Continuous Evolutions which Connect Quasilattices with Lattices

M. TORRW et al. : Evolutions which Connect Quasilattices with Lattices 439

phys. stat. sol. (b) 154, 439 (1989)

Subject classification: 61.50

Instituto de Cieneia de Materiales, Sede A (a), Instituto de Eleetrdnica de Comunicaciones ( b ) , and Instituto de Quimica-Fisica “Rocasolano” ( c ) , Consejo Superior de Investigaciones Cientificas, Madrid1)

Continuous Evolutions which Connect Quasilattices with Lattices BY M. TORRES (a), G. PASTOR (b), I. JIM~NEZ (b), and J. FAYOS (c)

A newly modified projection method is developed in order to generate continuous transformations which connect quasilattices with lattices. The evolution and collapse of a 2D octagonal quasilattice, B continuous evolution of the 3D Penrose tiling which leads to a cubic primitive lattice on a side and to an f.c.c. vertices arrangement on the other side and, finally, the evolution and collapse of the historical 2D Penrose tiling are studied in detail. Applications of these quasiperiodic tiling evolutions to quasicrystal-crystal transitions and to atomic decorations of quasicrystalline phases are suggested.

Eine erneut modifizierte Projektionsmethode wird entwickelt, um kontinuierliche tfbergknge zu generieren, die Quasigitter mit Gittern verbinden. Die Entwicklung und der Zusammenbruch eines 2D-oktogonalen Quasigitters, eine kontinuierliche Entwicklung der 3D-Penrose-Verkippung, die zu einem primitiv kubischen Gitter auf der einen Seite und zu einer k.f.z.-Vertexanordnung auf der anderen Seite fiihrt und schliel3lich die Entwicklung und der Zusammenbruch der historischen 2D-Penrose-Verkippung werden ausfiihrlich untersucht. Anwendungen dieser quasiperiodischen Verkippungsentwicklungen zu QuasikristalI-ICristall-vberg&ngen und zu atomaren Dekorationen von quasikristallinen Phasen werden vorgeschlagen.

1. Introduction

Starting from the historical works of Penrose [l, 21 (see also Gardner [3]), de Bruijn [4] and Mackay [5, 61, a lot of nice and fertile ideas have enriched the crystallography with the notion of quasiperiodic translational order [7] and with the so-called projection method [8 to 121. This mathematical technique is an elegant and powerful geometric medium to generate potential quasilattices for the recently found quasicrystalline phases [13] (see also [14] for more up-dated bibliography). On the other hand, a quasicrystal-crystal close relationship has been both theoretically and experimentally pointed out [15 to 221 (“ah initio”), [23 to 341 (recently) and [35 t,o 391 (where a continuous range of intermediate metastable phases is suggested).

I n this work, a new version of the projection method is developed in order to describe continuous evolutions from quasilattices to lattices. In our method, the hypercubic lattice strip [ lo to 121 (defined in the hyperspace Ep) is projected on a rotary subspace E m , p > n. So, a variable tiling is obtained. This evolutionary tiling connects the standard nonperiodic quasilattice with periodic lattices in a continuous way. Some examples of tiling evolutions are developed here. As we shall see, these geometric transformations seem to be coherent with the experimental observations

Serrano 144, E-28006 Madrid, Spain,

440 M. TORRES, G. PASTOR, I. JIM~NEZ, and J. FAYOS

about quasicrystal-crystal coexistence and transitions and, also, with the recently proposed AI-Mn quasicrystal phase atomic decoration [40].

2. A Newly Modified Projection Method

Elser and Henley modified the projection method to study the connection between crystal and quasicrystal structures [16]. These authors tilted the strip with respect to the hypercubic lattice but they fixed the projection hyperplane (or projection subspace). So, different hypercubic roofs were projected in such a way that the quasicrystal structure was the limit of a discontinuous sequence of periodic structures (or rational approximants).

