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Formation of quasicrystals
Physica 140A (1986) 298-305 North-Holland, Amsterdam
F O R M A T I O N O F QUASICRYSTALS Leonid A. BENDERSKY* and Robert J. SCHAEFER Metallurgy Division, National Bureau of Standards, Gaithersburg MD 20899, USA
It has recently been discovered that "quasicrystalline" intermetalhc phases can form which have point group symmetries not allowed in periodic crystals; icosahedral, with six five-foldaxes, and decagonal, with one 10-fold axis. These phases have long-range orientational order, but they must have translational quasiperiodicity. They usually form at alloy compositions close to those of intermetallic compounds which have crystal structures containing icosahedral groups of atoms. The quasicrystal phases nucleate preferentially in supercooled melts, thus replacing the equilibrium phases.
1. Introduction One can define two classes of solids according to their structure. One is glassy (amorphous) structures, where the distinct order of local atomic arrangement does not have a long-range correlation. The glassy structures are essentially frozen in undercooled liquids1). The other class is crystal structures where both short-range order and long-range translational periodicity are present2). The reciprocal space of a crystal is its reciprocal Bravais lattice, where nodes consist of Bragg reflection modulated according to the motif structure. Different deviations from ideal crystalline order (interfaces, chemical disorder, dislocations, etc.) appear as peak broadening and diffuse scattering. The ordered structures which have Fourier transforms consisting of sharp peaks are not necessarily periodic; for example, incommensurate systems containing interacting subsystems with different lattice periods. These systems can be described as periodic only in higher than 3-dimensional space 3) and usually have crystallographic point groups. Another class of structures with non-periodic Fourier transform is quasicrystals, in which incommensurability is a consequence of geometrical properties of their non-crystallographic 3-D point group4). Quasicrystals have an underlying quasiperiodic lattice which can be generated by cut and projection from higher-dimensional periodic lattice (see references in ref. 4). The existence of particular types of quasicrystal-icosahedral 3-D Penrose tiling *Also with the Center for Materials Research, The Johns Hopkins University, Baltimore, MD 21218, USA. 0378-4371/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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was first speculated by Mackay 5) and later by Levine and Steinhardt4). The discovery of the A1-Mn icosahedral phase by Shechtman et al. 6) seems to be the first experimental evidence that such quasicrystals exist. Additional support to the idea of quasicrystal came when another type of quasicrystal, decagonal phase, was found in A1-Mn and some other systemsT).
2. Crystallography of icosahedral and decagonal phases All selected area electron diffraction (SAD) patterns of the icosahedral phase 6) are incommensurate with the golden mean being the parameter of incommensuration (fig. la-c). There is no 3-D periodic Bravais lattice, but the positions and intensities of the reflections correspond perfectly to the Fourier transform of the icosahedral quasilattice (e.g., a 3-D Penrose tiling) which is a special projection of a 6-D primitive cubic lattice (references in ref. 4). The SAD patterns of the decagonal phase (fig. l d - f ) have 1-D periodicity, except for one (fig. ld) which has 10-fold rotation symmetry and can be described as the Fourier
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Fig. 1. Selected area electron diffraction patterns from icosahedral (a)-(c) and decagonal (d)-(f) phases. (a) Five-fold; (b) two-fold; (c) three-fold; (d) ten-fold; (e) pseudo-five-fold; (f) pseudo-threefold.
