Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell model of quasicrystals

PHYSICAL REVIEW B 68, 064102 ~2003!

Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell modelof quasicrystals

Hyeong-Chai Jeong1 and Paul J. Steinhardt2

1Department of Physics, Sejong University, Seoul 143-747, Korea2Department of Physics, Princeton University, Princeton, New Jersey 08544, USA

~Received 14 December 2002; published 7 August 2003!

The quasi-unit-cell picture describes the atomic structure of quasicrystals in terms of a single, repeatingcluster that overlaps neighbors according to specific overlap rules. In this paper, we discuss the preciserelationship between a general atomic decoration in the quasi-unit-cell picture of atomic decorations in thePenrose tiling and in related tiling pictures. Using these relations, we obtain a simple, practical method fordetermining the density, stoichiometry, and symmetry of a quasicrystal based on the atomic decoration of thequasi-unit-cell taking proper account of the sharing of atoms between clusters.

DOI: 10.1103/PhysRevB.68.064102 PACS number~s!: 61.44.Br

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I. INTRODUCTION

Quasicrystals are solids with quasiperiodic translatioorder and rotational symmetry that is disallowed for periocrystals, such as tenfold symmetry in the plane or icosadral symmetry in three dimensions.1 This paper focuses onthe quasi-unit cell description2 of decagonal quasicrystalthat are quasiperiodic in thexy plane and periodic along thz direction.3,4

The conventional description of decagonal quasicrystaa periodic stacking of quasiperiodically ordered layers, eof which consists of a tiling formed from two tile shapes thfill the layers without any overlaps or gaps. Each type ofcorresponds to some specific atom cluster. In general, mohave been considered in which the tiles within the layerarranged randomly5 or in a perfectly quasiperiodic~Penrose-like! pattern, depending on whether the solid is entropica6

or energetically stabilized. The perfectly quasiperiodic ptern can be viewed as a projection from a five-dimensioperiodic hypercubic lattice of points onto an incommensursubspace7 or a cut through a hypercubic lattice of ‘‘atomsurfaces’’~where the intersection between the projection sspace and the atomic surfaces is a point!.8

We have proposed an alternative description in whichquasicrystal structure is viewed in terms of a single repeaunit called a ‘‘quasi-unit-cell.’’2,9,10 For tenfold symmetriclayers, the quasi-unit-cell is typically chosen to be a decagcorresponding to a decagonal atomic cluster. A key diffence between quasi-unit-cells and tiles is that the quasi-ucells overlap. The overlap corresponds physically to the sing of atoms by neighboring decagonal clusters, a feathat is observed in real quasicrystals.11–13 The overlaps arerestricted to certain relative positions and orientations ofquasi-unit-cells according to ‘‘overlap rules.’’ The overlarules replace the Penrose rules for joining tiles. In real qsicrystals, these overlap rules are automatically enforcedthe atomic decoration, which only permits sharing corsponding to the allowed overlaps. For the decagonal cthis means that the decoration breaks tenfold symmetry.

The tiling picture and the quasi-unit-cell picture are maematically equivalent10,14–16, but the latter has numerous a

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vantages. First, the quasi-unit-cell picture encodes the enstructure, both the symmetry and the detailed atomic dection, within the single decagonal cluster.2,17 This simplifiesthe problem of finding the atomic structure based on empcal data. Second, the quasi-unit-cell suggests simple eneics that may explain why quasicrystals form.9,10,18The quasi-unit-cell structure can arise by maximizing the density ochosen cluster of atoms. If the atomic cluster is energeticpreferred, the quasicrystal could be the ground state confiration, which has maximum density of lowest enerclusters.10

In this paper, our aim is simply efficiently characterizinand solving for the structure of the solid with arbitraatomic decoration using the quasi-unit-cell picture withoany consideration of the energetics.~Once the structure issolved, a detailed understanding of the atomic interactican, in principle, be used to determine if the structure forbecause the atom cluster is energetically preferred orother energetic reasons, but this is beyond the scope ofpaper.! We first discuss the precise relationship betweatomic decorations in the quasi-unit-cell and the rhombPenrose tiling pictures, which is rather subtle, and to presa method for computing the density and stoichiometry oquasicrystal given the atomic decoration of the quasi-ucell. The latter is a simple but powerful tool for comparindata to proposed atomic structural models. For periodic ctals, the analogous calculation is trivial since the density astoichiometry are the same as those of the unit cell becathe unit cells join face-to-face. The technical complicatifor quasicrystals is finding a method that properly accoufor the overlap and avoids double counting or undercountiThis paper presents a straightforward method for handthe problem that has already been used in analyzing dat17

Section II discusses the relationship between the overping decagon quasi-unit-cell picture, the rhombus Penrtiling, and related tilings as ideal geometrical constructio~without atomic decoration!. Section III discusses the relationship between their atomic decorations. Section IV dcusses how this relation can be used to construct a mefor computing the density and stoichiometry based onatomic decoration of a quasi-unit-cell. Section V discussesinteresting subclass of quasi-unit-cell decorations, ‘‘alm

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HYEONG-CHAI JEONG AND PAUL J. STEINHARDT PHYSICAL REVIEW B68, 064102 ~2003!

occupied decorations’’ for which we have a very simpmethod of computing density and stoichiometry. Sectiondiscusses how to classify the symmetry of the atomic strture based on the decoration of the quasi-unit-cell.

