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Structure of the AlRhCu decagonal quasicrystal: I. A unit-cell approach
Physica B 240 (1997) 330-337
ELSEVIER
Structure of the A1-Rh-Cu decagonal quasicrystal: I. A unit-cell approach X.Z. Li*, K. Hiraga, K. Yubuta Institute./'or Materials Research, Tohoku UniversiO', Katahira, Aoba-ku, Sendai 980, Japan Received 12 May 1997
Abstract
Structure of the A1-Rh Cu decagonal quasicrystal has been studied by high-resolution electron microscopy. The high-resolution structure image shows an aperiodic tiling composed of three kinds of subunits, namely flattened hexagon, crown and five star. Therefore, a structural model of the A1-Rh-Cu decagonal quasicrystal has been constructed in a unit-cell approach, in which the atom arrangements in the subunits have been proposed. It is known that the phase has two layers in a period of 0.4 nm along the unique tenfold axis according to the previous study by electron diffraction method. The ideal model of the A1-Rh-Cu decagonal quasicrystal is proposed as periodic stacking of the layers with quasiperiodic tessellation of the three kinds of subunits, in each layer the two-colour Penrose tiling is obtained if different atom decorations for the same shape subunits are distinguished by white and black colours. Calculated images reproduces well the contrast features of the observed images, which means that the present model is basically correct. Structural relationship between the A1-Rh Cu decagonal quasicrystal and the previously reported AI-N%Co decagonal quasicrytsal, which has also a period of 0.4 nm, has also been discussed.
PACS: 61.44; 61.16.B Keywords: A1-Rh-Cu alloy; Decagonal quasicrystal; HREM
1. Introduction
Decagonal phase is a kind of polygonal quasicrystal, which shows a periodic translation order in a unique tenfold axis and quasiperiodic lattice in
* Corresponding address: Centre for Materials Science, Faculty of Mathematics and Natural Sciences, University of Oslo, Gaustadalleen 21, N-0371 Oslo, Norway. Tel.: + 472295 87 32; fax: +4722958749.
the perpendicular plane in reciprocal space. The decagonal phase (or decagonal quasicrystal, briefly DQC) was firstly observed in a rapidly solidified A1-Mn alloy, which has period of 1.2 nm along the unique axis [1]. Later the D Q C s with different periods were also observed, e.g. the A1-Ni D Q C with a period of 0.4 nm I-2] and the A1-Pd D Q C with a period of 1.6 nm [3]. The structures of the D Q C s have been extensively studied, especially after the thermodynamically stable D Q C s were found [4-6]. High-resolution electron microscopy
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and single-crystal X-ray analysis have been successfully used to reveal the structures of the DQCs at atomic level, e.g., on the basis of the high-resolution electron micrographs of the A1-Ni-Co DQC, the atom arrangement in columnar cluster with 2.0 nm in diameter has been proposed [7], which agrees quite well with the result of the single-crystal X-ray analysis [8]. The two methods have their own advantages in the study of the structures of the DQCs. Though atom arrangement in the cluster obtained from the single-crystal X-ray analysis is more preciser than that from the high-resolution electron microscopy. However, the single-crystal X-ray analysis of the DQCs coupled with a high-dimensional approach always provide only the average structures of the DQCs with quasiperiodic lattice. On the other hand, the high-resolution electron micrographs provide the real arrangement of atom clusters. The latter point becomes more important in distinguishing the DQCs with closely structural relationship. For instance, the single-crystal X-ray analyses of the A1-Mn and AI-Mn Pd DQCs show that the two phases have similar structure I-9, 10]. However, the high-resolution electron microscopic studies show clearly that the structures of the two DQCs are different [11, 12]. It has been found that the local configuration of atom arrangement is similar in the two structures and the result of the single-crystal X-ray analysis is better for the structure of the AI-Mn-Pd DQC than the structure of the A1-Mn DQC. A new structural model of the AI Mn DQC was then proposed by combining the results of high-resolution electron microscopy and single-crystal X-ray analysis [13, 14]. The coexistence of a DQC and crystalline phases in a conventional solidified A165Rh20Cu15 alloy was early reported in 1989 [15]. Recently the A1Rh-Cu DQC was confirmed in A163Rhls.sCu18.5 alloy by electron diffraction method. The structure of the A1-Rh Cu DQC has been studied the highresolution electron microscopy and the preliminary result has been briefly reported in Ref. [16]. The A1-Rh-Cu DQC has a periodicity of about 0.4 nm along its unique tenfold axis. However, the phase is different from the previously reported DQCs with a period of 0.4 nm, e.g. the A1-Ni-Co DQC since large decagonal subunits have been found in the structure of the AI Ni-Co DQC but not in the
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structure of the A1-Rh-Cu DQC. In the present paper, we propose a structural model of the AI-Rh-Cu DQC in detail on the basis of the highresolution electron microscopic study. First, we derive the structural model in a unit-cell approach in this paper as part I and then describe the structure in a high-dimensional approach in a separated paper as part II. The content of this paper is organized as follows. In Section 2, we analyze the arrangement of atom clusters in the structural image of the A1-Rh-Cu DQC and show the resemblance of the image contrasts of the atom clusters in structural images of the A1-Rh-Cu and the A1-Ni-Co DQCs. In Section 3, atom arrangements in the subunits of the A1-Rh-Co DQC are proposed and checked by image simulation. The divergence of the structural images of the A1-Rh-Cu and the A1-Co-Ni DQCs has been discussed in Section 4 and conclusion is given in Section 5.
