- Email: [email protected]
1 approximant
Journal of Non-Crystalline Solids 334&335 (2004) 161–166 www.elsevier.com/locate/jnoncrysol
Structure of the Al65Rh27Si8 2/1–2/1–2/1 approximant T. Takeuchi a
a,*
, N. Koshikawa b, E. Abe c, K. Kato d, U. Mizutani
a
Department of Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan b National Space Development Agency of Japan, Tsukuba 305-8505, Japan c Materials Engineering Laboratory, National Institute of Materials Science, Tsukuba 305-0047, Japan d Japan Synchrotron Radiation Research Institute, Sayo-gun, Hyogo 679-5198, Japan
Abstract Atomic arrangements in the thermodynamically stable Al65 Rh27 Si8 2/1–2/1–2/1 approximant were determined by means of synchrotron radiation Rietveld analysis. It was found that the Al65 Rh27 Si8 2/1–2/1–2/1 approximant possesses a cubic unit cell containing 564 atoms. We found that the atomic structure of this approximant is characterized by a CsCl-type structure with a large and a small icosahedral cluster of 6.8 A in diameter. The presence of chemical and structure icosahedral atomic cluster of 19.76 A disordering between (Al, Si) and Rh was also found in the structure as well as in the Mackay-type 1/1–1/1–1/1 approximants. Ó 2004 Elsevier B.V. All rights reserved. PACS: 61.44.Br; 61.10.Eq; 61.14.)x
1. Introduction The structure of icosahedral quasicrystals is generally understood as that appearing in a three-dimensional hyperplane sliced from a six-dimensional periodic lattice decorated with atomic surfaces. This three-dimensional hyperplane in which the structure of quasicrystal is constructed is called the Ôphysical space’. The slope of the physical space in the six-dimensional space is characterized by a setpof three irrational numbers ð s s s Þ, where s ¼ ð1 þ 5Þ=2 is known as the golden ratio. A quasiperiodic structure with icosahedral symmetry is consequently constructed by this slope of the irrational number s. Similarly, the structures of crystalline approximants are produced by the six-dimensional periodic lattice with a physical space slightly tilted from that of the quasicrystals [1]. The slope of the physical space for an approximant turns out to have ð p1 =q1 p2 =q2 p3 =q3 Þ instead of ð s s s Þ, where pn =qn (n ¼ 1, 2, 3) is continued-fraction approximant to s (pn =qn ¼ 1=0; 1=1; 2=1; 3=2; 5=3; . . . ; s). This rational slope leads to periodicity rather than quasiperiodicity in the resulting structure. *
Corresponding author. Tel.: +81-52 789 5620; fax: +81-52 789 3720/3724. E-mail address: [email protected] (T. Takeuchi). 0022-3093/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2003.11.032
Closely related slopes lead essentially to the same local atomic arrangements in quasicrystals and approximants. Thus the approximants can be regarded as crystalline materials possessing the same local atomic arrangements of the quasicrystals. In order to investigate the structure of the Mackaytype quasicrystals and approximants, we performed in the last decade the structure analysis for the Mackaytype 1/1–1/1–1/1 approximants [2–5]. The periodic structure of the approximants allowed us to make full use of the well-developed Rietveld analysis. We found, as a result of structure analysis, that these 1/1–1/1–1/1 approximants are characterized by the chemical disordering among simple metal elements and late-transition metal elements, and that this chemical disordering has a strong influence on the electron transport properties [5,6]. Although the presence of the disordering in the local atomic structure of the 1/1–1/1–1/1 approximants and its effect on the transport properties were clearly revealed in our previous work, it is still uncertain whether these features are commonly adapted for all quasicrystals and approximants. A structure analysis of the higher-order approximants would provide us a hint on this point. Thus, in this work, we analyzed the structure of the Mackay-type Al65 Rh27 Si8 2/1–2/1–2/1 approximant by performing the Rietveld analysis on the synchrotron radiation powder diffraction spectrum.
162
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
2. Experimental Al65 Rh27 Si8 alloy ingots were prepared by melting pure elements of Al (purity 99.99%), Rh (99.99%), and Si (99.999%) in an arc-furnace under a pressurized Ar gas atmosphere. Mother ingots were annealed at 700 °C for 24 h and crushed into fine powders of less than 5 lm in mean diameter. These small grains play an essential role in preventing the preferred orientation of the powders that is unfavorable for the precise powder diffraction measurement. Indeed absence of the diffraction spots in the Debye–Scherrer ring was confirmed in the Laue pictures of the powder samples stuffed in a quartz capillary of 0.2 mm in diameter. Synchrotron radiation diffraction spectrum was accumulated with a large Debye–Scherrer camera placed at the beam line BL02B2 in SPring-8, Hyogo, Japan. We employed an incident beam with a wave length of 0.7 A, an imaging plate as the detector, and 90 K nitrogen gas blowing on the quartz capillary with the powdered samples stuffed in it. A Rietveld analysis was performed on the measured synchrotron radiation diffraction spectrum with the program RIETAN2000 developed by Izumi [7]. Since the difference in the number of electrons between Al and Si is less than 10% and both numbers are much smaller than that of Re, it is very difficult to distinguish Si atoms from Al atoms in the Al–Rh–Si 2/1–2/ 1–2/1 approximant, having a complicated structure. Thus all Si atoms in this approximant were regarded as Al in the present Rietveld analysis. We used an overall isotropic atomic displacement parameter Biso: in the present Rietveld analysis instead of assigning an independent atomic displacement parameter to each atomic site to reduce the number of fitting parameters.
