Borates—Crystal Structures of Prospective Nonlinear Optical Materials: High Anisotropy of the Thermal Expansion Caused by Anharmonic Atomic Vibrations

Abstract

In the present study the thermal structure evolution is reviewed for known nonlinear optical borates such as β-BaB₂O₄, LiB₃O₅, CsLiB₆O₁₀, Li₂B₄O₇, K₂Al₂B₂O₇, and α-BiB₃O₆, based on single-crystal and powder X-ray diffraction data collected over wide temperature ranges. Temperature-dependent measurements of further borates are presented for the first time: α-BaB₂O₄ (295–673 K), β-BaB₂O₄ (98–693 K), LiB₃O₅ (98–650 K) and K₂Al₂B₂O₇ (98–348 K). In addition to the established criteria for nonlinear optical (NLO) properties of crystals, here the role of the anisotropy and anharmonicity of the thermal vibrations of atoms is analysed as well as changes in their coordination spheres and the anisotropy of the thermal expansion of the crystal structure. Non-centrosymmetric borates, especially those that have NLO properties, often show distinct anisotropies for each cation in comparison to centrosymmetric borates. All considered NLO borates contain BO₃ triangles, which are the principal cause of the strong anisotropy of the thermal expansion.

1. Introduction

In recent years, there has been extensive research and continuous development on second-order nonlinear optical (NLO) materials due to their potential applications. The revelation of the NLO effect on quartz crystals by Blombergen in 1962 [1] and the development of solid lasers in the early 1960s initiated huge progress in laser science and technology. In this context, the search for new NLO materials with optimized properties continues to be of special interest. Studies have shown that the second-order NLO properties of crystalline materials are closely related to their structures. Tens of inorganic and organic ([2,3,4,5], and Refs therein) crystals with NLO properties were identified, and their crystal structures, micro-structures, and various properties were studied. Nevertheless, there is growing interest in the search for new, prospective NLO materials due to the widening technical applications as a result of the replacement of gaseous and ionic laser sources with solid lasers. Basic principles of NLO material requirements, conditions of single crystal growing, studies of NLO properties, and applications are described in books, reviews, and multiple works [6,7,8,9].

There is a range of semi-empirical search criteria for NLO crystals. For example, the development of the Kurtz–Perry method [10] for instantaneous diagnostics of the second optical harmonic generation intensity considerably accelerated the screening for new NLO materials. Fast progress in the discovery of new NLO crystals was related with the works of Chen’s team who developed the theory of anionic groups [11,12]. According to this theory, a certain type of anionic groups makes, as a first approximation, the main contribution to the second harmonic generation, although the contributions of the cations cannot be neglected either, according to the authors. The analysis of NLO properties, in particular of the NLO susceptibilities, and of structural data makes some prediction possible in the search for prospective crystals in one or another class of chemical compounds [13].

After the discovery of the NLO effect of KB₅O₈·4H₂O by Dewey and co-authors [14], the search for new NLO materials has focused on borates, due to the possibility of their application in UV and deep-UV regions. It turned out that NLO borates show many other properties that are necessary for a significant second harmonic generation. Borates have a wide spectral range of transparency combined with a high laser damage threshold, as well as good chemical and mechanical stability [3,12,15]. These properties make borates crucial materials for the generation of the second optical harmonic in UV and deep-UV regions. The theory of anionic groups made a considerable contribution to the revelation of new NLO borates [11,12]. Using this approach, NLO borates with different anionic groups were found: β-BaB₂O₄ [16], LiB₃O₅ [17], CsB₃O₅ [18], Sr₂Be₂B₂O₇ [19], CsLiB₆O₁₀ [20], and K₂Al₂B₂O₇ [21,22]. Later on, α-BiB₃O₆ [23], BaBiBO₄ [24] and many other NLO crystals were discovered [12]. For this reason, the structural behaviour of anionic groups consisting of boron and oxygen atoms is discussed in this review in detail.

