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Atomic structure of AlMn quasicrystal
Journal of Non-Crystalline Solids 117/118 (1990) 765-768 North-Holland
765
ATOMIC STRUCTUREOF AI-Mn QUASICRYSTAL T. Yamaguchi and N. Fujima Faculty of Engineering, Shizuoka University, Hamamatsu 432, Japan An atomic structure of AI-Mn quasicrystal is presented. Two unit cells A6 and 06 are decorated as follows: Mn atoms are put at a l l vertices of A6 and 06. Al atoms are put at the face-center which divides the long-diagonal of a l l golden rhombuses into the ratio l:T (the golden r a t i o ) . Two Al atoms are put at points which divide the body-diagonal of A6 into the ratio l:v~J:l. The structure factor of the decorated 3-dimensional quasicrystal is calculated and compared with the experiment. The l a t t i c e constant is obtained as 4.60 A which is T times of AI-Al bond length. ed from higher-dimensional periodic l a t t i c e s .
I . INTRODUCTION Non-periodicity, s e l f - s i m i l a r i t y , and the existence of a few u n i t cells are the most remarkable properties of quasicrystals.
The 3-
He showed that Bragg peaks are the a-function peaks and f i l l
densely the reciprocal space.
Fortunately, most of these peaks are extremely
dimensional Penrose transformation was presented
weak, making i t possible to distinguish i n d i v i d -
by Ogawa.l
ual ones. Whenthe vectors ei line up with the
He assumed two unit cells of golden
rhombus whose two diagonals have the golden
six f i v e - f o l d symmetry axes of the icosahedron,
ratio.
a general Bragg vector is written as
The solid angle around a main vertex is
x/5 for A6 and 77/5 for 06.
He obtained the
expanded A6 and 06 which are composed of 55 A6 and 34 06, and of 34 A6 and 21 06, respectively.
6 s = (~/a) Z n i e i , i=l
(I)
He obtained quasicrystals by recursive trans-
where the integers ni are incices and a repre-
formations.
sents the edge length of the rhombohedral unit-
In his transformation, the icosa-
hedral star-shaped polyhedra with 60 golden
cells A6 and 06.
rhombuses ($60) are located at a l l vertices of
by the Fourier transform of the density which
the expanded A6 and 06. of 20 A6.
This $60 is composed
There is a golden icosahedron F20
The structure factor is given
is constant inside a triacontahedron and f a l l s abruptly to zero outside of i t .
He identified
with 20 golden rhombuses between two $60 along
a theoretical odd-parity peak of the two-fold
each edge of the expanded A6 and 06.
pattern with the experimental peak and obtained
In the
i n t e r i o r of an expanded A6, there are two golden triacontahedra K30 with 30 golden rhombuses
the result that a = 4.60 A. On the basis of Ogawa's transformation and
which are surrounded by six $60 and share an 06.
Elser's value of quasilattice constant, many
Two S6O at the main vertices of an expanded 06
attempts have already been made to determine
share an A6.
There are 6 06 between the two $60
of an expanded 06 .
An expanded A6 is composed
the atomic structure of quasicrystals and to answer questions such as: Are atoms at the ver-
of l S6O, 2 K30 and 3 F2O but minus l 06 , and
tices of A6 and 06 , on the edge, on the face,
an expanded 06 is composed of l S6O, 3 F20 and
and/or on the body-diagonal in the interior?
6 06 but minus l A6. Elser 2 discussed characteristic features of the d i f f r a c t i o n pattern of quasicrystal project-
0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)
We use two parameters t = I / ~ T = (5t+I)/2 (=I.6180).
(=0.4472) and
T. Yamaguchi, N. Fujima / Atomic structure of AI-Mn quasicrystal
766
where f is the atomic scattering factor, ~ the
2. DECORATION OF A6 AND 06 As shown in Figure l , we put Mn atoms ( f i l l e d circles) at a l l vertices of A6 and 06.
We put
scattering vector, and r j - r k the relative atomic coordinate-vector of the j - t h and k-th atoms.
Al atom at the face-center which divides the
The ratio of f(O) for Al and Mn is equal to the
long-diagonal of a l l golden rhombuses into the
ratio of atomic numbers which is nearly I/2.
ratio l:T (open c i r c l e ) .
We also put two Al
atoms at points which divide the body-diagonal of A6 into the ratio l : 5 t : l (shaded circles). The distance between Mn and the neighboring
3.1. Two-dimensional cluster and quasicrystal A 2-dimensional quasicrystal is obtained by the projection from the 5-dimensional simple cubic l a t t i c e , and shown in Figure 2.
