Lattice correspondence and fivefold twins of the orthorhombic (2/1, 1/1) and (1/0, 2/1) approximants in a Ga-Fe-Cu-Si al
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TRODUCTION
IT had been shown earlier
that the icosahedral quasicrystal, the decagonal quasicrystal (DQC), and the orthorhombic (2/1, 1/1) decagonal approximant, with a 5 2.00 nm, b 5 1.25 nm, and c 5 1.40 nm, formed successively during the rapid solidification of the Ga46Fe23Cu23Si8 alloy. Here, 2/1 and 1/1 are ratios of two consecutive Fibonacci numbers used to substitute for the irrational t in two quasiperiodic directions of a DQC, to recover the lattice parameters a and c, respectively, of the orthorhombic approximant. Sometimes, five DQC ribbons occur at about 36 deg apart, and the (2/1, 1/1) approximant appears between two of these ribbons, with an orientation difference of 36 deg between two neighboring (2/1, 1/1) approximants. In fact, this corresponds to fivefold rotational twins of an orthorhombic lattice, since a rotation of 3 3 72 5 216 deg is equivalent to 36 deg, and this is also valid for an orthorhombic structure with a center of conversion. At a lower temperature or when annealed at 630 7C, another orthorhombic approximant with a 5 0.76 nm, b 5 1.25 nm, and c 5 2.40 nm occurs, sometimes in intimate intergrowth with the (2/1, 1/1) approximant. This is the Ga-Mn (1/0, 2/1) approximant, whose a parameter is t2 times smaller, and whose c parameter is t times larger, than the corresponding ones in the (2/1, 1/1) approximant. The GaMn (1/0, 2/1) approximant has about the same lattice parameters as the orthorhombic Al60Mn11Ni4 compound reported first by Robinson[2] and later, for various Al-transition metal alloys, by many others.[3–6] [1]
S.P. GE, formerly Graduate Student, and K.H. KUO, Professor, Department of Materials Physics, University of Science and Technology Beijing, 100083 Beijing, China, are Research Staff Member and Professor, respectively, Beijing Laboratory of Electron Microscopy, Institute of Physics, Chinese Academy of Sciences, 100080 Beijing, P.R. China. Manuscript submitted June 29, 1998. METALLURGICAL AND MATERIALS TRANSACTIONS A
Figure 1 shows the set 1 Penrose tiles[7,8] consisting of pentagons, 36-deg pentagonal stars, 36-deg (thin) rhombi, and ‘‘boats.’’ The boat tile is an incomplete 36-deg pentagonal star with three instead of five angles of 36 deg. The aperiodic tiling of these tiles, namely, the set 1 Penrose tiling, has generally been accepted as the quasilattice of the two-dimensional DQC. If the centers of the pentagons are connected, a set of larger tiles of 72-deg pentagonal stars, flattened hexagons with an apex of 72 deg, and octagons with three angles of 72 deg (the ‘‘crown’’ subunit[10]) will result, as outlined by thick lines in Figure 1. These tiles are called binary tiles,[11] and all angles are multiples of 72 deg. Moreover, these tiles can be tiled aperiodically, periodically, or randomly. If the edge length of the pentagon is 0.47 nm, the edge length of the binary tiles will be close to 0.66 nm. In the three-dimensional structure consisting of layers of binary tiles such as those in Figure 1, either in a quasicrystal or an approximant, the vertices of these bi
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