Since the 1930s, when the first borate crystal structures were determined at ambient conditions by Zachariasen, Goldschmidt, Hauptmann and others, more than 2500 (re-)determined crystal structures of hydrous and anhydrous borates have been listed in the ICSD Database (ICSD-2016) up to now [25]. Modern descriptors of borate rigid groups, fundamental building blocks (FBBs) and finite clusters were introduced in [26,27,28,29,30,31]. The nomenclature of crystal structures and several classifications of borates have been described in a large number of review papers ([26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43] and Refs therein). As a result, the basic crystal chemistry principles of borates were established: (1) Boron atoms do occur equiprobably in both triangular and tetrahedral coordination to oxygen atoms and hydroxyl groups in the structures of crystals and glasses. (2) The BO₃ triangles or/and the BO₄ tetrahedra are connected via common corners (oxygen atoms) to form rigid cyclic 3B-groups composed from three of such polyhedra; several such groups can also be linked via shared BO₄ tetrahedra, thus forming multiple cyclic rigid groups. BO₄ tetrahedra scarcely share edges. These ways of condensation lead to the formation of boron–oxygen entities that do not change significantly in various crystals and glasses. (3) The rigid groups or their combinations linked by shared oxygen atoms constitute the fundamental building blocks (FBB) of the structure.

Due to the increased interest in borates, the thermal behaviour of these materials is currently being intensively investigated. The knowledge temperature-dependent changes of solids are required for the further development of crystal chemistry, solid state physics and chemistry, especially for the synthesis of materials and their applications. Our group has researched many years to identify new borates and study their crystal structures and thermal behaviour. In particular, we studied thermal expansion in a wide range of temperatures of more than 70 borates using powder X-ray diffraction [42,43,44,45], etc. and of dozens of borates using single-crystal X-ray diffraction [46,47,48,49,50,51,52,53,54,55,56,57,58]. Basic principles of the high-temperature crystal chemistry of borates were developed [26,42,43,44]. It was shown that BO₃ triangles, BO₄ tetrahedra and multiple cyclic 3B-groups remained practically invariable over a wide range of temperatures. We demonstrate that most of the borates show very strong anisotropies of the thermal expansion, even, in some cases, negative linear thermal expansion in certain directions [26,42,43,44], etc.

In this work the temperature-dependent structure evolution is reviewed for several NLO borates [47,49,50,51,52,53,54,55,56,57]. For anharmonic approximation [47,50,54] and behaviour under high pressure [59], several NLO borates are given as examples. Furthermore, new data from single crystal and powder X-ray diffraction studies at low and high temperatures are presented for the first time: β-BaB₂O₄, LiB₃O₅ and K₂Al₂B₂O₇, measured within the intervals 98–693, 89–298 and 98–348 K, respectively. Finally, for the first time structural data on α-BaB₂O₄ at 295 and 673 K are discussed in an anisotropic approximation—unlike in the literature, where an isotropic approximation was chosen [60]. The main question is how the crystal structures of NLO borates change when the temperature increases. NLO borates studied in this work contain various rigid and non-rigid groups, and the dimensionality of the borate anions varies from 0D to 3D. Based on our observations, we derive trends for the high-temperature crystal chemistry of NLO borates.

Furthermore, we discuss the most eminent challenge within this research field—how to predict aprioristically the NLO properties of compounds. Accordingly, criteria are derived from crystal chemistry that allow for the identification of promising materials among a huge variety of compounds. We presume that the anisotropy and anharmonicity of thermal vibrations of atoms are very relevant, as well as the high anisotropy of the thermal expansion of crystals, amongst other reasons. Causes for the high anisotropy of NLO borates, as discussed in [44], are considered, too.

2. Thermal Evolution of the Structures of NLO Borates

Before discussing the main topic of this section, which is the structural variation of NLO borates that happens with temperature changes—we will give some general information about the crystal structures of borates under ambient conditions. This includes a brief description of the triangular BO₃ group and the tetrahedral BO₄ polyhedron in borates, the possibilities for polymerizing polyhedra and the formation of rigid and non-rigid B–O groups. Furthermore, we show their diversity, the notations and systematics, and we comment on how frequent polyanions of different dimensionality do occur.

