Non-locally derivable sublattices in quasi-lattices

Journal of Alloys and Compounds, 209 (1994) 139-143 JALCOM 1041

139

Non-locally derivable sublattices in quasi-lattices R . Lfick* a n d K. L u National Key Laboratory for Rapidly Solidified Alloys, Institute of Metal Research, Chinese Academy of Science, Wenhua Road 72, Shenyang 110015 (China)

(Received November 5, 1993)

Abstract The concept of sublattices in quasi-lattices is applied to the two-layer structure of decagonal quasi-crystals such as A165Cu20C015and m17oNi15Co15.The Penrose pentagon pattern is chosen as the parent lattice for this procedure. This application shows that non-locally derivable sublattices and non-locally derivable patterns play an important role in the crystallography of ordered quasi-crystals. The relationships between the sublattices in question are discussed.

1. Introduction The idea of sublattices of quasi-lattices was presented in a previous paper [1]. Sublattices are generally formed by a subset of sites of the parent lattice; the same holds if quasi-lattices are considered. If the quasi-lattices are described by non-periodic patterns, such as Penrose patterns, the sublattices are formed by a subset of vertices. Several different procedures to derive sublattices from a parent non-periodic pattern are listed in ref. 1. There also may be equivalent sublattices not sharing any vertex. It is possible that some equivalent sublattices make up all of the parent lattice. The sublattice can be defined by a sublattice in high-dimensional space or by special atomic surfaces in the complementary space [2]. If points other than lattice sites are taken into account in deriving new lattices, the new lattices will not be sublattices, but they may have a strong structural relationship. A similar relationship exists between parent non-periodic tilings and tilings composed of a subset of vertices of the parent tiling and other sites determined by decorations of the parent tiles. It is the aim of this paper to give an example for this purpose. The present paper will be organized as follows. After this introductory description of a quasisublattice, locally and non-locally derivable patterns will be distinguished in Section 2. Section 3 gives a description of a two-layer decagonal structure based on the sublattice concept as an example, which is closely *On leave from Max-Planck-Institutfor Metallforschung,Institut fiir Werkstoffwissenschaft,Seestrasse 75, D-70174 Stuttgart, Germany.

related to experimentally determined structures. In Section 4, the ordering of the decagonal structure is explained as an example of non-locally derivable sublattices, and the corresponding atomic surfaces are demonstrated in Section 5. In Section 6, tilings with other symmetries are discussed and a remark on phasons in ordered quasi-crystals is made.

2. Locally and non-locally derivable patterns The derivation of a non-periodic pattern from another pattern is an interesting, often used procedure (see, for example, ref. 3). If this derivation can be carried out locally, i.e. with the knowledge of only a small surrounding patch within a limiting boundary, the structural relationship is unique and usually is simple. If the procedure of derivation can be carried out in both directions in this simple manner, the patterns are called mutually locally derivable. Baake et al. [4] introduced a new kind of class. In such a class, all the patterns are mutually locally derivable, which is called an MLD class. The most famous MLD class is formed by the Penrose patterns, i.e. the pentagon pattern (P1 [3]), the dart-and-kite pattern (P2), the rhomb pattern (P3) and the corresponding Ammann bar grid. Since the inflation patterns as well as the deflation patterns are mutually derivable, there can be uncountable different patterns in an MLD class. Naturally, there are sublattices which are locally derivable from the parent lattice. For example, these are sublattices defined by special vertex configurations. Hence, the inflation patterns are locally derivable sublattices.

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140

R. Lack, K. Lu I Non-locally derivable quasi-sublattices

The description of ordered quasi-crystals is analogous to the description of periodic crystals with a chemical long-range order. For the description of ordered quasicrystals, two (or more) kinds of atoms as well as the structural vacancies generally have to be taken into account. In a perfectly ordered quasi-lattice, each species occupies its own sublattice. The determination of the parent lattice and of its sublattices are required. Assuming that the parent lattice can be described by a non-periodic tiling, the sublattices may be demonstrated by several methods [1], however, the sublattice is not necessarily locally derivable [4]. Therefore, non-local procedures are required. The present paper will demonstrate these relationships using the recently determined crystallographic structure of two-layer decagonal quasi-crystals.

1

1

\l 2\

I'

\ 2 ~ 2, ,

-7

/

I, / ,,5"~X2

I/'~4XI XI/4~I/!~ /4~ I 2 z/I

,2

l,

2

3. Structure of the two-layer decagonai phases Steurer and Kuo [5] and Steurer [6] have described the structure of decagonal A165Cu2oCo15, applying the Patterson function method. Using the maximum entropy method, Steurer et al. [7] determined the structure of decagonal A17oNi15Co15. The structures consist of two equivalent layers. It has been reported [5] that, in both

Fig. 1. Patch of the Penrose pentagon pattern. Five different equivalent subsets of vertices are marked by numbers.

layers, the transition metal atoms form a sublattice. For the present analysis, it is assumed that this is the pentagon pattern; if another pattern is assumed, the

Fig. 2. (a) "Tie-and-navette" pattern and (b) "Tie-and-navette" pattern with two superimposed pentagon tilings forming sublattices.

