0.0989(13). 0.007(3). Zn2b 2b. 1. - 1/3. - 2/3. -0.0922(13) 0.005(3). Gd. Zn2a 2b. 1. 1/3. 2/3. 0.1020(13). 0.010(3). Zn2b 2b. 1. - 1/3. - 2/3. -0.0878(12) 0.016(4). Tb.
Syntheses and structural studies of quasicrystal approximants: RECd6 and RE13Zn∼58 Licentiate Thesis
Shuying Piao
Department of Inorganic Chemistry Stockholm University 2005
Front page: The structure of Ce13Zn58 compound.
Akademisk avhandling som för avläggande av filosofie licentiatexamen i ooganiskkemi vid Stockholm uniersitet framlägges till offentlig granskning i Magnélisalen, kemiska övningslaboratoriet, Svante Arrhenius väg 12, onsdagen den 14 December, 2005, kl. 13:00. Extern granskare: Docent Torbjörn Gustavsson, Uppsala Universitet Intern granskare: Docent Mats Johnsson, Stockholm universitet
© Shuying Piao, December 2005
Abstract After the first reports on the structures of the rare earth-Zn 13:58 phases in 1967, such phases have been identified in a large number of rare/alkaline earth-zinc/cadmium systems mainly by X-ray powder methods. Since A. P. Tsai and J. Q. Guo reported the presence of quasicrystals in the Ca-Cd and Yb-Cd systems, a variety of quasicrystalline phases in binary systems have been studied. So far, however, the actual structures of most quasicrystals are little investigated. On the other hand, the structures of the quasicrystal approximants play a key role in understanding quasicrystals, since they are expected to display the same local arrangement as in the quasicrystals, their long-range order make their structural determination possible by standard methods. In this work a series of quasicrystal approximants in the RE13Zn58 and RECd6 (RE = rare earth elements) systems were synthesized and structural studies were performed by single crystal X-ray diffraction methods. This licentiate thesis is based on study of these two systems. For the RE13Zn58 quasicrystal approximants (RE = Ce, Pr, Nd, Sm, Gd, Yb, Dy, Ho, Er, Tm, Yb and Lu), single crystals were prepared from the elements and the structures have been refined from single crystal X-ray diffraction data. Some of the systems show measurable compositional variations coupled to subtle structural differences, as refined from singlecrystal x-ray diffraction data. The crystal structures are generally rather more complex than previously reported, and exhibit a number of different ordering modes. The study of the RECd6 quasicrystal approximants (RE = Tb, Ho, Er, Tm and Lu), is a continuation of a previous study of MCd6 approximants (M = Pr, Nd, Sm, Eu, Gd, Dy, Yb) in which the different types of disorder of the central Cd4 tetrahedra located in the dodecahedral cavities were examined. The structures of the title compounds are all similar to previously reported GdCd6 and disorders was observed in all these compounds.
I
List of papers 1. S. Y. Piao, C. P. Gómez and S. Lidin. Complexity of hexagonal approximants in the RE13Zn∼58 system (RE = Ce, Pr, Nd, Sm, Gd, Tb and Dy) Z. Kristallographie (2005). Accepted. 2. S. Y. Piao, C. P. Gómez and S. Lidin. Structural study of the disordered RECd6 quasicrystal approximants (RE = Tb, Ho, Er, Tm and Lu) Z. Naturforsch. 60b (2005). In press.
II
Table of Contents Abstract List of Papers Table of Contents 1. Introduction ............................................................................................................................ 1 1.1 Aim and Scope ................................................................................................................. 1 1.2 Quasicrystals and Approximants...................................................................................... 2 1.2.1. The Fibonacci series ................................................................................................ 2 1.2.2 The Fibonacci chain (The Fibonacci sequence) ....................................................... 5 1.2.3 Penrose Tilings.......................................................................................................... 6 1.2.4 Quasicrystals and approximants............................................................................... 7 1.3 Intermetallic compounds .................................................................................................. 7 1.4 Prototype structures.......................................................................................................... 8 1.4.1 Fundamental building blocks .................................................................................... 8 1.4.2 The prototype structure of RECd6 phases................................................................ 10 1.4.3 The prototype structure of RE13Zn58 phases............................................................ 11 1.4.4 Relation between RECd6 and RE13Zn58 phases........................................................ 12 2. Experimental ........................................................................................................................ 14 2.1 Preparation ..................................................................................................................... 14 2.2 Elemental analysis.......................................................................................................... 15 2.3 Single crystal X-ray data collection and structural refinement ...................................... 16 3. Results and discussions ........................................................................................................ 17 3.1 RECd6 phases ................................................................................................................. 17 3.2 RE13Zn~58 phases............................................................................................................ 23 3.2.1. Basic structural considerations.............................................................................. 23 3.2.2 Structure description ............................................................................................... 25 3.2.3 Trends and Phase distribution in RE13Zn～58 system ............................................... 32 4. Summary and conclusions.................................................................................................... 34 5. Future work .......................................................................................................................... 35 Acknowledgements ................................................................................................................... 36 References ................................................................................................................................ 38 Appendixes ............................................................................................................................... 40
III
1. Introduction 1.1 Aim and Scope Quasicrystals are intermetallic compounds of specific stoichiometry, which often exhibit conventionally forbidden rotational symmetries in their diffraction patterns
[ 1 , 2 , 3 ]
.
Compositions, structures, and physical properties have been studied intensively for a variety of quasicrystalline phases in binary and ternary intermetallic systems
[ 4, 5, 6, 7]
. So far,
however, the actual structures of most quasicrystals are little investigated. On the other hand, some crystalline intermetallic compounds are considered "quasicrystalline approximants"; because their building blocks contain high-symmetry polyhedra that can be used as possible models for components of quasicrystalline structures
[2,3, 8 , 9 , 10 ]
. The structures of the
quasicrystal approximants play a key role in understanding quasicrystals, since they are expected to display the same local arrangement as in the quasicrystals, their long-range order make their structural determination possible by standard methods. Thus they provide a link to the underlying mechanism of quasicrystal formation. The reassessment of the approximant CeCd6 to Ce6Cd37
[11]
revealed a previously unreported
order in the central tetrahedron and prompted the re-examination of the structures in the RECd6 system. Pay-Gómez et al. have extensively studied some of the RECd6 approximants (RE = Pr, Nd, Sm, Eu, Gd, Dy, Yb) in the rare earth–cadmium system [12] and found different types of disorder with the central tetrahedra located in the dodecahedral cavities. The recent discovery of stable icosahedral quasi crystals in phases of similar composition in the RE-Cd systems [13,14] together with the elucidation of the complex order-disorder behavior of the cubic approximants [15,16,17,18] has sparked a renewed interest for more detailed understanding of the hexagonal 13:58 phases. From a study of the Dy-Zn
[19]
system, it is apparent that the 13:58
phases show similar complexity. In deed the binary systems between rare earth and zinc group metals have been studied since the 1960ies. After the first reports on the structures of the 13:58 phases in 1967[20 21], such phases have been identified in a large number of rare/alkaline earth-zinc/cadmium systems
1
[ 22 , 23 ]
. But most reports on the phases deal with characterization by powder diffraction
methods and frequently no refinements are reported. The structural and geometrical relations among the types of structure occurring were examined for all RE-Cd intermetallic compounds in 1973 [24] by G. Bruzzone et al. An interesting early example published without structural information deals with the system Ce-Cd where Roof and Elloit
[25]
report on Ce13Cd58 as a
collection of micro phases, where the variation of Cd content is coupled to the position of a set of satellite reflections. The precession images from the original paper are strongly indicative of a modulated structure. A closer examination of the early literature reveals many ambiguities in the structural descriptions of these related compounds. One of the aims of this work is to complete the study on the RECd6 family. Another, larger part of this work deals with the RE13Zn58 family.
1.2 Quasicrystals and Approximants 1.2.1. The Fibonacci series Leonardo Fibonacci, who was born in the 12th century, studied a sequence of numbers with a different type of rule for determining the next number in a sequence. He began the sequence with 0, 1, ... and each new number in the series is simply the sum of the two before it. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ….. This sequence of numbers is called the Fibonacci Numbers or Fibonacci series. The Fibonacci series is interesting in that it occurs throughout both nature and art. Especially of interest is what occurs when we look at the ratios of successive numbers. One property of the Fibonacci series is that the ratio of an element ƒn+1 and its preceding element ƒn rapidly approaches the irrational number τ , which is known as “the Golden Mean” or “the Golden Ratio”. The higher the number defining the ratio, the better the approximation is.
limƒn+1/ƒn→
2
τ
τ can be derived with a number of geometric constructions, each of which divides a line segment at the unique point where: the ratio of the whole line (AB) to the greater segment (AO) is the same as the ratio of the greater segment (AO) to the smaller segment (OB). See Figure 1.1.
O
A 1
B X-1
Figure 1.1. The golden ratio.
In an equation, we have
AB AO = , the Golden Ratio is the ratio of AO to OB. If we AO OB
arbitrarily set the value of AO to be 1, and use x to represent the length of entire line AB, then x 1 = 1 x −1 1+ 5 = τ ≈ 1.618 2 1− 5 1 x2 = =− 2 τ x1 =
Obviously, only the positive root has a physical meaning. The number τ has been used by mankind for centuries. It is closely associated with 5-fold symmetry and the occurrence of icosahedral coordination. The distance between nearest and next nearest corner in an icosahedron is in the proportion 1 to τ . See Figure 1.2. The number τ is frequently observed to relate the d-values of strong spots in quasicrystal and approximant
3
diffraction patterns. It has several remarkable mathematic relations, of which the power series is especially useful in geometric calculations concerning the icosahedron. τ 0=1
τ 1= τ τ 2= τ +1 τ 3= τ 2+ τ =2 τ +1 τ 4= τ 3+ τ 2=3 τ +2 τ 5= τ 4+ τ 3=5 τ +3 τ 6= τ 5+ τ 4=8 τ +5 τ 7= τ 6+ τ 5=13 τ +8
Figure 1.2. Relation between distances in the icosahedron. The ratio of bond distances between next neighbour and neighbour is τ.
4
1.2.2 The Fibonacci chain (The Fibonacci sequence)
The Fibonacci chain plays a central role in the structures of icosahedral quasicrystal. It is an example of a 1 dimensional quasicrystal. It can be constructed from two different segments, one long and one short which are hereafter denoted L and S, respectively. The constructing rule is to replace S with L and L with LS, in the following way. The first eight cycles of the Fibonacci chain are shown in Table 1.1. Table 1.1. The first eight cycles of Fibonacci chain Cycle number
Sequence
Ratio of L/S
0
S
0/1
1
L
1/0
2
LS
1/1
3
LSL
2/1
4
LSLLS
3/2
5
LSLLSLSL
5/3
6
LSLLSLSLLSLLS
8/5
7
LSLLSLSLLSLLSLSLLSLSL
13/8
8
LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS
21/13
As seen in the Table 1.1, the whole chain can be constructed with the simple constructing rule and the resulting Fibonacci chain of atoms is quasiperiodic. If the chain was grown infinitely, we would see that it has no repetition distance and it is impossible to describe the long-range order with only one unit cell. The ratio of L/S in the Fibonacci chain quickly converges on Golden Ratio τ . After the 40th cycle in the series, the ratio is accurate to 15 decimal places: 1.618033988749895 . . . .We can also see that the total number of generated segments L+S= F for a given cycle n equals the sum of generated segments of two preceding cycles; this can be expressed as: Fn+1=Fn +Fn-1 Furthermore, we can observe that not only the total sum F, but also the resulting sequence for a given cycle n, is obtained by concatenating the sequences of L and S for the two preceding cycles n-1 and n-2, in that order.
5
If we start with Fn-1=0 and Fn=1, the first ten elements of the sequence are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34….. We have already been familiar with these numbers, the Fibonacci Series, which were described above. 1.2.3 Penrose Tilings
Before quasicrystals were discovered in 1984, the British mathematician Roger Penrose devised a way to cover a plane in a nonperiodic fashion using two different types of tiles (rhombi) (Figure 1.3 (a). An example can be seen in Figure 1.3 (b) below. The tiles are arranged in such a way that they obey certain matching rules. The rules themselves are: When constructing a Penrose tiling, two adjacent vertices must both be blank or must both be black. If two edges lie next to each other they must both be blank, or both have an arrow. If the two adjacent edges have arrows, both arrows must point in the same direction.
(a)
(b)
Figure 1.3. (a) Two different types of rhombi. The thin rhomb has angles of 36 and 144 degrees. The thick rhomb has angles of 72 and 108 degrees. (b) Part of a 2-dimensional Penrose tiling constructed by two building blocks by matching rules.
It is possible to put the blocks together without the matching rule. This method is called random tiling. The long range order will be preserved but there will be some space unfilled by the building blocks. It has recently been suggested that the structure can be described by one unit cell, which is repeated periodically in the structure [26, 27]. The cells are allowed to overlap
6
with a certain pattern. This model has some experimental verification in the decagonal quasicrystal Al72Ni20Co8 [28]. In short, we can regard quasiperiodic tilings as frameworks that give quasicrystal structures when filled up with atoms in an appropriate way. 1.2.4 Quasicrystals and approximants
A quasicrystal is a material which shows diffraction patterns with rotational symmetries, that are "forbidden" by classical crystallography. The structure of a quasicrystal, quasiperiodic in the physical three-dimensional space, can be described as a periodic structure in a higher dimensional hyperspace. Quasicrystal structures show long-range order, but no translational periodicity and they can be approximated by filling an appropriate quasiperiodic tiling with atoms. In the compositional vicinity of quasicrystalline phases one often finds periodic phases with large unit cells, having cell parameters that can be expressed as a function of the hyperspace cell parameter associated with the corresponding quasicrystal and the golden mean
(
)
τ = 1 + 5 / 2 . These phases are termed approximants. Crystalline approximants are structures which are periodic, but which are very similar to quasicrystals. These approximants play an important role in solving quasicrystal structures.