II. RELATION BETWEEN DECAGON TILINGSAND PENROSE TILINGS

Figure 1 shows the overlap rule introduced by P. Gummthat forces a perfect quasicrystalline structure with a sinrepeating, decagonal quasi-unit-cell, the marked decashown in Fig. 1~a!. The overlap rule allows two decagonsoverlap only if the shaded regions overlap and if the overarea is greater than or equal to the hexagon overlap regioFig. 1~b!. This permits five types of pairwise overlaps, fotypes ofA overlaps and one type ofB overlap as shown inFig. 1~b!. The two types of overlaps correspond to two serations between clusters whose ratio is the golden ratio.infinite arrangement of decagons according to the aboveis called the decagonal ‘‘covering,’’ to distinguish it from‘‘tiling’’ in which the units join edge-to-edge without overlap. The covering can be mapped into a conventional edto-edge tiling in at least three ways:

~1! Convert the covering to a rhombus Penrose tilingdividing up central area of each decagon into a Jack confiration ~known in the Penrose- tiling literature to mean 5and 2 skinny rhombi surrounding a Jack vertex!. This con-struction leaves three skinny rhombus-shaped regions inthe decagon. When neighboring decagons overlap, thesgions are filled in by skinny Penrose tiles or a fraction of twfat Penrose tiles joined together, depending on the overSee Fig. 2~a!.

~2! Convert the covering to a three-tile ‘‘core-area tilingusing the core-area assignment. The core areas correspoan area assigned to a given decagon in a decagon covaccording to whether there areA or B overlaps on the sidesThe core-area tiling is useful in various computations woverlapping decagon tilings and is a quasiperiodic tilingits own right. The fundamental decagon of Fig. 1~a! ismarked with two ‘‘rocket’’ shapes and one ‘‘star’’ shape. Ththree types of core-area tiles are determined by the overof the star-shaped region. They are a large rhombus~corre-sponding to twoB overlaps!, a trapezoid~a B overlap on oneside and anA overlap on the other!, and a large kite~corre-sponding to twoA overlaps!. See Fig. 2~b!. If we draw only

FIG. 1. ~Color online! Marked decagon and overlapping rulethat force a quasiperiodic tiling.~a! A marked decagon can forcequasiperiodic tiling when it is arranged by the Gummelt overlaping rules.~b! Gummelt overlapping rules demand that two decgons may overlap only if shaded regions overlap and if the oveis no smaller than the hexagon overlap region inA.

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the long edges of the core areas, we get a hexagon-boatiling.13,19

~3! Convert to the original Penrose tiling20 consisting ofstar, ship, pentagon, and rhombus tiles by joining the cenof decagons. See Fig. 2~c!.

The mapping can be reversed to produce a decagonalering from a conventional tiling. The equivalence betwethe lattice structures of the decagonal covering and thosPenrose tiling has been shown explicitly.10,14 However, therelationship between the atomic models constructed byquasi-unit-cell decorations and those obtained by Penrtile decorations is a different issue and has not been careinvestigated previously. We will call the former atomic moels as decagonal quasi-unit-cell models and compare twith the latter Penrose-tile models. For Penrose-tile modwe will consider rhombus Penrose-tile decorations.

III. ATOMIC DECORATIONS OF QUASI-UNIT-CELLSAND PENROSE TILES

We define a Penrose atomic model to correspond to derating each fat rhombus identically, each skinny rhombidentically, and then joining them to form a Penrose tiling.quasi-unit-cell model corresponds to decorating a fundamtal decagon and then covering the plane with the identdecorated decagon according to the overlap rule. The dration must satisfy the condition that, where two atoms ovlap, they are the same atom type. The rule is that this resents a single atom shared by the two decagon clus~There can be more than two overlaps, too, in which caseatom is assigned to be shared by three decagons.! If an atomin one decagon overlaps an unoccupied site in a neighsite, the atom is still assigned to the spot and shared. Forgiven site, the associated ‘‘image’’ sites are positions infundamental decagon that can overlay that site when dgons are joined according to the Gummelt overlap rules. Tnumber of image sites varies, depending on where the onal site is in the fundamental decagon.