2. High-resolution electron microscopy Fig. 1 shows (a) the unique tenfold electron diffraction pattern and (b) the corresponding highresolution electron micrograph of the A1-Rh-Cu DQC. The distribution of the sharp diffraction spots in the diffraction pattern indicates that the sample under examination is a high-quality quasicrystal. From Fresnel fringe at the edge of an amorphous contamination layer adjacent to the observed region, it was confirmed that the electron micrograph was taken under nearly Scherzer defocus, Af= - 4 5 nm. The electron micrograph in a rather thin area (less than 10 nm) can be considered as a so-called high-resolution structure image, which faithfully reflects projected potentials and thus is useful in the structure modelling. Such an area has been marked with rectangle and enlarged in Fig. 2(a). In the structure image, typical image contrast can be noticed as five bright spots, in a more or less triangular shape, forming a pentagon with a ring contrast in the centre. For illustration, some selected pentagons have been outlined with thin lines. Three kinds of local aggregated forms of the pentagonal motifs have been observed and they are labelled as a flattened hexagon (H), a crown (C) and a fivefold star (S) in Fig. 2(b)
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(a)
>I<>r~., <.,
Fig. I. (a) The tenfold electron diffraction pattern and (b) the corresponding high-resolution electron micrograph of the AIRh Cu DQC. An area marked with rectangle is believed to be a high-resolution structure image and is enlarged in Fig. 2(a).
according to the shapes of the tiles (edge length of about 0.65 nm) formed by connecting the centres of the pentagonal motifs with lines. These tiles are considered as the structure subunits of the AIR h - C u DQC. It should be pointed out that the formation of the tiles sometime became ambiguous to a certain extent because the two-ring constrasts side by side in some parts of the image have been obsevered, e.g., such a pair of ring constrasts has been marked by arrow in Fig. 2(a). In this case, we just choose one of them without thinking of the priority since both choices can fit to other tiles in the images, as shown in Fig. 2(c) in dash lines. We should discuss the reason for this phenomenon in next section. The structure image is covered by a tessellation of these subunits without any gaps. Such a tiling corresponding to the enlarged part of the structure image is given in Fig. 2(c). Thus, the determination of atom arrangement in the three
(b)
(c)
Fig. 2. (a) Structure image of the AI-Rh Cu DQC; (b) three kinds of subunits deduced from the structure image; (c) an aperiodic tessellation of the subunits according to the structure image. For details, see text.
kinds of subunits leads to a solution for the structure of the A1 Rh Cu DQC. The ring contrast, surrounded by five triangles, is believed relating to the columnar shape of atom cluster in the structure of the A1-Rh-Cu DQC. Similar image contrast associated with the columnar cluster can be found out from the structural image of the A1 N i - C o DQC, see Fig. 9. Therefore, atom arrangement in the columnar cluster can be directly derived from the structural models of the Al Ni Co DQC. The structure of the A I - N i - C o D Q C has two layers in the 0.4 nm period along the unique axis and the atom arrangements in the two layers are related by a 105 screw axis. Fig. 3 shows the large cluster of the A1-Ni-Co DQC, (a) and (b) reproduced from a structural model based on a high-resolution electron microscopic study
)dZLietal./PhysicaB240(1997)330-337
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centred atoms in shadowed and equivalent areas are exchanged in the two models. The atom arrangement in the shadowed area corresponds the ring contrast in the high-resolution structure images. Therefore, the basic columnar cluster can be derived in order to construct the structural model of the A1 R h - C u DQC. Fig. 4 shows the columnar cluster in two adjacent layers in the structure of the A1-Rh-Cu DQC, for convenience, hereafter named as layer A and layer a. In Fig. 4, only the atom positions have been pointed out and the species of the atoms will be discussed in next section.
~
)
3. Structural model (d)
Fig. 3. Atom cluster of the A1-.Ni-Co DQC in two adjacent layers, (a) and (b) reproduced from a structural model based on a high-resolution electron microscopic study [7]; (c) and (d) reproduced from a structural model based on a single-crystal X-ray analysis [8].
Layer A
~
Layera
C~
Layer A
~)
(~ C~
Projection Fig. 4. Columnar shape of atom cluster of the A1 Rh Cu DQC and atom positions in two adjacent layers, layer A and layer a.
