A Mero-Plesiotype Series of Vanadates, Arsenates, and Phosphates with Blocks Based on Densely Packed Octahedral Layers as Repeating Modules

: The family of layered vanadates, arsenates, and phosphates is discussed in terms of a modular concept. The group includes minerals v é sign é ite and bayldonite, and a number of synthetic analogous and modiﬁcations which are not isotypic, but their crystal structures comprise similar blocks (modules) consisting of a central octahedral layer ﬁlled by atoms of d elements (Mn, Ni, Cu, or Co) and adjacent [VO 4 ], [AsO 4 ], or [PO 4 ] tetrahedra. The octahedral layers are based on the close-packing of oxygen atoms. Within these layers having the same anionic substructure, the number and distribution of octahedral voids are different. In the crystal structures of compounds participating in the polysomatic series, these blocks alternate with various other structural fragments. These circumstances deﬁne the row of structurally-related vanadates, arsenates, and phosphates as a mero-plesiotype series. Most of the series members exhibit magnetic properties, representing two-dimensional antiferromagnets or frustrated magnets.


Introduction
The modular approach for interpreting the crystal structures of minerals is well known. Its intensive use in modern crystal chemistry has become possible after the pioneer works of Thomson [1][2][3] and Veblen [4]. The modular concept operates with large segments (modules), which represent stable polyhedral complexes of definite topology and size, and can differ in structure and/or composition. Within this concept, the crystal structure can be interpreted as a derivative of diverse modules; then it is meant as a polysome. The polysomatism considers a particular crystal structure in required connection with other structures, assembled from the same modules. This method allows distinguishing series of non-isotypic, but crystal chemically-related phases, so-called polysomatic series, with structures built by fragments of the same topology taken in various combinations. The stability of repeating fragments is controlled by the energetic advantage of the associations of polyhedra forming these modules. Obviously, one can use the modular approach only in the case of several compounds (at least two) containing similar substructural segments (modules), with clearly pronounced rules for their spatial arrangement and alternation.
To expand the field of structurally-related compounds within the polysomatic series, Makovicky [5] proposed to adopt a possibility of variations both in the structure and in the chemical composition of modules still keeping their topological similarity. These advanced sequences of compounds were called mero-plesiotype series. In the framework of this concept, a "common" fragment for all of the structures shows clear variability; besides, modules of another type are dissimilar for different members of the series [6,7].
Through efforts of numerous scientific schools, this methodology has allowed the introduction of many families of inorganic compounds, including minerals, as polysomatic ( Figures 1 and 2). However, the way of the empty octahedra arrangement in the monoclinic reppiaite and isotypic Ni arsenate structures on one hand, and in the structure of triclinic cornubite on the other, are different, providing a diverse topology of their cationic substructures, as dual-width stripes of the triangular net separated by honeycombs or dual-width stepped stripes of the triangular net separated by honeycombs ( Figure 3). Note that both arrangements, as well as other diagrams shown in Figure 3 may be obtained as vacancy-modified triangular nets. The tetrahedral groups are placed on both sides of the layer just above and under empty octahedra with the formation of the {Me5(OH) 4[TO4]2} block. In all three structures, the similar blocks are stacked together in a direction normal to the plane of the layer through oxygen-bridging contacts. The neighboring modules are semi-translationally displaced in one direction to form an AAʹ sequence with a period value of about 9.4 Å (Table 1). Accordingly, it can be concluded that these phases with the crystal structures built exclusively from basic modules, on which the polysomatic series of vanadates, arsenates, and phosphates is based, represent the archetype structures.  ( Figures 1 and 2). However, the way of the empty octahedra arrangement in the monoclinic reppiaite and isotypic Ni arsenate structures on one hand, and in the structure of triclinic cornubite on the other, are different, providing a diverse topology of their cationic substructures, as dual-width stripes of the triangular net separated by honeycombs or dual-width stepped stripes of the triangular net separated by honeycombs ( Figure 3). Note that both arrangements, as well as other diagrams shown in Figure 3 may be obtained as vacancy-modified triangular nets. The tetrahedral groups are placed on both sides of the layer just above and under empty octahedra with the formation of the {Me5(OH)4[TO4]2} block. In all three structures, the similar blocks are stacked together in a direction normal to the plane of the layer through oxygen-bridging contacts. The neighboring modules are semi-translationally displaced in one direction to form an AAʹ sequence with a period value of about 9.4 Å (Table 1). Accordingly, it can be concluded that these phases with the crystal structures built exclusively from basic modules, on which the polysomatic series of vanadates, arsenates, and phosphates is based, represent the archetype structures.