There
Al is Doa = ~ 7 - ~ - ~ a = 0.5628a, that i s , the
are many clusters composed of 5 acute and 5 ob-
main diagonal of 06.
tuse parallelograms which are similar to K30 in
The distance between
the 3-dimension and shown in the insertion of
neighboring two Al atoms is also Doa.
Figure 3(a). Structure factors for the cluster and the quasicrystal along the S2 direction are shown in Figure 3(a) and 3(b), respectively. They are quasiperiodic as a function of sa.
Here,
the quasicrystal is composed of 77 atoms. The characteristic features such as the peak position of Figure 3(b) are well reproduced by Figure 3(a).
This means that the interval between
l a t t i c e planes of the quasicrystal coincides with that of the cluster.
The shaded region in
Figure 2 shows l-dimensional sequences of the acute and obtuse parallelograms whose edges are
FIGURE I Decoration of two unit cells A6 and 06 .
3. STRUCTURE FACTOR The observed electron-diffraction pattern of the quasicrystals has the icosahedral symmetry.3 As is well known4, the diffraction intensity is proportional to the structure factor F given by F:
~ fj(~)fk(~)exp[i~.(r-~j-~k)], jk
(2)
FIGURE 2 A 2-dimensional quasicrystal obtained by the projection from the 5-dimensional simple cubic l a t t i c e . Sequences of the acute and obtuse parallelograms along an S2 direction are shown by shaded area. Filled circles show vertices which do not belong to the sequences.
T. Yamaguchi, N. Fujima / A tomic structure of AI-Mn quasicrystal
perpendicular to the direction S2.
The compsi-
767
the quasiperiodic structure factor in Figure
tion rate of the obtuse and acute parallelograms
4(a).
in each i n f i n i t e sequence tents to T.
only 2 atoms are located f o r the decorated $60
the f i r s t
We have
peak of the d i f f r a c t i o n at the posi-
I f we neglect l a t t i c e planes on which
in Figure 5(b), the l a t t i c e spacings are equal
tion (sa = 1.266×2~) corresponding to the in-
to those of $60 but reduced to I / T .
verse of the average l a t t i c e spacing.
have the f i r s t
Note that
r e l a t i v e positions of the sequences and those of
Then, we
peak of the decorated $60 at the
position T times of that of $60.
In Figure 5,
atoms between the sequences are located coherent-
the l a t t i c e spacings of K30 and Mackay 54 are
l y and that the structure factor is clear-cut.
also shown for comparison.
3.2. Three-dimensional icosahedral clusters
As shown in Figure 6(a), the l a t t i c e spacings of $60 in C5 direction are t and t/T 3.
and quasicrystal Structure factors f o r $60 and the decorated
Then, we
have the structure factor in Figure 4(b) which
$60 along two-fold symmetry (C2) and C5 direc-
has peaks at positions multiple of I / t but
tions are shown in Figure 4.
shifted by I / t T 3 at around 4.5.
Note that Figure
A 3-dimensional quasicrystal is obtained by
4(a) is quasiperiodic and s i m i l a r to Figure 3. As shown in Figure 5(a), the l a t t i c e spacings of $60 in C2 direction are I : T .
Then, we have
the projection from the 6-dimensional simple cubic l a t t i c e .
This quasicrystal is decorated
F
1-
F
(a)
(a)
~.0-
0.5-
//
f
OF
F
1
(b)
1.0-
(b)
0,5-
o
sa/2~
,'o
FIGURE 3 Structure factor along S2 d i r e c t i o n . (a) For the cluster shown in the insertion. (b) For the 2-dimensional quasicrystal.
0
5
sa/2~
FIGURE 4 Structure factor for $60 shown by the blank region and for the decorated $60 by the shaded region. (a) Along C2 d i r e c t i o n . (b) Along C5 direction.
T. Yamaguchi, N. Fujima / Atomic structure of AI-Mn quasicrystal
768
in the way described in Sec.2. Structure factor for C5 direction is shown in Figure 7.
that in crystal Al, 2.86 A.
This is
similar to Figure 4(b).
