L. S. Levitov- Icosahedral quasicrystals with continuous phasons

8/3/2019 L. S. Levitov- Icosahedral quasicrystals with continuous phasons

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Icosahedral quasicrystals with continuous phasons

L. S. Levitov

L. D. Landau Institute for Theoretical Physics, Moscow

(Reu le 5 juin 1989, accept le 6 juillet 1989)

Rsum. 2014 Des familles de modles de quasi cristaux icosadriques sont construites. Dans cesmodles, les atomes peuvent se dplacer continment sous laction des dplacements danslespace des phasons R3* leurs distances mutuelles restant toujours plus grandes quun diamtrede coeur dur h > 0. Ces modles possdent la symtrie de licosadre et leurs proprits dediffraction sont celles de structures parfaitement quasi priodiques.

Abstract. 2014A variety of models for icosahedral quasicrystals is constructed such that atoms movecontinuously under displacements along the phason space R3*,while their positions are separatedby distances larger than some h > 0, a hard core diameter. The models exhibit icosahedralsymmetry and the diffraction properties of perfect quasiperiodic structures.

J. Phys. France 50 (1989) 3181-3190 1er NOVEMBRE 1989,

Classification

PhysicsAbstracts61.50E

1. Introduction.

For one-dimensional

quasiperiodicstructures it is known that

phasonscan be either

continuous or discontinuous, depending on the strength of the coupling [1]. These tworegimes are separated by a line of analyticity breaking transition. In almost all incommensu-rate structures studied so far (Hg3 _ sAsF6, TTF-TCNQ, etc.) phasons are continuous.Arecontinuous phasons possible for quasicrystals ? All models constructed for an ideal quasicrystal structure [2-4] lead to discontinuous phasons, i.e. each atom jumps at certainvalue of phason shift. It was also suggested [5] that large symmetry groups of quasicrystalsalways forbid continuous phasons (theorems 1, 2 of Ref. [5]). However, a counterexample tothese theorems was reported recently : an icosahedrally symmetric structure with continuousphason transformations [6].Here 1 return to this problem and construct many other structures having continuous

phasons. Results of reference [6] seem to be not completely satisfactory : a) The structuresuggested in [6] looks as a very artificially designed, so one might suspect it is unique. b) Themodel [6] is constructed for only one of 11 possible icosahedral space groups [7-9], but thesituation for other 10 groups is also of interest. c) The method of reference [6] refers strongly

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198900500210318100
http://www.edpsciences.org/http://dx.doi.org/10.1051/jphys:0198900500210318100http://dx.doi.org/10.1051/jphys:0198900500210318100http://www.edpsciences.org/


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to the readers geometrical intuition, so for some people a more algebraic approach isdesirable.

Let us describe shortly the structure of the paper. Main results of this work areconcentrated in sections 4, 5, 6. Models with continuous phasons are defined in section 2 as

periodicfamilies of

non-intersectingcontinuous atomic surfaces. The contents of section 2 is

not new, it almost entirely overlaps with references [5, 6]. Icosahedral symmetry groups arediscussed in section 3. It is shown that for constructing continuous atomic surfaces for six ofthese groups one needs to do this for only one of them. In sections 4, 5 we derive conditionswhich are necessary for the existence of continuous atomic surfaces.A wide class of solutionsof these conditions is found. In section 6 the solutions are used to construct continuous atomicsurfaces.

An important distinction of the structures constructed here and those described inreference [6] is that here we get straight linear atomic surfaces (hyperplanes in R 6) andperiodic interpenetrating lattices of atoms in 1R3, while in reference [6] the atomic surfaces arecurved and, thus, atomic positions in (iB3 form quasiperiodically modulated lattices.

Notation : trying to make the matter more transparent 1 almost everywhere use Gothicletters for algebraic objects (groups, (sub-)lattices, subspaces) in order to distinguish themfrom geometic ones (surfaces, hyperplanes, vectors). By the word hyperplane 1 denotelinear manifolds not necessarily containing the origin, in contrast with subspaces thatalways contain 0-vector.

2.Atomic surfaces, lattices and subspaces.