The Mero-Plesiotype Series of Structurally-Related Phases
All crystal structures of the series can be represented as an assembly of alternating twodimensional core module A and other slabs of diverse composition and design ( Table 1).
The crystal structure of the "mineralogically probable" divanadate carbonate K 2 Mn 3 [VO 4 ] 2 (CO 3 ) [23] is formed by two types of modules alternating along the c axis of the hexagonal unit cell (Figure 4a). Gibbsite-like layer parallel to the ab plane has a honeycomb arrangement of Mn 2+ cations (Figure 3a) at the centers of octahedra sharing cis and trans edges (Figure 5a). The [VO 4 ] tetrahedra at both sides of the dioctahedral layer complete the main block to the Mn 2 [VO 4 ] 2 composition and provide its linkage along the c axis with the second {K 2 MnCO 3 } module formed by Mn-centered trigonal bipyramids, [CO 3 ] triangles, and K atoms. The MnCO 3 cellular layer is topologically identical in the ab plane to the kalsilite network built of Si and Al tetrahedra (Figure 5b). From both sides, these layers are attached by sheets of K nine-vertex polyhedra sharing vertices. Thus, the structure can be represented as an alteration of negatively charged {Mn 2 [28]. Moreover, the Mn and Co containing formula analogues are isotypic, while the Rb, Mn variety possess trigonal symmetry. The authors of [28] noted the effect of the larger size of Rb + in comparison with the size of K + on the structure transformation that occurs when the [VO 4 ] tetrahedra adjacent along the c axis rotate around the

The Mero-Plesiotype Series of Structurally-Related Phases
All crystal structures of the series can be represented as an assembly of alternating two-dimensional core module A and other slabs of diverse composition and design (Table 1).
The crystal structure of the "mineralogically probable" divanadate carbonate K2Mn3[VO4]2(CO3) [23] is formed by two types of modules alternating along the c axis of the hexagonal unit cell (Figure 4a). Gibbsite-like layer parallel to the ab plane has a honeycomb arrangement of Mn 2+ cations (Figure 3a      The same blocks built of octahedral layers of the gibbsite-type and adjacent [VO 4 ] tetrahedra, form the crystal structure of BaNi 2 [VO 4 ] 2 [29] with similar parameters of the trigonal unit cell, but with the R Bravais lattice (  Synthetic phases Na2Ni3(OH)2[PO4]2 [24] and K2Mn3(OH)2[VO4]2 [36] present an example of isotypic compounds from different chemical classes. A virtual exchange of all cations (except H + ) at four symmetrically independent structural positions (Na + , Ni1 2+ , Ni2 2+ , and P 5+ ) in the Na2Ni3(OH)2[PO4]2 structure for cations in the same oxidation state of a larger radius, (K + , Mn1 2+ , Mn2 2+ , and V 5+ ) leads to the formation of the isostructural phase, K2Mn3(OH)2[VO4]2, of the same symmetry (space group C2/m), but with obviously   [36] present an example of isotypic compounds from different chemical classes. A virtual exchange of all cations (except H + ) at four symmetrically independent structural positions (Na + , Ni1 2+ , Ni2 2+ , and P 5+ ) in the Na 2 Ni 3 (OH) 2 [PO 4 ] 2 structure for cations in the same oxidation state of a larger radius, (K + , Mn1 2+ , Mn2 2+ , and V 5+ ) leads to the formation of the isostructural phase, K 2 Mn 3 (OH) 2 [VO 4 ] 2 , of the same symmetry (space group C2/m), but with obviously increased values of the unit cell parameters and the monoclinic angle (Table 1). In both cases, the layers of edge-sharing MeO 4 (OH) 2 octahedra include 1 4 of vacancies giving rise to the cationic substructure formed of stripes of the triangular net separated by honeycombs (Figure 3c). Distances between the modules with central layers built of Ni or Mn octahedra are 4.2 and 4.8 Å accordingly, and depend mainly on the sizes of [PO 4 ] and [VO 4 ] tetrahedra and the hydrogen bond lengths. The naturally smaller space between neighboring {Ni 3 (OH) 2 [PO 4 ] 2 } blocks in the phosphate structure suits perfectly for being filled by Na atoms (Figure 7). The similar modules alternate with the layers of KO 7 or NaO 7 polyhedra of the same topology (Figures 7 and 8). The described above lattice (Figure 3c) stands between the honeycomb lattice (Figure 3a), characteristic of the recently discussed divanadate carbonates, and the kagomé lattice (Figure 3b), inherent in the isostructural synthetic phase BaNi3(OH)2[VO4]2 [34] and mineral vésignéite, BaCu3(OH)2[VO4]2 [33]. In the vésignéite structure, the octahedral voids of the central layers of the main modules are also filled for ¾ by the Ni or Cu