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Cluster packing geometry for Al-based F-type icosahedral alloys Nobuhisa Fujita,1,∗ Hikari Takano,1 Akiji Yamamoto,2 and An-Pang Tsai1,2 1Institute of Multidisciplinary Research for Advanced Materials, Tohoku University, Sendai 980-8577, Japan 2National Institute for Materials Science, Tsukuba 305-0044, Japan (Dated: January 30, 2013) ThispaperpresentsanewhighlystableperiodicapproximanttotheAl-basedF-typeicosahedral quasicrystals, i-Al-Pd-TM (TM=transition metals). The structure of this intermetallic Al-Pd-Cr- Fe compound is determined ab initio using single-crystal X-ray diffraction, where the space group 3 is identified to be Pa¯3 and the lattice constant 40.5˚A. The structure is well described as a dense 1 packing of clusters of two kinds, which are known in the literature as the pseudo-Mackay typeand 0 the Bergman type clusters. The clusters are centered at the vertices of a canonical cell tiling, in 2 whichtheparityofeachvertexdeterminesthekindoftheassociated cluster. Adjacentclusterscan n bemarkedly interpenetrated, while thestructurerequires no glue atoms to fill in thegaps between a the clusters. It is shown that the crystal can be designated as a 2 2 2 superstructure of the × × J ordinary cubic 3/2 rational approximant. The superlattice ordering is shown to be of a different 9 kind from the P-typesuperlattice ordering previously reported in i-Al-Pd-Mn. The present results 2 willgreatly improvetheunderstandingofatomicstructuresofF-typeicosahedral quasicrystalsand theirapproximants. ] i PACSnumbers: 61.44.Br,61.66.Dk,61.50.Ah,61.05.cp c s - l r I. INTRODUCTION 1/1and2/1approximantstotheF-typeicosahedralqua- t sicrystal based on their lattice constants of 12.28˚A and m 20.21˚A5,6. Thestructuralinformationofthetworefined . The Al based F-type icosahedral quasicrystals, such t structures has been utilized as the basis for building a a as i-Al65Cu20Fe15 and i-Al70Pd20TM10 (TM=transition six-dimensional structure model of the relevant F-type m metal,e.g.,Mn,Re),poseanumberofquestionsastothe icosahedral quasicrystal7. However, the reliability of the detailed atomic arrangements. Researchershave tried to - d extract as much structural information as possible from two structures has remained controversial because their n materialsofthiskindthroughstate-of-the-arttechniques constituent clusters were totally different despite their o compositional similarity. for structure analysis using X-ray, electron or neutron c diffraction. These studies indicated that there existed a Ini-Al Pd TM anditscloseassociates,Al,Pdand [ 70 20 10 few different kinds of clusters as the basic building units TMarelikelytomaintaintheirdistinctrolesinthestruc- 1 that construct the quasicrystals. ture. While no favorableapproximantto the icosahedral v A direct measurement of a quasicrystalline sample, quasicrystal has been reported for TM = Mn or Re, an 3 2 however, entails a well known disadvantage in terms of ample room remains for the choice of TM elements. In 8 structure determination. Conventional methods devel- particular, a fine adjustment of the electron concentra- 6 opedforanalyzingcrystalstructuresareinherentlyinap- tion (e.g., e/a ratio) could be achieved by mixing two 1. plicabletoquasicrystalsduetotheabsenceofperiodicity. transition metal elements for TM. In this report, a syn- 0 However,thesemethodscouldstillprovideinvaluablein- thesis of a new stable approximant is accomplished by 3 formation regarding the constituent clusters as well as blending Cr and Fe, the two immediate neighbors to Mn 1 their local packing in quasicrystals through analyzing in the periodic table, for TM. An optimal ratio between v: their rationalapproximants. In particular,the latter ap- Cr and Fe is searched experimentally. The stability of i proach has so far been taken successfully in elucidating the new phase is such that fine single crystals may grow X local characteristics of P-type icosahedral quasicrystals, through a simple slow cooling method. Single crystal X- r such as i-Al Mn Si 1,2 and i-Cd Yb3. raydiffractionisthenperformedtoanalyzethestructure. a 73 21 6 5.7 The first part of this paper is devoted to a presentation Unfortunately, for the case of F-type icosahedral qua- of the synthesis and the crystal structure analysis. sicrystals, a stable approximant phase has scarcely been reported. Hence,localcharacteristicsofthestructurere- The present structure analysis offers a source of infor- mained to be uncertain to a large extent. It appears mationthatisessentialinunderstandingthelocalatomic as if Al-based F-type quasicrystals are so stable that arrangements in the F-type icosahedral quasicrystals. It no subordinate approximant phase could be obtained turns out that the structure is composed of two kinds via a slight change of the composition. To date, only of clusters, called the pseudo-Mackaytype and Bergman two possibly related cubic approximants have been pre- type clusters. The clusters are interconnected in such a viouslyreportedinarapidlyquenchedalloywithanom- way that significant interpenetrations are allowed, while inal composition of Al Pd Mn Si after annealing at no glue atom between them is required. Moreover, the 69 20 8 3 above 1000◦C4. By way of single crystal X-ray diffrac- centersoftheclustersaregivenasthenodes(orvertices) tion, these approximants were determined to be cubic ofathree-dimensionaltilingwithfourkindsofpolyhedra 