A new approach to quasicrystals

Solid State Communications, Vol 65, N o 7, pp. 637-641, 1988 Prmted in G r e a t Britain

0038-1098/88 $3 00 + 00 Pergamon Journals Ltd

A NEW APPROACH TO QUASICRYSTALS G Coddens* Laboratolre L6on Bnllomn, Laboratolre C o m m u n C E A -C N R S , F91191 Glf-sur-Yvette Cedex, France

(Accepted for pubhcatton 5 October 1987 by S Amehnckx) A new method IS gwen to describe quaslcrystals It IS very easy to calculate atomic positions with this new approach The relation with the projection methods and Penrose tiling is discussed 1 INTRODUCTION A F T E R T H E D I S C O V E R Y of a five fold symmetry axis in the electron diffraction pattern of AI-14 at % M n by Shechtman et al [1] several authors tried to solve the problem of what was felt to be a paradoxal situation the experimental evidence seemed to defy a mathematical theorem that the only types of rotational symmetry compatible with translational invanance are 2, 3, 4, or 6-fold In the mean time other types of "forbidden" symmetries, e g with a 12-fold axis [2] have been observed, and the sohds displaying these special symmetry features have been bapUsed quasicrystals The mathematical tools that have been most successful in describing these quaslcrystals are Penrose tding and projection methods [3] In the present Communication we are proposing a third method that has several interesting features First o f all it gives us a good insight in how we can get around the mathematical theorem mentioned above Secondly it is easier to handle than the other ones The method is illustrated with two toy models, one with 8 fold and one with 12 fold symmetry The relation with other methods is discussed 2 METHOD First of all it should be pointed out that it is absurd to take the mathematical theorem to the letter It has been proved e g mathematically that it is impossible to construct a regular heptagone with compasses and a ruler Nevertheless there exist constructions with a precision meeting all practical needs In a lattice it will be allowed to violate the mathematical theorem provided the errors revolved are small compared to the atomic displacements due to lattice vibrations I f we can achieve this then real non forbidden lattices are not doing any better than the forbidden ones * Also at I I K W Unlversltalre Instelhng Antwerpen, Physics Department B 2610 Wllrljk, Belgium

In m a n y text books the mathematical theorem is " p r o v e d " by showing that frustration occurs If the rotation symmetry is not 2, 3, 4, or 6-fold when we are trying to tile the entire space eventually we will always end up with two unit cells that are overlapping This Is however not prohibitive If the decorations of the two overlapping tiles match in the region m c o m m o n then nothing is wrong Then the frustration is irrelevant in the sense that it concerns only the tiling (which is a mathematical artifact) and not the physical occupation of the space by the atoms Both arguments are present In Figs 1 and 2 which show toy models of 2D quaslcrystals with 8-fold and 12-fold rotational symmetry respectively Figure 3 shows how Fig 1 was generated Figure 1 is nothing else than an exploded view of a region inside zone A of Fig 3 Figure 2 was produced in a similar way by superimposing two hexagonal lattices Since zone A IS a continuation both of zone B and C we can see that It will act very much hke a twin It is well-known that twins can display diffraction patterns with forbidden symmetries We are proposing the term "intertwined structure" for this type of structure Let us show why this method works We will do this for Fig 1 The argument can be generahsed to other cases The intertwined structure consists of two sublattIces that are superimposed (In other cases there m a y be more). We will call the basis vectors of these sublattlces gt, g2 and g'~, g~ respectively We suppose that they have unit length and that the two sublattlceS (S and S') are infinite Within each o f these two sublattlceS we have perfect translational lnvarlance for tl E S, t2 ~ S =~ tj + t2 e S, with the same for S ' However the translations that leave S invanant do not apply to S ' and vtce versa the coordinates of a point of S with respect to S ' involve the number x/~ which is incommensurable with 1 However for an arbztranly small positive number e we can find an infinite number o f vector pairs t~ e S, t2' e S ' such that It~ - t2"l < e This means that up to an error e, t~ or t2' are translation vectors for the whole lattice S US" They will m a p

