A Structural Model for Octagonal Quasicrystals Derived from Octagonal Symmetry Elements Arising in β -Mn Crystallization of a Simple Monatomic Liquid
Måns Elenius, Fredrik H. M. Zetterling, Mikhail Dzugutov, Daniel C. Fredrickson, Sven Lidin
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y A Stru tural Model for O tagonal Quasi rystals Derived fromO tagonal Symmetry Elements Arising in β -Mn Crystallization ofa Simple Monatomi LiquidMåns Elenius, Fredrik H. M. Zetterling and Mikhail DzugutovDepartment of Numeri al Analysis and Computer S ien e,Royal Institute of Te hnology, SE(cid:21)100 44 Sto kholm, SwedenDaniel C. Fredri kson and Sven LidinDepartment of Inorgani Chemistry, Arrhenius Laboratory,Sto kholm University, SE(cid:21)106 91 Sto kholm, Sweden(Dated: November 18, 2018)1bstra tWhile performing mole ular dynami s simulations of a simple monatomi liquid, we observedthe rystallization of a material displaying o tagonal symmetry in its simulated di(cid:27)ra tion pattern.Inspe tion of the atomi arrangements in the rystallization produ t reveals large grains of the β -Mnstru ture aligned along a ommon 4-fold axis, with 45 ◦ rotations between neighboring grains. These45 ◦ rotations an be tra ed to the inter ession of a se ond rystalline stru ture fused epitaxiallyto the β -Mn domain surfa es, whose primitive ell has latti e parameters a = b = c = a β − Mn , α = β = 90 ◦ , and γ = 45 ◦ . This se ondary phase adopts a stru ture whi h appears to have noknown ounterpart in the experimental literature, but an be simply derived from the Cr Si andAl Zr stru ture types. We used these observations as the basis for an atomisti stru tural modelfor o tagonal quasi rystals, in whi h the β -Mn and the se ondary phase stru ture unit ells serveas square and rhombi tiles (in proje tion), respe tively. Its di(cid:27)ra tion pattern down the o tagonalaxis resembles those experimentally measured. The model is unique in being onsistent with high-resolution ele tron mi ros opy images showing square and rhombi units with edge-lengths equal tothat of the β -Mn unit ell. Energy minimization of this on(cid:28)guration, using the same pair potentialas above, results in an alternative o tagonal quasiperiodi stru ture with the same tiling but adi(cid:27)erent atomi de oration and di(cid:27)ra tion pattern.PACS numbers: 61.44.Br, 31.15.xv, 61.72.Mm 2. INTRODUCTIONQuasi rystals are materials whose di(cid:27)ra tion patterns show what at (cid:28)rst glan e is a strik-ing ontradi tion: the sharp peaks normally arising from long-range periodi ity, arranged inpatterns of i osahedral, de agonal, dode agonal or o tagonal symmetries(cid:22)all fundamentallyin ompatible with periodi ordering. Sin e the dis overy of these puzzling and wonderfulobje ts,1 signi(cid:28) ant progress has been made in hara terizing the atomi arrangements un-derlying their di(cid:27)ra tion patterns. For instan es of both the i osahedral and de agonalfamilies, detailed rystal stru tures have been solved from X-ray di(cid:27)ra tion data,2,3 whilethe solution of a series approximant stru tures to the Ta . Te quasi rystal, also from X-raydi(cid:27)ra tion experiments, has brought insights into the atomi geometries in the dode agonalfamily.4,5,6In ontrast, only limited stru tural information is available for the o tagonal family ofquasi rystals. Just a handful of examples have been experimentally realized, those in theV-Ni-Si, Cr-Ni-Si, Mn-Si and Mn-Si-Al systems.7,8,9 In all of these ases, their preparationinvolves the rapid ooling of an alloy melt, a method not ondu ive to the formation ofhigh quality rystals for stru tural analysis with X-rays. Prolonged heating, the most likelypath to the growth of su h rystals, inevitably leads to de omposition of the quasi rystal,usually into a β -Mn-type phase,7,8,10 but the formation of a AuCu -type phase has alsobeen observed.11 Be ause of this metastability, the te hniques for investigating these phasesexperimentally have been limited mainly to high resolution ele tron mi ros opy (HREM)and ele tron di(cid:27)ra tion.Su h studies revealed that these stru tures an be viewed as o tagonal tilings of squareand rhombi units (with 45 ◦ angles at their a ute orners), whi h is one of several possibletilings giving o tagonal symmetry,12,13 and an be embedded into higher-dimensional spa esusing superspa e methods.14,15 It was also observed that the edges of these units are met-ri ally equal to that of a β -Mn-type stru ture.7 However, the positions of individual atomshave not been thus far resolvable. This gap between the experimental investigations of o -tagonal quasi rystals and their detailed stru tures at the atomi level pla es obvious limitson our ability al ulate theoreti ally their physi al properties. In attempts to al ulate theirele troni stru ture16 and vibrational properties,17 resear hers were for ed to resort to anunrealisti model in whi h a single atom is pla ed at every vertex of an o tagonal tiling of3quares and rhombi.Several attempts have been made to (cid:28)ll this gap. Two di(cid:27)erent stru ture models, thoseof Kuo et al.15,18,19 and Hovmöller et al.20 have been proposed making use of the apparently lose stru tural relationship between the o tagonal quasi rystals and the β -Mn stru ture.Both models extra t square and rhombus units from the β -Mn stru ture, and then use theseas tiles in an o tagonal tiling, the di(cid:27)eren e being in the de(cid:28)nition of the square and rhombi units. While these models exhibit di(cid:27)ra tion patterns with similarities to the experimentallymeasured ones, they are in ompatible with the tile-edge lengths measured from HREMimages. A third model proposed by Hovmöller et al. derived from 3D re onstru tions ofHREM images also is in onsistent with this observation.21In this paper, we use the results of mole ular dynami s simulations in an attempt tobring understanding of the stru tures of o tagonal quasi rystals. This approa h has beensu essfully employed by ourselves and others to observe the formation of dode agonal andde agonal quasi rystals from monatomi liquids.22,23,24,25In our ontinued work with su h simulations, we ame a ross a