2 called the canonicalcells8; this finding is of fundamental FIG. 1. Powder X-ray diffraction diagrams for importance as it allows a systematic description of the basicskeletonofthisaswellasotherrelatedcompounds. Al70Pd20CrxFe10−x (x = 3, 5 or 7). For x = 3 and 5, the peaks marked with filled circles are associated with an The main body of the work comprises a full account of approximant,whileforx=7thosemarkedwithfilledsquares the crystal structure and related discussions. are associated with a quasicrystal. Note that unassigned The structural description can be generalized in a peaks associated with impurity phases become prominent as straightforward manner, leading to the proposal of a x is increased number of hypothetical structures which could form as real approximants. Although a full description of the F- type icosahedral quasicrystals awaits future endeavors, our understanding on the local atomic arrangements in the F-type icosahedral quasicrystals and their approximants can now be revised significantly based on firm experimental evidences. This yet opens a way to further attempt to synthesize variantsof stable approximantsas well as to gain insights on the superlattice ordering phe- nomenon which was reported previously in the F-type icosahedral quasicrystal i-Al Pd TM 9. 70 20 10 This paper is organized as follows. Section II illus- § trates how a simple compositional search culminates in the discovery of a new approximant phase. The initial samples were polycrystalline, yet powder X-ray diffraction as well as electron diffractionproves the presence of The starting materials were Al (Kojundo Chemical a cubic approximant with a large unit cell. An electron Lab.; purity, 99.9%), Pd (Tanaka Kikinzoku; purity, micro-probe analyzer (EPMA) is used to evaluate the 99.95%), Cr (Furuuchi Chemical; purity, 99.9%) and Fe fine chemical composition of the approximant. Then an (Nilaco; purity, 99.5%). An alloy ingot with a nomi- optimizedsynthesisisperformed,resultinginthegrowth nal composition of Al70Pd20CrxFe10−x (x = 3, 5 or 7) of single crystals exceeding 100 µm in diameter. Single waspreparedbythearc-meltingmethodunderargonat- crystal X-ray diffraction and the structure analysis are mosphere. After sufficienthomogenizationwasachieved, performed in Section III. The crystal structure is de- the as-solidified ingot was fragmented, put into a Tam- § scribedindetailinSection IVinaconstructivemanner: mann crucible made of Al O and sealed altogether into § 2 3 After introducing the geometrical templates of the two a quartz tube with pure argon gas of about 0.08 MPa. kinds of clusters, the packing geometry of the clusters is Inanelectricfurnace,thesamplewasannealedat850◦C described in detail. It is shown that the global arrange- for48hoursandcooledswiftly ( 1hour)downtoroom ment of the clusters is determined based on a canoni- temperature. ∼ cal cell tiling with F-type ordering. In an early part of In Figure 1, powder X-ray diffraction patterns taken Section V, discussions are given of the relationship be- from the samples x = 3, 5 and 7 are shown, where § tweenasuperlatticeorderingofthepresentapproximant the characteristic X-ray of Cu Kα (λ = 1.543˚A) is used and that in the quasicrystal. A likelihood of anti-phase withBragg-Brentanodiffractiongeometry(Mac Science, boundariesasapossiblesourceofdisorderinthepresent diffractometer M03XHF22). Slightly abovethe strongest material is also discussed. Then the rest of this section ◦ Braggpeak at 2θ 44 , a clear sub-peak is observedfor examinesthe possibilityofintroducingatiling modelfor ∼ the two samples x = 3 and 5. The latter peak feature the atomic arrangement. Section VI is devoted to con- is usually associated with the cubic 2/1 approximant to § cluding remarks. an icosahedralquasicrystalwith the Miller indices being hkl=10 0 0. The corresponding lattice constant is calcu- latedtobec.a. 20.3˚A.Thesamplex=3appearstocon- II. SYNTHESIS AND CHARACTERIZATION tain the largestamount of the approximantphase, while the suppression of the sub-peak as well as the growth of ◦ Our synthesis of the new approximant was performed an extra feature at 2θ 43 clearly indicates the preva- ∼ intwosteps. Firstly,asearchforapossibleapproximant lence of impurity phase(s) as x is increased. Hence, it is phasewasmade byvaryingthe ratiobetweenCrandFe. expectedthatAl70Pd20Cr3Fe7 is the closestcomposition Afewcharacterizationtechniqueswereusedtolocatethe to that of the approximant phase. composition of the approximant phase as well as to ex- Back-scattered electron images taken from the three tract the basic crystallographic information. Secondly, samples using a scanning electron microscope (SEM) starting from the right composition a new synthesis was are shown in Figure 2. The SEM apparatus used is performed with a slow cooling. High quality single crys- JXA-8621MX(JEOL),inwhichanelectron-probemicro- talsoftheapproximantweresuccessfullyobtainedinthis analyzer (EPMA) is implemented for chemical compo- way. These steps are described in the following. sition analysis. A few domains of different gray lev- 3 FIG. 2. SEM back-scattered electron images for (a) FIG. 3. TEM diffraction patterns taken from the x=3, (b) 5 and (c) 7. The estimated composi- Al70Pd20Cr3Fe7 sample along (a) the two-fold axis ( < k tions are: (a) Al Pd Cr Fe (light gray) and 100 >), (b) the three-fold axis ( < 111 >) and (c) one of 69.1 22.0 2.1 6.8 k Al Pd Cr Fe (darkgray),(b)Al Pd Cr Fe thepseudo five-fold axes. 68.4 20.9 1.1 9.5 69.7 22.5 2.3 5.4 (light gray) and Al Pd Cr Fe (dark gray) and (c) 72.5 11.8 11.6 4.0 Al Pd Cr Fe (light gray) and Al Pd (white). 