In discussion, note that we can obtain quasicrystal taking Mackay 54 as the composition
I f we identify the f i r s t peak at sa = 2x/td (= 2.237x2~) as the strong diffraction peak
element. Note also that the structure factor of the decorated quasicrystal is approximated by a
(211111) in 6-dimensional index or (lO0000) in
convolution of those of the composion element
icosahedral index, we have the same value of the
and the un-decorated quasicrystal.4 As shown
l a t t i c e constant as that by Elser2, i . e . a =
in Figures 5 and 6, the l a t t i c e spacings of
4.594 A.
Then, we have the Mn-Al and AI-Al bond
length, Doa = 2.586 A.
This is just the Mn-Al
bond length in alloy but a l i t t l e shorter than 2
Mackay 54 are similar to those of $60. Then, we have the structure factor of Mackay 54 similar to that of $60.4
I f we identify the calcu-
lated peak as the experimental one, we have the
2
l a t t i c e constant 4.594 A.
2d-
The Mn-Al and AI-Al
bond length of Mackay 54 is T times of those of 1
3
8
12
3 z.
4
REFERENCES
2 9
2
I . T. Ogawa, J.Phys.Soc.Jpn. 54(1985)3205. 2. V. Elser, Phys.Rev. B32(1985)4892; Acta Cryst. A42(1986)36.
8
18
6
6
6
2 16
2
6
4
6
~6 2 6
14
crystal.
(b) (c) (d) FIGURE 5 Lattice spacing along C2 direction in the unit of l a t t i c e constant along C5 direction. Numerical values show the number of atoms. The parameter d is given by v ~ (= 1.0515). (a) $60. (b) Decorated $60. (c) K30. (d) Mackay 54.
3. D. Shechtman et a l . , Phys.Rev.Lett. 53(1984) 1951. 4. T. Yamaguchi and N. Fujima, J.Phys.Soc.Jpn. 57(1988)4206.
(a)
P
c5
5
5
5
lO
1
11
5
I
I
10 5 5 10
5 0
5
10
(a)
(b)
5
5
5 5 (c)
(d)
FIGURE 6 Lattice spacing along C5 direction. (a) $60. (b) Decorated $60. (d) K30. (d) Mackay 54.
11 ,/I1 . . , Ill II 5
i0
15
sa/2~ FIGURE 7 Structure factor of the decorated quasicrystal along C5 direction. Note that the scale of the horizontal axis is twice that in Figures 3 and 4
765
ATOMIC STRUCTUREOF AI-Mn QUASICRYSTAL T. Yamaguchi and N. Fujima Faculty of Engineering, Shizuoka University, Hamamatsu 432, Japan An atomic structure of AI-Mn quasicrystal is presented. Two unit cells A6 and 06 are decorated as follows: Mn atoms are put at a l l vertices of A6 and 06. Al atoms are put at the face-center which divides the long-diagonal of a l l golden rhombuses into the ratio l:T (the golden r a t i o ) . Two Al atoms are put at points which divide the body-diagonal of A6 into the ratio l:v~J:l. The structure factor of the decorated 3-dimensional quasicrystal is calculated and compared with the experiment. The l a t t i c e constant is obtained as 4.60 A which is T times of AI-Al bond length. ed from higher-dimensional periodic l a t t i c e s .
I . INTRODUCTION Non-periodicity, s e l f - s i m i l a r i t y , and the existence of a few u n i t cells are the most remarkable properties of quasicrystals.
The 3-
He showed that Bragg peaks are the a-function peaks and f i l l
densely the reciprocal space.
Fortunately, most of these peaks are extremely
dimensional Penrose transformation was presented
weak, making i t possible to distinguish i n d i v i d -
by Ogawa.l
ual ones. Whenthe vectors ei line up with the
He assumed two unit cells of golden
rhombus whose two diagonals have the golden
six f i v e - f o l d symmetry axes of the icosahedron,
ratio.
a general Bragg vector is written as
The solid angle around a main vertex is
x/5 for A6 and 77/5 for 06.
He obtained the
expanded A6 and 06 which are composed of 55 A6 and 34 06, and of 34 A6 and 21 06, respectively.
6 s = (~/a) Z n i e i , i=l
(I)
He obtained quasicrystals by recursive trans-
where the integers ni are incices and a repre-
formations.
sents the edge length of the rhombohedral unit-
In his transformation, the icosa-
hedral star-shaped polyhedra with 60 golden
cells A6 and 06.
rhombuses ($60) are located at a l l vertices of
by the Fourier transform of the density which
the expanded A6 and 06. of 20 A6.
