3.1. Unit-cell and space-group determination
There are several ways to identify crystallographic features from electron microscopy. For example, by comparing the net of reflections between ZOLZ (zero-order Laue zone) and FOLZ (first-order Laue zone) that emerge at precession angles > 0.5°, the Bravais lattice and glide planes can be inferred (Morniroli
et al., 2007
![[triangle]](/corehtml/pmc/pmcents/rtrif.gif)
). The point group may also be deduced from convergent-beam electron-diffraction (CBED) patterns. Here we determine the space group of
![[sm epsilon]](/corehtml/pmc/pmcents/x220A.gif)
16 by combining SAED and HRTEM techniques. Four electron-diffraction patterns of
16 along the [001], [010], [100] and [120] directions are shown in Fig. 1. From these, the unit cell was determined as B-centered orthorhombic with
a = 23.5,
b = 16.8 and
c = 32.4 Å.
16 has almost the same
a and
b parameters as
6 (unit-cell parameters
a = 23.5,
b = 16.8,
c = 12.3 Å), but
c is τ
2 times that of
6 (τ
![[similar, equals]](/corehtml/pmc/pmcents/sime.gif)
1.618 is the golden ratio).
The reflection conditions of
16 deduced from the SAED patterns are
hkl:
h +
l = 2
n.
According to the
International Tables for Crystallography (Hahn, 2002
), four space groups –
B222 (No. 21),
Bm2
m (No. 35),
B2
mm (No. 38) and
Bmmm (No. 65) – fulfill these reflection conditions. These four space groups are possible for
16 and cannot be distinguished by the reflection conditions only. Fortunately, the projection symmetries along the
b axis are different for these space groups:
cm for
B2
mm and
cmm for the others. It is possible to determine the projection symmetry of
16 by analyzing the phases extracted from the Fourier transform of the HRTEM images taken along the
b axis. This was performed by the program
CRISP (Hovmöller, 1992
). The projection symmetry of
16 was determined as
cm since the HRTEM image gave a much lower average phase error for
cm (phase residual = 19.5°) than that for
cmm (phase residual = 44.8°). This is also directly confirmed from the HRTEM image taken along [010] shown in Fig. 2. Only one mirror perpendicular to the
c axis appears in the [010] image, and no mirror perpendicular to the
a axis was found. Thus, the projected symmetry along [010] was determined as
cm but not
cmm and the only possible space group for
16 is
B2
mm.
3.2. Deducing a structure model
Once the space group has been determined, we are ready to deduce the structure model of
16 from the structure of
6 using the strong-reflections approach. The most important condition for the strong-reflections approach is that the intensity distribution of the strongest reflections between the known and unknown structures should be similar. Thus, the first step is to identify and relate the corresponding strong reflections of
16 to those of
6 using electron diffraction. For such a purpose, precession electron-diffraction patterns were taken as shown in Fig. 3, since they are less dynamical and show higher resolution (about 0.9 Å) than those of the SAED patterns in Fig. 1. The less-dynamical diffraction intensities obtained by precession electron diffraction make the identification of the corresponding strong reflections easier. As can be seen from the experimental precession electron-diffraction patterns of
6 and
16 in Fig. 3, all the strong reflections in
16 coincide with the strong reflections in
6. Since the
a and
b parameters are similar for
16 and
6, and the
c parameter of
16 (32.4 Å) is about τ
2 times that of
6 (12.3 Å, 32.4/12.3 = 2.634), the (
hkl) indices of the strong reflections in these two approximants are related by (
h k l
16) = (
h k τ
2
l
6). Here, the golden number τ = (1 + 5
1/2)/2 is associated with fivefold rotational symmetry of an icosahedral quasicrystal. Elser
et al. (Elser & Henley, 1985
) used the rational ratio of two successive Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, …,
F
n, …;
F
n =
F
n − 1 +
F
n − 2) as an approximation to substitute for the irrational τ to obtain the crystalline approximant of an icosahedral quasicrystal. According to the Fibonacci series, (
h k 8) in
16 is related to (
h k 3) in
6, (
h k 10) in
16 is related to (
h k 4) in
6, (
h k 26) in
16 is related to (
h k 10) in
6, and so on.
To confirm the similarity of the
16 and
6 structures, the
R value against the number of corresponding strongest reflections was plotted in Fig. 4 without any corrections (such as absorption correction, Lorenz correction and so on). The detailed reflection lists are given as supporting information.