69.5 23.5 3.6 3.4 3 2 els are observed in each sample (voids are shown as black regions), indicating the existence of a few alloy phases with different compositions. It is observed that alightgrayregioninFigure2(a)representsthe majority phase for x = 3 with the estimated composition being Al Pd Cr Fe . It is reasonable to assume that 69.1 22.0 2.1 6.8 this region corresponds to the approximant, which con- stitute the main body of the sample. For x = 5, a light vided by selected-area electron diffraction. Here two gray regionin Figure 2(b) has an estimated composition transmissionelectronmicroscopes(TEM),JEM-2000EX of Al69.7Pd22.5Cr2.3Fe5.4, and one can still associate it (JEOL) and JEM-2000EXII (JEOL), operating at 200 with the approximant for the compositional difference is kV were used. Electron diffraction patterns taken from insubstantial. However,allthephasesobservedforx=7 the Al Pd Cr Fe sample (Figure 3) clearly indicate 70 20 3 7 aredeviatedsignificantlyfromthe approximantinterms the existence of 2-fold, 3-fold and pseudo 5-fold axes, of chemical composition, implying no relevant approxi- where strong Bragg reflections are arranged in a similar mant phase exists in the sample. wayto the caseof the quasicrystal10. In Figure 3(a)and Firmer evidence of the approximant phase was pro- (b), periodic arrays of spots can be indexed as a cubic 4 0.320mm. The diffraction experiments were performed FIG. 4. SEM micrograph taken from the single crys- usinga BrukerSMART APEXdiffractometer,mounting talline sample obtained with a nominal composition of CCD area detector, with Mo Kα radiation and graphite Al Pd Cr Fe . 69.4 22.3 2.2 6.1 monochromator(wavelength=0.71073˚A).Indexingand empiricalabsorptioncorrectionwereperformedusingthe Bruker software package (SMART for WNT/2000 5.625 Bruker AXS and SAINT 6.45 SADABS). In collecting the reflection intensities, the crystal was identified as a primitive cubic crystalwith a lattice constantof 40.54˚A. Inspecting the original reflection data, it was found that a set of symmetrically equivalent reflections often included one or a few members showing exceptional de- viation in the intensity. This is due to the dynamical ef- fect (or multiple scattering) by which an extra intensity is added to the ‘kinematical’ Bragg intensity. Although the extra intensities are weak in contrast to those of the strongest Bragg reflections, they could still be harmful forweakreflections;notethatthestructuralinformation carried by weak reflections plays a crucial role in deter- mining long-range characteristics of a complex structure such as a quasicrystal and an approximant with a large crystal with a lattice constant of about 40˚A. Note that, unit cell. however,this is twice as large as the one estimated from In principle, a reflection intensity affected by multi- the above powder X-ray diffraction pattern by assum- ple scattering could be effectively screened out from the ing a conventional 2/1 approximant. Moreover, extinc- databasedontheintensitydistributionamongthe setof tions are observed at 0kl with k=odd and at 00l with equivalenthklentries. Inacommonstructurerefinement l=odd,indicatingthatthecrystalhasanon-symmorphic software (e.g., JANA2006), the procedure can be per- space group. Therefore, the pre-existing model of a 2/1 formed when equivalent reflections are averaged. How- approximant6,11 clearly fails to describe the present ma- ever, if the number of equivalent hkl entries is small, the terial. Thismotivatesustocarryoutathoroughinvesti- statistics is simply not enough to discern the ill-entry. gationofthecrystalgeometrybasedonanabinitiostruc- Hence, we took a precaution to remove those hkl entries ture determinationusing singlecrystalX-raydiffraction. which have less than six symmetrically equivalent asso- In order to obtain a single crystal to be used for the ciatesfromtheoriginaldatasetinordertoavoidthepos- structureanalysis,anewsamplewaspreparedinthe fol- sible source of error. The preprocesseddataset was used lowing way. A refined composition given by the arith- for the structure analysis as described in the following metic mean, Al Pd Cr Fe , of the two composi- subsection. 69.4 22.3 2.2 6.1 tionsevaluatedfromthesamplesx=3and5wasusedfor thestartingcomposition. Thepreparationstepswerethe sameasthosedescribedaboveexceptthatattheanneal- B. Structure analysis ingstepwithinthefurnacethesamplewasfullymeltedat ◦ 1160 Cforonehourandthencooleddownslowlytoroom OurstructureanalysisreliedentirelyontheJANA2006 ◦ temperature with a cooling rate of 10 C/hour. Figure softwarepackage12. TheLauegroupandthespacegroup 4 shows an SEM micrograph taken from the new sam- was determined unambiguously to be m¯3 (T ) and Pa¯3, h ple using JCM-5100 (JEOL), where faceted crystalline which turned out to be the only non-symmorphic space grains roughly in the shape of a truncated octahedron group with minor contradictions with the data. Out of are wrapped partially with debris (impurities). It con- thewhole95213observedreflectionssatisfyingI >3σ(I), firms that single crystals a few hundred µm’s in diame- 302symmetricallyextinctreflectionswereincluded. The ter were successfully grown, which also implies that the symmetry averaging