1.3 Intermetallic compounds The term intermetallic is used for compounds between metals. Intermetallic compounds comprise a highly diverse set. They possess structures and physical properties that may be very different from the constituent elements. They are also different from alloys from a structural point of view. When a alloy is formed, one of the participating reactants functions as the host and its atomic structure will persist after the reaction, The guest reactants will be inserted into the host matrix as either substitutional or interstitial atoms. This is not the case in intermetallic compounds, where the created structure is different from the parent structures. Many have a fixed stoichiometric composition, but some exist over a very large compositional range. There are several types of intermetallic compounds and one way of
7
classifying them is by the difference in electronegativity between the constituents. The intermetallic compounds investigated as part of this licentiate thesis fall into the category of polar intermetallics, since the valence electrons of the more electropositive RE metals to a larger extent are localized around the more electronegative Cd or Zn atoms. All reported quasicrystals are intermetallic compounds.
1.4 Prototype structures The sequence of related phase can be often described as a succession of symmetry reductions of a certain common parent structure called prototype structure. The reduction of the symmetry of the prototype leads to the occurrence of domain structure in the lower-symmetry phases. 1.4.1 Fundamental building blocks
The different compounds in RECd6 and RE13Zn58 systems are structurally related to each other and can to a certain extent be described by the same structural building blocks [19]. One of the fundamental building blocks of both phase types is the double pentagonal anti-prism (DPAP) formed by 15 Zn atoms around RE atom (Figure 1.4). The unique arrangements of different RE atoms surrounded by Cd/Zn or RE atoms are shown in Figure 1.5 (a)-(d). The pentagonal faces of the DPAPs are capped, either by Cd/Zn or by RE atoms. The different arrangements seen in the figure are the Zn1 mono-capped, Zn2 bi-capped, RE1Zn1 bi-capped and RE2 bicapped DPAP. In 1:6 phases, only case (a) and (b) can be seen; in 13:58 phases, on the other hand, all kinds of DPAPs exist in the different compounds. But it is not necessary for them to exist simultaneously. According to the number of Cd or Zn atoms which are
8
around the different rare earth atoms, they are called RET15, RET16, RET17 polyhedra respectively, T = Cd or Zn).
Figure 1.4. A double pentagonal anti-prism (DPAP) formed by 15 Zn atoms around one RE atom in Ce13Zn58 compound.
(a)
(b)
(c)
(d)
Figure 1.5. Capping schemes of DPAPs. Figure (a) to (d) show the DPAPs capped (a) by a Zn atom solely on one side (in HoCd6 compound); (b) on both pentagonal faces by Zn atoms (in HoCd6 compound); (c) by a RE atom on one side and a Zn atom on the other side (in Ce13Zn58 compound); (d) on both pentagonal faces by RE atoms (in Ce13Zn58 compound).
A second fundamental building blocks of both phase types are T8 cubes. The T8 cubes are the interstitial cavities formed in between the RET15, RET16, RET17 polyhedra. They exist in all the 1:6 and 13:58 phases. In some of the compounds, additional T atoms were observed within the T8 cubes. The third fundamental building block which is unique for the 13:58 phases is the rows of alternating RE and Zn atoms residing the hexagon-shaped tunnel.
9
1.4.2 The prototype structure of RECd6 phases
Three different type structures have been assigned as prototypes structures in the system RECd6; YCd6 [29], YbCd6 [30], and Ru3Be17 [31]. The skeletal networks of these three types of structures are identical; their structures can all be described as a simple body centred cubic (bcc) packing of partially interpenetrating defect triacontahedral cluster units as shown in Figure 1.6[12]. Another alternative representation of the triacontahedral cluster units is by building block of 12 RECd16 polyhedra and 8 Cd8 cubes as shown in Figure 1.7 [12].
Figure 1.6. The structure of the RECd6 phases can be displayed as a bcc packing of partially interpenetrating triacontahedral cluster units.
Figure 1.7. The basic building block of the RECd6 phases. The cluster unit is built up of 12 RECd16 polyhedra and eight Cd8 cubes.
10
The difference between three different type structures lies solely in their description of the species residing inside the central dodecahedral cavity that is found in all the RECd6 phases. In the case of Ru3Be17 the cavity is reported to be empty, while in the other prototype structures it contains a Cd4 tetrahedron exhibiting various types of disorder. The disorder of that tetrahedron is in YbCd6 modelled by a cube with one-half occupancy of all vertices. In the case of YCd6 the model is an icosahedron with one-third occupancy of all vertices. 1.4.3 The prototype structure of RE13Zn58 phases
Two different prototype structures have been used for the 13:58 phases. One, Gd13Cd58 [32], was actually first determined for the compound Pu13Zn58 [20] in the centro-symmetric space group P63/mmc. The structure is built up of Gd centred DPAPs and Cd centred Cd-cubes; together they form hexagon-shaped tunnels that are inhabited by rows of alternating Gd and Cd atoms as shown in Figure 1.8 (a). The other prototype structure was first given for Gd13Zn58
[20]
[Figure 1.8 (b)] in the non centro-symmetric space group P63mc. The type
structure is built up of Gd centred DPAPs and Zn centred Zn–cubes, together they form starshaped tunnels that are inhabited by rows of alternating Gd and Zn atoms. The two prototypes are very similar; the symmetry breaking is manifested in the movement of several atoms away from the mirror plane perpendicular to the rotation hexad.
(a)
(b)
Figure 1.8. RE13Zn58 prototype structures. (a) Gd13Cd58. (b) Gd13Zn58.
11
1.4.4 Relation between RECd6 and RE13Zn58 phases
The structures of different compounds in RECd6 and RE13Zn58 can to a certain extent be described by the same fundamental building blocks as described in the section 1.4.1. Figure 1.9 (a) and (b) show two similar building units which are very commonly seen in the structures for both systems. Both of them are combined by 8 RET16 polyhedra and 2 cubes. They differ only in the orientation of some of the RET16 polyhedra.
(a)
(b)
Figure 1.9. Similar building unit existing in both RECd6 and RE13Zn58 systems. (a) RECd6. (b) RE13Zn58.
An alternative way of describing the structures of both systems in a similar way is illustrated in Figure 1.10 (a) and (b). As seen in the figure, the network of the RECd6 phase is assembled by fusion of 2 building units and RE13Zn58 phase is by 3 building units. Therefore the RE13Zn58 structure is really a chemical 3-fold twin from the cubic structure. It has been found that the network of the RECd6 phases and the RE13Zn58 phase are obtained simply by changing the arrangement of these building units. Only unique feature of RE13Zn58 phase is the hexagon-shaped channel which we don’t see in RECd6.
12
(a)
(b) Figure 1.10. (a) The network of the RECd6 phase is assembled by fusion of 2 building units. (b) RE13Zn58 phase assembled by fusion of 3 building units.
13
2. Experimental 2.1 Preparation All materials were handled in an Ar-filled glove box in which the concentrations of water and oxygen were lower than 1 ppm. The single crystals used for the structural determination were obtained by mixing chips of target rare earth metal (STREM and CHEMPUR 99.9%) with Cd (Baker Chemicals 99.9%) splinters from a rod of pure metal in the molar proportions of 1:6 or with Zn ingot (Baker Chemicals 99.9%) in the molar proportions about 13:58 to 17:58. The mixtures were enclosed in sealed stainless steel ampoules for the Cd-containing samples. For the Zn-containing samples, using stainless steel ampoules may result in an enrichment of Ni in the final products. Therefore Nb ampoules were used instead. The ampoules were heated in regular muffle furnaces for the Cd-containing samples or in the vacuum furnaces for the Zncontaining ones to avoid the oxidization of Nb metal at high temperature. In order to obtain the desired phase in a well crystallized state, it was found effective to anneal the ampoules for a period of longer than 48 hours. The annealing temperatures were chosen at about 20 K below the reported melting points of each phase; the idea is let the more low-melting metal (Cd or Zn) act as a flux for the high-melting RE metals. For those systems where the pertinent information was lacking, several different annealing temperatures were tried to optimize the yield of the target phase. It was found that for some of the RE13Zn58 phases, namely Gd, Tb, Dy and Yb, it was possible to distinguish two phases, belonging to two different space groups. This may be inferred that this could be occurred by a solid solution field for the phase at elevated temperature. On cooling such a solution field could bifurcate into two separate phases. After annealing, the furnaces were turned off with the samples left inside to cool down slowly to ambient temperature (cooling rate 1~3 ˚C/min). Single crystals could easily be isolated from the resulting samples. All products were silvery and brittle, and sensitive to moisture and air. All pertinent details of the experiments are given in Table 2.1 and 2.2.
14
Table 2.1 Experimental details for syntheses of RECd6 Rare earth elements
Initial molar ratio (RE : Cd )
Reaction temperature [°C]
Reaction time [h]
Cooling rate [°/min]
Diffractometer
Tb
1:6
660
96
∼3
Stoe IPDS
Ho
1:6
780
48
∼3
Stoe IPDS
Er
1:6
800
48
∼3
Stoe IPDS
Tm
1:6
800
48
∼3
Stoe IPDS
Lu
1:6
820
72
∼3
Xcalibur
Table 2.2 Experimental details for syntheses of RE13Zn～58 Rare earth elements
Initial molar ratio (RE : Zn)
Reaction temperature [°C]
Reaction time [h]
Cooling rate [°/min]
Diffractometer
Ce
15:58
850
48
∼3
Xcalibur
Pr
17:58
880
48
∼3
Xcalibur
Nd
17:58
830
72
∼3
Xcalibur
Sm
17:58
850
96
∼3
Xcalibur
Gd[I]
13:58
870
72
∼3
Stoe IPDS
Gd[II]
13:58
835
48
∼3
Xcalibur
Tb[I]
13:58
830
90
∼3
Stoe IPDS
Tb[II]
15:58
83
90
∼3
Stoe IPDS
Dy[I]
15:58
870
48
∼3
Xcalibur
Dy[II]
13:58
870
87
∼3
Stoe IPDS
Yb[I]
15:58
720
48
∼3
Xcalibur
Yb[II]
17:58
720
48
∼3
Xcalibur
[I] form hexagonal phases; [II] form non hexagonal phases.
2.2 Elemental analysis Elemental compositions were ascertained with EDX (Energy Dispersive X-ray) analysis, using a LINK AN10 000 system mounted in a JEOL 820 scanning electron microscope,
15
operated at 20 kV. These analyses usually were in good agreement with the final compositions obtained from the refinements. Elemental impurity levels in the samples were below the detection limit of EDX. In all reactions the target structures were the major crystalline products.
2.3 Single crystal X-ray data collection and structural refinement Single crystals suitable for X-ray diffraction analysis were selected from small amounts of crushed products. Crystals showed irregular shapes and were mounted on the tips of glass fibers. The single-crystal X-ray diffraction data for detailed structural analyses were collected at 298K either on a Stoe IPDS single-crystal X-ray diffractometer with a rotating anode Mo Kα X-ray source operated at 50 kV and 90 mA or on an Oxford Diffraction Xcalibur CCD diffractometer with a graphite monochromatized Mo Kα radiation (λ = 0.71073Å) operated at 50 kV and 40 mA, and a detector to crystal distance of 80 mm. The range of 2θ values was 3.0-57.0ο. The intensities of the reflections were integrated using the machine specific software. Due to the twinning, absorption correction on the crystal by the actual measurement of the shape is inapplicable. Therefore, a numerical absorption correction, based on a shape obtained by optimizing the equivalence for symmetry related reflection, was performed with the programs X-RED [33] and X-SHAPE [34]. The refinements of the structures were performed using the program JANA2000 [35]. The structural analysis Electron-density isosurfaces were generated using the program JMap3D[ 36 ]. The images were rendered using the programs Diamond, version 2.1c [37] and Truespace, version 5.2 [38]. A detailed summary of the crystallographic data is given in Appendix 1 for RECd6 phases and in Appendix 2-3 for RE13Zn∼58 phases.
16
3. Results and discussions The results presented here are based on structural analyses on families of compounds where the members crystallize in similar structural build-ups. The overall structural relation between all the different compounds is described. Some of the results were presented in the Papers 1 [39]
and paper 2 [40].
3.1 RECd6 phases The results presented in this section are based on paper 1. Five compounds in RECd6 (RE = Tb, Ho, Er, Tm and Lu) phases which assigned to be approximants of quasicrystal RECd5.7 phases has been investigated in this work. The structures of the topic compounds are essentially similar to the structure of other members in the family as described in previous work [12]. The cluster unit of the basic building block of the RECd6 phases contains 12 RECd16 polyhedra and eight Cd8 cubes. Figure 3.1 shows the network of the RECd6 phases displayed by the cluster units of Cd atoms. The dodecahedra sitting in the corner and the centre serve as cages for the disordered Cd4 tetrahedra. The next shell of the dodecahedron in the centre is an icosidodecahedron, and the outermost shell in the centre is a defect triacontahedron.
Figure 3.1. The network of the RECd6 phases displayed by the cluster units of Cd atoms.
17
For the compounds studied previously, the innermost unit cluster (disordered tetrahedra) was described by two isotropic atomic positions, Cd1a and Cd1b, except for the case of the GdCd6 phase, where the irregular shape of the atoms was modelled solely by the position Cd1a with anisotropic displacement parameters. In the present structure refinements the electron density located in the innermost cluster was modelled using a single, anisotropic atomic position. The additional Cd atoms occasionally located inside the Cd8 cubes (cf. Ce6Cd37[11], Pr3Cd11-18[12] compounds) and the absence of the entire Cd4 tetrahedron in one case (cf. Ru3Be17[31]) are the main factors that result in the deviations from the ideal 1:6 stoichiometry of RECd6 phases. But neither of these factors was observed for the title compounds. The electron density isosurface at the 7 e/Å3 level for HoCd6 (Figure 3.2) shows that there is no electron density inside the cube, and none of the title compounds in the study show any signs of occupation of the cubes. Thus the ideal 1:6 stoichiometries of all the topic RECd6 phases were maintained. This result is consistent with the hypothesis that the compounds containing an RE atom smaller than Pr are not able to accept filled cubes [41].
Figure 3.2. Electron-density isosurfaces at the 7 e/Å3 level for HoCd6, generated from Fobs data originating from the individual single-crystal measurements, show that there is no electron density inside the cube. None of the title compounds shows any signs of occupation of the cubes.
