CRYSTALLOGRAPHY OF ICOSAHEDRAL CRYSTALS

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CRYSTALLOGRAPHY OF ICOSAHEDRAL CRYSTALS

P. Bak

To cite this version:

P. Bak. CRYSTALLOGRAPHY OF ICOSAHEDRAL CRYSTALS. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-135-C3-141. �10.1051/jphyscol:1986313�. �jpa-00225724�

JOURNAL DE PHYSIQUE

Colloque C3, supplément au n07, Tome 47, juillet 1986

CRYSTALLOGRAPHY OF ICOSAHEDRAL CRYSTALS

P. BAK

Physics Department, Brookhaven National Laboratory, Upton, NY 11973, U.S.A.

Abstract - The crystallography of icosahedral crystals is constructed. The actual three-dimensional crystal is represented by a three-dimensional cut in a regular six-dimensional periodic crystal with symmetry described by a six-dimensional space group, and the positions of atoms correspond to an arrangement of hypersurface segments. The resulting crystal can not in general be viewed as a space-filling arrangement of a small number of dif- ferent "Penrose tiles." The intensities of Bragg spots are given directly as the intensities of Bragg spots of the six-dimensional crystal.

1. INTRODUCTION

Despite a complete lack of proof, one of the most fundamental assumptions in con- densed matter physics is that crystalline ordering must necessarily be based upon a regular three-dimensional translational invariant Bravais lattice; as a consequence of this, fivefold rotation symmetries are not permitted in conventional crystallog- raphy. The exciting electron diffraction experiment by Shechtman et al. /1/ on an alloy of manganese and aluminum showing a diffraction pattern with icosahedral sym- metry (including fivefold axes) therefore ignited an unprecedented uproar of disbe- lief in the world of crystallographers 121. Soon after the Shechtman experiment, Poon et al. /3/ discovered an alloy of palladium, uranium and silicon also showing icosahedral symmetry. Has nature indeed invented a more elaborate way of arranging atoms in space with long range ordering (as indicated by the sharpness of the dif- fraction spots), but no regular discrete lattice (as indicated by the presence of the fivefold symmetry)? From a mathematical point of view there are no obstacles to the concept of icosahedral crystals, since it has been shown that there exist space-filling arrangements of a small number of different building blocks (Penrose tiles) /4,5/ with icosahedral diffraction patterns. Much of the effort in under- standing the atomic arrangement has been centered around constructing such tilings or "quasicrystals" 161. Obviously, more general atomic structures with icosahedral symmetry can be generated by decorating each tile by an arrangement of atoms, in analogy with regular crystal structures which are formed by decorating each unit ce11 with a "basis" of atoms /7,8/. So far, however, nobody has succeeded in reproducing the experimental diffraction patterns. Does this lack of success for the Penrose tilings indicate that icosahedral crystals have not been observed, in agreement with the belief of Pauling /2/?

In the following we shall use the term "icosahedral crystal" to denote any spatial arrangement of atoms which produces a diffraction pattern consisting of sharp spots with icosahedral symmetry, i.e. a pattern which is invariant under the 120 symmetry operations which leave a regular icosahedron invariant. Obviously the symmetry of whatever underlying structure produces such a pattern can not be described by any of the conventional 230 space groups which are based on the assumption of discrete translational invariance.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986313

C3-136 JOURNAL DE PHYSIQUE

The object of this lecture is to demonstrate how the concept of space groups can be generalized to icosahedral crystals. It will be seen that the symmetry of al1 such structures can be described by six-dimensional space groups, composed of a six- dimensional discrete translational group and a point group which is isomorphous with the icosahedral group. The point group includes twofold planes, threefold planes, and fivefold planes. Fivefold symmetries are allowed for lattices in six dimensions, and this is the explanation for the fivefold symmetry observed in experiment.

Once the symmetry group has been constructed, the question naturally arises as to which atomic arrangements are consistent with the symmetry. It turns out that the most general atomic structure is obtained by decorating the six-dimensional unit ce11 with an arrangement of hypersurfaces which must obey the point group symme- tries, and not by decorating the Penrose tiles with an atomic basis. The Penrose tiling appears as a special limit where the positions of the atoms are given as projections of lattice points in the 6D lattice /9,10,11/. Unfortunately, the six-dimensional symmetry is rather low so it is not possible to forcefully suggest a structure on the basis of the symmetry alone; the actual structure must be deter- mined from a six-dimensional crystallographic analysis of a diffraction experi- ment. In the simplest possible structure the position of the hypersurface is spec- ified by three continuous functions of three variables which must be fitted to experiment.

For a more complete treatment on the crystallography of icosahedral crystals con- sult the original ref erences /12,13/.