rystallization produ twhi h showed an o tagonal symmetry axis in its al ulated di(cid:27)ra tion pattern. In the ourseof this paper, we will examine the origins of this o tagonal di(cid:27)ra tion pattern in the atomi arrangements of the sample. As we will see below, this o tagonal symmetry arises from a45 ◦ di(cid:27)eren e in orientation between large grains of the β -Mn stru ture type, rather thanthe o urren e of a true o tagonal phase. However, the geometri al reasons for these 45 ◦ relative rotations form the roots of a new stru tural model for o tagonal quasi rystals. Thismodel a(cid:27)ords not only a di(cid:27)ra tion pattern with orresponden e to the experimental onesdown the o tagonal axis, but also is unique in having the β -Mn unit ell repeat period asthe length of its square and rhombus units, in a ord with HREM images. Central to this onstru tion are 8 s rew heli es implemented in Hovmöller et al.'s 1991 model20(cid:22)this timeused in a di(cid:27)erent de oration of the square and rhombi tiles.II. β -MN FORMATION IN MOLECULAR DYNAMICS SIMULATIONSOur investigations into the stru tures of o tagonal quasi rystals began with some sim-ulations of a monatomi liquid, using a pair potential designed to en ourage i osahedralordering. We started this simulation by (cid:28)rst equilibrating a system of atoms in a high4emperature liquid state. We then began a stepwise ooling at onstant density ( ρ = 0.84atoms/unit volume, in redu ed units, see Appendix) with intermediate equilibration. At apoint in this ooling, the system showed a marked drop in potential energy, pressure anddi(cid:27)usivity, indi ating rystallization (for further details of the simulation, see Appendix).To identify the rystallization produ t, we (cid:28)rst analyzed the radial distribution fun tion g ( r ) and the stru ture fa tor S ( Q ) of the sample. These on(cid:28)rmed that the stru ture hadrelatively long-range order, indi ative of a rystalline stru ture. We then turned to a moredetailed analysis of the di(cid:27)ra tion pattern. The di(cid:27)ra tion intensities for a on(cid:28)guration ofN point parti les are al ulated as: S ( Q ) = N − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X i =1 e i Qr i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (1)where ve tors r i denote the parti le positions. First we determined the orientations of the rystallographi axes of the phase in re ipro al spa e. To do this, we plotted the valuesof S(Q) on the sphere in re ipro al spa e orresponding to the maximum of S ( Q ) . Whatwe found was surprising: a symmetry axis of eight-fold symmetry, with eight perpendi ulartwo-fold axes. Having found this o tagonal symmetry axis, we then al ulated di(cid:27)ra tionintensities in the plane perpendi ular to it (Figure 1). In this re ipro al spa e plane, theeightfold symmetry is apparent. Hen e we arrive at the, now lassi al, ontradi tion seenin quasi rystals of sharp di(cid:27)ra tion spots indi ative of long-range order and rotational sym-metry in ompatible with periodi ordering.This en ouraged us to look at the real spa e atomi on(cid:28)guration. Instead of the expe tedo tagonal atomi arrangements, we found large domains of a rystalline stru ture. Figure2 shows approximately 7000 atoms in the simulation box, with two grains of this stru turehighlighted in green and red, respe tively. Unit ell edges for these rystallites are drawnin with yellow squares. Further inspe tion of these grains revealed them to be of the β -Mnstru ture type. A look at these ell edges reveals that the rystallographi axes of the twograins have di(cid:27)erent orientations. In the domain marked in green, these are aligned withthe horizontal and verti al dire tions of the page. In the red one, however, they are orienteddiagonally. The green and red domains are, in fa t, rotated by 45 ◦ relative to ea h other.This 45 ◦ rotation a ounts for the 8-fold hara ter of the di(cid:27)ra tion pattern al ulated forthis geometry.The o urren e of these beta-Mn grains together with the interstitial spa es between them5IG. 1: Cal ulated di(cid:27)ra tion intensities for our simulation rystallization produ t in the planeperpendi ular to the o tagonal axis. orrelates with density heterogeneities, the rystalline grains oin iding with the denserregions. This result an be rationalized in terms of energeti s: model al ulations showthat the β -Mn stru ture is an energy minimum for this potential, with the ideal densitybeing ρ =0.96, ompared to ρ =0.84 in the simulation. Indeed, the di(cid:27)eren e in unit ell sizebetween the β -Mn ells in the sample and that of the ideal rystal at ρ = 0 . is less than .The small β -Mn rystallites are not the only geometri al regularity that an be re ognizedin Figure 2. At the edges of the green and red domains, near the enter of the (cid:28)gure, twosets of ross-linked rows of atoms are highlighted in blue. Similar rows re ur frequently asterminating features at the β -Mn domain surfa es. As shown with yellow lines overlaid onthese atoms in blue, a unit ell an be identi(cid:28)ed for this pattern. In other words, these surfa efeatures orrespond to small domains of a se ond rystalline phase in the simulation result.The unit ell for this phase appears as a rhombus in this proje tion, and for this reason wewill refer to it as the rhombi phase. As we will see in the next se tion, a loser inspe tionof this stru ture provides insights into the prominen e 45 ◦ -twinning in this sample, and therelationship between the β -Mn stru ture and o tagonal quasi rystals.6IG. 2: (Color) Two regions of the β -Mn stru ture (green and red) and their surroundings in the rystallized sample resulting from our mole ular dynami s simulation (see text). Perpendi ular tothe plane of the page, the rystallographi order of these grains propagates in(cid:28)nitely (the sampleperiodi along this dire tion, with a repeat period orresponding to eight β -Mn ells). Smallfragments of a se ond rystalline phase an be per eived between these grains (blue). Yellow linestra e out unit ells of these two stru tures. As the se ondary phase unit ell appears as a nearly45 ◦ rhombus in this proje tion, we'll refer to this as the