[7]; (c) and (d) reproduced from a structural model based on a single-crystal X-ray analysis [8]. The two results resemble quite well although the different tilings are shown together with the atom arrangements, a rhombic Penrose tiling in Fig. 3(a) and Fig. 3(b) while in atoms were connected referring to partial structure of the Alx3Co 4 phase in Fig. 3(c) and Fig. 3(d). The only divergence between them is that the positions and species of the
Atom arrangements in the subunits of the AI Rh Cu D Q C can be obtained by using the basic columnar atom cluster and the following construction rules, (i) the distribution of atoms in each layer forms three kinds of local configurations as the hierarchical shapes of the subunits; (ii) the distribution of heavy atoms (Rh, Cu) are arranged in such a way that five of them form a pentagon with an edge length of 0.47 nm and with five or three A1 atoms inside each pentagon. The rules can be viewed as a summary from the atom arrangement in the large a t o m cluster of the A1-Ni Co D Q C (see Fig. 3). Fig. 5 shows the atom arrangements in the three kinds of subunits of the A1-Rh-Cu D Q C in two adjacent layers as a pair, the flattened hexagon in (a) and (e), the crown in (b) and (f), and the fivefold star in (c) and (g). The open circles indicate the A1 atoms, the filled circles the Rh/Cu atoms and the double circles the co-occupancy of A1 and Rh/Cu atoms. Positions marked with points indicate atom sites with partial occupancy and one set of possible occupation is applied in Fig. 5 to show the local atom configuration. Basic columnar atom clusters are important parts of these subunits, they situate in the vertices of the subunits, therefore there are six columnar clusters in the H subunit, eight in the C subunit and ten in the S subunit. A large circle is used to highlight one of the basic columnar clusters in the subunits. The formation of these subunits is also understood as the closepacking of the basic columnar clusters with fivefold symmetry in two-dimensional plane for the
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)LZ Li et al. / Physica B 240 (1997) 330 337
(a)
(e)
(b)
(t)
(c)
(d)
(g)
(h)
Fig. 5. The atom arrangements in the three kinds of subunits of the Al Rh-Cu DQC in two adjacent layers as a pair, the flattened hexagon in (a) and (e), the crown in (b) and (f), and the fivefold star in {c) and (g). In addition, the atom arrangement in a decagonal subunit is shown in two adjacent layers as a pair in (dt and (h}.
S subunit and as parts of the S subunit for the C and H subunits. A decagonal shape of subunit shown in Fig. 5(d) and (h) is obtained as another close-packing type, which is resemblance to that shown in Fig. 3(c) and (d), hereafter referred as the D subunit. The D subunit has not been observed in the structure of the A1-Rh-Cu DQC, but it does exist in the structure of the A1 Ni Co DQC. This shows that the two phases are different types of DQCs, more discussions are given in Section 4. The basic columnar atom clusters appear in the edges of the subunits play a role as a "matching rule" for packing these subunits in two-dimensional plane. The proposed model has been checked by image simulation. In order to carry out the image simulation, hypothetical periodic structure composed of the subunits has been used. Fig. 6 shows a part of a periodic structure in (a) layer A and (b) layer a; (c) the calculated image. The calculation was carried out by using the MacTempas package under
a defocus value of - 45 nm and a sample thickness of 3 nm at an accelerating voltage of 400 kV and spherical aberration of 1.0 mm. Triangular and ring bright contrasts can be observed in the calculated image in Fig. 6(c). The calculated image, in our belief, reproduces well the image contrasts in the experimental one. The special image contrast mentioned in Section 2 can be explained by considering that the basic columnar clusters may shift locally in certain layers. Fig. 7 shows the tessellation of the subunits change due to the shift of the atom clusters, (a) layer A and (b) layer a without shifting; (c) layer A and (d) layer a with shifting. It is clear that the arrangement of the atoms only change slightly within a decagonal area. We suggest that such changes may happen in the structure of the A1-Rh-Cu DQC. Fig. 7(e) shows the calculated image of another hypothetical periodic model with superposing of the atom arrangement in Fig. 7(a)Fig. 7(d). The simulation shows that a pair of ring
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q
(a)
(c)
(b)
Fig. 6. Part of a hypothetical periodic structure in (a) layer A and (b} layer a; (c) the calculated image.
Layer A
(a)
(c)
Layer
a
(b)
(e)
(d)
Fig. 7. Part of a hypothetical periodic structure with the shifts of the atom clusters, (a) layer A and (b) layer a without shifting; (cl layer A and (d) layer a with shifting. (e) The calculated image.
