Crystal Chemistry of High-Temperature Borates

Abstract

In recent years borate-based crystals has attracted substantial interest among the research community. The overall importance of this family of materials is reflected in miscellaneous articles and several reviews that have been published over the years. Crystalline borate materials exhibit numerous interesting physical properties, which make them promising for further practical applications. Diversity of functional characteristics results from their high structural flexibility caused in the linkage of planar/non–planar BO₃ groups and BO₄ tetrahedra, which can occur as isolated or condensed structural units. This report is a brief review on crystal chemistry and structure features of anhydrous/high-temperature borates. Polymorphism of boron-oxygen radicals has been considered basing on cations’ nature and synthesis conditions. Analysis of the laws governing borates structures and general principles of their systematics was discussed. As a result, an alternative classification of anhydrous compounds has been considered. It is based on four orders of their subdivision: (1) by the variety of anion formers, (2) by the cation charge, (3) by the N = N_M:N_B, i.e., ratio of metal atoms number to the ratio of boron atoms number (N-factor) value indicating the borate structural type (if it is known), (4) by the cation type and size.

1. Introduction

A critical analysis of the existing literature data on melting diagrams of borate systems, hydrothermal synthesis and mineralogical field studies indicates that more than 1300 high-temperature (anhydrous) borates and their structural derivatives have been synthesized and also found in Nature [1]. Their crystal structures and structural types have been studied using X-ray diffraction methods. Most of them belong to orthorhombic, monoclinic or trigonal/hexagonal systems. As a rule, the symmetry of borate materials decreases with decreasing cation and the ratio between the number of metal and boron atoms in the compound [2].

Isolated BO₃ triangles predominate in the structures of anhydrous borates, especially, those of binary and more complex compounds (about 65%) [3]. Among the remaining borates, almost one-half are represented by framework structures containing three-dimensional boron-oxygen polyanions (exclusively monocationic compounds), followed by insular layered and chain structures, i.e., those with dispersed individual elements (pyrogroups of two BO₃–triangles, rings of three triangles, groups of two BO₄–tetrahedra). In these cases, BO₃– and BO₄– groups are joined by sharing common O atoms. Polyanions of compounds with cations of alkali and alkali-earth metals are most diverse. Then, it is followed by isolated (pyroborates and ring metaborates), network and chain-forming structures. There are about fifty types of boron-oxygen anions and polyanions proper (including the BF₄ group) and a dozen of borosilicate, boroaluminate and boroberyllate anions (Figure 1).

As shown by our experimental studies [3,4], the regularities of the crystallization of borates with trivalent metals are in many respects determined by the specificity of their structure and the composition of crystallization medium. An increase in the content of anhydrous boric oxide in the melts facilitates the synthesis of borates with a higher polymerization of boron-oxygen radicals [2,3,4]. It should be also noted the tendency of boron atoms to tetrahedral coordination, and BO₃–and BO₄–groups-to polycondensation, when they prevail over other cations in the structures of compounds synthetized. Similar trends were also observed for some other groups of borates [5,6,7]. It is, therefore, worth assessing the extent to which the above peculiarities are typical for anhydrous borates in general.

2. Hierarchy of Basic Structural Clusters

This section provides a concise and precise description of the experimental results, their interpretation as well as the experimental conclusions that can be drawn. In polymeric boron-oxygen constructions of different composition one can easily identify, except for BO₃–triangles (∆) and BO₄–tetrahedra (t), comparatively small groups formed by them, most of which are represented in many structures. First this clusters (radicals), but only isolated, were found in anhydrous potassium and sodium metaborates [8,9], and then in three-dimensional frameworks as well [10]. These comparatively small structural components (usually single or double rings), containing 2–5 triangles and tetrahedra, are designated below as combined basic structural units CSU in contradistinction to the fundamental (elementary) structural units (FSU), i.e., BO₃–triangles and BO₄–tetrahedra.

In polyanions of anhydrous borates, it is expedient to discern one more type of structural units. Structural units of this category are, as a rule, more complex and characterize the structures and the structural types of compounds. It is appropriate to designate them, as complete radicals of polyanions (CRP). CRP can contain more than one CSU and/or additional triangles and tetrahedra. Thus, it represents a full repeating fragment of a polyanion (made up of 2–9 FSU) corresponding in composition or multiple of the boron-oxygen anion part in the structural formula of the compound.

2.1. Fundamental Structural Units

The B-O bond lengths in FSU vary within considerable limits, the range of variation being greater in complex structures. Their average values in triangles and tetrahedral amount to 1.37 Å and 1.48 Å, respectively. These values for O-O distances in the sides of triangles equal to 2.38 Å, and in the edges of tetrahedral they are 2.43 Å [1,5] (Figure 2).

2.2. Combined Structural Units

The structures of alkali metal anhydrous monocationic borates contain a rather limited number of CSU of the same kind and their different combinations. In this framework, layered and isolated borate structures of this group most often are encountered single or paired triple rings of triangles and tetrahedra: 1) 3∆, 2) 2∆ + 2t, 3) 2∆ + 1t and 4) 4∆ + 1t (Figure 3a–d), sometimes, they are variously mentioned in the literature as boroxol, diborate, triborate and pentaborate groups, respectively [9,10,11,12,13]. Occasionally, in framework K₂O·2B₂O₃ [14] and layered α-Na₂O·2B₂O₃ [11], were single and double 1∆ + 2t and 3∆ + 2t rings (Figure 3e,f) encountered, not very aptly called ditriborate and dipentaborate groups. This is because the ditriborate group together with the additional BO₃– triangle in K₂O·2B₂O₃ structure represents configurationally a strongly deformed diborate group.

In borates of divalent metals, namely CaB₂O₄, BO₃–triangles and BO₄–tetrahedra can form endless chains [15,16,17]. Apart from these, the most borates of divalent and trivalent metals also have other CSU. First of all, there is the pyroborate containing isolated pair of BO₃ triangles [18,19] (Figure 3g). The next one, a ring of three tetrahedra (Figure 3h) has been found in boracite, hausenite and their analogues [20], in CaB₂O₄ (III) [21], SrB₂O₄ [22], SrB₄O₇ and PbB₄O₇ [23] structures. The same rings, but without additional FSU were discovered in framework CaB₂O₄ (IV) [24]. Less widespread are quadruple (SrB₄O₇ and PbB₄O₇ [23]), sextuple (Zn₄OB₆O₁₂ [25]) and octuple (CuB₂O₄ [26]) rings of tetrahedra (Figure 3i–k). Overall, CSU here are more diversified than in the borates of alkali metals. Among them, the tetrahedral coordination of boron atoms predominates over the triangular one.

In the borates of trivalent metals, CSU differing from the others have also been found, e.g., quaternary rings of two BO₃–triangles and two BO₄–tetrahedra in corrugated metachains of rare earth borates, R(BO₂)₃ with R = La − Tb [27] (Figure 3l). In such chain, every BO₄ tetrahedron belongs to two adjacent rings, i.e., the oxygen corners of all the tetrahedra are shared with the triangles, while each of the triangles has one corner unbonded inside the chain. According to [28], isolated triple rings of BO₄ tetrahedra represent the anionic part of the structure of low-temperature GdBO₃ and its rare earth analogues. Borate with tetravalent cations are represented by ThB₂O₅ and ThB₄O₈ compounds. Thorium diborate, ThB₂O₅, exhibits α- and β- polymorphs. In the structure of β-ThB₂O₅ two corner sharing BO₃ triangles are linked forming an isolated B₂O₅ dimer [29]. Whereas, in the structure of α- modification boron atoms occupy corner sharing BO₄–tetrahedra and BO₃–triangles forming 1D zigzag chain [30]. ThB₄O₈ structure exhibits infinite chains, which consist of [B₂O₄]⁵⁻ tetrahedral borate groups that are connected through one common oxygen atom [31]. Monocationic borates of pentavalent elements have only isolated BO₄ tetrahedra [32].

The structures of binary and more complex borates predominantly contain isolated BO₃ triangles, less frequently tetrahedra or pyrogroups (Figure 4a). Only in four types of layered structures, i.e., in jochachidolite, synthetic RCo(BO₂)₅ with R = La − Ho, Na₂Zn₂MnB_4.67O₁₁ and RAl₂B₄O_10.5 with R = La − Nd [33,34,35,36,37], one can identify sextuple, quadruple and binary three-membered rings (Figure 4b–f).

In the R-aluminum metaborates synthetized by the authors of [38], the Al atoms are located in five-corner oxygen polyhedra, they can formally be regarded as a transitional compound between the borates proper and aluminoborates. Another type of “transitional” compounds can be represented by A1₅(BO₃)O₆ = Al₂[Al₃O₆BO₃], in this structure 60% of Al atoms are located in AlO₄–tetrahedra and 40% of them form AlO₆–octahedra [39].

