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Universal mechanism of tetrahedral metal cluster formation in structures with breathing pyrochlore sublattices
M. V. Talanov1, V.M. Talanov2
1 Research Institute of Physics, Southern Federal University, Rostov-on-Don, Russia 2 South-Russian State Polytechnical University, Novocherkassk, Russia
tmikle-man@mail.ru, valtalanov@mail.ru PACS 61.50.Ks; 36.40.-c DOI 10.17586/2220-8054-2017-8-5-677-687
A universal mechanism of tetrahedral metal cluster formation in crystal with geometrically frustrated pyrochlore sublattices is proposed. It has been shown that the critical irreducible representation t, which generated the formation of metal clusters in non-centrosymmetrical F43m-phases from high symmetry phases with Fd3m space group, is a one dimensional irreducible representation K11 (T4(A2u)) (in Kovalev notation). The structural theory of metal cluster formation based on group theoretical calculations was published earlier for the case of A-ordered spinel. In this work, the theory is generalized in the case of any high symmetry Fd3m structures that include pyrochlore sublattices. We presented a brief review of such structures and mechanisms of the tetrahedral metal cluster formation. The existence of so called "breathing" pyrochlore sublattices in ordered phases is predicted theoretically. The groups of atoms, between which bond clusters, are found. These groups of atoms define electron correlation effects. Examples of tetrahedral metal cluster formation in ordered spinels, ordered lacunar spinels, ordered Laves phases (MgCu4Sn structural type) and ordered pyrochlore are considered. The theoretical results are confirmed by the known experimental facts.
Keywords: tetrahedral clusters, ordered spinel, ordered Laves phases, ordered pyrochlore, geometrical frustration, breathing pyrochlore sublat-tices.
Received: 10 August 2017 Revised: 1 September 2017
1. Introduction
Geometric frustration is an old topic of physics and solid state chemistry [1-4]. L. Pauling, using the example of crystalline ice, has shown the existence of different orientations of water molecules, which do not lead to a change in the energy of the system [5,6].
The paradigm of geometric frustration is used in different scientific fields: for the description of the nontrivial packing of polyhedra in three-dimensional space in systems such as quasi-crystals; for amorphous metals and nematic disclination networks in densely packed 3D colloidal lattices; in the theory of magnetism when the arrangement of spins on a lattice does not satisfy all interactions at the same time; for the study of the orientation order in the proton configurations in ordinary hexagonal ice and for spins in magnetic molecules and artificial magnetic architectures.
The term "geometrical frustration" describes the structures with local order generated by the lattice geometry. Frustration arises when the geometry of any system allows for a set of degenerate ground states. Such highly degenerate systems are extremely sensitive to thermal and quantum fluctuations, and thereby intriguing classical and quantum ground states may emerge via "order by disorder".
The important physical properties of many inorganic crystals are connected with a structural feature - a three-dimensional network of the tetrahedra formed by cations. A network of these tetrahedra is called a pyrochlore sublattice. Some such pyrochlore sublattices are in the following: spinels AB2X4 (where the B site displays a pyrochlore sublattice); pyrochlores A2B2O7 (where both A and B sites form corner-sharing tetrahedra): and Laves phases (C15) AB2 (where B-metal atoms form a pyrochlore sublattice). There are hundreds of compounds crystallizing in these structural types. The geometrical frustration of spins results in the formation of exotic electronic and structural state in these substances. One of these states is spin liquid.
P. W. Anderson was the first to show that B-sublattice (pyrochlore sublattice) in spinel structure is geometrically frustrated and causes the unusual physical properties of this class of materials [4].
In this paper, we theoretically establish the universal structural mechanism for the formation of metal clusters namely metal tetrahedra in inorganic crystals with F43m space group. This short paper is written from a solid state chemist's point of view, with an emphasis on crystalline materials and structural chemistry aspects.
2. Symmetry of order parameter of Fd3m F43m phase transition
The crystal structures of spinels, pyrochlores and Laves phases (C15) are characterized by the space group Fd3m. The subject of our investigation is the ordered phases of these structural types with the F43m space group. By using the concept of one critical irreducible representation (irrep) we will derive theoretically the structures of low-symmetry ordered cubic F43m-phases from structures of high-symmetry disordered Fd3m-phases. The structure of low-symmetry F43m-phase is determined by the ordering of atoms and their displacements in the initial (parent) structure. The structural mechanism of F43m-phase formation is generated by critical and noncritical (improper) irreps of Fd3m space group.