contrast is reproduced in calculated image, as shown in Fig. 7(e) indicated by an arrow. The projected a t o m arrangements in each cases can be obtained by superposing the individual layers shown in Fig. 6(a) and (b) and Fig. 7(a)-(d), respectively, it is obvious that the a t o m positions cortes-
p o n d dark regions in the structure images. The agreement of experimental and calculated images shows that the proposed structural model is basically correct. As shown in Fig. 2, the real structure of the AI R h - C u D Q C is c o m p o s e d of three kinds of
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)dZ Li et al. / Physica B 240 (1997) 330 337
(a)
(b)
Fig. 8. Two-colour Penrose tilings in pair. subunits in an aperiodic tessellation. In fact, the three kinds of subunits are also allowed for quasiperiodic or periodic tessellation. The quasiperiodic tessellation of the three kinds of subunits yields the two-colour Penrose tiling if different atom decorations for the same shape subunits are distinguished by white and black colours. Thus the ideal model of the A1 R h - C u D Q C can be described as a structure of a periodic stacking of the two-colour Penrose tilings. The twocolour Penrose tiling in pair is shown in Fig. 8, which can be used to represent the two consecutive layers in the structure of the A1 R h - C u DQC. For more about the ideal model of the A1-Rh-Cu DQC, see part II of the present paper.
4. Comparison quasicrystal
with the AI Ni-Co
decagonal
Hiraga and his collaborators [7, 17] have systematically studied the structure of the A1-Ni-Co D Q C by the high-resolution electron microscopy. Aperiodic tessellations of decagonal clusters with 2.0 nm in diameter have been found in the structure image of the AI N i - C o D Q C and the atom arrangement in the decagonal cluster has been proposed, as shown in Fig. 2(a) and Fig. 2(b). Carefully examining the structure image, we found that the structure of the A I - N i - C o D Q C is composed of also the subunits in the shapes of flattened hexagon,
crown, and fivefold star besides mainly the decagonal subunit. Fig. 9 shows a structure image of the A1 N i - C o DQC. Decagonal subunits have been marked with the decagons in thick lines and the "true decagonal subunits ''~ have also been labelled with a letter D. On the other hand, two decagons are better explained as the combination of two H subunits and one C subunit than a single decagonal subunit. In the meantime, the S subunit has also been found in the structure image. This shows the structure of the A1 N i - C o D Q C in the observed sample is composed by mainly the decagonal subunits and also the subunits in the shape of the flattened hexagon, crown and fivefold star. Moreover, the special image contrasts with a pair of rings has also observed and marked with an arrow. By comparing the A1 Ni Co and the A1-Rh Cu DQCs, we found that (i) both of them are composed of the basic columnar cluster, which can be aggregated to form four kinds of subunits; (ii) the structure of the A1-Ni-Co D Q C is composed of mainly the D subunit and also the H, C and S subunits while the structure of the AI Rh Cu D Q C is composed of the H, C and S subunits. It has been found that the core part of the decagonal subunit of the AI- Ni Co DQC has fivefoldsymmetry according to the Ref. [18]. Here we use "true decagonal subunits" to describe some subunits with tenfold geometrical shape in order to distinguish to the others with decagonalshape, but actually composed of two H subunits and one C subunit.
X.Z Li et al. / Physica B 240 (1997) 330-337
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ture of the A1-Rh-Cu D Q C is different from that of the A1-Ni-Co DQC.
Acknowledgements This work was carried out as part of a research project, financially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan.
References Fig. 9. The structure image of the A1 Ni Co DQC. Some subunits have been outlined as the decagonal subunits (D), the flattened hexagonal subunits (H), the crown subunits (C) and the fivefold star subunit (S).
5. Concluding remarks The structure of the A1-Rh-Cu D Q C can be described as an aperiodic tiling composed of three kinds of subunits, namely flattened hexagon, crown and fivefold star according to the high-resolution electron microscopic study. The atom arrangements in the subunits have been proposed on the basis of the structure of the A1-Ni Co DQC. The image simulation shows that the calculated images reproduce well the contrast features of the observed images, which means that the present model is basically correct. Quasiperiodic tessellation of the three kinds of subunits generate the two-colour Penrose tiling if different atom decorations for the same shape subunits are distinguished by white and black colours. Thus, the ideal model of the A1 RhCu decagonal quasicrystal is proposed as periodic stacking of the two-colour Penrose tilings. By comparing the structure images, we show that the struc-
[1] [2] [3] [4]
L. Bendersky, Phys. Rev. Lett. 55 (1985) 1461. X.Z. Li, K.H. Kuo, Phil. Mag. Lett. 58 (1988) 167. P.A. Bancel, P.A. Heiney, J. Phys. Paris 47 (C3) (1986) 341. L.X. He, Z. Zhang, Y.K. Wu, K.H. Kuo, Proc. European Congr. of Electron Microscopy 1988, Institute of Physics Conf. Series No. 93, Bristol: Institute of Physics, p. 501. [5] A.P. Tsai, A. Inoue, T. Masumoto, Mater. Trans. JlM 30 (1989) 463. [6] C. Beeli, H.-U. Nissen, J. Robadey, Phil. Mag. Lett. 63 (1991) 87. [7] K. Hiraga, F.J. Lincoln, W. Sun, Mater. Trans. JIM 32 (1991) 308. [8] W. Steurer, T. Haibach, B. Zhang, Acta Crystallogr. B 49 (1993) 661. [9] W. Steurer, J. Phys.: Condens. Matter 3 (1991) 3397. [10] W. Steurer, T. Haibach, B. Zhang, C. Beeli, H.-U. Nissen, J. Phys.: Condens. Matter 6 (1994) 613. [11] M. Hirabayashi, K. Hiraga, Mater. Sci. Forum 22/24 (1987) 45. [12] K. Hiraga, W. Sun, Phil. Mag. Lett. 62 (1993) 117. [13] X.Z. Li, Acta Crystallogr. B 51 (1995) 265. [14] X.Z. Li, F. Frey, Acta Crystallogr. B 51 (1995) 271. [15] A.P. Tsai, A. Inoue, T. Masumoto, Mater. Trans. JIM 30 (1989) 463. [16] X.Z. Li, K. Hiraga, K. Yubuta, Phil. Mag. Lett. 74 11996) 247. [17] K. Hiraga, in: D.P. DiVincenzo, P. Steinhardt (Eds.), Quasicrystals: The State of the Art, World Scientific, Singapore, 1991, pp. 95-100. 1-18] K. Tsuda, Y. Nishida, K. Saitoh, M. Tanaka, A.P. Tsai, A. Inoue, T. Masumoto, Phil. Mag. A 74 (1996) 697.
