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Structural characteristics ofpentagonal Al—Fe—Cu phasesMichel Duneau a & Marc Audier ba Centre de Physique Théorique, Ecole Polytechnique, CNRSUPR 14, 91128, Palaiseau, Franceb Laboratorie de Thermodynamique et Physico-ChimieMétallurgiques, Ecole Nationale Supérieure d'Electrochimieet d'Electrométallurgie de Grenoble, BP 75 38402, SaintMartin d'Hères, France

Version of record first published: 12 Aug 2009.

To cite this article: Michel Duneau & Marc Audier (1998): Structural characteristics ofpentagonal Al—Fe—Cu phases, Philosophical Magazine A, 77:3, 675-688

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PHILOSOPHICAL MAGAZINE A, 1998, VOL. 77, No. 3, 675-688

Structural characteristics of pentagonal Al-Fe-Cu phases

By MICHEL DUNEAU Centre de Physique ThCorique, Ecole Polytechnique, CNRS UPR 14,

91 128 Palaiseau, France

and MARC AUDIER Laboratoire de Thermodynamique et Physico-Chimie MCtallurgiques,

Ecole Nationale SupCrieure d’Electrochimie et d’ElectromCtallurgie de Grenoble, BP 75, 38402 Saint Martin d’Heres, France

[Received 14 April 1997 and accepted in revised form 2 July 1997

ABSTRACT The vector basis and super spacegroup of the pentagonal Al-Fe-Cu phases,

observed as a transient state of reversible icosahedral-to-rhombohedra1 trans- formation, are precised in the six-dimensional representation. The proposed theoretical description of the pentagonal structures is shown to be in a good agreement with experimental observations. As a result of this study, a scheme for the next step of transformation, pentagonal-to-rhombohedral, can be suggested.

9 1. INTRODUCTION The formation of a mixture of two pentagonal phases occurs as a transient state

of a reversible icosahedral-to-rhombohedra1 transformation in the Al-Fe-Cu system (Menguy et al. 1993). These pentagonal structures have been identified as icosahedral approximants because of the use of a theory proposed by Qiu and Jaric (1989) where an icosahedral-to-pentagonal transformation can be obtained by means of a linear phason strain field. In a six-dimensional (6D) hyperspace description, this corre- sponds to a linear shear in a complementary subspace (or three-dimensional (3D) perpendicular space El) such that the 6D fivefold vector indexing the pentagonal period is brought into the physical space Ell. Simulations of Fourier transforms of high-resolution electron microscopy (HREM) images of both these pentagonal phases have proved to be in agreement with this theory. However, questions remained concerning their exact structural characteristics, for instance whether their superspace group is related to some observed conditions of reflection and a periodicity equal to a fifth of the cell parameter. Further studies were then required in order to advance the understanding of this transformation, in particular for the final transformation step between the pentagonal and rhombohedra1 phases.

In this paper, we develop and complete an interpretation of features experimen- tally observed on the mixed state of pentagonal phases. In § 2, a brief summary of the icosahedral-to-rhombohedra1 transformation occurring via different transient states in an A163.5Fe12.5C~24 alloy? is reported. In addition to the matrices representing linear phason strain fields identified in previous work (Menguy et al. 1993, Duneau

t This icosahedral-phase composition has been accurately determined by electron probe microanalysis (accuracy of less than 0.5% for each element).

0141-8610/98 $12.00 Q 1998 Taylor & Francis Ltd.

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and Audier, 1996), the vector basis for each pentagonal structure is determined from autocorrelation functions of HREM images. In $ 3, a theoretical analysis of the periodic correlation between fivefold quasiperiodic layers, made in both complemen- tary and physical spaces, leads to a model whose simulated HREM images, diffrac- tion patterns and autocorrelation functions are in agreement with our experimental observations. This yields an explanation of the apparent extinction rules mentioned above. In $4, we discuss which other linear phason strain fields could be considered in order to achieve the final transformation step between pentagonal and rhombohedral phases.