An individual group can be made up of CSU (in an overwhelming majority of cases - ring-shaped) in boroberyllates, boroaluminates and borosilicates where Be, Al and Si atoms with a tetrahedral coordination partially substitute boron (Figure 5) (see, for example, Ref. [40,41,42,43,44,45,46]). Unusual illustrations of trivalent Al and divalent Be in the role of boron are found in the structures of CaAlOBO₃ [42], SrAlBO₄ [43] and Y₂AlBeBO₇ [44]. In the first case, double (2t_Al + 2∆_B) rings in the chains by their configuration resemble the diborate groups very often found in the polyanions of mono- and divalent metal borate structures. The alumoborate metachain, i.e., one-dimensional (1D) chain of [Al₂B2O8]1D composition in SrAlBO₄, where (2t_Al + 1∆_B) rings can be identified, is similar to the metaborate chain in calciborite - [B4O8]1D, if the BO₄ tetrahedra in it are replaced by those of aluminum, i.e., by AlO₄ groups.

2.3. Complete Radicals of Polyanions

CRP in the structures of anhydrous borates are similar to some CSU shown in Figure 3 and Figure 4 and they are identical with FSU in orthocompounds. The CRP have a comparatively complex structure in monovalent metal borates, for example, in boracite, as well as in a number of CaB₂O₄ modifications [16,17,21], SrB₂O₄, Ca₂B₆O_l1, CaB₄O₇, BaO·2B₂O₃, BaO·4B₂O₃ [23,47,48] (Figure 6 Figure 7 and Figure 8) and in some others. A peculiar CRP is found in aluminum orthotriborate, A1₅(BO₃)O₆ [39] or, to be more precise, in aluminum-alumoborate Al₂(AlO₂)₃BO₃.

Thus, despite the great diversity of structural types of anhydrous borates, most of them contain only three configurations of the basic structural units: FSU, CSU, and CRP (Figure 6, Figure 7 and Figure 8)

3. Polymerization of Boron-Oxygen Radicals

Polymerization degree of B_nO_m anions can be defined by the ratio of N = N_M:N_B (called as N factor), i.e., by the relation between the number of atoms of a metal (metals, M) and B atoms in the formula of a compound. Generally it is equivalent to the N’ = N_MxOy:N_B2O3 ratio. For monocationic borates with cations of odd valence, N = N’, but for those with even valence N = N’/2. The effect of the magnitude of N–factor on the type of boron-oxygen radicals and the degree of radicals polymerization should be analyzed for compounds with the same cations, i.e., separately for Li, Na, K, etc.

The most comprehensive X-ray diffraction data have been accumulated on alkali metal borates. A decrease in the value of N in these compounds leads polyanion complication and changing of structural type. For Li borates, obtained at normal pressure, isolated BO₃–triangles (N = 3) in α-Li₃BO₃ [61] transform to one-dimensional chains of triangles (N = 1) in LiBO₂ [62], then to three-dimensional chains of (4∆ + 3t) groups (N = 4/7) [44] or (2∆ + 2t) groups for N = 1/2 in Li₂O·2B₂O₃ [10]. In Na borates isolated groups of 3∆ at N = 1 in Na₃(B₃O₆) [8] transform into two-dimensional nets of dipentaborate or triborate clusters with n = n_∆:n_t = 5/3 (N = 1/2) in Na₂O·2B₂O₃ [11]. Then one can observed transformation to: 1) a double two-dimensional net of pentaborate, triborate rings with additional tetrahedra (n = 2) in a metastable low-temperature modification or 2) twinned three-dimensional framework of penta- and diborate groups with the same ratio of BO₃–triangles and BO₄–tetrahedra in the stable high-temperature form for N = 1/3 in α-Na₂O·3B₂O₃ and β-Na₂O·3B₂O₃ [49,50]. Finally, twinned three-dimensional network of penta- and triborate rings with n = 3 at N = 1/4 are formed in [54]. For potassium borates isolated rings of 3∆ (N = 1) in K₃B₃O₆ [9,63] change over to three-dimensional networks: a) of diborate, ditriborate groups having additional BO₃–triangles with n = 1 at N = 1/2 in K₂O·2B₂O₃ [14]; b) of penta-, triborate rings, additional BO₃–triangles and BO₄–tetrahedra with n = 14/5 at N = 5/19 in 5K₂O·19B₂O₃ [52]; or c) of pentaborate CSU with n = 4 (N = 1/5) in K₂O·5B₂O₃ [13]. In rubidium and cesium borates isolated rings of 3∆ (N = 1) transform into a three-dimensional framework of triborate groups with n = 2 (N = 1/3) in Cs₂O·3B₂O₃ [12], then into twinned two-dimensional network of triborate and boroxol rings with n = 8 (N = 1/9) [64]. In the Ag-tetraborate AgO·4B₂O₃ the polyanion structure is similar to potassium one [65].

No isolated BO₄ groups have been found in monovalent metal borates. Isolated BF₄ tetrahedra are known only in the structures of fluoroborates NaBF₄ and NH₄BF₄ with N = 1 [66,67]. The maximal fraction of BO₄–tetrahedra (n = 1) is found in the framework lithium (Li₂O·2B₂O₃) and potassium (K₂O·2B₂O₃) borate structures with N = 1/2 but 1/n = 3/5 [10,14], and in cesium borate Cs₂O·3B₂O₃ it is even smaller (1/n = 1/2) and shifts to N = 1/3 [12]. No clear-cut relationship was established between the value of n, the type of BO radical and the degree of the deformation of BO₃–triangles and BO₄–tetrahedra, with the exception that isolated FSUs are more regular.

With a decrease in the N number in borates of divalent metals, the same tendency is observed as in monovalent metal borates. At N ≥ 3/2 (i.e., N_MO:N_B2O3 ≥ 3), only isolated BO₃–triangles are observed in the structures of monocationic and binary compounds. In pyroborates (N = 1), including binary M²⁺M³⁺B₂O₅ compounds, anions are represented by isolated B₂O₅ pyrogroups. Metaborate anions (N = 1/2) are more condensed: a) infinite metachains of BO₃–triangles (CaB₂O₄-I) [17], b) cyclic groups of three triangles bonded at common corners (BaO·B₂O₃ [68], c) ring-shaped three-membered rings of BO₄–tetrahedra in CuB₂O₄ [26]. Thus, in divalent metal metaborates besides BO₃–triangles, BO₄–tetrahedra also appear under normal pressure.

Compounds with an even smaller N–factor exhibit only 3D frameworks. The amount of triangular and tetrahedral boron in the structures with N = 1/4 is the same, with the exception of SrB₄O₇ and PbB₄O₇ in which the three-dimensional anionic framework consists only of BO₄–tetrahedra [23]. In the three-dimensional boron-oxygen net of BaB₈O₁₃ (N = 1/8) triangularly coordinated boron (n = 3) predominates [48]. Although no clear-cut regularity is observed in the variation of the limits of B-O interatomic distances depending on the N value in divalent metal borates; they are more isometric in orthoborates.

In the structures of borates with trivalent metals at N ≥ 1, only isolated BO₃–triangles (or isolated BO₄–tetrahedra in Fe₃BO₆ [69] and sinhalite MgAlBO₄ [70]) have been found. In simple R metaborates (N = 1/3), the polyanions are chain-like with n = 2. Simultaneously, in binary borates of tri- and divalent metals, for example, johachidolite CaAlB₃O₇ [34], synthetic RCo(BO₂)₅ [71] and RAl₂B₄O_10.5 [35] the polyanions are represented by layers of BO₄–tetrahedra or BO₄–tetrahedra together with BO₃–triangles, even if the N value is equal to 2/3, 2/5 and 3/4, respectively. Most likely, the nature of metals plays a significant role here. Both simple and binary pentavalent metal borates are not numerous and are not distinguished by any special diversity in the anion structure. For monocationic compounds, tetrahedral configuration of boron is preferable, but for binary compounds, it is triangular.

Decrease of the N ratio in borosilicates (in this case, it implies the ratio of the sum of metals to the sum of B and Si atoms) results to increasing of polymerization of the silicon-boron-oxygen motif. Thus, in the structure of grandidierite (Mg,Fe)Al₃SiBO₉ [72] (N = 2), the anion consists of isolated BO₃–triangles and SiO₄–tetrahedra, but in stillwellite LaBSiO₅ (N = 1/2), it is represented by BO₄–metachain encrusted with discrete SiO₄–tetrahedra [45]. In danburite CaB₂Si₂O₈ (N = 1/4) a silicon-boron-oxygen framework of Si₂O₇– and B₂O₇–diorthogroups is observed [46]. If the sum of metals exceeds the total amount of Si and B atoms, the boron is coordinated by three oxygen atoms. Taking into account the stillwellite structure, boron-oxygen FSU are more predisposed to polymerization in comparison with silicon-oxygen ones. It is noteworthy that there are practically no minerals of anhydrous borosilicates with the amount of boron predominating over the silicon content (with the exception of cappelenite BaY₆(Si₃B₆O₂₄)F₂ [73]), whereas borosilicates enriched with silicon are quite numerous. The shortage of adequately interpreted structures for boroaluminates (with the exception of synthetic A1₅(BO₃)O₆ [39]) makes it impossible to disclose their regularities. Even smaller amount of structural data are published for boroberyllates.