In this work we develop an alternative method and we describe a lattice as an evolved quasilattice. We rotate the projection hyperplane (or rotary subspace) with respect to the hypercubic lattice but we fix the particular strip which generates the standard quasiperiodic tiling. In our method, the orthogonal projection of a unique hypercubic roof (i.e. the standard tiling) continuously evolves. The tiles also evolve

'\ A B,C D E,F G,H I 1 K L L \ G \

B JE

U b - I

I 1 I I 1 7 I 1 K L

Fig. 1. The rotary projection method. The staircase fragment A, B, ... , L is orthogonally projected 011 a rotary line and so an evolutionary tiling is generated: 1,2, ... ,8. The original standard quasiperiodic tiling is obtained on 3. The tilings 1, 4, and 6 are atrophycally periodic (some tiles vanish on 1 and 6, and the tiles become equal to themselves on 4). Tilings 7 and 8 show tile overlapping. For 8, the infinite staircase projection collapses

Continuous Evolutions which Connect Quasilattices with Lattices 44 1

in a continuous way but the quasiperiodic general assembly [7, 411 of the original tiling is preserved. Some types of tiles vanish and other ones become equal to themselves for some special orientations of the projection hyperplane. In these tilings, as we shall see, a periodic arrangement of vertices appears. A range of hyperplane orientations generates transitional quasiperiodic tilings which connect in a continuous way the original tiling with periodic ones. There is another range of hyperplane orientations that generate tilings where the tiles overlap. When the projection hyperplane become orthogonal to the strip, the infinite tiling collapses. The idea of the method is shown in Fig. 1 (in the simple case of p = 2 and n = 1).

In order to develop the present method, we need to find evolutionary vector half- stars in the rotary subspace En (or projection hyperplane) which represent the orthogonal projections of the p hypercubic lattice basic vectors (half-cross) defined in the hyperspace E p . So, in the light of Hadwiger’s theorem [42, 431, we look for variable half-stars with p vectors which preserve the equation

where (wil, vi2, ... , ~ i , ~ ) i = ~ ,,.., are the components of the p vectors rotary subspace En (see Appendix).

,..., in the

3. Evolution and Collapse of a 2D Octagonal Quasilattiee

The a-variable 2D four-radiation (V4R): v1 = ( c , -s), v2 = (c, s), v3 = (-8, c ) , v4 = (s, c ) , c = cosa, s = sina, preserves (1) (with p = 4 and n = 2). When LX = = 22.5”, V4R coincides with the four generating directions of a perfect 2D octagonal quasilattice with eightfold rotational symmetry [44 to 461. When a varies, V4R and the octagonal quasilattice evolve also in a continuous way, There are three types of a-variable rhombi which are defined by vl, v2 (rhombi A), vl, v3 (rhombi B), and vl, v4 (rhombi C which are always squares). When a = OD, rhombi A vanish and rhombi B become equal to squares C . When 01 = 22.5”, rhombi A become equal to rhombi B (45” rhombi). When a = 45”, rhombi B vanish and rhombi A become equal to squares C . The orthogonal projection of a 4D hypercube y4 on the rotary subspace E2 is an &-variable octagon filled with = 6 a-variable rhombi (which are “visible”

projections on E2 of 2-facets of y4) . There are other Z4-2 - 1 = 3 configurationsof six “hidden” 2-facets. In the infinite tiling, the relative frequencies of rhombi are fa = fB = 1/(2 + i2) and f c = i 2 / ( 2 + f2). So, the potential square and octagonal phases must have a density ratio equal to 1.1715. Pig. 2a schematically shows the continuous evolution from a 2D octagonal quasilattice to two square lattices rotated by 45”. The coexistence of both octagonal quasicrystalline and 45” twinned square crystalline phases has been experimentally observed (in rapidly solidified V-Ni-Si and Cr-Ni-Si alloys) just a t the discovering moment [as].

As we shall see in the next section, vertices of “hidden” 2-facets in addition to the “visible” vertices can play an important role in the atomic decoration propositions of these evolutionary tilings. Fig. 2a shows some of these “hidden” vertices which are placed on the square diagonal, (12 - l )L far from the vertex, and on the rhombus diagonal, (2 - I/2)1/2L far from the vertex, where L is the length of the tiling edge.