$ 2. EXPERIMENTAL CHARACTERIZATIONS

2.1. Icosahedral-to-rhombohedra1 transformation The icosahedral A163.5Fe12.5C~24 phase is stable only at high temperatures above

T M 675°C. When very slowly cooled from 820°C (or annealed at T < 675°C after a normal cooling to room temperature) this icosahedral phase undergoes a phase transition towardso a final crystalline multidomain structure of a rhombohedral phase (aR = 32.16 A; cr = 32"). Domains of about 200 A in size are coherently oriented along each of the ten threefold axes of the icosahedron and they produce altogether an overall icosahedral symmetry (Audier and Guyot 1989). The relative orientations of the rhombohedral cell correspond to those observed on a stellate polyhedron of icosahedral symmetry (e.g. an arrangement of 20 prolate Ammann rhombohedra). When annealed at 500°C < T < 675°C from its metastable state at room temperature, different transient states are successively formed, that is a modulated icosahedral structure, with an average wavelength of modulation changing with time and tem- perature which transforms into a mixture of two pentagonal structures. Although both the initial icosahedral and the final rhombohedral singlestates are chemically homogeneous, transformations into modulated icosahedral and pentagonal struc- tures imply not only a lattice change but also a chemical composition variation as found from X-ray energy-dispersive spectroscopy analyses (Menguy et al. 1992).

2.2. Arrangement of pentagonal phases Both the pentagonal phases of periodic parameters 52.25 A and 84.53 A

(hereafter called PI and P2 respectively) form a multidomain platelet structure with orientational relationships yielding an overall icosahedral symmetry. The axes perpendicular to their large interfacial platelet planes correspond to fivefold symmetry pentagonal axes which have remained strictly parallel to fivefold axes of the preceding icosahedral state. The linear phason fields relating their structure to the icosahedral phase (Menguy et af . 1993) have been determined to be the result of 6D fivefold vectors of the type (5 3 3 3 3 3) for PI and (10 4 4 4 4 4) for P2 (see Cahn et af . (1986) for the indexing) brought into El, from shears of the 6D lattice. A direct link with the preceding modulated icosahedral state can be emphasized from an analysis of a sequence of PI and P2 along a fivefold direction (fig. 1). The fringes at 52.25 A15 x 10.45 A and 8453 A/5 NN 16.9 A observed in the HREM image in fig. 1 are characteristic of both these pentagonal structures. Then knowing how PI and P2 structures can be obtained from the icosahedral structure, the magnitude of the successive shears corresponding to the observed sample can be represented on a section plane of the 6D lattice. As a result, shown in the right-hand part of fig. 1, the successive shears appear to oscillate around Ell with an average wavelength of

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Structural characteristics of pentagonal Al-Fe-Cu phases

Fig. 1

677

HREM image of PI and P2 platelets sharing a common five-fold axis (left) and 6D representation of the successive approximant periods in the corresponding fivefold plane (right). Fivefold axes indicated in the top left-hand part of the figure correspond to icosahedral axes observed before the icosahedral-to-pentagonal transformation. One of these icosahedral axes becomes a fivefold axis of the pentagonal phases.

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Fig. 2

2D section of tbe 6D lattice showing the periods of PI and P2. The fringes at 10.45A and 16.9A observed in the HREM images cannot be indexed in this plane.

about 370 A. Therefore, if chemical composition changes have normally to be expected from linear phason strain fields introduced by shears, the fact that the successive shears oscillate around the physical space can be interpreted as due to an average composition of the PI + P2 phases equal to that of the icosahedral phase.