II. DERIVATION OF THE 6D SPACE GROUPS

The most general structure with icosahedral symmetry (apart from certain phase shifts to be discussed later) can be written as a density

where qi, i = 1,2,.

. ^.

B(0000-ll) resulting structure arises quite naturally from a phenomenologica * B(-0-1000). Similar relations hold for the phases J,

Pi

.

and the symmetry of the diffraction spectrum matches experiments. The expression (1) defines a function po of six variables:

where the density in the real r-space is obtained by setting

The function Po is periodic in al1 its arguments with period 2%; hence it repre- sents a six-dimensional crystal. The six-dimensional function po contains precise- ly the same information as the three-dimensional function p: In six dimensions the crystal gives diffraction peaks at Q = (nl,n2,n3,n4,n5,n6) which have the same intensity as the diffraction peaks of the real 3D crystal at Xniqi. This can be seen by comparing the Fourier transform of p(F) in 3D with the Fourier transform of p(0) in 6D. The crystallographer should think of the structure as a six-dimen- sional one and perform a "traditional" analysis of the intensities of the diffrac- tion spots in order to determine the content of the unit cell.

The permutations of *qi which leave the icosahedron invariant define permutations of '01 which leave (2a) invariant. For instance, the fivefold axis along ql

Fig. 1 Fundamental wave vectors qi of the simple icosahedral (si) structure.

The body-centered icosahedral (bci) structure is formed by the 15 pairs of edge vectors Ti.

permutes the vectors q2, q 3 , 91+, q5, 6 6 cyclically, and hence PO should be invari- ant under the operation + 81, 02 + 03 + + 05 + 06 + 02. This is a fivefold plane in the 6D space, leaving the lines (100000) and (011111) invariant. Thus, the fivefold symmetry of the diffraction experiment reflects a fivefold plane'-in the underlying 6D space. The possibility of having fivefold symmetry emerges because fivefold operations are consistent with a discrete lattice in 6D. The 120 operations form the noint group of the 6D crystal. In addition to the fivefold planes the point group has threefold and twofold planes. For instance, there is a threefold plane mg spanned by the vectors (111000) and (0001-11) and a twofold plane m2 spanned by (110000) and (001001). Eq. (2b) States that the actual density in 3D space is the density along the 3D plane P in 6D space which is traced out as 7 traverses real space. This plane forms an invariant subspace since al1 symmetry operations in 6D space correspond to rotations of t. Because of the irrationality of a, ^{the plane} P samples al1 the information contained in the 6D function. The various possible icosahedral structures correspond to the various ways of choosing the periodic function (2a) or, equivalently, the basis to associate with the 6D unit cell. This basis has to obey the icosahedral symmetry.

The idea of representing a quasiperiodic function by a periodic function in a high- er dimensional space is not new. It was done first by the mathematician H. Bohr (a brother of Niels Bohr) in his study of "alrnost-periodic" functions /14/. He noted that the function LA{n)cos(Lniqix), i 1,

...

periodic function of 'the d variables qix and can thus be understood as a cut along a line in a d-dimensional space. Later, the idea was taken up by Janner and Janssen in their study of the mathematics of incommensurate crystals / 1 5 / . It is well known that incommensurate structures have long range order and produce sharp Bragg peaks although they have no Bravais lattice, so they can not be described by traditional space groups. It is thus somehow surprising that the possibility of having ordered structures not based on a periodic lattice has such difficulty in being accepted

.

C3-138 JOURNAL DE PHYSIQUE

In addition to the "simple icosahedral (si)" structure represented by eq. 1 there is a possibility of having slightly more complicated icosahedral structures. For instance, one could have a structure characterized by the 15 fundamental wave vec- tors ti forming the edges of the icosahedron

wliere the B's and the 9's obey symmetry constraints as before. The edge vectors can be written as the sum of two vertex vectors, ri = qi1fqi2, so the sum (3) can be written in the form (1) with the constraint that only terms where the q- vectors occur in pairs are present. Obviously the res!lting function is invariant under the operation qi*F + gi*P t n so the function p(8) is invariant under the translations 8 + 8 + 2 n ( + 1 / 2 , ~ 1 / 2 , + 1 / 2 , ~ 1 / . 2 , ~ 1 / 2 , ~ 1 / 2 ) . Thus, one might conve- niently denote the edge-vector icosahedral structure a "body-centered icosahedral (bci)" structure.

Until now we have not made any assumptions about the objects which form the basis.