3. Results Fig. 1(a)–(c) show electron diffraction patterns of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant measured along
[1 0 0], [1 1 0], and [1 1 1] directions. The symmetry of the patterns along [1 0 0], [1 1 0], and [1 1 1] directions are 2mm, 2mm, and 6, respectively. All diffraction spots were indexed with those of a simple cubic lattice. These features of the diffraction patterns indicate that the 3 or space group of the Al65 Rh27 Si8 2/1–2/1–2/1 is Pm P 23. If an icosahedral cluster of m35 symmetry is located at the center of the cubic cell, the lattice turns out to have Pm3 symmetry. Indeed, many 1/1–1/1–1/1 approximants and the Al–Pd–Mn–Si 2/1–2/1–2/1 approximants were reported to have the Pm3 symmetry [2–5,8]. Thus, we assumed that the Al65 Rh27 Si8 2/1–2/1– 2/1 approximant has a cubic lattice with the Pm 3 symmetry. Before performing the Rietveld analysis, we constructed a structure model of the Al65 Rh27 Si8 2/1–2/1–2/ 1 approximant with Pm3 by means of the high-dimensional cut method with a simplified atomic surfaces on the basis of the Katz–Gratias model [9]. The numbers of atomic sites in the initial model were modified (removed or newly added) during the structure refinement, and the resulting structure was fairly different from the initial one. Thus, we avoid making any detailed comments on the initial model in order to save the limited space and concentrate to discuss the obtained structure. Fig. 2 shows the diffraction spectrum calculated using the refined parameters together with the measured one. Five hundred and two independent diffraction peaks were employed for the determination of the structure parameters. The resulting parameters and the reliable factors (R-factors) are listed in Table 1. The calculated and measured spectra show an extremely good agreement, and the resulting composition became almost the same as the nominal one. The reliable factors for the weighted pattern (Rwp ) and the structure (RI ) were sufficiently reduced to be ðRwp ; RI Þ ¼ ð3:21%; 1:32%Þ. All these results strongly indicate that the atomic structure in the Al65 Rh27 Si8 2/1–2/1–2/1 approximant was successfully determined by the present Rietveld analysis. Least-square parameter fitting methods such as the Rietveld analysis sometimes lead to wrong results be-
Fig. 1. Electron diffraction patterns measured along the [1 0 0], [1 1 0], and [1 1 1] directions possessing 2mm, 2mm, and 6 symmetries, respectively. All diffraction spots can be indexed with those of a simple cubic lattice. The space groups possessing these features are limited to Pm3 or P 23. With considering possible symmetry for the approximant of icosahedral quasicrystals, we determined the space group of this sample to be Pm3.
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
163
Fig. 2. X-ray diffraction spectrum measured by the synchrotron radiation together with the calculated spectrum as a result of the Rietveld analysis. in wave length. Diffraction spectrum was accumulated at the beam line BL02B2 in SPring-8, Hyogo, Japan, with a incident radiation of 0.7 A
cause of the presence of local minima in the meansquare error. In order to avoid such wrong results, we observed atomic arrangements in the real space by using the HAADF-STEM technique. The HAADF-STEM is powerful tool to determine the projected position of the heavy atoms because the contribution of elements on the HAADF-STEM images is proportional to the square of their atomic numbers. Measured HAADF-STEM image along [1 0 0] direction and that calculated with the refined parameters are shown in Fig. 3(a) and (b), respectively. The bright portions in the images correspond to the projected position of Rh atoms. The dark areas, on the other hand, indicated the projected positions of Al and/or Si or vacancies, which are not individually imaged. Obviously, the calculated image reproduces very well the observed one, lending a strong support to the validity of the refined structure parameters.
4. Discussion The high-dimensional method, with which the structures of quasicrystal and approximants are constructed, requires that the lattice constant of the p=q–p=q–p=q cubic approximant pffiffiffiffiffiffiffiffiffiffiffi satisfies the condition of ap=q–p=q–p=q ¼ 2aR ðps þ qÞ 2 þ s, where aR indicates the quasi-lattice constant [1]. One may notice that the lattice constant of the 2/1–2/1–2/1 approximant can be evaluated from that of the corresponding 1/1–1/1–1/1 approximant by transforming the above formula to a2=1–2=1–2=1 ¼ s a1=1–1=1–1=1 . The lattice constant of the Al65 Rh27 Si8 2/ that 1–2/1–2/1 approximant is determined to be 19.76 A, is indeed s times larger than 12.3–12.7 A of the 1/1–1/1–1/ 1 approximant [2,3,5,6,10]. If the local structure is kept unchanged between these two kinds of approximant, the number of atoms in the unit cell of the 2/1–2/1–2/1 approximant (N2=1–2=1–2=1 ) should be evaluated by using
that of the 1/1–1/1–1/1 approximant as N2=1–2=1–2=1 ¼ N1=1–1=1–1=1 s3 . The number of atoms in the unit cell of the 2/1–2/1–2/1 approximant obtained as a result of the Rietveld analysis was N2=1–2=1–2=1 ¼ 563. This number lies in the vicinity of ð135–144Þ s3 ¼ 571–609, where 135– 144 means the number of atoms in the Mackay-type 1/1– 1/1–/1/1 approximant [2,3,5,6,10]. It is found that a large icosahedral cluster with quintuple stacked atomic shells exists in the cubic unit cell of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant. The cluster in the unit cell is shown in Fig. 4. The 1st shell, the 2nd shell, the 3rd shell, the 4th shell, and the 5th shell consist of 14, 24, 92, 132, and 212 atoms, respectively. All atomic shells except for the 1st one has the icosahedral symmetry of m35. Note here that the symmetry of each icosahedral atomic shell is always distorted because the periodic lattice never has fivefold symmetry. The distortion in the icosahedral cluster can be evaluated by using edge lengths in the icosahedrons. If perfect icosahedral symmetry is present, all edge lengths are identical. However, once the icosahedral cluster is distorted within the restriction of the m 3 symmetry in the approximants, it turns out to have two different edge lengths, r1 and r2 . We calculated the r1 =r2 for all icosahedral shells using the refined structure parameters. Surprisingly, all r1 =r2 ratios calculated for the 2nd, 3rd, 4th, and 5th shells turned out to be 0.99. Thus, it is strongly argued that the icosahedral order almost perfectly persists in this large icosahedral cluster. A small icosahedral cluster with triply stacked atomic shells also exists with its center at the vertices of the cubic unit cell. The second and the third shell in this cluster have the icosahedral symmetry. In sharp contrast to the large icosahedral cluster, this small icosahedral cluster at the vertices is fairly distorted. That was confirmed by the r1 =r2 ratio described above; those for the second and third shells are 0.90 and 1.15, respectively.