141

R. Liick, K. Lu / Non-locally derivable quasi-sublattices

arguments are the same but the quantitative description would be slightly different. In a previous structural proposal on a decagonal structure [8], it was pointed out that the vertices of the Penrose pentagon pattern can be divided into five subsets. These subsets are indicated by numbers in Fig. 1. The subsets form sublattices and the pattern of these sublattices is composed of Robinson's golden triangles [1]; however, this pattern differs from Robinson's [1-3]. If the sublattices are distributed on five equivalent layers, a decagonal structure related to the icosahedral rhombohedral tiling is constructed [8]. We searched for a relationship between the twolayer structure of A165Cu20Co15 and the old five-layer proposal. This relationship was found in the following manner. Layer 1 is generally omitted - only some remnant densities are found at special sites [4-6]. Layers 2 and 3 as well as layers 4 and 5 condense into one layer; the corresponding tiling was described in a previous paper [1] before the determination of this structure, and it was called a "tie-and-navette" pattern. It is composed of two tiles shaped as a long bow-tie and as a navette (French expression for shuttle, also used for ceramic tiles) respectively. The vertices of this pattern correspond closely to the density of one layer determined by Steurer and Kuo [5]. For comparison, the "tie-and-navette" pattern is depicted in Fig. 2(a). It is obvious that the density given by Steurer and Kuo [5] is a mean structure - it is not very probable that both vertices with the short distance on the "ties" are simultaneously occupied by atoms. In the density maps, some split positions are found. In the paper by Steurcr and Kuo [5], both layers exhibited high densities, which were attributed to transition metal atoms. As mentioned above, these transition metal atoms might form a pentagonal pattern in both layers. Therefore, we suppose that there is at least one sublattice of the "tie-and-navette" pattern identical to the Penrose pentagon pattern. In fact, there are two (see Fig. 2(b)). In the next section, the relationship between this Penrose pentagon sublattice and the parent Penrose pentagon pattern will be discussed.

An extensive description of the substitution matrices of the inflation--deflation procedure of self-similar tilings was given in a previous paper [9]. The present matrix describes the composition of the units of the inflated generation by the units of the preceding generation. A power series of the matrix describes higher generations. Generally, the product of the two eigenvalues A1 and A2 of each of the matrices is + 1 or - 1 in the patterns, which can be uniquely inflated. There are other sublattices which also form the pentagon pattern, but their inflation factor is not a power of ~'. If this procedure is characterized by a matrix, the product of its eigenvalues will not be + 1. There are two different types of tiling with this property [10]. If IAII> 1 and IA21> 1, a non-Piso-type tiling is obtained; however, if IAl[> 1 and 0<121 <1, a Piso-type tiling is obtained. Usually only the Piso-type tiling is regarded as a quasilattice. The non-Piso-type tiling has been investigated recently by different authors. Figure 3(a) shows a composition of several inflation factors. One diagonal corresponds to multiplication by ~- and the other to multiplication by 51=r. The inflation by 51/21"2=4";'+ 1 of the matrix

/

1

=][+2

][2 =][+I

/

][2~ =3-[-+I

i_2.5

=5][+5

][3 =2-["'1 (o)

=Z~][÷3

I

=10][÷5

IJ numbers

=20T+15

3

2Jt~.numbers 4. Novel self-similar sublattice of the pentagon pattern

An inflation procedure is usually described by the inflation factor A, which is given for the Penrose MLD class by A = r = (5 lz2+ 1)/2. r and - ~- 1 are the eigenvalues of the matrix

I1 (b) Fig. 3. Genealogy of different inflation procedures of the Penrose pentagon pattern: (a) factors; (b) matrices. The scaling factor and the corresponding matrices are indicated. The steps of procedures are given together with arrows.

142

R. Lack, K~ Lu / Non-locally derivable quasi-sublattices

exists in at least five different patterns. Therefore, this inflation pattern is not locally derivable and, according to the eigenvalues, it is of the Piso-type. Figure 3(b) shows the matrices corresponding to Fig. 3(a). The components of the matrices increase according to the Fibonacci and Lucas series, as indicated in Fig. 3. Inflation by 5~/2z3=4~-+3 of the matrix

also exists. There are 25 different equivalent pentagon patterns, which do not share any vertex. These can be deflated to another scale by a scaling factor of (4~-+ 3) ~ combined with a rotation by ~r/10. Therefore, from the parent pattern, this rotated pattern can be gained by inflation by (4z+3) ~ combined with a rotation by ~'/10. A second operation of the same kind produces the

v

<" "'"/

~

"~"'l

if - - -(

(: ;)

k

-~

x

..

xyi

/

patterns. Inflation by (4~'+3) ~ with rotation by ~r/10 is the procedure to gain the sublattices in the decagonal

/

/

~.~

.~,

Fig. 5. Forced tiling of the (4 + 3r) ~/2 deflation of the Penrose pentagon tiles.

two-layer structure, the lattice sites of which are occupied by transition metal atoms. Figure 4 represents the derivation of the one sublattice (see also Fig. 1). Tests have shown that there are no simple rules to decorate patches of small tiles with larger tiles. They cannot be derived by local rules. The inverse procedure of non-locally derivable inflation is deflation. This procedure is also non-locally derivable. Figure 5 shows the four tiles of the Penrose pentagon pattern, together with their forced deflation tiles inside. The outside forced tiles cannot be depicted in total, since there are partly forced infinite patches. Figure 4 proves that there are undetermined regions. These regions will be tiled when the patches of the parent pattern are enlarged.

