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Structure of the AlRhCu decagonal quasicrystal: II. A higher-dimensional description
N
Physica B 240 (1997) 338 342
ELSEVIER
Structure of the A1-Rh-Cu decagonal quasicrystal: II. A higher-dimensional description Hiraga
X.Z. Li*, K.
Institute jor Materials" Researeh, Tohoku Universio,, Katahira, Aoba-ku, Sendai 980, Japan
Received 12 May 1997
Abstract A higher-dimensional description has been applied to the structural model of the AI-Rh-Cu decagonal quasicrystal described in a unit-cell approach in the previous paper (Part I), which was proposed on the basis of a high-resolution electron microscopic study. The structure of the A1 Rh-Cu decagonal quasicrystal consists of two layers in a period, which are related by 105 screw axis. The independent layer is described by two hyper-atoms in the five-dimensional structure. Hyper-atoms of the AI-Rh Cu DQC are at special positions with the site symmetry 5 m, the acceptance domain of the A1-Rh Cu decagonal quasicrystal in the internal (perpendicular) space can be viewed as the extensions of those for the Penrose tiling. The geometric and chemical structures of these hyper-atoms are described in detail. A projection of the five-dimensional structure into the external (physical) space generates the ideal quasiperiodic structure of the A1-Rh-Cu decagonal quasicrystal mentioned in the previous paper (Part I). PACS: 61.44 Keywords. Decagonal quasicrystal; High-dimensional space; A1-Rh Cu alloy
1. Introduction H i g h e r - d i m e n s i o n a l c r y s t a l l o g r a p h y was originally i n t r o d u c e d in 1974 for m o d u l a t e d crystals [-1]. The discovery of i c o s a h e d r a l phase in 1984 [2] e x t e n d e d the a p p l i c a b i l i t y of the h i g h e r - d i m e n sional c r y s t a l l o g r a p h y to quasicrystals [3, 4]. The application of the higher-dimensional crystallography * Correspondence address: Centre for Materials Science, Faculty of Mathematics and Natural Sciences, University of Oslo, Gaustadalleen 21, N-0371 Oslo, Norway, Tel.: + 47 22 95 87 32; fax: +4722958749.
is based on the fact that the diffraction p a t t e r n s of these structures are indexable with n ( > 3) vectors as h = Xhib*, where h is the diffraction vector in t h r e e - d i m e n s i o n a l (3D) space, b* (i -- 1, 2 , . . . , n) are the basis vectors to index the diffraction vector with generalised Miller indices hi, h2 . . . . . h,. This implies that the l o c a t i o n of diffraction spots is related to an n - d i m e n s i o n a l lattice, i.e., the diffraction spots can be r e g a r d e d as the p r o j e c t i o n for the nD reciprocal-lattice p o i n t s o n t o 3D space. This relation suggests a periodic structure in nD space. F r o m the p r o p e r t i e s of F o u r i e r t r a n s f o r m a t i o n , the intersection of electron density at the s u b s p a c e n o r m a l to
0921-4526/97,/$17.00 :~" 1997 Elsevier Science B.V. All rights reserved PII S0921 -45 2 6 ( 9 7 ) 0 0 4 4 4 - 4
X.Z Li, K. Hiraga / Physica B 240 (1997) 338-342
one direction is obtained from the diffraction pattern projected onto the subspace along that direction. The irrational (gradient) intersection of the periodic electron density leads to an aperiodic structure in the intersection. Thus, the quasiperiodic structure can be expressed as an irrational 3D intersection of a periodic structure in nD space
[5]. Besides the single-crystal X-ray diffraction coupled with higher-dimensional crystallography, high-resolution electron microscopy (HREM) is another important tool for studying the structures of the quasicrystals, especially for the two-dimensional quasicrystals, e.g., the decagonal quasicrystals (DQCs). In the latter case, the structures of the DQCs deduced from the H R E M images are usually described in 3D space, in which the a t o m arrangements in the subunits (or atom clusters) and aperiodic (including quasiperiodic) tessellation of subunits (or atom clusters) are defined [6-9]. This kind of description can be called as unit-cell approach. Nevertheless, this structural description in the unitcell approach is quite complex and not easy for quantitative description since atom arrangements are presented in subunits (or atom clusters) with special shapes. On the contrary, higher-dimensional crystallography provides a way to describe the structures of the D Q C s concisely and precisely, in which a finite set of parameters in analogy to usual crystal structure are defined. By the above consideration, a new structural model of the A1-Mn D Q C was proposed on basis of the H R E M images [9] and then has been converted into a higher-dimensional description [10]. In our previous paper (Part I) 1-11], a 3D structural model of the Al Rh Cu D Q C has been constructed on the basis of a H R E M study. In this paper (Part II), we describe this model in a higherdimensional description. The contents are arranged as follows: in Section 2, the hyper-atoms of the A1 Rh Cu D Q C are deduced, the co-ordinates are given in 5D space and the geometric shape and chemical distribution are defined in internal (perpendicular) space; in Section 3, the layer structures of the A1-Rh-Cu D Q C are obtained by the projection of the higher-dimensional structure onto external (physical) space and the concluding remarks are given in Section 3.