One might suppose that atomic decorations of the qu

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FIG. 2. ~Color online! Decagon covering can be mapped intoconventional tiling by decorating decagon properly.~a! RhombusPenrose tiling by dividing up central area of decagon into Jconfiguration.~b! Three-tile core-area tiling using the core area asignment of Jeong and Steinhardt~Ref. 10!. ~c! Original Penrosetiling ~Ref. 20! of four-tile tiling by joining the centers of decagons

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RULES FOR COMPUTING SYMMETRY, DENSITY, AND . . . PHYSICAL REVIEW B68, 064102 ~2003!

unit-cell models form a subset of the Penrose-tile modelsto there being a single unit and its being subject to overdecoration constraints. In contrast, as we will show next,set of atomic decorations of the decagonal quasi-unit-models is greater than the set of atomic decorations oPenrose tiling model with the same edge length.

Lemma I. Every rhombus Penrose-tile model is equivaleto a decagonal quasi-unit-cell model with the same elength.

Proof. A fundamental decagon can be subdivided intoJack configuration of fat and skinny rhombi plus thrskinny-shaped surrounding rhombus areas. Decorate eacand skinny rhombus in the Jack configuration accordingthe Penrose decoration. Place no atoms in the three skishaped surrounding rhombus areas. This defines a decorof the fundamental decagon. When the decagons are joby the overlap rules, the Jack configurations join to formfull Penrose tiling with no gaps or overlaps. This rescomes from the proof of equivalence between Penrose tiand decagon tiling from Steinhardt and Jeong.10 ~The skinnyoutlying regions are always overlapped by Jack regionsneighboring decagons and thereby resolve into a skirhombus or parts of a pair of fat rhombi.! Hence, the decagondecoration of the fundamental decagon obtained by this cstruction produces the same result as the Penrose decorobtained by joining identically decorated fat and skinny Prose tiles.

Lemma II. Some decagonal quasi-unit-cell models areequivalent to the rhombus Penrose-tile model with the saedge length.

Proof. This lemma asserts that the converse of Lemmanot true. Consider a configuration obtained by placing oatom at the center of the fundamental decagon as showFig. 3~a!. When the decagon is resolved into a Jack confiration, one fat rhombus has an atom, whereas the otherdo not. When the decagons are joined to form an ovetiling, many obtuse rhombi remain unoccupied despiteoverlap@see the upper part of Fig. 3~a!#. Hence, the decago

FIG. 3. ~Color online! Quasi-unit-cell models and Penrose-timodels.~a! Decagon decoration at the top is not a decorationoriginal Penrose tile~upper part! but a decoration of single inflatetiles ~lower part!. ~b! This decagon decoration is not a decorationoriginal ~upper left! Penrose tile, nor single inflated~upper right!tile. Neither doubly~middle! nor triply ~bottom! inflated Penrosetiles are decorated identically.~c! Decagon decoration of~b! is aPenrose decoration of quadratically inflated Penrose tiles.

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decoration is not equivalent to a Penrose-tile decorabased on these fat and skinny rhombi with the same elength.

We have shown that the set of atomic decorations of degons is strictly greater than the set of atomic decorationsPenrose tiles ofidenticaledge length. However, the relationship between the two is more subtle than one might suppLet us next compare the atomic decorations of the decagto decorations ofinflatedPenrose tiles whose edge lengthgreater than that of the decagons. For example, the quunit-cell model of Fig. 3~a! is not a Penrose-tile model withthe same edge size but it is a Penrose-tile model with~single! inflated rhombi as shown in the lower part of Fi3~a!. Each inflated fat rhombus is decorated identically aso is each inflated skinny rhombus. Is this true for a geneatomic decoration of the quasi-unit-cell model? The answis no, as can be seen from the example of Fig. 3~b!. Theinflated fat rhombi at the right upper corner are decorawith zero, one, or two atoms depending on the context. Nther doubly ~at the middle of the panel! nor triply ~at thebottom of the panel! inflated Penrose rhombi are decoratidentically. For example, some triply inflated fat tiles hasix atoms while some others have five. Therefore, the quunit-cell decoration is not generally equivalent to a Penrtiling even when one considers Penrose tiling edge lengthat aret3 times that of the decagon.

However, an equivalence does emerge if one considePenrose-tile model with fourfold inflated Penrose tilesshown in Fig 3~c!. In fact, any quasi-unit-cell model can breinterpreted as a Penrose-tile model with fourfold inflasupertiles.21 Here, we present an outline of the proof.~For arigorous proof, see Jeong, Ref. 21.! First, note that we canget a decoration of inflated Penrose tiles with decagons ifoverlay the inflated Penrose tiles on a decagon coveringshown in Fig. 4~b!. Due to the equivalence between the decgon covering and Penrose tiling,10,14 the atomic decorationsof the inflated tiles would be identical~independent of con-text! if every decagon~in the covering! were decorated identically. However, in general, there can be nine distinct psible types of decagon decorations corresponding to theconfigurations of overlapping neighbor decagons as showFig. 4~a!. Yet, even if the decagons are not identically decrated, the inflated tiles can be decorated identically ifdecagon decoration at equivalent positions in the inflatiles is the same independent of the context. One can sthat this is always the case for the fourfold inflated tiles usthe deflation rule for decorated decagons.21 The lower rightpanel of Fig. 4~b! illustrates the result in a finite part odecagon covering. We put the overlap type of each decain a decagon covering and overlay them with the inflatedtile. We see that the decagon arrangement of all inflated tof the same shapes is identical but their types are not idtical for doubly and triply inflated tiles. However, for thquadratically inflated fat tiles, we see that the types of degons are also identical as shown in the lower right panel