rhombi phase stru ture.III. A SECOND PHASE: RHOMBI WITH THE SQUARESUpon identifying the rhombi phase, we read o(cid:27) approximate atomi oordinates fromthe positions in the simulation result and generated a fully periodi model. Prompted bythe lose mat h in ell parameters to those of the β -Mn stru ture, and the near 45 ◦ angleof the rhombus, we idealized the unit ell to a = b = c , α = β = 90 ◦ , γ = 45 ◦ . We thenperformed a steepest des ent energy minimization to obtain more a urate atomi positions.The resulting rhombi phase stru ture is shown in Figures 3 and 4.As far as we an tell, this rystal stru ture has no ounterpart in the experimental litera-ture. However, it shows stru tural similarities to the Frank-Kasper family of stru tures, and an be viewed as a derivative of the Cr Si and Al Zr stru ture types. This is illustratedin Figure 3, in whi h we show that the stru tures of both Al Zr and the rhombi phase inour simulations an be derived by introdu ing variations into the Cr Si stru ture type. The7IG. 3: (Color) Stru tural relationships between the se ond phase dete ted alongside β -Mn inmole ular dynami s simulations (the rhombi phase), and two well-known intermetalli stru turetypes: those of Cr Si and Al Zr . In these proje tions, atoms in red and blue are at heights 0 and1/2, respe tively. Those in light gray o ur at both heights 1/4 and 3/4. Green bars: shear planesa ross whi h regions of lo al Cr Si stru ture type geometry are interrupted by shifts in height of1/2, with the deletion of some atoms whi h are brought into near oin iden e with others followingthese shifts. A bla k arrow shows the orientation of the o tagonal axis o urring in the simulationresults.Cr Si-type an be viewed (along with many other ways) as a sta king of nets: nets builtfrom hexagons and triangles (red and blue) alternate with more open square ones (grayspheres, dotted lines). The Al Zr stru ture, when viewed down the [100℄ dire tion, showssimilar features.26 In fa t, a simple way to generate the Al Zr stru ture is to begin with theCr Si-type and introdu e shear planes in rows parallel to one set of lines in the Cr Si-type's8quare nets, as shown with green lines in the top right orner of Figure 3. A ross ea h ofthese planes we introdu e a shift of half of a unit ell out of the page (a ompanied by thedeletion of some atoms that ome into near oin iden e with atoms on the other side ofthe shear plane following the shift). If we introdu e one su h shear plane for every row ofsquares in the proje tion of Figure 3, we arrive at the Al Zr stru ture type.The rhombi phase in our simulations an be derived from the Cr Si stru ture in asimilar manner. At the bottom of Figure 3, we show a view of the rhombi phase stru tureperpendi ular to that of Figure 2. In this view, similarities to the Cr Si stru ture an beseen in the square nets (with some distortion) and the (cid:28)lling of these squares with atoms inred and blue. The details of the stru ture an be derived, just as for the Al Zr stru turetype, by inserting shear elements into Cr Si-type parent stru ture. While in the Al Zr stru tures these shear elements o ur as regularly spa ed planes parallel to one set of linesin the square nets, in the rhombi phase the shears o ur diagonally a ross the square net,moving through the square nets in a zigzag pattern. The rhombi phase stru ture an beseen then as a variation on the Al Zr stru ture.A look at this stru ture from a di(cid:27)erent viewpoint, down the same dire tion as in Figure2, helps us to understand why this phase so frequently appears at the surfa es of β -Mn rystallites in the simulation. In Figure 4 we illustrate this for a single unit ell of therhombi phase, with the heights of the atoms given with white numbers (in fra tions of theunit ell repeat ve tor oming out of the page). In this view, the rhombi phase unit ellappears as a rhombus with ir ular 8-atom wreaths of atoms at ea h orner (and the very enter of the ell). A loser inspe tion of the heights of the atoms in one of these wreathsreveals that they tra e out heli es oming out of the plane of the page, with a lo al 8 symmetry axis passing through the enter of the ring.These 8 heli es have lose ounterparts in the β -Mn stru ture. In the left and rightof Figure 4 we draw unit ells of the β -Mn with their unit ell edges aligned with thoseof the rhombi phase ell. The orners of the β -Mn ells show similar heli es as in therhombi phase. In fa t, if we ompare the heights of the atoms at the ell edges of the twostru tures types, we (cid:28)nd a one-to-one orresponden e of atoms at nearly identi al sites. Anearly perfe t epitaxial mapping exists between the sides of the β -Mn unit ell, and thoseof the rhombi phase. It is not surprising, then, that grains of β -Mn should be terminatedby layers of the rhombi phase. 9IG. 4: (Color Online) Epitaxial mat hing between the {100} planes of the β -Mn stru ture andthe (100) and (010) planes of the rhombi phase stru ture. β -Mn grains interfa ing at these twodi(cid:27)erent planes of the rhombi phase di(cid:27)er in their orientation by 45 ◦ . Atoms to be merged at theinterfa e are drawn in blue in the online version.This epitaxial mapping an be used to understand the prevalen e of β -Mn grains orientedrelative to ea h other by a 45 ◦ rotation. Note from Figure 4 that the rhombi phase aninterfa e with β -Mn rystallites at both its (100) and (010) sets of planes. As the γ angle ofthis unit ell is 45 ◦ , these two sets of planes are also in lined relative to ea h other by 45 ◦ .A β -Mn rystallite interfa ing with a (100) plane will thus be misaligned by 45 o relative toanother rystalline interfa ing with a (010) plane. We posit that the 45 ◦ -twinning in oursimulations results from the presen e of this rhombi phase in the intersti es between β -Mngrains.Several fa tors in the energeti s of the β -Mn and rhombi phase stru tures elu idate theobservations we've made in our simulations. The β -Mn is energeti ally preferable with the urrent potential, onsistent with the fa t that we see more atoms in the β -Mn domains thanin the rhombi phase one in our simulation results. More importantly, the two stru tureshave