Z. Kristallogr. 219 (2004) 391–446 391 # by Oldenbourg Wissenschaftsverlag, Mu¨nchen
Twenty years of structure research on quasicrystals. Part I. Pentagonal, octagonal, decagonal and dodecagonal quasicrystals
Walter Steurer*
Laboratory of Crystallography, Department of Materials, ETH Zurich and MNF, University of Zurich, CH-8092 Zurich, Switzerland
Received November 11, 2003; accepted February 6, 2004
Pentagonal quasicrystals / Octagonal quasicrystals / 5.2.4 High-pressure experiments on decagonal Decagonal quasicrystals / Dodecagonal quasicrystals / phases ...... 434 Quasicrystal structure analysis 5.2.5 Surface studies of decagonal phases ..... 434 5.3 Dodecagonal phases ...... 435 6 Conclusions ...... 437 Contents References...... 437 1 Introduction ...... 391 2 Occurrence of axial quasicrystals ...... 392 Abstract. Is quasicrystal structure analysis a never-end- ing story? Why is still not a single quasicrystal structure 3 Crystallographic description of axial known with the same precision and reliability as structures quasicrystals ...... 393 of regular periodic crystals? What is the state-of-the-art of 3.1 Pentagonal phases ...... 395 structure analysis of axial quasicrystals? The present com- 3.2 Octagonal phases ...... 395 prehensive review summarizes the results of almost twenty 3.3 Decagonal phases ...... 396 years of structure analysis of axial quasicrystals and tries 3.4 Dodecagonal phases...... 397 to answer these questions as far as possible. More than 4 Limits and potentialities of methods for 2000 references have been screened for the most reliable quasicrystal structure analysis ...... 398 structural models of pentagonal, octagonal, decagonal and dodecagonal quasicrystals. These models, mainly based on 5 The structure of axial quasicrystals ...... 401 diffraction data and/or on bulk and surface microscopic 5.1 Octagonal phases ...... 402 images are critically discussed together with the limits and 5.2 Decagonal phases ...... 403 potentialities of the respective methods employed. 5.2.1 4A and 8A periodicity ...... 406 5.2.1.1 Al––Co...... 406 5.2.1.2 Al––Ni ...... 407 1 Introduction 5.2.1.3 Al––Co––Cu ...... 408 5.2.1.4 Al––Co––Ni ...... 411 At the end of 2003 more than 22 million different chemi- 1 5.2.1.5 Al––Co––Pd ...... 421 cal substances were known . For approximately 400.000 2 5.2.1.6 Al––Fe ––Ni ...... 422 of them the crystal structure was determined . Apart from 5.2.1.7 Al––Me1––Me2 (Me1 ¼ Cu, Ni, Me2 ¼ Rh, Ir) 423 the existence of a few hundred incommensurately modu- 3 5.2.1.8 Zn––Mg––RE (RE ¼ Y, Dy, Ho, Er, Tm, Lu) 423 lated structures (IMS) and composite structures (CS) , 5.2.2 12 A periodicity ...... 424 there was no reason to doubt that the ground state (i.e. the 5.2.2.1 d-Al––Mn...... 424 thermodynamic equilibrium state at 0 K) of all these com- 5.2.2.2 d-Al––Mn––Pd...... 426 pounds and of condensed matter in general is represented 5.2.3 16 A periodicity ...... 430 by a periodic crystal structure. 5.2.3.1 d-Al––Pd...... 431 5.2.3.2 Al––Fe...... 432 1 see CASTM at http://www.cas.org/ 5.2.3.3 d-Al––Me––Pd (Me ¼ Fe, Mg, Mn, Os, Ru) . 432 2 see data bases: organic and metalorganic structures, CSDTM TM 5.2.3.4 d-Al––Ni––Ru...... 434 at http://www.ccdc.cam.ac.uk/; inorganic structures, ICSD at http://www.fiz-informationsdienste.de/en/DB/icsd/; intermetallic phases, CRYSTMETTM at http://www.tothcanada.com/; Pauling FileTM at http://www.paulingfile.ch/ 3 partly contained in the data base ICSDB at * e-mail: [email protected] http://www.cryst.ehu.es/icsdb/ 392 Walter Steurer
On April 8th, 1982, D. Shechtman discovered a novel –– Structures of real QC (i.e. the experimentally studied phase with icosahedral diffraction symmetry in rapidly samples) closely resemble quasiperiodic structures at solidified Al86Mn14. This was the discovery of quasicrys- least at high temperatures and on time/space average. tals, which fundamentally changed our understanding of –– The structure of some QC can be quite well structural order on atomic scale. Quasicrystals (QC) rang described by cluster decorated quasiperiodic tilings, in a paradigm change (Kuhn, 1962) in crystallography. coverings and/or by the higher-dimensional ap- The completely new thing on QC is that they cannot be proach. described properly by a periodic basis structure and a peri- –– Many QC seem to be thermodynamically stable at odic modulation with incommensurate ratio of length scales least at high temperature. such as IMS, or as an incommensurate intergrowth of two The present review updates the article “The structure of qua- or more mutually modulated periodic structures such as CS. sicrystals” published in Zeitschrift fu¨r Kristallographie four- However, QC, IMS and CS have in common that they can teen years ago (Steurer, 1990). Since then the number of be described as three-dimensional (3D) irrational sections of publications on QC raised from approximately 1800 to nD(n > 3) translationally periodic hypercrystal structures more than 8000. Several books have been published as well (see Steurer, Haibach, 2001, and references therein). All either focusing on physical properties of QC (Goldman, three of them form the class of known aperiodic crystals. Sordelet, Thiel, Dubois, 1997; Stadnik, 1999; Dubois, Even twenty years after the first publication on icosahe- Thiel, Urban, 1999; Belin-Ferre, Berger, Quiquandon, 2000; dral Al––Mn (Shechtman, Blech, Gratias, Cahn, 1984) and Suck, Schreiber, Ha¨ussler, 2002; Trebin, 2003) or on mathe- more than 8000 publications later, there is still not a single matical aspects (Axel, Gratias, 1995; Axel, Denoyer, Ga- QC structure known with the reliability that is normal in zeau, 1999; Baake, Moody, 2000; Kramer, Papadopolos, standard structure analysis. This is reflected, for instance, in 2003). A number of comprehensive review articels appeared the long-lasting and still ongoing discussion about the struc- in the last years on general aspects of QC (Yamamoto, ture of decagonal Al––Co––Ni, the best studied decagonal 1996b), decagonal QC (Ranganathan, Chattopadhyay, QC model system so far (see chapter 5.2.1.4). Singh, Kelton, 1997), surface studies on QC (Diehl, Ledieu, Despite the more than 2000 papers on structural prob- Ferralis, Szmodis, McGrath, 2003; McGrath, Ledieu, Cox, lems of QC, only a few out of the approximately 50 stable Diehl 2002), electron-microscopic studies of QC (Hiraga, ternary and binary QC structures have been studied quanti- 2002), and many more. However, there was never a review tatively so far. The accurate knowledge of the structure since Steurer (1990) critically discussing the progress in and dynamical properties of at least one QC as a function structure analysis of QC in a comprehensive way. of temperature and pressure, however, is one of the prere- The review on QC structures will be published in two quisites for answering fundamental questions such as: parts. The present part I is dedicated to axial QC. In part II –– What governs formation and stability of QC? the structure of icosahedral QC will be reviewed. These –– Are QC entropy-stabilized high-temperature phases two reviews are aimed at taking stock of two decades of or are they a ground state of matter (thermodynami- combined effort of the QC community to solve the struc- cally stable at zero K)? ture of QC and to understand their formation and stability. –– Is the structure of QC quasiperiodic in the strict sense? –– Why only 5-, 8-, 10-, and 12-fold symmetries have been observed in QC (so far)? 2 Occurrence of axial quasicrystals What makes QC structure analysis so demanding is that it comprises the determination on atomic scale of both the QC, i.e. phases with diffraction patterns closely resem- short-range order (SRO) and long-range-order (LRO). bling those of quasiperiodic structures in the strict sense, SRO mainly refers to the atomic arrangement inside a unit have been found in approximately one hundred metallic tile or cluster (i.e. a recurrent structural building unit), systems. According to their diffraction (Fourier space) LRO to the ordering of the unit tiles or clusters them- symmetry, they are classified as icosahedral (i-), pentago- selves. The LRO of 3D periodic structures can definitely nal (p-), octagonal (o-), decagonal (d-) or dodecagonal be described by one of the 14 Bravais lattice types. In (dd-) phases. Also a few QC with crystallographic symme- case of QC, there are infinitely many different “quasilat- try, such as cubic, are known (Feng, Lu, Ye, Kuo, With- tices”, i.e. strictly quasiperiodic arrangements of unit tiles ers, van Tendeloo, 1990; Donnadieu, Su, Proult, Harmelin, or clusters, not to speak about other types of aperiodic Effenberg, Aldinger, 1996; Donnadieu, Harmelin, Su, Sei- structures (see, for instance, Axel, Gratias, 1995). Another fert, Effenberg, Aldinger, 1997). Quite a few QC were problem is that the inherent (?) disorder has to be consid- found to be stable at least at high temperature as far as it ered in structure analyses as well. A satisfactory QC struc- can be experimentally proved. Besides, there is a large ture solution will be based on experimental data from elec- number of metastable binary and ternary QC known, which tron microscopy, spectroscopy, surface imaging methods, can only be prepared by rapid solidification techniques. high-resolution diffraction methods and will include quan- Stable QC can reach a very high degree of perfection (sev- tum-mechanical calculations as well. eral micrometers correlation length). Large, millimeter- or Based on the structural studies published to date one even centimeter-sized single crystals have been grown of can summarize that: i-Al––Mn––Pd, i-Al––Pd––Re, i-Cd––Yb, i-Zn––Mg––Dy, –– The degree of perfection of several QC is compar- i-Zn––Mg––Ho, d-Al––Co––Cu and d-Al––Co––Ni. The able to that of silicon (i.e. several micrometer corre- elements known to be involved in QC formation are lation length). marked in the periodic table of elements (Fig. 2-1). Twenty years of structure research on quasicrystals 393
123456789101112131415161718 decagonal QC in this way due to reflection-profile over- 1 2 H He lapping and limited resolution. It is also not sufficient to 3 4 5 6 7 8 9 10 check electron diffraction patterns of annealed samples Li Be B C N O F Ne 11 12 13 14 15 16 17 18 (low resolution, only a few grains are investigated, the Na Mg Al Si P S Cl Ar sample may be in an intermediate transformation state). 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr The best proof of the stability of any phase is to study 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 reversible phase transformations by all techniques avail- Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 55 56 57* 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 able, from thermal analysis and dilatometry to in situ Cs Ba La Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 87 88 89+ 104 high-resolution electron microscopy and X-ray or neutron Fr Ra Ac Ku diffraction methods.
