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1 cubic approximant of the Ag42In42Ca16 icosahedral quasicrystal
Journal of Alloys and Compounds 366 (2004) L1–L5
Letter
The 2/1 cubic approximant of the Ag42 In42 Ca16 icosahedral quasicrystal B.B. Deng, K.H. Kuo∗ Key Laboratory of Electron Microscopy, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, 100080 Beijing, PR China Received 10 June 2003; accepted 1 July 2003
Abstract Using equal amounts of Ag and In to replace Cd in the stable Cd85 Ca15 icosahedral quasicrystal, Guo and Tsai obtained the stable Ag42 In42 Ca16 icosahedral quasicrystal. However, the crystalline approximant of this quasicrystal is still not known yet. In the present study, ¯ a = 2.496 nm), displaying pseudo-icosahedral symmetry, has been found as the dominant phase in the the simple cubic Ag42 In45 Ca13 (Pa3, ¯ a = 1.564 nm) and Cd76 Ca13 (Pa3, ¯ a = 2.5339 nm), ternary Ag3 In3 Ca alloy by electron and X-ray diffraction. Since the cubic Cd6 Ca (Im3, respectively, are known to be the 1/1 and 2/1 cubic approximants of the Cd85 Ca15 icosahedral quasicrystal, this cubic Ag42 In45 Ca13 may be considered the 2/1 approximant of the Ag42 In42 Ca16 icosahedral quasicrystal. © 2003 Elsevier B.V. All rights reserved. Keywords: Ag–In–Ca; Crystalline approximant; Quasicrystal
1. Introduction
2. Cubic crystal with a = 2.49 nm
Recently, an icosahedral quasicrystal in Ag alloys was reported for the first time by Guo and Tsai [1]. It was deduced from the stable Cd85 Ca15 icosahedral quasicrystal [2,3] by substituting equal amounts of Ag and In for Cd in alloys with a composition of Cd85 Ca15 . A single phase of the stable, primitive icosahedral quasicrystal was thus obtained in Ag42 In42 Ca16 [1]. It is well known that a quasicrystal and its crystalline approximant have similar composition and local structure so that a study of the approximant is helpful in the understanding of the structure of this quasicrystal [4]. The present investigation is a study of the crystalline approximant of the Ag42 In42 Ca16 icosahedral quasicrystal by means of electron and X-ray diffraction methods. Agx Inx Ca16 alloys (32 ≤ x ≤ 50) were made from pure metals of Ag (99.99%), In (99.99%), and Ca (98.5%) in an arc furnace under Ar atmosphere, followed by annealing at 823 K for 55 h. Thin foils prepared by ion milling were examined in a transmission electron microscope. Phase identification of samples was performed by powder X-ray diffraction using Cu K␣ radiation.
Selected-area electron diffraction was made on all Agx Inx Ca16 samples and the Ag42 In42 Ca16 alloy was found to consist almost completely of the icosahedral phase, as reported earlier by Guo and Tsai [1]. With an increase of the Ag–In content, however, a new phase appeared to coexist with this icosahedral quasicrystal and it became the dominant phase in the Ag48 In48 Ca16 or rather the Ag3 In3 Ca alloy. Its composition determined by the energy dispersive spectrometry made in an electron microscope corresponded to Ag42 In45 Ca13 . A comprehensive electron diffraction analysis was made on this new phase by tilting it through large angles in a transmission electron microscope. The electron diffraction pattern displaying two-fold symmetry is shown in Fig. 1(a). Tilting about 45◦ around the vertical as well as the horizontal axes yielded the same electron diffraction patterns (Fig. 1(b) and (e)), but rotated 90◦ from each other. After tilting about 45◦ around the vertical axis to yield Fig. 1(b), a further tilt about 45◦ around this axis gave an electron diffraction pattern identical to Fig. 1(a) but also rotated 90◦ from it (not shown). From Fig. 1(b), the crystallite under examination was tilted then about 33◦ around the horizontal axis to yield Fig. 1(d). This is a typical electron diffraction pattern of six- or three-fold symmetry. In the cubic system, one of the angles between {1 1 0} and {1 1 1} is 35.26◦ . All these imply that the new
∗ Corresponding author. Tel.: +86-10-8264-9359; fax: +86-10-6256-1422. E-mail address: [email protected] (K.H. Kuo).