energy minima at nearly equal densities ( . for β -Mn, . for the rhombi phase),10nd these densities give almost identi al unit ell lengths, making intergrowth of the twostru tures parti ularly fa ile.With the ease with whi h unit ells of β -Mn and the rhombi phase (cid:28)t together, it'stempting to imagine other ways of arranging them in spa e to generate new stru tures.After all, o tagonal quasi rystals are usually viewed as tilings of squares and rhombi justlike these. It's indeed tempting, and, as we shall see in the following se tions, this is atemptation we an't resist.IV. TILING WITH THE STRUCTURES OF β -MN AND THE RHOMBIC PHASEIn the previous two se tions, we identi(cid:28)ed two rystalline stru ture types in the resultsof our mole ular dynami s simulations of a simple monatomi liquid, those of β -Mn and apreviously unobserved stru ture, whi h we'll all the rhombi phase. The unit ell dimensionsand atomi arrangements in these two stru tures are propitious to the formation of epitaxialinterfa es between them. In the simulations, this is re(cid:29)e ted in the appearan e of rhombi phase layers terminating grains of the β -Mn stru ture. Many more possibilities be omeevident when we imagine the unit ells of these stru tures as square and rhombi tiles withwhi h we an over the plane (taking the periodi sta king of the ells in the third dimensionas a given), the square and rhombi tiles being derived from the β -Mn and rhombi phasestru tures, respe tively. In the following paragraphs, we will show that the unit ells of thesetwo stru tures are ompatible with any tiling of squares and rhombi.Several interfa e types are geometri ally feasible using these two tiles. We've alreadyseen that the squares join naturally at the orners and edges with the rhombi ( orners andedges referring to the 2D proje tion in Figure 4 and those to ome; in 3D, they are ell edgesand fa es, respe tively). Interfa es between two square tiles and between two rhombi tilesare seen, of ourse, in the rystal stru tures from whi h they are derived. In Figure 5 weshow a fourth interfa e type, between two rhombi tiles di(cid:27)ering in orientation by 45 ◦ . Thelo al pseudo 8 symmetry axes passing through the orners of the rhombi phase unit ellmakes it possible to fuse two su h tiles. The two tiles simply need to be related by an 8 operation, i.e. the 45 ◦ di(cid:27)eren e in orientation is a ompanied by a 3/8 shift in the heightsof the atoms between tiles. As an be seen in the (cid:28)gure, applying su h a 8 operation toone rhombi tile reates a se ond tile whi h is well-suited to edge-sharing with the original11IG. 5: (Color Online) Epitaxial interfa e between one rhombi phase unit ell, and a se ond ellgenerated from the (cid:28)rst using an 8 symmetry operation. Atoms to be merged at the interfa e aredrawn in blue in the online version.one. While there is some degree of mismat h in the oordinates within the plane, due tothe approximate nature of the 8 axis, there is a one-to-one orresponden e between theatoms at the edge to be shared between the two tiles. Also, all orresponding atoms at theedge mat h in height within one per ent of the unit ell repeat distan e out of the plane.In summary, the pseudo 8 symmetry of the rhombi tiles' orners means that two rhombi tiles related by an 8 symmetry element at a orner an be edge-fused together.Similar geometri al arguments an be used to derive a simple mat hing rule for buildinglarger stru tures with square and rhombi tiles. Both tiles have pseudo 8 -symmetry axesat their orners. This means that when two tiles ome together, they an share orners oredges, as long as their di(cid:27)eren e in orientation is a ompanied by an appropriate relativetranslational shift of their atoms out of the plane. Spe i(cid:28) ally, lo kwise rotations of a tileby 45 ◦ must be a ompanied by translations in height of 3/8. These translational shifts alignthe phases of the heli es at their orners so that they meet in register. Adheren e to thisrule an be seen in the interfa e (cid:28)gures presented so far, Figures 4 and 5. In both (cid:28)gures,ea h 8 helix at a tile orner o urs with the same orientation and phase.The key role played by the 8 axes in this mat hing rule suggests a simple notation for12IG. 6: (Color) A simpli(cid:28)ed notation for the 8 heli es o urring in the β -Mn and rhombi phasestru tures. (a) an idealized 8 helix proje ted down its heli al axis. Heights for the atoms aregiven in both de imal notation and as multiples of 3/8. (b) A olor wheel representation of thishelix. The wheel is divided into eight wedges, ea h olor- oded a ording to the height of the atomo urring in the orresponding o tant in the proje tion of (a).des ribing stru tures based on these tiles. In Figure 6a, we show an idealized 8 helix,viewed down its long axis, with the heights of the atoms drawn in both de imal and fra tionnotations. The height of ea h atom an be expressed as a di(cid:27)erent multiple of the fra tion3/8. We an simplify this pi ture by representing the helix as a ir le divided into 8 equalsli es (Figure 6b), with ea h sli e olor- oded a ording to the height of the atom o upyingthat o tant in the view of Figure 6a.An example of how this notation an be used in stru tural des riptions is given in Fig-ure 7a, in whi h we take another look at the β -Mn/rhombi phase/ β -Mn interfa e of Figure 4.Here, rather than seeing simple geometri al (cid:28)gures de orated by omplex arrangements ofatoms at various heights, we now see a rhombus meeting two squares, with olorful wheels atall of the orners. At both interfa es the ir les at the orners mat h in their olor patterns,meaning that they an merge in a fa ile manner.The olor-wheel notation also simpli(cid:28)es the investigation of relationships between tiles indi(cid:27)erent orientations. In Figure 7b, we show a simple graphi al demonstration that the twosquare tiles are related by a 8 − operation. This symmetry relationship means that whilethe olor patterns di(cid:27)er within the square interiors of these two tiles, they represent thesame rystal stru ture. They di(cid:27)er only in their orientations and by a relative shift in theiratomi