2.1. Rigity and Flexibility of B–O Groups

B–O groups at ambient conditions. As mentioned in the introduction, borate anions are able to polymerize and form rigid groups. In the past, when more and more crystal structures were resolved, it became evident that rigid and almost uniform boron–oxygen groups are found in the structures of different crystalline and vitreous borates, as observed by Krogh-Moe [61,62]. Although the term “rigid boron–oxygen group” was introduced by Krogh-Moe half a century ago [61,62], the definition of the term has been given relatively recently [26,43]. Rigid groups are single polyhedra of BO₃ and BO₄ as well as cyclic triborate B–O rings composed of three polyhedra with common oxygen atoms at the corners and combinations of these 3B-rings. The latter may be connected through shared tetrahedra or common edges of the tetrahedra (at high-pressure synthesis [63,64,65]). Hence, these rigid groups can be considered as multiple (single, double, triple and so on) triborate (3B) groups, i.e., they are formed by condensation of single 3B-groups and share one or two common tetrahedra, independently of the multiplicity of the group. So, the simplest rigid group is a BO₃ triangle or a BO₄ tetrahedron like the TO₄ tetrahedron [66,67] in silicates, phosphates and others. Over the last few decades, high-density borates with unusual groups were synthesized by Huppertz and co-authors; their crystal structures show edge-sharing tetrahedra and were prepared by high pressure/high temperature synthesis [63,64,65].

If multiple rings are condensed via common oxygen atoms, they form flexible clusters. Such combinations are considered FBBs. “The FBB, by definition, should be the simplest unit that can reflect the basic structural information of an assigned crystallographic frame” [30]. The repeat unit of a borate can consist of one or several FBBs.

Notations of B–O groups. By the mid-1970s the main boron–oxygen groups containing one to six polyhedra were derived by Christ and Clark [68] and others. The history of the development of notations for B–O groups and FBBs was described in [26,28,29,30,31,43]. Boron–oxygen polyhedra start to polymerize when there is no lack of these polyhedra. Therefore, the anions of borates with a low content of B₂O₃ are mainly represented by isolated BO₃ triangles and BO₄ tetrahedra or non-cyclic diortho groups (dimers), e.g., two corner-sharing triangles in [B₂O₅] and two corner-sharing tetrahedra in [B₂O₇]. Isolated BO₃ and BO₄ polyhedra [30,35] are the most frequent anions. Starting from n = 3 (n is the number of boron atoms), the main groups appear to be cyclic.

Burns, Hawthorne and Grice [28,31] developed modern descriptors for the connectivity of borate polyhedra in every FBB after Christ and Clark [68], in particular in a ring. Borate polyhedra are written as ∆ and □ depending on the coordination of the boron atom by three or four oxygen atoms, respectively. The elementary groups consisting of isolated Bϕ₃ triangles and isolated Bϕ₄ tetrahedra (ϕ = O, OH) are denoted as 1B:1∆:∆ and 1B:1□:□ (1B:[(1:1T)], respectively (Figure 1, 1B:1∆:∆). The presence of more than one polyhedron in the descriptor indicates polymerization of the polyhedra by sharing corners; hence the group consisting of a triangle and a tetrahedron is written as 2B:1∆1□:∆□. The delimiters (< >) indicate that borate polyhedra form a single cyclic ring (Figure 1, 3B), and the signs ‘–’ or ‘=’ between two rings indicate that they share one or two tetrahedra, respectively (Figure 1, 4B, 5B). If an oxygen atom or any other anion (ϕ), polyhedron, or ring of polyhedra is bonded to more than two boron atoms, the ϕ is enclosed in the delimeters.

Symbols of the most frequently occurring triborate, tetraborate and pentaborate groups are 3B:3∆:<3∆>, 4B:2∆2□:<∆2□>=<∆2□> and 5B:2∆3□:<∆2□>–<∆2□>, respectively. According to the notation after Touboul, Penin, and Nowogrocki [29], these triborate, tetraborate and pentaborate groups are symbolized as 3:[3:(2∆ + T)], 4:[4:(2∆ + 2T)] and 5:[(3∆ + 2T)], respectively. On the basis of both of these approaches, modified descriptors for the description of rigid groups were worked out by Guan and Xue [30]. Modified descriptors after Burns, Grice and Hawthorne [28,31] are used in this review.