This relationship gives precise quantification of the dgree to which the quasi-unit-cell picture simplifies the prolem of defining~and finding! atomic structures of quasicrystals. That is, the quasi-unit-cell description and the Penrtiling description~allowing for fourfold inflation! are equiva-

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IV. COMPARING DENSITY AND STOICHIOMETRYIN QUASI-UNIT-CELL AND TILING MODELS

We define the number density ofx-type atom,nx , as theaveragenumber of atoms of typex per unit volume in themodel, wherex5a, b, g, . . . represents the atomic typof interest.~Greek letters,a, b, g, . . . signify the differentatom types.! For the purposes of this discussion, we will moout the periodic direction in three-dimensional~3D! decago-nal quasicrystals, which can be handled trivially. Therefonx can be considered as the average number ofx-type atomsper unit area in the quasicrystalline plane. In this plane,take the edge of the decagonal quasi-unit-cell as thelength.

FIG. 4. ~Color online! ~a! Nine ways of surrounding a decagofound in a Gummelt decagon covering~Ref. 14!. The numbers rep-resent the surrounding configuration types.~b! Typed decagon ar-rangements in the decagon covering. The decagon arrangemeall inflated tiles of the same shapes are identical but their typesnot identical for doubly and triply inflated tiles. For the quadracally inflated fat tiles, the types of decagons are also identicashown in the lower right panel.

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Note that the number density ofx-type atoms in any finiteregion is, in general, different fromnx due to the quasiperi-odicity of the quasicrystalline lattice. We definen(x,C) as thenumber density ofx-type atoms in a finite areaC for a givenlocal context. For example,n(x,Ci )

represents the numbe

density ofx-type atoms in the core area@Fig. 2~b!# of thedecagon at the center of thei th-decagon [email protected]~a!#, where i 51, . . . ,9. Thestoichiometry of thex-typeatom,sx , is the ratio of thex-type atoms to the total numbe

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Analogously, the mass densityrx of the x-type atom withatomic massmx is is rx5mxnx , wherer5 (xmxnx is thetotal mass density.

To compute the number density in a decagon tiling,can first convert to a conventional tiling, where the calcution is trivial because tiles join edge-to-edge. The only coplication for tilings is for atoms at the boundaries where tconventional approach works. Namely, if each boundatom is treated as a disk with small finite radius, the atomthe boundaries are assigned a fractional weight accordinthe fraction of the disk that lies within the tile.

In the preceding section, we show that a quasi-unit-cmodel is a rhombus Penrose-tile model with edge lengtht4.Each fourfold inflated fat (F4) and skinny (S4) tile is deco-rated identically. Therefore, in principle, we can calculatenumber density ofx-type atomsnx in a quasi-unit-cellmodel in the following way. First, find the decorations ofF4andS4 that are equivalent to the given quasi-unit-cell modThen, count the number ofx-type atoms in aF4 , N(x,F4) ,

and in aS4 , N(x,S4) , including the fractional atoms at th

boundaries. Since the area ofF4 and S4 are AF4

5t8 sin (2p/5) and AS45t8 sin (2p/10), respectively, the

number densities ofx-type atoms inF4 andS4 are given by

n(x,F4)5N(x,F4) /t8 sin~2p/5!,

n(x,S4)5N(x,S4) /t8 sin~2p/10!. ~3!

The number density ofx-type atoms in the modelnx is nowobtained by considering the fractional area occupied byF4andS4 tiles:

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aret.Another way to calculate the density and stoichiometry

using core areas,10 which have the advantage that they anot so large. We first assign the core tile to each decagothe decagon covering as described in Fig. 2~b! and then workout how the decagon decoration translates into a decoraof core-area tiles. In general, the same shaped core-areacan be decorated differently due to the overlap. Therenine different decorated core-area tiles due to the nine difent ways of surrounding a decagon of Fig. 4~a!. The compu-tational method is to determine the number density for eof the nine configurations and then use the density of enine decagon-surrounding configurations in a decagon Prose tiling. Using the perpendicular-spacer-map volume, onecan calculate the density of each nine configurations:

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and the ratios ofn(x) give relative stoichiometry.