Coordination metal-oxygen polyhedra is more regular in structures with large N value, i.e., when they constitute the base of these structures. At small N value, not numerous metal atoms adapt themselves to the anionic motif. In other words, coordination oxygen polyhedra of metals in highly polymerized borates seem to be mostly determined by the nature of free spaces in the boron-oxygen base, but not by the directed bonds.

Among the borates having high N–factor values, the tendency to anion polymerization increases in compounds with smaller cations. For example, the difference in B-O interatomic distances inside the ring of potassium metaborate K₃B₃O₆ [63] is smaller than in sodium compound Na₃(B₃O₆) [8]. However, in lithium analogue LiBO₂ [62], the ring already becomes energetically disadvantageous orates. In the meantime, for highly condensed compounds the tendency of B-O anions to polymerize somewhat increases with an increase in the cation sizes.

An increase in the cation charge facilitates the weakening of the polymerization of BO₃–triangles and BO₄–tetrahedra and makes the tetrahedral coordination of boron atoms more preferable. In borates with comparatively weak singly charged and large divalent cations, the stability of the structure is ensured by the delocalization of anion charge, i.e., by a decrease in its formal specific charge during the polymerization process. In the case of cations with relatively high charge, which capable to form around themselves strong coordination polyhedral, making up the base of the structures, and the boron-oxygen anion stability is no longer of decisive importance. In such structures, therefore, not only isolated BO₃–triangles are commonly widespread, but BO₄–tetrahedra with an even higher negative charge (−5) as well.

4. Polymorphism of B_nO_m Polyanions

Metaborate radical of [BO₂]_n composition, mostly widespread in polyborates, was found in isolated (0D), chain (1D), ribbon (1D), layered (2D) and three-dimensional (3D) borates (

Table 1. Structural motifs based on the [BO₂]_n metaborate radical.

Type of Metaborate Anion	Composition of [BO2]n Polyanion	Compound
Isolated rings	[B3∆O6] = [B3O6]	α-Na2O·B2O3 K2O·B2O3 Rb2O·B2O3 Cs2O·B2O3 β-BaO·B2O3
Chain (one-dimensional, 1D)	[B2∆O4]1D = [B2O4]1D	α-Li2O·B2O3 CaO·B2O3 (I) SrO·B2O3 (I)
	[(B∆O2)2B2tO4]1D = [B4O8]1D	Calciborite CaO·B2O3 (II)
	[(B∆O2)4BtO2]1D = [B6O12]1D	Ln2O3·3B2O3 (Ln = La, Ce, Pr, Nd, Sm, Eu, Gd, Tb)
	[(B∆O2)2Al2tO4](1D) = [B2Al2O8]1D	2CaO·Al2O3 B2O3
	[(BO2)2Al2tO4]1D = [B2Al2O8]1D	2SrO·B2O3·Al2O3
	[B3tO6](1D) = [B3O6]1D (exc. Si)	Stillwellite Ce2O3·B2O3·2SiO2
	[B2∆O4](D) = [B2O4]1D (exc. Mo, W)	La2O3·B2O3·2MoO3 La2O3·B2O3·2WO3
Ribbon (one-dimensional, 1D)	[(B∆O2)4(AltO2)2]1D = [B4Al2O12]1D [(B∆O2)4(GatO2)2]1D = [B4Ga2O12]1D	3Li2O·Al2O3·2B2O3 3Li2O·Ga2O3·2B2O3
Layer (two-dimensional, 2D)	[(B∆O2)4B6tO12]2D = [B10O20]2D	Ln2O3·2CoO·5B2O3 (Ln = La-Nd, Sm-Ho)
Framework (three-dimensional, 3D)	[BtO2]3D = [BO2]3D	γ-Li2O·B2O3
	[B3tO6]3D = [B3O6]3D	CaO·B2O3 (II) SrO·B2O3 (II)
	[B3tO6]3D = [B3O6]3D	CuO·B2O3
	[B6tO12]3D = [B6O12]3D	4ZnO·3B2O3
	[(B6∆O2.5)(B∆O1.5)B4O8]3D = [B6O12]3D	CaO·B2O3 (III) SrO·B2O3 (III)
	[(B2tO4)(Si2O4)](3D) = [Si2B2O8]3D	Danburite—CaO·B2O3·2SiO2

). Index n in the formula [BO₂]_n for the currently known varieties of polyanions in metaborates assumes all the values from 1 to 6. With increasing polymerization of this metaradical, the triangular coordination of boron atoms regularly changes into BO₄–tetrahedra as one passes from the insular to chain, layered and 3D structures.

Pressure significantly affects the formation of metaborates’ structural motifs. Thus, in lithium borate obtained at low pressure and temperature, the boron atoms are located in oxygen triangles that are condensed into chains [62], but γ-LiBO₂ crystals synthesized at 950 °C and a pressure of 15 kbar have 3D structures with tetrahedral coordination of boron atoms [74,75]. In the Ca metaborate structure, stable at normal temperature, i.e., CaB₂O₄ (I), B and Ca atoms have triangular and eightfold coordination, respectively [17]. With the transition of this modification into CaB₂O₄ (II) synthetized at 12–15 kbar, half of the boron atoms increase their coordination number to four [16]. In CaB₂O₄ (III), which can be obtained at 900 °C and 15–25 kbar pressure, the fraction of boron atoms that preserve their triangular coordination is already only 1/3 [21]. In this case, for one-third of Ca atoms the coordination number increases to ten. In the structure of CaB₂O₄ (IV) all the B atoms are located in oxygen tetrahedra, and the coordination number of Ca increases to 9–12 [24]. A similar situation was observed in Sr metaborate [23,76].

A distinguishing feature of [B₄O₇]_n polyanions (n = 1 and 2) is their high degree of polymerization. Only one of them, α-Na₂O·2B₂O₃ is layered, in all others representatives were found 3D anionic motifs (

Table 2. Polyanions with [B₄O₇]_n and [B₈O₁₄]_n radicals.

Type of polyanion	Composition of B4O7 Based Polyanion	Compound
Layers (two-dimensional, 2D)	[(B2∆BtO5.5)(B3∆B2tO8.5)]2D = [B8O14]2D	α-Na2O·2B2O3
Frameworks (three- dimensional, 3D)	[B2∆B2tO7]3D = [B4O7]3D	LiO·2B2O3
	[B2∆B2tO7]2(3D) = [B4O7]2(3D)	MgO·2B2O3, MnO·2B2O3, ZnO·2B2O3, CdO·2B2O3
	[(B2∆B2tO7)(B∆B2tO5.5)(B∆O1.5)]3D = [B8O14]3D	K2O·2B2O3
	[(B2∆B2tO7)(B2∆BtO5)(BtO2)]3D = [B8O14]3D	CaO·2B2O3
	[B4tO7]3D = [B4O7]3D	SrO·2B2O3, PbO·2B2O3
	[(B3∆B2tO8.5)(B∆B2tO5.5)]3D = [B8O14]3D	BaO·2B2O3

). In all borates having [B₄O₇]_n radicals, with the exception of α-Na₂O·2B₂O₃, SrO·2B₂O₃ and PbO·2B₂O₃, the anions contain equal amounts of triangular and tetrahedral boron-oxygen coordination. The ratio of n_∆: n_t = 5/3 in α-Na₂O·2B₂O₃ is accompanied by part of BO₃ triangles being the corners unshared with other FSU, and in the exclusively tetrahedral frameworks of SrO·2B₂O₃ and PbO·2B₂O₃, part of oxygen atoms is coordinated with three boron atoms [23].

The 3D polyanion of [B₄O₇]_n composition in borates with comparatively small cations is made up of 2∆ + 2t diborate groups only. For larger M²⁺ cations the architectural principle was found in the twinning of 3D boron-oxygen nets, designated in Table 2 by doubling the contents of the square brackets. An increase in cation size leads to a strong deformation of diborate groups in K₂O·2B₂O₃. In CaO·2B₂O₃ triborate rings with additional tetrahedra are already stable, but a low-symmetry 3D net made up of dipentaborate and ditriborate CSU [47,76] is represented in the borate with the largest cation, barium.

The third and the fourth most widespread boron-oxygen clusters are B₃O₅ and B₈O₁₃, respectively (

Table 3. Polyanions based on [B₃O₅]_n and [B₈O₁₃]_n radicals.