Non-cyclic groups (linear and branched complexes). The simplest non-cyclic groups are single polyhedra followed by groups of two and three polyhedra, called dimers and trimers. Their variety is caused by the substitution and rearrangement of tetrahedra and triangles (Figure 1, 1B). Single polyhedra and dimers are the most frequent.

Cyclic groups. Cyclic B–O groups are the result of polymerization and start to form at a considerable boron content above M_xO_y:B₂O₃ = 1 (modifying agent to boron oxide ratio). This means that they are forming from approximately the centre of the M_xO_y–B₂O₃ series for univalent and bivalent metals [34,43].

Double and multiple cyclic groups formed by condensation of triborate rings via shared polyhedra. Further condensation of single rings to form larger rigid B–O groups is realized via shared BO₄ tetrahedra. Several multiple groups are formed by condensation of single triborate groups consisting of three boron–oxygen polyhedra. In [28,31], all variants of polymers of boron–oxygen polyhedra (triangles and tetrahedra) that are possible were theoretically derived (n < 6, where n is the number of polyhedra in a group) (e.g., Figure 1).

Many of these groups were actually observed experimentally, some occurring often, some seldom. A unique example is the cyclic four-fold quad-tetraborate group 4B:{<3B>=<3B>}≡{<3B>=<3B>} (Figure 1, 4B:4□). It is formed by condensation of four triborate rings in a certain way: two groups are linked via two shared tetrahedra. This group of four BO₄ tetrahedra forming a large B₄O₁₀ tetrahedron was first derived theoretically by Bokii and Kravchenko in 1966 [69]. All boron atoms in this 4B-group are tetrahedrally coordinated because each tetrahedron belongs to two 3B-rings. Recently, this large B₄O₁₀ tetrahedron was experimentally found for the first time by Wu et al. in a borosilicate, Cs₂B₄SiO₉ (# 425583-ICSD), which has NLO properties in the deep-ultraviolet range [70]. In 2016, the 4B-group known as “supertetrahedron” was found in the new indium borate In₁₉B₃₄O₇₄(OH)₁₁, synthesized under high-pressure/high-temperature conditions by Huppertz et al. [71]. Furthermore, it is worth noting that In₁₉B₃₄O₇₄(OH)₁₁ is the first borate containing a supertetrahedron unit.

Non-rigid cyclic groups. Non-branched single rings consisting of a greater number of corner-sharing polyhedra (n > 3) seldom occur. When the number of corner-sharing polyhedra in a cyclic group is more than three, the group becomes non-rigid. A few structures are known that contain four polyhedra, mainly tetrahedra, for example finite 4B:4□:<4□> rings-clusters in Ca₄Mg(CO₃)₂[B₄O₆(OH)₆] borcarite (80438-ICSD), infinite 4B:2∆2□:∞¹<2∆2□>—chains in La(BO₂)₃ (# 23609-ICSD), 4B:4□:∞²<4□>—layers in CaAlB₃O₇ johadachidolite (# 10245-ICSD), and CuTm₂(B₂O₅)₂ (# 401327-ICSD). There exist six-membered rings of tetrahedra in the MAl₂(B₄O₁₀)O_0.5 family, for example NdAl_2.07(B₄O₁₀)O_0.6 (# 200666-ICSD) and Bi_0.96Al_2.37(B₄O₁₀)O (# 250428-ICSD), and eight-membered rings of alternative triangles and tetrahedra (Figure 1, 8B) in a layered polymorph of NLO α-BiB₃O₆ [72].

Occurrence of B–O polyanions of different dimensionality. Becker in 2001 as well as Guan and Xue in 2007 examined how often B–O polyanions of different dimensionality (the degree of polymerization—0D, 1D, 2D, 3D) do occur in 460 anhydrous [35] and 841 hydrous and anhydrous borates, respectively [30]. It was shown that the frequency of occurrence for borate anions in 460 anhydrous borates [35] is the following: isolated triangles and tetrahedra—52%, layers—15%, frameworks—12%, finite B–O groups—12%, infinite chains—9%. The distribution of boron–oxygen groups in 841 borates [30] is as follows: isolated anions—55%, chains—12%, layers—11%, frameworks—22.6%. The frequency of occurrence of different FBBs was also represented in [30] with differentiation between non-centrosymmetric and centrosymmetric hydrated and anhydrous borates.