V. SIMPLE COMPUTATION OF DENSITYAND STOICHIOMETRY

In the preceding section, we presented two methodscalculating the density and stoichiometry for general quaunit-cell models. However, both methods are somewhat cbersome to use in practice. To calculate the density andichiometry for a given quasi-unit-cell decoration using tfirst method, we must figure out the atomic decorationsthe fourfold inflated fat and skinny tiles. The second methrequires the atomic decoration of the center decagon in eof the nine different configurations of nearest-neighbor ovlapping decagons. A computer code is necessary to dobookkeeping.

However, there is a subclass of decorations that incluthe cases of most practical interest that can be calculatehand. This class, which we call the almost occupied dection ~AOD!, encompasses all decorations where core aare decorated the same for any of the nine configurationswe can forego treating each of the nine separately. This cincludes most decorations with physically reasonable deties. It still includes and is larger than the Penrose-tile derations~with same edge length!.

First, let us consider the even simpler fully occupi

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decoration~FOD! in which overlap does not add any atomto the core areas.

Definition. A fully occupied decoration~FOD!: A singledecagon decoration is a fully occupied decoration if, for eery atom in the decagon, the image siteslying within the‘‘big kite’’ core area are all occupied by the same atom typ

Here, the ‘‘big kite’’ core area is the kite core area showin Fig. 5~a!. Note that images outside the big kite core armay be unoccupied. The special property of FOD is toverlaps can be ignored in computing stoichiometry. Thekite core area is the largest of the core areas and all ocore areas are subregions. If the decoration is fully occupthen everyatom anywhere in the decagon has all occupimage sites inside the big kite~and, therefore, inside each othe core areas!. Therefore, overlapping neighbor tiles donot add any new atoms inside core areas in any case. Thfore, the stoichiometry can be computed by considering othree core-area types.

Let three Roman lettersA, B, C signify the three types ofdecagons:

~a! A represents decagons with twoA overlaps,~b! B represents decagons with twoB overlaps,~c! C represents decagons with oneA and oneB overlap.Here, we refer only to overlaps on the two sides of de

gon. ~In Fig. 6, the center decagons in the configurationand 2 areA type, 3 – 6 areC type-and 7-9 areB type.!Associated with each type of decagon above, we can ass‘‘core area’’ of the decagon,

~a! A: kite shape~b! B: rhombus shape~c! C: trapezoid shape

FIG. 5. ~Color online! ~a! The big kite for FOD. For an FODmodel, image sites lying in the shaded big kite region mustfundamentally decorated.~b! The regions for the number densitweights for the FOD models. Weightsr

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II5t21r

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assigned to the atoms in the regions I and II, respectively.~c! Thebig and the tiny kites for AOD. For an AOD model, image sitlying in the shaded region must be fundamentally decorated.~d!The regions for the number density weights for the AOD modeThe region II is the same as in~b!. Weights for the region I and IIare the same as the FOD case. For atoms in the regions III, we0, r

III, or r

III8 can be assigned~see text for details!.

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as shown in Fig. 2~b!, and these core areas join to completely cover the plane with no gaps.

Let N(x,X) represent the number of atoms of typex lyingwithin the core area of anX decagon whereX5A, B, andC.Note thatN(x,X) can be counted directly from the fundametal decoration~the decagon decoration before overlap! sinceoverlaps with the neighbor decagons do not add any natoms inside core areas. The number densitynx for atomtype x is now given by

nx5rAN(x,A)1r

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CN(x,C) , ~6!

whererA52t23r

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Equivalently, we can divide the fundamental decagon iregions and assign the weight for each region as showFig. 5~b!. Since the the core areaA ~kite shape! andC ~trap-ezoid! include the core areaB ~rhombus shape!, the atoms inthe region I should be counted fully when we calculatenumber density. Region II does not belong to the core areBbut 50% of it belongs to the core areaC and the whole areabelongs to core areaA. Therefore, the number densitweights,r

Iandr

IIassigned with regions I and II are give

by

rI5r

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B1r

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rII5r

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1

2r

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0. ~7!

Now, we introduce the AOD class models, for which tsame shape of core areas are decorated identically as foFOD class models but the core-area decoration~after over-lap! may be different from that of the fundamental decotion ~before overlap!.

FIG. 6. ~Color online! Overlaps between the core area of tcentral decagon and the surrounding decagons. The region IFig. 5~d! can overlap with the tiny central kite area of the centdecagon for configurations 3–9.

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Definition. An almost occupied decoration~AOD!: Asingle decagon decoration is an almost occupied decoraif, for every atom in the decagonoutside of the ‘‘tiny centralkite’’ area shown in Fig. 5(c), the image sites lying within the‘‘big kite’’ core area are all occupied by the same atom typ

Lemma IV. All core areas of the same shape havesame decoration after overlap. This lemma means thatwith the FOD, we do not have to consider each of the nlocal arrangements of decagons separately.