atoms, but the distribution of the empty voids inside the octahedral layers corresponds to the regular kagomé lattice (Figure 9a). Along the [001] direction, these modules (A) alternates with layers formed by the large Ba-centered polyhedra presenting the second-type slabs B (Figure 9b). The AB sequence of their repetition along the [001] direction defines the The described above lattice (Figure 3c) stands between the honeycomb lattice (Figure 3a), characteristic of the recently discussed divanadate carbonates, and the kagomé lattice (Figure 3b), inherent in the isostructural synthetic phase BaNi3(OH)2[VO4]2 [34] and mineral vésignéite, BaCu3(OH)2[VO4]2 [33]. In the vésignéite structure, the octahedral voids of the central layers of the main modules are also filled for ¾ by the Ni or Cu atoms, but the distribution of the empty voids inside the octahedral layers corresponds to the regular kagomé lattice (Figure 9a). Along the [001] direction, these modules (A) alternates with layers formed by the large Ba-centered polyhedra presenting the second-type slabs B (Figure 9b). The AB sequence of their repetition along the [001] direction defines the The described above lattice (Figure 3c) stands between the honeycomb lattice (Figure 3a), characteristic of the recently discussed divanadate carbonates, and the kagomé lattice (Figure 3b), inherent in the isostructural synthetic phase BaNi 3 (OH) 2 [33]. In the vésignéite structure, the octahedral voids of the central layers of the main modules are also filled for 3 4 by the Ni or Cu atoms, but the distribution of the empty voids inside the octahedral layers corresponds to the regular kagomé lattice (Figure 9a). Along the [001] direction, these modules (A) alternates with layers formed by the large Ba-centered polyhedra presenting the second-type slabs B (Figure 9b). The AB sequence of their repetition along the [001] direction defines the value of the c axis close to 8 Å for both compounds (Table 1). A unique distribution of the filled and empty octahedral voids inside the closest packing of oxygen atoms is established in the crystal structure of the synthetic oxyfluoride vanadate Cu13(OH)10F4[VO4]4 [38]. Here, the core module A includes ¼ of the empty octahedra, but differently to the structures discussed above, an exclusive cationic arrangement within the layer arises. It can be described as stepped stripes of the triangular net separated by honeycombs (Figure 3e). The negatively charged A modules {Cu3(OH)2[VO4]2} 2-formed by such octahedral layers with adjacent vanadate tetrahedra, alternate along the b axis with B fragments of the brucite-type structure with a triangular sublattice (Figure 3d)  A unique distribution of the filled and empty octahedral voids inside the closest packing of oxygen atoms is established in the crystal structure of the synthetic oxyfluoride vanadate Cu13(OH)10F4[VO4]4 [38]. Here, the core module A includes ¼ of the empty octahedra, but differently to the structures discussed above, an exclusive cationic arrangement within the layer arises. It can be described as stepped stripes of the triangular net separated by honeycombs (Figure 3e). The negatively charged A modules {Cu3(OH)2[VO4]2} 2-formed by such octahedral layers with adjacent vanadate tetrahedra, alternate along the b axis with B fragments of the brucite-type structure with a triangular sublattice (Figure 3d). These positively charged B-modules {Cu3.5(OH)3F2} 2+ are based on the densely packed oxygen and fluorine atoms, with all octahedral voids occupied by Cu. A unique distribution of the filled and empty octahedral voids inside the closest packing of oxygen atoms is established in the crystal structure of the synthetic oxyfluoride vanadate Cu 13 (OH) 10 F 4 [VO 4 ] 4 [38]. Here, the core module A includes 1 / 4 of the empty octahedra, but differently to the structures discussed above, an exclusive cationic arrangement within the layer arises. It can be described as stepped stripes of the triangular net separated by honeycombs (Figure 3e). The negatively charged A modules {Cu 3 (OH) 2 [VO 4 ] 2 } 2− formed by such octahedral layers with adjacent vanadate tetrahedra, alternate along the b axis with B fragments of the brucite-type structure with a triangular sublattice (Figure 3d). These positively charged B-modules {Cu 3.5 (OH) 3 F 2 } 2+ are based on the densely packed oxygen and fluorine atoms, with all octahedral voids occupied by Cu. The AB repetition of the modules fixes the value near 10.2 Å of the b parameter (Figure 11) of the triclinic unit cell (Table 1).