heights. We see, then, that this parti ular arrangement of tiles is ompatible witha simple intergrowth of the β -Mn and rhombi phase stru ture types. No new geometri al13rrangements have been for ed within the tiles.FIG. 7: (Color) Illustration of tiling a spa e with ells of the β -Mn and rhombi phase stru tures,appearing respe tively as square and rhombi tiles in proje tion. (a) Simpli(cid:28)ed representation ofthe interfa es in Figure 4, in whi h the 8 heli es at the tile orners are drawn with the olor-wheelnotation of Figure 6. The remaining atoms in the ells are not shown in this representation for thesake of larity. (b) A graphi al demonstration that the two square tiles in (a) an be generatedfrom ea h other via 8 symmetry operations. ( ) An o tagonal luster built from β -Mn and rhombi phase tiles. Six distin t tiles o ur in this pattern; one instan e of ea h is shaded in gray. Thiso tagonal luster an be generated using the four basis ve tors given in the lower left orner. Seetext for a proof that the six distin t tiles indi ated here an be used to de orate any tiling of squaresand rhombi.More omplex patterns of squares and rhombi an be reated, and their feasibility he ked,using the same notation. In Figure 7 , we've generated an o tagonal luster, starting at the enter with an eight-fold star built from eight rhombi sharing a entral point. The ni hesof this star were then (cid:28)lled-in with squares, with the nas ent joints serving as so kets foradditional rhombi. The atomi geometries required by this o tagonal (cid:28)gure an be probedby pla ing olor-wheels, again representing 8 -heli es, at ea h orner with the same phase,and inspe ting the resulting tile olorings. As an be seen by inspe tion of the tiles, theentire pattern is built up from only six distin t tile types. One example of ea h is shadedin with gray. Two are square tiles, whi h are identi al to those found in Figure 7a-b. As14e saw above, they both represent a single unit ell of the β -Mn stru ture. The remainingfour are rhombi tiles. The (cid:28)rst of these is oriented verti ally as in the entral rhombusof Figure 7a, and the remaining three are generated from this by the appli ation of one,two, and three 8 operations. As ea h of these rhombi tiles are related to the others by8 operations, we an see that their de orations are single unit ells of the same stru turetype, the rhombi phase type. No new geometri al arrangements in the atomi de orationsof the tiles have been enfor ed by this tiling pattern. Furthermore, as these 8 operationsleave the olor patterns on the olor-wheels invariant, these rhombi tiles join naturally attheir orners and edges.From this o tagonal (cid:28)gure, we an generalize these observations to show that any tilingpattern of squares and rhombi an be de orated by single unit ells of the β -Mn and rhombi phase stru ture types. This an be done in three steps: (1) We note that while the patternis fairly intri ate, inspe tion of the pattern shows that it is based on only four basis ve tors: a , a , a , and a as drawn in the lower left orner of Figure 7 . Starting with the entral olor-wheel, the full o tagonal pattern an be generated by taking linear ombinations ofthese basis ve tors. (2) In our se ond step, we observe that there are a limited number ofsquares and rhombi that an be generated by making rings with these translation ve tors.They are in fa t those six distin t tiles that we have highlighted in Figure 7 . Any tiling ofsquares and rhombi tra ed out by linear ombinations of these four basis ve tors will ontainonly these six tiles. (3) In our third and (cid:28)nal step, we re ognize that these basis ve tors aresu(cid:30) ient to reate any tiling of squares and rhombi that we may desire.To summarize these arguments, we have found that the pseudo 8 symmetry elementsat the orners of the unit ells of both the β -Mn and rhombi phase stru tures allow usto use these as square and rhombi tiles, respe tively, with whi h we an tile the planein any arrangement. This would of ourse in lude quasiperiodi patterns su h as thoseindi ated by the ele tron mi ros opy studies of V-Ni-Si, Cr-Ni-Si, and Mn-Si-Al alloys byKuo and oworkers. Our β -Mn/rhombi phase intergrowth tilings provides the possibilityof proposing atomisti models for su h alloys to supplement the limited resolution of su hmi ros opy investigations. We will explore this possibility in the next and last major se tionof this paper. 15. QUASICRYSTAL MODELLINGO tagonal tilings using rhombi and squares have already been extensively des ribed.12Su h tilings rely on spe i(cid:28) rules that enfor e an aperiodi o tagonal tiling. They also in ludein(cid:29)ation rules making it possible to go from a given tiling to one with more tiles per unitarea, while preserving the aperiodi ity and o tagonal nature. Of parti ular importan e forour work, whi h uses simulation programs requiring periodi stru tures, is the observationthat applying these in(cid:29)ations to periodi approximant stru tures leads to better and betterapproximants to the full aperiodi tiling.27 Zijlstra has onstru ted one periodi approximantwhi h is well-suited to this appli ation;16 it has the advantage of ontaining only one vertexwhere the tiling rules leading to quasiperiodi ity are broken. In the following we will usea one step in(cid:29)ation of this tiling as the basis for the onstru tion of a stru tural model ofo tagonal quasi rystals. This in(cid:29)ation gives a tiling of 239 tiles per unit ell, as seen inFigure 8a, in whi h the tiles are delineated with bla k edges.Having onstru ted an approximate o tagonal tiling of squares and rhombi, we now de -orate the tiles following the onsiderations of the previous se tion. We pla e unit ells ofthe rhombi phase and β -Mn stru ture on the rhombi and square tiles, respe tively, withthe appropriate translational shifts to preserve the lo al 8 symmetry at the tile orners.This de oration leaves room for signi(cid:28) ant freedom in hoosing the exa t atomi positionswithin the tiles, parti ularly if we remove the requirement that these ells obey the sym-metries of their native latti es. The only requirement is that the -s rews at the