* Lanthanide 58 59 60 61 62 63 64 65 66 67 68 69 70 71 metals Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu + Actinide 90 91 92 93 94 95 96 97 98 99 100 101 102 103 metals Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 3 Crystallographic description of axial Fig. 2-1. Elements involved in the formation of binary, ternary or quasicrystals quaternary stable or metastable quasicrystals with icosahedral, penta- gonal, octagonal, decagonal or dodecagonal diffraction symmetry Pentagonal, octagonal, decagonal and dodecagonal QC (shaded cells). possess structures being quasiperodic in two dimensions and periodic in the third one. Their symmetry can be To date, stable axial QC have been found in more than equally well described in 3D reciprocal space (Rabson, twenty ternary and quaternary systems (see Tables 5.2-1 Mermin, Rokhsar, Wright, 1991) and 5D direct space and 5.3-1). Metastable axial QC are known in approxi- (Janssen, 1988; Yamamoto, 1996b, and references therein), mately forty binary and ternary systems (see Tables 5.1-1, respectively. For the detailed description of quasiperiodic 5.2-2, 5.3-1). Most of the stable QC were found under the structures (“where are the atoms?”), however, the nD assumption that they are electronically stabilized (Hume- description in direct space is more appropriate. In the Rothery phases), i.e. that they are stable only at a particu- following a short introduction is given into the 5D crystal- lar valence electron concentration such as e=a ¼ 1.75 or 2 lographic description of pentagonal, octagonal, decagonal (Tsai, 2003). This approach was particularly successful in and dodecagonal QC based on Steurer, Haibach (1999a, metallic systems where already approximants were known. 2001, and references therein). (Crystalline) approximants are phases with periodic struc- tures, which are locally similar to that of QC. Indexing Several chances were missed to identify novel interme- The set of all diffraction vectors H forms a Fourier mod- tallic phases as QC long before D. Shechtmann’s discov- ule M* of rank five in physical (parallel) space, ery. As early as 1939, the aluminum rich part of the () Al––Cu––Fe system was studied and a new phase, denoted P5 * k * w, reported (Bradley, Goldschmidt, 1939). Since at that M ¼ H ¼ hia i j hi 2 Z ; i¼1 time only X-ray powder diffraction was used for phase characterization, the icosahedral symmetry of this quasi- which can be decomposed into two submodules crystalline phase could not be seen. The same happened in M* ¼ M*1 M2*. M*1 ¼ h1a*1 þ h2a2* þ h3a3* þ h4a4* the system Al––Cu––Li, where Hardy, Silcock (1956) corresponds to a Z-module of rank four in a 2D subspace, found a phase T2 with weak, “fairly simple”, but not cubic M2* ¼fh5a*5g corresponds to a Z-module of rank one in a powder pattern, later identified as QC (Saintfort, Dubost, 1D subspace. Consequently, the first submodule can be Dubus, 1985; Ball, Lloyd, 1985). Lemmerz, Grushko, considered as a projection of a 4D reciprocal lattice, k Freiburg, Jansen (1994) identified the unknown stable M*1 ¼ p ðS*Þ, while the second submodule is of the form phase Al10FeNi3, discovered in a study of the phase dia- of a regular 1D reciprocal lattice, M2* ¼ L*. gram Al––Fe––Ni (Khaidar, Allibert, Driole, 1982), as dec- Due to scaling symmetry, indexing is not unique agonal phase. The Z phase in the system Mg––Zn––Y (cf. Mukhopadhyay, Lord, 2002; Lord, 2003). Nevertheless (Padezhnova, Melnik, Miliyevskiy, Dobatkina, Kinzhibalo, an optimum basis (low indices are assigned to strong 1982) also corresponds to a quasicrystalline phase (Luo, reflections) can be derived: not the metrics, as for regular Zhang, Tang, Zhao, 1993). Recently, the “unknown” com- periodic crystals, but the intensity distribution (related to pounds Cd5.7Y (Palenzona, 1971) and Cd17Ca3 (Bruzzone, the Patterson peaks in direct space) characterizes the best 1972) in the binary systems Cd––Y and Cd––Ca, respec- choice of indexing (see, e.g., Cervellino, Haibach, Steurer, tively, have also been identified as QC (Tsai, Guo, Abe, 1998). An appropriate set of reciprocal basis vectors can Takakura, Sato, 2000; Guo, Abe, Tsai, 2000b). be identified experimentally in the following way: There are only a very few QC for which there is some –– Find directions of systematic absences or pseudoab- experimental evidence for thermodynamic stability at least sences determining the possible orientations of the at high temperature (d-Al––Fe––Ni, d-Al––Co––Ni, reciprocal basis vectors (cf. Rabson, Mermin, Rokh- i-Al––Cu––Li, i-Al––Cu––Fe, d- and i-Al––Mn––Pd). In the sar, Wright, 1991); case of most other QC, their stability has not been studied –– Find pairs of strong reflections whose physical space sufficiently or not at all. It is certainly not sufficient to diffraction vectors are related to each other by the check X-ray powder diffraction patterns because high-or- scaling factor; der approximants or 1D QC cannot be distinguished from –– Index these reflections by assigning an appropriate 394 Walter Steurer
Table 3-1. Point group symmetry in recipro- 3D order of conditions n ¼ 5 n ¼ 8 n ¼ 10 n ¼ 12 cal space for the general case as well as for point the group trigonal tetragonal hexagonal dodecagonal pentagonal, octagonal, decagonal and dodeca- group type type type type gonal quasicrystals (see Rabson, Mermin, Rokhsar, Wright, 1991). The corresponding n 2 2 8 2 2 10 2 2 12 2 2 4nneven (if any) periodic crystal symmetry type is gi- m m m m m m m m m m m m ven as well. n 2 m 2nneven 882 m 102 m 122 m 2 2 n 4nnodd 5 m m nmm 2nneven 8mm 10mm 12mm nm 2nnodd 5 m n 22 2nneven 822 1022 12 22 n 22nnodd 52 n 2nneven 8 10 12 m m m m nnn neven 8 10 12 2nnodd 5 nn 5 8 10 12
Table 3-2. 3D decagonal point groups of order k and corresponding value to a*. This value should be derived from the 5D decagonal space groups with reflection conditions (cf. Rabson, shortest interatomic distance and the edge length of Mermin, Rokhsar, Wright, 1991). the unit tiles expected in the structure; 3D point Order 5D space reflection condition –– A reciprocal basis is appropriate if all observable group k group Bragg reflections can be indexed with integers. 10 2 2 10 2 2 40 P no condition Diffraction symmetry m m m m m m The diffraction symmetry of pentagonal, octagonal, deca- 10 2 2 h1h2h2h1h5 : h5 ¼ 2n P gonal and dodecagonal QC can be described by the Laue m c c h1h2h2h1h5 : h5 ¼ 2n groups 5 m or 5 and n/mmm or n/m with n ¼ 8, 10 and 12, 105 2 2 respectively. The axial point groups are listed in Tab. 3-1, P h1h2h2h1h5 : h5 ¼ 2n m m c the decagonal space groups in Tab. 3-2 (for pentagonal, 105 2 2 P h1h2h2h1h5 : h5 ¼ 2n octagonal and dodecagonal space groups see Rabson, Mer- m c m min, Rokhsar, Wright, 1991). These space groups are a 10 10 subset of all 5D space groups that fulfill the condition that 20 P no condition m m their 5D point groups are isomorphous to the 3D point 10 groups describing the diffraction symmetry. The P 5 0000h : h ¼ 2n m 5 5 orientation of the symmetry elements in the 5D space is defined by the isomorphism of the 3D and 5D point 10 2 2 20 P10 2 2 no condition groups. However, the action of the n-fold rotation is differ- P10j 2 2 0000h5 : jh5 ¼ 10n ent in the subspaces Vk and V?: a rotation by 2p/n in Vk is correlated to a rotation by 2 pk=n in V? (coupling fac- 10 mm 20 P10 mm no condition tor k ¼ 2 for n ¼ 5, k ¼ 3 for n ¼ 8 and 10, k ¼ 5 for P10 cc h1h2h2h1h5 : h5 ¼ 2n n ¼ 12). The reflection and inversion operations are h1h2h2h1h5 : h5 ¼ 2n equivalent in both subspaces.