Depending on whether the Cd8 cubes are filled or not, different mechanism on the disorders of the Cd4 tetrahedra comes into play. To describe the tetrahedral disorder mechanism, a general model based on the information gathered from the Fourier maps was used
18
[12]
. The model
relies on combinations of mainly two types of disorder: Type-1 disorder consist of a 90° rotational disorder of the Cd4 tetrahedron along its inherent two-fold axis resulting in an image of a semi occupied cube in the electron density maps [Figure 3.3 (a)]. Type-2 disorder is a triple split of the tetrahedral corner positions as a consequence of misalignment between the inherent three-fold axis of the Cd4 tetrahedron, and the three-fold axis of the cubic unit cell, see Figure 3.3 (b). The type-2 disorder is variable; when the splitting is sufficiently large, an intermediate state will be reached where the obtained coordination polyhedron will have the appearance of a cube octahedron. At this stage it is impossible to determine whether only the type-2 disorder is present, or both type 2 and type 1. If the triple spit is further increased, it will give the effect of pseudo rotation. The whole procedure is schematically showed in Figure 3.3 (c).
(a)
(b)
(c) Figure 3.3. Tetrahedral disorder mechanisms. In (a) the type-1 disorder, in (b) the type-2 disorder, and in (c), the effect of pseudo-rotation[12].
19
The atomic positions, isotropic displacement parameters and occupancies are given in Appendix 4. Basically the refinements reported in Appendix 4 are in good agreement with the result of the GdCd6 case, only type II disorders exist throughout all the topic compounds. The four atoms residing in the dodecahedral cavities of the title compounds form almost perfectly shaped cuboctahedra. The electron-density isosurfaces at the 9 e/Å3 level at the location corresponding to the centre of the dodecahedral cavity (Figure 3.4) clearly show the differences in the disorder of the Cd4 tetrahedron among the title compounds. As mentioned before the disordered tetrahedra for the phases containing Pr, Nd, Sm, and Dy were described with two atoms Cd1a and Cd1b isotropically, but it is possible to refine the positions by one anisotropic Cd1a position for the Nd, Sm and Dy compounds. For the Pr compound, this is not possible because of the complex shape of the electron density[12].
(a) TbCd6
(b) HoCd6
(d) TmCd6
(c) ErCd6
(e) LuCd6
Figure 3.4. Electron-density isosurfaces at the 9 e/Å3 level, from measured electron density corresponding to the disordered tetrahedra located inside the dodecahedral cavities of the different RECd6 phases, show the differences in the disorder of the Cd4 tetrahedron among the title compounds.
20
In order to correlate the disorder of the tetrahedra to the size of the rare earth element, the largest anisotropic component, U11, was compared to the unit cell dimensions. In Figure 3.5 the magnitude of U11 is represented as a function of the unit cell dimension of the RECd6 phases. A well defined trend is observed, wherein the motion of the atoms making up the tetrahedron inside the dodecahedral cavity depends strongly on the size of the unit cell, and consequently on the effective size of the rare earth atoms. In conclusion the studies on the RECd6 phases show that the structure of the topic compounds closely resembles that of GdCd6. While 1:6 compounds formed with large rare earth elements show a remarkable diversity of disorders, the smaller rare earth elements lead to smaller
Figure 3.5. The magnitude of U11 is represented as a function of the unit cell dimension of the RECd6 phases. A well defined trend shows that the motion of the atoms making up the tetrahedron inside the dodecahedral cavity depends strongly on the size of the unit cell, and consequently on the effective size of the rare earth element.
central cavities that do not allow for such behaviour. Further, none of the topic compounds shows any occupancy of the Cd8 cubes. It would seem that large rare earth elements tend to expand the Cd-network, leading to the creation of larger cavities; the Cd8 cubes are able to host additional Cd atoms. The network formed by the Cd8 cubes and the dodecahedral cavities is a strongly correlated system, and the presence of Cd atom inside the Cd8 cubes results in a
21
deformation of the dodecahedral cavity, making the environment much less spherical. Thus, for compounds with ordered cube occupancies, there is a strong tendency towards oriental ordering of the central tetrahedra as well. It is the case that we see in Eu and Ce compounds. Figure 3.6 shows the Cd4 tetrahedral residing in the dodecahedral cavity surrounded by cubes. In Figure 3.6 (a) half of the cubic interstices in Ce6Cd37 compound are filled by Cd atoms. Note how the filled cube in the circle displaces one of the vertices of the dodecahedron towards the centre. To avoid short Cd-Cd distances, all vertices of the tetrahedra are oriented towards vacant cubes and all the faces towards occupied cubes. In Figure 3.6 (b) for cases when the cubes are all empty, the dodecahedron is undistorted, and as a consequence, there is no preferred orientation of the tetrahedron. Therefore the tetrahedron appears to be a cuboctahedron. As the size of the rare earth element goes down from intermediate to small sizes, the displacement ellipsoids of the Cd atoms shrink. Thus there is not enough space inside the cubes. Since there are no filled cubes, the central tetrahedron has a much more isotropic environment, and the tendency for ordering disappears. The shrinking of the dodecahedral cavity does, however, lead to a more restricted motion of the tetrahedron as it goes down in size. The decreasing free volume in the central cavity is clearly displayed in the dependence of the anisotropic displacement parameters (U11) of the tetrahedra atoms on the unit cell dimensions. It is interesting to speculate on the behavior of the atoms in this position as the radius relation between the minority and majority atoms is decreased further, and there are reports on empty dodecahedral cavities [31], but these need further substantiation.
22
(a) Ce6Cd37
(b) HoCd6
Figure 3.6. Within the dodecahedral cavity surrounded by cubes, a Cd4 unit resides. The disorder of this entity is determined by the interstitial positions in the surrounding. (a) Half of the cubic interstices in Ce6Cd37 compound are filled by Cd atoms. (b) All the cubic interstices in HoCd6 compound are empty.
3.2 RE13Zn~58 phases The results presented in this section are based on paper 2 and deal with RE13Zn~58 (RE = Ce, Pr, Nd, Sm, Gd, Tb and Dy) phases. Further the results from Yb13Zn∼58 compounds have been added in this thesis. Besides a brief introduction on Ho, Er, Tm, Lu containing compounds is given.
3.2.1. Basic structural considerations
With the notable exception of the initial structural solutions[21,
29, 32]
, previous reports are
based on X-ray powder work, and all the compounds concerned have been assigned to one of the two space groups P63/mmc or P63mc. In this study, the superior sensitivity of single crystal methods to weak intensities revealed superstructure ordering in many of the compounds. Further, in some systems the existence of superstructure ordering depended on composition, and possibly on thermal history, so that different batches from the same system
23
might exhibit different ordering. For crystals from the same batch, however, the diffraction pattern was always invariant. The main features of the diffraction patterns found for the different systems are given in Table 3.1. Same elements but with different crystal systems are separately shown in the left and right columns respectively in the table. The Ce, Pr, Nd, Sm containing compounds which have larger metallic radii in the family show superstructure reflection neither on hexagonal base planes hk0 nor on hexagonal hk1 planes. Therefore hexagonal symmetries were kept for these compounds. On the other hand, Gd, Tb and Dy containing compounds with intermediate metallic radii in the family in addition to Yb containing compound have both situations of with (right side columns) or without superstructure reflections (left side columns) in the diffraction patterns. Super-structure reflections on hexagonal hk0 and/or hk1 plane resulted in the symmetry lowering to Pnma (Dy), P212121 (Gd, Tb) and Pc (Yb). The symmetry breaking down to P63mc for Sm, Gd, Tb, and Dy containing compounds is caused by the movement of capping atoms away from the mirror plane perpendicular to the rotation hexad. Although the prototype structures are in fact quite rare, it is useful first to consider the P63/mmc structure as a reference frame for the deviations that occur in all other structures.
Table 3.1. Experimental Details from single crystal x-ray diffraction patterns Superstructure reflection in hexagonal hk1 section
Atomic number
metallic radii(Å)
space group
Superstructure reflection in hexagonal hk0 section
Superstructure reflection in hexagonal hk1 section
RE element
Pc
+
+
Yb[II]
RE element
space group
Superstructure reflection in hexagonal hk0 section
Yb[I]
P63/mmc
-
-
70
1.9400
Ce
P63/mmc
-
-
58
1.8250
Pr
P63/mmc
-
-
59
1.8200
Nd
P63/mmc
-
-
60
1.8140
Sm
P63mc
-
-
62
1.7895
Gd[I]
P63mc
-
-
64
1.7865
P212121
+
+
Gd[II]
Tb[I]
P63mc
-
-
65
1.7625
P212121
+
+
Tb[II]
Dy[I]
P63mc
-
-
66
1.7515
Pnma
-
+
Dy[II] ∗
Hexagonal system
Orthorhombic and Monoclinic system
[I] form hexagonal phases; [II] form non hexagonal phases. ∗
The compounds Dy13Zn57 was not synthesized as part of this study. The data for it was taken from the previously published work [19].
24
3.2.2 Structure description
The relation between the high symmetry Gd13Cd58 prototype and the cubic RECd6 phases has been elaborated previously in section 1.4.4, and it has been shown how the hexagonal 13:58 type is generated by chemical twinning from the 1:6 phases. Here we focus on the role of the special coordination polyhedron that is the fundamental building block of both phase types; the double pentagonal anti-prism (DPAP) formed by Zn atoms around RE atoms (Figure 3.7). The pentagonal faces of the DPAPs are capped, either by Zn atoms or by RE atoms. The DPAP of RE1 atom is capped on both pentagonal faces by RE3 atom [conf. Figure 1.5 (d)], while the DPAP of RE2 atom is capped by RE3 atom on one side and by Zn10 atom on the other side [conf. Figure 1.5 (c)]. In the 13:58 phases the DPAPs surrounding the positions RE1 and RE2 form a regular, and rather rigid, network (Figure 3.8). Figure 3.8 (a) shows hexagonal channels and cubic interstices formed by RE2 centred DPAPs. RE4 atom and Zn11 atoms reside in the hexagonal tunnels alternately and the Zn12 atom occupies cubic interstices between the DPAPs. In Figure 3.8 (b) RE1 centred DPAPs (light grey) fill in the space in between RE2 centred DPAPs and form a rigid network. Apart from the RE atoms, this network contains the Zn positions Zn1-Zn9 (conf. Figure 3.7). The atomic positions, isotropic displacement parameters and occupancies (when applicable) are given in Appendix 5-8.
(a)
(b)
Figure 3.7. The double pentagonal anti-prism formed by Zn around RE atoms. (a) Coordination polyhedron of the position RE1. (b) Coordination polyhedron of the position RE2.
25
(a)
´
(b)
Figure 3.8. Assembly of the Network of DPAPs surrounding RE1 and RE2. View along [001] (a) RE2 centered DPAPs (dark grey) form hexagonal channels and cubic interstices. (b) RE1 centered DPAPs (light grey) fill in the space in between RE2 centered DPAPs and form a rigid network.
This is a complete picture for the compounds Ce13Zn58 and Pr13Zn58. The sequence of RE13Zn~58 compounds exhibits a large variability in local ordering and composition, and the archetype structure of Gd13Cd58 could only be found for the Ce and Pr compounds. For these structures refinements were possible directly from the prototype model. For Nd13Zn∼58 the RE4 position is not fully occupied, but partially replaced by a Zn2 dumbbell, Zn13 in the coordinate list. The Nd compound is the first where this rather simple exchange disorder mechanism (Mechanism I) is observed. It is the only compound in which this mechanism has been observed isolated. In hexagonal systems the Zn13 position becomes increasingly highly occupied as the size of the rare earth decreases through Nd, Sm, Gd, Tb and Dy (the behaviour of the Yb compound is in violation of this trend) (conf. Table 3.2 ). The partial (disordered) replacement of the RE4 position by a Zn2 dumbbell has clear consequences for the immediate surroundings. A typical electron density situation along the hexad (Nd13Zn∼58) is shown in Figure 3.9. The isotropic displacement parameter of the Nd atoms is unnaturally large. Ideally, the presence of the Zn2 dumbbell should lead to a displacement of the nearest neighbors along the hexad; position Zn11, away from the dumbbell. Because of the disorder, this position is, however, superimposed on the corresponding negative displacement, and,
26
further on the undisplaced position resulting from a Zn11 locally surrounded by RE4. Figure 3.9 (a) shows the electron density along the hexad. A large extent of the electron density corresponding to Zn11 is clear seen. In Figure 3.9 (b) disordered model used in the refinement is shown. The transparencies of the atoms are proportional to their refined occupancies. In the case of Nd compound, the occupancy for the Zn2 dumbbell is only 17.6% (conf. Table 3.2). It can be interpreted as that in some case, there is no pair exchange occurring, which means that the Nd and Zn atom were arranged alternately as shown in Fig 3.9 (c). And in some other cases, one of the Nd atoms was replaced by a pair of Zn atoms causing a short distance between Zn and Zn atoms and resulting in a movement of the neighbor Zn atoms towards its neighbor Nd atoms [Figure 3.9 (d)]. The arrows in the figure indicate expected deformation. This is why we see Zn13 atoms have very large displacement parameters. This causes another more important difference between prototype structure and the Nd compound, which is the non-stoichiometry. Therefore the refined composition for Nd compound is Nd12.82Zn58.35, rather than Nd13Zn58.
The pair exchange mechanism is well known from other systems, in ordered form notably in the Th2Zn17 and Th2Ni17 structure types[42,43] the large heteroatom is replaced by a pair of the small homoatoms in an ordered fashion and there are numerous of such replacements that take place in a disordered form in related structures, i.e. the Cu5Tb family[44,45,46].
Table 3.2. The occupancies of Zn pair in the tunnel and extra capping atoms for the hexagonal RE13Zn～58 compounds Compound
Refined occupancy of Zn pair (Zn13) in the tunnel
Occupancy of the extra capping atom(Zn14)
Ce13Zn58
—
—
Pr13Zn58
—
—
Nd12.82Zn58.35
0.176
—
Sm12.72Zn59.08
0.283
0.172
Gd12.71Zn59.50
0.291
0.310
Tb12.64Zn59.10
0.361
0.127
Dy12.60Zn59.22
0.403
0.139
Yb12.61Zn59.82
0.387
0.130
27
Zn11
Zn13 Nd4 Zn13
Zn11
Zn13 Nd4 Zn13
Zn11
(a) (b)
(c) (d) Figure 3.9. Arrangement along the hexad in Nd12.82Zn58.35 compound. The transparencies of the atoms are proportional to their refined occupancies. (a) Electron density along the hexad. Note the large extent of the electron density corresponding to Zn11. (b) Disordered model used in the refinement. The degree of transparency codes for occupancy. (c) Environment of a local sequence Zn11-RE4Zn11-RE4-Zn11. (d) Environment of a local sequence Zn11-RE4-Zn11-Zn13-Zn13-Zn11. Arrows indicate expected deformation.