We now want to describe structures which can be formed by arrangements of atoms in 3D space. The objects to be inserted into the unit ce11 must intersect the plane P in O-dimensional points representing the positions of atoms. Only 3D hypersur- faces, or hypersurface segments, have this property. There is a one-to-one rela- tion between the possible ways of arranging hypersurfaces on a 6D lattice and the possible ways of arranging atoms in 3D icosahedral crystals. The size and density of the atoms can be represented by associating a width to the hypersurface.

Figure 2 illustrates various periodic arrangements of hypersurfaces.

The arrangement of hyperplanes nnist obey the icosahedral space group symmetry defined above. If a surface is placed at a general position in the unit ce11 it must be accompanied by 119 others generated by applying the point group operations R to the surface, causing messy unphysical arrangements with overlapping atoms. To avoid this the surface must be placed at positions with special symmetry. The sim- plest structures correspond to hypersurfaces which are invariant under R

.

case one hypersurface in each unit ce11 will suffice. In the case of the bci structure one must place two hypersurface segments in the unit cell: one through (000000) and one through (112 112 112 1/2 112 1/2 112). The simplest invariant hypersurface is a 3D hyperplane P' perpendicular to P (Fig. 2a). Clearly the resulting crystal structure can be viewed as arising from a projection ont0 P of 6D lattice points within a distance defined by the size of the hyporplane segment.

The symmetry planes mg, mg, and y defined above intersect both P and P' along sym- metry axes. Figure 2 shows a fivefold plane intersecting P and P' along fivefold axes. In principle, the hyperplane may extend arbitrarily along Pt. If the plane segment is chosen to be the projection of the 6D unit cube onto P' the hyperplane segment becomes a three-dimensional rhombic triacontahedron and the Penrose tiling structure is recovered as shown by Duneau and Katz 1101, and Elser /Il/. The mode1 of Kalugin et al. 191 is recovered by choosing the 3D plane segment to be a

sphere. We have thus demonstrated that the tiling structure arise as special limits of much more general structures.

A rigid displacement of the 6D crystal can not change any bulk properly (such as the energy) of the crystal since the 3D plane traverses the entire unit ce11 ergodically. To any desired accuracy, the displacement can be represented by a shift of origin. The six orthogonal displacements thus define six hydrodynamic variables. The three orthogonal shifts along the 3D plane correspond to rigid dis- placements of the icosahedral crystal in 3D space. The remaining three shifts per- pendicular to the real-world plane describe certain rearrangements of the icosahe- dral structure. Since the topology of the hypersurfaces defined above usually is such that the surface can not continue indefinitely in the perpendicular direction, the rearrangements involve discontinuous displacement of some atoms, despite the fact that the energy is invariant. In the projection limit, Fig. 2a, the three phase shifts describe rearrangements of the Penrose tiles 1121.

In general the diffraction pattern can be derived simply as the Fourier transform of the content of the 6D unit cell, in analogy with regular 3D crystals. In the quasicrystal limit (where the unit ce11 contains only a plane segment) the

intensities of Bragg peaks obviously depend only on the component of the correspond vector Q perpendicular to P. Unfortunately this simple result does not apply to the general case.

The general structures can not be formed as projections, even in the simple case of one invariant surface. Symmetry does not restrict the invariant hypersurface to be a plane. It is possible to distort the-plane without destroying the invariant properties. Consider a general point (xl,O) on the plane segment along P', where the second ~ o g i n a t e indicates the component along P. A distorted plane is given by points (x',f(xl)) (see Fig. 2b).- kpplying any point group operation R to this point-yielgs the-point ( R (XI), R f(xl)). The surface is invariant as long as R (f(xl))=f( R (x')) for al1 x'. The distorted surface can be generated by rotat- ing the surEace within an irreducible ection of the plane segment along P'. The distortion f(x') is restricted only if x' is along one of the n-fold rotation axis in P' where R (X1)=X', since then the distortion must be along the corresponding n-fold axis in P to assure invariance. Figure 2b shows the intersection of the hypersurface arrangement with a fivefold plane. The surface must be symmetric around the center because of the-twofold plane perpendicular to the plane. In order to specify the position f(x1) of just one plane one must determine three functions f of three variable x'. A physical restriction on f is that the distance between planes along x must exceed a minimum atomic distance (the Delaunay condi- tion), but this is not a very severe limitation. In a realistic model there must be several surfaces corresponding to the different constituents of the alloy.

Clearly the intensities of the Bragg spots do not depend on the wave vector Q in a simple way as it wouïd for the Penrose tilings.