164
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
Table 1 Refined structure parameters of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant Sites
Wickoff notation
Occupancy
x
y
z
Sites constructing the body center cluster First shell M00:Al M01:Al/Rh
6e 8i
1.0 0.71/0.28 (2)
0.0 0.0837 (5)
0.0 0.0837
0.081 (1) 0.0837
Second shell M10:Al a M11:Al/Rh
12j 12j
0.70 (3) 0.83/0.17 (1)
0.070 (2) 0.0
0.0 0.1000 (8)
0.188 (2) 0.1636 (7)
Third shell M20:Al M21:Al M22:Al M23:Rh M24:Rh a M25:Al/Rh
24l 12j 24l 12j 8i 12j
1.0 1.0 1.0 1.0 1.0 0.17/0.83 (1)
0.1363 0.0 0.0654 0.1146 0.1934 0.0
0.2222 (7) 0.069 (1) 0.2402 (8) 0.00000 0.1934 0.1853 (4)
0.1085 (7) 0.261 (1) 0.2142 (7) 0.3127 (3) 0.1934 0.2960 (3)
Fourth shell M40:Al M41:Al M42:Al M43:Al M44:Al M45:Al a M46:Al
12j 24l 24l 24l 24l 12j 12j
1.0 1.0 1.0 1.0 1.0 1.0 0.80 (3)
0.283 (1) 0.1283 (9) 0.2409 (8) 0.0770 (8) 0.1950 (9) 0.079 (1) 0.0
0.0 0.3874 (8) 0.3436 (8) 0.3834 (8) 0.3075 (9) 0.0 0.248 (2)
0.315 (1) 0.0730 (8) 0.1035 (8) 0.1992 (8) 0.2383 (9) 0.430 (1) 0.404 (2)
Fifth shell M50:Rh M51:Al/Rh M52:Rh a M53:Rh M54:Rh M55:Al/Rh M56:Al/Rh M57:Al M58:Al/Rh M59:Al/Rh M60:Al M61:Al M62:Al/Rh
6g 24l 24l 6g 6f 8i 24l 24l 24l 12j 12k 24l 12k
1.0 0.16/0.84 (1) 1.0 1.0 1.0 0.38/0.62 (2) 0.924/0.076 (8) 0.72 (2) 0.892/0.108 (9) 0.81/0.19 (1) 1.0 0.61 (2) 0.82/0.18 (1)
0.0 0.1869 (3) 0.3800 (2) 0.0 0.0 0.3106 (4) 0.0653 (6) 0.266 (1) 0.3128 (7) 0.0 0.128 (1) 0.418 (1) 0.1211 (7)
0.0860 (5) 0.1926 (3) 0.3116 (2) 0.3052 (5) 0.5 0.3106 0.3782 (7) 0.432 (1) 0.3835 (8) 0.4303 (8) 0.096 (1) 0.300 (1) 0.5
0.5 0.4250 (3) 0.1196 (2) 0.5 0.1950 (5) 0.3106 0.4305 (8) 0.139 (1) 0.1932 (7) 0.310 (1) 0.5 0.231 (1) 0.2662 (8)
Glue sites G01:Al/Rh G02:Rh G03:Al/Rh G05:Al
12k 6h 6g 12k
0.89/0.11 (1) 1.0 0.79/0.21 (2) 0.27 (3)
0.0593 (8) 0.5 0.0 0.066 (4)
0.5 0.5 0.427 (1) 0.207 (4)
0.4002 (8) 0.1084 (5) 0.5 0.5
(7) (7) (3) (3)
Sites constructing the vertex cluster
Center atom N00:Rh
1b
1.0
0.5
0.5
0.5
First shell N10:Al/Rh N11:Al/Rh
6h 8i
0.80/0.20 (2) 0.82/0.18 (2)
0.355 (1) 0.4268 (6)
0.5 0.4268
0.5 0.4268
Second shell N20:Al/Rh a N21:Rh N22:Al
24l 12k 6h
0.855/0.115 (9) 1.0 0.60 (5)
0.4243 (6) 0.2981 (3) 0.5
0.3131 (6) 0.5 0.5
0.3760 (6) 0.3882 (4) 0.275 (3)
Third shell N30:Al a N31:Rh N32:Al/Rh
12k 12k 12k
1.0 1.0 0.72/0.28 (1)
0.435 (1) 0.3080 (3) 0.3869 (6)
0.5 0.5 0.1559 (6)
0.172 (1) 0.2374 (3) 0.5
2 , Rwp ¼ 3:21%, S ¼ 4:6178, Nominal composition: Al65 Si8 Rh27 , Resulting composition: (Al,Si)72:8 Rh27:2 , Space group: Pm3, a ¼ 12:7628ð1Þ, Biso: ¼ 0:58 A RI ¼ 1:32%, RF ¼ 0:61%. a Atomic sites at the vertices of the icosahedral clusters.