Further, for the Sm-Dy containing compounds and for the Yb compound, a second disordering mechanism (Mechanism II) comes into play; the distance between the pentagonal face of the DPAP and the capping RE3 atom is rather large, and as the size of the RE atom decreases, this leads to the creation of an increasingly larger interstitial position for a zinc atom, Zn14. Mechanism II was identified as the cause of the symmetry breaking. To study the different ordering mechanisms in the structures, it is instructive to draw a section of the electron density map including the hexagonal tunnel and the RE3 capping rare earth atom. Figure 3.10 (a) shows a template for describing the disorder mechanisms for the topic compounds. In Figure 3.10 (b)-(l) the electron density-maps (calculated from Fobs) for the topic compounds were shown. In Figure 3.10 (b) and (c), there are no unaccounted for electron densities or conspicuous sign of under occupancies, indicating there is no disorder in Ce and Pr containing compounds. For the Nd compound, the situation changes and the exchange mechanism I becomes apparent. Note how the occupancy of the Zn13 position gradually increases as the size of the rare earth decreases from Nd to Dy for the hexagonal phases [conf. Figure 3.10 (d) - (h) and Table 3.2], and the conspicuous absence of electron density for this position in the ordered orthorhombic Gd, Tb ones [conf. Figure 3.10 (j), (k)] and monoclinic Yb one [conf. Figure 3.10 (l)]. The behaviour of the Yb compound is
28
anomalous. Further note how the shape of the electron density corresponding to the position Zn11 changes from circular for the cases where Zn13 is missing [conf. Figure 3.10 (a), (b)] to pronouncedly elliptic for high occupations of Zn13 [conf. Figure 3.10(h) and (i)]. The same observation may be made for the presence/absence of the atom in the position Zn14. The shape of the electron density for the position RE3 is only slightly elliptic for Ce, Pr and Nd compounds, perhaps a manifestation of the rather long RE-Zn distances for this atom. For the disordered hexagonal Sm, Gd, Tb, Dy and Yb phases, this electron density is strongly elliptic. For the ordered form of Gd, Tb and Yb the shape is almost perfectly circular. Generally the inclusion of Zn14 leads to a lowering of the symmetry to P63mc and a deformation of the RE3 position away from the mirror plane perpendicular to the hexad for the hexagonal phases. However, for Yb compound, lowering symmetry to P63mc made the refinement worse. Therefore P63/mmc symmetry was kept [conf. Figure 3.10 (i)]. The effect of deformation is slight for Sm but quite pronounced for Gd, Tb and Dy. Further, at least the compounds containing Gd, Tb, Dy and Yb also allow for some compositional flexibility. Under zinc-rich conditions, the synthesis results in superstructure ordering manifest in satellite reflections compatible with a primitive orthorhombic and monoclinic cell. For the Gd, Tb, Dy and Yb compound it was able to find single (twinned) crystals of excellent quality and to solve the superstructure. This is clearly the effect of the ordering of the position Zn14 to one, fully occupied, independent position in the space group P212121 for the Gd, Tb cases, and space group Pc for Yb case. This corresponds to a 1/3 occupancy of the equivalent position in the hexagonal structures. Interestingly, for this compound the zinc dumbbell, (position Zn13) is unoccupied, and consequently, the position RE4 is fully occupied. While the occupancy of the position Zn13 varies monotonically with the size of the RE, the occupancy of the capping atom Zn14 shows no obvious trend (conf. Table 4). It is quite conceivable that both positions allow for some compositional flexibility, as is obviously the case for the Gd, Tb and Yb compound(s). For the superstructure (ordered) Gd, Tb, and Yb compound, the inclusion of a fully occupied Zn14 leads to a major movement of the atom in position RE3. In Figure 3.12 (a) and (b), two different situations of the capped DPAPs of the ordered orthorhombic Tb-compound are compared. Note how the atomic position RE3 distorted due to the insertion of Zn14 stom. Figure 3.13 shows an overview of the ordered Tb compound.
29
(a) Template
(d) Nd12.82Zn58.35
(g) Tb12.64Zn59.10
(j) Gd13Zn58
(b) Ce13Zn58
(e) Sm12.72Zn59.08
(h) Dy12.60Zn59.82
(k) Tb13Zn59
(c) Pr13Zn58
(f) Gd12.71Zn59.50
(i) Yb12.61Zn59.82
(l) Yb13Zn59
Figure 3.11 Electron density-maps (calculated from Fobs) showing the disorder mechanisms. (a) template for all the compounds. (b) and (c) show that there is no sign of disorder in Ce and Pr compounds. Figure (d) to (i) show the two types of disorder mechanism in the hexagonal phases. Figure (j) to (l) show the ordered Gd, Tb and Yb compound.
30
(a)
(b)
Figure 3.12. Comparison between the capped DPAP of RE2 atoms in the various situations that arises in the Tb-compounds. The images are constructed form the ordered orthorhombic phase. (a) RE Capped DPAP. (b) RE and Zn14 capped DPAP. Note the mutual distortion between the atomic positions RE3 and Zn14.
Figure 3.13. Overview of the structure of Tb13Zn59.
31
3.2.3 Trends and Phase distribution in RE13Zn～58 system
Generally the rare earth elements exhibit a large but smooth decrease in size in the series from La to Lu with the exception of the elements Eu and Yb. If we group the RE13Zn~58 compounds in the order of decreasing sizes of the rare earth elements as shown in Table 3.3, the largest element Eu does not form any 13:58 compound. The larger rare earth elements (Yb, Ce, Pr, and Nd) form the centro-symmetric structures in the space group P63/mmc. These structures show either perfect ordering (Ce, Pr) or mechanism I disorder (Yb, Nd). As we move to rare earth elements of intermediate size (Sm, Gd, Tb, Dy), the disorder becomes much more pronounced with both mechanism I and mechanism II playing a role. The occurrence of mechanism II is accompanied by the insertion of a Zn capping atom (Zn14), which breaks the original P63/mmc symmetry, lowering it to P63mc. When the occupancy of the Zn14 atom reaches 1/3, the disorder turn into the order and the symmetry is lowered to P212121. The monoclinic Pc form of Yb is an outlier in this sequence. For the smaller elements beyond Dy the situation becomes much more complex. From Ho to Lu, hexagonal symmetries were broken down to orthorhombic system (centred orthorhombic structures for the Ho compound and non-centred orthorhombic structures for Er, Tm, Lu compounds). The details of studies on Ho, Er, Tm and Lu containing phases will be reported in future work. Herein I would like to mention that there is quite a nice similarity for these four compounds. They all have very complex structures and show anomalous continuous electron densities in the electron density maps. Figure 3.14 shows the electron density map for Ho containing compound. We see a beads-on-astring-like electron density which indicates additional disorder in the structure. The Er, Tm and Lu structures are strictly disorder, but the Ho compound shows additional reflexions indicating incommensurate ordering.
32
Figure 3.14. Electron density map of incommensurate Ho13Zn59.16 compound.
Table 3.3. Trends and Phase distribution in RE13Zn～58 system RE
Radius (Å)
Crystal system of RE13Zn~58
Eu
1.9945
#
Yb
1.9400
Hexagonal + Monoclinic
La
1.8695
#
Ce
1.8250
Hexagonal
P63/mmc
Pr
1.8200
Hexagonal
P63/mmc
Nd
1.8140
Hexagonal
P63/mmc
Sm
1.7985
Hexagonal
P63mc
Gd
1.7865
Hexagonal + Orthorhombic
P63mc + P212121
Tb
1.7625
Hexagonal + Orthorhombic
P63mc + P212121
Dy
1.7515
Hexagonal + Orthorhombic
P63mc+ Pnma
Ho
1.7430
Orthorhombic
Er
1.7340
Tm
1.7235
Orthorhombic Orthorhombic
Pnma + Incommensurate Pnma Pc21n
Lu
1.7175
Orthorhombic
Space Group P63/mmc + Pc
Pc21n Pc21n
# doesn’t form 13:58 phas
33
4. Summary and conclusions The single crystals for RECd6 and RE13Zn~58 quasicrystal approximants were prepared and their structures were investigated by single crystal X-ray diffraction. The previously reported prototype structures of RECd6 and RE13Zn~58 phases have been proven to be insufficient. The crystal structures of Ce13Zn58 and Pr13Zn58 compounds are generally similar to those previously reported. However several types of higher order, disorder and nonstoichiometry that were not taken into account by the prototype structures have been observed in the structures of the compound formerly reported as RE13Zn58. The nonstoichiometry in the hexagonal compounds arises as a consequence of the exchange of one rare earth atom by Zn2 pairs of atoms in the hexagon-shaped channels of the structures or introducing an interstitial zinc atom between the pentagonal face of the DPAP and the capping RE3 atom. Both mechanisms appear to be disordered. In RECd6 system the disorder of the tetrahedra inside the dodecahedral cavity depends strongly on the size of the unit cell, and consequently on the effective size of the rare earth atoms. RECd6 compounds formed with large rare earth elements show a remarkable diversity of disorders, the smaller rare earth elements lead to smaller central cavities that do not allow for such behaviour. Further, none of the topic compounds shows any occupancy of the Cd8 cubes. It would seem that large rare earth elements tend to expand the Cd-network, leading to the creation of larger cavities; the Cd8 cubes are able to host additional Cd atoms. The hexagonal RE13Zn~58 approximants system shows a clear reversal of the trend from the cubic RECd6 approximants system; the size of the unit cell changes very little with the size of the rare earth element. Further, for the hexagonal systems, the concentration of interstitial Zn increases with decreasing rare earth element size. This may be understood in terms of the size determining species in the two cases; for the cubic system, the unit cell size is determined by the rare earth element, and hence larger rare earth elements lead to larger cavities. In the hexagonal systems the unit cell
34
size is instead determined by the Zn-network, and hence, smaller rare earth element leads to an increase in space for interstitial atoms.
5. Future work I will continue the research on quasicrystal approximants. Specifically I will focus my study on what are listed below: 1. I will investigate some synthetic methods for updating single crystal data for the compounds with incommensurate structures and finish the writing on overview of study in RE13Zn∼58 system. 2. Sr and Y are frequently included as pseudo rare earths; their sizes pace them in the vicinity of Dy and Lu respectively. Therefore I am going to do some single crystal studies on Sr/Y-Cd/Zn structures and make some comparison with RE-Cd/Zn structures. 3. There is a similarity between the structure of RE13Cd58 and RE13Zn58. From the study on Ce13Cd58 it is known that there are many ambiguities on the previous study on RE13Cd58 system which were based on X-ray powder diffraction. I am going to start systematic study on RE13Cd∼58 system by single crystal dirraction method.
35
Acknowledgements I wish to express my sincere gratitude to all the people who have helped me within these two years. Especially I would like to thank the following people: My supervisor, Sven Lidin, for accepting me as your Ph.D student and for being an excellent guide to the subject of intermetallic compound and practical crystallography. Thanks for your numerous suggestions of possible experiments. Thank you also for encouraging me and always looking onto the bright side on all kinds of results. I have been enjoying the time that we work together. Cesar Pay-Gomez, for the help with experiment set up and teaching me how to use many programs. Also for your encouragement and all fruitful discussions. I really enjoyed talking with you about everything other than science as well. My co-supervisor Prof. Osamu Terasaki, for your interest in my work and all valuable discussions. And also for caring me and my family. Andreas Flemström, for your great help in many different ways through these years. You made my life in Sweden easier and joyful. My deepest thanks to you. Hanna Lind, for your help in the laboratory and for all discussions on the incommensurate structure. Lars Eriksson, for always being patient and taking time to solve the problems that I had on my single crystal measurement. And also for helping me with the synchrotron measurement in Max-Lab. Magnus Boström, thanks for all your kindness and for all the help in Max-Planck Institute. I have got many beautiful CuIn single crystals with your help. Xiaodoing Zou and Sven Hovmöller, for the course you had and for all the fruitful discussion on quasicrystal approximants. Mats Johnsson, Jekabs Grins, Kjell Jansson and Margareta Sundberg, for the interesting courses you gave. I learned a lot from you. Lars Göthe, for the help with XRPD analysis. Rolf Eriksson, Per-Erik Persson, Ann-britt Rönell, Hellevi Isaksson and Eva Pettersson for always being so helpful. My roommate and my group member for creating such a stimulating and pleasant atmosphere to work in.
36
All my Chinese friends, for all the help you have given and for the enjoyable time we had together. My sincere thanks also go to: All of my colleagues at FOOS, for the supporting from different way and for providing such a nice working atmosphere. It’s my pleasure to work with you all. I would also like to thank my parents and my sisters, for their endless support and encouragement through these years. Finally, I am very thankful to my husband, for the support, encouragement and all love. And I thank my lovely daughter, Mimi, for being such a kind, positive and helpful girl and giving me another side of life than science. I love you so much. I will remember your teaching “To be positive” forever.
37
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[25] R. B. Roof and G. R. B. Elliott, Inorg. Chem. 4 (1965) 691-697. [26] P. J. Steinhardt, H. C. Jeong, Nature 382 (1996) 431-433. [27] H. C. Jeong, P. J. Steinhardt, Phys. Rev. B 55 (1997) 3520-3532. [28] P. J. Steinhardt, H. C. Jeong, K. Saitoh, M. Tanaka, E. Abe, A. P. Tsai, Nature 396 (1998) 55-57.
[29] A. C. Larson, D. T. Cromer, Acta Crystallogr. 27B (1971) 1875-1879. [30] A. Palenzona, J. Less-Common Met. 25 (1971) 367-372. [31] D. E. Sands, Q. C. Johnson, O. H. Krikorian, K. L. Kromholtz, Acta Crystallogr. 15 (1962) 1191-1195.