One feature, observed for each pentagonal structure, is not, bowever, e5plained. It is related to the fact that the observed periodicities of 10-45A and 16.9A, which are equal to one fifth of the periodic cell parameter of PI and P2 respectively, cannot be justified from the theory of Qiu and Jaric (1989). From Fourier transforms of HREM images of these phases, this implies considering a condition 1 = 5n (n integer) for 00001 reflections in order to interpret both the observed and the simulated pat- terns (Menguy et al. 1993). Evidence of this problem appears in the representation of the 6D fivefold vectors (5 3 3 3 3 3) and (10 4 4 44 4) on the fivefold symmetry section plane of the 6D lattice as given in fig. 1. From an enlargement of this section plane (fig. 2), a division of these 6D vectors into five equal parts cannot be justified by any parallel components of 6D fivefold vectors. Therefore both a complement of analysis of experimental results and a little more theory are required in order to understand the characteristics of these pentagonal phases.

2.3. Autocorrelation functions of pentagonal phases The purpose of this new analysis of experimental results was to characterize the

translational order between each quasiperiodic layer of the pentagonal structures. Such a translational order is related not only to periodic translations along the fivefold axis, which can easily be determined by diffraction, but also to non-periodic translations, which are features directly observable in a direct space as HREM images but not in the reciprocal space. However, if vectors or vector components for a non-periodic translational order are actually observed in the HREM images, all

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Structural characteristics of pentagonal Al-Fe-Cu phases 679

of them cannot be unambiguously identified (i.e. their direction and length). Therefore an appropriate way for determining these vectors is to calculate auto- correlation functions C ( t ) = F ( t ) * F ( t ) (where F ( t ) is the image function). C ( t ) can be determined from a Fourier transform of the intensity distribution observed on the diffraction pattern of a HREM image. Note that such a type of analysis is quite commonly applied to characterizations of the remaining local order of disordered systems. For instance, it has been shown that both translational and orientational order in a two-dimensional (2D) lattice of hard spheres assembled at random can be analysed from different types of correlation function (Nelson et al. 1982).

Autocorrelation functions were calculated for both PI and P2 from HREM images showing a set of periodic layers viewed along a twofold zone axis of orien- tation parallel to the direction of the electron beam. A computer program called ‘NIH Image 1.60, was used (Rasband 1996). Figures 3(a) and (b) show both these

Fig. 3

. . . * . . . , .. . . * . * - . . . , , . .

. . * . . .

. . . . . ., . - . . - * , .

. .

PI and (b) P2 with a twofold zone axis and ling the rhombic icosahedra built by the reen two slabs of PI) .

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680 M. Duneau and M. Audier

HREM images and their corresponding autocorrelation functions for PI and P2 respectively. Axes corresponding to the relative orientation of the perfect icosahedral state, observed before the transformation into pentagonal phases, are indicated at the top left of fig. 3(a). The different peaks of the autocorrelation functions were revealed after subtracting the important background which is directly related to a lack of definition of the HREM images. Through a scaling factor of T (the golden mean (1 + 5 ' 1 2 ) / 2 ) , a similar distribution of peaks can be observed for both these functions. For each structure, the periods of PI and P2 appear to correspond to a sum of five basis vectors, tl + t2 + t3 + t., + tS and rtl + rt2 + 7f3 + rt4 + rts respectively, but which can be realized in different ways (fig. 3). From comparisons between the directions of these vectors with those of the perfect icosahedral state, only that of the vectors t3 and rt3 correspond to a twofold direction of the icosahe- dral symmetry. In other words, the angle formed by the fivefold pentagonal axis (parallel to fivefold icosahedral axis) with the vector t3 (or 7f3) remains invariant through the icosahedral-to-pentagonal transformation. Also, we verified that the vector lengths of t3 and 7-t3 correspond to those of the parallel components of twofold 6D vectors of the type (1 T 0 2 0 2) for t3 and (22 0 3 0 3) for 7-t3. As permuta- tions of the ti vectors or rti ( i = 1-5) were also verified through each 72" rotation of the fivefold symmetry, it can be concluded that such vectors are all twofold vectors of the same length for PI or P2. Moreover, as the fivefold vector indexing the pentagonal period can be obtain through different orders of summation of these five twofold vectors, these last were found to define the edges of a rhombic icosahe- dron (fig. 3). Finally, it can be checked from translations of this rhombic icosahedron by the vectors ti or rti that all the peaks observed on the autocorrelation functions are justified.