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
165
Fig. 3. (a) HAADF-STEM image measured along [1 0 0] direction and (b) simulated image calculated by using the Rietveld refined structure parameters. The simulated image reproduces well the measured one. The largest icosahedral cluster existing in the unit cell is drawn by the white lines in the calculated image.
75 -100% Rh 50 - 75% Rh 25 - 50% Rh 0 - 25% Rh
Fig. 4. Atomic clusters exist at the body center (top panel) and vertices (bottom panel) in the unit cell of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant. Atomic arrangements on the triangular faces of each icosahedral atomic shell are shown below each atomic shell. Open circles and filled circles indicate (Al, Si) and Rh atoms, respectively. The atomic sites in which Al and Re coexist together are expressed as gray circles. The depth of the gray color indicates the Re occupancy in each mixed site; the darker the color is, the more the Re exist. The triangles decorated by circles indicate the atom arrangements on the triangular faces of the icosahedral cluster shells. Note that the seven atoms on the faces along the [1 1 1] direction in the fifth shell of the larger cluster centered at the body center are shared by the third shell in the small cluster centered at the vertices. Seven atoms constructing hexagons in the triangular surfaces of the outermost shells in the large cluster at body center of the unit cell are shared with the outermost shells in the small icosahedral clusters at vertices.
Note here that these large and small icosahedral clusters construct a CsCl-type structure with some glue atoms between them. The structure of the 2/1–2/1–2/1 approximant was first determined for the Al–Pd–Mn–Si 2/1–2/1–2/1 ap-
proximant by Sugiyama et al. [8]. Although the detailed structures are different, the clusters in the Al–Rh–Si 2/ 1–2/1–2/1 approximant are similar to those in the Al– Pd–Mn–Si 2/1–2/1–2/1 approximant. One of the most pronounced differences between the Al–Pd–Mn–Si and
166
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
Al–Rh–Si 2/1–2/1–2/1 approximants is that the vertices of the largest cluster shell are fully occupied by Rh atoms in the Al–Rh–Si 2/1–2/1–2/1 approximant, while the presence of this particular atomic site was not found in the Al–Pd–Mn–Si 2/1–2/1–2/1 approximant. Now we discuss the role of the Rh atoms in the quasicrystals and approximants. Previous structure analyses on the 1/1–1/1–1/1 approximants led us to conclude that half-filled transition metal elements such as Fe, Ru, Mn, and Re do not share the atomic sites with any other elements, while the late-transition metal element Cu coexists together with simple metal elements (Al and Si) in many atomic sites [2,4–6]. In the present Al–Rh–Si 2/1–2/1–2/1 approximant, some atomic sites are exclusively occupied by the Rh atoms, while (Al, Si) and Rh coexist together in some other atomic sites. Rh atoms in this approximant behave partly as the halffilled transition metal and partly as Cu. This is most likely because Rh lies in a column between the half-filled transition metal elements and Cu in the periodic table. It is important to claim here again that the chemical disordering between Rh and (Al, Si) takes places in many atomic sites in the Al–Rh–Si 2/1–2/1–2/1 approximant. Structure disordering was also observed in the resulting parameters. The structures of the Mackay-type 1/1–1/1– 1/1 approximants were characterized by the heavily introduced chemical and structure disordering [2,4–6]. Therefore, the presence of the disordering should be a common feature in the Mackay-type rational approximants and perhaps in their corresponding quasicrystal.
5. Conclusion As a result of the present synchrotron Rietveld analysis, the structure of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant was successfully determined. The structure of the 2/1–2/1–2/1 approximant is characterized by the CsCl-type structure with two icosahedral clusters centered at body center and vertices. Chemical and structure disordering exist in the clusters as in the 1/1–1/1–1/1 approximants. We conclude that the presence of the disordering is a common feature among the rational approximants and the corresponding quasicrystal.
References [1] V. Elser, C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [2] T. Takeuchi, H. Yamada, M. Takata, T. Nakata, N. Tanaka, U. Mizutani, Mater. Sci. Eng. 294–296 (2000) 340. [3] U. Mizutani, T. Takeuchi, E. Banno, V. Fournee, M. Takata, H. Sato, Mater. Res. Soc. Symp. Proc. 643 (2001) K13.1.1. [4] U. Mizutani, W. Iwakami, T. Takeuchi, M. Sakata, M. Takata, Philos. Mag. Lett. 5 (1997) 34. [5] T. Takeuchi, U. Mizutani, J. Alloy. Compd. 342 (2002) 416. [6] T. Takeuchi, E. Banno, T. Onogi, T. Mizuno, T. Sato, V. Founrnee, U. Mizutani, Mater. Res. Soc. Symp. Proc. 643 (2001) K13.4.1. [7] F. Izumi, T. Ikeda, Mater. Sci. Forum 321–324 (2000) 198. [8] K. Sugiyama, N. Kaji, K. Hiraga, T. Ishimasa, Z. Kristallogr. 213 (1998) 168. [9] A. Katz, D. Gratias, J. Non-Cryst. Solids 153&154 (1993) 187. [10] M. Cooper, K. Robinson, Acta Crystallogr. 20 (1996) 614.