Thermal invariability of rigid B–O groups. Another common rule for borates (not only for those with NLO properties) is the invariability of rigid groups under conditions of variable temperatures. During the last two decades, the high-temperature crystal chemistry of borates has been developed intensively. Earlier, the thermal invariability of SiO₄ tetrahedra was shown and systematic variations of individual Si–O bond lengths in a tetrahedron as a function of the ligand oxygen atoms (bridging or nonbridging) were analysed at different temperatures [66,67], etc. There are not only BO₃ and BO₄ polyhedra, but also rigid B–O groups built up from these polyhedral, that are invariable on heating at first approximation, e.g., they maintain their configuration and size with changing temperature, as it was assumed earlier [34]. The suggestion was based on the high bond valence of considerably covalent B³⁺–O bonds in BO₃ triangles (1 v. u. per each B–O bond in average) and BO₄ tetrahedra (3/4 v. u.). Our conclusion is also based on the invariability of rigid boron–oxygen groups in crystal structures of different borates.

The thermal invariability of rigid groups has been proven by single crystal HTXRD and LTXRD studies (HT, high temperature; LT low temperature) of about a dozen of borates [46,47,48,49,50,51,52,53,54,55,56,57,58], etc. These results have been summarized in [26,34,42,43] and are briefly described here. In this respect, the rigidity of a group means that the group cannot change its configuration as well as its size with varying temperature. It can be said that a rigid group is devoid of internal degrees of freedom. Therefore, borate structures constructed of rigid groups can react to temperature changes only by means of external degrees of freedom. An external degree of freedom of a rigid group is its ability to shift or to rotate as a unit relative to other groups around a mutual oxygen atom that serves as a hinge. Shear and hinge deformations are sharply anisotropic in nature [44].

Non-rigid cyclic groups. Corner-sharing BO₃ triangles and BO₄ tetrahedra can tilt and rotate around common oxygen atoms in a ring if the number of polyhedra exceeds three. These ways of sharing do not lead to rigid B–O groups and such “non-rigid” groups are not stable at high-temperature conditions. This can be illustrated by the examples of the 6B-ring in K₂Al₂B₂O₇ [21,22] and the 8B-ring in α-BiB₃O₆ [72], as described in the following paragraph.

2.2. Review of Temperature-Dependent Structural Studies of NLO Borates from Single-Crystal LTXRD and HTXRD Data

2.2.1. NLO Borates under Study

Here, we review the experimental results of single-crystal X-ray diffraction studies of NLO-borates in anisotropic and anharmonic approximation at low- and high-temperature conditions in the range of about 98 to 700 °C. To date, the thermal behaviour of the crystal structures of half a dozen NLO borates of different dimensionality from isolated BO₃ triangles and finite groups (0D) to double-framework (3D) were studied (

Table 1. List of low- and high-temperature single-crystal XRD experiments for NLO borates.

Formula, Space Group, Z	FBB	Temperatures	References
Rigid groups
BaBiBO4, Pnma, Z 4	1B:1Δ:Δ		[58]
β-BaB2O4, R3c, Z 6	3B:3Δ:<3Δ>	98, 123, 173, 223, 295, 323, 693 K	this work
		163, 293 K	[49]
α-BaB2O4, R–3c, Z 6	3B:3Δ:<3Δ>	295, 673 K	this work
LiB3O5, Pna21, Z 4	3B:2Δ1□:∞3<2Δ□>	98, 123, 148, 173, 198, 223, 248, 273, 298	this work
		293, 500, 650 K	[47]
CsLiB6O10, I–42d, Z 4	3B:2Δ1□:∞3<2Δ□>	173, 193, 203, 213, 243, 293 K	[55]
Li2B4O7, I41cd, Z 8	4B:2Δ2□:∞3<Δ2□>=<Δ2□>	293, 473, 673, 773 K	[50]
		123, 148, 173, 198, 223, 248, 298 K	[51]
		10–290 K with steps of 10 K	[52]
		3.4–300 K	[53] *
		293–1203 K with steps of 20 K	[54] *
Non-rigid groups
α-BiB3O6, C21, Z 2	8B:4Δ4□:∞2<Δ□Δ□Δ□Δ□>	100, 140, 160, 180, 240, 295 K	[57]
K2Al2B2O7 **, P321, Z 3	6B:3Δ3□:∞3<Δ□Δ□Δ□>	98, 123, 173, 223, 298, 348 K	this work