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transform of a 2-D Penrose tiling. The reciprocal lattice of the decagonal phase is thus a periodic stacking of reciprocal quasilattices of the 2-D Penrose tiling along the 10-fold axis. The results of powder X-ray and neutron diffraction are in perfect agreement with electron microscopy for the icosahedral phase6,8-1°), whereas for the decagonal phase only the preliminary results are available11). The point groups of both quasicrystals were determined by using convergent beam electron diffractionT,12): m5-3 for the icosahedral phase and 10/m (or 10/mmm) for the decagonal phase. Neither of these two point groups is a subgroup of the other 13); therefore a second order transformation (of ordering or displacements) between these two phases is not possible. There is no space group for the icosahedral phase in 3-D, since a translational group does not exist. However, there are 6-D space groups of 5 possible cubic Bravais lattices, and therefore different projections of the 3-D icosahedral lattice. That of the icosahedral phase was shown to be the simple cubic~3), according to the scaling properties of the diffraction pattern. Different decagonal phases can have different parameters of periodicity, e.g., - 1.24 nm for A1-Mn phase and - 1.65 nm for A1-Pd and A1-Fe phases (ref. 7 and its references). SAD patterns of both types of the decagonal phase show a systematic absence of odd reflections along the 10-fold c* axis 7) (also supported by kinematic X-ray diffractionll). Therefore, the space group P105/m with a ten-fold screw axis was proposed for the decagonal phasesT). The described quasiperiodic features of the reciprocal space of both phases result from a truly quasiperiodic arrangement of atoms and not from either multiple twinning or approximate icosahedral symmetry of a crystal with a large unit cell. The conventional crystallographic explanations are unable to account for a large bulk of experimental evidence, as was demonstrated in ref. 14. The most direct evidence comes from field ion microscopy where definitely no twin boundary and no periodic structure are observedlS). The same conclusions are obtained from high-resolution electron microscopy with - 2 ,~ resolution (ref. 16 and its references), where no domains of periodic twinned structure were observed. However, the images show a high regularity of its projected 3-D quasiperiodic structure. There is an excellent agreement between the experimental and computer-simulated structural images, where for the model we used the icosahedral quasiperiodic lattice with the nodes occupied by single and complex scatterers 16). If quasicrystals have ideal, continuously occupied quasilattices, the diffraction will have B-function Bragg peaks. Real icosahedral phase materials show peak broadening corresponding to a mean positional correlation range on the order of ten's of nmS'9). This limited positional correlation length can be due to the presence of growth defects or to the intrinsic properties of the quasicrystals. A defect substructure with 40-100 nm domains was observed in icosahedral
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A1-Mn-Si9). However, with this type of defect each diffracted peak should have the same width, which is not soS). An attempt has been made to correlate the width of peaks with G± (the phason momentum, or the counterpart of the real scattering vector G of 6-D G)17). The observed trend was found to be roughly consistent with the current theories either of phason strain (refs. 17, 20 and its references) or random icosahedral packing18'19). The anisotropic phason strain can also produce a shift of diffracted spots which was experimentally observed20,21).
3. Physical metallurgy of phase formation 3.1. Relation between crystalline and quasicrystalline phases It has recently been reported 22) that in the A1-Li-Cu system one of the equilibrium phases is icosahedral, and this phase can be formed by conventional slow solidification or solid state precipitation processes. In all other systems where icosahedral phases have been found, they have been metastable and were formed only by special processing methods, usually within particular groups of alloys thought to be particularly susceptible to their formation. There are many intermetallic crystals in which atoms are arranged in local icosahedral groupings. These include, among others, the Frank-Kasper phases which contain combinations of polytetrahedral configurations of atoms with coordination numbers 12, 14, 15 and 16. Ramachandrarao and Sastry 23) demonstrated that when one such alloy [MgaE(A1,Zn)49] was rapidly solidified, a quasicrystal with long-range icosahedral symmetry was formed instead of the equilibrium phase, which is body-centered cubic with an extended icosahedral grouping of atoms at each lattice point. A similar approach has lead to the formation of icosahedral quasicrystals in Mg4CuA1624), Cd-Cu25), and (Ti, V)2Ni26). The other major group of alloys which has been formed into icosahedral and decagonal quasicrystals is the aluminum-transition metal series6,2°). In the binary ahiminum-transition metal alloy systems, there are several cases where different icosahedral groups of atoms are present in equilibrium crystal structures. However, evidence from EXAFS27), Mossbauer spectroscopy 28) and N M R 29) all indicate that the icosahedral phase of A1-Mn is not formed from groups of atoms with a central Mn atom surrounded by a single icosahedral group of 12 aluminum atoms as found in the Al12Mn phase. Instead, such evidence points to larger icosahedral clusters of atoms such as are found in the ternary a phase of A1-Mn-Si.