Structure of the Al65Rh27Si8 2/1–2/1–2/1 approximant T. Takeuchi a
a,*
, N. Koshikawa b, E. Abe c, K. Kato d, U. Mizutani
a
Department of Crystalline Materials Science, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan b National Space Development Agency of Japan, Tsukuba 305-8505, Japan c Materials Engineering Laboratory, National Institute of Materials Science, Tsukuba 305-0047, Japan d Japan Synchrotron Radiation Research Institute, Sayo-gun, Hyogo 679-5198, Japan
Abstract Atomic arrangements in the thermodynamically stable Al65 Rh27 Si8 2/1–2/1–2/1 approximant were determined by means of synchrotron radiation Rietveld analysis. It was found that the Al65 Rh27 Si8 2/1–2/1–2/1 approximant possesses a cubic unit cell containing 564 atoms. We found that the atomic structure of this approximant is characterized by a CsCl-type structure with a large and a small icosahedral cluster of 6.8 A in diameter. The presence of chemical and structure icosahedral atomic cluster of 19.76 A disordering between (Al, Si) and Rh was also found in the structure as well as in the Mackay-type 1/1–1/1–1/1 approximants. Ó 2004 Elsevier B.V. All rights reserved. PACS: 61.44.Br; 61.10.Eq; 61.14.)x
1. Introduction The structure of icosahedral quasicrystals is generally understood as that appearing in a three-dimensional hyperplane sliced from a six-dimensional periodic lattice decorated with atomic surfaces. This three-dimensional hyperplane in which the structure of quasicrystal is constructed is called the Ôphysical space’. The slope of the physical space in the six-dimensional space is characterized by a setpof three irrational numbers ð s s s Þ, where s ¼ ð1 þ 5Þ=2 is known as the golden ratio. A quasiperiodic structure with icosahedral symmetry is consequently constructed by this slope of the irrational number s. Similarly, the structures of crystalline approximants are produced by the six-dimensional periodic lattice with a physical space slightly tilted from that of the quasicrystals [1]. The slope of the physical space for an approximant turns out to have ð p1 =q1 p2 =q2 p3 =q3 Þ instead of ð s s s Þ, where pn =qn (n ¼ 1, 2, 3) is continued-fraction approximant to s (pn =qn ¼ 1=0; 1=1; 2=1; 3=2; 5=3; . . . ; s). This rational slope leads to periodicity rather than quasiperiodicity in the resulting structure. *
Corresponding author. Tel.: +81-52 789 5620; fax: +81-52 789 3720/3724. E-mail address: [email protected] (T. Takeuchi). 0022-3093/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2003.11.032
Closely related slopes lead essentially to the same local atomic arrangements in quasicrystals and approximants. Thus the approximants can be regarded as crystalline materials possessing the same local atomic arrangements of the quasicrystals. In order to investigate the structure of the Mackaytype quasicrystals and approximants, we performed in the last decade the structure analysis for the Mackaytype 1/1–1/1–1/1 approximants [2–5]. The periodic structure of the approximants allowed us to make full use of the well-developed Rietveld analysis. We found, as a result of structure analysis, that these 1/1–1/1–1/1 approximants are characterized by the chemical disordering among simple metal elements and late-transition metal elements, and that this chemical disordering has a strong influence on the electron transport properties [5,6]. Although the presence of the disordering in the local atomic structure of the 1/1–1/1–1/1 approximants and its effect on the transport properties were clearly revealed in our previous work, it is still uncertain whether these features are commonly adapted for all quasicrystals and approximants. A structure analysis of the higher-order approximants would provide us a hint on this point. Thus, in this work, we analyzed the structure of the Mackay-type Al65 Rh27 Si8 2/1–2/1–2/1 approximant by performing the Rietveld analysis on the synchrotron radiation powder diffraction spectrum.
162
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
2. Experimental Al65 Rh27 Si8 alloy ingots were prepared by melting pure elements of Al (purity 99.99%), Rh (99.99%), and Si (99.999%) in an arc-furnace under a pressurized Ar gas atmosphere. Mother ingots were annealed at 700 °C for 24 h and crushed into fine powders of less than 5 lm in mean diameter. These small grains play an essential role in preventing the preferred orientation of the powders that is unfavorable for the precise powder diffraction measurement. Indeed absence of the diffraction spots in the Debye–Scherrer ring was confirmed in the Laue pictures of the powder samples stuffed in a quartz capillary of 0.2 mm in diameter. Synchrotron radiation diffraction spectrum was accumulated with a large Debye–Scherrer camera placed at the beam line BL02B2 in SPring-8, Hyogo, Japan. We employed an incident beam with a wave length of 0.7 A, an imaging plate as the detector, and 90 K nitrogen gas blowing on the quartz capillary with the powdered samples stuffed in it. A Rietveld analysis was performed on the measured synchrotron radiation diffraction spectrum with the program RIETAN2000 developed by Izumi [7]. Since the difference in the number of electrons between Al and Si is less than 10% and both numbers are much smaller than that of Re, it is very difficult to distinguish Si atoms from Al atoms in the Al–Rh–Si 2/1–2/ 1–2/1 approximant, having a complicated structure. Thus all Si atoms in this approximant were regarded as Al in the present Rietveld analysis. We used an overall isotropic atomic displacement parameter Biso: in the present Rietveld analysis instead of assigning an independent atomic displacement parameter to each atomic site to reduce the number of fitting parameters.
