information on the intensity of diffraction reflections.
DOI: 10.7868/'S0044451014110091
1. INTRODUCTION
The description of the quasicrystallino phase structure is a nontrivial problem of material science. This is a consequence of the impossibility to select a unit cell reflecting the overall structure of a lattice fl 5]. Different approaches and methods possessing both the virtues and shortcomings are commonly used for this. For example, we note the Ammann Breaker tiling, which is an eight-fold sibling of the more famous, fivefold Penrose rhombus tiling [6,7], and the Burkov method fl], which consists in structure constructing with overlapping clusters. Unfortunately, essential difficulties arise in indexing the diffraction patterns from such phases. In this case, it is necessary to use more than three basis vectors, and moreover, the number of vectors depends on a symmetry of the quasilattice. For example, in the case of icosahedral quasicrystals, six basis vectors could be used [8 10], whereas in the case of a decagonal quasilattice, the number of basis vectors is five or six [11]. Accordingly, more than three indices should be used for the diffraction reflection indexing. The feasible values of these indices are not quite obvious, because the limitations specified by the quasilattice symmetry should be held. In terms of the
E-mail: asmolyakov'fflmail.ru
projection method, which is often used for the modeling of quasi-lattices [12,13], this limitation is, evidently, equivalent to forbidding projection of the sites of a periodic hyper-lattice with the dimension higher than three, which are sufficiently far from the physical space. Thus, the complications related to both the identification and indexing of the reflection arise. It should be noted that the method of solving the above problem using only two indices (AT,M) is now well developed [14]. The indexing procedure of the diffraction reflections for decagonal quasicrystals using three indices was proposed in Ref. [15]. A similar procedure of indexing for other types of quasicrystalline phases (octagonal and dodecagonal) has not yet been considered. Therefore, the aim of this work is the generalization and extension of the approach developed previously for icosahedral and decagonal quasilattices to other types of quasicrystals, in particular, those possessing octagonal symmetry.
2. MODELING THE QUASI-LATTICE WITH OCTAGONAL SYMMETRY
The method of modeling a quasicrystalline lattice with a tenfold axis of symmetry has been proposed and described in detail in Ref. [15]. This method consists in the multiplication of geometric groups (sites) according
qs
qi
U Q4
Fig. 1. The variants of choosing the basis vectors
to one of the three possible algorithms. These algorithms could be formally expressed as Dn = Dn_i + + {rn-2qi}Dn_2, Dn = Dn_2 + {rn-2qi}Dn_u and Dn = Dn-i + {rn~2ç\i}Dn-i. Here, Dn is a geometric group of sites of n; the order q^ is a set of (±qi, ±q2, ±q3, ±q4, ±qs) vectors, which are the vertices of a regular decagon* the expression Dn — Dn_i + + {rn~2(\i}Dn-i corresponds to adding the geometric groups shifted by the rn_2q^ vectors to a preceding geometric group; and r = 2cos(7r/5) = (1 + \/5 )/2 is the irrational number expressing the so-called "golden ratio".
We show that this algorithm is applicable to qua-sicrystalline lattices of the octagonal symmetry. In this case, the system of basis vectors can be specified by two methods differing by the mutual orientation of four basis vectors (Fig. 1):
and
qi = (li + 0j), q2= __i+__jl
q3 = (0i + lj), q4 = I
qi = (li + 0j), q2= __i+__jl
(1)
(2)
qs
2"i+ j > q4 = (oi-ij).
Fig. 2. Illustration of the geometry groups, construction in the case of an octagonal lattice
02 = 01+{qi}01, On=On-1+{S^~zcii}On-l. (3)
Here, we use the irrational number Ss = 1 + \/2, known as "silver ratio", as a counterpart for the "golden ratio" r [16]. One of the features of the silver ratio is that it allows expressing the powers of Ss in the form
6? = Kn6. + Kn-1,
(4)
where Kn are the Pell's numbers (0, 1; 2; 5; 12; 29; 70; 169; 408; ... ) satisfying the condition Kn = 2Kn-i + + Kn_2 [17].
We emphasize that the following relation between the basis vectors (1) exists:
qi + q2 + q3 = ¿sq2-With Eqs. (4) and (5), we can write
(5)
<^q2 = Kn{ qi + q2 + q3) + i^n-iq2 =
= Ên{ qi + q 3)(Kn + Ên-i)q2. (6)
Hence, an ambiguity in the selection of a basis arises. Accordingly, if q^ are considered as the reciprocal lattice vectors, then indexing the diffraction reflections for an octagonal quasicrystal is also ambiguous. For defi-niteness, system (1) is adopted as a basis.