339
2. Hyper-atoms in five-dimensional space We use a decagonal coordinate system in 5D space defined below, which was first adapted by Y a m a m o t o and Ishihara in 1988 [12]. The five basic vectors di(i <<,5) of the decagonal lattice are given by dl = SjMijaj (i, j <~ 5), where the vectors a~.2.5 are the orthogonal unit vectors in the external (physical) space VE and a3.4 in the internal (perpendicular) space V~. The M u is written as 'cl - 1
sl
c2 - 1
s2
0
1
s2
c4- 1
s4
0
1
s3
c I -- 1
sl
0
c4- 1
s4
c3- 1
s3
0
0
0
0
0
(51/2c)/2a
c2
2a/51/2
c3
-
-
-
-
where cj = cos(27zj/5), sj = sin(2rtj/5) (j = 1, 2, .... 4); a = 0.277 nm and c = 0.41 nm. Hereafter, we use the xs-coordinates for describing the positions of the hyper-atoms in 5D space and also the corresponding z-coordinates for the 3D structure. It has been known that the A1 Ni Co D Q C has a superspace group of P10s/mmc, which leads to a systematic extinction of the electron diffraction patterns of the A1-Ni-Co DQC. Similar systematic extinction rules were also observed in the electron diffraction patterns of the A1 Rh Cu DQC, which indicates a superspace group of P105/mmc for the A1 Rh Cu DQC. F r o m the symmetry operators, we know that there are two sets of hyper-atoms related by theglide plane, which corresponds to the two layers in the structure of the A1-Rh-Cu DQC. Since the model of the ideal structure of the A I - R h Cu D Q C is assumed to be the periodic stacking of the two-colour Penrose tiling, then all hyper-atoms of the A1-Rh-Cu DQC, like those for the Penrose tiling, are at special positions with the site symmetry 5 m. Thus, the shapes of the hyper-atoms in V~ is restricted to a fivefold or higher symmetry. This restriction has been used in the determination of the shape and the chemical distribution of the hyper-atoms in V~. The hyper-atom coordinates of the A1-Rh-Cu D Q C are listed in Table 1. The hyper-atoms of the A1 Rh Cu D Q C were constructed by lifting the partial atom positions in
X.Z. Li, K. Hiraga / Physica B 240 (1997) 338 342
34O
Table 1 The hyper-atom coordinates of the AI-Rh-Cu decagonal quasicrystal Hyper-atoms'\ coordinates
X1
1
-~
2 3 4
-~ -~ 0
-g2
X3
X4
-~ -~ -4 0
--5
0
X5
-~ -~
4
--4 5
0
0 0 i 2
;
the independent layers of the 3D structure onto the 5D space. The hyper-atoms in internal space are usually called as acceptance domains. Fig. 1 shows the acceptance domains for the A1-Rh-Cu D Q C in the lower part and those for the Penrose tiling in the upper part for comparison. In fact, the acceptance domains for the A1-Rh-Cu DQC are the extension of the acceptance domains for the Penrose tiling. Fig. 2 shows the geometrical size and chemical distribution of two independent acceptance domains in detail. In Fig. 2(a), the inner pentagon marked with 1 is co-occupied by Rh and Cu elements; the trapezoid marked with 3 and equivalent ones by A1 element; area marked with 2 and equivalent ones by the co-occupation of the Rh/Cu and A1 elements; the triangle marked with 5 and equivalent ones by A1 element partially. In Fig. 2(b), the AI
element is distributed in an inner area marked with 1, which is formed by the two superposed pentagons; partially occupied A1 element in an outside area marked with 2 and equivalent ones. The sizes of the two acceptance domains are defined with their geometry and two parameters rl and r2, where rl = 4.618, r 2 = 3.236 in Fig. 2(a) and rl = 5.618, r 2 = 4.236 in Fig. 2(b). The complex shape of the acceptance domains are necessary, which is associated to the atom distribution in the 3D structure in the external (physical) space. It is clear that the further from the centres of the acceptance domains the intensity becomes lower. Therefore, the sharp edges may not be re-constructed directly from electron or X-ray diffraction methods and only the approximate shapes of areas concerning to higher intensity can be determined experimentally. The composition of the structural model was calculated to be A166.57 (Rh/Cu)33.43 by using the acceptance domains of the AI Rh-Cu D Q C illustrated above.