was performed while at the same approximant formed congruently from the melt. timethosedataentriesshowingsignificantdeviationfrom the averages of their equivalent reflections (I I > av | − | 10σ(I )) were eliminated from the data. The averaged av III. AB INITIO STRUCTURE data containing 8454 independent reflections satisfying DETERMINATION I >3σ(I) were obtained with R =7.91%. int TheinitialstructuremodelwasgeneratedwiththeSU- A. Single crystal X-ray diffraction PERFLIP program, which is an implementation of the charge flipping algorithm in JANA2006. Then Fourier A single crystal was taken from the new sample with synthesis and least squares fitting were iterated. After the estimated dimensions of 0.272mm 0.178mm severaliterations,the automatedpeaksearchturnedun- × × 5 TABLE I. Crystallographic data FIG. 5. The idealized templates for the two kinds of cluster. (a) M-cluster consists of a central site (M ), a dodecahedral 0 shell(M )andacompositeshellwithanicosahedral subshell 3 Formula Al72.515 Pd22.498 Cr4.928 Fe7.853 (M5) and an icosidodecahedral subshell (M2). (b) B-cluster Molar mass 5045.2 g/mol consistsofacentralsite(B ),anicosahedralshell(B )anda 0 5 Temp. of data collection room temp. dodecahedral shell (B ). Spacegroup Pa¯3 (No. 205) 3 Lattice constant, a 40.5405 ˚A lat Cell volume, Ω 66629.6 ˚A3 Z 40 Calculated density 5.028 g/cm3 Absorption coefficient 9.294 mm−1 ◦ Rangeof 2θ 1.74 53.2 ∼ Independentreflections 21526 Obs. reflections (I >3σ(I)) 8454 R (obs/all) 7.91/13.37 int A. Cluster templates Num. of parameters 783 R(F) 11.96 Rw(F) 12.13 The geometrical templates for M- and B-clusters are S 4.59 shown in Figure 5; both have an atomic site at the cen- ∆ρmax, ∆ρmin 8.01, 5.38 e/˚A3 ter and two shells having the full icosahedral symmetry, − ∆/e.s.d. 0.0002 ¯5¯32/m (I ). The central sites are symbolized as M and h 0 B for M-cluster and B-cluster, respectively. The inner 0 shellofM-clusterconsistsoftwentysitesformingthever- tices of a regulardodecahedron. This shellis symbolized successful in locating new atoms. Then we carefully in- as M because the relevant sites are on the three-fold 3 spectedthetentativestructure,findingincompleteicosa- rotation axes of the icosahedral point group; the same hedral clusters orderly packed in the unit cell. We principle also applies when symbolizing the remaining identified two kinds of clusters, which in the literature shells. The outer shell of M-cluster is a composite of are known as the pseudo-Mackay type clusters and the two subshells, twelve sites forming a regular icosahedron Bergman type clusters. It was observed that adjacent (M )andthirtysitesformingaregularicosidodecahedron 5 clusterswereconnectedalongeitheratwo-foldorathree- (M ). Ontheotherhand,theinnershellofB-clustercon- 2 fold symmetry axis of the reference icosahedron; a two- sists of twelve sites forming a regular icosahedron (B ), 5 fold linkage connects clusters of the same kind and a while the outer shell twenty sites forming a regular do- three-fold linkage connects ones of different kinds. Af- decahedron (B ). 3 ter the central atoms of all the clusters were identified, Let us now define the six icosahedral basis vectors a the missing atoms within each cluster were located by j (j =1, 2, ... and 6) as inspecting the charge density plot. Finally the refinement converged with the reliability index of R = 11.96 (%) or R = 12.13 (%). A detailed presentation of the a a a a a a w 1 2 3 4 5 6 refinementaswellasatableofatomicparametersispro- (cid:0) (cid:1) τ 0 1 1 τ 0 vided in Appendix B. The basic crystallographic data  −  § :=ℓ 1 τ 0 0 1 τ , (1) are summarized in Table I. −  0 1 τ τ 0 1  − where τ = (1+√5)/2 is the golden mean and ℓ repre- IV. CRYSTAL STRUCTURE sents an appropriate scale. The norm of the basis vectors is hereafter denoted a(:= a ). Note that the num- j | | The refined structure contains 4728 atomic sites per bering, j = 1, 2, ... and 6, of the six basis vectors is unitcell,wherein204sitesaresymmetricallyunique. Im- taken such that they are arrangedconcentrically around portantly, the structure can be pictured as one formed a three-fold rotation axis13. Taking the central site of through dense packing of clusters, which are allowed to a cluster template as the origin, the position vector x overlap with each other across their peripheries. There of every site within the cluster template is written as is no need for glue atoms to fill in the gaps between x = Pjcjaj =: [c1c2c3c4c5c6], in which the indices cj the clusters, meaning each of the atomic sites belongs (j = 1, 2, ... and 6) are all integers or half-integers. To to at least one cluster. Two kinds of cluster called the bemorespecific,eachoftheshells,M andB (x=0,2,3 x x pseudo-Mackay type and the Bergman type clusters are or5),isobtainedasanorbitofitsrepresentativemember identified; these are henceforth referred to as M- and B- with respect to the point group ¯5¯32/m (I ); the repre- h clusters, respectively. sentative indices and the radius are listed in Table II. 6 a two-fold or a three-fold axis with a distance of about TABLE II. The indices given for each shell represent the 7.7˚A or 6.7˚A, respectively; these are the shortest two vector from the center to a representative site of the shell. distances between cluster centers. An analogousfeature, Representative vectors for b- and c-linkages as well as the with the distances being appropriately scaled, has been