5. Atomic surface

Fig. 4. One of the five self-similar sublattices of the Penrose pentagon pattern with scaling factor ( 4 + 3 r ) ~n.

In previous papers [2, 11], it was shown that phason flips, and a second type of phason flips which result in an exchange of two different atoms, are required by a "closeness condition" to have very special shapes of the atomic surfaces for the quasi-lattice and the quasi-sublattice in the complementary space. Therefore, in Fig. 6, the atomic surfaces are depicted. These surfaces are simplified in that the split positions are not corrected.

R. Liick, K. Lu / Non-locally derivable quasi-sublattices

143

...@ @ @ @ 1

2

3

~

5

Fig. 6. Atomic surfaces of the supposed structure model.

The important point is that the atomic surface of the sublattice is in no way fixed in the general atomic surface. It is assumed that the centrosymmetric position is the most probable, but small movements from the centred position are not obstructed. The general atomic surface can be subdivided in regions responsible for special vertex configurations, but the boundary of the atomic surface of the sublattice has no connection with the boundaries of these regions. This fact can be regarded as proof that this sublattice is not locally derivable from the parent lattice.

7. Conclusions

Sublattices can describe chemical long-range order in quasi-crystals. The sublattices can be locally or nonlocally derivable from the parent lattice. If the sublattices are non-locally derivable, there is generally more than one equivalent sublattice, though not all of these are realized by the quasi-crystal's order. The sites of the transition metal atoms in the two-layer decagonal quasicrystals form a sublattice in both layers.

Acknowledgments

6. Discussion

6.1. Non-locally derivable patterns with other symmetries There are several other inflation procedures which produce non-locally derivable inflation patterns. The same holds for other symmetries. For example, in the square and rhombus pattern with eightfold symmetry, inflation by 21/2+ 2 produces four different, equivalent non-locally derivable patterns which do not share vertices. Janssen reported in a recent paper [12] that a pattern with 12-fold symmetry exhibits an inflation pattern with the scaling factor pl/2 (p=2+3a/2) combined with a rotation by ~-/12. The pattern was called "shield tiling" in the review given by Nissen [13] and has been described in detail by G~ihler [14, 15]. It should be stressed that the inflation by p~/z is not unique and, therefore, is not locally derivable. Nevertheless, the inflation by p seems to be locally derivable.

6.2. Phasons in ordered quasi-crystals Phasons also may exist in the sublattices; however, their physical meaning is different from the phasons in the parent lattice. The consequences of these differences have been discussed in previous papers [2, 11]. A phason's flip in the sublattice discussed will require an exchange of a transition metal atom with an aluminium atom [2].

One of the authors (RL) is grateful to the United Nations Development Programme (UNPD), managed by the China International Centre for Economic and Technical Exchanges and by the National Science Foundation China, for financial support to make the visit in Shenyang possible, and to the National Key Laboratory for RSA, Institute of Metal Research, CAS for the hospitality.

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Imperfectly Ordered Crystals, Chengde, China, 1993, Mater Sci. Forum, in press. 3 B. Griinbaum and C.G. Shephard, Tilings and Patterns, Freeman, New York, 1987. 4 M. Baake, M. Schlottmann and P.D. Jarvis, J. Phys. A, 24 (1991) 4637. 5 W. Steurer and K.H. Kuo, Philos. Mag. Lett., 62 (1990) 175-182. 6 W. Steurer, J. Non-Cryst. Solids, 153-154 (1993) 92-97. 7 W. Steurer, T. Haibach, B. Zhang, S. Kek and R. LOck, Acta Crystallogr. B, 49 (1993) 661-675. 8 R. LOck, Mater. Sci. Forum, 22-24 (1987) 231-246. 9 R. Liack, Int. Z Mod. Phys. B, 7 (1993) 1437-1453. 10 K. Niizeki, J. Phys. A, 22 (1989) 193, 205. 11 R. LOck, J. Non-Cryst. Solids, 156-158 (1993) 940-943. 12 T. Janssen, Acta Crystallogr. A, 47 (1991) 243-255. 13 H.-U. Nissen, in I. Hargittay (ed.), Quasicrystals, Networks, and Molecules of Fivefold Symmetry, VCH, Weinheim, 1990, pp. 181-199. 14 F. G~ihler, in Ch. Janot and J.M. Dubois (eds.), Proc. ILL~ CODEST Workshop, Grenoble, World Scientific, Singapore, 1988, pp. 272-284. 15 F. G~ihler, J. Non-Cryst. Solids, 153-154 (1993) 160-164.