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The addition of Si to A1-Mn alloys is known to stabilize the icosahedral phase3°m), and possible similarities between the icosahedral phase and the a phase of A1-Mn-Si have been pointed 0ut32,33). a(A1MnSi) phase has an almost body-centered cubic lattice with each lattice point being occupied by a large double-layered icosahedral cluster of atoms, usually described as a Mackay icosahedron. Thus formation of the icosahedral phase in A1-Mn-Si may be attributed to the same principle which was used to explain its formation in Mg 32(A1, Zn) 49There are also several Al-transition metal phases with crystal structures which are not known in detail, especially in the A1-Mn and A1-Cr systems. Recent diffraction evidence 34) indicates that some of the A1-Mn phases of composition close to A14Mn contain icosahedral clusters. According to this work, it seems possible to describe the structure of quasicrystalline and crystalline AI-Mn phases roughly as an agglomeration of the same type of icosahedral clusters, joined in different ways such that either a periodic or a quasiperiodic lattice is formed. The cluster was shown to be oriented uniquely for the icosahedral phase, in two orientations for decagonal and in three for a hexagonal phase.
3.2. Formation of quasicrystalphases In contrast to the formation of amorphous solids, the formation of quasicrystals is a first-order nucleation and growth process somewhat similar to that of conventional crystals. Thus the initial TEM images of icosahedral AI-Mn 7a9) formed by rapid solidification of a molten alloy showed that the icosahedral phase had grown radially outward from a central point of origin with a branched or dendritic structure typical of diffusion-controlled crystal growth into an undercooled melt. Rapid solidification from the liquid state can be carried out by several different processes, but a common feature is that small volumes of liquid are used so that heat can be extracted rapidly and the probability of the presence of heterogeneous nucleants for the equilibrium phases is small. Under these conditions, the alloy melt can become highly supercooled and it is found that the nucleation rate of the icosahedral phase is large. Increasing cooling rates lead to greater rates of nucleation and correspondingly smaller grain sizes, until in extreme cases 35) the size of the individual grains is so small that the structure is indistinguishable from glassy structures with certain local order. This material must be contrasted to true metallic glasses, however, in which an amorphous structure is formed by completely avoiding nucleation in the liquid; in the present case the structure is formed by an exceedingly high rate of nucleation. It has been speculated 35) that preexisting icosahedral clusters in the undercooled melt can promote the observed homogeneous nucleation, being both centers for nuclei and reducing icosahedral phase/melt interface energy.
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The growth of quasicrystals following their nucleation is in many ways similar to that of conventional crystals in a supercooled melt. In most cases the quasicrystals grow with a composition different from that of the melt, thus requiring solute diffusion to occur in the liquid. This is a slow process and leads to the formation of a dendritic (branched) structure, in which the dendrites grow preferentially in the directions of the three-fold symmetry axes3°). The selection of the preferred dendrite growth direction must be attributed to the anisotropic characteristics of the solid-liquid interface, but there is no direct evidence to indicate whether this is an anisotropy of the surface energy or of the atomic attachment kinetics. Surface energy effects are expected to dominate with unfaceted solid-liquid interfaces while faceted interfaces indicate the importance of kinetic effects. Facets are only occasionally seen9) on the planes of five-fold symmetry. If these are the most densely packed planes of the solid-liquid interface, they would have the lowest surface energy and would also be the most difficult surfaces on which to add new layers of crystal. Either effect would result in preferred growth along the three-fold directions. Quasicrystals have also been formed in the A1-Mn system by several processes other than rapid solidification of liquids, the success of which indicates that although the quasicrystal phases are not equilibrium phases of this system, they are not far from being stable and are frequently favored by kinetic factors. These processes include ion beam mixing36), ion irradiation37), ion implantation3S), solid state interdiffusion 39'4°) and rapid pressurization41). There have been several cases in which icosahedral crystals are reported to form by devitrification of metallic glasses. This occurs within a very narrow composition range in the P d - U - S i system42). It is also reported to occur in A1-Mn, using amorphous alloys formed by evaporation or ion beam mixing with cold substrates36,3v). In such cases, extremely fine-grained structures are obtained, again indicating very high rates of nucleation.
4. Some conclusions (and more questions)
We presented experimental results on the structure and formation of quasicrystals, which lead to some conclusions but raise many questions. 1. The following empirical rules can be suggested for formation of quasicrystals: (a) Suitable alloy systems contain intermetallic compounds where icosahedral clustering is extensively present. (b) The alloy melt has to be undercooled to nucleate the usually metastable quasicrystals without forming more stable phases. This is most readily accom-
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plished if the alloy composition is off stoichiometry of the intermetallic compounds. (c) Phase diagrams with peritectic reactions are more favorable than eutectics, since condition (b) can be fulfilled without forming glass.