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the o n g m to a point that therefore will have the same symmetry as the origin, ] e 8-fold symmetry In these points we wdl postulate only one a t o m p o m l o n instead of two F o r each such point there are seven equivalent ones obtained by rotations around the origin by tn/4 (t = 1, 2, 7) So we find to an arbitrarily accurate precision long distance translational and rotational lnvanance The set of eight equivalent points could be taken as the corners of a "unit cell" for this structure By tflmg the space w~th this "unit cell' however the errors e will accumulate the set of translation vectors with precision e is lnfimte but not closed for the operations sum and multiplication by an integer The reasoning given here is readdy verified m Fig 1 it is m the points where two points more or less coincide and the two sublatUces are hooking rot• each other that we have 8-fold rotational symmetry Another way of looking at this structure Is saying that ~t Is a cubic lattice and that its dec•ration shows a modulation The second argument given above is well seen m Fig 2 I f we use the dodecagon ABC as a tile the system gets frustrated already between the first neighbourmg tdes but the overlapping is doing no harm smce the overlapping dec•rations are matching So

(Penrose) tiling 1s not doing entirely justice to this structure since it hides part of the symmetry 3 RELATION WITH OTHER METHODS Bak [3] has pointed out that one could understand the structure with lcosahedral symmetry as the projection of a perfectly p e n o & c structure with a five fold axis in R 6 to the real physical space R 3 However if one projects the whole space R 6 to R 3 the lattice becomes infinitely dense throughout Some rules have to be established to heal this SltuaUon It is also not straightforward how one should reconstruct the real structure from a diffraction pattern passing through R 6 m the reasoning In a simple case good guessing might work An example where this succeeded is in [2] The authors mention that they have &fliculties w~th the indexing and m estabhshlng what the smallest cell would be There is a close relationship between this projection method and Penrose Uhng which we will repeat here When we look at a Penrose tiling, e g m [4] this relation jumps at ones eyes the tdmg looks as a photograph of a complicated 3D construction completely made up of cubes (or rhombohedra) each vertex with 3 tdes in the Penrose pattern looks like a drawing m

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Fig 3 Generation of the quaslcrystal of Fig 1 two cubic sublattlces are rotated over an angle of 45 ° with respect to each other

perspective of a 3D cube That there is really something more behmd this ~s explained with the help of Fig 4 This is a 2D representation of a 4D hypercube Such a hypercube has eight "sides" which are 3D cubes and which are readily seen in the Figure, e g A B C D E F G H All relations of mutual touching,

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parallelhsm, etc of these cubes are present in this Figure Since a 4D hypercube can tile ~4 the projection of such a tiling will represent a tiling for R 2 The reason why all this works becomes apparent If one considers the vectors AB, AD, AE, and AI as the projections onto R 2 of the basis vectors el, e2, ej, and e4 of R 4 a s shown In the Figure The trick is in the choice of the angles between the projected vectors which is such that one ends up with 8-fold symmetry A similar trick can be designed for any other polygone The rhombs (e g A B C D or ABFE) in Fig 4 are nothing else than the Penrose t]les for tlhng R 2 with 8-fold symmetry The observation that some of the lines in the Figure intersect, l e that the octagone can be tded in several ways brings us back to the problem that too many points are projected from R4 to R 2 One can remove superfluous points by applying e g a cnterlum of visibility one Interprets the 2D figure as the photograph of a 3D construction, this time of rhombohedra, and rejects the points that are invisible since they are hidden behind a plane as shown in Fig 5 However without any further rules the structure derived may become rather disordered and it is not straightforward to derive rules to make the result

Vol 65, N o 7

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A NEW APPROACH TO QUASICRYSTALS

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F~g 5 3D interpretation o f a 2D Penrose tdmg symmetrical and to avoid frustration One gets more or less the impression that there ~s an mfinlty of choices and th~s m turn gwes the impression that it is only poslble to describe the structure along the mare hnes In our own method finally we took g, = e,, g2 = e3, g~ = e2, g~ = e4, without taking all hnear combinations o f e,, e2, e3, e4 to belong to the quasilattice as m the projection method This seems to be viable m the sense that ~t y~elds the h~ghest symmetry

than the former ones and which gives a better insight into the symmetries inside a quaslcrystal Apparently their symmetry can be so high that m a physical sense they are real crystals

REFERENCES

4 CONCLUSION

D Schechtman, I Blech, D G r a u a s & J W Cahn, Phys Rev Lett 53, 1951 (1984) Q B Yang & W D Wel, Phys Rev Lett 58, 1020 (1987) P Bak, Neutron Scattering, p 296, m Proc of

We have proposed a new method to describe quaslcrystals which is much easier to handle to users

the Int Conf on Neutron scattering held m Santa Fe, NM, USA on August 19-23, (1985) D Nelson, Sc American, 255, 232 (1986)