0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0925-8388(03)00690-X
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B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
Fig. 1. (a)–(e) Selected-area and (f) convergent-beam electron diffraction patterns of the 2/1 cubic Ag42 In45 Ca13 approximant. Tilting angles are shown and those in parentheses are calculated values.
phase is possibly a cubic one, provided that alternative, horizontal rows of diffraction spots in Fig. 1(a) belong to systematic extinctions. These electron diffraction patterns are thus indexed and the indices of some strong diffraction spots are given in Fig. 1. All 0kl spots in the [1 0 0] zone
pattern shown in Fig. 1(a) are absent for k = 2n + 1. The 0k0 spots in the [1 0 1] zone pattern shown in Fig. 1(b) and the 00l spots in the [1 1 0] zone pattern shown in Fig. 1(e) are either absent or very weak for k, l = 2n + 1. They were possibly also systematic extinctions but appeared owing
B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
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to multiple diffraction. Indeed, this was the case, since the 00l spots with l = 2n + 1 were completely absent in Fig. 1(c) when the crystallite was tilted off a main zone axis so that no such secondary diffraction source existed. Combining with the observations in other main zone patterns, the general reflection conditions for this new phase are: 0kl : k = 2n h00 : h = 2n h0l : l = 2n 0k0 : k = 2n and hl0 : h = 2n 00l : l = 2n
The agreement between the X-ray and electron diffraction results is good. The peaks {3 3 3}, {5 2 0}, {5 3 2}, {6 4 0}, {8 5 0}, {8 5 3}, {10 0 0}, {10 6 0} and {11 2 0} in Fig. 2(b) all appeared as medium to strong spots (indexed) in the electron diffraction patterns in Fig. 1. However, it should be mentioned that some peaks (marked with circles and triangle) in Fig. 2(b) do not belong to this cubic phase. Checked with the ICDD database, they possibly belong to AgIn2 , In4 Ag9 , and Ag3 In, respectively. In fact, these phases and Ag8 Ca3 were found to be the decomposition products of the cubic Ag42 In45 Ca13 after long exposure in air.
Looking up the International Crystallographic Table [5], both the space groups Pa3¯ (No. 205) and the Pbca (No. 61) have these reflection conditions. If this new phase is a simple cubic crystal, its lattice parameter a will be 2.49 nm. Since the accuracy of selected-area electron diffraction is not high, it is impossible to differentiate a cubic crystal of large lattice parameter from an orthorhombic one with about equal a, b, and c parameters, such as in the case of Mg51 Zn20 (a = 1.4083, b = 1.4486, c = 1.4025 nm) [6] and Hf54 Os17 (a = 1.3856, b = 1.4104, c = 1.450 nm) [7]. For this reason, convergent-beam electron diffraction pattern along the [1 1 1] direction, Fig. 1(f), as well as powder X-ray diffraction pattern, Fig. 2(b), were examined. In Fig. 1(f), the discs in the zero-order Laue zone show a six-fold distribution, whereas those of the first-order Laue zone prove the three-fold symmetry (marked). Thus, the electron diffraction evidences confirm the new phase a cubic ¯ one and its space group being Pa3. Fig. 2(a) and (b) show the powder X-ray diffraction spectra of the alloys Ag42 In42 Ca16 and Ag3 In3 Ca, respectively. The former is identical to that of the icosahedral quasicrystal reported by Guo and Tsai [1] and is similarly indexed (compare with that shown in Fig. 2(c) for the Al15 Mg44 Zn41 icosahedral quasicrystal [8]). The latter is different from the former, but the strong peaks appear at positions close to those of the former, implying similar local structure in both the icosahedral quasicrystal and the new phase. The peaks in Fig. 2(b) are sharp. Though there are some overlapping, no systematic splitting due to a lowering of the cubic symmetry can be detected. In order to index the powder X-ray diffraction pattern of large cubic crystals, weak reflections at low angles are very important. In a slow-scanned pattern, a very weak peak at 16.22◦ can be discerned. Including this tiny peak (later, it indexed the (4 2 1) reflection), the diffraction pattern in Fig. 2(b) can thus be indexed using the Dicvo191 program [9] and the lattice parameter a thus obtained is 2.4955(5) nm. The indices in Fig. 2(b) agree with the indexing scheme of the 2/1 cubic approximant (Fig. 2(d)) of the Al15 Mg44 Zn41 icosahedral quasicrystal reported earlier by Takeuchi and Mizutani [8]. The space group of the cubic Al15 Mg43 Zn42 is also Pa3¯ and its a = 2.306 nm, somewhat smaller than 2.496 nm of Ag42 In45 Ca13 . Therefore, in comparison with Fig. 2(a) and (b), the peaks in both Fig. 2(c) and (d) are shifted to higher angles.