The phases with F43m symmetry can be generated by one-dimensional irrep kh(t4(A2u)) and can also be generated by two four-dimensional irreps k9(t1 ) and k9(t4) [7-10]. Below, we will only discuss the phase induced by the irrep kh(t4) of group Fd3m. The expression Kj(t) means the star of wave vectors Kj, where i is number of corresponding irrep t for given star j [7]. Critical irrep kh(t4) determines the symmetry and structure of low-symmetry phase near point of phase transition.
The Wyckoff positions splitting takes place in the low-symmetry phase as a result of phase transition. To find the splitting of Wyckoff positions in the structure of the highly-symmetric phase it is necessary to analyze the composition of permutation and mechanical representations of high-symmetry structure. We have found that the critical one dimensional irrep kh (t4) enters into the mechanical representation on 16c, 16d, 32e, 48f Wyckoff positions and enters into the permutation representation on 8a, 8b, 32e, 48f Wyckoff positions [11-15].
Therefore the low-symmetry F43m-phases formation is accompanied by simultaneous displacements of atoms spaced on 16c, 16d, 32e, 48f Wyckoff positions as well as by ordering of atoms spaced on 8a, 8b, 32e, 48f Wyckoff positions.
3. B4-clusters in A-ordered spinel structures
The structural formula of cubic centrosymmetric Fd3m-spinel AB2X4 is (A)8a [B2]16dX|2e. The formation of a low-symmetry non-centrosymmetric F43m-phase is accompanied by ordering of tetrahedral cations (1:1 order type) and anions (1:1 order type), and also by displacements of octahedral B-cations and anions. The calculated structure of the ordered F43m-phase is shown in Fig. 1. The results of the calculations demonstrate that the tetrahedral cations in the ordered phase occupy nonvariant 4a and 4c Wyckoff positions (site symmetry 43m). The octahedral cations are displaced by equal distances along the four threefold axes and occupy monovariant Wyckoff positions 16e with local symmetry 3m in the ordered structure (Figs. 1, 2). The free parameter x1 is approximately equal to 0.675. It is interesting that the B-cations and anions form specific clusters (Fig. 2(a, c, e)). The anionic lattice is separated into two sublattices in the ordered phase. The anions occupy the16c Wyckoff positions (site symmetry 3m). The anions' arrangement in each sublattice is described by the free parameters x2 « 0.875 and x3 « 0.375. The anions are displaced along the diagonals of the octants in the [111]-directions (Fig. 2a). It has been stated by calculation that general structural formula of the A-ordered spinel is A'4aA4cB46eX' 16eX46e (Fig. 3) [16].
Fig. 1. Calculated structure of an A-ordered spinel (space group F43m). Atom presentation of the structure (a) and projections along (001) (b) and along (111) (c) of the ordered spinel structure
Features of the F43m-phase structure are discussed in [14,16]. The metal clusters are the most interesting feature of this structure. Two neighboring groups of octahedral cations and anions form the expanded (Fig. 2(c, d)) and contracted (Fig. 2(b,e)) regular tetrahedra. Net of such alternative tetrahedra (expanded and contracted) forms the unusual pyrochlore sublattice with two different B-B distances [16] (Fig. 3, Table 1). In the work [17] this was termed "breathing" pyrochlore sublattice. Also it is interesting that the structural feature of ordered phase is "breathing" ring of tetrahedra (Fig. 4).
Fig. 2. Structural mechanism of atom cluster formation. Displacements of octahedral cations and anions in the adjacent octants of the spinel structure (a); compressed cluster B^6e (b), expanded cluster X^6e (c), expanded cluster B^6e (d) and compressed cluster X^6e (e)
Fig. 3. Part of the crystal structure of spinel type with space group F43m (a); expansion and contraction of the B4 tetrahedra are overdrawn (b). The designation of atoms is the same as in Fig. 1
The expanded and contracted tetrahedra of B-cations are considered as a metal cluster. Contracted tetrahedra have the linear size (21/2/4)a and are located in the distance (61/2/4)a, where a is the parameter of a cubic elementary cell of a spinel structure.
The chemical bonding between metal tetrahedra is realized with the help of bridge from two atoms Xх 16c (Fig. 5). This bond influences the correlation of electron movement and the participant in the hopping mechanism of electrical conductivity for A-ordered spinels.