Questions.
In the crystals of the A1-Mn system there are 3 types of icosahedral clusters; a shell of 12 aluminum with a central Mn in Alt2Mn G phase, possible Mackay icosahedron in A14Mn and truncated icosahedra in A16Mn. Why is only one realized as quasicrystal? Do icosahedral phases of different metallurgical systems have the same local order? 2. The projection of a quasiperiodic lattice from higher dimension seems to be a well established description for the crystallography and structure of quasicrystals.
Questions. Should quasicrystals be considered as decorated tiles? Or does the quasilattice have to be rearranged (i.e., disordered) to allow symmetric grouping of atoms to dominate in a structure which is still quasiperiodic but not deterministic? What determines formation of quasicrystals: energetics of duster formation and entropy of their packing, or long-range quasiperiodic interaction? Is the observed position disorder an intrinsic property of the structure, or is it due to metastable growth defects? The answers are expected from the presumably equilibrium A 1 - L i - C u icosahedral phase. 3. Nucleation of the icosahedral phase is homogeneous and its subsequent growth is relatively slow and proceeds in the 3-fold directions.
Questions.
Does the nucleation behavior of the icosahedral phase mean preexistence of the icosahedral dusters in liquid? How do atoms become attached to the growing quasicrystal surface? Is it an atom-by-atom transfer or clusterby-cluster transfer? What is the critical nucleus of the icosahedral phase and how is it related to the duster size? References
1) 2) 3) 4) 5) 6) 7) 8)
S. Sachdev and D.R. Nelson, Phys. Rev. B 32 (1985) 1480 and references. B.K. Vainstein, Modern Crystallography(Springer, Berlin, 1981), vol. 1. A. Janner and T. Janssen, Acta Cryst. A (1980) 408. D. Levine and P.J. Steinhardt, Phys. Rev. Lett. 53 (1984) 2477; Phys. Rev. B 34 (1986) 596. A.L. Mackay, Acta Cryst. 15 (1962) 916; Physica lI4A (1982) 609. D. Shechtman, I. Blech, D. Gratias and J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. L. Bendersky,Phys. Rev. Lett. 55 (1985) 1461; J. de Phys. Colloq. (:3 (1986) 457. P.A. Bancel, P.A. Heiney, P.W. Stephens, A.I. Goldman and P.M. Horn, Phys. Rev. Lett. 54 (1985) 2422.
FORMATION OF QUASICRYSTALS 9) 10) 11) 12) 13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31) 32) 33) 34) 35) 36) 37) 38) 39) 40) 41) 42)
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J.L. Robertson, M.E. Misenheimer, S.C. Moss and L. Bendersky, submitted to Acta Met. B. Mozer, J.W. Calm, D. Gratias and D. Shechtman, J. de Phys. Colloq. C3 (1986) 351. Chr. Janot, J. Dannetier and A. Pianelli, submitted to Phys. Lett. A. L.A. Bendersky and M.J. Kaufman, Phil. Mag. B 53 (1986) L75. LW. Cahn, D. Shechtman and D. Gratias, J. Mater. Res. 1 (1986) 13. J.W. Cahn, D. Gratias and D. Shechtman, Nature 319 (1986) 1478; P.A. Bancel, P.A. Heiney, P.W. Stephens and A.I. Goldman, ibid. A.J. Melmed and R. Klein, Phys. Rev. Lett. 56 (1986) 1478. M. Cornier, R. Portier and D. Gratias, Inst. Phys. Conf. Ser. 78 (1985), chap. 3, p. 91; M. Cornier, K. Zhang, R. Portier, and D. Gratias, J. de Phys. Colloq. C3 (1986) 447. W. Malzfeldt, P.M. Horn, D.P. Di Vincenzo, B. Stephenson, R. Gambino and S. Herd, J. de Phys. Colloq. C3 (1986) 379. P.W. Stephens and A.I. Goldman, Phys. Rev. Lett. 56 (1986) 1168. D. Shechtman and I+ Blech, Met. Trans. 