3. Results Fig. 1(a)–(c) show electron diffraction patterns of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant measured along
[1 0 0], [1 1 0], and [1 1 1] directions. The symmetry of the patterns along [1 0 0], [1 1 0], and [1 1 1] directions are 2mm, 2mm, and 6, respectively. All diffraction spots were indexed with those of a simple cubic lattice. These features of the diffraction patterns indicate that the 3 or space group of the Al65 Rh27 Si8 2/1–2/1–2/1 is Pm P 23. If an icosahedral cluster of m35 symmetry is located at the center of the cubic cell, the lattice turns out to have Pm3 symmetry. Indeed, many 1/1–1/1–1/1 approximants and the Al–Pd–Mn–Si 2/1–2/1–2/1 approximants were reported to have the Pm3 symmetry [2–5,8]. Thus, we assumed that the Al65 Rh27 Si8 2/1–2/1– 2/1 approximant has a cubic lattice with the Pm 3 symmetry. Before performing the Rietveld analysis, we constructed a structure model of the Al65 Rh27 Si8 2/1–2/1–2/ 1 approximant with Pm3 by means of the high-dimensional cut method with a simplified atomic surfaces on the basis of the Katz–Gratias model [9]. The numbers of atomic sites in the initial model were modified (removed or newly added) during the structure refinement, and the resulting structure was fairly different from the initial one. Thus, we avoid making any detailed comments on the initial model in order to save the limited space and concentrate to discuss the obtained structure. Fig. 2 shows the diffraction spectrum calculated using the refined parameters together with the measured one. Five hundred and two independent diffraction peaks were employed for the determination of the structure parameters. The resulting parameters and the reliable factors (R-factors) are listed in Table 1. The calculated and measured spectra show an extremely good agreement, and the resulting composition became almost the same as the nominal one. The reliable factors for the weighted pattern (Rwp ) and the structure (RI ) were sufficiently reduced to be ðRwp ; RI Þ ¼ ð3:21%; 1:32%Þ. All these results strongly indicate that the atomic structure in the Al65 Rh27 Si8 2/1–2/1–2/1 approximant was successfully determined by the present Rietveld analysis. Least-square parameter fitting methods such as the Rietveld analysis sometimes lead to wrong results be-
Fig. 1. Electron diffraction patterns measured along the [1 0 0], [1 1 0], and [1 1 1] directions possessing 2mm, 2mm, and 6 symmetries, respectively. All diffraction spots can be indexed with those of a simple cubic lattice. The space groups possessing these features are limited to Pm3 or P 23. With considering possible symmetry for the approximant of icosahedral quasicrystals, we determined the space group of this sample to be Pm3.
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
163
Fig. 2. X-ray diffraction spectrum measured by the synchrotron radiation together with the calculated spectrum as a result of the Rietveld analysis. in wave length. Diffraction spectrum was accumulated at the beam line BL02B2 in SPring-8, Hyogo, Japan, with a incident radiation of 0.7 A
cause of the presence of local minima in the meansquare error. In order to avoid such wrong results, we observed atomic arrangements in the real space by using the HAADF-STEM technique. The HAADF-STEM is powerful tool to determine the projected position of the heavy atoms because the contribution of elements on the HAADF-STEM images is proportional to the square of their atomic numbers. Measured HAADF-STEM image along [1 0 0] direction and that calculated with the refined parameters are shown in Fig. 3(a) and (b), respectively. The bright portions in the images correspond to the projected position of Rh atoms. The dark areas, on the other hand, indicated the projected positions of Al and/or Si or vacancies, which are not individually imaged. Obviously, the calculated image reproduces very well the observed one, lending a strong support to the validity of the refined structure parameters.
4. Discussion The high-dimensional method, with which the structures of quasicrystal and approximants are constructed, requires that the lattice constant of the p=q–p=q–p=q cubic approximant pffiffiffiffiffiffiffiffiffiffiffi satisfies the condition of ap=q–p=q–p=q ¼ 2aR ðps þ qÞ 2 þ s, where aR indicates the quasi-lattice constant [1]. One may notice that the lattice constant of the 2/1–2/1–2/1 approximant can be evaluated from that of the corresponding 1/1–1/1–1/1 approximant by transforming the above formula to a2=1–2=1–2=1 ¼ s a1=1–1=1–1=1 . The lattice constant of the Al65 Rh27 Si8 2/ that 1–2/1–2/1 approximant is determined to be 19.76 A, is indeed s times larger than 12.3–12.7 A of the 1/1–1/1–1/ 1 approximant [2,3,5,6,10]. If the local structure is kept unchanged between these two kinds of approximant, the number of atoms in the unit cell of the 2/1–2/1–2/1 approximant (N2=1–2=1–2=1 ) should be evaluated by using
that of the 1/1–1/1–1/1 approximant as N2=1–2=1–2=1 ¼ N1=1–1=1–1=1 s3 . The number of atoms in the unit cell of the 2/1–2/1–2/1 approximant obtained as a result of the Rietveld analysis was N2=1–2=1–2=1 ¼ 563. This number lies in the vicinity of ð135–144Þ s3 ¼ 571–609, where 135– 144 means the number of atoms in the Mackay-type 1/1– 1/1–/1/1 approximant [2,3,5,6,10]. It is found that a large icosahedral cluster with quintuple stacked atomic shells exists in the cubic unit cell of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant. The cluster in the unit cell is shown in Fig. 4. The 1st shell, the 2nd shell, the 3rd shell, the 4th shell, and the 5th shell consist of 14, 24, 92, 132, and 212 atoms, respectively. All atomic shells except for the 1st one has the icosahedral symmetry of m35. Note here that the symmetry of each icosahedral atomic shell is always distorted because the periodic lattice never has fivefold symmetry. The distortion in the icosahedral cluster can be evaluated by using edge lengths in the icosahedrons. If perfect icosahedral symmetry is present, all edge lengths are identical. However, once the icosahedral cluster is distorted within the restriction of the m 3 symmetry in the approximants, it turns out to have two different edge lengths, r1 and r2 . We calculated the r1 =r2 for all icosahedral shells using the refined structure parameters. Surprisingly, all r1 =r2 ratios calculated for the 2nd, 3rd, 4th, and 5th shells turned out to be 0.99. Thus, it is strongly argued that the icosahedral order almost perfectly persists in this large icosahedral cluster. A small icosahedral cluster with triply stacked atomic shells also exists with its center at the vertices of the cubic unit cell. The second and the third shell in this cluster have the icosahedral symmetry. In sharp contrast to the large icosahedral cluster, this small icosahedral cluster at the vertices is fairly distorted. That was confirmed by the r1 =r2 ratio described above; those for the second and third shells are 0.90 and 1.15, respectively.