Therefore the vector basis for each pentagonal structure is formed by five two- fold vectors corresponding to parallel components of icosahedral twofold 6D vectors. The angles between adjacent vectors are all 60". Both vector bases of the PI and P2 pentagonal structures are related by a scaling of T.

6 3. PENTAGONAL APPROXIMANT PHASE The crystallography of pentagonal, decagonal, octagonal and dodecagonal

quasicrystals in three dimensions was extensively studied by Gahler (1990). In the particular case of pentagonal quasicrystals the space groups were determined from a five-dimensional approach. There are five point groups denoted 5 , 5 2 , 5 , 5 m and 5m. The first three have only one symmorphic space group. The last two also have non- symmorphic space groups but none of them contains a screw axis. Since pentagonal approximants of an icosahedral quasicrystal are naturally described and constructed in the usual 6D space, we present first a direct 6D analysis of their Fourier spectrum. Then a rough theoretical model is examined in order to understand the autocorrela- tion functions presented in the above section.

3.1. Fourier spectrum of pentagonal phases The Bravais lattice corresponding to the Al-Cu-Fe icosahedral phase is the

6D lattice L = of index 2 of the primitive lattice a6DZ6, where the lattice parameter is a6D = 6.31 A. The standard basis of a6DZ6 is denoted B = { E ~ , E ~ , & 3 , s4, E ~ , & 6 } . With respect to B, L is the set of points with integer coordinates which sum up to an even number. Let V denote the 2D plane of R6 spanned by

= ( 1 0 0 0 0 0) and the diagonal 6 = (1 1 1 1 1 T ) (indexing of Cahn et al. (1986)).

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Structural characteristics of pentagonal Al-Fe-Cu phases 68 1

The orthogonal four-dimensional (4D) space W = V’ is spanned by four vectors such as (0 1 1 0 0 0). It is well known that the particular orientation of the physical space Ell with respect to the 6D lattice implies that 611 = (5Il2 + l)E1ll. Therefore the projection of Vinto Ell is the one-dimensional(1D) line spanned by that we shall assume to be the five-fold axis of the pentagonal phase. For similar reasons, the projection of W into Ell is the 2D plane orthogonal to

The dual bases of B is denoted B* = {ET, E;, E:, E:, ET, €2). so that the dual lattice L* of L is the set of points with coordinates either all integers or half-integers. The Fourier spectrum of the icosahedral phase is carried by the parallel projection Lf = nll(L*) of L* which is a 2 module of rank equal to six. The intersection of L* with the 2D space V is a 2D lattice denoted L*y which is spanned by E: and 6*/2 = (1 1 1 1 I f)/2. Similarly, the intersection of L* with the 4D space W is a 4D lattice denoted L*,. One checks that the sum lattice Lo* = L t @ L$ is of index five in L*: the unit cell of &* contains five nodes of L* which build a finite set F. Accordingly, any reciprocal vector Q in L* can be written Q = + 0, with Q, in Lo* and where Q1 only takes five possible different values in F. Thus L* can be viewed as the union of five translates of Lo*, which we can write as

L* = L$+ F .

Different sets F can be found, depending on the choice of a unit cell of Lo*. We can check that F = (0, E;, E; + E:, E; + E: + E:, E ; + E: + E: + E T } is a proper decoration of the unit cell of Lo* and that its parallel projection builds a screw about the five-fold axis 11.

The parallel projection of L*y is carried by the five-fold axis and the parallel projection of L*, is dense in the perpendicular plane.