3. Structures of the layers in three-dimensional space The 3D structure of the A1-Rh Cu D Q C has been generated from the 5D structural model by
A c c e p t a n c e d o m a i n s o f the P e n r o s e tiling E
A
B
C
D
E
Hyper-atoms in internal space o f the AI-Rh-Cu decagonal quasicyrstal
.it L a y e r (Xs=0)
4
L a y e r (xs= 1/2)
Fig. 1. The geometrical shape and positions of the acceptance domains for the AI Rh Cu DQC (lower part) and the acceptance domains for the Penrose tiling with rhombic tiles (upper part) for comparison.
341
)LZ Li, K. Hiraga / Physica B 240 (1997) 338-342
•..
'
l 2 3 4
(a)
::~:#~::
paper (Part I) can be noticed by the framework in solid lines, which are generated by two acceptance domains of the Penrose tiling, B and C in Fig. 2. Undoubtedly, the atom configurations in these subunits are identical to those deduced in a unitcell approach. As explained earlier in the previous paper (Part I), in one kind of the S subunits, possible atom positions are shown by ten points, but actually only three A1 atoms exist in these positions. The similar situations can be found in one kind of the C subunits. As the ideal structure of the A1-Rh-Cu DQC, the layers can be represented by the two-colour Penrose tilings if the different atom configurations in the subunits with the same shape are distinguished by white and black colours, and also the colours are inverse in the corresponding subunits in the adjacent layers. Fig. 4 shows the projection of the layers in Fig. 3, which can be used to compare the structure image of the A1Rh-Cu DQC shown in the previous paper (Part I). The atom distribution in the structure projection shows that all the vertices of the subunits are individually surrounded by ten atoms forming a decagon, which corresponds to the ring contrast in the highresolution structure image regardless of a quasiperiodic tessellation of the subunits in this
\x
i~iiii~ii!iiiiii::iiiii::ii::~iii~::~"'"
Rh,Cu AI +Rh,Cu AI partial AI
I AI
2 partial AI
(b)
Fig. 2. The acceptance domains for the A1-Rh-Cu with the details on geometrical size, subdivisions and chemical composition.
using the cut-and-projection (or section) method. Fig. 3 shows the 3D structure of the A1-Rh-Cu DQC in (a) the layer with x5 = 0 and (b) the layer with x5 = ½, which correspond to the hyper-atoms shown in Fig. 2. Rh and Cu atoms are indicated by filled circles; A1 atoms are indicated by open circles; positions co-occupied by Rh/Cu and A1 atoms are indicated by double circles; and the positions occupied partially by A1 atoms are indicated by points. The subunits mentioned in the previous
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Fig. 3. The 3D structure of the AI Rh Cu D Q C generated from the 5D structural model by the cut-and-projection method:
(a) the layer with x5 = 0 and; (b) the layer with x5 = ½.
342
X.Z. Li. K. Hiraga / Physica B 240 (1997) 3 3 8 342
Shape, size and chemical distribution of two independent hyper-atoms are presented in details. A projection of the higher-dimensional structure into the physical space generates the ideal quasiperiodic structure of the A1-Rh-Cu decagonal quasicrystal mentioned in the previous paper (Part I).
Acknowledgements The authors thank Dr. K. Yubuta and Prof. A. Yamamoto for discussions. 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.
Fig. 4. The structure projection of the AI-Rh-Cu DQC along the unique tenfold axis in 3D space.
structure projection and a random tessellation of subunits in the structure image.
4. Concluding remarks We show that the higher-dimensional description of the structural model of the A1 Rh Cu decagonal quasicrystal, which was proposed on the basis of a high-resolution electron microscopic study in the previous paper (Part I). The coordinates of the hyper-atoms are given in 5D space.