ELSEVIER
Structure of the A1-Rh-Cu decagonal quasicrystal: I. A unit-cell approach X.Z. Li*, K. Hiraga, K. Yubuta Institute./'or Materials Research, Tohoku UniversiO', Katahira, Aoba-ku, Sendai 980, Japan Received 12 May 1997
Abstract
Structure of the A1-Rh Cu decagonal quasicrystal has been studied by high-resolution electron microscopy. The high-resolution structure image shows an aperiodic tiling composed of three kinds of subunits, namely flattened hexagon, crown and five star. Therefore, a structural model of the A1-Rh-Cu decagonal quasicrystal has been constructed in a unit-cell approach, in which the atom arrangements in the subunits have been proposed. It is known that the phase has two layers in a period of 0.4 nm along the unique tenfold axis according to the previous study by electron diffraction method. The ideal model of the A1-Rh-Cu decagonal quasicrystal is proposed as periodic stacking of the layers with quasiperiodic tessellation of the three kinds of subunits, in each layer the two-colour Penrose tiling is obtained if different atom decorations for the same shape subunits are distinguished by white and black colours. Calculated images reproduces well the contrast features of the observed images, which means that the present model is basically correct. Structural relationship between the A1-Rh Cu decagonal quasicrystal and the previously reported AI-N%Co decagonal quasicrytsal, which has also a period of 0.4 nm, has also been discussed.
PACS: 61.44; 61.16.B Keywords: A1-Rh-Cu alloy; Decagonal quasicrystal; HREM
1. Introduction
Decagonal phase is a kind of polygonal quasicrystal, which shows a periodic translation order in a unique tenfold axis and quasiperiodic lattice in
* Corresponding address: Centre for Materials Science, Faculty of Mathematics and Natural Sciences, University of Oslo, Gaustadalleen 21, N-0371 Oslo, Norway. Tel.: + 472295 87 32; fax: +4722958749.
the perpendicular plane in reciprocal space. The decagonal phase (or decagonal quasicrystal, briefly DQC) was firstly observed in a rapidly solidified A1-Mn alloy, which has period of 1.2 nm along the unique axis [1]. Later the D Q C s with different periods were also observed, e.g. the A1-Ni D Q C with a period of 0.4 nm I-2] and the A1-Pd D Q C with a period of 1.6 nm [3]. The structures of the D Q C s have been extensively studied, especially after the thermodynamically stable D Q C s were found [4-6]. High-resolution electron microscopy
0921-4526/97/$17.00 ~ 1997 Elsevier Science B.V. All rights reserved PII S 0 9 2 1 - 4 5 2 6 ( 9 7 ) 0 0 4 4 3 - 2
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and single-crystal X-ray analysis have been successfully used to reveal the structures of the DQCs at atomic level, e.g., on the basis of the high-resolution electron micrographs of the A1-Ni-Co DQC, the atom arrangement in columnar cluster with 2.0 nm in diameter has been proposed [7], which agrees quite well with the result of the single-crystal X-ray analysis [8]. The two methods have their own advantages in the study of the structures of the DQCs. Though atom arrangement in the cluster obtained from the single-crystal X-ray analysis is more preciser than that from the high-resolution electron microscopy. However, the single-crystal X-ray analysis of the DQCs coupled with a high-dimensional approach always provide only the average structures of the DQCs with quasiperiodic lattice. On the other hand, the high-resolution electron micrographs provide the real arrangement of atom clusters. The latter point becomes more important in distinguishing the DQCs with closely structural relationship. For instance, the single-crystal X-ray analyses of the A1-Mn and AI-Mn Pd DQCs show that the two phases have similar structure I-9, 10]. However, the high-resolution electron microscopic studies show clearly that the structures of the two DQCs are different [11, 12]. It has been found that the local configuration of atom arrangement is similar in the two structures and the result of the single-crystal X-ray analysis is better for the structure of the AI-Mn-Pd DQC than the structure of the A1-Mn DQC. A new structural model of the AI Mn DQC was then proposed by combining the results of high-resolution electron microscopy and single-crystal X-ray analysis [13, 14]. The coexistence of a DQC and crystalline phases in a conventional solidified A165Rh20Cu15 alloy was early reported in 1989 [15]. Recently the A1Rh-Cu DQC was confirmed in A163Rhls.sCu18.5 alloy by electron diffraction method. The structure of the A1-Rh Cu DQC has been studied the highresolution electron microscopy and the preliminary result has been briefly reported in Ref. [16]. The A1-Rh-Cu DQC has a periodicity of about 0.4 nm along its unique tenfold axis. However, the phase is different from the previously reported DQCs with a period of 0.4 nm, e.g. the A1-Ni-Co DQC since large decagonal subunits have been found in the structure of the AI Ni-Co DQC but not in the