We specify now mainly to the PI approximant although a similar discussion holds for P2. In view of the previous analysis, we define the pentagonal basis by the following five vectors of L which are parallel to two-fold axes: TI = (1 2 2 0 0 I ) , T 2 = (1022fO), T3=(1T0202), T 4 = ( 1 0 i 0 2 ? ) and T5=(120T20) which verify Till =ti. The sum of these five vectors gives the 6D period along the five-fold axis:

5

C T i = C = (533333) E V , i= 1

so that the projection CII = c gives the period a 3 ~ ( 5 + 3 x 5 ’ ” ) M 52.25 A of P I . One can also check that Ti = $C + wi where wi E W. It follows that the projections of the pentagonal basis in parallel and perpendicular space build pentagons about the five-fold axis such that (c,ti) = $c2 and (CI,Til) = $C: respectively.

In the case of the P2 approximant, the pentagonal basis is given by a T inflation of {TI, T2, T3, T4, T,} in such a way that TI is replaced by (2 3 3 0 0 2). The sum of the P2 basis vectors is indexed by (10 4 4 4 4 4) and projects in parallel space on a vector of norm a3D( 10 + 4 x 5 ’ / * ) x 84.54A which gives the period of Pz.

The description of the pentagonal approximant involves a shear transformation S which brings the 6D period C in parallel space. If C = CII + CI denotes the split- ting of C in parallel and perpendicular components, then the 6D shear can be simply written as

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682 M. Duneau and M. Audier

which means that for any 6D vector x we have

in such a way that SC = CII as required. We shall denote the 6D sheared lattice A = SL and similarly Av = SLv and

A, = SL,. Note that A , = L , since vectors in W are not changed by S. We shall use a Greek letter to represent the image 6 = Sx of a point x. We define in particular ti = STi and the above decomposition of Ti yields ti = fC1, + wi. It follows that (CLl ti,) = 0, which means that til lies in the plane orthogonal to C, in perpendicular space. Thus the action of S amounts to cancelling the components of all Ti, along the fivefold axis cIl so that the vectors til build a regular pentagon around the origin. In the parallel space we have till =.ti.

The Fourier spectrum of the pentagonal phase is carried by the orthogonal projection of the dual lattice A* = (S-')'L* with

which means that

We can transpose the previous decomposition of L* to the transformed lattice A*. We get a similar decomposition

A* = A! i- @,

with A: = A*, @ A%, A*, = (S-')'L*, and A*, = L*,. @ = (S- ' ) 'F is a set of five vectors which gives a decoration of the unit cell of A:.

Finally, the Fourier module in physical space can be constructed along the same lines by orthogonal projection A! = rll(A*):

";i = .ll(&) + .II(@).

The parallel projection rl~(L*,) gives the Bragg peaks lying in the quasiperiodic plane perpendicular to the fivefold axis. The Bragg peaks lying on the fivefold axis are generated by the projection r i~ (A* , ) which must be a 1D lattice. This can be checked as follows. A reciprocal vector 0 in L: can be written 0 = rn6*/2 + nat with rn, n integers. (S-')'Q belongs to A*, and projects on qI1 = rll(S-')tQ. Then we have

(qll Cll) = ((S-')%17 Cll) = (Q, S-'C,I) = (Q, C) = lorn + 5n = 51.

Thus we see that the 'extinction rule' (qI1 , Cll) = 51 in fact follows from the very definition of the dual lattice of the pentagonal structure, as does a similar 'extinction rule' in the more familiar trigonal lattice. Note that this description fully agrees with the results of Gahler (1990) concerning the pentagonal phases.

Finally, the pentagonal reciprocal lattice A; can be viewed as a periodic stack of strata such as rll(L*,) + TI , (@). Each stratum is made of five horizontal layers such as q(L*,) translated by the five vectors of rll(@) which build a screw about the fivefold axis. The simulations of the diffraction patterns were found to be identical with the experimental patterns presented by Menguy et al. (1993).