Radical	Type of Polyanion	Composition B3O5 and B8O13 Based Polyanions	Compound
B3O5	Layers (two-dimensional, 2D)	[(B4∆BtO8)(B2∆BtO5)(BtO2)]2(2D) = [B9O15]2(2D)	β-Na2O·3B2O3
	Frameworks (three-dimensional,3D)	[(B4∆BtO8)(B2∆B2tO7)]2(3D) = [B9O15]2(3D)	α-Na2O·3B2O3
		[B2∆BtO5]3D = [B3O5]3D	Cs2O·3B2O3
B8O13	Frameworks (three-dimensional, 3D)	[(B4∆BtO8)(B2∆BtO5)]2(3D) = [B8O13]2(3D)	α-Na2O·4B2O3
		[(B4∆BtO8)(B2∆BtO5)]2(3D) = [B8O13]2(3D)	0.6Ag2O·0.4Na2O·4B2O3; BaO·4B2O3

). The first one takes part in the formation of layered and 3D structures, and the second was found in 3D borates only. Common features of structures with these two radicals are their comparative complexity, predominance of BO₃–triangles over BO₄–tetrahedra, low symmetry and the twinning of boron-oxygen nets. On the whole, with an increase in the ratio of the total number of boron atoms to that of oxygen atoms in the polyanions their composition and structure become more complex.

5. Isostructural Series

Isostructural series are widely encountered among high-temperature borates. Two dozen borates M²⁺M³⁺[BO₃]O₃ with warwickite-type structure and a considerable range of M²⁺:M³⁺ ratio are limited both by the sizes of trivalent cations close to Al³⁺ and lanthanides [77,78]. In this case, Ca²⁺ has the maximum radius value among M²⁺cations.

Isostructural with respect to one another are numerous borates of the ludwigite-vonsenite group, M²⁺Fe³⁺[BO₃]O₃, where M = Mg, Fe, Cu, Co, Ni, partially Sn [77]. This structure is also preserved in Co compounds, where Fe³⁺ is substituted by Cr, Ga, V, Sc, as well as in Fe²⁺ borate [78]. Trivalent iron is almost half substituted by aluminium in aluminoludwigite. However, the substitution of Fe³⁺ by Mn³⁺ in pinakiolite and orthopinakiolite, as well as a substantial inclusion of Sn⁴⁺ in hulsite, cause a considerable reduction of the symmetry of the structures [79].

Another group of di- and tetravalent metal borates and binary borates of trivalent elements (is headed by nordenskioldine CaSn[BO₃]₂ [80] which is isostructural to dolomite. More than fifty compounds compose the boracite group [56], with six minerals among them: boracite, stassfurtite, gauesinite, congolite, ericaite and chambersite. It is not only with various divalent cations anhydrous boracites were synthesized, but with chromium and lithium as well. One can also mention here the isostructural groups of MB₄O₇, where M = Mg, Mn, Zn, Cd; M₂B₂O₅ with M = Mg, Mn, Fe, Co, Cd; and M₃B₃O₆ (M = Na, K, Rb, Cs) in which the size of cations differs considerably.

Structural peculiarities of rare-earth borates should be specifically noted. Analysis shown that isostructural along the whole R series are only those compounds where the mutual linkage of these cations is comparatively small.

Orthoborates with the general formula M³⁺BO₃ can be structurally subdivided into three groups: (1) isostructural to calcite; (2) isostructural to aragonite; (3) compounds with a structure close to the third modification of CaCO₃–vaterite. Of them only LuBO₃, or, to be more specific, its low-temperature modification [81], belongs to the first group. The size of trivalent cation and the c:a ratio in it are obviously close to the maximum limiting values at which borates with the structure of calcite are stable (these values seem to be minimal in AlBO₃ [82], synthesized only at high pressures [83]). It is thus seen that in monocationic orthoborates (N = 1) the characteristics of each R ion (electron shell structure, its radius) exert a decisive effect on the formation of the structure.

In R(BO₂)₃ metaborates, although the influence of R is less pronounced here, the replacement of cations by smaller ones still leads to the deformation of B-O metachains. Therefore, two structural types for them are known. The situation is approximately the same with RCo(BO₂)₅ metaborates.

Double orthoborates of the R₂Sr₃(BO₃)₄, R₂Ba₃(BO₃)₄ and R₂Ca₃(BO₃)₄ families are already isostructural with one another, although in some compounds the intensities of X-ray reflection do not coincide [84].

In RM₃(BO₃)₄ (M = Al, Ga, Sc, Fe, Cr) borates with the huntite-type structure, the R-oxygen polyhedra are essentially simplified and constitute trigonal prisms deformed to a different extent. In these structures, RO₆–polyhedra are isolated from each other, and the base of the motif is formed by columns of Al, Ga, Sc, Fe, or Cr octahedra connected by isolated BO₃–triangles [85,86,87,88,89]. When Al is substituted by Fe, and then by Ga, in the NdM₃(BO₃)₄ compound, the symmetry of MO₆–octahedron is increased, with some increase in the size of RO₆–prisms. The relative sharing of R ions is not great here, and their specific properties tell even less on the formation of the structure. It seems that in this family of double trivalent metal borates the critical values of the ratio of cation sizes are close to Al³⁺/Nd³⁺ on the one side and to Fe³⁺/Sc³⁺ on the other. Also, there is monoclinic structural modification of NdAl₃(BO₃)₄. In the case of large R cations of the cerium subgroup, layered double RAl-metaborates with N = 3/4 are also synthesized [38]. Despite out numerous attempts it was impossible to obtain LaAl₃(BO₃)₄, and ScFe₃(BO₃)₄ was also crystallized with difficulty in a narrow range of conditions [90].

6. Structural Formulas of Polyborates

In the above text formulas of all anhydrous borates are given in the form taken from the literature, i.e., mainly as a ratio of metal oxide to boron oxide. Although they clearly reflect the N′ ratio, their structural formulas are undoubtedly more informative for polyborates with known crystal structure. Showing the ratio of cations and anions in the compounds, the CRP composition structure type (isolated anions, clusters, chains and ribbons, layers, 3D motifs), they can show, if necessary, the polyanion nature (simple or twinned), the amounts of triangularly and tetrahedrally coordinated boron atoms. Since most of the polyanions are built of single and double rings, it is useful to note this in the formula too. It could be provided them even more information, but because of the polyanions complexity, their structural formulas would become too bulky.

Table 4. Structural formulas of monocationic polyborates.

Formula in Oxides (Bulk Composition)	Structural Formula
	Complete		Abbreviated
α-Li2O·B2O3	α-Li2[B∆2O4]1D	α-Li2[B△2O4]1D	α-Li2[B2O4]
γ-Li2O·B2O3	γ-Li[BtO2]3D	γ-Li[BtO2]3D	γ-Li[BO2]
γ-3Li2O·7B2O3·2LiCl	γ-Li4Cl[(B∆3Bt3O10.5)(B∆O1.5)]3D	γ-Li4Cl[B△4Bt3O12]3D	γ-Li4Cl[B7O12]
Li2O·2B2O3	Li2[(B∆2Bt2O7)]3D	Li2[B△2Bt2O7]3D	Li2[B4O7]
MI2O·B2O3 (MI = Na-Cs)	MI3[(B∆3O6)]	MI3[B△3O6]	MI3[B3O6]
α-Na2O·2B2O3	α-Na4[(B∆2BtO5.5)(B∆3Bt2O8.5)]2D	α-Na4[B△5Bt3O14]2D	α-Na4[B8O14]
α-Na2O·3B2O3	α-Na6[(B2Bt2O7)(B∆4BtO8)]2(3D)	α-Na6[B△6Bt3O15]2(3D)	α-Na6[B9O15]2
β-Na2O·3B2O3	β-Na6[(B∆2BtO5)(B∆4BtO8)(BtO2)]2(2D)	β-Na6[B△6Bt3O15]2(2D)	β-Na6[B9O15]2
α-Na2O·4B2O3	α-Na4[(B∆2BtO5)(B∆4BtO8)]2(3D)	α-Na4[B△6Bt2O13]2(3D)	α-Na4[B8O13]2
K2O·2B2O3	K4[(B∆Bt2O5.5)(B∆2Bt2O7)(B∆O1.5)]3D	K4[B△4Bt4O14]3D	K4[B8O14]
5K2O·19B2O3	K2.5[(B∆2BtO5)(B∆4BtO8)(B∆O1.5)(Bt0.5O)]3D	K5[B△14Bt5O31]3D	K5[B19O31]
α-K2O·5B2O3	α-K2[(B∆4BtO8)]2(3D)	α-K2[B△4BtO8]2(3D)	α-K2[B5O8]2
β-MI2O·5B2O3 (MI = K, Rb)	β-MI2[(B∆4BtO8)]2(3D)	β-MI2[B△4BtO8]2(3D)	β-MI2[B5O8]2
Cs2O·3B2O3	Cs[(B∆2BtO5)]3D	Cs[B△2BtO5]3D	Cs[B3O5]
Cs2O·9B2O3	Cs2[(B∆3O4.5)2(B∆2BtO5)]2(3D)	Cs2[B△8BtO14]2(3D)	Cs2[B9O14]2
0.6Ag2O·0.4Na2O·4B2O3	Ag2.4Na1.6[(B∆2BtO5)(B∆4BtO8)]2(3D)	Ag2.4Na1.6[B△6Bt2O13]2(3D)	Ag2.4Na1.6[B8O13]2
5MIIO·7B2O3·MIIAI2—rhomb. and trig. Boracites (MII = Mg, Mn, Zn, Cd, Co, Ni, Cu; AI = Cl, NO3)—rhomb. (MII = Mg, Mn, Zn, Fe, Co, Ni, Cu; AI = F, Cl)—trig.	M3IIAI[(B∆O1.5)2(Bt6O11.5)]3D	M3IIAI[B△Bt6O13]3D	M3IIAI[B7O13]
5MIIO·7B2O3·MIIAI2—cub. Boracites (MII = Mg, Mn, Fe, Ni, Co, Cu; AI = Cl, Br, NO3)	M3IIAI[(Bt7O13)]3D	M3IIAI[Bt7O13]3D	M3IIAI[B7O13]
MIIO·2B2O3 (MII = Mg, Mn, Zn, Cd)	M2II[(B∆2Bt2O7)]2(3D)	M2II[B∆2Bt2O7)]2(3D)	M2II[B4O7)]2
MIIO·B2O3-I (MII = Ca, Sr)	MII[(B∆2O4)]1D	MII[B∆2O4]1D	MII[B2O4]
CaO·B2O3-II (calciborate)	Ca2[(B∆O2)2Bt2O4)]1D	Ca2[B∆2Bt2O4]1D	Ca2[B4O8]
MIIO·B2O3-III (MII = Ca, Sr)	M3II[(B∆O1.5)(B∆O2)(BtO2)(Bt3O6)]3D	M3II[B∆2Bt4O12]3D	M3II[(B6O12)]
MIIO·B2O3-IV (MII = Ca, Sr)	M3II[(Bt3O6)2]3D	M3II[(Bt3O6)2]3D	M3II[(Bt3O6)2]
2CaO·3B2O3	Ca2[(B∆2Bt3O9)2(BtO2)]3D	Ca2[B∆2Bt4O11]3D	Ca2[B6O11]
CaO·2B2O3-II	Ca2[(B∆2BtO5)(B∆2Bt2O7)(BtO2)]3D	Ca2[B∆4Bt4O14]3D	Ca2[B8O14]
MIIO·2B2O3 (MII = Sr, Po)	MII[(Bt4O7)]3D	MII[Bt4O7]3D	MII[B4O7]
β-BaO·B2O3	Ba3[(B∆3O6)2]	Ba3[(B∆3O6)2]	Ba3[(B3O6)2]
BaO·2B2O3	Ba2[(B∆Bt2O5.5)(B∆3Bt2O8.5)]3D	Ba2[B∆4Bt4O14]3D	Ba2[B8O14]
BaO·4B2O3	Ba2[(B∆2BtO5)(B∆4BtO8)]2(3D)	Ba2[B∆6Bt2O13]2(3D)	Ba2[B8O13]2
4ZnO·3B2O3	Zn4O[(Bt6O12)]3D	Zn4O[Bt6O12]3D	Zn4O[B6O12]
CuO·B2O3	Cu3[(Bt3O6)2]3D	Cu3[(Bt3O6)2]3D	Cu3[(B3O6)2]
5Al2O3·B2O3	Al2[(Alt3O6)(B∆O3)]3D	Al2[Alt3B∆O9]3D	Al2[Al3BO9]
R2O3·3B2O3 (R = La-Tb)	R2[(B∆2Bt2O8)(B∆O2)2]1D	R2[B∆4Bt2O12]1D	R2[B6O12]