For small distortions f the resulting crystal structure can be viewed as a modula- tion of the Penrose tiles and Q spots with large perpendicular component will be weak. For larger distortions new classes of structures appear, for instance because it is possible to connect surfaces corresponding to different cells without violating the symmetry. Figure 2c shows a cut along a fivefold plane of such a surface. A detailed analysis reveals that it is not possible to connect the sur- faces to form extended surfaces with the topology of 3D planes. In general the resulting surfaces are multiply connected (for a lower dimensional analogue think for instance of the multiply connected Fermi surface in a cubic material like cop- per; the cubic symmetry forbids extended planes). A variety of interesting topolo- gies are possible. The connectedness implies that the crystal structure can be formed by stacking a continuous distribution of unit cells; these icosahedral crys- tals can be thought of physically as caused by the frustration due to the inability of some natural unit ce11 to fil1 space: as a compromise the unit ce11 adjusts to the local environment by distorting. Nevertheless the long range order is strictly preserved. The tiling or quasicrystal picture looses its meaning completely! The connected topology has interesting consequences for the continuous phase degrees of freedom associated with a shift of P along P': sorne atoms are shifted continuously whereas in the tiling limit the phase shifts describe rearrangement of tiles /13/.

The fact that the general icosahedral crystals can not be formed as a simple tiling should not come as no surprise. We are well aware of this phenornenon from

incommensurate crystals, where no two interatomic lengths are identical. Only in a rather unphysical limit does the incommensurate structure reduce to a tiling struc- ture composed of two different unit tiles /16/.

So where are the atoms in the particular icosahedral crystals discovered by Shechtman et al. and by Poon et al.? Unfortunately, the conclusion is that there is no simple way to solve the crystal structure since the icosahedral synmietry is sufficiently low to allow for complicated continuous possibilities for arranging the atoms. No simple mathematic model can possibly describe the structure in every detail. One might speculate that in the Mn alloy there is an Mn-hypersurface through (000000) and six Al-surfaces through (1/2 00000), but this 1s merely a

JOURNAL DE PHYSIQUE

Fig. 2 Periodic arrangements of 3D hypersurface segments in 6D space. The figures show cuts along fivefold planes. The plane P represents real space, the plane P' is the invariant suhspace orthogonal to P. The positions of atoms are at the points where the surface segments intersect P. (a) The 3D surfaces are plane seg- mente perpendicular to P. The resulting structure is a Penrose tlling, and the construction is equivalent to a projection of -6D lattice points onto P, (b) the surfaces are distorted planes and (c) the surface segments form a multiply con- nected extended surface. The right hand sided figures show the hasis associated with the unit ce11 in each of the three cases.

guess. Even if this were thecase it would be a monumental job to determine the surfaces. The stoichiometry is not of much help: the extensions of the hypersur- faces and therefore the concentrations can Vary continuously. The surfaces can even be fractals. A one-dimensional example of this is the Frenkel-Kontorowa mode1 where some of the equilibrium structures can be represented by fractal "curve seg- ments" in 2D space (Cantori). If one really wants to know where the atoms are there is no way out: one has to perform acomplete six-dimensional crystal analysis to determine the positions of the hypersurfaces within the unit cell.

Supported by the Division of Materials Sciences U.S. Department of Energy under contract DE-AC02-76CH00016.

References

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/2/ See for instance L. Pauling, Nature

x,

/3/ S. J. Poon, A. J. Drehman, and K. R. Lawless, Phys. Rev. Lett. E, ²³²⁴

(1985).

141 R. Penrose, Bull. Inst. Math.

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1. 110 (1977).

P. Kramer and R. Neri, Acta Cryst. _{A s ,} 580 (1985).

D. Levine and P. J. Steinhardt, Phys. Rev. Lett. E, 2477 (1984).

V. Elser and C. L. Henley, Phys. Rev. Lett. 55, 2883 (1985).

P. Guyot and M. Audier, Philos. Mag. B E , ~lT(1985).

P. Kalugin, A. Kitayev, and A. Levitov, Pisma ZhETF 41, 119 (1985); J. Phys.

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M. Duneau and A. Katz, Phys. Rev. Lett. 2, 2688 (1985).

V. Elser, Phys. Rev. B32, 4892 (1985).

P. Bak, Phys. Rev. ~ e t K 2 , 1517 (1985); Phys. Rev. B z , 5764 (1985).

P. Bak, Phys. Rev. Lett. 2, 861 (1986).

H. Bohr, Acta Mathematica 2, 29 (1925).

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P. Bak, in "Scaling Phenomena in Disordered Systems", p. 197, edited by R. Pynn and A. Skjeltorp (Plenum, New York and London 1985).

COMMENTS AFTER THE P. BAK TALK :

M. V. J A R I C . -

In your argument, you required that for a given star al1 phases are equal. In fact, your argument is valid even without this asumption.