Metal tetrahedra form hyper-tetrahedra (Fig. 6). The hyper-tetrahedra are formed by 4 B-tetrahedra. The hyper-tetrahedra formed by B-cations may participate in the formation of magnetic properties of the ordered phase.
Metal clusters were experimentally observed in ordered spinels LiXY4O8 (X = Ga, Fe, In; Y = Cr, Rh), Cr4GaLiOg, Cr4InLiO8, Rh4InLiO8, Ago.5Ino.5Cr2S4 and Cuo.5Ino.5Cr2S4 [18-20] and their existence, apparently, causes unusual magnetic properties of these compounds. If such metal clusters have a magnetic moment, then it
M. V. Talanov, V. M. Talanov Table 1. Interatomic distances in the ordered structure of the spinel Al/2B2X4
Atom pair Distances
F43m from Fd3m F43m
A-X 31/2 (1/8 + 5 - d) 31/2(1 - X2)
A' - X 31/2 (1/8 + 5 - d) 31/2 (X2 - 1/4)
A-X [19/64 + 3(5 - d)2 + 5/4(d - 5)]1/2 [3/2 + 3x2]1/2
A' - X [11/64 + 3(5 - d)2 + 1/4(5 - d)] 1/2 [11/16 - (3/2)x2 +3x2]1/2
B-X [(1/4 - 5 + d - b)2 + 2(d + b - 5)2] 1/2 [(X2 - x)2 + 2(x2 + x - 1/2)2]1/2
B-X 31/2 (1/4 + d - 5 - b) 31/2 (x2 - x)
B-X [2(3/8 + 5 + d - b)2 + (-1/4 + 5 + d - b)2] 1/2 [2(x - x1 - 1/2)2 + (x - x1 )2]1/2
A - A; A' - A' (1/2)21/2 (1/2)21/2
A-A' (1/4)31/2 (1/4)31/2
B -B 23/2 (1/8 - b) 23/2 (3/4 - x)
B -B 23/2 (1/8 + b) 21/2 (2x - 1)
X-X 23/2 (1/8 - 5 + x) 21/2(2x2 - 1/2)
Note: x = 5/8 + b; x1 = u + d; x2 = 1/4 — u + d, where b, d - displacements of B-cations and X-anions,
u = 3/8 + 5 - anion parameter (free parameter).
a b
Fig. 4. B-cations in ordered spinel form "breathing" ring of tetrahedra. The designation of atoms is the same as in Fig. 1
is possible to expect that distribution of the magnetic moments on space of a crystal should be chaotic. In the results, the crystal which does not have a long ferromagnetic order, but has clusters with a certain degree of the local magnetic order, is formed. Such magnetic state is named spin glass. Thus, structural ordering of cations in spinel tetrahedral sites naturally causes an opportunity for the existence of a magnetic spin-glass state. In the case of magnetic interaction of metal tetrahedra unusual magnetic states of ordered spinels can be observed.
4. B4-clusters in some lacunar ordered spinel structures
The structure of some lacunar phases can be considered as a particular case of ordered spinel structures. The structure of these phases can be represented as a structure of defect spinels with the 1:1 ordering of cations in
Fig. 5. The chemical bridge between two metal tetrahedra
Fig. 6. The hyper-tetrahedra formed by B-catons
Table 2. The distribution of atoms in Wyckoff positions of some lacunar phases
Examples of the chemical compounds Structural formula Wyckoff positions of F43m-phase References
A' 4(a) A4(C) B16(e) X16(e) Y' 16(e) X 4
Re4 S 4 Te4 iDl6e v' 16e v 16e B4 X 4 X4 □ □ Re4 S4 Te4 [21,22]
Gai.33 Cr4 Sg A' 4a A4c Bl6e X' 16e X^6 Ga1.33 + □ Cr4 S4 S4 [23-25]
GaMo4 O8 A4C Bl6e X' 16e Xl6e □ Ga Mo4 O4 O4 [26,27]
Re4 As6 S3 A4c B16e X' 16e X16e □ As Re4 As4 AsS3 [28]
tetrahedral sites. Therefore, the lacunar structures and their properties are inherently associated with the structure of A-ordered spinels. All varieties of possible compositions of lacunar phases are theoretically deduced from the structural formula of A-ordered defect spinel with space group F43m (Table 2). Lacunar phases of different types are obtained depending on the types of atoms that are absent in the structural formula of the A-ordered spinel. For example, Table 2 presents the experimentally revealed distributions of atoms on Wyckoff positions for some lacunar phases as particular cases of defect A-ordered spinel [21,22]. Defects in the table are designed by a symbol □.