References [1] P.M. de Wolff, Acta Crystallogr. A 30 (1974) 777. [2] D. Shechtman, I. Blech, D. Gratias, J.W. Cahn, Phys. Rev. Lett. 53 (1984) 1951. [3] P. Bak, Phys. Rev. Lett. 56 (1986) 861. [4] T. Janssen, Acta Crystallogr. A 42 (1986) 261. [5] A. Yamamoto, Acta Crystallogr. A 52 (1996) 509. [6] K. Hiraga, W. Sun, F.J. Lincoln, Jpn. J. Appl. Phys. 30 (19911 L302. [7] K. Hiraga, F.J. Lincoln, W. Sun, Mater. Trans. JIM 32 [1991) 308. [8] K. Hiraga, W. Sun, Phil. Mag. Lett. 62 (1993) 117. [93 x.z. Li, Acta Crystallogr. B 51 (1995) 265. [10] X.Z. Li, F. Frey, Acta Crystallogr. B 51 (1995) 27l. [I13 X.Z. Li, K. Hiraga, K. Yubuta, preceding article in this issue, Physica B 240 (1997) 330. [12] A. Yamamoto, K.N. Ishihara, Acta Crystallogr. A 44 (1988) 707.
Physica B 240 (1997) 338 342
ELSEVIER
Structure of the A1-Rh-Cu decagonal quasicrystal: II. A higher-dimensional description Hiraga
X.Z. Li*, K.
Institute jor Materials" Researeh, Tohoku Universio,, Katahira, Aoba-ku, Sendai 980, Japan
Received 12 May 1997
Abstract A higher-dimensional description has been applied to the structural model of the AI-Rh-Cu decagonal quasicrystal described in a unit-cell approach in the previous paper (Part I), which was proposed on the basis of a high-resolution electron microscopic study. The structure of the A1 Rh-Cu decagonal quasicrystal consists of two layers in a period, which are related by 105 screw axis. The independent layer is described by two hyper-atoms in the five-dimensional structure. Hyper-atoms of the AI-Rh Cu DQC are at special positions with the site symmetry 5 m, the acceptance domain of the A1-Rh Cu decagonal quasicrystal in the internal (perpendicular) space can be viewed as the extensions of those for the Penrose tiling. The geometric and chemical structures of these hyper-atoms are described in detail. A projection of the five-dimensional structure into the external (physical) space generates the ideal quasiperiodic structure of the A1-Rh-Cu decagonal quasicrystal mentioned in the previous paper (Part I). PACS: 61.44 Keywords. Decagonal quasicrystal; High-dimensional space; A1-Rh Cu alloy
1. Introduction H i g h e r - d i m e n s i o n a l c r y s t a l l o g r a p h y was originally i n t r o d u c e d in 1974 for m o d u l a t e d crystals [-1]. The discovery of i c o s a h e d r a l phase in 1984 [2] e x t e n d e d the a p p l i c a b i l i t y of the h i g h e r - d i m e n sional c r y s t a l l o g r a p h y to quasicrystals [3, 4]. The application of the higher-dimensional crystallography * Correspondence address: Centre for Materials Science, Faculty of Mathematics and Natural Sciences, University of Oslo, Gaustadalleen 21, N-0371 Oslo, Norway, Tel.: + 47 22 95 87 32; fax: +4722958749.