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Structural characteristics of pentagonal Al-Fe-Cu phases 683

3.2. A simulation of the pentagonal high-resolution electron microscopy images We present now a rough numerical simulation of the pentagonal phase P1 which

will give a convincing interpretation of the HREM images presented in 4 2. Owing to the periodicity along the fivefold axis we only need to represent the five layers on planes Po, P I , P3 and P4 perpendicular to the fivefold axis. The relative translations between these planes are such that P1 = Po + t l , P2 = PI + t2, P3 = P2 + t3 and P4 = P3 + t4. Now, the lattice points 5 E A which fall in Po necessary belong to the 4D lattice A , = Lw. Those which fall in P I belong to L , + el and, more generally, the nodes which fall on Pi belong to Lw + ai, where ai = el +. . . + ti (recall that eill = ti so that sill = tl + . . . +ti).

The selection of nodes is operated by a window W which we assume to have the shape of a triacontahedron slightly deformed as a consequence of the pentagonal shear S. The subwindow associated with the Po layer is then the equatorial section of W , which turns out to be a regular decagon denoted Wo. In this respect the Pi layer is made of the projections of nodes to + ai (with b E L,) such that b1 + ail belongs to W . This condition can be written kol E (W -,aiL); however, since (C1, ejL) = 0 ( j = 1, . . . , 5 ) the translation ail is horizontal and, finally, the condi- tion is that toL belongs to the translated decagon Wo - ail. The subwindows of the five layers are therefore equal up to a translation and consequently the layers are equivalent in the sense they satisfy the local isomorphism property: any finite pattern of a layer can be found with the same frequency in any other layer.

Now, the correlation between layers Po and Pi can be seen as follows. The points toll and bl1 + sill belong to the Po and Pi layers respectively if and only if tol belongs to intersection of Wo and Wo - ail; the probability of such a correlation is given by the ratio of the window areas IWo - aill/lWol. Figure 4 illustrates this situation in the case i = 1 (al = z I ) . This analysis shows that vertical screws such as

and Po + CII on base points (oll such that k1 belongs to the intersection of the five decagons Wo n (Wo - all) n ( WO - azl) n (Wo - 031) n (WO - ~ 4 ~ ) .

Another feature of the HREM images can be understood along the same lines. The HREM image in fig. 3 (a ) shows that many ‘periodic’ sequences of white dots of type {tNll, tMll + tc1. . . , kNll + do} can be identified. They occur on base points bll such that tol belongs to the intersection of the n + 1 translated decagons Wo n (W, - tml) n . . . n (W, - \eml). The maximal length n of these sequences

(6011, kO11 + @ I ( ) I 5011 + a211, 5011 + a3)1 > ‘%I1 + 6411~ 6011 f CII 1 Occur between layers PO

Windows in perpendicular space describing the Po and the P I layers are equal up to a translation tll . Their intersection is the subwindow which determines the correlated pairs of sites { ~ o l l , ~ l l + t,} in parallel space.

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684 M. Duneau and M. Audier

Fig. 5

Finite periodic sequences {so,, + .“fl}, n = 0, 1,2, . . . , occur in parallel space if tol belongs to the intersection of translated windows {Wo + nrll}.

Fig. 6

. .

. . . . - . r - . . . . . . . - . . . . . . . .

- . .

Simulation of the pentagonal structure: (a) projection of nodes in the same plane as the HREM image in fig. 3 and (b) its autocorrelation function where a rhombic icosahedron spanned by the pentagonal basis i s superimposed.

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Structural characteristics of pentagonal Al-Fe-Cu phases 685

corresponds to the vanishing of the intersection Wo n [Wo - (n + 1)rI1] and thus depends on the relative scale between rI1 and Wo as can be seen in fig. 5.

The simple model presented above allows a simulation of the 3D structure. In order to compare with the HREM image in fig. 3 (a), the projection of the calculated structure on the same plane is given in fig. 6 ( a ) . The density of nodes on the lines perpendicular to this plane is taken into account and a Gaussian convolution is applied. The calculated autocorrelation of the projected structure is given in fig. 6 (b) and can be successfully compared with the autocorrelation function of the HREM image shown in fig. 3(b).

0 4. DISCUSSION In this last section we examine some features related to the pentagonal-to-rhom-

bohedral transformations and we present some results concerning the analysis of the corresponding shears.