2.1. Approximant Five-fold rotational symmetry is associated with the √ golden number τ = (1 + 5)/2 or cos 72◦ = (τ − 1)/2. Elser and Henley [4] used the rational ratio of two successive Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, . . . , Fn , . . . ; Fn = Fn−1 + Fn−2 ) as an approximant to substitute for the irrational τ to obtain the crystalline approximant of an icosahedral quasicrystal. The lattice parameters of the successive cubic aproximants are a, τa, τ 2 a, . . . and the pseudo-five-fold axis of the approximant is in a direction close to the τ 1 0 direction of the icosahedral quasicrystal, such as [2 1 0], [3 2 0], [5 3 0], etc. The cubic (Al, ¯ a = 1.416 nm) is the 1/1 approximant Zn)49 Mg32 (Im3, of the Al–Mg–Zn icosahedral quasicrystal. According to ¯ a = 1.568 nm Takakura et al. [10], the cubic Cd6 Ca (Im3, [11]) is the 1/1 approximant of the Cd85 Ca15 icosahedral quasicrystal, but the concentric atom shells are different from those of the 1/1 approximant of the Al–Mg–Zn icosahedral quasicrystal. These authors considered the Cd85 Ca15 possibly being a new kind of icosahedral quasicrystal. Recently, Gómez and Lidin [12] studied the crystal structure ¯ a = 2.5339 nm) with a lattice of the cubic Cd76 Ca13 (Pa3, parameter about τ times of that of Cd6 Ca. This is the 2/1 approximant of the Cd5.7 Ca icosahedral quasicrystal. ¯ a = 2.28 nm) as well The 2/1 cubic approximant (Pa3, ¯ a = 3.69 nm) were first reas the 3/2 approximant (Pa3, ported by Spaepen et al. [13] in Ga20 Mg37 Zn43 . Later, Edagawa et al. [14] found the 2/1 Al14.35 Ga6.15 Mg39.5 Zn40.0 approximant with a = 2.300 nm and Takeuchi and Mizutani [8] found the 2/1 Al15 Mg43 Zn42 approximant. Recently, Sugiyama et al. [15] have studied the crystal structure of the 2/1 Al17 Mg46 Zn37 cubic approximant (a = 2.3064 nm) by means of single crystal X-ray diffraction. The atomic species and positions of the atoms in the first six successive atom shells of icosahedral symmetry have been determined. Fig. 1(c) is a pseudo-five-fold diffraction pattern of the cubic Ag42 In45 Ca13 consisting of diffraction spots of the [2 1 0], [3 2 0], and [5 3 0] zones, which are the approximate directions of [τ 1 0]. The diffraction spots no longer lie on a single cross-grid and the two decagons of strong diffraction spots (marked by arrows) are located at about the same positions as those in the five-fold diffraction pattern of an icosahedral quasicrystal.
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B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
Fig. 2. X-ray diffraction spectra of (a), (c) stable icosahedral quasicrystals and (b), (d) their 2/1 cubic approximants, respectively. (c), (d) are taken from Takeuchi and Mizutani [8].
B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
The metallic radii of Ag, Cd, and In are 0.144, 0.151 and 0.158 nm, respectively. It is of interest to note that the average atomic radius of Ag–In is 0.151 nm, exactly the same as that of Cd. The ratio of the lattice parameter of the cubic Ag42 In45 Ca13 is comparable to that of Cd76 Ca13 and therefore the cubic Ag42 In45 Ca13 may be considered the 2/1 approximant of the Ag42 In42 Ca16 icosahedral quasicrystal. The 1/1 Ag2 In4 Ca approximant also exists (a = 1.5454 nm) [16] and these atomic shells of both the 1/1 and 2/1 approximants should be of great interest for a further study. Acknowledgements The authors would like to thank Professor S.S. Xie, Messrs D.F. Liu and Y.X. Liang for assistance in preparing the alloys. Drs. A.P. Tsai, J.Q. Guo, and W. Sun are thanked for their constructive comments. This study was generously supported by the National Natural Science Foundation of China.