Magnetic Behavior of the Series Members
Recently, studies of quantum-spin systems have been of considerable interest. On the basis of low-dimensional magnetic systems, exotic quantum ground states of matter can be realized including gapped and gapless spin liquids [42], spin ice [43,44], spin glass, and various peculiar forms of long-range magnetic order [45][46][47]. If the way of the distribution of spin carriers can be described by a special lattice constructed of triangles sharing vertices or edges, such as kagomé and triangular lattices [48][49][50], the three-dimensional magnetic order tends to be suppressed due to spin frustration [38]. The presence of ions with an open shell Mn 2+ , Ni 2+ , Co 2+ , or Cu 2+ inside the octahedral layers causes the appearance of two-dimensional antiferromagnetic or frustrated magnetic properties for the compounds considered in this work. Anionic tetrahedra [PO4] 3− , [VO4] 3− , or [AsO4] 3− serve as ideal non-magnetic separators between transition-metal 2D planes and contribute to the octahedral environment of oxygen atoms around cations. Moreover, these units can prevent distortions of the Me-centered octahedra to preserve the planar geometry of the magnetic substructures. The way in which magnetically active ions fill octahedral voids inside the closely packed oxygen/fluorine atoms determines the type of magnetic structure (Figure 3) that inevitably contains geometric frustrations.
The kagomé lattice should provide a much stronger frustration than the simplest triangular lattice (due to the chirality degeneration), and, as a consequence, the absence of long-range magnetic order [51]. The classical kagomé lattices describe the cationic substructures of the vésignéite BaCu3(OH)2[VO4]2 and BaNi3(OH)2[VO4]2. At 53 K, vésignéite demonstrates a strong antiferromagnetic interaction between spins of nearest-neighbors without long-range magnetic order up to 2 K, and its ground state is assumed to be a gapless spin liquid [52]. In BaNi3(OH)2[VO4]2, magnetic frustration arises due to the competition between ferromagnetic and antiferromagnetic ordering, which leads to a glassy transition at 19 K [34]. According to [51], in the honeycomb-type substructure should be no frustration, except in the presence of next-nearest-neighbor exchange interactions. The honeycomb lattices characterize isostructural BaMe2[TO4]2 (Me = Co, Ni; T = P, V, As) compounds, which display quite different magnetic properties. Thus, BaNi2[VO4]2 demonstrates the onset of antiferromagnetic long-range ordering near 50 K [29]. BaCo2[PO4]2 is a rare highly frustrated, quasi-2D magnetic material with the honeycomb substructure that displays competing for short-range magnetic ordering below TN1 ∼ 6 K and TN2 ∼ 3.5 K, but resists long-range magnetic order and spin freezing [44]. The complex magnetic ordering and spin dynamics in BaCo2[AsO4]2 has been in-

Magnetic Behavior of the Series Members
Recently, studies of quantum-spin systems have been of considerable interest. On the basis of low-dimensional magnetic systems, exotic quantum ground states of matter can be realized including gapped and gapless spin liquids [42], spin ice [43,44], spin glass, and various peculiar forms of long-range magnetic order [45][46][47]. If the way of the distribution of spin carriers can be described by a special lattice constructed of triangles sharing vertices or edges, such as kagomé and triangular lattices [48][49][50], the three-dimensional magnetic order tends to be suppressed due to spin frustration [38]. The presence of ions with an open shell Mn 2+ , Ni 2+ , Co 2+ , or Cu 2+ inside the octahedral layers causes the appearance of two-dimensional antiferromagnetic or frustrated magnetic properties for the compounds considered in this work. Anionic tetrahedra [PO 4 ] 3− , [VO 4 ] 3− , or [AsO 4 ] 3− serve as ideal non-magnetic separators between transition-metal 2D planes and contribute to the octahedral environment of oxygen atoms around cations. Moreover, these units can prevent distortions of the Me-centered octahedra to preserve the planar geometry of the magnetic substructures. The way in which magnetically active ions fill octahedral voids inside the closely packed oxygen/fluorine atoms determines the type of magnetic structure (Figure 3) that inevitably contains geometric frustrations.