From previous works (Menguy et al. 1992, 1993), it has been found that the single state of the rhombohedral phase is recovered by annealing treatments of the mixture of pentagonal phases, for instance at 600°C for about 1 month. Samples of different intermediate states, in between those of the PI +P2 and rhombohedral phases, were also prepared for intermediate times of annealing treatments. However, no attempt of interpretation of transmission electron microscopy observa- tions on such samples was made because of the complexity of the observed features and the different hypotheses which could be put forward concerning this step of the transformation. Now, the above theoretical interpretation yields information on vectors remaining, a priori, invariant through this step of transformation which can be used as a guide for experimental identifications. For instance, as mentioned in $ 3 , the period of the P2 approximant is (1044444) and the basis involves five vectors such as ( 2 3 3 0 0 2 ) along twofold axes which correspond precisely to the basis vectors of the rhombohedral phase. More precisely, the orientations of the rhombohedral domains with respect to the pentagonal phase P2 show that one basis vector, say All[, belongs to the pentagonal basis whereas the two remaining vectors, A211 and A311, are perpendicular to the fivefold axis (see below). Therefore, on account of the fivefold symmetry, it might be assumed that transformation of the Pz- phase platelets should give rise to the formation of five orientations for rhombohe- dral domains, on the ten observed. These five rhombohedral orientations should exhibit a common axis parallel to the fivefold pentagonal axis (i.e. the parallel component of the vector (104 4 4 4 4)). The five other rhombohedral orientations should then result from the transformation of the PI -phase platelets. In that case too, it might be interesting to point out that the five twofold vectors (such as (1 2 2 0 0 1)) involved in the basis of PI correspond precisely to vectors defining the small diagonals of the rhombic faces of the rhombohedral cell, five other similar vectors being perpendicular to the fivefold axis of PI. The inset of fig. 7 shows how both these sets of rhombohedral orientations can be distinguished in a stellate poly- hedron of icosahedral symmetry, built with 20 rhombohedral cells of angular para- meter a = 36". Along one of its six fivefold axes, ten rhombohedral cells exhibit five orientations with a same irrational axis [TTT] parallel to this fivefold axis and the ten other rhombohedral cells exhibit five other orientations with a same rational axis [T 111 parallel to this fivefold axis. Both these groups of rhombohedral orientations would result in the transformation of P2 and PI pentagonal platelets respectively.

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686 M. Duneau and M. Audier

Fig. 7

Pentagonal-to-rhombohedra1 step of transformation: HREM image indicating the different sets of orientation of rhombohedral domains which would correspond to the transformation of PI and Pz platelets respectively; these rhombohedral orientations are gathered on the stellate polyhedron of icosahedral symmetry shown in the inset.

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Structural characteristics of pentagonal Al-FeCu phases 687

This point of view seems effectively supported by observations showing both a remaining trace on a sequence of the PI and P2 pentagonal phases along a fivefold axis (i.e. as previously shown in fig. 2) and the beginning of formation of rhombo- hedral domains. For instance, the alternating sets of fringes at 10.45 and 16.9A observable at a glance along the arrow on the HREM image in fig. 7 can be con- sidered as the remaining trace of a sequence of the PI and P2 pentagonal phases. The icosahedral orientation, indicated at the top right-hand corner of this figure, cor- responds to this in the preceding icosahedral state, before transformation. It also corresponds to that of the stellate polyhedron shown in the inset. Thus from a direct comparison between both the image and the stellate polyhedron, one can easily verify one rational orientation of rhombohedral domains within a transformed P2 slab. We have checked that only two other orientations of rhombohedral domains might exist within this slab. These orientations of rhombohedra, drawn on the image, are in accordance with the proposed scheme of transformation of P2. For the transformed PI slabs, there are variations in the fringe contrast which at least indicate the beginning of transformation. Although such variations mainly result in moirt fringes due to superimposed domains, it seems that orientations of rhombo- hedral domains might also be assumed to be in accordance with those observed on the stellate polyhedron for the proposed scheme of the PI transformation.