P105 mc h1h2h2h1h5 : h5 ¼ 2n
P105 cm h1h2h2h1h5 : h5 ¼ 2n Structure factor The structure factor of a QC in the 5D description corre- 10m 220P10m 2 no condition sponds to the Fourier transform of the content of the 5D
P10c 2 h1h2h2h1h5 : h5 ¼ 2n unit cell P10 2m no condition PN FðHÞ¼ f ðHjjÞ T ðHjj; H?Þ g ðH?Þ e2piHrk P chh h h h : h n k k k 102 1 2 2 1 5 5 ¼ 2 k¼1 P5 10 10 P10 no condition with 5D diffraction vectors H ¼ hid*i, N hyperatoms i¼1 per 5D unit cell, parallel space atomic scattering factor 10 10 P10 no condition k k ? fk(H ), temperature factor Tk(H , H ), and perpendicular P10j 0000h5 : jh5 ¼ 10n ? k space geometrical form factor gk(H ). Tk(H , 0) is equiva- Twenty years of structure research on quasicrystals 395
? P5 lent to the conventional Debye-Waller factor, Tk(0, H ) de- k k All vectors H 2 M* can be represented as H ¼ hia*i scribes random fluctuations along the perpendicular space i¼1 coordinate. These fluctuations cause in physical space on the basis (V-basis) a*i ¼ a*i (cos 2pi=5, sin 2pi=5, 0), characteristic jumps of vertices (phason flips). All phason i ¼ 1; ...; 4 and a*5 ¼ a*5 (0, 0, 1) (Fig. 3.1-1). The com- flips map the vertices on positions, which can be de- ponents of a*i refer to a Cartesian coordinate system in P5 jj physical space. The reciprocal basis d*i, i ¼ 1; ...; 5, in scribed by physical() space vectors of the type r ¼ nia . i the 5D embedding space (D-space) reads P5 i¼1 jj 0 1 The set M ¼ r ¼ niai j ni 2 Z of all possible vec- cos ð2pi=5Þ i¼1 B C B C tors forms a Z-module. The shape of the atomic surfaces B sin ð2pi=5Þ C B C corresponds to a selection rule for the actually occupied d*i ¼ a*iei ; with e ¼ B 0 C ; i B C positions out of the infinite number given by M (de @ c cos ð6pi=5Þ A Bruijn, 1981). The geometrical form factor g (H?)is k c sin ð6pi=5Þ equivalent to the Fourier transform of the atomic surface, V i.e. the 2D perpendicular space component of the 5D 0 1 0 hyperatoms. B C B 0 C i ¼ 1 ...4 and d* ¼ a* B 1 C : Point density 5 5 B C @ 0 A The point density rPD (number of vertices per unit area) 0 V of a tiling with edge length ar can be calculated from the ratio of the relative number of unit tiles in the tiling to The coupling factor can be k ¼ 2ork ¼ 3 without chan- their area or from the 5D description ging the metric tensor. With the condition di d*j ¼ dij a P i basis in direct 5D space is obtained WAS 0 1 i 1 rPD ¼ B C WUC 0 2 B C d ¼ ðe e Þ; with e ¼ B 0 C ; i i * i 0 0 B C with WAS and WUC the areas of the atomic surfaces and 5a i @ 1 A the volume of the unit cell, respectively. 0 V 0 1 3.1 Pentagonal phases 0 B 0 C 1 B C An axial QC with pentagonal diffraction symmetry is i ¼ 1 ...4 and d ¼ B 1 C : 5 * B C called pentagonal phase. Its holohedral Laue symmetry a 5 @ 0 A * group is K ¼ 5m. The set M of all reciprocal space 0 vectors Hk remains invariant under the action of the V symmetry operators of the point group 5 m and its sub- Without loss of generality c can be set to 1. The metric * * groups (Table 3-1). M is alsopffiffiffi invariant under scaling tensors G, G are of the type SmM* ¼ smM* with s ¼ð1 5Þ=2. The scaling matrix 0 1 ABBB0 reads B C 0 1 B BABB0 C 010 10 B C B C B BBAB0 C B 011 10C @ A B C BBBA0 S ¼ B 111 00C : 0000C @ 101 00A 2 1 2 2 000 01 with A ¼ 2a*1 ; B ¼ =2a*1 ; C ¼ a*5 and A ¼ 4= 2 2 2 ð5a*1 Þ, B ¼ 2=ð5a*1 Þ; C ¼ 1=a*5 for the reciprocal and directp space,ffiffiffi respectively. Therewith we obtain * * * * * d i ¼ a i 2, i ¼ 1 ...4, pdffiffiffi5 ¼ a 5, a ij ¼ 104:47, i; j ¼ * * 1 ...5 and di ¼ 2=ð 5 a iÞ; i ¼ 1 ...4; d5 ¼ 1=a 5; a*ij ¼ 60; a*i5 ¼ 90; i; j ¼ 1 ...4 for the magnitudes of the lattice parameters in reciprocal and direct space, respec- tively.pffiffiffiffiffiffiffi The volumepffiffiffi of the 5D unit cell amounts to 4 V ¼ jGj ¼ 4=ð5 5 a*1 a*5Þ.
3.2 Octagonal phases An axial QC with octagonal diffraction symmetry is called Fig. 3.1-1. Star of reciprocal basis vectors of the pentagonal quasi- octagonal phase. Its holohedral Laue symmetry group is crystal and the decagonal quasicrystal (setting I), respectively. At left, * the projection along a*5 is shown, at right, a perspective view is K ¼ 8=mmm. The set M of all reciprocal space vectors given. Hk remains invariant under the action of the symmetry 396 Walter Steurer
2 2 2 2 with A ¼ 2a*1 ; B ¼ a5* and A ¼ 1=ð2a*1 Þ; B ¼ 1=a*5 for the reciprocalp andffiffiffi direct space, respectively. Thus we * * * * * obtain d i ¼ a i 2; i ¼p1ffiffiffi...4; d 5 ¼ a 5; a ij ¼ 90; * * i; j ¼ 1 ...5 and di ¼ 1=ð 2 a iÞ; i ¼ 1 ...4; d5 ¼ 1=a 5; a*ij ¼ 90; i; j ¼ 1 ...5 for the magnitudes of the lattice parameters in reciprocal and direct space, respectively.p Theffiffiffiffiffiffiffi volume of the 5D unit cell amounts to V ¼ jGj 4 ¼ 1=ð4a*1 a*5Þ.
Fig. 3.2-1. Star of reciprocal basis vectors of the octagonal quasicrys- 3.3 Decagonal phases tal. At left, the projection along a*5 is shown, at right, a perspective view is given. An axial QC with decagonal diffraction symmetry is called decagonal phase. Its holohedral Laue symmetry operators of the point group 8/mmm and its subgroups group is K ¼ 10=mmm. The set M* of all vectors Hk re- * (Table 3-1). M is alsop invariantffiffiffi under scaling mains invariant under the action of the symmetry opera- m m S M* ¼ s M* with s ¼ 1 2. The scaling matrix tors of the point group 10=mmm and its subgroups reads * n (Table 3-1). M is also invariantpffiffiffi under scaling by t 0 1 1 (n 2 Z), with t ¼ =2ð1 þ 5Þ. All reciprocalP5 space 110 10 k B C vectors H 2 M* can be represented as H ¼ hia*i on B 111 00C B C i¼1 S ¼ B 011 10C : the basis (V-basis) a*i ¼ a*i (cos 2pi=5, sin 2pi=5, 0), @ A 101 10 i ¼ 1; ...; 4 and a*5¼ a*5 (0, 0, 1) (Fig. 3.1-1) (Setting I). 000 01 Another possible choice (setting II) of basis vectors is P5 ai*¼ a*i (cos 2pi=10, sin 2pi=10, 0), i ¼ 1; ...; 4 and k * * All vectors H 2 M can be represented as H ¼ hia i a*5 ¼ a*5 (0, 0, 1). The vector components refer to a Carte- i¼1 sian coordinate system in physical space. on the basis (V-basis) a*¼ a* (cos 2pi=8, sin 2pi=8, 0), i i In case of setting I, which we will use in the following, the i ¼ 1; ...; 4 and a*5 ¼ a*5 (0, 0, 1) (Fig. 3.2-1). The com- ponents of a* refer to a Cartesian coordinate system in reciprocal and direct bases in the 5D embedding space i (D-space) and the scaling symmetry are the same as for the physical space. The reciprocal basis d*i; i ¼ 1; ...; 5, in the 5D embedding space (D-space) reads pentagonal case and coupling factor k ¼ 3 (see chapter 3.1). 0 1 cos ð2 pi=8Þ Indexing B C B sin ð2 pi=8Þ C There are several indexing schemes in use. Rarely employed B C d*i ¼ a*iei ; with e*i ¼ B 0 C ; are six-membered indices based on either a distorted icosa- @ A c cos ð6 pi=8Þ hedral basis vector set derived from the icosahedral basis c sin ð6 pi=8Þ V vector set or on a pentagonal-pyramidal basis vector set. For 0 1 a comparative discussion see Choy, Fitz Gerald, Kalloniatis 0 (1988) and references therein. Most frequently applied are B C B 0 C the schemes of Yamamoto, Ishihara (1988), YI-scheme, and B C i ¼ 1 ...4 and d*5 ¼ a*5 B 1 C : Steurer (1989), S-scheme, respectively.pffiffiffi The reciprocal basis @ A 0 vectors are related by: a*YI ¼ 5=t aS* (for a*i ¼ a*, i ¼ 1; ...;4), the coupling factors are kYI ¼ 2, kS ¼ 3, and 0 V the magnitudes of the physical space projections of thep 5Dffiffiffi jj * * The coupling factor amounts to k ¼ 3. With the condition reciprocal basis vectors amount to kp ðd iÞkYI ¼ a YI= 5 * jj * * di d j ¼ dij a basis in direct 5D space is obtained and kp ðd iÞkS ¼ aS , respectively. The transformation 0 1 matrices for the indices are just the scaling matrices 0 B C 0 1 B 0 C 010 10 B C B C 1 1 B C B 011 10C di ¼ ei; with i ¼ 1 ...4 and d ¼ B 1 C : 2a* 5 a* B C B 111 00C and i 5 @ A B C 0 @ 101 00A 0 V 000 01S )YI 0 1 Without loss of generality c can be set to 1. The metric 110 10 * B C tensors G, G are of the type B 001 10C B C 0 1 B 110 00C : A 0000 @ A B C 101 10 B 0 A 000C B C 000 01YI )S B 00A 00C @ A 000A 0 For instance, a* and a ¼ 1=a* amount to 0.26360 A 1 and 0000B 3.794 A in the S-scheme (Steurer, Haibach, Zhang, Kek, Twenty years of structure research on quasicrystals 397
Lu¨ck, 1993), and to 0.36429 A 1 and 2.745 A , respectively, Periodic average structure (PAS) in the YI-scheme (Yamamoto, Kato, Shibuya, Takeuchi, The periodic average structure of a decagonal phase can 1990). be obtained by oblique projection of the five-dimensional Fung, Yang, Zhou, Zhao, Zhan, Shen (1986) named hypercrystal structure (Steurer, Haibach, 1999b; Steurer, the two symmetrically inequivalent 2-fold axes D- and 2000). The lattice parameters of the most important side- P-type. The SAED images perpendicular to a P-type axis face centered orthorhombic PAS (IMS-setting), which is (shortly P-pattern) contain systematic extinct reciprocal closely related to the CsCl-type structure of the b-phase, layer lines. The directions are denoted A2D, A2P and can be calculated from A10. PAS 5ar PAS 5ar PAS a ¼ ; a ¼ pffiffiffiffiffiffiffiffiffiffiffi ; a ¼ a5 ; 1 3 2 2 3 Approximants t t 3 t