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structure of the A1-Rh-Cu DQC. In the present paper, we propose a structural model of the AI-Rh-Cu DQC in detail on the basis of the highresolution electron microscopic study. First, we derive the structural model in a unit-cell approach in this paper as part I and then describe the structure in a high-dimensional approach in a separated paper as part II. The content of this paper is organized as follows. In Section 2, we analyze the arrangement of atom clusters in the structural image of the A1-Rh-Cu DQC and show the resemblance of the image contrasts of the atom clusters in structural images of the A1-Rh-Cu and the A1-Ni-Co DQCs. In Section 3, atom arrangements in the subunits of the A1-Rh-Co DQC are proposed and checked by image simulation. The divergence of the structural images of the A1-Rh-Cu and the A1-Co-Ni DQCs has been discussed in Section 4 and conclusion is given in Section 5.
2. High-resolution electron microscopy Fig. 1 shows (a) the unique tenfold electron diffraction pattern and (b) the corresponding highresolution electron micrograph of the A1-Rh-Cu DQC. The distribution of the sharp diffraction spots in the diffraction pattern indicates that the sample under examination is a high-quality quasicrystal. From Fresnel fringe at the edge of an amorphous contamination layer adjacent to the observed region, it was confirmed that the electron micrograph was taken under nearly Scherzer defocus, Af= - 4 5 nm. The electron micrograph in a rather thin area (less than 10 nm) can be considered as a so-called high-resolution structure image, which faithfully reflects projected potentials and thus is useful in the structure modelling. Such an area has been marked with rectangle and enlarged in Fig. 2(a). In the structure image, typical image contrast can be noticed as five bright spots, in a more or less triangular shape, forming a pentagon with a ring contrast in the centre. For illustration, some selected pentagons have been outlined with thin lines. Three kinds of local aggregated forms of the pentagonal motifs have been observed and they are labelled as a flattened hexagon (H), a crown (C) and a fivefold star (S) in Fig. 2(b)
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(a)
>I<>r~., <.,
Fig. I. (a) The tenfold electron diffraction pattern and (b) the corresponding high-resolution electron micrograph of the AIRh Cu DQC. An area marked with rectangle is believed to be a high-resolution structure image and is enlarged in Fig. 2(a).
according to the shapes of the tiles (edge length of about 0.65 nm) formed by connecting the centres of the pentagonal motifs with lines. These tiles are considered as the structure subunits of the AIR h - C u DQC. It should be pointed out that the formation of the tiles sometime became ambiguous to a certain extent because the two-ring constrasts side by side in some parts of the image have been obsevered, e.g., such a pair of ring constrasts has been marked by arrow in Fig. 2(a). In this case, we just choose one of them without thinking of the priority since both choices can fit to other tiles in the images, as shown in Fig. 2(c) in dash lines. We should discuss the reason for this phenomenon in next section. The structure image is covered by a tessellation of these subunits without any gaps. Such a tiling corresponding to the enlarged part of the structure image is given in Fig. 2(c). Thus, the determination of atom arrangement in the three
(b)
(c)
Fig. 2. (a) Structure image of the AI-Rh Cu DQC; (b) three kinds of subunits deduced from the structure image; (c) an aperiodic tessellation of the subunits according to the structure image. For details, see text.
kinds of subunits leads to a solution for the structure of the A1 Rh Cu DQC. The ring contrast, surrounded by five triangles, is believed relating to the columnar shape of atom cluster in the structure of the A1-Rh-Cu DQC. Similar image contrast associated with the columnar cluster can be found out from the structural image of the A1 N i - C o DQC, see Fig. 9. Therefore, atom arrangement in the columnar cluster can be directly derived from the structural models of the Al Ni Co DQC. The structure of the A I - N i - C o D Q C has two layers in the 0.4 nm period along the unique axis and the atom arrangements in the two layers are related by a 105 screw axis. Fig. 3 shows the large cluster of the A1-Ni-Co DQC, (a) and (b) reproduced from a structural model based on a high-resolution electron microscopic study
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centred atoms in shadowed and equivalent areas are exchanged in the two models. The atom arrangement in the shadowed area corresponds the ring contrast in the high-resolution structure images. Therefore, the basic columnar cluster can be derived in order to construct the structural model of the A1 R h - C u DQC. Fig. 4 shows the columnar cluster in two adjacent layers in the structure of the A1-Rh-Cu DQC, for convenience, hereafter named as layer A and layer a. In Fig. 4, only the atom positions have been pointed out and the species of the atoms will be discussed in next section.