Atomic surfaces describe the dependence of atoms positions on phason variables. For a d-

dimensional quasiperiodic structure with D = d + d incommensurate frequencies atomicsurfaces live in the space IRD = V 0153 V * (V = IR d represents the physical space, V * = R d isthe phason space). The surfaces have dimension d. We denote them by si (i labels atoms ofthe structure). Three characteristic properties of atomic surfaces should be mentioned :

(i) (periodicity) The family of the surfaces si is periodic and has D independent periodsgenerating a D-dimensional lattice 2 in RD. This periodicity in RD reflects quasiperiodicity inthe physical space V.

(ii) (conservation of atoms) Each atomic surface si has one-to-one projection on the phasonspace V *. Thus si can be described by a function fi: V * -+ V as a set of points( f (y ), y) e RD@ where standard coordinates in RD are used : (x, y), x e V, y e V *.

(iii) (hard core condition)Atoms cannot be closer to each other than by some constanth >- o. In terms of functions fi this restriction reads :1 fi (y ) - f j (y ) 1 >- h for all i,je Z, y eV*.Atomic surfaces define positions of atoms in the physical space as follows. Choose some

a e V* (phason parameter) and consider a d-dimensional hyperplane Va consisting of all

points (x, a) e IRD,@ i. e. the space V shifted by the vector (0, a). The hyperplaneVa represents the physical space.Atoms positions are given by intersection points of thesurfaces si and Va.A family of atomic surfaces satisfying conditions (i), (ii), (iii) together with the rule offinding atoms positions constitute the most general way of describing models for quasiperio-dic arrangements of atoms. By varying the phason parameter a in the above given definitionon obtains different configurations of atoms. If the space V and the lattice 2 areincommensurate ( [incommensurability ] - [2 n V = 0 ]), then all configurations of atomsproduced by varying a are physically equivalent. In this case phason shifts are included in the


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group of Goldstone equivalence transformations of the structure (like translations forordinary crystals).

It is important that atoms positions as functions of phason shifts can be either continuousor discontinuous. Since here we are interested in continuous phasons, let us introduce the

condition(iv) (continuity)All atomic surfaces si (and corresponding functions fi) are continuous.The continuity condition enables one to introduce lattices of periods of single atomic

surfaces. For an atomic surface si its lattice li is defined as a sublattice ouf 2 consisting of alltranslations transforming si into itself. Simple analysis (presented in Ref. [5]) shows thatdim (l i) = d. Besidesi we need to introduce d-dimensional subspaces ri : ri is a real linearenvelope of vectors of li. The subspace ri is parallel to si: si remains within a fixeddistance of ri. Both lattices 1i and subspaces ri will be used below.

3. Icosahedral symmetry.

Icosahedral group acts in (R6 = V (f) V* (V == (R3, V* = (R3*) according to its 3 + 3representation (see [2-4]). Beginning from here we are interested only in icosahedrallysymmetric structures, i. e. we demand that the symmetry group of the family of the atomicsurfaces si coincides with one o f 11 icosahedral space groups.All possible icosahedral groupswere found in [7-9] (our notation of symmetry groups and related objects coincides with thatof Ref. [8]). There are 2 point groups ffi : Y (60 elements, inversion is not included) andYI (120 elements, inversion is present). Their 6-dimensional representations 3 + 3 are pointsymmetry groups of the structures considered (thus, d = d = 3). Three Bravais lattices arepossible : DSC, Fcc, 2BCC. For each pair [0, 2], ((5 = Y, Y,, 2 = 2sc, Spcc. Bcc) two

space groups exist-

symmorphic and non-symmorphic-

except for the case (5 = YI,.2 = Spcc which yields only a symmorphic group. Thus get 11 groups.For two groups H1, h2 we say that h1 includes H2 if H2 is isomorphic to a subgroup of

h1.Proposition 1 : If two space symmetry groups H1, H2 are such thata) -51 includes H2 ;b) the group -51 allows for continuous atomic surfaces, then the group H2 also allows for

continuous atomic surfaces.

Proo f : Given a family of atomic surfaces with the space symmetry group H1, one easily getsa family of surfaces with the symmetry group H2 by lowering the symmetry fromH1 to H2 (this can be done by a proper continuous small deformation of the surfaces).