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References [1] J.Q. Guo, A.P. Tsai, Phil. Mag. Lett. 82 (2002) 349. [2] A.P. Tsai, J.Q. Guo, E. Abe, H. Takakura, T.J. Sato, Nature 408 (2000) 537. [3] J.Q. Guo, E. Abe, A.P. Tsai, Phys. Rev. B 62 (2000) R14605. [4] V. Elser, C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [5] T. Hahn, International Tables for Crystallography, vol. A, fifth ed. (revised), Kluwer Academic Publishers, Dordrecht, 2002. [6] I. Higashi, N. Shiotani, M. Uda, T. Mizoguchi, H. Katoh, J. Solid State Chem. 36 (1981) 225. [7] K. Cenzual, B. Chabot, E. Parthé, Acta Crystallogr. 41 (1985) 313. [8] T. Takeuchi, U. Mizutani, Phys. Rev. B 52 (1995) 9300. [9] A. Boultif, D. Louer, J. Appl. Cryst. 24 (1991) 987. [10] H. Takakura, J.Q. Guo, A.P. Tsai, Phil. Mag. Lett. 81 (2001) 411. [11] G. Bruzzone, Gazz. Chim. Ital. 102 (1972) 234. [12] C.P. Gómez, S. Lidin, Angew. Chem. Int. Ed. 21 (2001) 4037. [13] F. Spaepen, L.C. Chen, S. Ebalard, W. Ohashi, Quasicrystals, in: M.V. Jaric, S. Lundqvist (Eds.), World Scientific, Singapore, 1990, pp. 1–18. [14] K. Edagawa, N. Naito, S. Takeuchi, Phil. Mag. B 65 (1992) 1011. [15] K. Sugiyama, W. Sun, K. Hiraga, J. Alloys Comp. 32 (2002) 139. [16] L.V. Sysa, Ya.M. Kalychak, Ya.V. Galadzhun, V.L. Zaremba, L.G. Akselrud, R.V. Skolozdra, J. Alloys Comp. 266 (1998) 17.
Letter
The 2/1 cubic approximant of the Ag42 In42 Ca16 icosahedral quasicrystal B.B. Deng, K.H. Kuo∗ Key Laboratory of Electron Microscopy, Institute of Physics, Chinese Academy of Sciences, P.O. Box 603, 100080 Beijing, PR China Received 10 June 2003; accepted 1 July 2003
Abstract Using equal amounts of Ag and In to replace Cd in the stable Cd85 Ca15 icosahedral quasicrystal, Guo and Tsai obtained the stable Ag42 In42 Ca16 icosahedral quasicrystal. However, the crystalline approximant of this quasicrystal is still not known yet. In the present study, ¯ a = 2.496 nm), displaying pseudo-icosahedral symmetry, has been found as the dominant phase in the the simple cubic Ag42 In45 Ca13 (Pa3, ¯ a = 1.564 nm) and Cd76 Ca13 (Pa3, ¯ a = 2.5339 nm), ternary Ag3 In3 Ca alloy by electron and X-ray diffraction. Since the cubic Cd6 Ca (Im3, respectively, are known to be the 1/1 and 2/1 cubic approximants of the Cd85 Ca15 icosahedral quasicrystal, this cubic Ag42 In45 Ca13 may be considered the 2/1 approximant of the Ag42 In42 Ca16 icosahedral quasicrystal. © 2003 Elsevier B.V. All rights reserved. Keywords: Ag–In–Ca; Crystalline approximant; Quasicrystal
1. Introduction
2. Cubic crystal with a = 2.49 nm
Recently, an icosahedral quasicrystal in Ag alloys was reported for the first time by Guo and Tsai [1]. It was deduced from the stable Cd85 Ca15 icosahedral quasicrystal [2,3] by substituting equal amounts of Ag and In for Cd in alloys with a composition of Cd85 Ca15 . A single phase of the stable, primitive icosahedral quasicrystal was thus obtained in Ag42 In42 Ca16 [1]. It is well known that a quasicrystal and its crystalline approximant have similar composition and local structure so that a study of the approximant is helpful in the understanding of the structure of this quasicrystal [4]. The present investigation is a study of the crystalline approximant of the Ag42 In42 Ca16 icosahedral quasicrystal by means of electron and X-ray diffraction methods. Agx Inx Ca16 alloys (32 ≤ x ≤ 50) were made from pure metals of Ag (99.99%), In (99.99%), and Ca (98.5%) in an arc furnace under Ar atmosphere, followed by annealing at 823 K for 55 h. Thin foils prepared by ion milling were examined in a transmission electron microscope. Phase identification of samples was performed by powder X-ray diffraction using Cu K␣ radiation.