The kagomé lattice should provide a much stronger frustration than the simplest triangular lattice (due to the chirality degeneration), and, as a consequence, the absence of long-range magnetic order [51]. The classical kagomé lattices describe the cationic substructures of the vésignéite BaCu 3 (OH) 2 [VO 4 ] 2 and BaNi 3 (OH) 2 [VO 4 ] 2 . At 53 K, vésignéite demonstrates a strong antiferromagnetic interaction between spins of nearest-neighbors without long-range magnetic order up to 2 K, and its ground state is assumed to be a gapless spin liquid [52]. In BaNi 3 (OH) 2 [VO 4 ] 2 , magnetic frustration arises due to the competition between ferromagnetic and antiferromagnetic ordering, which leads to a glassy transition at 19 K [34]. According to [51], in the honeycomb-type substructure should be no frustration, except in the presence of next-nearest-neighbor exchange interactions. The honeycomb lattices characterize isostructural BaMe 2 [TO 4 ] 2 (Me = Co, Ni; T = P, V, As) compounds, which display quite different magnetic properties. Thus, BaNi 2 [VO 4 ] 2 demonstrates the onset of antiferromagnetic long-range ordering near 50 K [29]. BaCo 2 [PO 4 ] 2 is a rare highly frustrated, quasi-2D magnetic material with the honeycomb substructure that displays competing for short-range magnetic ordering below T N1 ∼ 6 K and T N2 ∼ 3.5 K, but resists long-range magnetic order and spin freezing [44]. The complex magnetic ordering and spin dynamics in BaCo 2 [AsO 4 ] 2 has been investigated several times, but still remains an open problem. The BaCo 2 [AsO 4 ] 2 frustrated magnet with the honeycomb distribution of Co atoms sharply ordered at T~5.4 K with a probable formation of a helical design [53], but repeated research revealed its quasi-collinear incommensurate ground-state structure [54].
Another type of frustrated substructure, described as built from stripes of triangular nets separated by honeycombs, which is a compromise between honeycomb and kagomé lattices, should not have an antiferromagnetic order of the Néel-type, but should have geometric magnetic frustration [51]. The same lattice characterizes the isotypic crystal structures of Na 2 Ni 3 (OH) 2 (PO 4 ) 2 , Na 2 Co 3 (OH) 2 [VO 4 ] 2 , and K 2 Mn 3 (OH) 2 [VO 4 ] 2 ; they all demonstrate low-dimensional antiferromagnetic ordering of various origins. In the phosphate compound, it arises as a result of the presence and competition of ferro-and antiferromagnetic interactions [24]. Mn vanadate exhibits geometric frustration and lowdimensionality effects and spin-lattice coupling [51]. Cobalt phase shows evidence of spin-orbit coupling in Co 2+ ions with antiferromagnetic ordering at 4.4.K and highly anisotropic field-dependent behavior with multiple metamagnetic transitions [35]. Dualwidth stripes of triangular net are surrounded by honeycombs in the substructure of reppiaite, Mn 5 (OH) 4 [VO 4 ] 2 . This 2D magnet displays Curie−Weiss behavior above 100 K with significant antiferromagnetic coupling and canted antiferromagnetic order below 57 K [55].