We examine now how the pentagonal-to-rhombohedra1 transformations could be interpreted by means of 6 D shears. We first consider the case of P2 and we specify a particular rhombohedral basis {A,, A2, A3) with Al,,//tl so that this approximant is identified by A, = (2 3 3 0 0 2), A2 = (0 2 3 0 2 3) and A3 = (0 3 2 2 0 3) (fig. 8). As discussed in 0 3, the first structural transformation from the icosahedral phase to the P2 pentagonal phase involves a simple shear S = Z - ( C ~ ~ I ) - ~ ~ C ~ J ( C ~ ~ which brings the 6 D period C2 into parallel space. The rhombohedral basis is trans- formed into {a,, a2, a3} with ai = SAi, but the parallel projections remain invariant:

Fig. 8

P2 fivefold axis + (1044443)

\I

3 3 0 0 2 )

A3

A2

Relative orientation of a rhombohedral cell with respect to the fivefold basis of Pz.

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688 Structural characteristics of pentagonal Al-Fe-Cu phases

ail, = Ail,. The second shear transformation must bring {a l , a2, a3} into parallel space. It turns out that this can be obtained by the product S2S3 of only two shears S2 = ( I + lazl)(q2,,1) and S3 = (Z + 1a31)(q3111). This remarkable property follows from the fact that the three vectors ail lie in the horizontal plane of EL. The wave-vectors q2,, and q311 show a more complex orientation between fivefold and twofold axes q211 and q311 are parallel to (1 000 51/2 0)11 and (1 00 5’120 O ) , , respectively. The three shears S, S2 and S3 commute and the overall transformation between the icosahedral phase and the rhombohedral phase is described by 1 - (c2,,)-21c2*)(~2111 + Ia*i)(q2,11 + lasl)(q31,1-

The transformation between PI and the rhombohedral phases can be analysed in a similar way. However, the corresponding shear transformations do not show the remarkable simplicity of the previous transformations associated with P2. From a purely geometrical point of view, the occurrence of the intermediate pentagonal phase PI does not seem to favour the subsequent transformation to a rhombohedral phase. Further investigations should be carried in order to find whether this distinc- tion could be connected with experimental observations.

ACKNOWLEDGEMENT One of us (M. D.) is indebted to F. Gahler for fruitful discussions.

REFERENCES AUDIER, M., and GUYOT, P., 1989, Proceedings of the Anniversary Adriatico Research

Conference on Quasicrystals, Trieste, 1989, edited by M. V. Jaric and S. Lundqvist (Singapore: World Scientific), p. 74.

CAHN, J. W., SHECHTMAN, D., and GRATIAS, D., 1986, J. Mater. Res., I, 13. DUNEAU, M., and AUDIER, M., 1994, Lectures on Quasicrystals, Aussois, 1994, edited by

F. Hippert and D. Gratias (Les Ulis: Les Editions de Physique), p. 283. GAHLER, F., Proceedings of the Third International Meeting on Quasicrystals, Mexico, 1990,

edited by M. J. Yacaman, D. Romeu, V. Castaiio and A. Gomez (Singapore: World Scientific), p. 69.

QUI, S. Y., and JARIC, M. V., 1989, Proceedings of the Anniversary Adriatico Research Conference on Quasicrystals, Trieste, 1989, edited by M. V. Jaric and S. Lundqvist (Singapore: World Scientific), p. 19.

MENGUY, N., AUDIER, M., and GUYOT, P., 1992, Phil. Mag. Lett., 65, 7 . MENGW, N., AUDIER, M., GUYOT, P., and VACHER, M., 1993, Phil., Mug. B, 68, 595. NELSON, D. R., RUBINSTEIN, M., and SPAEPEN, F., 1982, Phil. Mag. A, 46,105. RASBAND, W., 1996, ‘NIH Image 1.60’ Computer Program. (National Institutes of Health)

Oyww: http: //rsb.info.nih.gov/nih-image/).

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