M^I—monovalent metal; M^II—divalent metal.

lists both complete and abbreviated structural formulas for some polyborates that have been studied.

The composition of repeating radical is enclosed in square brackets with the 1D, 2D or 3D symbols for various types of structures (one-, two- or three-dimensional polyanion, respectively). In the case of a twinned polyanion, the composition of one of the equivalent boron-oxygen nets is doubled (indicated by the figure of two after the square brackets). The maximum information is contained in the first complete version of the formula, where CSU are shown, as well as additional BO₃–triangles and BO₄–tetrahedra. The second version, as an alternative for writing out the formula, only shows a relationship between the triangularly and the tetrahedrally coordinated boron. For example, since the three-dimensional polyanion in Li₂O·2B₂O₃ consists of 2_∆ + 2_t diborate groups [10] its chemical composition can be written out as [(B₂^∆B₂^tO₇)]_3D or in the abbreviated form as [B₄O₇]_3D. The negative charge of one such radical is equal to two. The structural formula will, therefore, have the form of Li₂[B₄O₇]_3D. In contradistinction to Li₂[B₄O₇], in Mg, Mn, Zn and Cd borates with N^′ = 2, the polyanions consist of twinned three-dimensional boron-oxygen nets. Therefore, their structural formula is M₂^II[B₄O₇]_2(3D). In the α-Na₂O·2B₂O₃ structure, the two-dimensional polyanionic net consists of dipentaborate and triborate CSU, with the latter each containing a free O atom [11]. Polyanion composition of this sodium borate can be written as (B₃^∆B₂^tO_8.5) + (B₂^∆B^tO_5.5) = [B₅^∆B₃^tO₁₄]_2D = [B₈O₁₄]_2D; its negative charge is equal to four, and borate structural formula will be represented in the form of α-Na₄[B₈O₁₄]_2D. In K₂O·2B₂O₃, the composition of three-dimensional polyanionic network, consisting of (2∆ + 2t) diborate and (1∆ + 2t) triborate groups with additional BO₃ triangles (1∆) [14], is (B₂^∆B₂^tO₇) + (B^∆B₂^tO_5.5) + (B^tO_1.5) = [B₄^∆B₄^tO₁₄]_3D = [B₈O₁₄]_3D. The charge of this polyradical is four, the borate structural formula can be look like K₄[B₈O₁₄]_3D.

The anionic motif of metastable β-Na₂O·3B₂O₃ consists of double two-dimensional networks represented by pentaborate (4∆ + 1t), triborate (2∆ + 1t) rings and additional tetrahedra (1t) [50], i.e., polyanion composition is [(B₄^∆B^tO₈) + (B₂^∆B^tO₅) + (B^tO₂)] × 2 = [B₆^∆B₃^tO₁₅]_2(2D) = [B₉O₁₅]_2(2D); and structural formula of this borate β-Na₆[B₉O₁₅]_2(2D). The composition of three-dimensional twinned α-Na₂O·3B₂O₃ polyanion made up of triborate (4∆ + 1t) and diborate (2∆ + 2t) CSU is the same as that of β-Na₂O·3B₂O₃ [49]: [(B₄^∆B^tO₈) + (B₂^∆B₂^tO₇)] × 2 = [B₆^∆B₃^tO₁₅]_2(3D), and Na-triborate structural formula can be written as β-Na₆[B₉O₁₅]_2(3D). Cs₂O·3B₂O₃ has a simpler formula since its three-dimensional boron-oxygen network consists only of triborate groups (2∆ + 1t) [12] which build polymerized radical [B₂^∆B^tO₅]_3D in structural formula of this compound Cs[B₃O₅]_3D.

In the structures of α-Na₂O·4B₂O₃ (and silver borate of similar composition) the binary three-dimensional anionic motif is represented by pentaborate (4∆ + 1t) and tetraborate (2∆ + 2t) CSU [54]. Its polymerized radical [(B₄^∆B^tO₈) + (B₂^∆B^tO₅)] × 2 = [B₆^∆B₂^tO₁₃]_2(3D) = [B₈O₁₃]_2(3D) having charge four takes place in borate structural formula α-Na₄[B₈O₁₃]_2(3D). General formula, i.e., gross composition of 5K₂O·19B₂O₃ (K₂O·3.8B₂O₃, which has a similar boron content, a three-dimensional polyanion built of triborate (2∆ + 1t), pentaborate (4∆ + 1t) groups, additional tetrahedra (1/2t) and triangles (1∆) [52] corresponds to K₅[B₁₉O₃₁] containing (B₄^∆B^tO₈) + (B₂^∆B^tO₅) + (B_0.5^tO) + (B^∆O_1.5)] × 2 = [B₇^∆B_2.5^tO_15.5] × 2 = [B₁₄^∆B₅^tO₃₁]_3D = [B₁₉O₃₁]_3D radical. Structural formulas of all other borates can be derived in a similar way.

7. Classification

Any systematic is usually aimed at finding particular regularities in the system studied, in order to predict a variation of its characteristic features. Classification of numerous borates with their specific structural features is far from a simple matter, but in many respects, it implies the progress of their study. The chemical, crystallochemical and genetic systematics undertaken with the accumulation of factual data reflects a considerable extent of knowledge that had been acquired by the moment of classification. Each subsequent classification is, as a rule, superior to the previous one. The schemes proposed by 1966 were thoroughly and critically discussed in the review [91]. Therefore, there is a sense in dwelling shortly on some of them here.

In [92], by analogy with silicates, the following borate subclasses are described: (1) “nesoborates”, i.e., compounds with isolated BO₃ triangles and BO₄ tetrahedra, (2) “soroborates” (grouped), (3) “inoborates” (chainlike), (4) “phyloborates” (layered) and (5) “tectoborates” (three-dimensional). Its main shortcoming is the absence of a definite regularity in the systematics of isolated boron-oxygen polyanions, including a great number of other structural units. As a result, it was not possible to ascertain the relations between borates belonging to different groups and to include a number of synthetic borates.