This $60 is composed
There is a golden icosahedron F20
The structure factor is given
is constant inside a triacontahedron and f a l l s abruptly to zero outside of i t .
He identified
with 20 golden rhombuses between two $60 along
a theoretical odd-parity peak of the two-fold
each edge of the expanded A6 and 06.
pattern with the experimental peak and obtained
In the
i n t e r i o r of an expanded A6, there are two golden triacontahedra K30 with 30 golden rhombuses
the result that a = 4.60 A. On the basis of Ogawa's transformation and
which are surrounded by six $60 and share an 06.
Elser's value of quasilattice constant, many
Two S6O at the main vertices of an expanded 06
attempts have already been made to determine
share an A6.
There are 6 06 between the two $60
of an expanded 06 .
An expanded A6 is composed
the atomic structure of quasicrystals and to answer questions such as: Are atoms at the ver-
of l S6O, 2 K30 and 3 F2O but minus l 06 , and
tices of A6 and 06 , on the edge, on the face,
an expanded 06 is composed of l S6O, 3 F20 and
and/or on the body-diagonal in the interior?
6 06 but minus l A6. Elser 2 discussed characteristic features of the d i f f r a c t i o n pattern of quasicrystal project-
0022-3093/90/$03.50 (~) Elsevier Science Publishers B.V. (North-Holland)
We use two parameters t = I / ~ T = (5t+I)/2 (=I.6180).
(=0.4472) and
T. Yamaguchi, N. Fujima / Atomic structure of AI-Mn quasicrystal
766
where f is the atomic scattering factor, ~ the
2. DECORATION OF A6 AND 06 As shown in Figure l , we put Mn atoms ( f i l l e d circles) at a l l vertices of A6 and 06.
We put
scattering vector, and r j - r k the relative atomic coordinate-vector of the j - t h and k-th atoms.
Al atom at the face-center which divides the
The ratio of f(O) for Al and Mn is equal to the
long-diagonal of a l l golden rhombuses into the
ratio of atomic numbers which is nearly I/2.
ratio l:T (open c i r c l e ) .
We also put two Al
atoms at points which divide the body-diagonal of A6 into the ratio l : 5 t : l (shaded circles). The distance between Mn and the neighboring
3.1. Two-dimensional cluster and quasicrystal A 2-dimensional quasicrystal is obtained by the projection from the 5-dimensional simple cubic l a t t i c e , and shown in Figure 2.
There
Al is Doa = ~ 7 - ~ - ~ a = 0.5628a, that i s , the
are many clusters composed of 5 acute and 5 ob-
main diagonal of 06.
tuse parallelograms which are similar to K30 in
The distance between
the 3-dimension and shown in the insertion of
neighboring two Al atoms is also Doa.
Figure 3(a). Structure factors for the cluster and the quasicrystal along the S2 direction are shown in Figure 3(a) and 3(b), respectively. They are quasiperiodic as a function of sa.
Here,
the quasicrystal is composed of 77 atoms. The characteristic features such as the peak position of Figure 3(b) are well reproduced by Figure 3(a).
This means that the interval between
l a t t i c e planes of the quasicrystal coincides with that of the cluster.
The shaded region in
Figure 2 shows l-dimensional sequences of the acute and obtuse parallelograms whose edges are
FIGURE I Decoration of two unit cells A6 and 06 .
3. STRUCTURE FACTOR The observed electron-diffraction pattern of the quasicrystals has the icosahedral symmetry.3 As is well known4, the diffraction intensity is proportional to the structure factor F given by F:
~ fj(~)fk(~)exp[i~.(r-~j-~k)], jk
(2)
FIGURE 2 A 2-dimensional quasicrystal obtained by the projection from the 5-dimensional simple cubic l a t t i c e . Sequences of the acute and obtuse parallelograms along an S2 direction are shown by shaded area. Filled circles show vertices which do not belong to the sequences.
T. Yamaguchi, N. Fujima / A tomic structure of AI-Mn quasicrystal
perpendicular to the direction S2.
The compsi-
767
the quasiperiodic structure factor in Figure
tion rate of the obtuse and acute parallelograms
4(a).
in each i n f i n i t e sequence tents to T.
only 2 atoms are located f o r the decorated $60
the f i r s t
We have
peak of the d i f f r a c t i o n at the posi-
I f we neglect l a t t i c e planes on which
in Figure 5(b), the l a t t i c e spacings are equal
tion (sa = 1.266×2~) corresponding to the in-
to those of $60 but reduced to I / T .
verse of the average l a t t i c e spacing.
have the f i r s t
Note that
r e l a t i v e positions of the sequences and those of
Then, we
peak of the decorated $60 at the
position T times of that of $60.