1 The reflections from different orientations were merged using their common reflections. It shows that the intensities of the strong corresponding reflections are very close to each other (
R = 0.19) for the 30 strongest reflections. As more and more moderately strong reflections are included, the
R value increases, reaching 0.29 for the 146 strongest reflections. This kind of
R value is close to a typical internal
R value for elctron-diffraction data obtained from different particles of the same structure. Thus, we think there is obvious similarity in the
16 and
6 structures. Note that here we only compared the experimental strong reflections along the three main zone axes, but they constitute a major part of all data. For the 256 strongest reflections (the 45 strongest of which are listed in Table 2), the amplitudes along these three main zone axes sum up to 45% of the total amplitudes in
6.
| Table 2List of structure-factor amplitudes and phases of 45 strongest independent reflections of 6 (with Pnma) and 16 (with P1 and B2mm) |
The strong reflections in
16 are then deduced from the corresponding reflections in the known
6 according to the relations described above. In order to determine the number of independent strong reflections that are needed to obtain a sufficiently good electron-density map, we first checked the procedure on the
6 structure. Generally speaking, the structure-factor amplitude sum of the strong reflections must be more than 50% of the total amplitude sum in order to generate an electron-density map that represents the structure (Zhang, He
et al., 2006
). Here we choose the 256 strongest reflections of
6 among the total 2640 independent reflections within 1.0 Å resolution. Table 2 lists 45 of them, together with their structure-factor amplitudes and phases. The 256 reflections sum up to 57% of the total amplitude and they were expanded to 1590 reflections according to the symmetry of
6. A three-dimensional electron-density map was calculated from the structure-factor amplitudes and phases of these 1590 reflections by inverse Fourier transformation, using the program
eMap (Oleynikov, 2006
). All the atomic positions of
6 as determined by single-crystal X-ray diffraction (Boudard
et al., 1996
) could be found in the three-dimensional density map. This indicates that the 256 strong reflections are sufficient to obtain a correct structure model of
6. Since the strong reflections in
16 coincide with the strong reflections in
6, a reasonable structure model of
16 should be obtained using the 256 strong reflections of
16 deduced from those of
6.
The structure-factor amplitudes of
16 were assigned from those of the corresponding reflections in
6, calculated from the single-crystal X-ray structure model (Table 2). We did not use the amplitudes from the experimental PED data since it was difficult to collect a complete three-dimensional PED data due to mixed phases in the same particle. A new technique, called electron-diffraction tomography, is being developed in our department (Zhang
et al., 2010
) and may be applied in the future for collecting complete three-dimensional electron-diffraction data. Our earlier studies have shown that the amplitudes taken from the corresponding known approximants are enough to deduce the structure model.
Different from our previous studies (Christensen
et al., 2004
; Zhang, He
et al., 2006
; Zhang, Zou
et al., 2006
; He
et al., 2007
), the structure-factor phases for the strong reflections of
16 cannot be taken directly from those of
6, since the choice of origin for the space group of
16 (
B2
mm) is different from that of the space group for
6 (
Pnma). Consequently, the phase relations between the symmetry-related reflections in the two space groups are different. For the space group
Pnma, the relations are (Table 2):
- (i) If h + l = 2n and k = 2n
- (ii) If h + l = 2n and k = 2n + 1:
- (iii) If h + l = 2n + 1 and k = 2n:
- (iv) If h + l = 2n + 1 and k = 2n +1:
For the space group
B2
mm, the relations are simpler
One way to overcome the problem of the different phase relationships is to first assume the space group
P1 for
16. Starting from 256 strong reflections in
6, they are expanded into 1590 reflections using
Pnma symmetry. Based on the strong-reflection approach, each of these strong reflections in
6 structure has one corresponding reflection in the
16 structure in
P1 symmetry with the same phases and amplitudes but different indices [see Table 2, column Phase
6 (
Pnma) and
16 (
P1)]. This
16 structure in
P1 symmetry turns out to be very close to
B2
mm symmetry. Note that although the phase relations between the symmetry-related reflections in the two space groups
Pnma and
B2
mm are different, the phase relations for the strongest reflections in the two structures are almost the same.
The three-dimensional electron-density map of
16 in
P1 symmetry gives well resolved peaks that can be assigned to atomic positions, as shown in Fig. 5. From the density map viewed along the
b axis (Fig. 5
a), the banana-shaped tiles and pentagonal tiles (Balanetskyy, Grushko & Velikanova, 2004
) can be identified. The two types of tiles are alternating along the
a and
c directions and connected to each other. Similar tiles and connections are observed in the [010] HRTEM image in Fig. 2. The symmetries can be identified from the three-dimensional electron-density map as follows:
B-centering, 2 //
a,
m ![[perpendicular]](/corehtml/pmc/pmcents/x22A5.gif)
b,
m c, which agrees with the space group
B2
mm. An origin that is compatible with the space group
B2
mm was found at a 2
mm Wyckoff position, (0, 0.25, 0.15625). Thus, the origin was shifted to this position and the new structure-factor phases were calculated using the following equation
The new structure factor phases of 45 reflections from the 256 independent strong reflections are listed in Table 2, together with the symmetry-imposed phases. The average deviation of the phases from the symmetry
B2
mm is only 7.8°, and the largest phase error is 22.5°. The amplitudes together with the phases of the 256 reflections after imposing the symmetry
B2
mm were used to calculate a new three-dimensional electron-density map of
16. The three-dimensional density map is very similar to that with
P1 symmetry, but the electron densities at symmetry-related positions become exactly identical instead of just similar to each other.