Selected-area electron diffraction was made on all Agx Inx Ca16 samples and the Ag42 In42 Ca16 alloy was found to consist almost completely of the icosahedral phase, as reported earlier by Guo and Tsai [1]. With an increase of the Ag–In content, however, a new phase appeared to coexist with this icosahedral quasicrystal and it became the dominant phase in the Ag48 In48 Ca16 or rather the Ag3 In3 Ca alloy. Its composition determined by the energy dispersive spectrometry made in an electron microscope corresponded to Ag42 In45 Ca13 . A comprehensive electron diffraction analysis was made on this new phase by tilting it through large angles in a transmission electron microscope. The electron diffraction pattern displaying two-fold symmetry is shown in Fig. 1(a). Tilting about 45◦ around the vertical as well as the horizontal axes yielded the same electron diffraction patterns (Fig. 1(b) and (e)), but rotated 90◦ from each other. After tilting about 45◦ around the vertical axis to yield Fig. 1(b), a further tilt about 45◦ around this axis gave an electron diffraction pattern identical to Fig. 1(a) but also rotated 90◦ from it (not shown). From Fig. 1(b), the crystallite under examination was tilted then about 33◦ around the horizontal axis to yield Fig. 1(d). This is a typical electron diffraction pattern of six- or three-fold symmetry. In the cubic system, one of the angles between {1 1 0} and {1 1 1} is 35.26◦ . All these imply that the new
∗ Corresponding author. Tel.: +86-10-8264-9359; fax: +86-10-6256-1422. E-mail address: [email protected] (K.H. Kuo).
0925-8388/$ – see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0925-8388(03)00690-X
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B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
Fig. 1. (a)–(e) Selected-area and (f) convergent-beam electron diffraction patterns of the 2/1 cubic Ag42 In45 Ca13 approximant. Tilting angles are shown and those in parentheses are calculated values.
phase is possibly a cubic one, provided that alternative, horizontal rows of diffraction spots in Fig. 1(a) belong to systematic extinctions. These electron diffraction patterns are thus indexed and the indices of some strong diffraction spots are given in Fig. 1. All 0kl spots in the [1 0 0] zone
pattern shown in Fig. 1(a) are absent for k = 2n + 1. The 0k0 spots in the [1 0 1] zone pattern shown in Fig. 1(b) and the 00l spots in the [1 1 0] zone pattern shown in Fig. 1(e) are either absent or very weak for k, l = 2n + 1. They were possibly also systematic extinctions but appeared owing
B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
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to multiple diffraction. Indeed, this was the case, since the 00l spots with l = 2n + 1 were completely absent in Fig. 1(c) when the crystallite was tilted off a main zone axis so that no such secondary diffraction source existed. Combining with the observations in other main zone patterns, the general reflection conditions for this new phase are: 0kl : k = 2n h00 : h = 2n h0l : l = 2n 0k0 : k = 2n and hl0 : h = 2n 00l : l = 2n
The agreement between the X-ray and electron diffraction results is good. The peaks {3 3 3}, {5 2 0}, {5 3 2}, {6 4 0}, {8 5 0}, {8 5 3}, {10 0 0}, {10 6 0} and {11 2 0} in Fig. 2(b) all appeared as medium to strong spots (indexed) in the electron diffraction patterns in Fig. 1. However, it should be mentioned that some peaks (marked with circles and triangle) in Fig. 2(b) do not belong to this cubic phase. Checked with the ICDD database, they possibly belong to AgIn2 , In4 Ag9 , and Ag3 In, respectively. In fact, these phases and Ag8 Ca3 were found to be the decomposition products of the cubic Ag42 In45 Ca13 after long exposure in air.