[32] G. Bruzzone, M. L. Fornasini, F. Merlo , J. Less-Common Met. 25 (1971) 295301. [33] Computer code X-RED, version 1.22 (Stoe and Cie GmbH, Darmstadt, Germany), (2001). [34] Computer code X-SHAPE, version 1.06 (Stoe and Cie GmbH, Darmstadt, Germany), (1999). [35] V. Petřίček, M. Dusek, computer code JANA2000 (Institute of Physics AVCR, Praha, Czech Republic) (2002). [36] S. Weber, computer code JMAP3D (NIRIM, Tsukuba, Japan (1999). [37] K. Brandenburg, computer code DIAMOND, version 2.1c (Crystal Impact, Bonn, Germany) (1999). [38] R. Ormandy, computer code TRUESPACE, version 5.2 (Caligari Corporation, Mountain View, USA) (2000) [39] S. Y. Piao, C. P. Gómez, S. Lidin, Z. Naturforsch., 60b (2005).(In press) [40] S. Y. Piao, C. P. Gómez, S. Lidin, Z. Kristallogr., (2005). (Accepted). [41] C.P.Gómez, thesis, Stockholm University, (2003). [42] J. V. Florio, N. C. Baenziger, R.E. Rundle, Acta Crystallogr., 9 (1956) 367-372. [43] E. S. Makarov, L. S. Gudkov, Kristallografiya, 1 (1956) 650-656. [44] K.H.J. Buschow, A.S.v.d.Goot, Acta Crystallogr., B27 (1971) 1085-1088. [45] Y. Khan, Acta Crystallogr., B29 (1973) 2502-2507. [46] Z. Drzaga, J. Kork, H. Broda, A. Chelkowski, Acta Phys. Pol. A55 (1979) 849.
39
Appendixes Appendix 1. Crystal data, data collection and refinement parameters for the structures of RECd6 compounds Formula
TbCd6
HoCd6
ErCd6
TmCd6
LuCd6
Molar mass (g/mol)
833.4
839.4
841.7
843.4
849.4
Temperature of measurement (K)
293
293
293
293
293
Crystal system
cubic
cubic
cubic
cubic
cubic
Space group
Im 3
Im 3
Im 3
Im 3
Im 3
15.453(4)
15.332(4)
15.399(4)
15.423(1)
15.330(4)
Cell volume (Å )
3690.1
3604.1
3651.5
3668.6
3602.7
Z
24
24
24
24
24
8472
8520
8544
8568
8616
Calculated density (g/cm )
8.998
9.279
9.183
9.159
9.393
Absorption coefficient
31.5
33.7
34.0
33.1
37.0
Diffractometer
Stoe IPDS
Stoe IPDS
Stoe IPDS
Stoe IPDS
Stoe IPDS
Range of 2θ(°)
3.7-51.9
3.7-51.9
3.7-52.2
3.7-52.0
9.9-53.4
Radiation
MoKα
MoKα
MoKα
MoKα
MoKα
Observed reflections [I>3σ]
671
657
647
672
694
Independent reflections
670
656
549
651
598
Rint (obs/all)
9.41/9.41
4.58/4.58
12.36/12.60
5.55/5.56
6.20/6.31
Number of parameters
46
45
45
45
45
R1(obs)
0.0369
0.0299
0.0252
0.0249
0.0327
wR2 (all)
0.0661
0.0492
0.0300
0.0531
0.0366
Absorption correction
numerical, from shape
numerical, from shape
numerical, from shape
numerical, from shape
numerical, from shape
Tmin , Tmax
0.0170, 0.0746
0.0493, 0.1419
0.2810, 0.5050
0.2882, 0.5186
0.1664, 0.6645
Δρmax , Δρmin (e/A3)
6.44, -3.25
2.11, -2.91
1.81, -2.08
1.85, -2.09
2.71, -7.58
a Axis (Å) 3
F(000) 3
40
Appendix 2. Crystal data, data collection and refinement parameters for the RE13Zn～58 structures Formula
Ce13Zn58
Pr13Zn58
Nd12.82Zn58.35
Sm12.72Zn59.08
Gd12.71Zn59.50
Tb12.64Zn59.10
Dy12.60Zn59.22
Dy12.61Zn59.18
Yb12.61Zn59.82
Molar mass (g/mol)
5613.6
5623.8
5664.7
5774.5
5888.8
5872.8
5918.4
5918.4
6093.7
Temperature of measurement (K)
293
293
293
293
293
293
293
293
293
Space group
P63/mmc (194)
P63/mmc (194)
P63/mmc (194)
P63mc (186)
P63mc (186)
P63mc (186)
P63mc (186)
P63mc (186)
P63/mmc (186)
a Axis (Å)
14.638(1)
14.585(8)
14.514(8)
14.374(14)
14.335(13)
14.282(8)
14.226(16)
14.226(16)
14.266(2)
14.158(1)
14.133(7)
14.124(8)
14.052(13)
14.070(10)
14.031(8)
14.019(20)
14.019(20)
14.135(2)
2630.3 (4)
2602.2(2)
2576.7 (2)
2514.3(4)
2503.9(4)
2478.4(2)
2457.2(5)
2457.2(5)
4983.0
Z
2
2
2
2
2
2
2
2
2
F(000)
4988
5014
5040
5171
5170
5189
5216
5190
5355
Calculated density (g/cm3)
7.094
7.175
7.299
7.625
7.808
7.867
7.997
7.997
8.120
Absorption coefficient
36.9
38.0
39.2
42.1
44.4
45.7
47.1
47.1
51.5
Range of 2θ (°)
9.6 - 53.3
9.6 - 53.3
9.7 - 53.6
9.1 - 52.8
9.8 - 53.5
4.4 - 51.9
3.3 - 48.1
3.3-48.1
4.4-48.2
Observed reflections [I>3σ(I)]
1054
1034
1027
1770
1780
1813
1460
1460
787
Independent reflections
944
964
948
1290
1293
1600
902
902
544
Rint (obs/all)
6.33 / 6.35
5.08 / 5.10
5.12 / 5.14
6.45 / 6.60
9.65 / 9.80
6.43 / 6.57
11.52 / 12.85
11.52 / 12.85
11.29 / 12.28
Number of parameters
76
76
80
141
147
129
86
86
84
R1
3.35
3.17
2.86
3.94
4.38
4.01
6.22
6.21
4.13
wR2 (obs/all)
3.92 / 4.07
4.35 / 4.45
4.15 / 4.26
3.93 / 4.06
3.73 / 3.85
4.46 / 4.53
6.23 / 6.46
6.23 / 6.45
4.29 / 4.39
Absorption correction
Numerical From shape
Numerical from shape
Numerical, From shape
Numerical from shape
Numerical, from shape
Numerical, from shape
Numerical from shape
Numerical from shape
Numerical from shape
Tmin Tmax
0.0653 0.3500
0.1833 0.5699
0.0227, 0.4057
0.0479, 0.1160
0.4450, 0.6790
0.0177, 0.3283
0.1618, 0.5675
0.1618, 0.5675
0.2521, 0.5326
Δρmax , Δρmin (e/Å3)
1.99, -4.26
2.30, -2.12
1.88, -2.92
2.58, -2.03
2.12, -2.12
2.03, -3.28
5.37, -3.96
5.37, -3.96
5.09, -6.89
c Axis (Å) 3
Cell volume (Å )
41
Appendix 3. Crystal data, data collection and refinement parameters for the Orthorhombic and monoclinic RE13Zn59 structures Formula
Gd13Zn59
Tb13Zn59
Yb13Zn59
Molar mass (g/mol)
5901.7
5923.4
6106.9
Temperature of measurement (K)
293
293
293
Space group
P212121
P212121
Pc
a Axis (Å)
24.858(2)
24.803(12)
24.811(6)
b Axis (Å)
14.322(2)
14.304(7)
14.322(6)
c Axis (Å)
14.054(3)
14.008(7)
14.191(7)
Cell volume (Å3)
5003.6
4969.7(4)
5056.9
Z
4
4
4
F(000)
10408
10460
10653
Calculated density (g/cm3)
7.832
7.914
8.019
Absorption coefficient
44.6
46.0
51.1
Range of 2θ (°)
3.3 - 48.1
10.2 - 56.7
9.5 - 53.7
Observed reflections [I>3σ(I)]
10615
14266
17261
Independent reflections
6914
12503
11554
Number of parameters
651
649
698
Rint (obs/all)
13.30 / 14.24
5.49 / 5.51
8.84 / 8.96
R1
8.59
5.56
7.77
wR2 (obs/all)
9.48 / 9.71
4.88 / 4.96
7.62 / 7.97
Absorption correction
Numerical, From shape
Numerical, From shape
Numerical from shape
Tmin , Tmax
0.0020, 0.0212
0.2858, 0.7067
0.0507, 0.1205
Δρmax , Δρmin (e/Å3)
8.45, -10.02
4.58, -5.32
10.64, -8.19
42
Appendix 4. Atomic coordinates and equivalent isotropic atom displacement parameters of RECd6 phases (RE = Tb, Ho, Er, Tm, Lu). RE
Atom
Wyck.
Occ.
x
y
z
Uiso/Ueq(Å2)
Tb
Cd1a
24g
1/3
0
0.0827(5)
0.0763(5)
0.0950(3)
Ho
Cd1a
24g
1/3
0
0.0806(3)
0.0805(4)
0.0773(19)
Er
Cd1a
24g
1/3
0
0.0824(4)
0.0774(4)
0.080(2)
Tm
Cd1a
24g
1/3
0
0.0830(3)
0.0764(4)
0.088(2)
Lu
Cd1a
24g
1/3
0
0.0794(4)
0.0813(5)
0.0750(3)
Tb
Cd2
24g
1
0
0.09218(8)
0.24027(11)
0.0318(5)
Ho
Cd2
24g
1
0
0.09139(7)
0.23851(9)
0.0357(3)
Er
Cd2
24g
1
0
0.09203(7)
0.23958(10)
0.0315(4)
Tm
Cd2
24g
1
0
0.09219(6)
0.23985(8)
0.0297(3)
Lu
Cd2
24g
1
0
0.09106(10)
0.2382(12)
0.0394(6)
Tb
Cd4
16f
1
0.16075(6)
0.16075(6)
0.16075(6)
0.0218(3)
Ho
Cd4
16f
1
0.16039(13)
0.16050(6)
0.16050(6)
0.0203(2)
Er
Cd4
16f
1
0.16066(5)
0.16066(5)
0.16066(5)
0.0185(2)
Tm
Cd4
16f
1
0.16075(4)
0.16075(4)
0.16075(4)
0.0180(1)
Lu
Cd4
16f
1
0.16034(7)
0.16034(7)
0.16034(7)
0.0222(2)
Tb
Cd6
48h
1
0.20034(5)
0.34051(5)
0.11774(6)
0.0175(3)
Ho
Cd6
48h
1
0.19993(4)
0.34040(4)
0.11711(4)
0.0179(2)
Er
Cd6
48h
1
0.20037(5)
0.34038(5)
0.11784(5)
0.0160(2)
Tm
Cd6
48h
1
0.20039(4)
0.34041(4)
0.11801(4)
0.0147(2)
Lu
Cd6
48h
1
0.19981(6)
0.34050(6)
0.11706(6)
0.0198(3)
Tb
Cd7
12d
1
0.40574(11)
0
0
0.0291(6)
Ho
Cd7
12d
1
0.40520(8)
0
0
0.0255(4)
Er
Cd7
12d
1
0.40534(10)
0
0
0.0253(5)
Tm
Cd7
12d
1
0.40545(8)
0
0
0.0247(4)
Lu
Cd7
12d
1
0.40503(13)
0
0
0.0271(7)
Tb
Cd8
24g
1
0
0.34569(7)
0.40417(7)
0.0141(4)
Ho
Cd8
24g
1
0
0.34428(5)
0.40418(5)
0.0151(2)
Er
Cd8
24g
1
0
0.34512(6)
0.40417(6)
0.0128(3)
Tm
Cd8
24g
1
0
0.34541(5)
0.40414(5)
0.0114(2)
Lu
Cd8
24g
1
0
0.34398(8)
0.40437(8)
0.0176(4)
Tb
Cd9
12e
1
0.19033(9)
1/2
0
0.0160(5)
Ho
Cd9
12e
1
0.18827(8)
1/2
0
0.0177(3)
Er
Cd9
12e
1
0.18921(9)
1/2
0
0.0147(4)
Tm
Cd9
12e
1
0.18956(7)
1/2
0
0.0140(3)
Lu
Cd9
12e
1
0.18759(12)
1/2
0
0.0196(6)
Tb
Tb1
24g
1
0
0.29945(4)
0.18909(4)
0.0113(3)
Ho
Ho1
24g
1
0
0.29903(3)
0.18777(3)
0.0114(2)
Er
Er1
24g
1
0
0.29947(4)
0.18887(4)
0.0104(2)
Tm
Tm1
24g
1
0
0.29949(3)
0.18915(3)
0.0112(1)
Lu
Lu1
24g
1
0
0.29881(5)
0.18759(5)
0.0166(3)
43
Appendix 5. Fractional atomic coordinates, occupancies and isotropic ADPs for the hexagonal RE13Zn～58 structures RE Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy
atom Yb1 Ce1 Pr1 Nd1 Sm1 Gd1 Tb1 Dy1 Yb2 Ce2 Pr2 Nd2 Sm2a Sm2b Gd2a Gd2b Tb2a Tb2b Dy2a Dy2b Yb3 Ce3 Pr3 Nd3 Sm3 Gd3 Tb3 Dy3 Yb4 Ce4 Pr4 Nd4 Sm4 Gd4 Tb4 Dy4 Zn1 Zn1 Zn1 Zn1 Zn1a Zn1b Zn1a Zn1b Zn1a Zn1b Zn1a Zn1b Zn2 Zn2 Zn2 Zn2 Zn2a Zn2b Zn2a Zn2b Zn2a Zn2b Zn2a Zn2b
Wyck. 6h 6h 6h 6h 6c 6c 6c 6c 12k 12k 12k 12k 6c 6c 6c 6c 6c 6c 6c 6c 6h 6h 6h 6h 6c 6c 6c 6c 2a 2a 2a 2a 2a 2a 2a 2a 24l 24l 24l 24l 12d 12d 12d 12d 12d 12d 12d 12d 4f 4f 4f 4f 2b 2b 2b 2b 2b 2b 2b 2b
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.613 1 1 0.824 0.717 0.709 0.639 0.597 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x 0.54311(7) 0.54136(3) 0.54156(4) 0.54163(3) 0.54181(4) 0.54210(4) 0.54230(3) 0.54254(9) 0.20699(4) 0.20488(2) 0.20503(3) 0.2051(2) 0.20524(12) -0.20599(12) 0.2058(13) 0.2060(3) 0.2060(7) 0.2060(7) 0.2063(3) -0.2059(2) 0.12456(9) 0.12782(3) 0.12775(4) 0.12764(3) 0.12724(5) 0.12725(5) 0.12712(3) 0.12707(12) 0 0 0 0 0 0 0 0 0.3666(2) 0.3659(8) 0.3663(9) 0.3652(8) 0.3646(5) -0.3651(5) 0.3637(6) -0.3656(5) 0.3634(3) -0.3658(3) 0.3749(6) -0.3553(6) 1/3 1/3 1/3 1/3 1/3 - 1/3 1/3 - 1/3 1/3 - 1/3 1/3 1/3
y 0.0862(13) 0.0827(7) 0.0831(7) 0.0833(6) 0.08363(8) 0.0842(8) 0.0846(5) 0.08507(19) 0.7930(4) 0.7951(2) 0.7950(3) 0.7949(2) 0.79476(12) -0.79401(12) 0.7942(8) 0.7940(8) 0.7940(7) 0.7940(7) 0.7937(3) -0.7941(2) 0.87544(9) 0.87218(3) 0.87226(4) 0.87236(3) 0.87276(5) 0.87275(5) 0.87288(3) 0.87293(12) 0 0 0 0 0 0 0 0 0.0345(19) 0.0346(7) 0.0347(8) 0.0346(7) 0.0356(4) -0.0330(4) 0.0355(5) 0.0335(5) 0.0361(3) -0.0328(3) 0.0342(6) -0.0349(6) 2/3 2/3 2/3 2/3 2/3 - 2/3 2/3 - 2/3 2/3 - 2/3 2/3 2/3
44
z 1/4 1/4 1/4 1/4 0.2517(9) 0.2560(9) 0.2524(5) 0.2446(13) 0.05430(8) 0.05128(4) 0.05139(5) 0.0510(9) 0.0531(9) -0.0495(9) 0.0578(8) 0.0451(9) 0.0536(5) 0.0489(5) 0.0461(12) -0.0570(12) - 1/4 - 1/4 - 1/4 - 1/4 -0.2439(10) -0.2407(10) -0.2435(6) -0.2558(18) 0 0 0 0 0 0 0 0 0.0993(17) 0.1023(6) 0.1021(7) 0.10202(7) 0.1029(10) -0.0999(9) 0.1080(9) -0.0940(9) 0.1023(5) -0.0997(5) 0.0904(12) -0.1106(12) 0.0956(4) 0.0961(14) 0.0959(17) 0.0957(14) 0.0989(13) -0.0922(13) 0.1020(13) -0.0878(12) 0.0995(7) -0.0906(7) 0.0865(19) -0.105(2)
2
Uiso/Ueq(Å ) 0.0097(6) 0.0071(3) 0.0082(3) 0.0101(3) 0.0074(4) 0.0121(4) 0.0100(2) 0.0103(8) 0.0075(4) 0.0073(2) 0.0084(3) 0.0099(2) 0.0085(13) 0.0053(12) 0.0114(13) 0.0118(13) 0.0100(6) 0.0089(6) 0.012(2) 0.009(2) 0.0703(14) 0.0123(3) 0.0171(4) 0.0188(3) 0.0317(6) 0.0419(7) 0.0426(5) 0.0459(13) 0.013(2) 0.0097(4) 0.0110(5) 0.0115(6) 0.0084(8) 0.0122(12) 0.0118(6) 0.004(2) 0.0339(13) 0.0144(4) 0.0161(5) 0.0198(4) 0.022(3) 0.021(3) 0.034(3) 0.023(3) 0.0306(16) 0.0234(15) 0.023(2) 0.017(2) 0.0065(13) 0.0081(6) 0.0092(7) 0.0105(6) 0.007(3) 0.005(3) 0.010(3) 0.016(4) 0.0089(5) 0.0089(5) 0.002(5) 0.023(7)
Appendix 5. (continued) RE Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy
Atom Zn3 Zn3 Zn3 Zn3 Zn3 Zn3 Zn3 Zn3 Zn4 Zn4 Zn4 Zn4 Zn4a Zn4b Zn4a Zn4b Zn4a Zn4b Zn4a Zn4b Zn5 Zn5 Zn5 Zn5 Zn5 Zn5 Zn5 Zn5 Zn6 Zn6 Zn6 Zn6 Zn6 Zn6 Zn6 Zn6 Zn7 Zn7 Zn7 Zn7 Zn7a Zn7b Zn7a Zn7b Zn7a Zn7b Zn7a Zn7b Zn8 Zn8 Zn8 Zn8 Zn8a Zn8b Zn8a Zn8b Zn8a Zn8b Zn8a Zn8b
Wyck. 2d 2d 2d 2d 2b 2b 2b 2b 12k 12k 12k 12k 6c 6c 6c 6c 6c 6c 6c 6c 12i 12i 12i 12i 12d 12d 12d 12d 12j 12j 12j 12j 12d 12d 12d 12d 12k 12k 12k 12k 6c 6c 6c 6c 6c 6c 6c 6c 12k 12k 12k 12k 6c 6c 6c 6c 6c 6c 6c 6c
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x 2/3 2/3 2/3 2/3 2/3 2/3 2/3 2/3 0.09951(15) 0.09471(5) 0.09468(6) 0.09590(5) 0.0975(3) 0.0975(3) 0.0969(4) -0.0971(4) 0.0980(2) -0.0974(2) 0.0981(7) -0.0978(7) 0.1959(3) 0.20148(9) 0.20127(10) 0.19886(10) 0.1986(4) 0.1977(5) 0.1958(3) 0.1892(5) 0.0827(2) 0.08561(10) 0.08542(11) 0.08547(10) 0.08487(12) 0.08465(13) 0.08448(10) 0.0835(3) 0.43714(12) 0.43554(5) 0.43578(6) 0.43595(5) 0.4381(3) -0.4350(3) 0.4378(4) -0.4362(3) 0.4372(2) -0.4370(2) 0.4380(6) -0.4367(7) 0.60081(13) 0.60160(5) 0.60142(6) 0.60139(5) 0.6015(3) -0.6006(3) 0.6016(4) -0.6001(3) 0.6008(2) -0.6009(2) 0.6037(6) -0.5984(5)
y 1/3 1/3 1/3 1/3 1/3 1/3 1/3 1/3 0.1990(3) 0.18943(10) 0.18936(12) 0.1918(1) 0.1949(6) 0.1949(6) 0.1939(8) -0.1942(8) 0.1960(4) -0.1948(4) 0.1962(14) -0.1956(15) 0 0 0 0 0.0039(8) 0.0030(9) 0.0020(6) -0.0100(8 0.3686(2) 0.3681(1) 0.36790(12) 0.36799(10) 0.36737(12) 0.36686(13) 0.36691(10) 0.3663(3) 0.8743(2) 0.87109(10) 0.87157(11) 0.87191(10) 0.8762(6) -0.8700(6) 0.8756(7) -0.8723(6) 0.8743(4) -0.8740(4) 0.8759(13) -0.8734(14) 0.2016(3) 0.2032(1) 0.20285(11) 0.20279(10) 0.2030(6) -0.2011(6) 0.2032(7) -0.2003(7) 0.2016(4) -0.2018(4) 0.2075(12) -0.1967(9)
45
z 1/4 1/4 1/4 1/4 0.2479(17) 0.2550(19) 0.2513(10) 0.2530(19) 0.1588(3) 0.15618(9) 0.15650(11) 0.15684(10) 0.1614(10) 0.1614(10) 0.1658(10) -0.1491(9) 0.1598(6) -0.1555(6) 0.1515(18) -0.1642(18) 0 0 0 0 0.0056(11) 0.0079(11) 0.0039(7) -0.0196(13) 1/4 1/4 1/4 1/4 0.2499(9) 0.2555(10) 0.2498(6) 0.2447(15) 0.1441(2) 0.14167(9) 0.14186(10) 0.14175(9) 0.1455(10) -0.1383(10) 0.1496(9) -0.135(1) 0.1450(6) -0.1395(6) 0.1357(15) -0.1505(16) 0.0514(2) 0.05116(9) 0.05117(10) 0.05132(9) 0.0522(11) -0.0514(10) 0.0572(10) -0.0467(10) 0.0535(6) -0.0506(6) 0.0472(16) -0.0577(14)
2
Uiso/Ueq(Å ) 0.018(3) 0.0170(9) 0.0174(11) 0.0179(9) 0.0126(14) 0.0195(15) 0.0149(8) 0.011(3) 0.0310(15) 0.0151(5) 0.0170(6) 0.0195(5) 0.019(3) 0.019(3) 0.029(4) 0.022(3) 0.0264(19) 0.0224(18) 0.027(5) 0.028(5) 0.080(3) 0.0215(6) 0.0272(7) 0.0303(6) 0.046(2) 0.060(3) 0.0520(17) 0.0362(16) 0.0111(13) 0.0121(5) 0.0125(6) 0.0145(5) 0.0098(7) 0.0150(8) 0.0121(5) 0.0119(9) 0.0101(11) 0.0119(5) 0.0127(5) 0.0142(5) 0.013(2) 0.007(2) 0.020(3) 0.011(2) 0.0127(4) 0.0127(4) 0.009(3) 0.018(4) 0.0143(12) 0.0115(5) 0.0124(5) 0.0135(5) 0.013(3) 0.008(3) 0.018(3) 0.013(3) 0.0131(4) 0.0131(4) 0.023(4) 0.003(3)
Appendix 5. (continued) Element Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Ce Pr Nd Sm Gd Tb Dy Yb Nd Sm Gd Tb Dy Yb Sm Gd Tb Dy
Atom Zn9 Zn9 Zn9 Zn9 Zn9a Zn9b Zn9a Zn9b Zn9a Zn9b Zn9a Zn9b Zn10 Zn10 Zn10 Zn10 Zn10 Zn10 Zn10 Zn10 Zn11 Zn11 Zn11 Zn11 Zn11 Zn11 Zn11 Zn11 Zn12 Zn12 Zn12 Zn12 Zn12 Zn12 Zn12 Zn12 Zn13 Zn13 Zn13a Zn13b Zn13a Zn13b Zn13a Zn13b Zn13a Zn13b Zn14 Zn14 Zn14 Zn14 Zn14
Wyck. 12k 12k 12k 12k 6c 6c 6c 6c 6c 6c 6c 6c 6h 6h 6h 6h 6c 6c 6c 6c 2b 2b 2b 2b 2a 2a 2a 2a 6g 6g 6g 6g 6c 6c 6c 6c 4e 4e 2a 2a 2a 2a 2a 2a 2a 2a 12k 6c 6c 6c 6c
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0.387 0.176 0.283 0.283 0.291 0.291 0.361 0.361 0.403 0.403 0.170 0.172 0.310 0.127 0.139
x 0.76440(13) 0.76219(5) 0.76242(5) 0.76272(5) 0.7626(3) -0.7637(3) 0.7629(4) -0.7637(4) 0.7629(2) -0.7645(2) 0.7646(6) -0.7628(6) 0.27049(17) 0.27029(7) 0.27028(8) 0.27005(7) 0.26993(9) 0.26978(10) 0.26983(6) 0.2699(2) 0 0 0 0 0 0 0 0 1/2 1/2 1/2 1/2 0.4988(5) 0.4981(6) 0.5003(3) 0.5001(13) 0 0 0 0 0 0 0 0 0 0 0.8826(18) 0.8840(8) 0.8834(7) 0.8834(8) 0.886(2)
y 0.23560(13) 0.23781(5) 0.23758(5) 0.23728(5) 0.2374(3) -0.2363(3) 0.2371(4) -0.2363(4) 0.2371(2) -0.2355(2) 0.2354(6) -0.2372(6) 0.5410(3) 0.54058(14) 0.54055(16) 0.54009(13) 0.53986(17) 0.53956(19) 0.53965(13) 0.5398(4) 0 0 0 0 0 0 0 0 0 0 0 0 -0.0025(9) -0.0039(11) 0.0006(7) 0.000(3) 0 0 0 0 0 0 0 0 0 0 0.1174(18) 0.1160(8) 0.1166(7) 0.1166(8) 0.114(2)
46
z 0.1585(2) 0.15815(9) 0.1582(1) 0.15819(9) 0.1621(9) -0.1546(9) 0.1651(9) -0.1515(10) 0.1610(6) -0.1561(6) 0.1512(16) -0.1659(16) 1/4 1/4 1/4 1/4 0.2498(11) 0.2551(11) 0.2526(6) 0.2447(19) 1/4 1/4 1/4 1/4 0.251(2) 0.259(2) 0.2564(14) 0.2268(16) 0 0 0 0 0.0005(13) 0.0080(13) 0.0080(7) -0.008(2) -0.0899(11) -0.090(1) -0.090(4) 0.093(4) -0.091(3) 0.091(3) -0.0907(15) 0.0889(15) -0.0931(14) 0.0931(14) 0.081(3) 0.0854(17) 0.0802(15) 0.0840(13) 0.067(4)
Uiso/Ueq(Å2) 0.0106(11) 0.0085(4) 0.0096(5) 0.0113(4) 0.007(3) 0.009(2) 0.013(3) 0.015(3) 0.0104(4) 0.0104(4) 0.012(3) 0.014(4) 0.0091(16) 0.0105(6) 0.0110(7) 0.0119(6) 0.0072(8) 0.0118(9) 0.0100(5) 0.0098(13) 0.051(4) 0.0125(9) 0.0129(10) 0.0255(11) 0.0290(15) 0.037(2) 0.0397(15) 0.019(3) 0.069(3) 0.0196(7) 0.0223(9) 0.0277(8) 0.0324(12) 0.0415(15) 0.0424(10) 0.046(2) 0.017(6) 0.018(4) 0.023(2) 0.023(2) 0.010(4) 0.010(4) 0.0148(18) 0.0148(18) 0.016(5) 0.016(5) 0.09(2) 0.023(2) 0.13(3) 0.0148(18) 0.016(5)
Appendix 6. Fractional atomic coordinates, occupancies and isotropic ADPs for the Orthorhombic Gd13Zn59 structure 2
Element
Atom
Wyck. Occ.