Approximants in the wider sense of the word posses peri- with ar ¼ 2.456 A , the Penrose rhomb edge length in odic crystal structures consisting of the same atomic clus- d-Al––Co––Ni, for instance. The PAS shows the correspon- ters as QC. Rational approximants are the subset of struc- dence between the atoms of a quasiperiodic structure and tures that can geometrically be derived by a perpendicular- an underlying periodic structure. It can be very useful in space shear of the QC in the higher-dimensional descrip- understanding the geometry of continuous quasicrystal-to- tion (see Steurer, Haibach, 1999a; Ranganathan, Subrama- crystal transformations. It may also help to discover struc- niam, Ramakrishnan, 2001; Niizeki, 1991). Thereby, the tural relationships between QC and non-rational approxi- irrational number t is replaced by a rational number. mants such as the b-phase, for instance. According to the group/subgroup symmetry relationship The geometrical relationship between the (110)-layer of between the d-phase and its approximants, the approxi- the Al(Co, Ni) b-phase and the PAS of decagonal mants may exhibit orthorhombic, monoclinic or triclinic Al––Co––Ni (Steurer, Haibach, Zhank, Kek, Lu¨ck, 1993), symmetry. In the following, only the most frequent orthor- for instance, is: the ½1110 direction of the b-phase is paral- hombic approximants will be discussed. lel to the 10-fold axis, [110] and [111] are parallel to the The lattice parameters of the general orthorhombic two different 2-fold axes of the d-phase. Therefrom it fol- hip=q; r=s -approximant are PAS PAS PAS lows for the PAS: a1 jj½001 , a2 jj½110 and a3 jj½1110 . pffiffiffiffiffiffiffiffiffiffiffi 2 The translation period along [001] amounts to 2.88 A, 2ð3 tÞðt p þ qÞ 3 t ðtr þ sÞ a1 ¼ ; a2 ¼ ; along [110] and ½1110 to 4.08 A for a lattice parameter of 5a* 5a* ab ¼ 2.88 A for the b-phase. This fits nicely to the periods a3 ¼ a5 : in the respective directions of the PAS of d-Al––Co––Ni: PAS PAS PAS a1 ¼ 2:88A, a2 ¼ 3:99A, a3 ¼ 4:08A. For the special case p ¼ Fnþ2; q ¼ Fn, r ¼ Fn0þ1, 0 s ¼ Fn0 we obtain the orthorhombic hn; n i-approximants with lattice parameters 3.4 Dodecagonal phases pffiffiffiffiffiffiffiffiffiffiffi An axial QC with dodecagonal diffraction symmetry is 2ð3 tÞ nþ2 2 3 t n0þ1 a1 ¼ t ; a2 ¼ t ; a3 ¼ a5 : 5a* 5a* called dodecagonal phase (see also Ga¨hler, 1988). Its holo- K =mmm M* 2 hedral Laue symmetry group is ¼ 12 . The set This corresponds to a1 ¼ t aP and a2 ¼ taD in the terminol- of all reciprocal space vectors Hk remains invariant under ogy aP, aD, used by Zhang, Kuo (1990). A few examples the action of the symmetry operators of the point group are given in Table 3.3-1. The orthorhombic unit cells 12=mmm and its subgroups (Table 3-1). M* is also invar- 0 pffiffiffi are (110)-face centered if n mod ð3Þ¼ðn þ 1Þ mod ð3Þ. m * m * jj iant under scaling S M ¼ s M with s ¼ð2 3Þ 2. The reciprocal space vectors H ¼ðh1h2h3h4h5Þ are The scaling matrix reads transformed by the perpendicular-space shear to 0 1 Hjj ¼ ð½ pðh þ h Þ qðh þ h Þ ½rðh h Þ sðh þ h Þ h Þ. 110 10 2 3 1 4 1 4 2 3 5 B C B 111 00C B C S ¼ B 011 10C : @ 101 10A Table 3.3-1. Series of common rational hn, n0i-approximants of the d-phase. The reciprocal lattice parameter used is that of 000 01 d-Al––Co––Ni, a* ¼ 0.26352 A 1 (Steurer, Haibach, Zhang, Kek, P5 k jj Lu¨ck, 1993). All vectors H 2 M* can be represented as H ¼ hia*i i¼1 0 * * n, n a1 a2 on a basis (V-basis) a i ¼ a i(cos 2pi=12, sin 2pi=12, 0), i ¼ 1; ...; 4 and a*5 ¼ a*5(0,0,1) (Fig. 3.4-1). The vector 0 5.492 2.887 components refer to a Cartesian coordinate system in phy- 1 8.886 4.672 sical space. The reciprocal basis d*i i ¼ 1; ...; 5 in the 5D 2 14.378 7.559 embedding space (D-space) reads 3 23.264 12.230 0 1 cos ð2 pi=12Þ 4 37.641 19.789 B C B sin ð2 pi=12Þ C 5 60.905 32.020 * * B C d i ¼ a iei ; with ei ¼ B 0 C ; 6 98.546 51.809 @ c cos ð10 pi=12Þ A 7 159.451 83.829 c sin ð10 pi=12Þ V 398 Walter Steurer
grain boundaries, respectively. If it only has equilibrium defects it is called a perfect crystal otherwise an imper- fect crystal. Beside defects a crystal may also show in- herent structural disorder. A crystal is not a static ar- rangement of atoms or molecules. The atoms vibrate around their equilibrium positions due to the superposi- tion of all thermally excited lattice vibrations (phonons). Consequently, the ideal structure of a crystal is just a simplified model of the real structure. To fully describe the real structure of a crystal one needs a model for the Fig. 3.4-1. Star of reciprocal basis vectors of the dodecagonal quasi- ideal structure as well as one describing the deviations crystal. At left, the projection along a*5 is shown, at right, a perspec- tive view is given. from it (dynamics, disorder and defects). A real crystal is rarely in thermodynamic equilibrium. If crystallized from 0 1 the melt, the actual structure at ambient conditions al- 0 B C ways is a kind of quenched metastable state. Thermody- B 0 C B C namic equilibrium cannot be reached due to sluggish ki- i ¼ 1 ...4 and d*5 ¼ a*5 B 1 C : @ A netics at lower temperature. 0 There is no doubt that regular crystals possess transla- 0 V tionally periodic structures on time and space average. No- body ever doubted that really and tried to prove this or k The coupling factor amounts to ¼ 5. With the condition even thought that it would make sense to prove it. In the d d* i j ¼ dij a basis in direct 5D space is obtained case of QC the situation is different. At least on one single 1 example it has to be demonstrated how an ideal QC struc- d ¼ ð2e e Þ with i ¼ 1 ...4 ; i * i ðiþ2Þ mod ð4Þ ture looks like and what kind of disorder and defect struc- 3a i 0 1 0 ture is inherent or common. It is typical for X-ray diffrac- B C tion patterns of QC that they show sharp Bragg reflections B C B 0 C even if strong (phason) diffuse scattering is present. This 1 B C and d5 ¼ B 1 C : indicates long correlation lengths (micrometers) of the a*5 B C @ 0 A space and time averaged structure. Thus, QC show long- range order accompanied by short-range disorder. This is 0 V preferentially random phason disorder and, in particular for Without loss of generality c can be set to 1. The metric pseudoternary QC, in addition chemical disorder. tensors G, G* are of the type There are, however, many problems making QC struc- 0 1 ture analysis extremely difficult. The most serious problem A 0 B 00 is that only very limited data sets are experimentally ac- B C B 0 A 0 B 0 C cessible, be it diffraction data or microscopic data. This B C B B 0 A 00C makes it impossible to determine the “absolute order” of a @ 0 B 0 A 0 A macroscopic QC. A good fit to experimental data of a 0000C model is no proof that the global minimum was found and that the proposed model is the best possible one. Thus it 2 2 2 2 with A ¼ 2a*1 ; B ¼ a*1 ; C ¼ a*5 and A ¼ 2=ð3a*1 Þ; is very difficult to find out whether a QC is quasiperiodic 2 2 B ¼ 1=ð3a*1 Þ; C ¼ 1=a*5 for the direct and reciprocal in the strict meaning of the word, only on average, or not space, respectively. Thus we obtainp forffiffiffi the lattice at all; or to prove that a QC is energy or entropy stabi- parameters in reciprocal space d*i ¼ a*i 2, i ¼ 1 ...4, lized, whether its structure can be described by an ordered * * * * d 5 ¼ a 5, a ij ¼ 60, api5ffiffiffi¼p90,ffiffiffi i; j ¼ 1 ...4 and for those in tiling or rather by a random tiling. It will also be difficult * * direct space di ¼ 2=ð 3 a iÞ, i ¼ 1 ...4, d5 ¼ 1 a 5, to prove that QC modeling can be accurately done by the a*ij ¼ 120; a*i5 ¼ 90; i; j ¼ 1 ...4.p Theffiffiffiffiffiffiffi volume of the 5D nD approach. Probably, final modeling has to be per- 4 unit cell can be calculated to V ¼ jGj ¼ 1=ð4a*1 a5Þ. formed in 3D space to properly account for atomic relaxa- tion and disorder. Therefore, it is essential to know at least 4 Limits and potentialities of methods the maximum error one can make by using the one or the for quasicrystal structure analysis other method. There is a couple of publications on the potential and limits of QC structure analysis (cf. Beeli, What do we want to know about the structure of a QC Steurer, 2000; Beeli, Nissen, 1993). In the following, the and what can we know employing the full potential of tools, techniques and methods most frequently employed state-of-the-art methods? Why do we want to know that? in QC structure analysis are shortly commented. How does this compare with the information we have about periodic structures from small molecule crystals to Observation, interpretation, modeling virus crystals? Theoreticians are eager to find confirmation of their Any real crystal, be it a periodic or an aperiodic one, ideal(ized) model systems in the “lowly spheres” of the is finite and has equilibrium and non-equilibrium defects experiment. Experimentalists like their role as pioneers such as thermal vacancies as well as dislocations and who provide the proofs for important theories. Both of Twenty years of structure research on quasicrystals 399 them run the risk to use the experimental evidence some- are imaged in the same way as in the case of X-ray and what selectively to support their points of view. Evidence neutron diffraction. for this can be found not only in the QC literature. There are some other reasons as well for the difference Theories are ideal because reality is too complex to be in SAED and XRD patterns. Due to the small penetration fully mapped into model systems and computing power is depth of the electron beam in the sample (<1000 A ) and still too limited. Unfortunately, reality is imperfect, experi- its usually small diameter (500–5000 A ) (Tsuda, Nishida, mental resolution mostly too low, samples are not pure en- Tanaka, Tsai, Inoue, Masumoto, 1996), scattering informa- ough, annealing times too short, data finally too inaccurate, tion of submicroscopic parts of a sample can be obtained and funding never sufficient. It is particularly difficult to only (the volume of a sample for XRD is larger by about communicate experimental results to theoreticians and vice nine orders of magnitude). versa. There is a source of misunderstandings if a theoreti- cian him/herself