164
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
Table 1 Refined structure parameters of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant Sites
Wickoff notation
Occupancy
x
y
z
Sites constructing the body center cluster First shell M00:Al M01:Al/Rh
6e 8i
1.0 0.71/0.28 (2)
0.0 0.0837 (5)
0.0 0.0837
0.081 (1) 0.0837
Second shell M10:Al a M11:Al/Rh
12j 12j
0.70 (3) 0.83/0.17 (1)
0.070 (2) 0.0
0.0 0.1000 (8)
0.188 (2) 0.1636 (7)
Third shell M20:Al M21:Al M22:Al M23:Rh M24:Rh a M25:Al/Rh
24l 12j 24l 12j 8i 12j
1.0 1.0 1.0 1.0 1.0 0.17/0.83 (1)
0.1363 0.0 0.0654 0.1146 0.1934 0.0
0.2222 (7) 0.069 (1) 0.2402 (8) 0.00000 0.1934 0.1853 (4)
0.1085 (7) 0.261 (1) 0.2142 (7) 0.3127 (3) 0.1934 0.2960 (3)
Fourth shell M40:Al M41:Al M42:Al M43:Al M44:Al M45:Al a M46:Al
12j 24l 24l 24l 24l 12j 12j
1.0 1.0 1.0 1.0 1.0 1.0 0.80 (3)
0.283 (1) 0.1283 (9) 0.2409 (8) 0.0770 (8) 0.1950 (9) 0.079 (1) 0.0
0.0 0.3874 (8) 0.3436 (8) 0.3834 (8) 0.3075 (9) 0.0 0.248 (2)
0.315 (1) 0.0730 (8) 0.1035 (8) 0.1992 (8) 0.2383 (9) 0.430 (1) 0.404 (2)
Fifth shell M50:Rh M51:Al/Rh M52:Rh a M53:Rh M54:Rh M55:Al/Rh M56:Al/Rh M57:Al M58:Al/Rh M59:Al/Rh M60:Al M61:Al M62:Al/Rh
6g 24l 24l 6g 6f 8i 24l 24l 24l 12j 12k 24l 12k
1.0 0.16/0.84 (1) 1.0 1.0 1.0 0.38/0.62 (2) 0.924/0.076 (8) 0.72 (2) 0.892/0.108 (9) 0.81/0.19 (1) 1.0 0.61 (2) 0.82/0.18 (1)
0.0 0.1869 (3) 0.3800 (2) 0.0 0.0 0.3106 (4) 0.0653 (6) 0.266 (1) 0.3128 (7) 0.0 0.128 (1) 0.418 (1) 0.1211 (7)
0.0860 (5) 0.1926 (3) 0.3116 (2) 0.3052 (5) 0.5 0.3106 0.3782 (7) 0.432 (1) 0.3835 (8) 0.4303 (8) 0.096 (1) 0.300 (1) 0.5
0.5 0.4250 (3) 0.1196 (2) 0.5 0.1950 (5) 0.3106 0.4305 (8) 0.139 (1) 0.1932 (7) 0.310 (1) 0.5 0.231 (1) 0.2662 (8)
Glue sites G01:Al/Rh G02:Rh G03:Al/Rh G05:Al
12k 6h 6g 12k
0.89/0.11 (1) 1.0 0.79/0.21 (2) 0.27 (3)
0.0593 (8) 0.5 0.0 0.066 (4)
0.5 0.5 0.427 (1) 0.207 (4)
0.4002 (8) 0.1084 (5) 0.5 0.5
(7) (7) (3) (3)
Sites constructing the vertex cluster
Center atom N00:Rh
1b
1.0
0.5
0.5
0.5
First shell N10:Al/Rh N11:Al/Rh
6h 8i
0.80/0.20 (2) 0.82/0.18 (2)
0.355 (1) 0.4268 (6)
0.5 0.4268
0.5 0.4268
Second shell N20:Al/Rh a N21:Rh N22:Al
24l 12k 6h
0.855/0.115 (9) 1.0 0.60 (5)
0.4243 (6) 0.2981 (3) 0.5
0.3131 (6) 0.5 0.5
0.3760 (6) 0.3882 (4) 0.275 (3)
Third shell N30:Al a N31:Rh N32:Al/Rh
12k 12k 12k
1.0 1.0 0.72/0.28 (1)
0.435 (1) 0.3080 (3) 0.3869 (6)
0.5 0.5 0.1559 (6)
0.172 (1) 0.2374 (3) 0.5
2 , Rwp ¼ 3:21%, S ¼ 4:6178, Nominal composition: Al65 Si8 Rh27 , Resulting composition: (Al,Si)72:8 Rh27:2 , Space group: Pm3, a ¼ 12:7628ð1Þ, Biso: ¼ 0:58 A RI ¼ 1:32%, RF ¼ 0:61%. a Atomic sites at the vertices of the icosahedral clusters.