There were 150 unique peaks corresponding to the atomic positions of
16 identified from the three-dimensional density map and the atomic coordinates were determined. There were 33 of them assigned as Rh, and the remaining 117 were Al. The assignment of Rh positions was based on:
- (i) the peak height (the highest peaks) and
- (ii) chemical knowledge.
Since
16 and
6 are both members of the same series of icosahedral quasicrystal approximants, they are expected to have very similar local atomic structures. Atoms in similar clusters should have a similar environment, and thus atoms in
16 were assigned to form similar clusters as those in
6. In addition, three Al atoms were added to complete the structure based on the geometry and similarity to
6, see Table 3 in the
supplementary material. Most of the atoms have reasonable distances to their neighbors, ranging from 2.2 to 3.1 Å. The final composition of our
16 model is Al
340Rh
99, which fits the synthesis stoichiometry (Al
77Rh
23) very well.
The precession electron-diffraction patterns simulated using the derived
16 structure model (Figs. 3
g–
i) agree well with the experimental precession electron-diffraction patterns (Figs. 3
d–
f). A least-squares refinement was performed with only four refined parameters (overall scale factor, extinction parameters and isotropic atomic displacement parameters for Rh and Al) using the experimental
16 intensities from PED patterns (659 independent reflections within 1.0 Å resolution) in the
SHELXL program (Sheldrick, 2008
). The refinement converged with
R
1 = 0.33. Thus, the structure model deduced for
16 from
6 can be taken as a good preliminary model.
3.3. Structure description
The final structure of
16 viewed along the
c axis is shown in Fig. 6(
a). There are eight layers that are perpendicular to the
b axis in each unit cell, including four flat (
f) and four puckered (
p) layers. Five of the layers, at
y = 0 (
f
1), ~ 0.125 (
p
1), ~ 0.25 (
f
2), ~ 0.375 (
p
2) and 0.5 (
f
3), are independent (Figs. 6
b–
f). The other three can be generated by a mirror located at
y = 0.5. The atoms within the
f
2 layer deviate slightly from
y = 0.25. Since this deviation is very small (< 0.019),
f
2 is still considered as a flat layer. Therefore, the structure can be described as
, where
are related to the layers
p
1
f
2
p
2 by a mirror operation at
y = 0.5. Each layer contains three common basic tilings: a squashed hexagon (
h), a pentagonal star (
s) and a crown (
c) (marked in Fig. 6
d). Two
h tilings together with one
c tiling form a decagonal ring. Each
s tiling is surrounded by five inter-connected decagonal rings (marked in Fig. 6
f). An additional eight-ring tiling (
o) is also found in the layer
f
2 (marked in Fig. 6
e).
The structure of
16 can be described using two types of columns along the
b axis: a decagonal column (
Dc) and a pentagonal column (
Pc) (marked in Fig. 6
b). The construction of the decagonal and pentagonal columns is given in Fig. 7. Each puckered layer contributes a decagonal ring to the decagonal column. Between the decagonal rings are small pentagons (in
f
1 and
f
3) and irregular tilings (in
f
2) that are alternating along the
b axis (Fig. 7
a). The pentagonal column is constructed by pentagonal stars or pentagons that are stacked along the
b axis (Fig. 7
b). The centers of the decagonal columns are always occupied by the heavy Rh atoms. The decagonal columns are connected to each other through edge-sharing (
via two common atoms). Five decagonal columns form a pentagonal tile with a pentagonal column inside (Fig. 6
b). Nine decagonal columns form a banana-shaped tile with two pairs of intersecting pentagonal columns inside. The banana-shaped and pentagonal tiles are seen in the HRTEM images (see Fig. 2). Similar decagonal and pentagonal columns have also been observed in the decagonal quasicrystal d-Al–Pd (Li
et al., 1996
).
A comparison of the structure features of
6 and
16 is given in Fig. 8. Both structures can be described by the decagonal and pentagonal columns. All decagonal columns in
6 and
16 are connected by edge-sharing. The structure of
6 is constructed by only one type of hexagonal tile built from six decagonal columns with two intersecting pentagonal columns inside. The structure of
16 is constructed by two types of tiles: a banana-shaped tile and a pentagonal tile. The diameter of decagonal and pentagonal columns is about 7.6 Å in both structures. This is also the edge length (marked by thick lines in Fig. 8) of the different tiles. We designate this edge length as ‘S’, as the short distance in a Fibonacci series as found in quasicrystal structures. The diagonal distance of the pentagonal tile is τ times longer than S, and thus defined as ‘L’. As shown in Fig. 8, the
c parameters can be given by S and L as follows:
6:
c = L = τS
12.3 Å;
16:
c = S + L + L
32.2 Å. Consequently the
c parameters for the other approximants in the series are:
22:
c = S + L + L + L
44.5 Å;
28:
c = S + L + L + L + L
56.8 Å. They are related by 1: (1 + τ): (2 + τ): (3 + τ).
We expect that it will be possible to obtain the structures of the complete
series and describe them using the stacks of decagonal and pentagonal columns. Furthermore, a detailed structure model of the decagonal quasicrystal (d-Al–TM) can possibly be deduced from these common features.