The crystallochemical classification of borates suggested in [91] is based on two main characteristics: the structure of boron-oxygen polyanions and the manner in which they are combined. It also takes into account the peculiarities of the borate structures, consisting in a much greater variety of their polyanions, in comparison with silicates, phosphates, etc. In this classification, boron-oxygen isolated polyanions, as well as chainlike, layered, and 3D motifs are examined in detail. In addition, possible ways of combining anions and polyanions are analyzed and their general formulas are derived. All borates are subdivided into four orders (subclasses): insular, chainlike, layered and three-dimensional.

Insular borates are divided into eight suborders: (1) isolated non-ringed with isolated polyions; (1a) non-ringed with dimers of B(O,OH)₃ triangles and B(O,OH)₄ tetrahedra; (2) one-ringed triborates with isolated polyions; (3) two-ringed borates with isolated polyions; (4) three-ringed borates; (5) four-ringed borates; (6) borates with mixed polyions and (7) borosilicates. Three-ringed borates have no representatives, only one four-ringed borate was known, as well as one borosilicate. In all the other borosilicates whose structures were interpreted by that time, B and Si atoms constituted the general motif. In suborders (1), (1a) and (2), the families of oxygen-containing and hydroxyl borates were singled out, but tetra- and pentaborates were fallen into the third suborder.

Chainlike borates have been subdivided into following categories: (1) non-ringed; (2) one-ringed; (3) two-ringed; (4) three-ringed; (5) borates with mixed polyions and (6) borosilicates. For the fourth and the sixth suborders, there were no representatives, and only two borates were fallen into the fifth suborder. In non-ringed chainlike borates, the families of oxygen-containing and hydroxyl borates (one compound) were identified, and the families of tetra- and pentaborates represent the two-ringed compounds.

Layered and three-dimensional borates are also regarded as having six similar suborders. Three-ringed three-dimensional borates were yet unknown, but non-ringed three-dimensional representatives are divided into oxygen-containing and hydroxyl (only one compound) borates, and the tetra- and pentaborate families are fallen into the two-ringed suborder.

Within most of the suborders and families, the compounds were classified by the basicity of individual polyions (zero-, uni-, di-, tri-, tetra- and pentabasic) in [92].

As distinct from Tennyson’ systematic [93], in [91], borates with isolated BO₃–triangles and BO₄–tetrahedra are included into the section of insular borates, as well as borates with isolated “mono”-ions and dimmers. The classification described in Ref. [91] comprises practically all the borates and many borosilicates known at the time, both with interpreted and hypothetical structures, and reflects the regularities of the polymerization of borates (mainly, hydrated ones). A place was found in it for 119 of the more than six hundreds known by that time anhydrous borates and their polymorphic modifications. All of them were mainly attributed to the section of insular borates, a small part was assigned to three-dimensional and to chainlike compounds, but no comment is made on layered borates. Detailed systematics of boron-oxygen radicals can be considered as a progressive step in understanding the crystal chemistry of this unusual class of inorganic polymer compounds.

Nowadays, the number of anhydrous high-temperature borates with interpreted structures and structural types is several times in comparison with the middle of last century. It is, therefore, quite natural that some of them, mainly those with hypothetical and approximately solved structures, after the interpretation or refinement had to be moved from one section of the classification scheme to another. For this reason, for example, barium metaborate with crystal chemical formula of Ba[B₂O₄], included in the section of chainlike compounds, a year later proved to be insular [48]. Its structural formula should be written out as Ba₃[(B₃O₆)₂]. Also, the crystallographic characteristics of Mg[B₂O₄] were unknown before to be assigned to the same order on the strength of the data [92]. should be noted that in a later work [76] the authors doubt whether this compound actually exists. The situation is approximately the same with 2PbO⋅B₂O₃ appearing in the same group with B₂O₅ dimers among insular borates [76].

The subsequent refinements and interpretations of crystal structures have also revealed numerous inaccuracies in the classification of 3D borates. For instance, triborates with the general structural formula M¹⁺[B₃O₅], where M¹⁺ = Li, Na, K, Rb, as well as Mg[B₂O₅]₂ have been included in the group of ringed three-dimensional compounds. However, the polyanion in β-NaO·3B₂O₃ happened to be a twinned layered one made up of pentaborate (4∆ + 1t), triborate (2∆ + 1t) groups and the additional BO₄ tetrahedra bonding them [50], and therefore its structural formula should be β-Na₆[B₉O₁₅]_2(2D). The three-dimensional α-modification of Na₂O·3B₂O₃, also with a 3D twinned polyanion of diborate (2∆ + 2t) and pentaborate (4∆ + 1t) CSU [49], can probably be better described by a similar structural formula α-Na₆[B₉O₁₅]_2(3D). In K-triborate, cell parameters were determined only by that time [92]. For Li- and Rb- borates, no crystallographic data were known then and MgO·3B₂O₃ has was not synthesized yet, in our knowledge. Layered sodium borate with N = N_M:N_B = 1/2 and, as established later [11], with the anion of triborate (2∆ + 1t) and diborate (3∆ + 2t) CSU, has been placed by the authors of [90] into the section of two-ringed three-dimensional tetraborates with 3D polyanions. However, its more realistic structural formula might be written as α-Na₄[B₈O₁₄]_2D. By that time, there were no detailed structural data for some other 3D borates: Li₂[B₈O₁₃], Na[B₅O₈], Cs₂[B₄O₇], Cs₂[B₈O₁₃], Cs[B₅O₈] and α-Cs₂O·5B₂O₃, with exception of their lattice cell parameters [92]. As for K₂[B₈O₁₃], only the structure of a compound with close composition, 5K₂O·19B₂O₃ (K₂O·3.8B₂O₃) has been solved [52]. Taking into account the X-ray diffraction studies performed later, the crystallochemical formulas of K₂[B₄O₇], Ca[B₄O₇], K[B₅O₈], Rb[B₅O₈], Na₂[B₈O₁₃], Ag₂[B₈O₁₃] and Ba₂[B₈O₁₃] seem to be not so adequate in Ref. [92]. Probably, it would be better to write them as K₄[B₈O₁₄]_3D, Ca₂[B₈O₁₄]_3D, α- and β-K₂[B₅O₈]_2(3D), β-Rb₂[B₅O₈]_2(3D), α-Na₄[B₈O₁₃]_3D, Ag₄[B₈O₁₃]_2(3D), and Ba[B₈O₁₃]_2(3D), in correspondence with the composition of their polyanions. In addition, boracite was regarded as non-ringed, because of the insufficiently accurate interpretation of its structure [94].

Finally, it should be noted that in accordance with [23], the first SrB₄O₇ structure determination by the author of [95] was also incorrect. Namely, based on this example it was assumed possible for the polymerization to take place not only by the combining of the corners of BO₃–triangles and BO₄–tetrahedra but also the edges of the tetrahedral [91]. In this connection, it is unlikely that approximately interpreted in [7] layered Na₂Zn₂MnB_4.67O₁₁ structure with a very close proximity of triple-charged B³⁺ can be stable.

Another attempt to systematize borates (including organic compounds) was undertaken by G. Heller [96]. It was based on the cation type and the number of boron atoms in the polyanion structural unit. This classification schematically presents the possible polyanions and gives several examples of different structures (many of which later proved to be incorrect) set out in accordance with the number of boron atoms in the basic polyanion structural unit and the type of the anionic radical (isolated, chainlike, layered, three-dimensional). An attempt to encompass numerous anhydrous, hydrated borates and organoboron compounds has made it very cumbersome and led to a number of errors and discrepancies, including representation of the crystallochemical formulas of some compounds.

Christ and Clark [6] have proposed a rational crystallochemical classification of the anions of hydrated borates. The authors have identified the basic polyanion structural units, suggested an abbreviated notation, and the rules of their formation. They have also deduced crystallochemical formulas from the structural data at hand, and the other compounds were combined in a separate group. It was assumed that polymerization could be realized in the following schemes: (1) by the corners of BO₃–triangles and BO₄–tetrahedra being combined, (2) by elimination of water from isolated boron groups, (3) by complication of anions into additional groups. The most probable sequence was shown for the addition of protons to the oxygen atoms in hydrous borates.

Therefore, every systematics of borates has made a contribution to the development of the crystallochemistry of this class of compounds. New data on borate crystal structures require further refinement and the revision of existing classifications. This primarily concerns numerous anhydrous compounds the peculiarities of whose structure and crystallization have not been analyzed taking into account the latest data.

The major source of errors in all the systematics is associated with underestimating the regularities of the polymerization of boron oxygen anions. That is why some borates are often prematurely placed into certain sections of the classification schemes. Moreover, this is not surprising since in the course of their study a number of characteristic features previously not taken into account became known. For instance, all crystallochemical systematics did not take into consideration that a part of oxygen atoms in highly condensed 2D and 3D polyanions being cannot be not coordinated not only with two, which is usual, but also with one, three and even four boron atoms. All this, as well as a number of other factors, have introduced substantial uncertainties into the classification of compounds with unknown structures.