Proof. For an FOD class model, the same shape of careas have the same decoration since the overlap doebring any new image atoms to the core areas. For an Aclass model, new image atoms may exist in the tiny cenkite area. However, due to the way the decagons overlapimage atom arrangements in the central kite are the sameach type of core, areas as shown in Fig. 6. When the cecore area is the big kite shape~configurations 1 and 2!, over-lapping decagons cannot add any images on the centralarea. Therefore, the big kite-shaped core areas~for both con-figurations 1 and 2! have the same images. When the centcore area is the rhombus shape~configurations 7, 8, and 9!,two neighboring decagons overlap with the tiny central karea. For all three configurations, the relative position athe direction of the overlapping decagons are the saTherefore, the image sites in the tiny central kite area aresame for all three cases. Finally, when the central core arethe trapezoid shape~configurations 3, 4, 5, and 6!, only oneneighboring decagon can overlap the tiny central kite arThe position and the direction of the overlapping decagare the same for the same orientation of the trapezoid~con-figurations 3 and 4, and configurations 5 and 6 are the sorientation, respectively!. Therefore, the trapezoid-shapecore areas with the same orientation have the same imaAltogether, the image sites in the tiny central kite area issame decoration for the same shape of core areas.

Since the same shaped core areas are decorated idcally, the number densitynx for an AOD model can be calculated from Eq.~6! provided thatN(x,X) is the number ofatoms in the the core area X (X5A,B,C) including imageatoms. We can divide the fundamental decagon into regiwith the weight for each region for AOD case too as shoin Fig. 5~d!. Here, the atoms in region III may bring neimage atoms in the tiny central kite area and hence mustaken into account for the number density of the atoms.mentioned, region III does not add any image atoms totiny central kite area for theA decagons~configurations 1and 2! but always adds images for theB decagons~configu-rations 7, 8, and 9! and 50% for theC decagons~configura-tions 3, 4, 5, and 6!. Therefore, the number density weighare given by

rI5r

0,

rII5t21r

0,

rIII

5rB1

1

2r

C5t22r

0, ~8!

inl

2-6

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osw,

y

htp

ot

mta

tthanulgotio

h-oco

ticee

eocaim

tham

b

ing

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s.

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-

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-

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for the AOD class models. Note that the overlap does notthe new image atom if there was an atom at the image sitthe tiny central kite area of the fundamental decoratiTherefore, the weightr

IIIshould be assigned to the atoms

region III only if their image sites~in the tiny central area!are unoccupied. Also, it is possible that the image sites frthe left and the right part of region III coincide as in the caof the mirror symmetric decoration. For these atoms,should count them only once for the type-B decagons andtherefore, we should assign

rIII8 5

1

2~r

B1r

C!5S 1

2t231t24D r

0~9!

to the atoms in the regions III~whose image sites in the tincentral area are unoccupied!. While the description abovemay seem complex, a brief study will show that it is straigforward to apply the method using the back of an envelofor a wide class of useful decorations.

VI. SYMMETRY OF A QUASI-UNIT-CELL MODEL

The unit-cell construction in periodic crystals is used nonly to determine the density and stoichiometry, but alsoclassify the symmetry. In this section, we analyze the symetry of quasi-unit-cell models for decagonal quasicryswhose atomic sites can be interpreted as projections fromhigher-dimensional hypercubic lattices. In general,atomic positions in the fundamental decagon can be inlocation as long as they satisfy the overlap decoration rHowever, here we only consider the fundamental decadecorations whose atomic positions, relative to the posiof a vertex of the decagon, are given by

rW5 (k50

4

mk ek , ~10!

where mk is an integer and ek5 cos (2kp/5)x1 sin (2kp/5)y is a unit vector in the physical space. Matematically speaking, this quasilattice of points is a setmeasure zero compared to the entire 2D interior of the degon, but this does not impose any practical restrictionconstructing physical atomic models since the quasilatpoints are dense everywhere due to the irrationality betwthe basic vectorseks.

For these models, atoms can be lifted to the fivdimensional hypercubic lattice points and the informationthe correlations between positions of atoms in the modelbe encoded into the geometry of the perpendicular-spaceages, which is the set of points given by

rW'5 (k50

4

mk ek' , ~11!

whereek'5 cos (4kp/5)x'1 sin (4kp/5)y' is a unit vector in

a plane perpendicular to the physical space. Whenperpendicular-space images of two models have the sgeometrical shape, they are indistinguishable22 since the cor-relations between atomic positions are fully determined

06410

din.

mee

-e

to-

lsheey

e.nn

fa-nen

-nn-

ee

y

the shape of the perpendicular-space image. FollowMermin,22 we call an operationg a symmetry operation if thestate after the operation is indistinguishable from the sbefore the operation. That is, ifg is a symmetry operation foa model with density functionr(rW) then r(rW) and r8(rW)ªr(g21rW) should have the identical correlation functionTherefore, an operationg represented by

gek5 (j

gjk ej ~12!

is a symmetry operation if the corresponding perpendicuspace operationg� given by

g�ek'5 (

jgjk ej

' ~13!