Fig. 2 b shows the tiling range where the tiles overlap (a > 45”) and the tilings collapse (a = 112.5”). In this collapse all vertices of the infinite tiling fall inside the 4D hypercube shadow (i.e. inside the octagon marked in Fig. 2a) and, due to the

(3

442 M. TORRES, G. PASTOR, I. J I M ~ E Z , and J. FAYOS

Continuous Evolutions which Connect Quasilattices with Lattices 443

e

A

444 M. TORRES, 0. PASTOR, I. JIMEXEZ, and J. FAYOS

quasiperiodicity of the original tiling (i.e. due to the irrational character of the strip slopes), no vertices are superimposed.

4. Continuous Evolution of the 3D Penrose Tiling

The ol-variable 3D six-radiation (V6R): vl = (c, -s, 0) , v2 = (c , s, 0) , v3 = (0, c, s), v, = ( -3 , 0, c), w 5 = (0, -c, s), and v,, = (s, 0, c), connects icosahedral with cubic symmetries and orders [23, 47, 481 and preserves (1) (with p = 6 and n = 3). Essen- tially the same V6R was first suggested by Kramer in a pioneer work [23]. This author used group theory, especially Schur’s lemma, to find V6R. Through Had- wiger’s theorem we have also found V6R [47].

When v1 . v2 = w 2 . v, = v6 . v1 (i.e. when 01 = 31.7174”), V6R exhibits icosahedral symmetry and coincides with the six spatial directions of a 3D Penrose tiling (3DPT) [5 to 8, 10 to 12, 41, 48 to 501. When a varies, V6R and 3DPT evolve also in a continuous way. When LX = O ” , V6R is atrophied into the radial skeleton (centre to vertices directions) of an octahedron and it is related to the cubic primitive lattice, and, when OL = 45”, V6R becomes the radial skeleton of a cuboctahedron and i t is related to the f.c.c. order. In the &-variable 3D tiling there are four types of ol-variable rhombohedra1 tiles corresponding to the four vector triads: vl, v5, v6 (“green” rhombohedra, G), q, v2, v6 (“red” rhombohedra, R), el, v,, w6 (“blue” rhombohedra, B) and vl, v,, v6 (“yellow” rhombohedra, Y), with relative frequencies f G = 4r1/10, f R =

= 6 ~ - ~ / l O , fB = 4c2/10, f Y = 6 ~ - ~ / 1 0 (where t = (1 + t 5 ) / 2 is the golden ratio) and &-variable volumes V G = c3 + 83, VR = 2c2s, VB = c3 - 83, Vy = 2s2c. G and R are prolate rhombohedra and B and Y are oblate ones (G and B are equifacial tiles). This prolate and oblate rhombohedra doubling does not seem to be coherent with the Steinhardt and Ostlund consideration of two types of both of them, prolate and oblate rhombohedra, each with a different matching decoration [ 141. The basic zonohedron is an &-variable triacontahedron (T) which preserves m3 point symmetry (maximal subgroup of icosahedral m35 and cubic m3m point groups). The &-variable generalized

-

36

Fig. 3. Decomposition of the generalized t(riaconta1iedron T (for a = 17”) in their twenty rhombohedra1 tiles, grouped in seven structural blocks of three types. The central block 1 is formed by a core of 2R tiles surrounded by 4G and 4B tiles, where faces connect different colours. Blocks 2 are formed by 2R + 1Y tiles, and blocks 3 are Y tiles. To fill the generalized triacontahedron, first blocks 2a, 3a, 3b, and 3 c cover concavities of block 1 rotating as indicated. This fragment exhibits m point symmetry. To complete the generalized triacontahedron, the first block 3d rotates as indicated and after that block 2 b covers it. All the blocks rotate around a vertex


37.77"

Fig. 4. Continuous evolution of the a-variable generalized triacontahedron T of an evolutionary 3DPT. The angular values a and the face colours are indicated

T is the orthogonal projection of a 6D hypercube y6 on the rotary subspace E3 and it

is filled with (!) = 20 &-variable rhombohedra1 tiles (which are the "visible" ortho-

gonal projections on the rotary subspace E3 of 3-facets of y6). There are 0 t h e r 2 ~ - ~ - 1 = = 7 configurations of 20 "hidden" 3-facets. T is filled with 4G + 6R + 4B + 6Y;