Looking up the International Crystallographic Table [5], both the space groups Pa3¯ (No. 205) and the Pbca (No. 61) have these reflection conditions. If this new phase is a simple cubic crystal, its lattice parameter a will be 2.49 nm. Since the accuracy of selected-area electron diffraction is not high, it is impossible to differentiate a cubic crystal of large lattice parameter from an orthorhombic one with about equal a, b, and c parameters, such as in the case of Mg51 Zn20 (a = 1.4083, b = 1.4486, c = 1.4025 nm) [6] and Hf54 Os17 (a = 1.3856, b = 1.4104, c = 1.450 nm) [7]. For this reason, convergent-beam electron diffraction pattern along the [1 1 1] direction, Fig. 1(f), as well as powder X-ray diffraction pattern, Fig. 2(b), were examined. In Fig. 1(f), the discs in the zero-order Laue zone show a six-fold distribution, whereas those of the first-order Laue zone prove the three-fold symmetry (marked). Thus, the electron diffraction evidences confirm the new phase a cubic ¯ one and its space group being Pa3. Fig. 2(a) and (b) show the powder X-ray diffraction spectra of the alloys Ag42 In42 Ca16 and Ag3 In3 Ca, respectively. The former is identical to that of the icosahedral quasicrystal reported by Guo and Tsai [1] and is similarly indexed (compare with that shown in Fig. 2(c) for the Al15 Mg44 Zn41 icosahedral quasicrystal [8]). The latter is different from the former, but the strong peaks appear at positions close to those of the former, implying similar local structure in both the icosahedral quasicrystal and the new phase. The peaks in Fig. 2(b) are sharp. Though there are some overlapping, no systematic splitting due to a lowering of the cubic symmetry can be detected. In order to index the powder X-ray diffraction pattern of large cubic crystals, weak reflections at low angles are very important. In a slow-scanned pattern, a very weak peak at 16.22◦ can be discerned. Including this tiny peak (later, it indexed the (4 2 1) reflection), the diffraction pattern in Fig. 2(b) can thus be indexed using the Dicvo191 program [9] and the lattice parameter a thus obtained is 2.4955(5) nm. The indices in Fig. 2(b) agree with the indexing scheme of the 2/1 cubic approximant (Fig. 2(d)) of the Al15 Mg44 Zn41 icosahedral quasicrystal reported earlier by Takeuchi and Mizutani [8]. The space group of the cubic Al15 Mg43 Zn42 is also Pa3¯ and its a = 2.306 nm, somewhat smaller than 2.496 nm of Ag42 In45 Ca13 . Therefore, in comparison with Fig. 2(a) and (b), the peaks in both Fig. 2(c) and (d) are shifted to higher angles.
2.1. Approximant Five-fold rotational symmetry is associated with the √ golden number τ = (1 + 5)/2 or cos 72◦ = (τ − 1)/2. Elser and Henley [4] used the rational ratio of two successive Fibonacci numbers (0, 1, 1, 2, 3, 5, 8, . . . , Fn , . . . ; Fn = Fn−1 + Fn−2 ) as an approximant to substitute for the irrational τ to obtain the crystalline approximant of an icosahedral quasicrystal. The lattice parameters of the successive cubic aproximants are a, τa, τ 2 a, . . . and the pseudo-five-fold axis of the approximant is in a direction close to the τ 1 0 direction of the icosahedral quasicrystal, such as [2 1 0], [3 2 0], [5 3 0], etc. The cubic (Al, ¯ a = 1.416 nm) is the 1/1 approximant Zn)49 Mg32 (Im3, of the Al–Mg–Zn icosahedral quasicrystal. According to ¯ a = 1.568 nm Takakura et al. [10], the cubic Cd6 Ca (Im3, [11]) is the 1/1 approximant of the Cd85 Ca15 icosahedral quasicrystal, but the concentric atom shells are different from those of the 1/1 approximant of the Al–Mg–Zn icosahedral quasicrystal. These authors considered the Cd85 Ca15 possibly being a new kind of icosahedral quasicrystal. Recently, Gómez and Lidin [12] studied the crystal structure ¯ a = 2.5339 nm) with a lattice of the cubic Cd76 Ca13 (Pa3, parameter about τ times of that of Cd6 Ca. This is the 2/1 approximant of the Cd5.7 Ca icosahedral quasicrystal. ¯ a = 2.28 nm) as well The 2/1 cubic approximant (Pa3, ¯ a = 3.69 nm) were first reas the 3/2 approximant (Pa3, ported by Spaepen et al. [13] in Ga20 Mg37 Zn43 . Later, Edagawa et al. [14] found the 2/1 Al14.35 Ga6.15 Mg39.5 Zn40.0 approximant with a = 2.300 nm and Takeuchi and Mizutani [8] found the 2/1 Al15 Mg43 Zn42 approximant. Recently, Sugiyama et al. [15] have studied the crystal structure of the 2/1 Al17 Mg46 Zn37 cubic approximant (a = 2.3064 nm) by means of single crystal X-ray diffraction. The atomic species and positions of the atoms in the first six successive atom shells of icosahedral symmetry have been determined. Fig. 1(c) is a pseudo-five-fold diffraction pattern of the cubic Ag42 In45 Ca13 consisting of diffraction spots of the [2 1 0], [3 2 0], and [5 3 0] zones, which are the approximate directions of [τ 1 0]. The diffraction spots no longer lie on a single cross-grid and the two decagons of strong diffraction spots (marked by arrows) are located at about the same positions as those in the five-fold diffraction pattern of an icosahedral quasicrystal.