In what follows we work only with the symmorphic group .!5: (fj = Y, L = sc, andconstruct continuous atomic surfaces for it. One can check that this group includes all groups(both symmorphic and non-symmorphic) with 6) = Y, L = .2sc, .2Fcc, 2BCC (totally,6 groups).According to Proposition 1, the existence of continuous atomic surfaces for thesegroups is guaranteed by the existence of such surfaces for the group H.Other 5 icosahedral groups (based on (fi = YI) will not be discussed here. Due to the

presence of inversion not all of them allow for continuous phasons (proof will be reportedelsewhere).

4. Non-transversality condition and its solutions.

Consider two atomic surfaces sl, s2 and find their lattices ll, l2 and corresponding subspacesrl, r2. Calculate the dimension of r1 p r2, the linear envelope of rl and r2. Note that


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dim ( rl ) = dim ( r2 ) = d = 3, so if dim ( ri q) r2 ) = d + d = 6, then the subspaces r, andr2 are transversal, i.e. they have only 0-vector in common. In this case the surfacessi, S2 are also transversal : they intersect and, moreover, their intersection is topologicallyunremovable (see Refs. [6, 10]). But intersections of atomic surfaces are unphysical- theycannot exist due to the hard core condition. Thus, we obtain

Proposition 2 : For every two atomic surfaces si, sj the corresponding subspacesri rj are not transversal :

For moving further we need to write the non-transversality condition (4.1) in thecoordinates (x, y), x e V, y E V *.But first let us see how the subspaces ri can be described using these coordinates. Since for

any i e Z dim ( ri ) = 3 and ri has one-to-one projection on V*, one can find a linear

transformationi : V *- V

such that all points of riare

given by (i (y), y), ye

V *.Afterbases are chosen in V and V *, each subspace ri is given by 3 equations : xa = At/3 y/3

03B2

(03B1, f3 = 1, 2, 3 ), whereAt/3 is a matrix corresponding toi. Note that x and y live indifferent spaces (V and V * respectively). Consequently, transformation rules for the matrix

At{3 are non-standard : if coordinates are changed by (x, y ) _ (RI (x), R2 (y), then

(here R1, R2 are arbitrary 3 x 3 matrices).Non-transversality condition (4.1) means that ri and rj have a common nonzero vector. Let

it be (i (y), y ) _ (j(Y), y). We obtain y = y,i (y) =j (y), that can be finally rewrittenas

This is an algebraic form of the geometric restriction (4.1).Now we turn to the analysis of the action of the symmetry group -5 on the subspaces

ri. Beginning from here we use only orthonormal bases in V and V *. Operations of the pointgroup (5 are given by matrices 6 x 6:

where G and Gare 3 x 3 orthogonal matrices of the representations 3 and 3 respectively, is 0-matrix.The set of subspaces ri is symmetric under the point group (4.4), since the surfaces

si are symmetric under the space group .5 . Let i be a 3 x 3 matrix connected withri as explained above. Symmetry implies that not only ri but also other subspaces of the orbit

{Rg[ri]}are present. After applying transformation (4.4) to i and using (4.2) get

R9 [i] = Gi G-1. Non-transversality condition (4.3) thus reads


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Sixty equations (4.5) impose strong constraints on possible matrices i. One solution ofequations (4.5) was found in reference [6]. Here we get many other solutions.

Proposition 3 : The orthogonal matrices e SO (3 ) solve non-transversality equations (4.5).

Proof : If E SO (3 ) then det- det1 since an orthog-

onal 3 x 3 matrix with det = 1 always has at least one eigenvalue 1 (here is unit matrix).One could be interested whether there are other solutions of equations (4.5).Although the

whole set of solutions is not known, one can mention a) matrices proportional to orthogonal : , where E R, E SO (3 ) ; b) matrices corresponding to the lattices laf3Y introduced inreference [6]. Probably, other solutions also exist. We are not looking for them here, sinceorthogonal matrices suffice for our needs.