0

0

0

I

0

0

0

0

0

0

0

Fig. 5. F.c.c. arrangement of vertices in the a = 45' case. The four types of tiles are also shown

446 M. TORRES, G. PASTOR, I. JIM~NFZ, and J. FAYOS

a=O" I

Fig. 6. Continuous evolution of the 3DPT. An a-variable 3D tiling fragment is uncovered by a section perpendicdar to vs and the corresponding sectional roof is shown. The angular values a are indicated. When a = 31.7174", a generalized 2D Penrose tiling appears. When a = 45", the nonperiodic 2D tiling is the roof projection of a periodic 3D f.c.c. vertex arrangement. The pointed up decagon (in the case of a = 31.7174') shows the projections of some "hidden" vertices

there are sixteen external tiles (2B and 5Y with three faces on the surface, and 4G and 5R with one face on the surface) and four internal tiles (2B and 1Y with two edges on the surface, and 1R with only one vertex on the surface). This complex decomposition of T is shown in Pig. 3 (in the case of a = 17"). The a-variable T evolution generates a continuous transformation of the 3DPT. Whena = O", R and Y vanish, G = B become cubes, T degenerates also into a cube and the 3DPT atrophies into a cubic primitive lattice. When 01 = 31.7174", T is an ordinary triacontahedron, G = R become the well known prolate rhombohedra of the SDPT and B = Y become the oblate ones. When a = 45", B vanishes, R = Y, T degenerates into a truncated octahedron and the 3DPT becomes an f.c.c. vertex arrangement although being an aperiodic rhombohedron tiling. In this last case, the tiles which connect through an

Continuous Evolut,ioiis which Connect Quasilattices with Lattices 447

extinct B are not adjacent face to face, but half-face to half-face. All the above- described is schematically shown in Fig. 4, 5, and 6. The four SDPT basic bricks of Levine and Steinhardt (411 and Socolar and Steinhardt [49] become here six a-variable structural units : T, generalized rhombic icosahedron (I), generalized rhombic dodeca- hedra 1 and 2 (D1 and D2), G and R. I is filled with 2G + 3R + 2B + 3Y, there is only one B internal tile (or, alternatively, one Y internal tile) which has three edges on the surface. D1 is filled with 1G + 1R + 1B + lY, all of them with three faces on the surface. D2 is filled with 2R + 2P , all of them with three faces on the surface. T, I, and D1 have no internal m symmetry (in general), so there are two enantio- morphs of each one. When 01 = 31.7174", I becomes a regular rhombic icosahedron, D1 = D2 = D (regular rhombic dodecahedron) and G = R = P R (the well-known prolate rhombohedron of the 3DPT). The tiling ranges where the tiles overlap and the SDPT collapse are omitted in this work.

There are two potential candidates for physically near phases with a continuous transition between them (icosahedral, i.e. 01 = 31.7174", and f.c.c., i.e. a = 45", see Fig. 4c, 4d, 5 , and 6). In the framework of the above-described we deduce that the density ratio of these phases should be equal to 0.9218. Our SDPT evolution is directly related with the experimental observations and the ideas about the close similarity between &-cubic Al-Mn-Si and icosahedral i-Al-Mn-Xi phases [17, 19 to 211. There is experimental evidence of the slight differences in structures and density values for both above-mentioned phases [20, 211. Besides, the Guyot and Audier model consideration of two types of rhombohedra1 prolate tiles, a full and an empty one, takes sense taking into account our distinction between G and R tiles. Our B tile vanishing when 01 = 45" is also coherent with the fact that all oblate rhombohedra tiles are empty in the above-mentioned Guyot and Audier model. Moreover, these authors consider that there is a slight shape variation between the full prolate rhombohedra for both phases, in close agreement with our slight geometric differences for G tiles between 01 = 31.7174" and a = 45". We also point out that our geometric tiling evolution seems to be coherent with the following recent experimental observations : transformation from icosahedral to a cubic phase through a distorted inter- mediary phase [37], coexistence of both the icosahedral and the f.c.c. phases [29, 511, and existence of an orientation relation between icosahedral and f.c.c. phases [26, 291. That is expected because there are theoretical considerations that prove the possibility of continuous deformations from icosahedral to m3 point symmetry (T,,) structures [38]. In addition, the Landau general theory about second-order phase transitions [52] shows that the continuous transition from the hemihedral m3 point subgroup phase to the holohedral m3m point group phase is also favoured. Finally, we remark that our geometric models connect the not yet well-known quasiperiodic structures with well known periodic ones. These transformations can be useful in order to illuminate the atomic decoration problem (open subject today). Thus, Janot et al. have just published the first atomic decoration proposition of an icosahedral A1-Mn quasicrystal based on direct experimental observations by using a sophisticated neutron-diffraction technique [40]. In accordance with this work the subnetwork of Mn atoms are a t the vertices of a 3DPT but "the two vertices a t the short diagonals of the oblate rhombohedra are scarcely occupied simultaneously by Mn atoms". Our geometric model of quasicrystal-crystal transition is coherent with the above- said because B oblate rhombohedra are flattened in the mentioned transformation. I n relation to the subnetwork of A1 atoms, Janot et al. [40] have found that thesr atoms only occupy a few 3DPT vertex positions. However, they have also found othee A1 atoms on the long diagonals of the 3DPT prolate rhombohedra near positions situated a t 0.257 and 0.678 nm from the vertices (where 1.096 nm is the length of this