tries to interpret HRTEM images or elec- CBED (convergent beam electron diffraction) tron density maps to find confirmation for his/her hypoth- By this method a very small area of the thinned ( 100 A esis; or, if an experimentalist looks to his/her data through thickness) sample with 10–100 A diameter is probed by the filter of the higher-dimensional description and wants the convergent electron beam. This method allows, for in- to proof that a few ring contrasts on a HRTEM image are stance, deriving the full point group symmetry of the sam- sufficient to prove strict quasiperiodicity. ple (Tanaka, 1994) instead of just the Laue class. One can There is also a problem in the interpretation of qualita- even quantitatively refine the parameters of a trial struc- tive and quantitative structural information. It is state-of- ture model by fitting the line profiles of the high-order the-art in structure research to describe a crystal structure Laue-zone (HOLZ) reflections (see, for instance, Tsuda, quantitatively (coordinates, atomic displacement para- Tanaka, 1995). Thus, by this method it is possible to meters, etc.). Due to the easy access to HRTEM images, determine the structure of a cluster, for instance. structure solution is often reduced to a simplistic contrast interpretation of the projected structure. However, this HRTEM (high-resolution transmission electron microscopy) should be only a first step to a quantitative structure The contrasts visible on electron micrographs are related description. A similar problem occurs in X-ray structure to the projected structure (potential). They strongly depend analysis, which always is the refinement of a globally on sample thickness (>50 A ) and defocus value. Their in- averaged structure model. One never knows whether or terpretation is not straightforward and contrast simulations not a better model does exist. This uncertainty should be should confirm the models derived. For instance, Tsuda, considered in the discussion of the results. Nishida, Tanaka, Tsai, Inoue, Masumoto (1996) demon- The story of decagonal Al––Co––Ni (see chapter strated by computer simulations that a pentagonal cluster 5.2.1.4) is the perfect example for the long way scientists model can produce HRTEM images with local pentagonal have to go in QC research until observation, interpretation, as well as decagonal symmetry depending on the acceler- and modeling converge. Certainly, the end of this walk, ating voltage, 200 kV and 300 kV, repectively. fortunately not a random walk, is still out of sight. Look- The lateral resolution of HRTEM experiments is ing back, however, real progress becomes visible. approximately 1 to 2 A depending on the acceleration voltage of the electrons. However, it is not always possible Electron microscopy to work at highest resolution because even metallic sam- General overviews of the application of electron mi- ples may undergo structural changes under irradiation, in croscopic methods to QC are given by Hiraga (2002) and particular for voltages >400 kV (sometimes >250 kV). Beeli (2000), and in general by Smith (1997), for instance. An automated approach for the analysis of HRTEM images of QC was developed by Soltmann, Beeli (2001) SAED (selected area electron diffraction) based on previous work of Joseph, Ritsch, Beeli (1997). Due to multiple scattering and other interaction potentials, This technique should be used to match tilings to the SAED patterns significantly differ from X-ray diffraction observed contrasts in an unbiased way. (XRD) patterns. The reflection intensities are not propor- tional to the squares of the structure amplitudes as it is the HAADF-STEM (high-angle annular detector dark-field case if the kinematical theory applies. For SAED pattern scanning transmission electron microscopy) or Z-contrast calculation dynamical theory is needed, the powerful tools method of X-ray structure analysis (direct methods, e.g.) do not The image is formed by electrons scattered incoherently at work. However, the rapid progress in electron crystallogra- high angles ( 100 mrad) in a STEM. Dynamical effects phy is going to change this situation (see Dorset, Gilmore, and the influence of specimen thickness are less signifi- 2003). Multiple scattering generally leads to a relative cant compared to SAED and HRTEM. By a finely focused enhancement of weak reflections and diffuse scattering. electron beam ( 2 A diameter) as probe the specimen is However, compared to X-ray diffraction, the SAED expo- scanned illuminating atomic column by atomic column. sure time is usually much shorter, the intrinsic background The annular detector generates an intensity map of inco- much higher and the dynamical range of the SAED herently scattered electrons with atomic number (Z) con- patterns much smaller (2–3 orders of magnitude). X-ray trast. Therefore, this method is also called ‘Z-contrast intensities may be quantitatively collected within a dyna- method’ (see Saitoh, Tsuda, Tanaka, Tsai, 2000, and refer- mical range of eight orders of magnitude. Diffraction ences therein; Pennycook, 2002). Thus, in case of transi- symmetry (Laue class) as well as systematic extinctions tion-metal aluminides it allows an easy differentiation 400 Walter Steurer between contrasts originating from transition metal atoms space). Unfortunately, this class of reflections is intrinsi- or from aluminum atoms. Usually the contrast is reversed cally weak. Since it is crucial to include in the nD refine- compared with HRTEM micrographs. Image deformation ments as much as possible information about weak reflec- is possible due to the sample drift obscuring the symmetry tions, no threshold value ðI > 3s (I), for instance) should (distortion of decagonal clusters, for instance). be set as it is often done in the course of standard struc- ture analyses of periodic structures. It is also very impor- Electron and Ion beam irradiation tant to check the results of an nD refinement not only by By fast particle irradiation of a sample above a specific global reliability factors (R-factors) but also by F(obs)/ energy (20–30 eV) threshold radiolytic (ionization and F(clc)-plots and statistical analysis (cf. Cervellino, Steurer, bond breaking) or knock-on (collision and knocking out of Haibach, 2002, for instance). atoms from their sites) damage can take place. Radiolytic The potentialities and limits of XRD on QC have been effects predominantely occur at low energies, knock-on ef- discussed by Haibach, Cervellino, Estermann, Steurer fects only at high-energies. The induced defects accelerate (2000) with the focus on Bragg scattering and by Ester- atomic diffusion considerably (Gittus, 1978; Smith, 1997). mann, Lemster, Haibach, Steurer (2000) focusing on diffuse This may overcome the sluggish kinetics of low-tempera- scattering. Those of ND have been outlined by de Boissieu ture phase transformations. Irradiation by electrons in the (2000), Frey (2002) and Frey, Weidner (2003) with empha- electron microscope may induce structural defects already sis on Bragg and diffuse scattering, respectively. for energies >250 keV (Ritsch, Beeli, Nissen, Lu¨ck 1995). Inelastic and quasielastic neutron scattering XRD (X-ray diffraction) and Neutron diffraction (ND) The study of the dynamical properties of QC, i.e. the pho- In both cases, the kinematical theory can be applied to non and phason dynamics, can be performed in the usual describe the correspondence between structure and diffrac- way by inelastic and quasielastic NS, respectively (de tion pattern of a sample. A very powerful toolbox of struc- Boisssieu, 2000; Coddens, Lyonnard, Hennion, Calvayrac, ture determination techniques has been developed during 2000, and references therein). the past 80 years, which allow solving even virus structures in a more or less straightforward way. Some of these tools Coherent X-ray diffraction (Patterson analysis, least-squares refinement, method of With the advent of third generation synchrotron sources entropy maximization etc.) have been modified for higher- (partly) coherent X-ray radiation became available. It can be dimensional structure analysis. XRD on polycrystalline particularly useful in the interpretation of the diffuse scatter- materials is often used to characterize quasicrystalline ing. For instance, the study of the phason dynamics may samples and to check their quality. It has to be kept in mind, profit from this technique (Letoublon, Yakhou, Livet, Bley, however, that a powder XRD pattern is just a projection of de Boissieu, Mancini, Caudron, Vettier, Gastaldi, 2001). the full 3D reciprocal space information upon 1D. Superpo- sition of Bragg peaks and levelling out of structured diffuse Higher-dimensional approach scattering is the rule. High-resolution powder XRD may be This approach, first proposed by deWolff (1974) for in- used as a fingerprint of a known QC or for the accurate commensurately modulated structures, is very powerful in determination of lattice parameters. It must not be used as case of perfectly ordered quasiperiodic structures with the sole proof for quasiperiodicity or perfection of a QC. dense atomic surfaces of polygonal shape (Yamamoto, Single-crystal XRD or elastic ND are the diffraction 1996b; Steurer, Haibach, 2001; and references therein). It methods of choice. The Bragg reflections carry informa- is also the only way to apply crystallographic tools such tion about the globally averaged structure, the diffuse as the Patterson technique or direct methods. It allows intensities about the pair correlation functions of local ‘lifting’ atoms or clusters and refining easily infinite