In crystal structures of some compounds, magnetic cations form two types of magnetic arrangements. Thus, the complex structure of Cu 13 (OH) 10 F 4 [VO 4 ] 4 is described by the triangular lattice, which alternates with a unique substructure of dual-width stripes of the triangular net separated by honeycombs. The magnetic system of Cu 13 (OH) 10 F 4 [VO 4 ] 4 created in this way exhibits long-range antiferromagnetic ordering at ∼3 K, a strong spin-frustration effect, and a spin-flop transition at 5 T [38]. Each member of the vanadatecarbonate family is also characterized by two magnetic subsystems, honeycomb-and triangular-type in their crystal structures. They all order antiferromagnetically but at different temperatures due to diverse sorts of transition cations. K 2 Mn 3 [VO 4 ] 2 (CO 3 ) showed a two-step formation of long-range magnetic order at low temperatures [23].
The following study showed that triangular and honeycomb magnetic layers undergo sequential magnetic ordering and act as nearly independent magnetic subsystems. The honeycomb substructure orders at about 85 K in a Néel-type antiferromagnetic structure, while the triangular arrangement displays two consecutive ordered states at much lower temperatures of 3 and 2.2 K [56]. Likewise, the Rb-analogue Rb 2 Mn 3 [VO 4 ] 2 (CO 3 ) exhibits three magnetic transitions at 77 K, 2.3 K, and 1.5 K. At 77 K, it orders in the honeycomb layer in a Néel-type antiferromagnetic orientation, while the lower temperature spin structure has either a collinear or a canted magnetic structure for the triangular Mn lattice [28]. K 2 Co 3 (VO 4 ) 2 CO 3 displays a canted antiferromagnetic ordering below T N = 8 K [28].

Conclusions
We have shown [57] that the crystal structures of the mero-plesiotype series of natural and synthetic vanadates, phosphates and arsenates with first row transition metals are formed of similar 2D modules, which alternate in one direction with other structural fragments, diverse for different members of the series. The core modules are built of a central octahedral layer filled with d elements (Mn, Ni, Cu, or Co) and adjacent anionic (VO 4 , PO 4 , AsO 4 ) tetrahedra. The central layer is based on the closest packing of oxygen (and fluorine) atoms. Changed amounts of the octahedral voids and their varying distribution differ from one structure to another in a framework of the same anionic substructure. Seven "magnetic" topologies of the transition metal distribution within the layers have been identified: triangular, honeycomb, kagomé, and different combinations of fragments of the triangular and honeycomb lattices (Figure 3). The crystal structure constitution based on the same repeating modules establishes the 2D character design for all polysomes. Consequently, they all have similar translations inside the module plane equal of about 5 Å, while the values of the unit cell axis perpendicular to the sheet is defined by the size of the second module and the sequence of the module's alternation. The only basic type of modules with similar dimensions reproduces in one direction in the structures of minerals reppiaite Mn 5 [VO 4 ] 2 (OH) 4 and cornubite Cu 5 (OH) 4 [AsO 4 ] 2 , as well as synthetic arsenate Ni 5 (OH) 4 [AsO 4 ] 2 . Therefore, these phases with the crystal structures built exclusively from the main modules, on which the polysomatic series of vanadates, arsenates, and phosphates is based, represent the archetype structures.
When additional (B) slabs of the alkaline (or alkaline-earth) metals interrupt the basic (A) modules, a few combinations within the mero-plesiotype series are formed: (AB) for vésignéite BaCu 3 (OH) 2  The main block is a core module based on the gibbsite-type octahedral layer. The second one is the block formed by the kalsilite-like sheet of Mn bipyramids and CO 3 triangles that is sandwiched between two layers of alkaline metals.
It is important to note that the minerals and laboratory-synthesized compounds considered here exhibit magnetic properties, representing two-dimensional antiferromagnets or frustrated magnets. Obviously, their magnetic behavior is directly related to the crystal structure peculiarities. In particular, the way in which magnetically active Mn 2+ , Ni 2+ , Co 2+ , or Cu 2+ ions occupy the octahedral voids between the densely packed oxygen atoms is crucial.