16A (1985) 1005. P.A. Bancel and P.A. Heiney, J. de Phys. Colloq. C3 (1986)341; Phys. Rev. B 33 (1986) 7917. M. Tanaka, M. Terauchi, K. Hiraga, and M. Hirabayashi, Ultramicroscopy 17 (1985) 127. P. Sainfort and B. Dubost, J. de Phys. Colloq. C3 (1986) 321. P. Ramachandrarao and G.V.S. Sastry, Pramana 24 (1985) L225. G.V.S. Sastry, P. Ramachandrarao and T.R. Anantharaman, Script Met. 20 (1986) 191. L.A. Bendersky, unpublished work. Z. Zhang, H.Q. Ye and K.H. Kuo, phil. Mag. Lett. 52 (1985) L49. E.A. Stern, Y.A. Ma and C.E. Bouldin, Phys. Rev. Lett. 55 (1985) 2172. L.J. Swartzendruber, D. Shechtman, L. Bendersky and J.W. Calm, Phys. Rev. B 32 (1985) 1383. L.H. Bennett, J.W. Cahn, R.J. Schaefer, M. Rubinstein and G.H. Stauss, to be published. R.J. Schaefer, L.A. Bendersky, D. Shechtman, W.J. Boettinger and F.S. Biancaniello, to appear in Met. Trans. A (1986). C.H. Chen and H.S. Chen, Phys. Rev. B 33 (1986) 2814. V. Elser and C.L. Henley, Phys. Rev. Lett. 55 (1985) 1883. P. Guyot and A. Audier, Phil. Mag. 52B (1985) L15. L.A. Bendersky, submitted to J. Microscopy. L.A. Bendersky and S.D. Ridder, J. Mater. Res. 1 (1986) 405. D.M. FoUstaedt and J.A. Knapp, J. Appl. Phys. 59 (1986) 1756. D.A. Lilienfeld, M. Nastasi, H.H. Johnson, D.G. Ast and J.W. Mayer, Phys. Rev. Lett. 55 (1985) 1587. J.D. Burial and M.J. Aziz, Phys. Rev. B 33 (1986) 2876. D.M. Follstaedt and J.A. Knapp, Phys. Rev. Lett. 56 (1986) 1827. K. Urban, J. Mayer, M. Rapp, M. Wilkens, A. Csanady and J. Fidler, J. Phys. Colloq. (1986). J.A. Sekhar and T. Rajasekharan, Nature 320 (1986) 153. S.J. Poon, A.J. Drehman and K.R. Lawless, Phys. Rev. Lett. 55 (1985) 2324.
F O R M A T I O N O F QUASICRYSTALS Leonid A. BENDERSKY* and Robert J. SCHAEFER Metallurgy Division, National Bureau of Standards, Gaithersburg MD 20899, USA
It has recently been discovered that "quasicrystalline" intermetalhc phases can form which have point group symmetries not allowed in periodic crystals; icosahedral, with six five-foldaxes, and decagonal, with one 10-fold axis. These phases have long-range orientational order, but they must have translational quasiperiodicity. They usually form at alloy compositions close to those of intermetallic compounds which have crystal structures containing icosahedral groups of atoms. The quasicrystal phases nucleate preferentially in supercooled melts, thus replacing the equilibrium phases.
1. Introduction One can define two classes of solids according to their structure. One is glassy (amorphous) structures, where the distinct order of local atomic arrangement does not have a long-range correlation. The glassy structures are essentially frozen in undercooled liquids1). The other class is crystal structures where both short-range order and long-range translational periodicity are present2). The reciprocal space of a crystal is its reciprocal Bravais lattice, where nodes consist of Bragg reflection modulated according to the motif structure. Different deviations from ideal crystalline order (interfaces, chemical disorder, dislocations, etc.) appear as peak broadening and diffuse scattering. The ordered structures which have Fourier transforms consisting of sharp peaks are not necessarily periodic; for example, incommensurate systems containing interacting subsystems with different lattice periods. These systems can be described as periodic only in higher than 3-dimensional space 3) and usually have crystallographic point groups. Another class of structures with non-periodic Fourier transform is quasicrystals, in which incommensurability is a consequence of geometrical properties of their non-crystallographic 3-D point group4). Quasicrystals have an underlying quasiperiodic lattice which can be generated by cut and projection from higher-dimensional periodic lattice (see references in ref. 4). The existence of particular types of quasicrystal-icosahedral 3-D Penrose tiling *Also with the Center for Materials Research, The Johns Hopkins University, Baltimore, MD 21218, USA. 0378-4371/86/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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was first speculated by Mackay 5) and later by Levine and Steinhardt4). The discovery of the A1-Mn icosahedral phase by Shechtman et al. 6) seems to be the first experimental evidence that such quasicrystals exist. Additional support to the idea of quasicrystal came when another type of quasicrystal, decagonal phase, was found in A1-Mn and some other systemsT).