lattice translation vectors are also given for reference. The extensively discussed in the case of P-type icosahedral norm of each vector is given in the right column. Note that b = 2(τ3/√5)1/2a and b = √3/2c. Negative indices are rep- quasicrystalsandtheirapproximants8,14,15,inwhichonly resented by integers with top bars. a single kind of cluster would come into play. The two- and three-fold linkages are called b- and c-linkages, re- shell indices norm spectively, whereas a packing of clusters with this prop- erty is called a bc-packing. The b- and c-linkages are in- M [000000] 0 0 dexedwithintegersandtheirrepresentativesaregivenin M [110000] b/τ 2 Table II; the respective norms are denoted b and c. The M [111111]/2 c/τ2 M3 [1111¯1¯1]/2 τa skeleton of the crystal structure can thus be described 5 as a network of nodes, which correspond to the cluster B [000000] 0 0 B [¯1¯1¯1111]/2 c/τ centers, connected through b- and c-linkages. 3 B [100000] a (=1.902ℓ) F-type ordering (or F-centering) in icosahedral qua- 5 sicrystalsaswellastheirapproximantsoccursastheeven b-linkage [11001¯1] b (=5.236ℓ) and odd parities of the nodes in the relevant bc-packing c-linkage [111000] c (=4.535ℓ) are differentiated. Here, the parity of each node is de- R [6044¯60] 2bτ2 1 fined as that of the sum of the relevant indices. Note in R [46004¯6] 2bτ2 2 TableIIthateachb-linkageconnectsapairofnodeshav- R [046¯604] 2bτ2 3 ing the same parity,while that eachc-linkage connects a pair of nodes having different parities. Therefore, if the two kinds of cluster, M and B, are the entities that dif- ferentiate the two subsets of the nodes of a bc-packing, TABLE III. Centers of the clusters within the unit cell ex- it follows that every b-linkage connects the same kind of tracted from therefined parameters (Appendix B). § cluster(M-MorB-B),whilethateveryc-linkageconnects site symbol atom Wyckoff indices nodetype different kinds of cluster (M-B). The three combinations for an adjacent pair of clusters are depicted in Figure 6. B(01) Pd1 8c [111000] (67)333 The coordinates of the cluster centers given in the re- B(2) Pd6 8c [444¯1¯1¯1] (67) fined atomic parameters (Appendix B) can be used to 0 333 § B(3) Pd11 24d [33200¯1] (66) enumerate which of the b- and c-linkages connect every 0 4322 B(4) Pd24 24d [44300¯2] (67) site to its adjacent neighbors. The task of obtaining the 0 333 B(5) Pd37 24d [3121¯20] (76) indices for every cluster center is straightforward. First 0 433 B(6) Pd50 24d [4231¯2¯1] (76) we set the indices of the atomic site Cr/Al76 lying at 0 433 B(7) Pd63 24d [5331¯1¯2] (67) the origin (0,0,0), which is the center of an M-cluster, M0(1) Cr/Al76 4a [000000] (683)33 to [000000]. Then we recursively trace linkages to ob- 0 0 tain the indices of adjacent cluster centers until all the M(2) Cr/Al79 4b [3022¯30] (68) 0 0 cluster centers within the unit cell as well as the lattice M(3) Fe/Pd82 24d [43210¯2] (57) 0 3322 translation vectors are identified. The resulting lattice M(04) Fe/Pd90 24d [4122¯2¯1] (57)332 translation vectors R (j = 1, 2 and 3) are presented in MM(0(056)) FCer//PAdl19085 2244dd [[2425110002¯1¯4]] ((6667))′433323 Teqaubalel tIhI.eSeixnpceeritmheenltaajtltivcaeluceonosftaant,=|R4j0|.=54˚A2b,τt2heshboauslidc M(07) Fe113 24d [4330¯1¯1] (67)333 parameter b is immediately evalualatted as 7.74˚A. Nowweareequippedwithanappropriatescaleforex- aminingtheinter-atomicdistances. Letusfirstnotethat theidealedgelengthb/τ3 oftheinnerdodecahedralshell B. Packing geometry of M-cluster (M ; Figure 5(a)) is evaluated to be 1.83˚A, 3 andthat itis unrealisticallyshortforaninteratomicdis- In the refined structure (Appendix B), the unit cell tance. A naturalconsequence of this is that the relevant § accommodates a total of 264 clusters, which divide into shells cannot be occupied by more than 8 atoms at the 128M-and136B-clusters. InTableIII,theindependent same time. Next, note that the outer shell of M-cluster positionsoftheclustercentersaresummarized. Thecen- (includingthetwosubshells)hastwokindsofedges. The ters ofthe M-clusters(M0)areoccupiedmainly by Fe or edges connecting the adjacent pairs in the M2 subshell Cr, although better fit is attained by occupying some are parallel to the two-fold axes and have a length of of them partially with Pd or Al. The centers of the B- b′ :=b/τ2 =2.958˚A,while those connecting the M sub- 2 clusters (B0) are purely occupied by Pd. shell with the M5 subshell are parallel to the three-fold Adjacent clusters are mutually connected either along axesandhavealengthofc′ :=c/τ2 =2.562˚A.Thelatter 7 FIG.6. Thethreedifferentcombinationsforanadjacentpair FIG.7. Thepackingofclusters. Thesquaredrawnwithwhite of clusters: M-M (left), B-B (middle) and M-B (right). Two dashed lines indicate a face of the cubic unit cell, whereas clusters of the same kind are connected via b-linkage, while thehorizontal and vertical edges correspond to theprimitive those of the two different kinds are connected via c-linkage. lattice vectors, R and R , respectively. 