In order to avoid such ambiguities, borates with unknown structure should not be included in the crystallochemical systematics. On this way, however, one of the main objectives of classification will not be met, i.e., it will not serve as a basis for theoretical and experimental research, will not favor to forecast structures and properties of new materials. In the case of distribution and redistribution of numerous subsequently studied anhydrous borates among the sections of a latest crystallochemical systematics, similar (with the same cations and having close M_xO_y/B₂O₃ ratios) compounds and even borates with analogous structure will be placed into unsuitable for them units. This is because only the structure and composition of the anion (polyanion) were taken into account leaving aside the cation type, size and charge. At times it will be difficult to explain the difference between structures with polyanions of similar composition but with different cations, in order to understand the nature of boron-oxygen radicals polymerization, etc. As a result, it seems impossible to expect a tangible assistance from such systematics on the way of interpretation and refinement of structures, and predicting new compounds.

Ways of searching for a more flexible classification can be based on the general regularities in the structure of this class of borate materials, and on knowledge about the previous systematics of borates and other compounds [97]. Analysis of anhydrous borates structure, composition and conditions shows that there is a sense to examine them separately within the framework of the general classification of borates. In addition, this is because of the specificity of their structures. Thus, e.g., Christ and Clark have shown that in every known hydrated borate structure, in contradistinction to anhydrous borates, there is only one type of the basic structural unit in the anion [6]. The difference between them is clearly illustrated by the comparison of Ca₂B₆O₁₁ structures and the Ca₂B₆O₁₁·xH₂O series, where 1 ≤ x ≤ 15. Isolated or bonded into chains and layers triborate groups of one BO₃–triangle and two BO₄–tetrahedra represent anions of all the hydrated Ca borates. In the anhydrous 3D borates, usually regarded as the end member of this series, the paired rings of two BO₃–triangles and three BO₄–tetrahedra are bonded into a framework by additional tetrahedra [96]. For this reason G. Christ and J. Clark emphasized the difference in the structure of anhydrous and hydrated borates and suggested the necessity of their independent analysis.

When classifying anhydrous high-temperature borates, therefore, one should bear in mind the set of the following prerequisites, most of which are individually well known:

(1)

In crystal structures each boron atom is bonded with three or with four oxygen atoms in BO₃–triangles and BO₄–tetrahedra;

(2)

In one structure not only triangular or only tetrahedral coordination is possible, but both of them jointly as well;

(3)

Isolated BO₃–triangles and BO₄–tetrahedra are not found jointly, insular polyanions;

(4)

A decrease in the N = N_M/N_B ratio (N-factor), as well as an increase in cation size (although to a smaller extent), leads to an increase in the degree of polymerization of the anion and raises the n = n_∆/n_t number (at N ≤ 1), whereas an increase in cation charge causes the inverse tendency;

(5)

Polymerization, or the formation of chains, layers and frameworks, is actualizing by the sharing corners of triangles and tetrahedra (the sharing edges has up to now not been proved conclusively);

(6)

In 3D and 2D polyanions (less frequently in chainlike and insular ones), BO₃–triangles and BO₄–tetrahedra tend to combine into comparatively compact CSU, i.e., diborate (2∆ + 2t), triborate (2∆ + 1t), pentaborate (4∆ + 1t), boroxol (3∆), ditriborate (1∆ + 2t), dipentaborate (3∆ + 2t) and other single and double ringed boron-oxygen negative charged polymerized radicals;

(7)

Complex polyanions of anhydrous borates of uni- and divalent metals tend to twinning;

(8)

In most of complex polyanions, each oxygen atom is bonded with two boron atoms, for such compounds as M_xO⋅mB₂O₃ with m > 1, there is n = m − 1 relationship (where n = n_∆/n_t);

(9)

As an exclusion for 2D and 3D highly condensed polyanions, the coordination numbers of oxygen atoms (relative to boron) can be equal to one or three (in cubic boracite even to four).

The first (primary) classification level, successfully used in inorganic chemistry and mineralogy, is known to be based on the type of the anion-forming element (sulphides, halides, silicates, borates, phosphates, etc.). This reflects the characteristic common features of all classes of compounds, determined by the position of anion forming elements in the Periodic table of the elements. The second order (sublevel) represents the subdivision of classes and is usually also based on the composition of compounds or on their structure. For silicates, e.g., in their overwhelming majority natural and, therefore, of complex composition, in which it is sometimes difficult to identify the predominant cations, the crystallochemical classification reflecting the functional dependence between the composition and the structure of the anion has proved to be the most expedient. This, however, does not mean that this approach should be used for systematics onto the other classes of compounds as well.

When the main cations and the above nine prerequisites are considered together, it becomes possible to move the structural principle onto a higher rank (level) of the classification scheme. It is because the structure of polyanions, the polymerization degree, the ratio of the BO₃ triangles number to the amount of tetrahedra are determined, to a considerable extent, by the N-factor and the type of cation.

The first level of the systematics of anhydrous borates can be subdivided by the quantitative composition of anions and polyanions into the following sublevels:

(a)

Borates proper (also, there is a sense to divide this very numerous group, having diverse cations, into two subgroups: aI—monocationic or “simple” borates and all-binary and more complex compounds);

(b)

Borosilicates;

(c)

Boroaluminates;

(d)

Boroberyllates;

(e)

Borocarbonates;

(f)

Boromolybdates and borotungstates.

It is reasonable to carry out the next, second, order (sublevel) of classification by the value of cation charge. Then, inside these subdivisions, compounds can be ranked in accordance with their decreasing N factor, indicating its value and structural type, if the structure has been studied. Therefore, the third level is structural. Moreover, finally, the fourth order of this scheme should as far as possible represent the change in the type and size of cations having the same charge. It is also expedient to single out the isostructural and isomorphic series, that especially characteristic for borates with isolated BO₃–triangles.

An example of the scheme of classification of high-temperature anhydrous borates is given in

Table 5. Classification scheme of anhydrous borates.

By the Composition of Anion Formers		By Cation Valency		By Cation Type (and Size)	By the Value of N Factor (N = NM/NB)
					N > 1	N = 1	n = n∆/nt = m − 1, where m—coeff. from MxO·mB2O3 *
							1 < N ≤ 1/2	1/2 < N ≤1/3	N < 1/3
Borates	Monocationic (simple)	Monovalent	a	Li ↓ Cs	Orthoborates with isolated BO3 triangles	Metaborates, insular and chainlike **	Polyborates, 3D and seldom - layered	Polyborates, 3D and seldom - layered	Polyborates, 3D
			b	Ag Tl	Same	0	0	0	Same
		Divalent	a	Be ↓ Ba	Same	Pyroborates	Metaborates, insular and chainlike**	Polyborates, 3D	Same
			b	Zn, Cd, Mn, Fe, Co, Ni, Cu, Pb	Same	Same	Metaborates, 3D	Same	Same
		Trivalent	a	Al	Orthoborates with BO4 tetrahedra	Orthoborates with BO3 triangles	-	-	-
			b	Sc, Ti, V, Cr, Ga, In	-	Same	-	-	-
			c	Fe	Orthoborates with BO4 tetrahedra	Same	-	-	-
			d	Y, La-Nd, Sm-Yb	0	Same	-	Metaborates, chainlike	-
			e	Bi	Orthoborates with BO3 triangles	-	-	0	0
		Tetravalent		Th	-	-	0	-	-
		Pentavalent		P, As, Ta, Nb	-	Orthoborates with BO3 triangles	-	-	-
1		2		3	4	5	6	7	8
Borates	Binary and more complex	M+M2+			Orthoborates with BO3 triangles	-	-	-	-
		M+M3+			Same	0	0	-	-
		M+M5+			Same	Pyroborates	-	-	-
		M2+M2+			-	Same	-	-	-
		M2+M3+			Orthoborates with BO3 triangles and BO4 tetrahedra	-	Metaborates, layered (t)	Metaborates, layered (∆ + t)	-
		M2+M4+			-	Orthoborates with BO3 triangles	-	-	-
		M3+M3+			-	Orthoborates with BO3 triangles	Metaborates with B tetrahedrons	-	-
Boron silicates		M2+M3+, M3+, M2+			Orthocompounds	Metacompounds, chainlike	-	-	Polycompounds, 3D
Boron aluminates		M2+, M3+			-	Same	Polycompounds, layered and 3D	-	-
Boron beryllates		M+			-	-	Polycompounds, layered	Polycompounds, layered and 3D	-
Boron carbonates		M2+M3+			Orthocompounds	-	-	-	-
Boron molybdates and boron tungstanates		M M3+			-	-	Metacompounds, chainlike	-	-

* Simple mono- and divalent metal borates having polyanions with coordination number of oxygen atoms with respect to boron are equal to two obey this rule. ** 3D γ-LiBO₂, Ca and Sr metaborates obtained at high pressure are an exception. Note: Symbols “-” and “0” indicate that this compound is unknown (“-”) or it is known but its structure was not solved (“0”).

and

Table 6. Classification of mono- and divalent metal borates.