In Figure 5,
atoms between the sequences are located coherent-
the l a t t i c e spacings of K30 and Mackay 54 are
l y and that the structure factor is clear-cut.
also shown for comparison.
3.2. Three-dimensional icosahedral clusters
As shown in Figure 6(a), the l a t t i c e spacings of $60 in C5 direction are t and t/T 3.
and quasicrystal Structure factors f o r $60 and the decorated
Then, we
have the structure factor in Figure 4(b) which
$60 along two-fold symmetry (C2) and C5 direc-
has peaks at positions multiple of I / t but
tions are shown in Figure 4.
shifted by I / t T 3 at around 4.5.
Note that Figure
A 3-dimensional quasicrystal is obtained by
4(a) is quasiperiodic and s i m i l a r to Figure 3. As shown in Figure 5(a), the l a t t i c e spacings of $60 in C2 direction are I : T .
Then, we have
the projection from the 6-dimensional simple cubic l a t t i c e .
This quasicrystal is decorated
F
1-
F
(a)
(a)
~.0-
0.5-
//
f
OF
F
1
(b)
1.0-
(b)
0,5-
o
sa/2~
,'o
FIGURE 3 Structure factor along S2 d i r e c t i o n . (a) For the cluster shown in the insertion. (b) For the 2-dimensional quasicrystal.
0
5
sa/2~
FIGURE 4 Structure factor for $60 shown by the blank region and for the decorated $60 by the shaded region. (a) Along C2 d i r e c t i o n . (b) Along C5 direction.
T. Yamaguchi, N. Fujima / Atomic structure of AI-Mn quasicrystal
768
in the way described in Sec.2. Structure factor for C5 direction is shown in Figure 7.
that in crystal Al, 2.86 A.
This is
similar to Figure 4(b).
In discussion, note that we can obtain quasicrystal taking Mackay 54 as the composition
I f we identify the f i r s t peak at sa = 2x/td (= 2.237x2~) as the strong diffraction peak
element. Note also that the structure factor of the decorated quasicrystal is approximated by a
(211111) in 6-dimensional index or (lO0000) in
convolution of those of the composion element
icosahedral index, we have the same value of the
and the un-decorated quasicrystal.4 As shown
l a t t i c e constant as that by Elser2, i . e . a =
in Figures 5 and 6, the l a t t i c e spacings of
4.594 A.
Then, we have the Mn-Al and AI-Al bond
length, Doa = 2.586 A.
This is just the Mn-Al
bond length in alloy but a l i t t l e shorter than 2
Mackay 54 are similar to those of $60. Then, we have the structure factor of Mackay 54 similar to that of $60.4
I f we identify the calcu-
lated peak as the experimental one, we have the
2
l a t t i c e constant 4.594 A.
2d-
The Mn-Al and AI-Al
bond length of Mackay 54 is T times of those of 1
3
8
12
3 z.
4
REFERENCES
2 9
2
I . T. Ogawa, J.Phys.Soc.Jpn. 54(1985)3205. 2. V. Elser, Phys.Rev. B32(1985)4892; Acta Cryst. A42(1986)36.
8
18
6
6
6
2 16
2
6
4
6
~6 2 6
14
crystal.
(b) (c) (d) FIGURE 5 Lattice spacing along C2 direction in the unit of l a t t i c e constant along C5 direction. Numerical values show the number of atoms. The parameter d is given by v ~ (= 1.0515). (a) $60. (b) Decorated $60. (c) K30. (d) Mackay 54.
3. D. Shechtman et a l . , Phys.Rev.Lett. 53(1984) 1951. 4. T. Yamaguchi and N. Fujima, J.Phys.Soc.Jpn. 57(1988)4206.
(a)
P
c5
5
5
5
lO
1
11
5
I
I
10 5 5 10
5 0
5
10
(a)
(b)
5
5
5 5 (c)
(d)
FIGURE 6 Lattice spacing along C5 direction. (a) $60. (b) Decorated $60. (d) K30. (d) Mackay 54.
11 ,/I1 . . , Ill II 5
i0
15
sa/2~ FIGURE 7 Structure factor of the decorated quasicrystal along C5 direction. Note that the scale of the horizontal axis is twice that in Figures 3 and 4