The lacunar structures "inherits" from the structure of the ordered spinel four types clusters, which have been established as a result of theoretical research of the structural mechanism of the spinel formation with the type 1:1 order in 8a and 32e Wyckoff positions (Section 3). There are two types of metal clusters - extended and compressed metal tetrahedra B46(e) and two types of extended and compressed anion clusters X^6^ and X'46(e). We believe that the formation of these clusters is the main reason of the unusual physical properties of these phases. There are many experimental examples which confirm with the considered concept of the universal mechanism of cluster formation [21-28].
The Re4 clusters in the Re4 Te4S4 (Fig. 7). The arrangement of the ReS3Te3 octahedra is the same as that in the BX6 octahedra in the spinel structure AB2X4, where A is a tetrahedrally coordinated metal atom and B is an octahedrally coordinated metal atom. According to our concept the B atom is at the centre of the BX6 octahedron in the spinel structure, but the Re atom is shifted from the centre of the S3Te3 octahedron to form Re4 cluster. The sites which are occupied by A atoms in the spinel structure are empty in Re4Te4S4 structure (Table 1, Fig. 5). In this structure Re4S4 moieties are linked by bridging Te-groups. The part of rhenium atoms of this cluster may be replaced by molybdenum atoms, which lead to the formation of Re4-xMox heterometallic cluster in Re4-xMoxS4Te4 (0 < x < 1.25) structure [22,29].
Fig. 7. The calculated structural motif of Re4S4Te4 structure
The structure of the Re4 S4Te4 has been determined using X-ray diffraction [21,22]. The compound contains tetrahedral Re4 clusters with the Re-Re distance of 2.785 A [21]. The distance between the Re4 clusters is equal to 4.345 A [30]. The geometry and arrangement of these Re4 clusters corresponds exactly to the B4-clusters arrangement on Figs. 1,2. By introducing halogens into such rhenium-chalcogenide systems, related compounds, such as Re4Te4(TeBr2)4Br8 [31] are obtained. It is interesting to note that the structure of Re4Te4 S4 is isotypic with those of the chalcogenohalides of niobium and molybdenum, M4X4X'4 (M = Nb, Mo; X = S, Se; X' = C1, Br, I) [21,32,33].
The Re4 clusters in the Re4As6S3 structure [28]. The Re4As6 S3 structure is isostructural to GaMo4 S8 and may be written as AsRe4(Aso.25S0.75)4As4 according to the family of compounds AB4X4Y4. Anions in 32e Wyckoff position spinel splits into two 16-fold positions in the F43m-phase. These two positions are occupied by the same atoms in the thio-spinel AB2 S4; but they may also contain chemically different atoms in case of more general family of compounds with structural formula AAXB4X4X'4. In Re4As6 S3, one of these positions is occupied entirely by arsenic atoms (IV) whereas the other contains a random mixture of sulfur and arsenic atoms (II) in a 3:1 ratio. As a consequence, the simple close packed anion lattice in the thio-spinels now consists of two distinct sublattices, producing a distorted close packed anion lattice (Fig. 8).
Fig. 8. Calculated low-symmetrical anion ordered spinel structure with the space group F43m and with the structural formula A4(a)B46(e)X^6^X416(e). Atom presentation of ordered spinel structures (a), tetrahedra in ordered spinel structure (b), B16(e)-metallic tetrahedral (nanoclus-ters) (c). A substance with composition Re4As6S3 has a similar structure. In this structure there are Re4-clusters
The position of atom A in the spinel-type (previously the 8a position) splits into two different positions of multiplicity 4 each, one at (0, 0, 0), surrounded by four atoms from the first sublattice (in Re4As6S3, 4 arsenic atoms), the other at (3/4, 3/4, 3/4), surrounded by 4 atoms from the second sublattice (in Re4As6 S3 4 As/S atoms) [28]. A central arsenic is found in (0, 0, 0), producing an As5 clusters. The atom B, in the ideal spinel at the octahedral site 16d (at 3/8, 3/8, 3/8), is moved out of its ideal position, shifting along the body diagonal of the unit cell (coordinates (x, x, x)). This produces different interatomic distances, allowing contacts with other atoms in symmetry-related B positions [28]. In Re4As6S3, the rhenium atoms have short contacts at a distance of 2.776 A [28].