~
)
3. Structural model (d)
Fig. 3. Atom cluster of the A1-.Ni-Co DQC in two adjacent layers, (a) and (b) reproduced from a structural model based on a high-resolution electron microscopic study [7]; (c) and (d) reproduced from a structural model based on a single-crystal X-ray analysis [8].
Layer A
~
Layera
C~
Layer A
~)
(~ C~
Projection Fig. 4. Columnar shape of atom cluster of the A1 Rh Cu DQC and atom positions in two adjacent layers, layer A and layer a.
[7]; (c) and (d) reproduced from a structural model based on a single-crystal X-ray analysis [8]. The two results resemble quite well although the different tilings are shown together with the atom arrangements, a rhombic Penrose tiling in Fig. 3(a) and Fig. 3(b) while in atoms were connected referring to partial structure of the Alx3Co 4 phase in Fig. 3(c) and Fig. 3(d). The only divergence between them is that the positions and species of the
Atom arrangements in the subunits of the AI Rh Cu D Q C can be obtained by using the basic columnar atom cluster and the following construction rules, (i) the distribution of atoms in each layer forms three kinds of local configurations as the hierarchical shapes of the subunits; (ii) the distribution of heavy atoms (Rh, Cu) are arranged in such a way that five of them form a pentagon with an edge length of 0.47 nm and with five or three A1 atoms inside each pentagon. The rules can be viewed as a summary from the atom arrangement in the large a t o m cluster of the A1-Ni Co D Q C (see Fig. 3). Fig. 5 shows the atom arrangements in the three kinds of subunits of the A1-Rh-Cu D Q C in two adjacent layers as a pair, the flattened hexagon in (a) and (e), the crown in (b) and (f), and the fivefold star in (c) and (g). The open circles indicate the A1 atoms, the filled circles the Rh/Cu atoms and the double circles the co-occupancy of A1 and Rh/Cu atoms. Positions marked with points indicate atom sites with partial occupancy and one set of possible occupation is applied in Fig. 5 to show the local atom configuration. Basic columnar atom clusters are important parts of these subunits, they situate in the vertices of the subunits, therefore there are six columnar clusters in the H subunit, eight in the C subunit and ten in the S subunit. A large circle is used to highlight one of the basic columnar clusters in the subunits. The formation of these subunits is also understood as the closepacking of the basic columnar clusters with fivefold symmetry in two-dimensional plane for the
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(a)
(e)
(b)
(t)
(c)
(d)
(g)
(h)
Fig. 5. The atom arrangements in the three kinds of subunits of the Al Rh-Cu DQC in two adjacent layers as a pair, the flattened hexagon in (a) and (e), the crown in (b) and (f), and the fivefold star in {c) and (g). In addition, the atom arrangement in a decagonal subunit is shown in two adjacent layers as a pair in (dt and (h}.
S subunit and as parts of the S subunit for the C and H subunits. A decagonal shape of subunit shown in Fig. 5(d) and (h) is obtained as another close-packing type, which is resemblance to that shown in Fig. 3(c) and (d), hereafter referred as the D subunit. The D subunit has not been observed in the structure of the A1-Rh-Cu DQC, but it does exist in the structure of the A1 Ni Co DQC. This shows that the two phases are different types of DQCs, more discussions are given in Section 4. The basic columnar atom clusters appear in the edges of the subunits play a role as a "matching rule" for packing these subunits in two-dimensional plane. The proposed model has been checked by image simulation. In order to carry out the image simulation, hypothetical periodic structure composed of the subunits has been used. Fig. 6 shows a part of a periodic structure in (a) layer A and (b) layer a; (c) the calculated image. The calculation was carried out by using the MacTempas package under
a defocus value of - 45 nm and a sample thickness of 3 nm at an accelerating voltage of 400 kV and spherical aberration of 1.0 mm. Triangular and ring bright contrasts can be observed in the calculated image in Fig. 6(c). The calculated image, in our belief, reproduces well the image contrasts in the experimental one. The special image contrast mentioned in Section 2 can be explained by considering that the basic columnar clusters may shift locally in certain layers. Fig. 7 shows the tessellation of the subunits change due to the shift of the atom clusters, (a) layer A and (b) layer a without shifting; (c) layer A and (d) layer a with shifting. It is clear that the arrangement of the atoms only change slightly within a decagonal area. We suggest that such changes may happen in the structure of the A1-Rh-Cu DQC. Fig. 7(e) shows the calculated image of another hypothetical periodic model with superposing of the atom arrangement in Fig. 7(a)Fig. 7(d). The simulation shows that a pair of ring
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q
(a)
(c)
(b)
Fig. 6. Part of a hypothetical periodic structure in (a) layer A and (b} layer a; (c) the calculated image.