struc- structural deviations from this average structure. A 3D tures in a closed form. However, the nD approach does resolution better than 0.001 A can easily be obtained. not work properly for (strongly) disordered structures and XRD and ND are sometimes complimentary in scattering ‘non-quasicrystallographical’ degrees of freedom (continu- power. This can be used to distinguish, for instance, the ous shifts instead of phason flips etc.). It also does not ordering of Co and Ni. The different scattering power of work properly in case of QC with random-tiling related different isotopes of an element can be used to calculate structures (Henley, Elser, Mihalkovic, 2000). partial structure factors. In case of an nD structure analysis, the long-range or- Tiling decoration approach der of a QC structure is coded in the fine details of atomic This 3D approach (Mihalkovic, Mrafko, 1997, and refer- surfaces (occupation domains, hyperatoms). Consequently, ences therein) is much more flexible than the nD 5D structure analysis means determination of the detailed approach, in particular if the tiling is not fixed and if shape and internal structure of the atomic surfaces. Since beside vertex flips also continuous atomic shifts are possi- the atomic surfaces are mainly extended in perpendicular ble. It allows modeling easily even tilings that possess space, reflections with large perpendicular space compo- fractal atomic surfaces in the nD description, which are nents of the diffraction vectors are particularly important common for maximum-density disk packings (Cockayne, in the refinement of their detailed structure (shapes of 1995), for instance, or random tilings in the general mean- domains and subdomains, chemical composition, occu- ing. The geometrical degrees of freedom of each atom can pancy factors, parallel space shifts, mean square random vary continuously in this approach contrary to the phason displacement parameters in perpendicular and parallel coordinate shifts in the nD description, for instance. To Twenty years of structure research on quasicrystals 401 get a physically and crystal-chemically reasonable model describe the structure of a decagonal phase by a covering by the tiling decoration method, some constraints are was in the paper by Burkov (1992). He presented a model needed. Joint refinements minimizing the differences consisting of a random assembly of decagonal clusters. between calculated and observed diffraction data and the From a crystal-chemical point of view, the structure may energy of the system show promising results (Henley, be seen as optimum packing of energetically favorable struc- Mihalkovic, Widom, 2002). A drawback is that only a tural units such as coordination polyhedra. With other words, rather limited “box of atoms” can be used as model. the optimization of interactions between atoms determines the type of coordination polyhedron and its way of packing. Quasi-unit-cell approach The problem of QC structure formation could then be re- This term has been coined by Steinhardt, Jeong (1996). It duced to the question why a quasiperiodic structure forms refers to the description of quasiperiodic structures by cov- despite a large unit-cell approximant would locally have the erings. In a covering, one single structure motif (cluster) is same structure without sacrificing the benefits of periodicity. sufficient to build a quasiperiodic structure while for a til- ing at least two different unit tiles are needed. How such a Approximants quasi-unit-cell (“monopteros”) and its overlaps could look Crystalline approximants, i.e. phases with periodic struc- like has already been shown for d-Al––Co––Ni by Steurer, tures that are locally similar to those of QC, play a key role Haibach, Zhang, Kek, Lu¨ck (1993). A decagon with over- for QC structure analysis. Their close relationship to QC lap rules forcing a covering fully equivalent to a Penrose can easily be seen on SAED patterns. The reciprocal lattice tiling was discovered by Gummelt (1996). Steinhard, has basic vectors of characteristic length and the intensity Jeong (1996) showed that the Penrose tiling contains the distribution corresponds to the Fourier transform of the fun- maximum possible number of these Gummelt-decagons. If damental cluster building both the QC and the approximant. it were decorated with an energetically favorable cluster, For reviews see, for instance, Goldman, Kelton (1993), Gra- the quasiperiodic covering would have the lowest total tias, Katz, Quiquandon (1995), Tamura (1997). energy. However, this is not always true. The density of Domain structures of approximants, resulting from a 0 pentagons in the approximant O2 (Dong, Dubois, Song, phase transformation from QC obey the usual twin laws. The Audier, 1992), for instance, is with 0.854 higher than in lost symmetry elements relate approximant domains to each the pentagonal Penrose tiling where it amounts to other (orientational twinning). Approximants often form t/2 ¼ 0.809. Consequently, the total energy of a structure nanodomain structures with coherent domain boundaries. with energetically favorable pentagonal clusters would be lower for the approximant than for the quasicrystal. Disorder Disorder related to short-range correlations within the quasi- Clusters periodic plane of a superperiod along the tenfold axis is dis- Most recent models of quasiperiodic structures are based on cussed, for instance, by Tsai, Inoue, Masumoto (1995). For one or more unit clusters. However, the term ‘cluster’ is not general reviews on diffuse scattering in QC see Steurer, always used with the same meaning. The classical definition Frey (1998) and Frey (2002). For modeling the short-range distinguishes between ‘naked’ clusters as obtained and in- order in periodic crystal structures the Fourier transform of vestigated in mass spectrometers, for example, and em- the cluster is multiplied by the Fourier transform of the fi- bedded clusters, like in metallorganic compounds. In both nite crystal lattice. In case of a quasiperiodic structure, this cases it is clearly defined which atom belongs to a cluster has to be done in nD space using the cluster related atomic and which one does not. The chemical bond between an surfaces. In case of the Penrose tiling decorated with equal atom in the cluster to another atom in the same cluster point atoms, for instance, the Fourier transform of the atom- clearly differs from the bond to an atom outside the cluster. ic surfaces (four pentagons) is only a function of the perpen- In the QC community the term cluster is frequently used for dicular space components of the reciprocal space. “structure motif”, “structural unit”, “quasi-unit cell” or “coordination polyhedron”. For reviews on bare clusters see, for instance, Martin (1996), Wales, Munro, Doye (1996). 5 The structure of axial quasicrystals
Tilings, coverings There are alternative descriptions of quasiperiodic struc- Conventional crystal structures are crystallographically tures possible either as incommensurately modulated described by their (space group) symmetry and by the phases or as composite crystals (Elcoro, Perez-Mato, content of their unit cells. This has some analogies with 1996; Steurer, 2000). In these approaches, the special the decription of a QC structure in terms of a tiling or characteristics of QC such as scaling symmetry in recipro- covering decorated by atoms or clusters (see Kramer, Pa- cal space, for instance, are just special cases. padopolos, 2003, and references therein). A quasiperiodic In the case of octagonal and dodecagonal QC the alter- tiling can be built based on at least two unit tiles. The native description as 2D incommensurately modulated tiles are put together without any overlaps and gaps ac- structures may be more obvious than in the case of penta- cording to given matching rules (if there are any). The gonal or decagonal phases. The basic structures as well as quasi-unit cell of a covering, for instance a decagon in the average structures are just tetragonal and hexagonal, case of decagonal structures, forms the structure by cover- respectively. This has important consequences. Firstly, in ing the plane (space) without gaps but with well-defined both cases a periodic average structure does exist. Sec- overlaps. The first time a decagonal cluster was used to ondly, the quasiperiodic structure can be described as 2D 402 Walter Steurer modulation of a tetragonal and a hexagonal basic struc- ture, respectively. By continuous variation of the length and orientation of the satellite vectors a transformation from the octagonal and dodecagonal structures to their tetragonal and hexagonal approximants, respectively, can be easily described. Thirdly, the finding that a dodecago- nal QC can be nicely described as modulated phase (Uchida, Horiuchi, 1998a; 2000) is not an argument against the existence of the dodecagonal QC and its more appropriate, symmetry adapted description as quasicrystal. In the following, the different classes of axial QC will be described in detail. The very few pentagonal QC known will be dealt together with decagonal phases.
5.1 Octagonal phases A general introduction into simple octagonal tilings (Fig. 5.1-1), their generation and properties was given by Socolar (1989) and Ingalls (1993), for instance. The ther- mal and phason diffuse scattering for octagonal QC was discussed by Lei, Hu, Wang, Ding (1999). It is amazing that the only octagonal QC found ever were discovered Fig. 5.1-1. Octagonal tiling generated by the dual-grid method using within the two years 1987 and 1988 (Table 5.1-1). the JAVA applet JTiling (Weber, 1999).