The B4 and B2B'2 clusters in lacunar AB4X8 spinels. The structural formula of these compounds is A'4aBl6eX' 16eXl6e. The filling of half of the A sites with metal atoms gives the GaMo4 S8 type of structure, which is found for AB4X8 (A = Ga, Al, Ge; B = V, Nb, Na, Mo; X = S, Se) (Fig. 9) and AMo2Re2S8 (A = Zn, Fe, Co, Ni) [26,27].
Fig. 9. Calculated crystal structure of the AB4X8 type compounds containing B^6 clusters (a), the bridge between two B46e clusters mediated by two Xf 16e atoms (b)
The structure of these a cation-deficient ordered spinels consists of heterocubane-like B4units and AX^-tetrahedra arranged in a similar manner to the ions in the rock-salt structure (Fig. 9(a)). The 8a site (0, 0, 0) of the spinel splits into sites 4a (0, 0, 0) and 4c (1/4, 1/4, 1/4) in F43m-phase, where only 4a is occupied by atoms. The 16d site (5/8, 5/8, 5/8), occupied by the B atoms, becomes the 16e site (x, x, x) in F43m, which allows the B atoms in this site to dislocate along the body diagonal of the unit cell. Within these clusters the B-B distances are compatible with the formation of metallic bonds, while the large intercluster distances prevent metal-metal bonding. This peculiar topology leads to the formation of molecular-like electronic states within the clusters.
The formation of B4 clusters causes significant changes in the physical properties. The metal electrons not incorporated in M-S bonds localize in cluster molecular orbitals. Because the clusters are widely separated, orbitals of different clusters do not overlap and the compound becomes nonmetallic. Electric conduction takes place in case the electrons hop between the clusters. These materials are Mott insulators, in which electrical conduction occurs by the hopping of electrons between clusters separated by relatively large distances (« 4 A) and not between single atoms as in classical Mott insulators like NiO.
It has been suggested that GaV4 S8 behaves as an electron glass at low temperatures, while GaNb4 S8 as recently has been shown to undergo a pressure-induced transition to a superconducting state (Tc « 4 K at 23 GPa) [34]. The large separation of the tetrahedral metal B4 clusters is believed to be the origin of strong electron correlations (Fig. 9(b)).
Recently, several correlation effects have been reported for AB4X8 lacunar spinels, including pressure-induced superconductivity [35], bandwidth-controlled metal-to-insulator transition [36, 37], large negative magnetoresistance [38], a two-dimensional topological insulating state [39], resistive switching through electric field-induced transition [40-42], emergence of orbitally driven ferroelectricity [43], and an extended Neel-type skyrmion phase [44].
Contrary to the spinel structure, the B atoms of the cubes shift toward the cube center creating a B4 regular tetrahedral cluster with short metal-metal bond lengths (2.8 to 2.9 A), and the B atoms of the other type cubes shift toward the cube outside, leading to the metal-metal distances of about 4 A, which is too long to be considered as bonds.
Ordered spinels of AxB2X4 type, where A = Al, Ga, B = Mo, V, Cr and X=S, Se. The structural formula ordered spinels in this case is A2xD2(1-x)B^66X'46eX46e. The examples of such type ordered spinels are Ga0.67Mo2S4, Ga0.5V2S4, AI0 75Mo2S4 and Ga0.67Cr2S4 [23-25]. According to our concept, B4-clusters are also in the structures of these substances.
5. B4-clusters in ordered Laves phases with MgCu4Sn structure type
The crystal structure of Laves phases can be either a cubic C15 (MgCu2), hexagonal C14 (MgZn2) or dihexagonal C36 (MgNi2). The C15 phase has more than 1000 binary and ternary forms among the three types of Laves phases. The MgCu2, type structure (Strukturbericht symbol C15) is face-centered cubic, belonging to the space group Fd3m with 8 formula units per non-primitive cubic unit cell. The A sublattice of the MgCu2 has the structure of diamond net, while the B-sublattice can be described in terms of B-tetrahedra connected by their vertices. So far, more than 80 of such ordered compounds have been reported [45].
The structure formula of ordered Laves phases C15 may be obtained from structural formula ordered spinel A'4a A4c Bl6e X' 46e X46e if we will consider atoms positions X' 16e and Xfe as vacant. The structural formula of the ordered Laves phases is A'4aA4cB^6. The A'4aA4cB46e structure is presented as an example in Fig. 10. The B-atoms build up the three-dimensional network of slightly distorted corner-sharing B4-tetrahedra. The A and A' atoms fill larger cages with coordination number 16 (Frank-Kasper polyhedra [46,47]) within this network. The small distortion of the B-tetrahedra is due to the difference in size between A and A' atoms [48]. For a more detailed discussion on the crystal chemistry and chemical bonding in such Laves phases we refer to review articles [49-51] and references which present there in.