1 2 The sites in the constituent cluster templates are depicted as spheres. The drawing plane is perpendicular to a 2-fold symmetry axis. twoedgelengthsarereasonableforinteratomicdistances. Similarly, the edges connecting the adjacent pairs in the inner(B )ortheouter(B )shellofB-clusterareparallel 5 3 ′ to the two-fold axes and have a length of b. Interest- ′ ′ ingly, the two kinds of linkages, namely b and c, form the majority of the interatomic linkages in the idealized construction. This applies not only to the closest interatomic linkages within each shell of a cluster but also to sects with each other in their outer (sub)shells M5 and those connecting between the inner and the outer shells. B3,too,wheretherearethreecommonsitesbetweenthe It follows that the closest distances are b′ for the pairs, two. M -M , M -M , M -M , B -B and B -B , while c′ for The idealized atomic positions can be obtained by 3 3 2 2 3 5 5 5 3 3 the pairs M -M , M -M and B -B . Here we have in- replicating the relevant cluster templates at the posi- 2 5 3 2 5 3 cluded the pair M -M because the closest interatomic tions given in Table III. The arrangement of clusters in 3 3 distance that is allowedwithin the partially occupiedin- a unit cell is illustrated in Figure 7. As we have just ner shell (M ) of an M-cluster is b′. seen, an atomic position can be shared by two or more 3 TwoM-clustersconnectedthroughb-linkageinterpen- cluster shells that intersect with each other. Therefore, etrateintoeachother(Figure6,left)withtheoverlapbe- each individual site can be characterized by the set of tweentheouterpolyhedrabeingaflathexagonalbipyra- the cluster shells to which it belongs. Take, for instance, mid. The two tips of the bipyramid belong to the M2 a site in the inner shell (M3) of an M-cluster belong- shells and are very close to each other (1.828˚A), so that ing also to the outer shell (B3) of an adjacent B-cluster. they cannot be occupied at the same time. In the re- We simply assign to this site a class, M3,B3 . More h i finement, suchpositions have been fitted as splitting po- generally, if an atomic site belongs to n cluster shells, sitions for a single Al atom. The intersection between X1,X2,...,Xn, then the class associated with this site the two M-clusters consists of the six sites forming the would be X1,X2,...,Xn . Bear in mind that only the h i base hexagon of the hexagonal bipyramid; two of them combination of the cluster shells matters here, so that belong to the M subshells of the two M-clusters, while the order of the shell symbols in the angle brackets are 5 fourofthemtotheM subshells. Ontheotherhand,two irrelevant. It turns out that the idealized structure of 2 B-clustersconnectedthroughb-linkageshareanedgebe- the present crystal contains in total 16 different classes tween their outer shells (B , Figure 6, middle). In both of atomic sites, which are listed in Table IV. 3 cases, the intersection between the two clusters does not Itis obviousfromthe aboveargumentthatthecentral involve the inner shells. siteofeachclustercannotbesharedbyanyothercluster. The situation is somewhat more intricate in the case Hence, the symbols for the cluster centers are M and 0 h i of an M-B pair connected through c-linkage. Observe in B . In addition, the crystal structure includes three 0 h i Figure6(right)thattheoutershell(B )oftheB-cluster more classes of unshared sites, which are M , M and 3 2 3 h ih i penetratestotheinnershell(M )oftheM-cluster,where B . Importantly, the former two bear distinct roles 3 5 h i a site is shared by these shells. Similarly, the outer sub- in the physical construction of the structure: (i) M , 2 h i shell (M ) of the M-cluster penetrates to the inner shell the splitting positions given at the tip of the overlap 2 (B ) of the B-cluster, where a triangular face is shared hexagonalbipyramidassociatedwitheachinterpenetrat- 5 by these shells. One also finds that these clusters inter- ing pair of M-clusters and (ii) M , the vacant sites in 3 h i 8 enough and do not accommodate any additional atoms TABLEIV.The16different classes of idealized atomic sites. (called glue atoms). In Subsection VC, this feature is Their number frequencies per unit cell, presented in the sec- §§ described from a somewhat different viewpoint. ondcolumn,sumuptoatotalof4680. Theaveragecomposi- tion (in percentage) of each class is given in thelast column, where ‘Vc’ stands for vacancy. C. Canonical cell tiling with F-type ordering site class number average composition Recall that the atomic arrangement and the chemical M 128 (=F) Al Pd (Cr,Fe) 0 6 4 90 h i compositions within each individual cluster are subject M 720 (=G) Al Vc 2 50 50 hM i 1680 (=H) Vc to constraints posed by the existence of adjacent clus- 3 100 hB i 136 (=I) Pd ters (Subsection IVB). These geometrical constraints h 0i 100 §§ B 24 (=J) Al mayhardlyallowothercombinationsofclustersthanthe 5 100 h i M2,M2 72 (=K) Al100 threecasesshowninFigure6,thusenforcingaperfectF- h i M2,B5 576 (=L) Al100 type ordering. TheF-type icosahedralorderingobserved h i M3,B3 880 (=M) Al81Pd1(Cr,Fe)17Vc1 in more general Al-based alloys could also have a sim- h i M ,B 24 (=N) Pd h 5 3i 100 ilar geometrical origin, where an uneven distribution of B ,B 72 (=O) Pd h 3 3i 100 atomic species on different sublattices is understood to M ,M ,B 1032 (=P) Al h 2 2 5i 100 be a secondary consequence. M ,M ,M 112 (=Q) Al 2 2 2 100 h i The skeletal structure of the present approximant is M ,M ,M 24 (=R) Al (Cr,Fe) 5 5 5 1 99 hM ,B ,B i 408 (=S) Pd (Cr,Fe) described as the bc-packing, in which the cluster centers 5 3 3 98 2 hM ,M ,Bi,B 384 (=T) Pd (Cr,Fe) are represented as the nodes (Table III). The nodes can 5 5 3 3 65 35 h i M ,M ,M ,B 88 (=U) Pd (Cr,Fe) beconnectedtoeachotherthroughedgesofthetwokinds 5 5 5 3 3 97 h i (b- and c-linkages; Figure 