* Powder Rietveld refinement data; ** Aluminoborate K₂Al₂B₂O₇ consists of rings in which BO₃ triangles alternate AlO₄ tetrahedra.

). We also refined the structures of two NLO borates (LiB₃O₅ and Li₂B₄O₇) in anharmonic approximation, which had not been done for borates before [47,50]. Here these data are supplemented by new temperature-dependent structure data on the non-centrosymmetric β-BaB₂O₄ polymorph in comparison to centrosymmetric α-BaB₂O₄, recorded over a wide temperature range, and data for K₂Al₂B₂O₇ and LiB₃O₅ at low temperatures. Moreover, for the first time the crystal structure of α-BaB₂O₄ is refined in anisotropic approximation in addition to the isotropic one [60]. Structures of β-BaB₂O₄, K₂Al₂B₂O₇ and LiB₃O₅ are refined in anharmonic approximation, too. Temperature-dependent structure data and results of the thermal expansion tensor studied by the means of high-temperature powder X-ray diffraction are also discussed in parallel in Section 3.4 and Section 3.5 (see

Table 2. Main characteristics of the thermal expansion of borates.

Compound	System, Space Group	α × 106 K−1					Δt, °C	Refs
		α11	α22	α33	αV	αmax − αmin
Li–borates
Li2B4O7 *	Tetrag., I41cd	17	17	−13	21	30	−189–27	[51]
		16	16	−4	28	20	20–250	[50]
		16	16	12	44	4	250–750
LiCsB6O10 *	Tetrag., I–42d	20	20	−22	18	42	25–600	[55]
Li3B11O18	Monocl., P21/a	14	38	−16	36	54	20–180	[78]
LiB3O5 *	Orth., Pna21	101	31	−71	61	172	25–530	[47]
		66	29	−63	32	129	25–790	[79]
		108	34	−88	54	196	25–790	[80]
Na–borates
Na2B4O5(OH)4·3H2O	Trigon., R32	14	14	11	39	3	0–80	[81]
γ–Na2B4O7	Tricl., P–1	16	12	4	32	12	20–550	[56]
BaNaSc(BO3)2	Trigon., R–3	8	8	21	38	13	20–550	[82]
α–Na2B4O7	Tricl., P–1	25	20	9	54	16	20–700	[83]
BaNaY(BO3)2	Trigon., R–3	6	6	26	38	20	25	[82]
		19	19	11	49	8	300
Na3(NO3)B6O10	Orth., Pnma	8	9	39	56	31	20–700	[26]
α-Na2B8O13	Monocl., P21/a	34	11	1	46	33	300	[46]
β-Na2B8O13	Monocl., P21/c	15	38	−1	52	39	20–600	[26]
β-NaB3O5	Monocl., P21/c	33	−7	6	32	40	20–700	[83]
Na2B4O6(OH)2·3H2O	Monocl., P21/c	43	24	0	67	43	0–80	[81]
(K0.5Na0.5)3B9O15	Monocl., P21/c	49	5	7	61	44	20–700	[84]
Na2B4O5(OH)4·8H2O	Monocl., P2/c	77	−2	19	80	79	4–30	[81]
K–borates
K2Al2B2O7 *	Trigon., P32	8	8	17	33	9	30–295	[85]
K5B19O31	Monocl., P2/c	18	3	3	24	15	20–700	[34]
α-KB5O8	Orth., Pbca	12	12	−4	20	16	20–370	[86]
		12	12	5	29	7	370–550
KB3O5	Monocl., P21/c	37	3	6	46	34	20–650	[87]
K[B5O6(OH)4]·2H2O *	Orth., Aba2	7	18	44	69	37	20–115	[88]
K2NaB9O15	Monocl., P21/c	52	3	0	55	52	20–650	[84]
β-KB5O8	Orth., Pbca	60	20	−3	77	63	200–700	[34]
Rb–borates
Rb2B4O7	Tricl., P–1	20	25	12	57	13	20–700	[89]
Rb5B19O31	Monocl., P2/c	27	3	11	41	24	20–600	[90]
α-RbB5O8	Orth., Pbca	5	10	−14	1	24	150–300	[86]
		5	10	30	45	25	300–500
Rb3B7O12	Tricl., P–1	54	9	2	65	52	20–600	[91]
β-RbB5O8	Orth., Pbca	61	23	5	89	56	20–720	[92]