x
y
z
Uiso/Ueq(Å )
Gd Gd Gd Gd Gd Gd Gd Gd Gd Gd Gd Gd Gd Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn
Gd1a Gd1b Gd1c Gd2a Gd2b Gd2c Gd2d Gd2e Gd2f Gd3a Gd3b Gd3c Gd4 Zn1a Zn1b Zn1c Zn1d Zn1e Zn1f Zn1g Zn1h Zn1i Zn1j Zn1k Zn1l Zn2a Zn2b Zn3 Zn4a Zn4b Zn4c Zn4d Zn4e Zn4f Zn5a Zn5b Zn5c Zn5d Zn5e Zn5f
4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a
0.9813(3) 0.9764(3) 0.2924(3) 0.1473(3) 0.1448(3) 0.1457(3) 0.1488(3) 0.4550(3) 0.9570(3) 0.8140(3) 0.8125(3) 0.6236(3) 0.2537(4) 0.2499(7) 0.9163(7) 0.0820(7) 0.9180(7) 0.4504(6) 0.4474(7) 0.4499(7) 0.4480(8) 1.1477(6) 0.3481(7) 0.0039(9) -0.0085(8) 0.6145(8) 0.6144(6) 0.5199(7) 0.2103(7) 0.2077(8) 0.1942(8) 0.1986(7) 0.8465(8) 0.8463(7) 0.4136(7) 0.4206(6) 0.4200(7) 0.4081(7) 0.0660(7) 0.0649(6)
0.1874(4) -0.1868(4) -0.5053(5) -0.3063(5) 0.3096(4) 0.3123(5) -0.3071(5) -0.0026(5) -0.4911(5) -0.3051(4) 0.3124(5) 0.4972(6) 0.0004(5) 0.0024(13) 0.0048(11) -0.4921(11) 0.0017(11) -0.4036(10) 0.3966(11) 0.3966(11) -0.4093(12) 0.4982(13) 0.4984(12) 0.2528(14) -0.2583(15) 0.0909(11) -0.1008(9) -0.0067(13) -0.1448(13) 0.153(1) 0.1379(12) -0.1421(12) 0.5068(15) -0.5112(12) -0.2050(13) 0.1961(13) 0.1979(13) -0.1967(11) 0.1498(12) -0.1476(10)
-0.0032(5) 0.0082(5) 0.0064(5) 0.1989(5) 0.1996(4) 0.8014(4) 0.8071(5) 0.7987(5) 0.8038(5) -0.0038(6) 0.0334(5) 0.9772(5) 0.2519(6) 0.0019(15) 0.6573(8) -0.4991(13) 0.3479(8) 0.1982(10) 0.1980(11) 0.8028(8) 0.7965(15) 0.6991(11) 0.8024(11) 0.2507(17) 0.7397(15) -0.0023(14) -0.0017(9) -0.0049(12) 0.1017(13) 0.0966(10) 0.9069(11) 0.9143(9) 0.0962(13) 0.9076(12) 0.1388(10) 0.1565(9) 0.8477(11) 0.8511(11) 0.1512(12) 0.1470(9)
0.025(2) 0.028(2) 0.0259(18) 0.032(2) 0.024(2) 0.021(2) 0.032(2) 0.028(2) 0.023(2) 0.037(2) 0.038(2) 0.032(2) 0.0440(19) 0.030(4) 0.016(4) 0.020(4) 0.016(4) 0.016(5) 0.019(5) 0.016(5) 0.033(6) 0.023(5) 0.020(5) 0.034(5) 0.040(6) 0.026(5) 0.009(4) 0.025(5) 0.025(5) 0.019(5) 0.025(5) 0.016(5) 0.031(6) 0.023(5) 0.021(5) 0.017(5) 0.022(5) 0.023(5) 0.022(5) 0.008(4)
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
47
Appendix 6. (continued) Element Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn
Atom Zn6a Zn6b Zn6c Zn6d Zn6e Zn6f Zn7a Zn7b Zn7c Zn7d Zn7e Zn7f Zn8a Zn8b Zn8c Zn8d Zn8e Zn8f Zn9a Zn9b Zn9c Zn9d Zn9e Zn9f Zn10a Zn10b Zn10c Zn11 Zn12a Zn12b Zn12c Zn14
Wyck. 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a 4a
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x 0.0568(7) 0.0790(8) 0.2649(7) 0.2689(8) 0.2658(7) 0.2709(8) 0.0294(8) 0.0334(7) 0.0285(7) 0.0324(7) 1.1882(7) 0.8144(7) 0.3626(7) 0.8485(7) 0.8507(6) 0.8446(8) 0.2498(7) 0.2366(8) 0.1093(6) 0.1085(7) 0.2080(7) 0.2057(7) 0.5666(7) 0.5671(6) 0.3703(7) 0.5158(6) 0.5116(6) 0.7463(6) 0.0039(9) 0.3685(8) 0.3678(7) 0.8151(10)
y -0.1574(12) 0.1357(13) -0.3461(12) 0.3294(15) 0.3613(12) -0.3603(14) -0.6494(13) 0.6586(10) 0.6567(13) -0.6523(11) -0.4988(13) 0.0053(13) 0.0591(14) 0.3965(12) -0.4091(10) -0.3861(15) -0.202(1) 0.1987(15) -0.2182(11) 0.2316(9) -0.3269(12) 0.3231(11) 0.3976(10) -0.4056(10) -0.6426(10) -0.4985(11) 0.4994(12) 0.9854(12) 0.2528(14) 0.6354(11) 0.6473(12) 0.3360(12)
48
z 0.8416(14) 0.8589(13) 0.1461(9) 0.1457(13) 0.8447(11) 0.8509(17) 0.1027(14) 0.1098(10) 0.8948(12) 0.8914(11) 0.8921(12) 0.3925(10) 0.3065(13) 0.2440(16) 0.2621(8) 0.7778(16) 0.2589(12) 0.2828(16) 0.0042(13) 0.0017(10) 0.0018(10) -0.0024(14) 0.5060(14) 0.5003(8) 0.9093(9) 0.0932(9) 0.9099(10) 0.2655(11) 0.2507(17) 0.0920(12) 0.9092(10) 0.8250(16)
Uiso/Ueq(Å2) 0.028(5) 0.029(5) 0.016(5) 0.034(6) 0.019(5) 0.043(7) 0.029(6) 0.014(5) 0.022(5) 0.016(5) 0.023(5) 0.019(5) 0.036(6) 0.030(6) 0.009(4) 0.044(7) 0.022(5) 0.048(7) 0.020(5) 0.018(5) 0.023(5) 0.027(5) 0.025(5) 0.011(4) 0.012(4) 0.011(4) 0.016(5) 0.024(5) 0.034(5) 0.021(5) 0.020(5) 0.066(8)
Appendix 7. Fractional atomic coordinates, occupancies and isotropic ADPs for the Orthorhombic Tb13Zn59 structure Element Atom
Wyck. Occ.
x
y
z
Uiso/Ueq(Å2)
Tb
Tb1a
4a
1
0.98157(7)
0.18763(9)
-0.00327(11)
0.0071(4)
Tb
Tb1b
4a
1
0.97709(7)
-0.18496(9)
0.00831(11)
0.0070(4)
Tb
Tb1c
4a
1
0.29268(6)
-0.50529(11)
0.00579(10)
0.0065(4)
Tb
Tb2a
4a
1
0.14739(7)
-0.30471(11)
0.19717(10)
0.0065(5)
Tb
Tb2b
4a
1
0.14521(7)
0.31239(11)
0.19754(10)
0.0069(5)
Tb
Tb2c
4a
1
0.14560(7)
0.31320(11)
0.80081(10)
0.0067(5)
Tb
Tb2d
4a
1
0.14847(7)
-0.30602(11)
0.80547(10)
0.0075(5)
Tb
Tb2e
4a
1
0.45568(6)
-0.00181(12)
0.80033(10)
0.0064(4)
Tb
Tb2f
4a
1
0.95674(7)
-0.48968(12)
0.80374(10)
0.0081(4)
Tb
Tb3a
4a
1
0.81403(7)
-0.30528(9)
-0.00474(12)
0.0096(4)
Tb
Tb3b
4a
1
0.81256(7)
0.31228(10)
0.03766(10)
0.0100(4)
Tb
Tb3c
4a
1
0.62437(6)
0.49871(12)
0.97343(11)
0.0106(4)
Tb
Tb4
4a
1
0.25286(7)
0.00181(13)
0.25354(11)
0.0116(4)
Zn
Zn1a
4a
1
0.41960(17)
0.2016(3)
0.8476(2)
0.0105(12)
Zn
Zn1b
4a
1
0.40845(18)
-0.1967(3)
0.8520(3)
0.0135(12)
Zn
Zn1c
4a
1
0.05604(18)
-0.1578(3)
0.8427(3)
0.0134(12)
Zn
Zn1d
4a
1
0.08284(19)
0.1326(3)
0.8600(3)
0.0206(14)
Zn
Zn1e
4a
1
0.26697(17)
0.3585(3)
0.8455(3)
0.0099(12)
Zn
Zn1f
4a
1
0.26854(18)
-0.3608(3)
0.8540(3)
0.0126(12)
Zn
Zn1g
4a
1
0.41791(18)
-0.2103(3)
0.1425(3)
0.0126(12)
Zn
Zn1h
4a
1
0.42072(18)
0.1943(3)
0.1548(3)
0.0175(14)
Zn
Zn1i
4a
1
0.06435(17)
0.1533(2)
0.1494(3)
0.0089(12)
Zn
Zn1j
4a
1
0.06616(17)
-0.1472(3)
0.1506(3)
0.0099(12)
Zn
Zn1k
4a
1
0.26767(18)
-0.3472(3)
0.1488(3)
0.0133(13)
Zn
Zn1l
4a
1
0.26617(19)
0.3289(3)
0.1468(3)
0.0177(13)
Zn
Zn2a
4a
1
0.91723(16)
0.0044(3)
0.6543(2)
0.0078(10)
Zn
Zn2b
4a
1
0.08290(17)
-0.4960(2)
-0.4996(3)
0.0129(9)
Zn
Zn3
4a
1
0.91618(16)
0.0041(3)
0.3449(2)
0.0077(10)
Zn
Zn4a
4a
1
0.36393(19)
0.0563(4)
0.3084(3)
0.0336(17)
Zn
Zn4b
4a
1
0.84960(17)
0.4013(3)
0.2490(3)
0.0110(12)
Zn
Zn4c
4a
1
0.85085(17)
-0.4047(3)
0.2674(3)
0.0129(12)
Zn
Zn4d
4a
1
0.84489(17)
-0.3836(3)
0.7810(3)
0.0176(13)
Zn
Zn4e
4a
1
0.24959(15)
-0.2015(3)
0.2596(3)
0.0112(12)
Zn
Zn4f
4a
1
0.23386(18)
0.2006(3)
0.2871(3)
0.0156(12)
Zn
Zn5a
4a
1
0.2078(2)
-0.1407(3)
0.1012(3)
0.0142(13)
Zn
Zn5b
4a
1
0.21203(19)
0.1519(3)
0.0960(3)
0.0136(13)
Zn
Zn5c
4a
1
0.19457(18)
0.1381(3)
0.9045(3)
0.0134(13)
Zn
Zn5d
4a
1
0.20144(19)
-0.1408(3)
0.9177(2)
0.0121(12)
Zn
Zn5e
4a
1
0.84519(17)
0.5120(3)
0.0991(3)
0.0139(12)
Zn
Zn5f
4a
1
0.84699(16)
-0.5075(3)
0.9136(2)
0.0143(12)
49
Appendix 7. (continued) Element
Atom
Wyck.
Occ.