2. Crystallography of icosahedral and decagonal phases All selected area electron diffraction (SAD) patterns of the icosahedral phase 6) are incommensurate with the golden mean being the parameter of incommensuration (fig. la-c). There is no 3-D periodic Bravais lattice, but the positions and intensities of the reflections correspond perfectly to the Fourier transform of the icosahedral quasilattice (e.g., a 3-D Penrose tiling) which is a special projection of a 6-D primitive cubic lattice (references in ref. 4). The SAD patterns of the decagonal phase (fig. l d - f ) have 1-D periodicity, except for one (fig. ld) which has 10-fold rotation symmetry and can be described as the Fourier
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Fig. 1. Selected area electron diffraction patterns from icosahedral (a)-(c) and decagonal (d)-(f) phases. (a) Five-fold; (b) two-fold; (c) three-fold; (d) ten-fold; (e) pseudo-five-fold; (f) pseudo-threefold.
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transform of a 2-D Penrose tiling. The reciprocal lattice of the decagonal phase is thus a periodic stacking of reciprocal quasilattices of the 2-D Penrose tiling along the 10-fold axis. The results of powder X-ray and neutron diffraction are in perfect agreement with electron microscopy for the icosahedral phase6,8-1°), whereas for the decagonal phase only the preliminary results are available11). The point groups of both quasicrystals were determined by using convergent beam electron diffractionT,12): m5-3 for the icosahedral phase and 10/m (or 10/mmm) for the decagonal phase. Neither of these two point groups is a subgroup of the other 13); therefore a second order transformation (of ordering or displacements) between these two phases is not possible. There is no space group for the icosahedral phase in 3-D, since a translational group does not exist. However, there are 6-D space groups of 5 possible cubic Bravais lattices, and therefore different projections of the 3-D icosahedral lattice. That of the icosahedral phase was shown to be the simple cubic~3), according to the scaling properties of the diffraction pattern. Different decagonal phases can have different parameters of periodicity, e.g., - 1.24 nm for A1-Mn phase and - 1.65 nm for A1-Pd and A1-Fe phases (ref. 7 and its references). SAD patterns of both types of the decagonal phase show a systematic absence of odd reflections along the 10-fold c* axis 7) (also supported by kinematic X-ray diffractionll). Therefore, the space group P105/m with a ten-fold screw axis was proposed for the decagonal phasesT). The described quasiperiodic features of the reciprocal space of both phases result from a truly quasiperiodic arrangement of atoms and not from either multiple twinning or approximate icosahedral symmetry of a crystal with a large unit cell. The conventional crystallographic explanations are unable to account for a large bulk of experimental evidence, as was demonstrated in ref. 14. The most direct evidence comes from field ion microscopy where definitely no twin boundary and no periodic structure are observedlS). The same conclusions are obtained from high-resolution electron microscopy with - 2 ,~ resolution (ref. 16 and its references), where no domains of periodic twinned structure were observed. However, the images show a high regularity of its projected 3-D quasiperiodic structure. There is an excellent agreement between the experimental and computer-simulated structural images, where for the model we used the icosahedral quasiperiodic lattice with the nodes occupied by single and complex scatterers 16). If quasicrystals have ideal, continuously occupied quasilattices, the diffraction will have B-function Bragg peaks. Real icosahedral phase materials show peak broadening corresponding to a mean positional correlation range on the order of ten's of nmS'9). This limited positional correlation length can be due to the presence of growth defects or to the intrinsic properties of the quasicrystals. A defect substructure with 40-100 nm domains was observed in icosahedral
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A1-Mn-Si9). However, with this type of defect each diffracted peak should have the same width, which is not soS). An attempt has been made to correlate the width of peaks with G± (the phason momentum, or the counterpart of the real scattering vector G of 6-D G)17). The observed trend was found to be roughly consistent with the current theories either of phason strain (refs. 17, 20 and its references) or random icosahedral packing18'19). The anisotropic phason strain can also produce a shift of diffracted spots which was experimentally observed20,21).