RbB5O6(OH)4·2H2O *	Orth., Aba2	77	−20	21	78	97	20–100	[88]
α-RbB3O5	Orth., P212121	29	−27	74	76	101	20–600	[93]
β-RbB3O5	Orth., P212121	18	−11	89	96	100	20–700	[93]
Rb0.9Cs0.1B3O5	Orth., P212121	24	−40	73	57	113	20–700	[94]
Cs–borates
CsB3O5 *	Orth., P212121	23	11	48	82	37	20–800	[34]
β-CsB5O8	Orth., Pbca	53	16	14	83	39	20–540	[95]
α-CsB5O8	Monocl. P21/c	27	61	−8	80	69	20–600	[48]
CsB5O6(OH)4·2H2O	Monocl. A2/a	83	18	4	100	79	20–95	[88]
NH4–borates
(NH4)3[B15O20(OH)8]·4H2O	Monocl. C2/c	28	41	18	87	23	10–80	[96]
NH4B5O8	Orth., Pbca	39	6	20	65	33	20–330	[97]
NH4[B5O7(OH)2]·H2O	Monocl. P21/c	32	53	−3	82	56	20–90	[98]
Ca–borates
CaMg[B3O4(OH)3]2·3H2O	Monocl. P2/c	16	18	10	44	8	20–270	[99]
Ca[B3O4(OH)3]·H2O	Monocl. P2/a	29	29	−11	47	40	20–300	[100]
Sr–borates
SrB4O7	Orth., Pmn21	7	9	8	24	2	20–900	[101]
Sr2B16O26	Monocl. P2/c	21	10	4	35	17	20–740	[101]
γ-Sr2B2O5	Monocl. P2/c	20	7	1	28	19	20–292	[102]
SrB2O4	Orth., Pbcn	4	4	33	41	29	20–900	[101]
Sr3B2O6	Trigon., R–3c	10.5	10.5	44	65	33.5	20–900	[101]
Ba–borates
LiBaB9O15	Trigon., R–3c	7	7	−5	9	12	20–700	[103]
α-BaB2O4	Hex., R–3c	6	6	28	40	22	20–700	Here
BaB4O7	Monocl., P21/c	23	−12	5	16	35	20–700	[104]
β-BaB2O4 *	Trigon., R3c	3	3	45	51	42	20–700	[104]
Bi–borates
Bi24B2O39	Cub., I23	17	17	17	51	0	20–600	[43]
Ba3Bi2(BO3)4	Orth., Pnma	16	11	11	38	5	25	[105]
		30	12	10	52	20	500
Ba2Bi3B25O44	Trigon., R–3m	12	12	6	30	6	25	[106]
		12	12	0	24	12	700
Bi3B5O12	Orth., Pnma	12	12	3	27	9	20–700	[107]
SrBi2B4O10	Tricl., P–1	13	9	2	24	11	20–700	[108]
Bi4B2O9	Monocl., P21/c	20	15	6	41	14	20–500	[109]
BaBiBO4	Orth., Pnma	28	6	10	22	44	25	[58]
		42	−3	12	51	45	700
BaBi2B2O7	Hex., P63	6	6	20	32	14	25	[110]
		8	8	34	50	26	625
BaBi2B4O10	Monocl., P21/c	13	11	−3	21	16	20	[111]
		13	11	4	28	9	150
		31	11	9	51	22	600
Sr0.5Ba0.5Bi2B2O7	Hex., P63	3	3	22	29	19	25	[110]
		9	9	33	50	24	625
SrBi2B2O7	Hex., P63	4	4	22	30	18	25	[110]
		8	8	30	46	22	625
α-Bi2B8O15	Monocl., P21	2	49	−8	43	57	20–300	[112]
		8	30	2	40	28	300–700
α-BiB3O6 *	Monocl., C2	−28	54	8	34	82	−200–300	[113]
		−25	54	10	39	79	−253–525	[57]
REE–borates
π-NdBO3	Hex., P63/mmc	15	15	13	43	2	20	[114]
		14	14	2	30	12	540
λ-NdBO3	Orth. Pnma	22	15	17	54	7	20	[115]
		21	3	7	37	18	820
π-LuBO3	Monocl., C2/c	9	9	2	20	7	20	[114]
		8	8	1	17	7	540
β-LuBO3	Trigon., R–3c	2	2	13	17	11	20–600	[114]
Mixed borates
LuBa3B9O18	Hex., P63/m	3	3	39	39	36	20–900	[116]
Fe3BO6	Orth. Pnma	14	12	2	28	12	20	[117]
		11	9	10	29	2	800
KZnB3O6	Tricl., P–1	−1	1	45	45	46	100–740	[118]
Zn4B6O13	Cub., I–43m	0.3	0.3	0.3	1	0	−260–−163	[119]
		1	1	1	3	0	−163–−3
LiBeBO3	Tricl., P–1	−3.3	1.8	7.6	6.1	10.9	−200–−80	[120]
GdCa4O(BO3)3 *	Monocl., Cm	12 **	5 **	6 **	23 **		23–300	[12]
YCa4O(BO3)3 *	Monocl., Cm	13	4	25	43	21	27	[121]
		13	5	30	48	25	100
La2CaB10O19 *	Monocl., C2	9 **	8	2 **	19 **		25–300	[122]
Na3La9O3(BO3)8 *	Hex., P–62m	8	8	15	31	7	30–500	[123]