preserves the geometry of perpendicular-space images.As one can see from the definition ofek and ek

' , 2p/10rotation in the real space corresponds to 4p/10 rotation inthe perpendicular-space since it is equivalent toek→ek8 withk85k11~mod 5!. A tenth of 2p (p/10) rotation in the realspace is equivalent toek→2ek8 with k85k13 ~mod 5! and

therefore corresponds to7p

10rotation in the perpendicular

space. Since the 7p/10 rotation~with the modulus 2p) gen-erates all ten integer multiples ofp/10, the quasi-unit-cellmodel has tenfold symmetry if and only if the perpendiculspace image has tenfold symmetry. Also, the quasi-unit-model has reflection symmetry to anen axis if and only if theperpendicular-space image has reflection symmetry to theen

'

axis. This is because the the reflection to theen axis in realspace corresponds to the reflection to theen

' axis in the per-

pendicular space. They are equivalent toek→ek8 or ek'

→ek8' with k1k852n ~mod 5! in both real and perpendicu

lar spaces.The geometry of the perpendicular-space image fo

quasi-unit-model can be obtained from the perpendicuspace image of the atoms in the fundamental decagon. Lefirst consider a simple case where the fundamental decais decorated with only one atom. In this case, tperpendicular-space image of the atoms in the decagonsa certain direction is a simply connected triangle shape.@TheGummelt decoration of a decagon breaks the tenfold symtry of the decagon. We can define the ‘‘direction’’ of a~Gum-melt decorated! decagon from this. In Fig. 7, we define thdirection of a decagon along the mirror symmetry line, frothe intersection of two ‘‘rockets’’ to the center of the ‘‘star.The decagon in a Gummelt covering has 10 different psible directions and the perpendicular-space image ofJack vertices in the decagons with the same direction istriangle shown in Fig. 7~b!.# The perpendicular-space imageof all atoms in the model is then given by the union of ttriangles of ten different directions. Figure 7~a! shows such asimple decoration of the fundamental decagon, whichequivalent to a Penrose lattice decoration where atoms drate each Jack vertex. The perpendicular-space image ofquasi-unit-cell model is simply the perpendicular-space

2-7

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foel

ca

e

ge

e

ganud

el

thls?Fn

laththasul

oththd

trye

as

nit-asi-llyasi-usbe-

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aroseinalrels.

0-

2-

nnheino

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ithtedirrorage


cupation domain of Jack vertices23 and is given by Fig. 7~b!.Here Jk and Jk represent the perpendicular-space imageJack vertices in theek direction and the2ek direction, re-spectively. Since the perpendicular-space image has tensymmetry and the mirror symmetry, the quasi-unit-cmodel with the fundamental decoration of Fig. 7~a! hasD10symmetry.

The symmetry for a general fundamental decorationbe obtained similarly. If the relative position~from a Jackvertex! of a decorated atom is given byrW5 (k50

4 mk ek , then

the perpendicular-space image from the decagons with the0

direction will be the triangle given by therW'5 (k504 mk ek

'

translation of theJ0 triangle. The perpendicular-space imafrom the decagons with the6ek direction will be obtainedby rotating thee0

' direction triangle by the angle that th

6ek' and thee0

' axes make. The perpendicular-space imaof all decagons is then the union of these 10 trianglesalways has tenfold symmetry. Therefore we can conclthat all quasi-unit-cell models have tenfold symmetry.

How about the mirror symmetry? The quasi-unit-cmodel with the fundamental decoration of Fig. 7~b! has thethe mirror symmetry as well as the tenfold symmetry. Ismirror symmetry also general for quasi-unit-cell modeThe answer is no, as one can see from the example of8~a!. In this example, the perpendicular-space image hasmirror symmetry axis. Then when does the perpendicuspace image have the mirror symmetry? From the factthe reflection transforms the base vectors inperpendicular-space equivalently to the transform of bvectors in the real space, we can say that the perpendicspace image has the mirror symmetry if the decorationfundamental decagon has mirror symmetry. However,converse is not true. Some asymmetric decorations offundamental decagon can produce a quasi-unit-cell mowith the mirror symmetry as shown in Fig. 8~b!.

In sum, a quasi-unit-cell model has the tenfold symmeIf the decoration of fundamental decagon has mirror symm

FIG. 7. ~Color online! A decoration of the fundamental decago~a! and the perpendicular-space image of the resulting quasi-ucell model ~b!. When the decagon covering is converted to trhombus Penrose tiling by dividing up central area of decagona Jack configuration, the decorated atom position corresponds tJack vertex. In the perpendicular-space image,Jk andJk represent

the perpendicular-space image of Jack vertices in theek direction

and the2ek direction decagons, respectively.

06410

f

ldl

n

ede

l

e

ig.o

r-atee

ar-feeel

.-

try, then the quasi-unit-cell model has mirror symmetrywell as tenfold symmetry.