Discovery b-Mn by Huang, Hovmo¨ller (1991) and Jiang, Hovmo¨ller, The first octagonal phase was discovered during the Zou (1995). The structure model consists of a stacking of investigation of rapidly solidified samples of V––Ni––Si and four quasiperiodic layers with sequence ...ABAB0... and Cr––Ni––Si alloys by SAED and HRTEM (Wang, Chen, a period of 6.3 A . The layer A shows 8-fold symmetry Kuo, 1987). The contrasts resembled a square/rhomb octa- while the layers B and B0 are 4-fold symmetric only. B gonal tiling on a scale of several nanometers. The octagonal results by rotating layer B by 45 . Each layer corresponds phase was found to coexist with an orientationally twinned to a quasiperiodic tiling built from 45 rhombs and squares approximant. This is a cubic phase with b-Mn structure with edge length a ¼ 8.2 A . These unit tiles can be type (P4 32, a ¼ 6.3 A ) and pseudo-octagonal diffraction r 1 further decomposed to tilesffiffiffi with an edge length symmetry. The cube in the octagonal phase (i.e. the square p a0 ¼ 3.4 A , smaller by a factor 2 1. The local symme- in the octagonal tiling) has exactly the same size as the unit r try is 84/mmc. Based on this model, Ben-Abraham, Ga¨hler cell of the cubic phase, i.e. the edge lengths of the tiling is (1999) developed a covering-cluster description. The octa- equal to the lattice parameter of the cubic phase. gonal QC has the highest density of this prismatic cluster. The authors also determined to I84/mcm the 5D space HRTEM and SAED group of the ideal structure obtained in this way. A more perfect octagonal phase could be obtained in the A continuous change from metastable o-Cr––Ni––Si and system Mn––Si––Al (Wang, Fung, Kuo, 1988). Its symme- o-Mn––Si––Al to the cubic phase with b-Mn structure type try was determined by CBED to 8/m or 8/mmm. A qualita- was observed by moving the SAED aperture successively tive comparison of 17 kinematically calculated and ob- from the octagonal to the cubic area of the samples served SAED reflection intensities (o-Cr––Ni––Si) was (Wang, Kuo, 1990). The orientational relationship between performed by Wang, Kuo (1988). The calculations were the cubic and the octagonal phase resulted to based on the canonical octagonal tiling (4D hypercubic lattice decorated by octagons). [001]b-Mn || [00001]octagonal , A detailed structure model of o-Mn80Si15Al5 was derived from HRTEM images based on the structure of [100]b-Mn || [11000]octagonal . The transformation was explained by gradual introduction of Table 5.1-1. Timetable of the discovery of octagonal quasicrystals. a phason strain field (Mai, Xu, Wang, Kuo, Jin, Cheng, All phases have been obtained by rapid solidification and all are me- tastable. 1989). A theoretical model based on the Schur rotation (i.e. a one-parameter rotation in the nD description) for this transi- Year of Nominal Period along References tion was published by Baake, Joseph, Kramer (1991). A the- discovery Composition 8-fold axis oretical analysis of the transformation of o-Mn––Si––Al to the b-Mn structure and to the Mn3Si structure was performed 1987 o-V15Ni10Si 6.3 A Wang, Chen, Kuo (1987) 1988 o-Cr Ni Si 6.3 A Wang, Chen, Kuo (1987) by Xu, Wang, Lee, Fung (2000). The first transition takes 5 3 2 place if the metastable octagonal phase is heated rapidly, and 1988 o-Mn Si 6.2 A Cao, Ye, Kuo (1988) 4 the other if it is heated slowly. The relationship between 1988 o-Mn82Si15Al3 6.2 A Wang, Fung, Kuo (1988) o-Mn––Si––Al and the b-Mn structure was described in more 1988 o-Mn––Fe––Si Wang, Kuo (1988) detail by Li, Cheng (1996). Twenty years of structure research on quasicrystals 403
5.2 Decagonal phases
Contrary to octagonal and dodecagonal phases, decagonal phases have been discovered more or less continuously over the last fifteen years (Tab. 5.2-1 and -2). There are many theoretical papers on tilings, which may serve as quasilat- tices of decagonal phases, the most famous one is the Penrose tiling (Fig. 5.2-1). More information on pentagonal and decagonal tilings can be found in Ingalls (1992) and Pavlovitch, Kleman (1987), for instance. There are only a few “quantitative” X-ray and neutron structure analyses of decagonal QC (Tab. 5.2-3) and approximants (Tab. 5.2-4) beside a huge number of “qualitative” electron microscopic (HRTEM, HAADF, ALCHEMI, SAED, CBED etc.) studies and quite a few spectroscopic (EXAFS, NMR, electron or ion channeling etc.) structural investigations. In the last few years, an increasing number of surface structural studies, most of them by STM or AFM, has been carried out. The results of all these studies, i.e. in some way idea- lized 3D and/or 5D structure models, indicate that at least some classes of QC posses rather well ordered quasiperio- dic structures with correlation lengths up to several micro- Fig. 5.2-1. Penrose tiling generated by the dual-grid method using the meters. There remain, however, many uncertainties since JAVA applet JTiling (Weber, 1999). the amount of accessible experimental data is rather small compared to the complexity of the problem.
Table 5.2-1. Timetable of the discovery of stable decagonal quasicrystals. Only a few Year of Nominal alloy Period along References decagonal phases have been proved by differ- discovery Composition 10-fold axis ent groups to be thermodynamically stable. 1988 d-Al65Cu20Co15 4(8) A He, Zhang, Wu, Kuo (1988)
1989 d-Al70Ni15Me15 4(8) A Tsai, Inoue, Masumoto (1989a) (Me ¼ Co,Fe)
1989 d-Al65Cu15Rh20 4(8,12) A Tsai, Inoue, Masumoto (1989d)
1991 d-Al70Pd15Mn15 12 A Beeli, Nissen, Robadey (1991) a 1991 d-Al75Me10Pd15 16 A Tsai, Inoue, (Me¼Fe, Ru, Os) Masumoto (1991) b 1992 d-Al72Cu12Cr16 37.8 A Okabe, Furihata, Morishita, Fujimori (1992)
1995 d-Al70Ni20Rh10 4(8) A Tsai, Inoue, Masumoto (1995) c 1997 d-Ga33Fe46Cu3Si18 12.5 A Ge, Kuo (1997) c d-Ga43Co47Cu10 ? Ge, Kuo (1997) c d-Ga35V45Ni6Si14 ? Ge, Kuo (1997) d 1997 d-Al40Mn25Fe15Ge20 12 A Yokoyama, Yamada, Fukaura, Sunada, Inoue, Note (1997)
1997 d-Zn58Mg40Dy2 5.1 A Sato, Abe, Tsai (1997)
1997 d-Al65Cu20Ir15 Athanasiou (1997)
1998 d-Zn58Mg40RE2 5.1 A Sato, Abe, Tsai (1998) (RE¼Dy, Er, Ho, Lu, Tm, Y)
2000 d-Al70Ni20Ru10 16.7 A Sun, Hiraga (2000a)
a: sample seems to be stable at least for an annealing time of 48 h at 1100 K. At least, Al––Fe––Pd turned out to be unstable in later investigations (Balanetskyy et al., 2004). b: sample is stable in a very small temperature range around 1000 C at least for an annealing time of 100 h. However, according to Wu, Ma, Kuo (1996) it disappears after “long” annealing. c: sample is stable in a small temperature range above around 800 C at least for an annealing time of 48 h. It forms from the low-temperature crystalline approximant. d: sample with decaprismatic morphology (d > 0.1 mm) is stable above around 798 K, for instance at 1050 K at least for an annealing time of 100 h and seems to be retained up to the melting temperature Tm ¼ 1070 K. 404 Walter Steurer
Table 5.2-2. Timetable of the discovery of Year of Nominal Alloy Period along References metastable decagonal quasicrystals obtained discovery Composition 10-fold axis by rapid solidifaction.
1985 d-Al4Mn 12.4 A Bendersky (1985)
1985 d-Al5Ru 16 A Bancel, Heiney (1986) 1986 d-Al––Pd 16.5 A Bendersky (1986) 1986 d-Al––Pta 16 A Bancel, Heiney (1986)
1986 d-Al4.5Fe 16.4 A Fung, Yang, Zhou, Zhao, Zhan, Shen (1986)
1987 d-Al5Os 16 A Kuo (1987) 1987 d-Al––Cr(––Si) 12 A Kuo (1987) 1987 d-V––Ni––Si Kuo, Zhou, Li (1987)
1987 d-Al77Co22.5 16 A Dong, Li, Kuo (1987)
1987 d-Al79Mn19.4Fe2.6 Ma, Stern (1987)
1988 d-Al5Ir 16 A Ma, Wang, Kuo (1988)
1988 d-Al5Rh 16 A Ma, Wang, Kuo (1988)
1988 d-Al4Ni 4 A Li, Kuo (1988)
1988 d-Al6Ni(Si) 16 A Li, Kuo (1988)
1988 d-Al65Cu20Mn15 12 A He, Wu, Kuo (1988)
1988 d-Al65Cu20Fe15 12 A He, Wu, Kuo (1988)
1988 d-Al65Cu20Co15 8, 12, 16 A He, Wu, Kuo (1988)
1988 d-Al75Cu10Ni15 4 A He, Wu, Kuo (1988)
1988 d-Al40Cu10Ge25Mn25 Kimura, Inoue, Bizen, Masumoto, Chen (1988) 1989 d-Al––Cr––Si Zhang, Wang, Kuo (1989)
1990 d-Al5Os 4.2 A Wang, Gao, Kuo (1990)
1990 d-Al5Ru 4.2 A Wang, Gao, Kuo (1990)
1990 d-Fe52Nb48 4.7 A He, Yang, Ye (1990)
1990 d-Al78Co20Pd10 4 A Tsai, Yokoyama, Inoue, Masumoto (1990)
1993 d-Al72MgxPd28–x, 16.57 A Koshikawa, Edagawa, 5 x 10 Honda, Takeuchi (1993)
1995 d-Al70Ni20Ir10 4 A Tsai, Inoue, Masumoto (1995)
1995 d-Al72Co28 8 A Tsai, Inoue, Masumoto (1995)
1995 d-Al70Ni13Ir17 16 A Tsai, Inoue, Masumoto (1995)
1997 d-Ga52-63Mn48–37 12.5 A Wu, Kuo (1997)
2000 d-Al70Ni20Ru10 4 A Sun, Hiraga (2000a,b)
2002 d-Co(Al,Ga)3, Ellner, Meyer (2002) xGa 0.1 a: Could not be confirmed by Ma, Kuo, Wang (1990)
Table 5.2-3. Quantitative X-ray structure analyses of d-Al––Co––Cu, d-Al––Co––Ni. Whether R factors are based on structure amplitudes or on intensities is unclear in most cases (intensity based R factors are approximately by a factor two larger than the structure amplitude based ones). Dx ...calculated density, PD ...point density,p SGffiffiffi ...space group, PS ...Pearson symbol, NR ...number of reflections, NV ...number of vari- S Y S Y ables. The quasilattice parameter listed is ai ¼ 5 tai (ai is defined in Steurer, Haibach, 1999a; ai is defined in Yamamoto, Ishihara, 1988).