8(a)), whereas the edges form three kinds of polygons: an isosceles triangle formed by a b-linkage and two c-linkages (X-face; point symmetry, theM3shellsoftheM-clusters. Rememberthattheinner m), an equilateral triangle formed by three b-linkages shellofM-cluster(M3)cannotaccommodatemorethan8 (Y-face; 3m) and a rectangle formed by two b-linkages atomsduetotheconstraintimposedbytheshortnearest- and two c-linkages (Z-face; 2/m); Figure 8(b). These neighbordistance. Anditisclearlydemonstratedbyour polygonsarefurtherfoundtobethefacesoffourkindsof structure analysis that the only occupied M3 sites are polyhedracalledthe canonical cells8: a tetrahedronwith those represented by the symbol M3,B3 . The last un- four X-faces (A-cell; 2m), a pyramid with three X-faces, h i shared class B5 does not seem to differ from the other oneY-faceandoneZ-face(B-cell;m),atetrahedronwith h i types ofB5 sites, as allof them are fully occupied by Al. three X-faces and one Y-face (C-cell; 3m) and a trigonal Correlations are further found between the local com- prism with two Y-faces and three Z-faces (D-cell; 3m); positions at individual atomic sites and the site classes. Figure 8(c). The present bc-packing thus proves to be Take, for instance, the B shell, which involves six dif- representedasaperiodictilingofspacecalledacanonical 3 ferent site classes (Table IV). Whereas the sites sym- celltiling (CCT); that is, the whole spaceis dividedinto bolized as M ,B are primarily occupied by Al, the pieces congruent to the canonical cells (Figure 9). 3 3 h i sites in the remaining five classes, M5,B3 , B3,B3 , So far, suitable atomic decorations of periodic CCT’s h i h i M5,B3,B3 , M5,M5,B3,B3 and M5,M5,M5,B3 ,areoc- havebeencommonlyusedtodescribeavarietyofapprox- h i h i h i cupied by markedlyheavier elements in different degrees imantphasestoP-typeicosahedralquasicrystals8,15. On dependingontheirsiteclasses. Similarobservationssug- the other hand, the present compound demonstrates for gestthatthe physicalnatureofanindividualatomicsite the first time that the CCT construction extends natu- is profoundly affected by the manner how clusters inter- rally to approximant phases to F-type icosahedral qua- sect there. Therefore, the two kinds of clusters not only sicrystals. Here, the two parities of the vertices need to provide a handy means of describing the complex struc- be clearly distinguished to account for the F-type order- ture but also serve as the true physical units which play ing;thisisdonebyrepresentingeven(resp. odd)vertices a significant role in the formation of the structure. This with white (resp. black) spheres in Figure 9. The geo- underlying basic idea may also apply to a more general metricalcompositionforthepresentCCTinparticularis class of Al-basedalloys which exhibit F-type icosahedral summarizedin Table V. There are 11 congruence classes ordering. of objects, some of which are further divided into two The packing of the cluster templates as described subclasses (I and II) owing to the parities of the nodes, above generates in total 4680 idealized atomic positions leading to the 18 object classes in total. Remember that (excluding the vacant M sites), which agree remark- X-face as well as A-, B-, C- or D-cell has more than one 3 h i ably well with the refined atomic positions, although in unique corners (see the corner symbols in Table V), for the refinement we included 48 additional sites which are which the parities need to be specified explicitly. It fol- needed to explain minor irregularities in the real mate- lows that there are two ways in coloring the vertices of rial. Importantly,thegapsbetweentheclustersaresmall each kind of cell, leading to two different configurations 9 FIG.8. Thecanonicalcellsandtheirgeometricalcomponents. FIG.9. (color) Atopviewofacleavedsurfaceofthepresent (a) The two kinds of linkages, b and c, represented as thick CCT parallel to the (001) plane; the foremost cells are cho- and double bars, respectively. (b) The three kinds of faces, sen somewhat arbitrarily. The spheres representing the even X, Y and Z, whose sides are distinguishable in terms of the (resp. odd) nodesare colored white (resp. black). The white icosahedralsymmetry;where‘+’signisononeside,‘ ’signis dashedlinesindicateasquarefaceofthecubicunitcell,while − ontheotherside. Thereversesidesofthefacesareshownby thehorizontal and vertical edges correspond to theprimitive rotatingthemby180degreesaroundtheverticaldashedline. latticevectors,R andR ,respectively. Atthelowerleftcor- 1 2 (c) The four canonical cells, A, B, C and D, and a trigonal nerofthesquareliesanevennodecorrespondingtotheorigin. anti-prism unit, B2, which can be divided into two B-cells in The four kinds of canonical cells are colored yellow (A-cell), three ways. The unique corners are indicated by the corner red (B-cell), blue(C-cell) and green (D-cell), respectively. symbols, while the numbers on the edges correspond to the followingdihedralangles: (1)π/2,(2)π/3,(3)π ǫ,(4)π η, − − (5)η,(6)π 2η,(7)(π ǫ)/2,(8)ǫ,(9)η,(10)π/2and(11) − − π/3. TABLEV.Thenumberfrequenciesofthegeometricalobjects containedinasingleunitcellofthepresentCCT.The11con- gruenttypescanbefurtherdividedintosubclassesaccording totheparitiesofthenodesasspecifiedintheparenthesesus- ing the symbols, + (for even) and (for odd), leading to 18 − basic object types. Note that for the faces (X - Z) as well as thecells(A-D),theparitysymbolsarearrangedinthesame