verti esand dumbbell atoms at ea h tile edge are preserved. We followed two main routes whensele ting the exa t atomi oordinates. In one, we tried to optimize the (cid:28)t to the experi-mental di(cid:27)ra tion pattern.19 In doing this, we assumed perfe t -s rews at ea h vertex anddumbbells entered at the edges; we then varied the internal distan es in and rotations ofthese elements, while making sure reasonable interatomi distan es were preserved. We will all the result of this the idealized de oration. In our se ond approa h, we aimed at (cid:28)ndingthe stru ture with minimal energy with our potential. To do this, we performed a steepestdes ent energy minimization on the atomi on(cid:28)guration resulting from the (cid:28)rst approa h.The result of these two approa hes an be found in Figure 8, along with their simulatedele tron di(cid:27)ra tion patterns (putting a Mn atom at ea h atomi position, and adjusting thelength s ale a ordingly).42 In both di(cid:27)ra tion patterns we have inferred twinning with a16irror along the verti al axis in the ele tron di(cid:27)ra tion pattern to re on ile the prominen eof hiral -s rews present in our model with the lear mirror symmetry apparent in theexperimental di(cid:27)ra tion patterns, and the determination of the point symmetry of a Mn-Si-Al o tagonal quasi rystal as 8/ m or 8/ mmm through onvergent beam ele tron di(cid:27)ra tionmeasurements.8FIG. 8: O tagonal quasi rystal approximants built from ells of the β -Mn and rhombi phasestru tures using the onsiderations of Se tion IV (see text), and their simulated di(cid:27)ra tion patterns.(a) Idealized stru ture. (b) A simulated ele tron di(cid:27)ra tion pre ession photograph of the hk layer al ulated for this stru ture. ( ) The stru ture resulting from a steepest des ent energy minimizationof the stru ture in (a). (d) The simulated di(cid:27)ra tion pattern for this minimized stru ture.17ote that while the atomi arrangements resulting from these two approa hes show rathersmall di(cid:27)eren es (Figures 8a and ), these di(cid:27)eren es ause major hanges in the ele trondi(cid:27)ra tion patterns (Figures 8b and d). The most striking di(cid:27)eren e is that the di(cid:27)ra tionintensity in is rather uniformly distributed about o tagonal axis before the energy minimiza-tion (Figure 8b), but be omes spoked in appearan e afterwards (Figure 8d).This di(cid:27)eren e between the di(cid:27)ra tion patterns an be understood by a more detailed omparison of the atomi arrangements before and after the energy minimization. Over the ourse of the energy minimization, the ir ular form of the 8 heli es in the idealized stru tureof Figure 8a has morphed into more square like arrangements (in proje tion, Figure 8 ). Thissquaring of the 8 heli es is reminis ent of the orresponding heli es in the β -Mn stru ture.In fa t, during the minimization ea h tile has returned to a form mu h loser to the originalunit ells of β -Manganese and the rhombi phase. Clearly our potential favors the lo alenvironments of the β -Mn stru ture to those of the idealized de oration. Even so, theo tagonal nature of the minimized stru ture is still apparent in both real and re ipro alspa e images; hen e our simple spheri al potential gives a minimum for this ompli atedo tagonal stru ture.We are now in a position to understand the spoked appearan e of the di(cid:27)ra tion patternof the energy minimized stru ture (Figure 8 ). The spokes of intensity in the pattern lieparallel to the square tile edges in the real spa e stru ture. In the true β -Mn stru ture theseedges would lie parallel to 4 -s rew axes, whi h are known to reate systemati absen es indi(cid:27)ra tion patterns. For instan e, a 4 axis along a leads to the rule that for re(cid:29)e tions with k = 0 and l = 0, only re(cid:29)e tions with h = 4 n will give nonzero intensity. In the idealizedstru ture, our idealization pro edure has destroyed the 4 symmetry elements perpendi ularto the o tagonal axis, allowing a more uniform distribution of intensity about this axis. Withthe return of β -Mn type hara ter in the relaxed stru ture, these 4 axes have returned, atleast on a lo al level; thus the observed spoked appearan e of Figure 8d is not surprising.As we des ribed above, the idealized de oration was designed to give the best obtainable(by us) mat h to the experimentally measured ele tron di(cid:27)ra tion patterns. A detailed omparison between the simulated pattern for this stru ture to one of the highest qualityele tron di(cid:27)ra tion images measured to date19 an be found in Figure 9. Here, we use yellowspots to highlight key orresponden es between the two patterns. While there are somedi(cid:27)eren es between the two patterns, the orresponden e is still lear. The hief di(cid:27)eren es18re that (1) two peaks are missing, one rather strong outer peak and a less intense peak alongthe verti al axis and (2) some weak or moderate peaks present in the simulated pattern arenot visible in the experimental pattern. The extra peaks in the simulated pattern an easilybe attributed to the fa t that this pattern is al ulated for a perfe tly ordered stru ture;su h perfe tion is not expe ted to o ur in the experimental sample, parti ularly with the oarseness of the preparation method used. The missing peaks are harder to explain, but ould be attributed to dynami al s attering e(cid:27)e ts.FIG. 9: (Color Online) Comparison between the simulated ele tron di(cid:27)ra tion pattern of (a) ouridealized stru ture and (b) the pattern experimentally measured by Jiang et al. on a Mn-Si-Alo tagonal quasi rystal. As a guide to the eye, some re(cid:29)e tions in (a) orresponding to re(cid:29)e tionsin (b) are highlighted with yellow (light grey) ir les. (b) is reprodu ed from the paper of Jiang etal. with the kind permission of the Ameri an Physi al So iety.In fa t, the (cid:28)t between the simulation result and the experimental di(cid:27)ra tion patternis remarkable onsidering the simpli ity of our model. Further re(cid:28)nements ould in ludeattempts to go beyond an all-Mn o upation of the atomi sites, and allowing variations inthe atomi de orations of the tiles based on their lo