448 31. TORRES, G. PASTOR, I. JIMENEZ, and J. FAYOS

727.7'

90"

748.3"

Fig. 7. An evolutionary cycle of the historical 2DPT. This cycle contains a singular state of tiling collapse (a = 121.7'). A fragment evolution is shown and the angular values IX are indicated


long diagonal) and also on the long diagonals of SDPT prolate and oblate rhombohedra faces, dividing these long diagonals of the rhombi into segments of 0.298 and 0.483 nm (0.781 nm for the diagonal). Our quasicrystal-crystal transition model is also coherent with this atomic decoration if we suppose that some A1 atoms are a t the “visible” vertices of the 3DPT and the rest of A1 atoms are a t the hypercube centres and a t some vertices of “hidden” 3DPT rhombohedra. Fig. 6 shows the projections of some of these “hidden” vertices. These A1 positions fall on the long diagonals of the prolate rhombohedra, t - lD and r 3 D far from the vertex (where D is the length of this long diagonal) and also on the long diagonals of prolate and oblate rhombohedra faces, z-ld far from the vertex (where d is the diagonal length), in perfect agreement with the above-mentioned experimental observations. When the icosahedral tiling evolves to the f.c.c. vertex arrangement, the “hidden” vertices coincide with the “visible” vertices, many of which are unoccupied.

5. Evolution and Collapse of the Historical 2D Penrose Tiling

Now, we look for the orthogonal projection of V6R on the variable plane which remains perpendicular to the vector v6. By using this simple artifice we obtain an &-variable 2D five-radiation (V5R): v: = (c2, -s), V; = (c2, s), v: = (-9, c ) , v: = (-2sc, 0) , andvz = (-9, -c) , which preserves (1) (withp = 5 and n = 2). Whena = 31.7174”, V5R coincides with the five pentagonal directions of the historical 2D Penrose tiling (2DPT) [ 1 to 6, 14,41,49]. When a varies, V5R and SDPT evolve also in a continuous way. In the a-variable 2D tiling there are six types of a-variable parallelogram tiles which are defined by the pairs of vectors: v:, v$ (tiles t, which are rhombi), v?, v$ (or v;, vz, tiles t, which are rhombi), v:, v$ (or v?, v:, tiles t3), v;Z, vz (or v$, v:, tiles t4 which are rhombi), vg, v: (or v:, vz, tiles t,) and v:, v: (tiles t, which are rhombi). In the infinite tiling, the relative frequencies of tiles are ftl = t -1 /5, f t z = = f t3 = 2 r 2 / 5 , f t4 = f t5 = 2t-l/5, and fts = r 2 / 5 . When a = 31.7174”, tl = t4 = t, become the “fat” rhombi of the standard 2DPT and t, = t, = t, become the “skinny” ones. The basic zonogon is an a-variable decagon filled with It, + 2t, + 2t, + 2t4 + + 2t, + It, (i.e. (E) = 10a-variable tiles . Thisa-variable decagon is the orthogonal