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Fig. 2. X-ray diffraction spectra of (a), (c) stable icosahedral quasicrystals and (b), (d) their 2/1 cubic approximants, respectively. (c), (d) are taken from Takeuchi and Mizutani [8].
B.B. Deng, K.H. Kuo / Journal of Alloys and Compounds 366 (2004) L1–L5
The metallic radii of Ag, Cd, and In are 0.144, 0.151 and 0.158 nm, respectively. It is of interest to note that the average atomic radius of Ag–In is 0.151 nm, exactly the same as that of Cd. The ratio of the lattice parameter of the cubic Ag42 In45 Ca13 is comparable to that of Cd76 Ca13 and therefore the cubic Ag42 In45 Ca13 may be considered the 2/1 approximant of the Ag42 In42 Ca16 icosahedral quasicrystal. The 1/1 Ag2 In4 Ca approximant also exists (a = 1.5454 nm) [16] and these atomic shells of both the 1/1 and 2/1 approximants should be of great interest for a further study. Acknowledgements The authors would like to thank Professor S.S. Xie, Messrs D.F. Liu and Y.X. Liang for assistance in preparing the alloys. Drs. A.P. Tsai, J.Q. Guo, and W. Sun are thanked for their constructive comments. This study was generously supported by the National Natural Science Foundation of China.
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References [1] J.Q. Guo, A.P. Tsai, Phil. Mag. Lett. 82 (2002) 349. [2] A.P. Tsai, J.Q. Guo, E. Abe, H. Takakura, T.J. Sato, Nature 408 (2000) 537. [3] J.Q. Guo, E. Abe, A.P. Tsai, Phys. Rev. B 62 (2000) R14605. [4] V. Elser, C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [5] T. Hahn, International Tables for Crystallography, vol. A, fifth ed. (revised), Kluwer Academic Publishers, Dordrecht, 2002. [6] I. Higashi, N. Shiotani, M. Uda, T. Mizoguchi, H. Katoh, J. Solid State Chem. 36 (1981) 225. [7] K. Cenzual, B. Chabot, E. Parthé, Acta Crystallogr. 41 (1985) 313. [8] T. Takeuchi, U. Mizutani, Phys. Rev. B 52 (1995) 9300. [9] A. Boultif, D. Louer, J. Appl. Cryst. 24 (1991) 987. [10] H. Takakura, J.Q. Guo, A.P. Tsai, Phil. Mag. Lett. 81 (2001) 411. [11] G. Bruzzone, Gazz. Chim. Ital. 102 (1972) 234. [12] C.P. Gómez, S. Lidin, Angew. Chem. Int. Ed. 21 (2001) 4037. [13] F. Spaepen, L.C. Chen, S. Ebalard, W. Ohashi, Quasicrystals, in: M.V. Jaric, S. Lundqvist (Eds.), World Scientific, Singapore, 1990, pp. 1–18. [14] K. Edagawa, N. Naito, S. Takeuchi, Phil. Mag. B 65 (1992) 1011. [15] K. Sugiyama, W. Sun, K. Hiraga, J. Alloys Comp. 32 (2002) 139. [16] L.V. Sysa, Ya.M. Kalychak, Ya.V. Galadzhun, V.L. Zaremba, L.G. Akselrud, R.V. Skolozdra, J. Alloys Comp. 266 (1998) 17.