VII. CONCLUDING REMARKS

We have shown that a Penrose-tile model is a quasi-ucell model with the same edge size and a decagonal quunit-cell model is a Penrose-tile model with quadraticainflated rhombi. This means the set of all decagonal quunit-cell models is the same as the set of all rhombPenrose-tile models. However, mathematical equivalencetween the two sets does not imply that they are physicequivalent in constructing atomic models for quasicrystaWe need only one kind of atomic cluster as a building unitquasi-unit-cell models, while at least two basic building unare needed in Penrose-tile models. Furthermore, the builunit size is much smaller for quasi-unit-cell models. Forgeneral quasi-unit-cell decoration, the corresponding Pentiles, whose decorations are identical, have scores of origquasi-unit-cells. In other words, we have to look for moand larger building units when we use Penrose tile mode

ACKNOWLEDGMENTS

This work was supported by Grant No. R01-2001-0000025-0 from the basic research program of KOSEF~H.C.J.!and U.S. Department of Energy Grant No. DE-FG091ER40671~P.J.S.!.

it-

tothe

FIG. 8. ~Color online! ~a! A decoration of the fundamental decagon that results in a quasi-unit-cell model with no mirror symmetBoth the fundamental decoration and the perpendicular-space imhave no mirror symmetry axis.~b! An asymmetric decoration of thefundamental decagon that results in a quasi-unit-cell model wmirror symmetry. Although the fundamental decagon is decoraasymmetric, decagons in the covering are decorated with the msymmetry due to the overlap and their perpendicular-space imhas the mirror symmetry.

2-8

an

n

g

fnd


1D. Levine and P. J. Steinhardt, Phys. Rev. Lett.53, 2477~1984!.2P. J. Steinhardt, H.-C. Jeong, K. Saitoh, M. Tanaka, E. Abe,

A. P. Tsai, Nature~London! 396, 55 ~1998!.3S. Ritsch, C. Beeli, H.-U. Nissen, T. Godecke, M. Scheffer, a

R. Luck, Philos. Mag. Lett.74, 99 ~1996!.4A. P. Tsai, A. Fujiwara, A. Inoue, and T. Masumoto, Philos. Ma

Lett. 74, 233 ~1996!.5D. DiVincenzo and P. Steinhardt, inQuasicrystals, The State o

Art, edited by D. DiVincenzo and P. J. Steinhardt~World Scien-tific, Singapore, 1991!, pp. 1–16.

6V. Elser, Phys. Rev. Lett.54, 1730~1985!.7P. A. Kalugin, A. Yu. Kitaev, L. S. Levitov, Pisma Zh. E´ksp. Teor.

Fiz. 41, 145~1985! @JETP Lett.41, 145~1985!#; M. Duneau andA. Katz, Phys. Rev. Lett.54, 2688~1985!; V. Elser, Acta Crys-tallogr., Sect. A: Found. Crystallogr.42, 36 ~1985!.

8P. Bak, Phys. Rev. Lett.56, 861 ~1986!.9P. J. Steinhardt and H.-C. Jeong, Nature~London! 362, 431

~1996!.10H.-C. Jeong and P. J. Steinhardt, Phys. Rev. B55, 3520~1997!.11L. Pauling, Nature~London! 317, 512 ~1985!.

06410

d

d

.

12S. Burkov, Phys. Rev. Lett.67, 614 ~1991!.13C. L. Henley, inQuasicrystals, The State of Art, edited by D.

DiVincenzo and P. J. Steinhardt~World Scientific, Singapore,1991!, pp. 429–524.

14P. Gummelt, Geom. Dedic.62, 1 ~1996!.15P. Gummelt and C. Bandt, Mater. Sci. Eng., A294–296, 250

~2000!.16F. Gahler and M. Reichert, J. Alloys Compd.342, 180 ~2002!.17E. Abe, K. Saitoh, H. Takakura, A. P. Tsai, P. J. Steinhardt, a

H.-C. Jeong, Phys. Rev. Lett.84, 4609~2000!.18H.-C. Jeong and P. J. Steinhardt, Phys. Rev. Lett.73, 1943~1994!.19E. Cockayne, Mater. Sci. Eng., A294–296, 224 ~2000!.20R. Penrose, Bull. Inst. Math. Appl.10, 266 ~1974!.21H.-C. Jeong, Acta Crystallogr. Sect. A: Found. Crystallogr.59,

361 ~2003!.22N. D. Mermin, in Quasicrystals, The State of Art, edited by D.

DiVincenzo and P. J. Steinhardt~World Scientific, Singapore,1991!, pp. 133–184.

23N. G. de Bruijn, Proc. K. Ned. Akad. Wet., Ser. A: Math. Sci.43,39 ~1981!.

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Documents

Rules for computing symmetry, density, and stoichiometry in a quasi-unit-cell model of quasicrystals