5. Rationality of the subspaces ri.

We have constructed a wide class of solutions of non-transversality equations (4.5). However,besides non-transversality there exists another important restriction on the subspacesri- their rationality. Indeed, r1 is a linear envelope of li, a sublattice of 2. Thus,

ri is rational, i.e. it is commensurate with 2 ( [rationality ] - [commensurability ] -[dim (ri n ) = dim ( ri ) ] ). Let us study which of orthogonal matrices generate rationalsubspaces.For this study a special choice of bases in V and V * will be crucial. It turns convenient to

work with a cubic coordinate system in which the axes are aligned along 3 perpendicular 2-fold symmetry axes of the icosahedral group (for details see Ref. [11] where this system was

elaborated upon). The bases inV

and V* are chosen so that vectors of the lattice2sc projected on V and V * have the form

Here h, h, k, k,l, f are integers such that h + k, k + l and + h are even numbers ;T = (J5 + 1 )/2. Vectors commensurate with sc are given by (5.1) with arbitrary rationalh, h, k, k, f, f. It turns convenient to modify representation (5.1) and write vectors ofR6 commensurate with Ssc as

where a = (ai , a2, a3 ), b = (bl, b2, b3 ) E R3 have rational components : aa, ba e Q(a = 1, 2, 3 ). The connection of (5.2) and (5.1) is given by bi = h, b, = k, bi = l,

Let us explore rationality of ri using representation (5.2). Rationality of a vectora = (E1 (y ), y) means that equations

can be solved

simultaneously.From

(5.3)find a =

R (b ),where

Rationality of the subspace implies that R is a rational matrix (all its matrix


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elements are rational numbers).Another important observation is that R is an orthogonalmatrix : R E SO (3). This is a direct consequence of two facts :

a) Rand commute, hence they have common eigenvectors ;

b)Function z -

(T 2 Z+

1 )/ (z+

T 2)transforms the

circle[ z1 = 1 in the

complex z-planeinto itself.

Thus, we come to

Proposition 4 : Orthogonal matrices generating rational subspaces are parametrized byrational orthogonal matrices R :

In order to see how large is the set (5.5) of allowed matrices we prove

Proposition 5 : Matrices of the form (5.5) are dense in the group SO (3 ).For the proof of Proposition 5 we need to prove

Lemma : Rational orthogonal matrices so (3) f1 GL3(Q) are dense in the groupSO (3 ).

Proof of Lemma : Each matrix R E SO(3) can be written using Euler parametrization

Here TX (a ) (T y ({3 ), T z ( l is a matrix of rotation around the axis x (y, z ) by an angle

a (f3, y ). The matrix elements of Tx( a), Ty(f3), Tz( 1)are 1, 0, cos (a ), sin (a ),

cos (13 ), sin (f3), cos (y ), sin (y ). It is well known that the angles a for whichcos (a ) and sin (a ) are rational numbers are dense in the interval [0, 2 7T] (this is an easyconsequence of the existence of infinitely many Pithagoras triplets a, b, c E Z such thataz+ bz = C2). Since the matrix product (5.6) is continuous as a function of a, /3, y, rationalmatrices can be found in any vicinity of any matrix of SO (3 ).

Proof o f Proposition 5 : Consider a transformation p : SO (3) --+ SO (3 ), p () =

Applying the method used above to prove orthogonality of

R we find that p is a continuous one-to-one transformation. From Lemma we know thatrational matrices are dense in SO (3 ). But continuous one-to-one image of a dense set is adense set.

Thus we found a wide class (5.5) of orthogonal matrices yielding orbits of non-transversalrational subspaces. Matrices (5.5) will be used in section 6 for constructing a family oficosahedrally symmetric continuous atomic surfaces.

6. Continuous atomic surfaces.

The atomic surfaces discussed here are always linear, i.e. they are hyperplanes. Thus we usethe words surface and

hyperplane without

distinguishingbetween them. We

beginwith

describing our Basic construction. Choose an orthogonal matrix satisfying (5.5) and a vectora = (xo, yo ) E V # V *. We consider an atomic surface


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One checks easily that (6.6) combined with periodicity yields (6.5). Thus we have to choose and a so that the intersections of hyperplanes h (s ) of the family (6.2) are absent.