If the system of q^ vectors (±qi, ±q2, iqu) is chosen as the initial geometric group Oi, it is possible to express the algorithm of lattice construction in the form
Thus, any site of the On = On_i + {6™ 2qi}On_i geometric group can, evidently, be expressed as a linear combination of the basis vectors in the form Q = = niqi + ri2q2 + 7i3q3 + 714q4. The application of this algorithm to the O4 geometric group is illustrated in Fig. 2.
We note that algorithm (3) of the construction of a quasi-periodic lattice with an eightfold symmetry axis can be modified by substituting one or several
Fig. 3. Fragments of the octagonal lattices constructed according to different algorithms (the Oi group is distinguished): a) O2 = Oi + {q*}Oi, On = On-1 + {^-\i}On-i\ b) 02 = Oi + {q*}Oi, Os = 02 + {2qJ02, On = On-i + №"3qi}On_i; c) 02 = Oi + {q*}Oi, Os = 02 + {>/2q<}02, On = On-1 + 3q*}On-i;
d) 02 = Oi + {V^q<}Oi, On = On-i + {^~\i}On-i
Fig. 4. Two-dimensional colloidal quasicrystals organized with holographic optical traps [18]
numerical coefficients (Fig. 3). It is important that this coefficient is expressed in terms of a relation between the basis vectors similar to Eq. (5). In contrast to the known methods of modeling [1, 5-7,11,13], this method for multiplying groups of nodes allows classifying the quasicrystalline structures. For example, a two-dimensional dielectric quasicrystalline he-terostructure is shown in Fig. 4 [18]. It is evident that this structure corresponds to the model shown in Fig. 3d. This structure can be assigned to type 0(6S — 1 according to the numerical coefficients
involved in the algorithm. The structures obtained for other algorithms (Fig. 3a,b,c) can be denoted as 0(M?-2), 0(1,2,¿J"3), and 0(1,- M?"3)- It is easily seen that the algorithm changes; for example, the coefficients in 02 = Oi + {¿sq*}Oi, 03 = 02 + {q¿}02,
o4 = o3 + {2qjo3, and on = on_i + {¿r3qJon_i
can be reduced to the construction of structures such as 02 = Oi + {qJOi, 03 = 02 + {2qJ02, and On = = On_i+{(Jr3qi}°n-i. Hence, it is advisable to write the coefficients in the notation for the structural class in ascending order.
. q'J / q|
q"
Fig. 5. Mutual orientation of the basis vector projections in the physical and perpendicular spaces
It is known [5,19] that a quasicrystallino lattice can bo represented in terms of the projection of a periodic lattice in a space of dimension R onto the space of a lower dimension d. In the case of an octagonal planar lattice, the projection of the four-dimensional cubic lattice onto the plane can be proposed. If the basis of the four-dimensional lattice is represented in a form of orthogonal vectors
111 = [1010], u2 =
u3 = [010-1], u4 =
(7)
then the first two coordinates of each vectors correspond to basis vectors (1). The other two coordinates correspond to the vectors
qjL = (li + 0j)1 =
■1+—J
(8)
q^- = (0i-lj), qf=l_Yi+Yj
which are the projections of system (7) onto a perpendicular space. Mutual orientation of the basis vectors in the perpendicular space for the selected basis (1) in the physical space is presented in Fig. 5. Evidently, the vector qi + q2 + q:> in the physical space corresponds to the vector q^ + q.f + q.f in the perpendicular space. Moreover, the modulus of the latter vector is minimal for the random combination of three basis vectors.
We show that algorithm (3) corresponds to the projection of sites of the four-dimensional cubic lattice that are located close to the physical space, thereby proving the equivalence of the proposed method and the projection method. For this, it is sufficient to show that the radius of the On geometric group in the perpendicular space r^ (the maximal distance of the sites of the
four-dimensional lattice to the physical space) is a finite quantity. As is clearly seen from Fig. 5, the validity of the equality
qi +q2 =
i)q2=-fq2 (9)
directly follows from Eq. (5).
With Eqs. (6) and (9), it can be easily shown that the ultimate radii of the geometric groups r„^oo and
= i + 5>r2 = oc
n=2 oo
= i+£en = i+
(10)
n=2
l^1 2+ 2 •
Therefore, the distance of the sites of the four-dimensional lattice to the physical space does not exceed 2 + s/2/2. Hence, the proposed algorithm is quite valid.
3. MODELING THE RECIPROCAL OCTAGONAL LATTI
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