is based on the fact that the diffraction p a t t e r n s of these structures are indexable with n ( > 3) vectors as h = Xhib*, where h is the diffraction vector in t h r e e - d i m e n s i o n a l (3D) space, b* (i -- 1, 2 , . . . , n) are the basis vectors to index the diffraction vector with generalised Miller indices hi, h2 . . . . . h,. This implies that the l o c a t i o n of diffraction spots is related to an n - d i m e n s i o n a l lattice, i.e., the diffraction spots can be r e g a r d e d as the p r o j e c t i o n for the nD reciprocal-lattice p o i n t s o n t o 3D space. This relation suggests a periodic structure in nD space. F r o m the p r o p e r t i e s of F o u r i e r t r a n s f o r m a t i o n , the intersection of electron density at the s u b s p a c e n o r m a l to
0921-4526/97,/$17.00 :~" 1997 Elsevier Science B.V. All rights reserved PII S0921 -45 2 6 ( 9 7 ) 0 0 4 4 4 - 4
X.Z Li, K. Hiraga / Physica B 240 (1997) 338-342
one direction is obtained from the diffraction pattern projected onto the subspace along that direction. The irrational (gradient) intersection of the periodic electron density leads to an aperiodic structure in the intersection. Thus, the quasiperiodic structure can be expressed as an irrational 3D intersection of a periodic structure in nD space
[5]. Besides the single-crystal X-ray diffraction coupled with higher-dimensional crystallography, high-resolution electron microscopy (HREM) is another important tool for studying the structures of the quasicrystals, especially for the two-dimensional quasicrystals, e.g., the decagonal quasicrystals (DQCs). In the latter case, the structures of the DQCs deduced from the H R E M images are usually described in 3D space, in which the a t o m arrangements in the subunits (or atom clusters) and aperiodic (including quasiperiodic) tessellation of subunits (or atom clusters) are defined [6-9]. This kind of description can be called as unit-cell approach. Nevertheless, this structural description in the unitcell approach is quite complex and not easy for quantitative description since atom arrangements are presented in subunits (or atom clusters) with special shapes. On the contrary, higher-dimensional crystallography provides a way to describe the structures of the D Q C s concisely and precisely, in which a finite set of parameters in analogy to usual crystal structure are defined. By the above consideration, a new structural model of the A1-Mn D Q C was proposed on basis of the H R E M images [9] and then has been converted into a higher-dimensional description [10]. In our previous paper (Part I) 1-11], a 3D structural model of the Al Rh Cu D Q C has been constructed on the basis of a H R E M study. In this paper (Part II), we describe this model in a higherdimensional description. The contents are arranged as follows: in Section 2, the hyper-atoms of the A1 Rh Cu D Q C are deduced, the co-ordinates are given in 5D space and the geometric shape and chemical distribution are defined in internal (perpendicular) space; in Section 3, the layer structures of the A1-Rh-Cu D Q C are obtained by the projection of the higher-dimensional structure onto external (physical) space and the concluding remarks are given in Section 3.
339
2. Hyper-atoms in five-dimensional space We use a decagonal coordinate system in 5D space defined below, which was first adapted by Y a m a m o t o and Ishihara in 1988 [12]. The five basic vectors di(i <<,5) of the decagonal lattice are given by dl = SjMijaj (i, j <~ 5), where the vectors a~.2.5 are the orthogonal unit vectors in the external (physical) space VE and a3.4 in the internal (perpendicular) space V~. The M u is written as 'cl - 1
sl
c2 - 1
s2
0
1
s2
c4- 1
s4
0
1
s3
c I -- 1
sl
0
c4- 1
s4
c3- 1
s3
0
0
0
0
0
(51/2c)/2a
c2
2a/51/2
c3
-
-
-
-
where cj = cos(27zj/5), sj = sin(2rtj/5) (j = 1, 2, .... 4); a = 0.277 nm and c = 0.41 nm. Hereafter, we use the xs-coordinates for describing the positions of the hyper-atoms in 5D space and also the corresponding z-coordinates for the 3D structure. It has been known that the A1 Ni Co D Q C has a superspace group of P10s/mmc, which leads to a systematic extinction of the electron diffraction patterns of the A1-Ni-Co DQC. Similar systematic extinction rules were also observed in the electron diffraction patterns of the A1 Rh Cu DQC, which indicates a superspace group of P105/mmc for the A1 Rh Cu DQC. F r o m the symmetry operators, we know that there are two sets of hyper-atoms related by theglide plane, which corresponds to the two layers in the structure of the A1-Rh-Cu DQC. Since the model of the ideal structure of the A I - R h Cu D Q C is assumed to be the periodic stacking of the two-colour Penrose tiling, then all hyper-atoms of the A1-Rh-Cu DQC, like those for the Penrose tiling, are at special positions with the site symmetry 5 m. Thus, the shapes of the hyper-atoms in V~ is restricted to a fivefold or higher symmetry. This restriction has been used in the determination of the shape and the chemical distribution of the hyper-atoms in V~. The hyper-atom coordinates of the A1-Rh-Cu D Q C are listed in Table 1. The hyper-atoms of the A1 Rh Cu D Q C were constructed by lifting the partial atom positions in