orderasthecornersymbolsreferenced inthesecond column. The object type B can be divided into two B-cells in three 2 different ways, and we donot takeit as a basic object type. ofclustersandthustotwodifferentatomicarrangements object type within the cell (in the sense that the atomic species are (I) (II) disregarded). The present argument extends naturally to arbitrary node 128 (+) 136 ( ) − CCT’s, whereby a number of hypothetical approximants edge b 360 (++) 432 ( ) −− with F-type ordering can be constructed. From this edge c * 880 (+ ) − broader perspective, it is worthwhile making a general face X (XaXaXb) 1032 (++ ) 1176 ( +) − −− considerationon the statistics of the geometricalobjects face Y (YaYaYa) 112 (+++) 176 ( ) −−− face Z (Z Z Z Z ) * 176 (++ ) as well as that of the atomic sites. Still, readers who a a a a −− cell A (A A A A ) 384 (++ ) 336 ( ++) might want to avoid handling the mathematical deriva- a a b b −− −− cell B (B B B B B ) 104 ( +++) 152 (++ ) tions may safely skip the rest of this subsection and go a a b b c −− −−− cell B (B B B B B B ) * 80 (+++ ) 2 c c c c c c to Section V. −−− § cell C (CaCbCbCb) 88 ( +++) 168 (+ ) − −−− In the following argument, the number density of an cell D (D D D D D D ) 8 (+++ ) 24 ( +++) a a a b b b −−− −−− arbitraryobject type O among the 18 basic objecttypes is denoted n(O). In particular, the number densities of 10 theeighttypesofcellsareoffundamentalimportance,so unphysical consequences that is the number of edges of that we introduce the following parameters: α =n(A ), each type would be irrational, we need to constrain the I I α = n(A ), β = n(B ), β = n(B ), γ = n(C ), coefficients for ǫ and η to be zero. Therefore, the follow- II II I I II II I I γ =n(C ),δ =n(D )andδ =n(D ). Thefollowing ing equations need to be satisfied; II II I I II II sums are also used whenever it is convenient to do so: 4β +β +3γ =0, β 4β +3γ =0, I II I I II II − − α:=α +α , β :=β +β , I II I II 3β+3γ =0. (13) − γ :=γ +γ , δ :=δ +δ . (2) I II I II OnecanreadilycheckthatEqs.(4)-(7)and(13)canbe Bydefinition,thesumruleforthecellvolumesiswritten reduced to the following four universal equations, as β =γ, ∆α= ∆β, − v α+v β+v γ+v δ =1, (3) A B C D 3∆γ =5∆β, 3∆δ =∆β, (14) where vA = b3/12, vB = √5b3/12, vC = √5b3/24 and where v = 3b3/8 are the volumes of A-, B-, C- and D-cells, D respectively. In addition to this, these statistical param- ∆α:=αI αII, ∆β :=βI βII, − − etersholdasetofuniversalequationsduetogeometrical ∆γ :=γ γ , ∆δ :=δ δ . (15) I II I II constraints imposed by the shapes of the cells. − − Let us first consider the matching constraints across It follows from Eqs.(3) and (14) that there are only the faces. Take, for instance, a cell that has an X face three independent degrees of freedom to determine the I withits + sidefacing outward,thenanothercellwith an statistical properties of any CCT with F-type ordering. X facewithits sidefacingoutwardcanonlymatchthe Following ref.8, an independent parameter µ is defined I formercell. Sinc−ethesamerequirementisfulfilledacross as the volume fraction occupied by A- and D-cells: anyofthe X facesinthe structure,the+and sidesof thefacesoftIhistypemustappearequalnumber−oftimes. µ:=vAα+vDδ, (16) By counting the two distinct sides of XI faces for each while another ζ is defined so that µζ gives the volume typeofcells,Eq.(4)canbereadilyverified. Furthermore, fraction occupied by D-cells: similar considerations when applied to the X , Y and II I YII faces lead to Eqs.(5), (6) and (7), respectively. ζ :=vDδ/µ. (17) 2α +3γ =2α +2β +β , (4) It is readily shown that these two parameters determine I I II I II the number densities of the canonical cells through8 2α +3γ =2α +2β +β , (5) II II I II I β +δ =γ +δ , (6) α=12µ(1 ζ)/b3, (18) I I I II − β +δ =γ +δ . (7) β γ =(8/√5)(1 µ)/b3, (19) II II II I ≡ − δ =(8/3)µζ/b3. (20) Now we show that the matching constraints around the edgeswill leadto two more equations. Note that the The last independent parameter ν, which is necessary if dihedral angles contained in a B- or C-cell (see Figure F-type ordering is considered, can be defined as 8(c)) involve two irrational constant, ǫ and η, defined by8 ν :=∆β/β. (21) 1 As soon as the statistics is fixed for the cells, it is a ǫ arccos( ) 4π(0.1049), (8) ≡ 4 ≈ straightforward task to enumerate the objects of lower 1 dimensionalities (faces, edges and nodes) by taking the η arccos( ) 4π(0.0915). (9) ≡ √6 ≈ contributions from all the cells. The edges have already been enumerated via Eqs.(10)-(12), while for enumerat- Hereeachcellcontributesa fixedamountofdihedralan- ing the faces the double counting of each face must be gles associated with a fixed edge type. Hence, the con- taken care of. The nodes are enumerated by taking the tributions from all the cells sum up to the total number sum of the solid angles associated with the relevant cor- of the relevant edges times 2π; that is, ners of the cells8 and by dividing the results by 4π. The resulting formulae are 2πn(b )=(π/2)α+(3π 4η)β +ηβ I I II − +3ηγI+(3π/2)δ, (10) n(nodeI)=α/12+β/4+∆β/12+δ/4, (22) 2πn(bII)=(π/2)α+ηβI+(3π−4η)βII n(nodeII)=α/12+β/4−∆β/12+δ/4, (23) +3ηγII+(3π/2)δ, (11) n(bI)=α/4+3(β+∆β)/4+3δ/4, (24) 2πn(c)=(4π/3)α+3(π ǫ)β+3ǫγ+πδ. (12) n(b )=α/4+3(β ∆β)/4+3δ/4, (25) − II − Observe that in Eqs.(10)-(12) terms containing ǫ and η n(c)=2α/3+3β/2+δ/2, (26) donotcanceloutspontaneously. Hence,inordertoavoid n(X )=α+3(β+∆β)/2, (27) I