Layer A
(a)
(c)
Layer
a
(b)
(e)
(d)
Fig. 7. Part of a hypothetical periodic structure with the shifts of the atom clusters, (a) layer A and (b) layer a without shifting; (cl layer A and (d) layer a with shifting. (e) The calculated image.
contrast is reproduced in calculated image, as shown in Fig. 7(e) indicated by an arrow. The projected a t o m arrangements in each cases can be obtained by superposing the individual layers shown in Fig. 6(a) and (b) and Fig. 7(a)-(d), respectively, it is obvious that the a t o m positions cortes-
p o n d dark regions in the structure images. The agreement of experimental and calculated images shows that the proposed structural model is basically correct. As shown in Fig. 2, the real structure of the AI R h - C u D Q C is c o m p o s e d of three kinds of
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(a)
(b)
Fig. 8. Two-colour Penrose tilings in pair. subunits in an aperiodic tessellation. In fact, the three kinds of subunits are also allowed for quasiperiodic or periodic tessellation. The quasiperiodic tessellation of the three kinds of subunits yields the two-colour Penrose tiling if different atom decorations for the same shape subunits are distinguished by white and black colours. Thus the ideal model of the A1 R h - C u D Q C can be described as a structure of a periodic stacking of the two-colour Penrose tilings. The twocolour Penrose tiling in pair is shown in Fig. 8, which can be used to represent the two consecutive layers in the structure of the A1 R h - C u DQC. For more about the ideal model of the A1-Rh-Cu DQC, see part II of the present paper.
4. Comparison quasicrystal
with the AI Ni-Co
decagonal
Hiraga and his collaborators [7, 17] have systematically studied the structure of the A1-Ni-Co D Q C by the high-resolution electron microscopy. Aperiodic tessellations of decagonal clusters with 2.0 nm in diameter have been found in the structure image of the AI N i - C o D Q C and the atom arrangement in the decagonal cluster has been proposed, as shown in Fig. 2(a) and Fig. 2(b). Carefully examining the structure image, we found that the structure of the A I - N i - C o D Q C is composed of also the subunits in the shapes of flattened hexagon,
crown, and fivefold star besides mainly the decagonal subunit. Fig. 9 shows a structure image of the A1 N i - C o DQC. Decagonal subunits have been marked with the decagons in thick lines and the "true decagonal subunits ''~ have also been labelled with a letter D. On the other hand, two decagons are better explained as the combination of two H subunits and one C subunit than a single decagonal subunit. In the meantime, the S subunit has also been found in the structure image. This shows the structure of the A1 N i - C o D Q C in the observed sample is composed by mainly the decagonal subunits and also the subunits in the shape of the flattened hexagon, crown and fivefold star. Moreover, the special image contrasts with a pair of rings has also observed and marked with an arrow. By comparing the A1 Ni Co and the A1-Rh Cu DQCs, we found that (i) both of them are composed of the basic columnar cluster, which can be aggregated to form four kinds of subunits; (ii) the structure of the A1-Ni-Co D Q C is composed of mainly the D subunit and also the H, C and S subunits while the structure of the AI Rh Cu D Q C is composed of the H, C and S subunits. It has been found that the core part of the decagonal subunit of the AI- Ni Co DQC has fivefoldsymmetry according to the Ref. [18]. Here we use "true decagonal subunits" to describe some subunits with tenfold geometrical shape in order to distinguish to the others with decagonalshape, but actually composed of two H subunits and one C subunit.
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ture of the A1-Rh-Cu D Q C is different from that of the A1-Ni-Co DQC.
Acknowledgements This work was carried out as part of a research project, financially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan.
References Fig. 9. The structure image of the A1 Ni Co DQC. Some subunits have been outlined as the decagonal subunits (D), the flattened hexagonal subunits (H), the crown subunits (C) and the fivefold star subunit (S).
5. Concluding remarks The structure of the A1-Rh-Cu D Q C can be described as an aperiodic tiling composed of three kinds of subunits, namely flattened hexagon, crown and fivefold star according to the high-resolution electron microscopic study. The atom arrangements in the subunits have been proposed on the basis of the structure of the A1-Ni Co DQC. The image simulation shows that the calculated images reproduce well the contrast features of the observed images, which means that the present model is basically correct. Quasiperiodic tessellation of the three kinds of subunits generate the two-colour Penrose tiling if different atom decorations for the same shape subunits are distinguished by white and black colours. Thus, the ideal model of the A1 RhCu decagonal quasicrystal is proposed as periodic stacking of the two-colour Penrose tilings. By comparing the structure images, we show that the struc-
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