For some satisfying (5.5) we take a = 0. Obviously, in this case intersections are present.Try to remove them by varying a. Let two different surfaces si , so belonging to

intersect : Then where, according to theabove discussion, gl =F g2. Change a, then The intersection

cannot be removed if

We will see that condition (6.7) has a more convenient representation in terms of the subspaceWl , orthogonal complement to W12. We find

But,

Hence, W 2 has dimension -- 1, it is spanned by all vectors

where vectors u E V are such that always has a

solution since is an orthogonal 3 x 3 matrix). Rewritten in terms of vectors

au of the form (6.10), condition (6.7) reads : or

for all au E W 2 given by (6.10). Since for all a condition (6.11) is true (see (b.7)), we obtainTaking au from (6.10) and R9n1 from (4.4) get Gi 1 (u ) - GZ 1 (u ) = 0

and

-

or, finally,

Clearly, implies so conditions (6.12) mean that both u andare symmetry axes of the icosahedral group Y in the spaces V and

V* of its representations 3 and 3 respectively. Each of these representations has 15 2-fold

axes, 10 3-fold axes and 6 5-fold axes- totally, 31 symmetry axes. We introduce unit vectorsaligned along the axes (a = 1,..., 31 ). We impose the

following,restrictions on :

If (6.13) is true, then (d.l2.i) is not compatible with (6.12.ii) and, hence, for allg1 =P 92 one can choose a such that (6.11) is not true. This, in its turn, makes condition (6.7)


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false for all gl, 92 (g1 # g2). Thus, all intersections can be removed by a proper choice of a if satisfies (6.13).

Note that, due to periodicity, no new intersections will appear if a is sufficiently small.Each of 2 x 312 restrictions (6.13) erases a 1-dimensional curve from the group

SO (3), where lives. This is not crucial, since SO (3) is a 3-dimensional manifold and, thus,the forbidden regions have zero measure.We see that, although important, conditions (6.13) are not very restrictive. They enable

one to choose so that all intersections can be eliminated by a properly chosen vector a.To summarize the results, an orthogonal matrix can generate non-intersecting atomic

surfaces (6.1) (6.2) if it satisfies conditions (5.5), (6.13). Then hard core condition is

guaranteed by periodicity. The set of good matrices is dense in SO (3 ).

7. Discussion.

We have constructed a variety of models with continuous phasons for icosahedral quasicrys-tals. The main features of the obtained structures are the following.

(i) In the physical space V = (R3 each family of parallel surfaces (hyperplanes) (6.3)generates atoms positions arranged in a 3-dimensional periodic lattice. The whole family ofsurfaces (6.2) yields 60 interpenetrating lattices of atomic positions.

(ii) Diffraction properties of these structures coincide with those of conventional discon-tinuous models : icosahedral symmetry + perfect 3-peaks.

(iii) By lowering the symmetry of the atomic surfaces one can get a structure symmetricunder any of 6 icosahedral space groups having point groups (5 = Y.

The main distinction between continuous and discontinuous atomic surfaces is in thecharacter of phason modes. Phasons exhibit massless diffusive dynamics in the continuouscase (observed by scattering of neutrons in Hg3 _ sAsFb and other incommensurate structures).Probably, in the discontinuous case phasons are pinned and give contribution only to statisticsbut not to dynamics (at least at low temperatures).The results of this work show that both discontinuous and continuous phasons are possible

for icosahedral quasicrystals. This enables one to conclude that an analyticity breakingtransition between these regimes must exist.

Acknowledgements.

1 am grateful to P.A. Kalugin for his interest and discussions.


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[3] DUNEAU M., KATZA., Phys. Rev. Lett. 54 (1985) 1730.[4] ELSER V.,Acta Cryst.A 42 (1985) 36.[5] FRENKEL D., HENLEY C. L., SIGGIA E., Phys. Rev. B 34 (1986) 3649.[6] KALUGIN P.A., LEVITOV L. S., Int. Jor. Mod. Phys. B (1989) in print.[7] JANSSEN T.,Acta Cryst.A 42 (1986) 261.[8] LEVITOV L. S., RHYNER J., J. Phys. France 49 (1988) 1835.[9] ROKHSAR D. S., MERMIN N. D., WRIGHT D. C., Phys. Rev. B 35 (1987) 548.

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L. S. Levitov- Icosahedral quasicrystals with continuous phasons

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