Nominal SG Lattice Dx NR R Reference composition PS parameters PD NV wR d-Al65Co15Cu20 P105=mmc a1...4 ¼ 3:765ð3Þ A 4.4 259 0.167 Steurer, Kuo (1990a, b) a5 ¼ 4:1481ð3Þ A 11 0.098 d-Al70Co20Ni10 P10=mmm a1...4 ¼ 3:80 A 41 0.110 Yamamoto, Kato, a5 ¼ 4:08 A 0.0680 2 Shibuya, Takeuchi (1990) d-Al70Co15Ni15 P105=mmc a1...4 ¼ 3:794ð1Þ A 4.5 253 0.091 Steurer, Haibach, Zhang, a5 ¼ 4:0807ð3Þ A 0.0741 21 0.078 Kek, Lu¨ck (1993) d-Al70Co15Ni15 P105=mmc a1...4 ¼ 3:794ð1Þ A –– 253 0.092 Elcoro, Perez-Mato (1995) a5 ¼ 4:0807ð3Þ A 18 0.080 d-Al72Co8Ni20 P105=mmc a1...4 ¼ 3:758 A 3.94 449 0.063 Takakura, Yamamoto, a5 ¼ 4:090 A 0.0661 103 0.045 Tsai (2001) d-Al70.6Co6.7Ni22.7 P10 a1...4 ¼ 3:757ð2Þ A 3.89 2767 0.170 Cervellino, Haibach, a5 ¼ 4:0855ð1Þ A 0.0644 750 0.060 Steurer (2002) Twenty years of structure research on quasicrystals 405
Table 5.2-4. Quantitative X-ray structure analyses of d-phase approximants. The pseudodecagonal axis is underlined. Dx...calculated density, SG ...space group, PS ...Pearson symbol, NR ...number of reflections, NV ...number of variables.
Nominal SG Lattice Dx NR R Reference composition PS parameters NV wR
Co2Al5 P63=mmc a ¼ 7:6717ð4Þ A 0.072 Burkhardt, Ellner, Al71.5Co28.5 hP28 c ¼ 7:6052ð5Þ A Grin, Baumgartner (1998) c=a ¼ 0:9913
(Al,Cu)13Co4 Cm a ¼ 15:2215ð4Þ A 0.020 Freiburg, Grushko (1996) Al71.5Co23.6Cu4.9 mC102 b ¼ 8:0849ð2Þ A c ¼ 12:3908ð3Þ A b ¼ 108:081ð1Þ
Al11Co4 P2 a ¼ 24:661ð5Þ A 4.26 718 0.088 Li, Shi, Ma, Ma, Kuo (1995) Al73.24Co26.76 mP52 b ¼ 4:056ð9Þ A 240 c ¼ 7:569ð2Þ A b ¼ 107:88ð3Þ a Co4Al13 Cm a ¼ 15:183ð2Þ A 3.81 0.122 Hudd, Taylor (1962) b Al73.7Co26.3 mC102-5.2 b ¼ 8:122ð1Þ A 0.142 c ¼ 12:340ð2Þ A b ¼ 107:9ð1Þ
Co2NiAl9 Immm a ¼ 12:0646ð7Þ A 3.98 437 0.112 Grin, Peters, Burkhardt, Al74.8Co17Ni8.2 oI 96 b ¼ 7:5553ð7Þ A 0.117 Gotzmann, Ellner (1998) c ¼ 15:353ð1Þ A
Al13 x(Co1 yNiy)4 C2=m a ¼ 17:071ð2Þ A 3.96 963 0.053 Zhang, Gramlich, Steurer (1995) Al75.2(Co, Ni)24.8 mC34-1.8 b ¼ 4:0993ð6Þ A 58 0.034 c ¼ 7:4910ð9Þ A b ¼ 116:17ð1Þ o-Co4Al13 Pmn21 a ¼ 8:158ð1Þ A 839 0.062 Grin, Burkhardt, Al76Co24 oP102 b ¼ 12:342ð1Þ A Ellner, Peters (1994b) c ¼ 14:452ð2Þ A b ¼ 107:9ð1Þ a: For [110] projections b: For [010] projections
Most X-ray or neutron diffraction structure analyses of full data set would consist of ntot ¼ 11 464 952 Bragg decagonal QC have been based on a ridiculously small reflections for standard resolution (MoKa, qmax ¼ 30 ). number of Bragg reflections. Even for low-order rational The number of unique reflections would amount to approximants, much larger data sets have been used (see ntot=48 ¼ 238 853. In case of a decagonal QC with 4.17 A Tab. 5.2-3 and -4). How many reflections should be used periodicity and 10/mmm Laue symmetry we would expect and how many can be observed anyway? The total num- ntot=(30 40) ¼ 9554 unique reflections. This is much ber of Bragg reflections ntot we can expect for a diffrac- more than what has been used in the structure analyses of tion experiment on a periodic crystal can be calculated decagonal phases published so far. 3 from the formula ntot ¼ (2pV)/(3dmin ), with unit cell It is surprising, however, that we do not observe many volume V and resolution dmin ¼ l/(2 sin (qmax)). This num- more Bragg reflections for QC than for their low-order ber calculated for a high-order approximant gives also an approximants. For instance, along reciprocal lattice lines estimate of the number of reflections, which should be of QC corresponding to net planes with Fibonacci sequence- observable in case of the related quasicrystal. One should like distances rarely more than 30 reflections are observed 6 keep in mind, however, that such a diffraction data set (dynamic range of Imax=Imin 10 ,0 sin (q=l) 0.8). allows determining the structure of the QC only on a scale According to Steurer (1995) the expected number of comparable to the approximant unit cell size. Most infor- reflections for this dynamic range would be 3940; and still 4 mation about the type of quasiperiodic LRO is in the very 389 for a dynamic range of Imax=Imin 10 . This indicates weak reflections with large perpendicular-space compo- a significant deviation from perfect quasiperiodic order nents of the diffraction vectors. This would correspond to and makes impossible an accurate structure solution solely rational approximants with lattice parameters of the order by diffraction methods. Just for comparison, even such of micrometers and billions of Bragg reflections. imperfect crystals as those of biological macromolecules The smallest observed distance between Bragg reflec- exhibit often more than one million observable reflections. tions of decagonal Al––Co––Ni amounts to 0.008 A 1 This finding may be explained by the fact that QC are (Beeli, Steurer, 2000; Steurer, Cervellino, Lemster, Ortelli, usually annealed at high-temperatures and then rapidly Estermann, 2001). In case of a cubic rational approximant, quenched. By this procedure, a considerable amount of this would correspond to a lattice parameter of 125 A .A atomic disorder (correlated and random phason fluctua- 406 Walter Steurer tions, thermal vacancies etc.) is frozen in to some extent. 5.2.1 4 A˚ and 8 A˚ periodicity The diffraction experiment is performed on such a meta- 5.2.1.1 Al––Co stable sample that is in a partially relaxed but not very well-defined state. This is illustrated in Fig. 5.2.1.4-5 on Discovery the example of the temperature dependence of the diffuse Based on the similarity of the X-ray powder diffractogram interlayers in the diffraction pattern of decagonal of rapidly solidified Al86Co14 to that of d-Al––Mn, Dun- Al70Co12Ni18. In situ HT X-ray diffraction experiments lap, Dini (1986) concluded erroneously icosahedral quasi- show that the diffuse intensities become sharper with rais- crystallinity of that sample. This was confirmed by a ing the temperature from RT to 800 C. SAED study on Al74Co26 by which a metastable decago- The period along the decagonal axis results from the nal QC with 8 A periodicity was identified (Suryanar- stacking of structural units. In all decagonal phases, the ayana, Menon, 1987). Menon, Suryarayana (1989) discuss smallest structural unit is a pentagonal antiprism (PA) (i.e. the polytypism of this d-phase, which can exhibit 4, an icosahedron without caps, or the period of a stack of 8, 12 and 16 A periodicity. After heating the interpenetrating icosahedra) (see, for instance Steurer, rapidly solidified sample with 16 A periodicity to above Haibach, Zhang, Kek, Lu¨ck, 1993). This PA is coordi- 450K, a transformation to 12 A periodicity was nated by a puckered decagon yielding a decoration of observed in situ.Anin situ study of the formation of each one of the ten side planes by a tetrahedron. This d-Al––Co in Al––Co multilayers with starting composition structure motif is also common in approximants (Bostro¨m, Al13Co4 was performed by Bergman, Joulaud, Capitan, Hovmo¨ller, 2001). Clugnet, Gas (2001). The formation of Al9Co2 takes place