3. Physical metallurgy of phase formation 3.1. Relation between crystalline and quasicrystalline phases It has recently been reported 22) that in the A1-Li-Cu system one of the equilibrium phases is icosahedral, and this phase can be formed by conventional slow solidification or solid state precipitation processes. In all other systems where icosahedral phases have been found, they have been metastable and were formed only by special processing methods, usually within particular groups of alloys thought to be particularly susceptible to their formation. There are many intermetallic crystals in which atoms are arranged in local icosahedral groupings. These include, among others, the Frank-Kasper phases which contain combinations of polytetrahedral configurations of atoms with coordination numbers 12, 14, 15 and 16. Ramachandrarao and Sastry 23) demonstrated that when one such alloy [MgaE(A1,Zn)49] was rapidly solidified, a quasicrystal with long-range icosahedral symmetry was formed instead of the equilibrium phase, which is body-centered cubic with an extended icosahedral grouping of atoms at each lattice point. A similar approach has lead to the formation of icosahedral quasicrystals in Mg4CuA1624), Cd-Cu25), and (Ti, V)2Ni26). The other major group of alloys which has been formed into icosahedral and decagonal quasicrystals is the aluminum-transition metal series6,2°). In the binary ahiminum-transition metal alloy systems, there are several cases where different icosahedral groups of atoms are present in equilibrium crystal structures. However, evidence from EXAFS27), Mossbauer spectroscopy 28) and N M R 29) all indicate that the icosahedral phase of A1-Mn is not formed from groups of atoms with a central Mn atom surrounded by a single icosahedral group of 12 aluminum atoms as found in the Al12Mn phase. Instead, such evidence points to larger icosahedral clusters of atoms such as are found in the ternary a phase of A1-Mn-Si.
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The addition of Si to A1-Mn alloys is known to stabilize the icosahedral phase3°m), and possible similarities between the icosahedral phase and the a phase of A1-Mn-Si have been pointed 0ut32,33). a(A1MnSi) phase has an almost body-centered cubic lattice with each lattice point being occupied by a large double-layered icosahedral cluster of atoms, usually described as a Mackay icosahedron. Thus formation of the icosahedral phase in A1-Mn-Si may be attributed to the same principle which was used to explain its formation in Mg 32(A1, Zn) 49There are also several Al-transition metal phases with crystal structures which are not known in detail, especially in the A1-Mn and A1-Cr systems. Recent diffraction evidence 34) indicates that some of the A1-Mn phases of composition close to A14Mn contain icosahedral clusters. According to this work, it seems possible to describe the structure of quasicrystalline and crystalline AI-Mn phases roughly as an agglomeration of the same type of icosahedral clusters, joined in different ways such that either a periodic or a quasiperiodic lattice is formed. The cluster was shown to be oriented uniquely for the icosahedral phase, in two orientations for decagonal and in three for a hexagonal phase.
3.2. Formation of quasicrystalphases In contrast to the formation of amorphous solids, the formation of quasicrystals is a first-order nucleation and growth process somewhat similar to that of conventional crystals. Thus the initial TEM images of icosahedral AI-Mn 7a9) formed by rapid solidification of a molten alloy showed that the icosahedral phase had grown radially outward from a central point of origin with a branched or dendritic structure typical of diffusion-controlled crystal growth into an undercooled melt. Rapid solidification from the liquid state can be carried out by several different processes, but a common feature is that small volumes of liquid are used so that heat can be extracted rapidly and the probability of the presence of heterogeneous nucleants for the equilibrium phases is small. Under these conditions, the alloy melt can become highly supercooled and it is found that the nucleation rate of the icosahedral phase is large. Increasing cooling rates lead to greater rates of nucleation and correspondingly smaller grain sizes, until in extreme cases 35) the size of the individual grains is so small that the structure is indistinguishable from glassy structures with certain local order. This material must be contrasted to true metallic glasses, however, in which an amorphous structure is formed by completely avoiding nucleation in the liquid; in the present case the structure is formed by an exceedingly high rate of nucleation. It has been speculated 35) that preexisting icosahedral clusters in the undercooled melt can promote the observed homogeneous nucleation, being both centers for nuclei and reducing icosahedral phase/melt interface energy.