al environment.VI. CONCLUSIONSIn this paper, we began with the observation of the rystallization of β -Mn grains ina mole ular dynami s simulation of a simple monatomi liquid. A urious 45 ◦ twinning19 urred between these grains, imparting an o tagonal symmetry to the sample's al ulateddi(cid:27)ra tion pattern. This twin law ould be tra ed to the presen e of a se ondary rystallinephase, whose unit ell appears as a 45 ◦ rhombus in proje tion, the sides of whi h interfa e leanly with the fa es of β -Mn unit ells. This provided an explanation for the observed twinlaw, and also served as a basis for the onstru tion of hypotheti al stru tures using theseunit ells as tiles. We extended this to build a detailed stru tural model of an o tagonalquasi rystal. This model not only shows a reasonable mat h to the experimental ele trondi(cid:27)ra tion patterns, but is also the only stru tural model thus far pro(cid:27)ered that is onsistentwith the observation that the tile edges have the same length as the unit ell edge of the β -Mn stru ture.7,9We found that energy minimization of this model stru ture results in signi(cid:28) ant hangesto the tile de oration and di(cid:27)ra tion pattern, but the overall tiling remains un hanged.Energy minimization, then, yields an alternative o tagonal stru ture model. This (cid:28)nding(cid:22)that a single- omponent system of parti les intera ting via a spheri ally symmetri potentialexhibits a me hani ally stable energy minimum on(cid:28)guration with an o tagonal di(cid:27)ra tionpattern(cid:22)is of signi(cid:28) ant on eptual interest. It demonstrates that the stru tural omplexityof o tagonal quasiperiodi order experimentally observed in multi omponent intermetalli phases, with pronoun ed dire tional bonding, may be largely redu ed to a simple monatomi ar hetypal quasi rystal.We envision several avenues by whi h further insights into o tagonal quasi rystals may befollowed using mole ular dynami s simulations. The (cid:28)rst would be the pursuit of a modi(cid:28)edversion of our pair potential that is prone to the rystallization of an o tagonal phase. One ould also imagine studying possible de omposition routes for o tagonal phases, by annealingsamples of our ideal quas rystal stru ture model using the present potential. Insights intothe ways o tagonal tilings are stabilized ould be approa hed through the analysis of theenergeti relevan e, for our spe i(cid:28) tile de orations, of the luster overings proposed andanalyzed for o tagonal phases.28,29,30In advan e of these future endeavors, we an draw some stru tural on lusions from ourmodel. It rea(cid:30)rms the lose stru tural relationship between o tagonal quas rystals and the β -Mn stru ture inferred from experimental investigations. In terms of lo al geometries, 8 heli es play a prominent role in both, as Hovmöller et al. assumed in their (cid:28)rst stru turalmodel.20 However, a urious di(cid:27)eren e o urs in how these heli es pa k and interpenetrate20o form the stru tures of o tagonal phases and β -Mn. In β -Mn, symmetri ally equivalentheli es of this form propagate along a , b and c , in a ord with the stru ture's ubi symmetry.These heli es are tightly interpenetrating, with ea h atom lying simultaneously on at leasttwo heli es.The o tagonal phase stru ture is, however, de idedly uniaxial. The stru tures of thesematerials, in our model, an be onstru ted by pla ing 8 heli es at the verti es of a 2Do tagonal tiling of squares and rhombi (in proje tion) with the same handedness and phase.The remaining spa es are then (cid:28)lled to form unit ells of the β -Mn and rhombi phasestru tures for the square and rhombus tiles, respe tively. No referen e is made to any sort of oupling between neighboring heli es. This uniaxial hara ter is emphasized in our simulatedele tron di(cid:27)ra tion patterns. Our ideal stru tural model, designed to reprodu e the resultsof di(cid:27)ra tion experiments, shows an absen e of re(cid:29)e tion onditions onne ted with the 4 s rew symmetry of these heli es perpendi ular to the o tagonal axis. An intriguing questionis how this transformation from independent heli es in quasi rystalline phases to tightlyinter onne ting heli es in the β -Mn stru ture onne ts to the relative stabilities of thesestru tures, and the kineti s for de omposition of o tagonal phases into β -Mn-type ones.Conne tions between o tagonal phases and their twinned β -Mn-type de omposition prod-u ts an also be seen on a larger length s ale than their heli al building units. If, as we seein our simulations and was hypothesized by Kuo et al.,7 the twinning in the experimentallyobserved β -Mn phases is mediated by small rystallites of the rhombi phase, then both theo tagonal phase and their de omposition produ ts onsist of the same square and rhombi tiles. Indeed, one ould imagine this de omposition following an aggregation of square tilesinto domains, with rhombi tiles segregating to the domain surfa es or merging to formnew square tiles. Viewing this de omposition in rewind mode, we an portray o tagonalquasi rystals as derived from β -Mn via twinning on a progressively (cid:28)ner and (cid:28)ner lengths ale, until the domains onsist of only one or two unit ells. We see then that in the aseof o tagonal phases there is a ontinuity between quasi rystals and twinned rystals; theyrepresent the same interfa ial phenomena o urring on di(cid:27)erent length s ales.21 knowledgmentsThe authors gratefully a knowledge the support of the following Swedish Resear h Foun-dations: TFR, NFR and NTM and the Swedish National Infrastru ture for Computing(SNIC) and the Centre for parallel Computers (PDC). D.C.F. thanks the National S i-en e Foundation for a post-do toral fellowship (through Grant DMR-0502582). Finally, wethank Dr. Junliang Sun for elu idating onversations regarding the simulation of ele trondi(cid:27)ra tion patterns.APPENDIX: DETAILS OF POTENTIAL FUNCTION AND MOLECULAR DY-NAMICS SIMULATIONSPair intera tion appear to be su(cid:30) ient for modelling liquid metals.31,32,33 The pair poten-tial used in the present study was onstru ted to imitate the interioni intera tion in simplemetals, onsisting of a short-range repulsive ore and a longer-range os illatory part. Thelatter is meant to represent the Friedel os illations34 whi h are hara teristi of the e(cid:27)e tiveinterioni potentials in simple metals. The period of these os illations is determined by theFermi wave-ve tor K F whi h is a fun tion of the density of valen e ele trons. The fun tionalform of this potential is V ( r ) = A exp( αr ) cos(2 K F r ) r + Br P + V (A.1)We used parameters for this potential whi h were previously determined to en ourage i osa-hedral arrangements in the (cid:28)rst oordination shell.35 ,