I. Borates of Monovalent Elements
		N	Cations
			Li	Na (Ag)		K	Rb	Cs	Tl
Orthoborates (with isolated B triangles)		5	-	0		-	-	-	-
		3	α-Li3[BO3] β from – 0	0		-	-	-	Tl3[BO3]
		5/2	-	0		-	-	-	-
		2	-	0		-	-	-	-
		3/2	0	0		-	-	-	-
Fluoroborates (with isol. BF4 tetr.)		1	-	Na[BF4]		K[BF4]	Rb[BF4]	Cs[BF4]	Tl[BF4]
Metaborates (ins. with boroxol gr.) *		1	α-Li2[B2O4]1D γ-Li[BO2]3D	α-Na3[B3O6] β phase – 0		K3[B3O6]	α from - 0	Cs3[B3O6]	0
Polyborates (skeletal, sometimes compounds, layered with diborate – D, triborate – T, pentaborate – P, boroxol – B, ditriborate – DT, dipentaborate – DP rings and additional B triangles and tetrahedrons; n = n∆/nt = m − 1)		2/3	-	0		-	-	-	-
		4/7	α and β Cl, Br and I “boracites” – 0 γ-Li4Cl[B7O12]3D	-		-	-	-	-
		1/2	Li4[B4O7]2(3D)2(D)	(α)-Na4[B8O14]2D ** (T+DP) β and γ phase - 0		K4[B8O14]3D	0	0	0
		2/5	0	0		-	-	-	-
		1/3	0	α-Na6[B9O15]2(3D) 2(D + P) β-Na6[B9O15]2(2D) 2(P + T + t) γ-form – 0		0	0	Cs[B3O5](3D) (T)	0
		5/19	-	-		K5[B19O31]3D	-	-	-
		1/4	0	α-(Na,Ag)4[B8O13]2(3D) 2(T + P) β modification – 0		0	0	0	0
		1/5		α, β and γ phases – 0		α-K2[B5O8]2(3D) 2(P) β-K2[B5O8]2(3D) 2(P), γ phase - 0	α phase – 0 β-Rb2[B5O8]2(3D) 2(P)	α, β, γ phases - 0	0
		1/9	0	α, β, and γ phases – 0		0	0	Cs2[B9O14]2(3D) 2(B + T) β phase	-
II. Borates with Cations of Transition Metals
	N	Cations
		Zn	Cd	Mn	Fe	Co	Ni	Cu	Pb
Orthoborates (with isolated BO3 triangles)	3	-	-	Wiserite Mn3[BO3]F3	-	-	-	-	-
	2	-	-	Mn2[BO3]F	-	-	-	-	0 (α and β forms)
	3/2	α-Zn3[BO3]2	0 (α and β forms)	Jimboite Mn3[BO3]2	-	0	Ni3[BO3]2	0	-
	5/4	0 (α and β forms)	-	-	-	-	-	-	-
Pyroborates (with isol.2)	1	-	Cd2[B2O5]	Mn2[B2O5]	Fe2[B2O5]	Co2[B2O5]	0	-	0 (α and β forms)
Metaborates (skeletal)	1/2	0 (α and β forms) Zn4O[B6O12] * (t)	-	0	0	0	0	Cu3[(B3O6)2] (t)	0
Polyborates (skeletal)	3/7	cub., rhomb. and trig. F, Cl, Br, I and NO3 “boracites”	Cl, Br, I and NO3 “boracites”	cub., rhomb. and trig. F, Cl, Br, I and NO3 “boracites”	cub. and trig. F, Cl, Br, I and NO3 “boracites”	cub., rhomb. and trig. F, Cl, Br, I and NO3 “boracites”	cub. and rhomb. Cl, Br, I and NO3 “boracites”	cub. and rhomb. Cl, Br, I and NO3 “boracites”	-
	1/3	-	0	-	-	-	-	-	-
	1/4	Zn2[B4O7]2 2D	Cd2[B4O7]2 2D	Mn2[B4O7]2 2D	-	-	-	-	Pb[B4O7] (t)
	1/6	0	-	0	-	-	-	-	-

* 3D γ-LiBO₂ obtained at high pressure is an exception. ** By its n = 5/3 number is an exception; in a part of O atoms coordination number with respect to boron is equal to 1.

, where the example of systematics of monocationic mono- and bivalve anhydrous metal compounds is shown. This systematics allows to cover all known anhydrous borate compounds, and to develop an understanding of variations of their properties, limits of stability, as well as the possibility of synthesizing new compounds of these series. Following [98], e.g., Li₂O⋅4B₂O₃ borate seems to contain three-dimensional boron-oxygen nets with the ratio of n ≥ 1. The polyanions in Na₂O⋅5B₂O₃ and Na₂O⋅9B₂O₃ are most probably three-dimensional with n equal to 4 and 8, K₂O⋅3B₂O₃ possibly also contains three-dimensional nets of triborate groups with n = 2. The structure of polyanions in Rb-borates is close to that in the corresponding potassium compounds. The structures of Cs₂O⋅4B₂O₃ and Cs₂O⋅5B₂O₃ are evidently also three-dimensional with n = 3 and 4.

8. Structural Aspects of Acid-Base Properties

Understanding of growth kinetics and mechanism of borate crystals from melts and fluxed melts is still a problem and leads to deterioration in crystal quality. Thus, it is useful to consider a correlation between the polymerization of anions in the structures of anhydrous borates and their derivatives in order to explain the capability of these inorganic polymers to crystallize.

There were some attempts to estimate acid-base properties for oxide compounds, both solids and melts. The most popular of them is the Lux-Flood’s acid-base theory [99]. This concept seems to be more effective for assessment of the acid-base characteristics of anhydrous borates finding of promising solvents for the flux growth of high-temperature borate crystals. According to the Lewis-Lux’s equation: Acid + O²⁻ ↔ Base, the acid-base parameters of melts depend on the oxygen activity, thus, B₂O₃ + O²⁻ = B₂O₅⁴⁻, which means that the pyroborate anion (2∆) has higher oxygen activity in comparison with boron trioxide increasing its base component.

Therefore, the activity of O²⁻, and, correspondingly, the reactivity of boron-containing melts decreases with an increase in the N_B/N_O ratio, and a tendency to glass formation is observed due to the features of their structure, for which the B–O bond energy is 519 kJ/mol [100]. Since atoms and other particles in such viscous systems move slowly, the glasses obtained by rapid quenching retain pronounced traces of frozen processes.

Taking into consideration the above borate classification and the Lux-Flood’s concept, it is intuitively obvious that the simplest way to assess quantitatively acid-base properties of the anhydrous borates is to estimate the dependence of polymerization of anions in the borate structures on the sizes and valences of cations, and also on the N-factor (Figure 9). From the crystallochemical point of view, it can see that the increase of the N-factor increases the anion polymerization and the N_B/N_O ratio. This leads to a decrease in the oxygen activity factor and simultaneously to an increase in the acid component of these compounds. Also, the value of n = n_∆/n_t, i.e., the ratio of the number of BO₃–triangles to BO₄–tetrahedra in the structures of compounds increases.

9. Summary

This review is an alternative approach by the authors to present the structural aspects of high-temperature anhydrous borates in the way of synthesis and growth of crystals of new technologically attractive materials from this numerous family of borates. They can be described by only three types according to the level of complexity of structural units: (1) BO₃–triangles (∆) and BO₄–tetrahedra (t) as fundamental (elementary) structural units (FSU) constituting the anions of all borates (only triangles, only tetrahedra or both the triangles and tetrahedra); (2) the second level of structural units is represented by combined basic units (CSU) which usually built up of several FSU (from 2 to 5) joined by sharing common O atoms occurring in many structures; (3) the third type of borate structural units corresponds to complete radicals of polyanions (CRP) which constructed of 2–9 FSU, i.e., with a composition equal or aliquot to the anionic portions of the compound structural formulas. With a decrease in the N = N_M/N_B ratio, i.e., N-factor, as well as with an increase in the cation size (though to a smaller extent), the anion polymerization degree and the n = n_∆/n_t number (at N < 1) regularly increase. An increase in the cation charge causes the reverse tendency. It facilitates the attenuation of the polycondensation of BO₃–triangles BO₄–tetrahedra. In borates with highly charged cations, the boron atoms prefer tetrahedral coordination. Highly charged cations are, however, capable to form around themselves rigid coordination polyhedra, usually making up the basis of the structure. Stability of the boron-oxygen anion here loses its decisive importance. This encourages the formation of borate structures with isolated both BO₃–triangles and BO₄–tetrahedra having comparatively large charges, −3 and −5 respectively. Tendency to polymerization and, therefore, to more acidic properties makes it possible to forecast new phase systems for the synthesis of predicted borate structures. A new approach to borates classification is proposed, and an improved systematics of anhydrous compounds has been performed. The place of a borate in this scheme, including those with an unsolved structure, characterizes to a certain extent its structure and properties.