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The growth of quasicrystals following their nucleation is in many ways similar to that of conventional crystals in a supercooled melt. In most cases the quasicrystals grow with a composition different from that of the melt, thus requiring solute diffusion to occur in the liquid. This is a slow process and leads to the formation of a dendritic (branched) structure, in which the dendrites grow preferentially in the directions of the three-fold symmetry axes3°). The selection of the preferred dendrite growth direction must be attributed to the anisotropic characteristics of the solid-liquid interface, but there is no direct evidence to indicate whether this is an anisotropy of the surface energy or of the atomic attachment kinetics. Surface energy effects are expected to dominate with unfaceted solid-liquid interfaces while faceted interfaces indicate the importance of kinetic effects. Facets are only occasionally seen9) on the planes of five-fold symmetry. If these are the most densely packed planes of the solid-liquid interface, they would have the lowest surface energy and would also be the most difficult surfaces on which to add new layers of crystal. Either effect would result in preferred growth along the three-fold directions. Quasicrystals have also been formed in the A1-Mn system by several processes other than rapid solidification of liquids, the success of which indicates that although the quasicrystal phases are not equilibrium phases of this system, they are not far from being stable and are frequently favored by kinetic factors. These processes include ion beam mixing36), ion irradiation37), ion implantation3S), solid state interdiffusion 39'4°) and rapid pressurization41). There have been several cases in which icosahedral crystals are reported to form by devitrification of metallic glasses. This occurs within a very narrow composition range in the P d - U - S i system42). It is also reported to occur in A1-Mn, using amorphous alloys formed by evaporation or ion beam mixing with cold substrates36,3v). In such cases, extremely fine-grained structures are obtained, again indicating very high rates of nucleation.
4. Some conclusions (and more questions)
We presented experimental results on the structure and formation of quasicrystals, which lead to some conclusions but raise many questions. 1. The following empirical rules can be suggested for formation of quasicrystals: (a) Suitable alloy systems contain intermetallic compounds where icosahedral clustering is extensively present. (b) The alloy melt has to be undercooled to nucleate the usually metastable quasicrystals without forming more stable phases. This is most readily accom-
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plished if the alloy composition is off stoichiometry of the intermetallic compounds. (c) Phase diagrams with peritectic reactions are more favorable than eutectics, since condition (b) can be fulfilled without forming glass.
Questions.
In the crystals of the A1-Mn system there are 3 types of icosahedral clusters; a shell of 12 aluminum with a central Mn in Alt2Mn G phase, possible Mackay icosahedron in A14Mn and truncated icosahedra in A16Mn. Why is only one realized as quasicrystal? Do icosahedral phases of different metallurgical systems have the same local order? 2. The projection of a quasiperiodic lattice from higher dimension seems to be a well established description for the crystallography and structure of quasicrystals.
Questions. Should quasicrystals be considered as decorated tiles? Or does the quasilattice have to be rearranged (i.e., disordered) to allow symmetric grouping of atoms to dominate in a structure which is still quasiperiodic but not deterministic? What determines formation of quasicrystals: energetics of duster formation and entropy of their packing, or long-range quasiperiodic interaction? Is the observed position disorder an intrinsic property of the structure, or is it due to metastable growth defects? The answers are expected from the presumably equilibrium A 1 - L i - C u icosahedral phase. 3. Nucleation of the icosahedral phase is homogeneous and its subsequent growth is relatively slow and proceeds in the 3-fold directions.
Questions.
Does the nucleation behavior of the icosahedral phase mean preexistence of the icosahedral dusters in liquid? How do atoms become attached to the growing quasicrystal surface? Is it an atom-by-atom transfer or clusterby-cluster transfer? What is the critical nucleus of the icosahedral phase and how is it related to the duster size? References
1) 2) 3) 4) 5) 6) 7) 8)
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