Fig. 10. The crystal structure of A'4aA4cBl6e (the structural type of MgCu4Sn). The three-dimensional network of corner-sharing B4-tetrahedra is emphasized
6. A4 and B4-clusters in ordered pyrochlore structures
Compounds A2B2O6X with pyrochlore structure are notable for the significant variety of their physical properties: electric (semiconductors, metals, superionics), ferroelectric, magnetic (ferromagnetic), piezoelectric, catalitic. Recently in this family of materials superconductors with structures Cd2Re2O7 (Tc = 1 K) and KOs2O6 (Tc = 9.6 K) were opened [52,53]. Such a wide spectrum of properties is caused substantially by processes of the structural ordering of atoms.
The crystal structure of cubic pyrochlore-type oxides, which have the general formula A2B2O6X, is composed of a framework of corner shared BO6 octahedra. Here, the X site is partially or fully occupied by an oxygen atom. The space group of pyrochlore was originally suggested to be Fd3m [54,55], in which the X site is randomly and partially occupied. Later investigation by a powder neutron diffraction experiment [56] showed that the space group is F43m; in this model vacancies in the X site are ordered. In Fd3m-pyrochlore structure A26c B26d O48f X8a atoms occupy four nonequivalent crystallographic positions: cations A and B occupy 16c and 16d Wyckoff positions, and anions occupy 48f and 8a Wyckoff positions. The metal atoms forming clusters, occupy in pyrochlores, as well as in spinel, 16d Wyckoff positions of space group Fd3m. Also, clusters are formed by atoms occupying 16c Wyckoff positions. The frustrated geometry arises from sublattices of corner-sharing tetrahedra, which are present for both the A and B cations.
We have made the prediction of the existence of a new type of cluster materials on the basis of pyrochlore family with the ordered arrangement of anions O and X (type of order 1:1); we have also submitted a calculated structure of pyrochlore (Fig. 11).
i-A16c • -B16d
Fig. 11. A calculated elementary cell of pyrochlore, a network of B4- and A4-tetrahedra. Atoms in 48f and 8a Wyckoff positions are not shown
Metal clusters in the structure of ordered pyrochlore are formed due to displacement of cations in 16d and 16c Wyckoff positions. Thus, the theory predicts the existence of four types of metal clusters - compressed and expanded tetrahedra. These tetrahedra form two a three-dimensional network as it is schematically shown on Fig. 11. To explain the unusual physical properties of pyrochlore, it is essential to have direct interactions of metal - metal. These interactions are realized in two networks of A4- and B4-tetrahedra.
7. Summary
Various interesting phenomena concerning the spin, orbital and charge degrees of freedom on these sublattices have been observed in structures with pyrochlore sublattices caused by geometrical frustration. Typical examples are the Verwey transition in Fe3O4 [57,58], a heavy-fermion state in LiV2O4 [59], and a heptamer formation in AlV2O4 [60].
In this work we have established that the structural mechanism of B4-cluster formation in crystals with geometrically frustrated pyrochlore sublattices has universal character. The reason of universal character for B4-cluster formation is caused by the fact of one the same critical irreducible representation (k11 (t4)) which generates the same changes in the geometry of pyroclore sublattices of high-symmetry Fd3m-phases. This conclusion has been illustrated for crystals that belong to different structural types.
We have also showed that cluster formation accompany changes in the geometry of pyrochlore sublattices -by appearance of two different B-B-distances. It is so called "breathing" pyroclore sublattice which are in all structural types with low-symmetry F43m space group. The "breathing" pyrochlore lattice must be an important factor to explore interesting phenomena in frustrated magnets.
We believe that to understand of the origin of magnetic and electrical properties in a given class of materials are important theoretical results as well (hypertetrahedron formation, bridges between tetrahedral metal clusters, "breathing" rings of tetrahedra).
Future tasks will be connected with investigation of the symmetry reduction from space group Fd3m to F43m in pyrochlore family and the magnesium aluminum-chromium system with composition intermetallic compound Mg3 Cr2 AI18 [61] more detailed.
Acknowledgements
The reported study was funded by RFBR, according to research project no. 16-32-60025 mol_a_dk (Ta-lanov M. V.).
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