43 This was a hieved by putting anenergy penalty at the distan e of √ times that of the (cid:28)rst potential minimum, to dis- ourage the formation of ubi stru tures. The same liquid, at a higher density than thatexplored here, has also been observed to rystallize in the γ -brass stru ture.36The potential is shown in Figure 10, along with two other well-known potentials for om-parison, the IC potential of Dzugutov37 and the Lennard-Jones potential. The IC potentialalso indu es i osahedral short-range order and was used in simulations of a dode agonalquasi rystal22 and a σ -phase-type stru ture.38,39 All three potentials have nearly identi alshort-range repulsive parts.The simulations were performed using redu ed units. The redu ed units orrespond tothose of the Lennard-Jones potential. In pra ti e this means that the length unit is de(cid:28)ned22IG. 10: Pair-potentials, solid line: the urrent potential, dash-dotted line: IC potential37, dottedline: Lennard-Jones potential. The energy level of the IC potential and the urrent potential hasbeen s aled to give an energy of -1 at the (cid:28)rst minimum.as the onset of hard repulsion, orresponding to half the parti le radius. The mass unit isthe mass of a single parti le. The energy unit is equal to the depth of the (cid:28)rst minimum ofthe potential and the time unit is derived from the other three.The simulation was performed at the onstant number density ρ = 0 . parti les perunit volume with 16384 ( = 16 · · · ) parti les. Newtonian equations were integratedusing a se ond order (cid:28)nite di(cid:27)eren e method, the leap-frog version of the symple ti Verletalgorithm.40 Temperature adjustments of the system were performed by s aling the velo -ities of the parti les. After ea h hange in temperature an equilibration period followed.Crystallization o urs at a temperature of T=0.45 in our simulation units (note that the urrent potential in Figure 10 is s aled. In that s ale the rystallization temperature isT=0.64). After rystallization, the resulting parti le on(cid:28)gurations were treated with asteepest des ent algorithm to remove the stati of thermal motions and yield the inherentstru ture.411 D. She htman, I. Ble h, D. Gratias, and J. W. Cahn, Phys. Rev. Lett. 53, 1951 (1984).2 H. Takakura, C. P. Gómez, A. Yamamoto, M. de Boissieu, and A. P. Tsai, Nat. Mater. 6, 58(2007).3 S. Katry h, T. Weber, M. Kobas, L. Massüger, L. Palatinus, G. Chapuis, and W. Steurer, J.Alloys Compd. 428, 164 (2007). 23 M. Conrad, F. Krumei h, and B. Harbre ht, Angew. Chem. Int. Ed. 37, 1383 (1998).5 M. Conrad, F. Krumei h, C. Rei h, and B. Harbre ht, Mat. S i. Eng. 294-296, 37 (2000).6 M. Conrad and B. Harbre ht, Chem. Eur. J. 8, 3095 (2002).7 N. Wang, H. Chen, and K. H. Kuo, Phys. Rev. Lett. 59, 1010 (1987).8 N. Wang, K. K. Fung, and K. H. Kuo, Appl. Phys. Lett. 52, 2120 (1988).9 W. Steurer, Z. Kristallographie 219, 391 (2004).10 Z. H. Mai, L. Xu, N. Wang, K. H. Kuo, Z. C. Jin, and G. Cheng, Phys. Rev. B 40, 12183 (1989).11 L. Xu, N. Wang, S. T. Lee, and K. K. Fung, Phys. Rev. B 62, 3078 (2000).12 J. E. S. So olar, Phys. Rev. B 39, 10519 (1989).13 R. Ingalls, J. Non-Cryst. Solids 153-154, 177 (1993).14 Z. M. Wang and K. H. Kuo, A ta Cryst. A 44, 857 (1988).15 F. Li and Y. Cheng, Chin. Phys. Lett. 13, 199 (1996).16 E. S. Zijlstra, J. Non-Cryst. Solids 334-335, 126 (2004).17 Z. Liu, Z. Zhang, Q. Jiang, and D. Tian, J. Phys.-Condens. Mat. 4, 6343 (1992).18 K. H. Kuo, J. Non-Cryst. Solids 117/118, 756 (1990).19 J. C. Jiang, N. Wang, K. K. Fung, and K. H. Kuo, Phys. Rev. Lett. 67, 1302 (1991).20 Z. Huang and S. Hovmöller, Philos. Mag. Lett. 64, 83 (1991).21 J.-C. Jiang, S. Hovmöller, and X.-D. Zou, Philos. Mag. Lett. 71, 123 (1995).22 M. Dzugutov, Phys. Rev. Lett. 70, 2924 (1993).23 A. Quandt and M. P. Teter, Phys. Rev. B 59, 8586 (1999).24 M. Engel and H.-R. Trebin, Phys. Rev. Lett. 98, 225505 (2007).25 A. S. Keys and S. C. Glotzer, Phys. Rev. Lett. 99, 235503 (2007).26 B. G. Hyde and S. Andersson, Inorgani Crystal Stru tures (John Wiley and Sons, New York,1989).27 M. Duneau, R. Mosseri, and C. Oguey, J. Phys. A: Math. Gen. 22, 4549 (1989).28 S. I. Ben-Abraham and F. Gähler, Phys. Rev. B 60, 860 (1999).29 F. Gähler, Mater. S i. Eng. 294-296, 199 (2000).30 L. Liao and X. Fu, Solid State Commun. 146, 35 (2008).31 N. W. Ash roft, Nuovo Cimento So . Ital. Fis., D 12, 597 (1990).32 M. Dzugutov, Phys. Rev. A 40, 5434 (1989).33 M. Dzugutov, M. Alvarez, and E. Lomba, J. Phys.: Condens. Matter 6, 4419 (1994).244 D. G. Pettifor, Bonding and Stru ture of Mole ules and Solids (Oxford University Press, Oxford,1995).35 J. P. K. Doye, D. J. Wales, F. H. M. Zetterling, and M. Dzugutov, J. Chem. Phys. 118, 2792(2003).36 F. Zetterling, M. Dzugutov, and S. Lidin, MRS Symposium Pro . 643, K9.5.1 (2001).37 M. Dzugutov, Phys. Rev. A 46, R2984 (1992).38 S. I. Simdyankin, S. N. Taraskin, M. Dzugutov, and S. R. Elliott, Phys. Rev. B 62, 3223 (2000).39 J. Roth and A. R. Denton, Phys. Rev. E 61, 6845 (2000).40 L. Verlet, Phys. Rev. 159, 98 (1967).41 F. H. Stillinger and T. A. Weber, S ien e 225, 983 (1984).42 Ele tron di(cid:27)ra tion simulations were made using the program eMap: AnaliteX Crystallographi Computing Software, Version 1.0, produ ed by AnaliTEX, Sto kholm, Sweden. Di(cid:27)ra tion pat-tern images were exported from this program in gray-s ale mode, with dark spots on a lightba kground. A separate graphi s program was used to invert this s heme to aid omparisonwith experimentally measured patterns.43 Numbers used in simulation: A = 1 . , B = 0 . , K F = 4 . , α = − . , P = − , V = 0 .04682632