X.Z. Li, K. Hiraga / Physica B 240 (1997) 338 342
34O
Table 1 The hyper-atom coordinates of the AI-Rh-Cu decagonal quasicrystal Hyper-atoms'\ coordinates
X1
1
-~
2 3 4
-~ -~ 0
-g2
X3
X4
-~ -~ -4 0
--5
0
X5
-~ -~
4
--4 5
0
0 0 i 2
;
the independent layers of the 3D structure onto the 5D space. The hyper-atoms in internal space are usually called as acceptance domains. Fig. 1 shows the acceptance domains for the A1-Rh-Cu D Q C in the lower part and those for the Penrose tiling in the upper part for comparison. In fact, the acceptance domains for the A1-Rh-Cu DQC are the extension of the acceptance domains for the Penrose tiling. Fig. 2 shows the geometrical size and chemical distribution of two independent acceptance domains in detail. In Fig. 2(a), the inner pentagon marked with 1 is co-occupied by Rh and Cu elements; the trapezoid marked with 3 and equivalent ones by A1 element; area marked with 2 and equivalent ones by the co-occupation of the Rh/Cu and A1 elements; the triangle marked with 5 and equivalent ones by A1 element partially. In Fig. 2(b), the AI
element is distributed in an inner area marked with 1, which is formed by the two superposed pentagons; partially occupied A1 element in an outside area marked with 2 and equivalent ones. The sizes of the two acceptance domains are defined with their geometry and two parameters rl and r2, where rl = 4.618, r 2 = 3.236 in Fig. 2(a) and rl = 5.618, r 2 = 4.236 in Fig. 2(b). The complex shape of the acceptance domains are necessary, which is associated to the atom distribution in the 3D structure in the external (physical) space. It is clear that the further from the centres of the acceptance domains the intensity becomes lower. Therefore, the sharp edges may not be re-constructed directly from electron or X-ray diffraction methods and only the approximate shapes of areas concerning to higher intensity can be determined experimentally. The composition of the structural model was calculated to be A166.57 (Rh/Cu)33.43 by using the acceptance domains of the AI Rh-Cu D Q C illustrated above.
3. Structures of the layers in three-dimensional space The 3D structure of the A1-Rh Cu D Q C has been generated from the 5D structural model by
A c c e p t a n c e d o m a i n s o f the P e n r o s e tiling E
A
B
C
D
E
Hyper-atoms in internal space o f the AI-Rh-Cu decagonal quasicyrstal
.it L a y e r (Xs=0)
4
L a y e r (xs= 1/2)
Fig. 1. The geometrical shape and positions of the acceptance domains for the AI Rh Cu DQC (lower part) and the acceptance domains for the Penrose tiling with rhombic tiles (upper part) for comparison.
341
)LZ Li, K. Hiraga / Physica B 240 (1997) 338-342
•..
'
l 2 3 4
(a)
::~:#~::
paper (Part I) can be noticed by the framework in solid lines, which are generated by two acceptance domains of the Penrose tiling, B and C in Fig. 2. Undoubtedly, the atom configurations in these subunits are identical to those deduced in a unitcell approach. As explained earlier in the previous paper (Part I), in one kind of the S subunits, possible atom positions are shown by ten points, but actually only three A1 atoms exist in these positions. The similar situations can be found in one kind of the C subunits. As the ideal structure of the A1-Rh-Cu DQC, the layers can be represented by the two-colour Penrose tilings if the different atom configurations in the subunits with the same shape are distinguished by white and black colours, and also the colours are inverse in the corresponding subunits in the adjacent layers. Fig. 4 shows the projection of the layers in Fig. 3, which can be used to compare the structure image of the A1Rh-Cu DQC shown in the previous paper (Part I). The atom distribution in the structure projection shows that all the vertices of the subunits are individually surrounded by ten atoms forming a decagon, which corresponds to the ring contrast in the highresolution structure image regardless of a quasiperiodic tessellation of the subunits in this
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342
X.Z. Li. K. Hiraga / Physica B 240 (1997) 3 3 8 342
Shape, size and chemical distribution of two independent hyper-atoms are presented in details. A projection of the higher-dimensional structure into the physical space generates the ideal quasiperiodic structure of the A1-Rh-Cu decagonal quasicrystal mentioned in the previous paper (Part I).
Acknowledgements The authors thank Dr. K. Yubuta and Prof. A. Yamamoto for discussions. 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.
Fig. 4. The structure projection of the AI-Rh-Cu DQC along the unique tenfold axis in 3D space.
structure projection and a random tessellation of subunits in the structure image.
4. Concluding remarks We show that the higher-dimensional description of the structural model of the A1 Rh Cu decagonal quasicrystal, which was proposed on the basis of a high-resolution electron microscopic study in the previous paper (Part I). The coordinates of the hyper-atoms are given in 5D space.
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