x
y
z
Uiso/Ueq(Å2)
Zn
Zn6a
4a
1
0.10660(16)
-0.2199(2)
-0.0025(3)
0.0116(11)
Zn
Zn6b
4a
1
0.11106(16)
0.2286(2)
0.0017(3)
0.0118(11)
Zn
Zn6c
4a
1
0.20805(16)
-0.3238(2)
0.0004(3)
0.0113(11)
Zn
Zn6d
4a
1
0.20765(16)
0.3252(2)
-0.0032(3)
0.0120(11)
Zn
Zn6e
4a
1
0.56798(16)
0.3997(2)
0.5014(3)
0.0101(11)
Zn
Zn6f
4a
1
0.56678(16)
-0.4027(2)
0.5004(3)
0.0102(11)
Zn
Zn7a
4a
1
0.03103(19)
-0.6529(3)
0.1067(3)
0.0098(12)
Zn
Zn7b
4a
1
0.0320(2)
0.6575(3)
0.1094(3)
0.0112(13)
Zn
Zn7c
4a
1
0.03109(19)
0.6633(3)
0.8901(3)
0.0095(12)
Zn
Zn7d
4a
1
0.0300(2)
-0.6475(3)
0.8945(3)
0.0122(13)
Zn
Zn7e
4a
1
1.18820(15)
-0.4983(3)
0.8912(2)
0.0092(10)
Zn
Zn7f
4a
1
0.81415(16)
0.0030(3)
0.3944(2)
0.0108(11)
Zn
Zn8a
4a
1
0.45230(18)
-0.3998(3)
0.1982(2)
0.0079(11)
Zn
Zn8b
4a
1
0.44688(18)
0.4007(3)
0.1952(3)
0.0120(12)
Zn
Zn8c
4a
1
0.45123(18)
0.3939(3)
0.8005(3)
0.0095(12)
Zn
Zn8d
4a
1
0.44748(18)
-0.4097(3)
0.8013(3)
0.0102(12)
Zn
Zn8e
4a
1
1.14490(15)
0.5062(3)
0.6964(2)
0.0119(11)
Zn
Zn8f
4a
1
0.34656(14)
0.4972(3)
0.8000(2)
0.0081(10)
Zn
Zn9a
4a
1
0.36692(19)
-0.6591(3)
0.0906(2)
0.0109(12)
Zn
Zn9b
4a
1
0.36823(18)
0.6402(3)
0.0911(2)
0.0077(12)
Zn
Zn9c
4a
1
0.36747(18)
0.6409(3)
0.9082(2)
0.0080(12)
Zn
Zn9d
4a
1
0.37084(18)
-0.6439(3)
0.9072(2)
0.0074(12)
Zn
Zn9e
4a
1
0.51613(15)
-0.4992(3)
0.0912(2)
0.0094(11)
Zn
Zn9f
4a
1
0.51187(15)
0.4977(3)
0.9082(2)
0.0065(10)
Zn
Zn10a
4a
1
0.61484(16)
0.0924(2)
0.0001(3)
0.0090(11)
Zn
Zn10b
4a
1
0.61479(16)
-0.0993(2)
0.0003(3)
0.0099(11)
Zn
Zn10c
4a
1
0.51941(14)
-0.0060(2)
-0.0009(2)
0.0091(10)
Zn
Zn11
4a
1
0.25037(17)
0.0016(3)
-0.0005(3)
0.0134(9)
Zn
Zn12a
4a
1
0.0053(2)
0.2545(3)
0.2562(3)
0.0172(12)
Zn
Zn12b
4a
1
-0.01088(19)
-0.2655(3)
0.7311(3)
0.0248(15)
Zn
Zn12c
4a
1
0.74532(16)
0.9920(4)
0.2635(2)
0.0200(13)
Zn
Zn14
4a
1
0.8171(2)
0.3404(3)
0.8298(3)
0.0327(16)
50
Appendix 8. Fractional atomic coordinates, occupancies and isotropic ADPs for the Monoclinic Yb13Zn59 structure Element Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Yb Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn
Atom Yb1a Yb1b Yb1c Yb1d Yb1e Yb1f Yb2a Yb2b Yb2c Yb2d Yb2e Yb2f Yb2g Yb2h Yb2i Yb2j Yb2k Yb2l Yb3a Yb3b Yb3c Yb3d Yb3e Yb3f Yb4a Yb4b Zn1a Zn1b Zn1c Zn1d Zn1e Zn1f Zn1g Zn1h Zn1i Zn1j Zn1k Zn1l Zn1m Zn1n Zn1o Zn1p Zn1q Zn1r Zn1s Zn1t Zn1u Zn1v Zn1w Zn1x
Wyck. 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x 0.2273(3) 0.7297(3) 0.2706(3) 0.7721(3) 0.54512(17) 1.04404(18) 0.3966(3) 0.8975(3) 0.1031(3) 0.6041(3) 0.8975(3) 0.3970(3) -0.3965(3) 0.1022(3) 0.7065(2) 1.20525(19) 1.20183(19) 1.7018(2) 0.0630(3) 0.5650(3) 0.4348(3) 0.9361(3) 0.8795(2) 0.3816(2) 0.48511(17) 0.9868(2) 0.8172(7) 0.3156(6) -0.3157(6) 0.1837(6) 0.5183(7) 1.0161(6) -0.0174(7) 0.4807(6) 1.0183(7) 0.5191(6) -0.5204(6) -0.0200(6) 0.3241(6) 0.8142(6) 0.1774(6) 0.6725(6) 0.8405(6) 0.3327(6) -0.3333(7) 0.1650(6) 0.2848(6) -0.1515(9) 0.7115(6) 1.2060(7)
51
y 0.4330(7) 0.9311(7) -0.4354(7) 0.0672(7) -0.2518(8) 0.2484(8) -0.0615(6) 0.4394(7) 0.0596(7) 0.5598(7) 0.4415(6) 0.9421(7) -0.4440(7) 0.0578(6) 0.2491(7) 0.7488(7) 0.7485(7) 1.2493(7) -0.0600(7) 0.4400(6) 0.0580(7) 0.5601(7) -0.2510(7) 0.2495(7) 0.25000 0.7474(8) 0.8965(12) 0.3996(11) -0.9015(11) -0.4005(11) -0.1194(12) 0.3841(11) 0.1090(12) 0.6144(11) 0.4069(11) 0.9131(11) -0.4232(12) 0.0707(11) 0.4574(11) 0.9664(11) -0.4634(12) 0.0387(11) 0.9481(12) 0.4504(12) -0.9526(13) -0.4517(12) 0.4393(11) -0.1413(15) 0.0637(11) 0.5717(12)
z -0.2546(5) 0.7597(5) 0.2597(5) 0.2460(5) -0.2546(4) 0.7608(5) -0.0553(6) 0.5625(6) 0.0614(6) 0.4451(6) -0.0539(6) 0.5613(6) 0.0625(6) 0.4451(6) -0.0504(5) 0.5581(5) -0.0537(5) 0.5592(5) -0.2662(7) 0.7758(5) 0.2758(6) 0.2358(7) -0.2091(5) 0.7111(5) -0.02580 0.5188(5) -0.0919(14) 0.6007(14) 0.1000(13) 0.4041(12) -0.0913(14) 0.6040(13) 0.1007(14) 0.4054(13) -0.0964(13) 0.6015(13) 0.1063(14) 0.4023(13) 0.1029(13) 0.4161(12) -0.0962(13) 0.6065(14) 0.1001(13) 0.4050(13) -0.0927(14) 0.5985(13) -0.0623(12) 0.6120(17) 0.4310(12) 0.0897(14)
2
Uiso/Ueq(Å ) 0.016(2) 0.015(2) 0.016(2) 0.013(2) 0.0117(13) 0.0147(14) 0.0107(19) 0.013(2) 0.016(2) 0.013(2) 0.0120(14) 0.015(2) 0.0137(19) 0.0122(17) 0.0129(15) 0.0135(15) 0.0123(15) 0.0172(16) 0.037(3) 0.0130(14) 0.0163(17) 0.036(3) 0.0251(17) 0.0169(14) 0.0156(11) 0.088(3) 0.020(4) 0.015(4) 0.012(4) 0.010(4) 0.025(4) 0.016(4) 0.023(4) 0.013(4) 0.029(4) 0.015(4) 0.023(4) 0.029(4) 0.012(4) 0.036(4) 0.018(4) 0.021(4) 0.023(4) 0.012(4) 0.021(4) 0.012(4) 0.019(4) 0.103(7) 0.015(3) 0.057(5)
Appendix 8 (continued) Element Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn
Atom Zn2a Zn2b Zn3a Zn3b Zn3c Zn3d Zn4a Zn4b Zn4c Zn4d Zn4e Zn4f Zn4g Zn4h Zn4i Zn4j Zn4k Zn4l Zn5a Zn5b Zn5c Zn5d Zn5e Zn5f Zn5g Zn5h Zn5i Zn5j Zn5k Zn5l Zn6a Zn6b Zn6c Zn6d Zn6e Zn6f Zn6g Zn6h Zn6i Zn6j Zn6k Zn6l Zn7a Zn7b Zn7c Zn7d Zn7e Zn7f
Wyck. 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x 0.3320(4) 0.8318(4) 0.3318(5) 0.8327(5) 0.8342(5) 0.3338(5) 0.4487(7) 0.9490(7) 0.0521(7) 0.5521(8) 0.4736(6) -0.4753(7) 0.0401(8) 0.6001(5) 1.0968(6) 1.0992(5) 0.5948(5) 1.0316(8) 0.6295(6) 0.9094(8) 0.0896(10) -0.6308(7) 0.9025(6) 0.4030(6) -0.4045(6) 0.0953(7) 0.5057(7) 1.0042(8) -0.0105(8) 0.4948(6) 0.4596(8) 0.9608(8) 0.0398(8) 0.5401(7) 0.6868(7) 1.1818(8) -0.1834(8) 0.3172(8) 0.6484(8) 1.1468(8) -0.1452(8) 0.3534(8) 0.2777(6) 0.7770(8) 0.2205(8) 0.7215(7) 0.7815(7) 1.2798(8)
y -0.2502(18) 0.2486(16) -0.2520(14) 0.2483(14) 0.2478(14) 0.7472(14) 0.0935(11) 0.5966(13) -0.1060(12) 0.4055(12) 0.1058(10) -0.6109(12) -0.1045(14) 0.2470(15) 0.7427(15) 0.7469(15) 0.2506(15) 0.5983(14) 0.1570(11) 0.8840(13) -0.4000(17) -0.6620(13) 0.8420(11) 0.3447(11) -0.8509(11) -0.3501(12) 0.0446(12) 0.5469(13) -0.0499(15) 0.4528(11) -0.0847(14) 0.4188(14) 0.0772(14) 0.5781(13) 0.3531(13) 0.8519(13) -0.3556(14) 0.1466(14) 0.4673(14) 0.9705(14) -0.4741(14) 0.0281(14) -0.4018(11) 0.0955(13) 0.4046(12) 0.9027(12) 0.0895(12) 0.5915(13)
52
z -0.7481(8) 1.2681(7) -0.0942(9) 0.5996(9) -0.0919(9) 0.6010(9) -0.1826(12) 0.6841(13) 0.1702(12) 0.3186(14) 0.6385(12) 0.1373(13) 0.3554(16) -0.1508(10) 0.6532(12) -0.1711(9) 0.6680(9) 0.3618(16) 0.4110(12) 0.4448(15) -0.068(2) -0.0925(13) -0.0078(12) 0.5068(12) -0.0020(12) 0.5045(14) -0.0297(14) 0.5289(14) 0.0433(15) 0.4715(12) -0.2445(13) 0.7551(13) 0.2544(12) 0.2547(12) -0.2443(12) 0.7538(12) 0.2550(13) 0.2538(13) 0.2613(13) 0.2480(13) -0.2489(13) 0.7578(13) -0.1462(11) 0.6414(14) 0.1437(14) 0.3597(13) -0.1386(12) 0.6493(13)
2
Uiso/Ueq(Å ) 0.017(3) 0.014(2) 0.012(3) 0.012(2) 0.012(2) 0.011(2) 0.017(4) 0.023(4) 0.037(5) 0.023(4) 0.013(3) 0.023(4) 0.055(6) 0.018(3) 0.036(4) 0.021(3) 0.019(3) 0.044(6) 0.015(3) 0.049(5) 0.076(8) 0.022(4) 0.020(4) 0.011(4) 0.015(4) 0.026(4) 0.018(4) 0.035(5) 0.046(5) 0.008(3) 0.024(5) 0.021(5) 0.016(5) 0.011(4) 0.017(5) 0.014(5) 0.021(5) 0.018(5) 0.019(5) 0.015(5) 0.023(5) 0.018(5) 0.010(4) 0.021(4) 0.020(4) 0.014(4) 0.016(4) 0.018(4)
Appendix 8 (continued) Element Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn Zn
Atom Zn7g Zn7h Zn7i Zn7j Zn7k Zn7l Zn8a Zn8b Zn8c Zn8d Zn8e Zn8f Zn8g Zn8h Zn8i Zn8j Zn8k Zn8l Zn9a Zn9b Zn9c Zn9d Zn9e Zn9f Zn9g Zn9h Zn9i Zn9j Zn9k Zn9l Zn10a Zn10b Zn10c Zn10d Zn10e Zn10f Zn11a Zn11b Zn12a Zn12b Zn12c Zn12d Zn14a Zn14b Znx Znz
Wyck. 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a 2a
Occ. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
x -0.2812(8) 0.2186(7) 0.9355(6) 0.4357(6) 0.4392(5) 0.9397(6) 0.1985(7) 0.7032(8) 0.2995(8) 0.8016(6) 0.6985(8) 1.1976(8) -0.1988(8) 0.2986(6) 0.5969(5) 1.0933(5) 1.1048(4) 0.6090(5) 0.1172(7) 0.6201(7) 0.3814(8) 0.8826(7) 0.6191(7) 1.1183(7) -0.1178(7) 0.3800(7) 0.7629(5) 0.2641(5) 0.2722(5) 0.7719(6) 0.3636(9) 0.8632(7) 0.1356(8) 0.6355(8) 0.7312(4) 1.2299(5) 0.5031(5) 1.0001(5) 0.7482(9) 1.2495(10) 0.4923(6) 0.9927(7) 0.5808(6) 0.4172(7) -0.2228(6) 1.1291(8)
y -0.0971(12) 0.4024(12) 0.2501(15) 0.7470(16) 0.7479(14) 0.2495(16) 0.3519(12) 0.8538(13) -0.3506(12) 0.1564(10) 0.8438(13) 0.3441(12) -0.8471(13) -0.3555(10) -0.2517(15) 0.2505(15) 0.2506(14) 0.7515(17) 0.0950(11) 0.5974(12) -0.1013(12) 0.3976(12) 0.6089(12) 0.1048(11) -0.6112(12) -0.1120(12) -0.2513(15) 0.2510(14) 0.2470(15) 0.7461(15) -0.1548(16) 0.3460(13) 0.1551(14) 0.6510(13) 0.7494(16) 1.2485(18) 0.2485(18) 0.7465(17) -0.0051(14) 0.4982(15) 0.7417(13) 0.2434(15) 0.0985(11) 0.3963(13) -0.0513(12) 0.6618(14)
53
z 0.1462(13) 0.3577(12) -0.1448(10) 0.6496(11) -0.1419(9) 0.6472(11) -0.0447(12) 0.5499(14) 0.0545(14) 0.4485(10) -0.0464(14) 0.5538(13) 0.0524(14) 0.4574(11) -0.043(1) 0.5506(10) -0.0497(9) 0.5569(11) -0.1494(12) 0.6633(13) 0.1610(13) 0.3415(12) -0.1572(12) 0.6573(12) 0.1603(13) 0.3485(14) -0.1542(10) 0.6625(10) -0.1535(10) 0.6604(11) -0.2481(13) 0.7508(11) 0.2518(12) 0.2555(11) 0.2523(10) 0.2536(11) -0.2506(8) 0.7465(8) 0.0032(15) 0.5047(16) 0.0225(11) 0.4855(12) 0.5709(11) 0.0659(14) 0.5519(13) 0.0967(16)
Uiso/Ueq(Å2) 0.018(4) 0.011(4) 0.016(3) 0.021(3) 0.009(3) 0.022(3) 0.017(4) 0.031(5) 0.019(4) 0.006(3) 0.022(4) 0.020(4) 0.022(4) 0.012(4) 0.016(3) 0.016(3) 0.008(3) 0.026(3) 0.013(4) 0.015(4) 0.016(4) 0.014(4) 0.012(4) 0.009(4) 0.014(4) 0.019(4) 0.015(3) 0.011(3) 0.016(3) 0.023(3) 0.024(5) 0.011(4) 0.014(5) 0.010(4) 0.010(2) 0.019(3) 0.020(3) 0.022(3) 0.028(3) 0.026(3) 0.027(4) 0.039(4) 0.015(3) 0.027(4) 0.057(5) 0.043(5)
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