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
165
Fig. 3. (a) HAADF-STEM image measured along [1 0 0] direction and (b) simulated image calculated by using the Rietveld refined structure parameters. The simulated image reproduces well the measured one. The largest icosahedral cluster existing in the unit cell is drawn by the white lines in the calculated image.
75 -100% Rh 50 - 75% Rh 25 - 50% Rh 0 - 25% Rh
Fig. 4. Atomic clusters exist at the body center (top panel) and vertices (bottom panel) in the unit cell of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant. Atomic arrangements on the triangular faces of each icosahedral atomic shell are shown below each atomic shell. Open circles and filled circles indicate (Al, Si) and Rh atoms, respectively. The atomic sites in which Al and Re coexist together are expressed as gray circles. The depth of the gray color indicates the Re occupancy in each mixed site; the darker the color is, the more the Re exist. The triangles decorated by circles indicate the atom arrangements on the triangular faces of the icosahedral cluster shells. Note that the seven atoms on the faces along the [1 1 1] direction in the fifth shell of the larger cluster centered at the body center are shared by the third shell in the small cluster centered at the vertices. Seven atoms constructing hexagons in the triangular surfaces of the outermost shells in the large cluster at body center of the unit cell are shared with the outermost shells in the small icosahedral clusters at vertices.
Note here that these large and small icosahedral clusters construct a CsCl-type structure with some glue atoms between them. The structure of the 2/1–2/1–2/1 approximant was first determined for the Al–Pd–Mn–Si 2/1–2/1–2/1 ap-
proximant by Sugiyama et al. [8]. Although the detailed structures are different, the clusters in the Al–Rh–Si 2/ 1–2/1–2/1 approximant are similar to those in the Al– Pd–Mn–Si 2/1–2/1–2/1 approximant. One of the most pronounced differences between the Al–Pd–Mn–Si and
166
T. Takeuchi et al. / Journal of Non-Crystalline Solids 334&335 (2004) 161–166
Al–Rh–Si 2/1–2/1–2/1 approximants is that the vertices of the largest cluster shell are fully occupied by Rh atoms in the Al–Rh–Si 2/1–2/1–2/1 approximant, while the presence of this particular atomic site was not found in the Al–Pd–Mn–Si 2/1–2/1–2/1 approximant. Now we discuss the role of the Rh atoms in the quasicrystals and approximants. Previous structure analyses on the 1/1–1/1–1/1 approximants led us to conclude that half-filled transition metal elements such as Fe, Ru, Mn, and Re do not share the atomic sites with any other elements, while the late-transition metal element Cu coexists together with simple metal elements (Al and Si) in many atomic sites [2,4–6]. In the present Al–Rh–Si 2/1–2/1–2/1 approximant, some atomic sites are exclusively occupied by the Rh atoms, while (Al, Si) and Rh coexist together in some other atomic sites. Rh atoms in this approximant behave partly as the halffilled transition metal and partly as Cu. This is most likely because Rh lies in a column between the half-filled transition metal elements and Cu in the periodic table. It is important to claim here again that the chemical disordering between Rh and (Al, Si) takes places in many atomic sites in the Al–Rh–Si 2/1–2/1–2/1 approximant. Structure disordering was also observed in the resulting parameters. The structures of the Mackay-type 1/1–1/1– 1/1 approximants were characterized by the heavily introduced chemical and structure disordering [2,4–6]. Therefore, the presence of the disordering should be a common feature in the Mackay-type rational approximants and perhaps in their corresponding quasicrystal.
5. Conclusion As a result of the present synchrotron Rietveld analysis, the structure of the Al65 Rh27 Si8 2/1–2/1–2/1 approximant was successfully determined. The structure of the 2/1–2/1–2/1 approximant is characterized by the CsCl-type structure with two icosahedral clusters centered at body center and vertices. Chemical and structure disordering exist in the clusters as in the 1/1–1/1–1/1 approximants. We conclude that the presence of the disordering is a common feature among the rational approximants and the corresponding quasicrystal.
References [1] V. Elser, C.L. Henley, Phys. Rev. Lett. 55 (1985) 2883. [2] T. Takeuchi, H. Yamada, M. Takata, T. Nakata, N. Tanaka, U. Mizutani, Mater. Sci. Eng. 294–296 (2000) 340. [3] U. Mizutani, T. Takeuchi, E. Banno, V. Fournee, M. Takata, H. Sato, Mater. Res. Soc. Symp. Proc. 643 (2001) K13.1.1. [4] U. Mizutani, W. Iwakami, T. Takeuchi, M. Sakata, M. Takata, Philos. Mag. Lett. 5 (1997) 34. [5] T. Takeuchi, U. Mizutani, J. Alloy. Compd. 342 (2002) 416. [6] T. Takeuchi, E. Banno, T. Onogi, T. Mizuno, T. Sato, V. Founrnee, U. Mizutani, Mater. Res. Soc. Symp. Proc. 643 (2001) K13.4.1. [7] F. Izumi, T. Ikeda, Mater. Sci. Forum 321–324 (2000) 198. [8] K. Sugiyama, N. Kaji, K. Hiraga, T. Ishimasa, Z. Kristallogr. 213 (1998) 168. [9] A. Katz, D. Gratias, J. Non-Cryst. Solids 153&154 (1993) 187. [10] M. Cooper, K. Robinson, Acta Crystallogr. 20 (1996) 614.