projection of a 5D hypercube y, on the rotary subspace E2 and the a-variable parallelogram tiles are the orthogonal projections on the rotary E2 of 2-facets of y,. Pig. 7 shows the continuous evolution and the collapse of a 2DPT fragment. Whena = 0”, 31.7174”, and 45’, special tilings are obtained again. When 45’ < a < 180”, the tilings show tile overlapping. I n this range, when a = 90”, 135”, and 180”, periodic tilings appear. When a = 180“, the geometric cycle is rounded off. I n the range 180” <a < 360’, the anticlockwise V5R changes into a clockwise star. Whena = 121.7174’ or 301.7174” the infinite tiling collapses. This extreme folding (or reversible collapse) may be a new interesting topic of tiling theory.

)

Appendix

Let v be a vector along a fixed direction in the Euclidean nD space En and let u be a unit vector along an arbitrary direction in the same n-space. We wish to calculate the mean value ((v . u ) ~ ) when u scans all the hyperspace En. For that, we choose v = (0, 0 , ... , 0, v) and then we transform the commonly used Cartesian coordinate 29 physica (b) 154/2

450 M. TORRES, G. PASTOR, I. JIIKENEZ, and J. FAYOS

syst,em into the hyperspherical one as follows:

x, = r sin 8, sin 8, ... sin sin sin Q, , x2 = r sin 8, sin O2 . . . sin 8, - sin 8, -2 cos Q, , x3 = r sin 8, sin 8, . . . sin 8, - cos 8, -2 , x4 = r sin sin 8, ... cos

x , - ~ = r sin 8, cos O2 , X f i = r COB e l , x: + xt + ... + xi = r 2 ,

0 I r < co , 0 5 8,, 8,, ... , O n P 2 sn, Q 5 Q, S 2 n , where r is the hyperradius and el, O,, ... , On+, Q, are the angular coordinates. The Jacobian of the coordinates transformation is

J = rn-l(sin (sin 8,)n-3 ... (sin t9n-3)2 sin and

dOl do, ... dOn-2 dQ, J

dQn =

is the solid angle element. Now, by integrating over the angular ranges 0 5 el, 02, _.. , O n - , 5 z, 0 5 @ (= 2n, we obtain the simple result

where

is the “surface” of the general sphere with hyperradius r = 1 (P is the gamma function).

Based on (A4), we define an isotropic p-radiation in En [53] as theset { ~ i } g = I , . . . , ~ ,

p n, of p vectors in E n that verify the equation

In other words, we say that the p-radiation { v ~ } ~ = ~ , , , , , is isotropic when deviations of every value (vi . u),, with respect to the mean value vi/n, are compensated between them in the global sum (A6) independently of the chosen proof unit vector u in En.

Equation (A6) can be written by using the Hadwiger-Coxeter vector transformation

I

as

Continuous Evolutions which Connect Quasilattices with Lattices 45 1

Consequently, taking into account Hadwiger’s theorem [42, 431 and (A8), we can state that an isotropic p-radiation in En is an orthogonal projection on the subspace En, of a mutually perpendicular p-radiation defined in the p D hyperspace Ep (p 2 ?L) ; i.e. { v ~ } ~ = ~ , . . . , p is a eutactic half-star in En. The above mentioned (1) is a more useful way to state (A6) and (A8). So, (l), (A6), and (AS) arc: equivalent equations.

Now, from the set of vectors ( v ~ } ~ = ~ , . . . , ~ the orthogonal projection of a pD hypercube y p on the subspace En can be constructed. For n = 2, this projection is a zonogon

with 2p sides. Each zonogon is filled with “visible” 2D-tiles. For n = 3, this

projection is a zonohedron withp(p - 1) faces. This zonohedron is completely specified when vectors of the above-mentioned half-star are known. Each zonohedron is filled

with t) “visible” 3D-tiles [43].

(2”)

Acknowledgenients

We are indebted to Carmen Hurtado due to the construction of useful and nice 3D geometric models. Pinancial support from DGICYT (project PB 87/0291) and CICYT (project MAT 88/202) is also gratefully acknowledged.

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Continuous Evolutions which Connect Quasilattices with Lattices

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