Non-centrosymmetric borates are marked by bold. * NLO borates; ** Thermal expansion coefficients are given along the a-, b-, and c-axes (eigenvalues of thermal expansion tensor were not calculated for monoclinic crystals).

and comments).

For NLO borates like K₂Al₂B₂O₇, β-BaB₂O₄, LiB₃O₅, CsLiB₆O₁₀, and Li₂B₄O₇, thermal invariance of 1B-, 3B- and 4B-groups was observed, whilst for K₂Al₂B₂O₇ and α-BiB₃O₆ the thermal flexibility of non-rigid 6B- and 8B-groups was shown. Because NLO borates containing 5B-groups have not yet been studied at elevated temperatures, we discuss the data for 5B-groups based on the structures of α-Na₂B₈O₁₃ [46] and α-CsB₅O₈ [48]. First, we consider systematic variations in B–O bond lengths with temperature and then the temperature-dependent structural behaviour of rigid groups and cations in NLO borates. We place an emphasis on the new data on NLO borates (Table 1 and Section 4).

2.2.2. Systematic Variations with Temperature of Boron–Oxygen Bond Lengths in NLO Borates

Variations of B–O bond lengths at ambient conditions. The distribution of boron–oxygen bond lengths and angles is of special interest due to capacity of boron atoms to be in triangular and tetrahedral coordination by oxygen. Average B–O bond lengths for triangles and tetrahedra are equal to 1.370 and 1.476 Å [31], respectively. Systematic variations in individual bond lengths were examined at ambient conditions by Filatov and Bubnova [26,34]. In contrast to SiO₄ tetrahedra [66,67] where the oxygen atoms can be bridging and non-bridging only, borate groups show additionally bridging oxygen atoms in different environments for boron atoms in triangular and tetrahedral coordination: (1) the ^∆O^□ oxygen atom links tetrahedral and triangular boron polyhedral; (2) the ^∆O^∆ atom links a triangle with a triangle; (3) the ^□O^□ atom links a tetrahedron with a tetrahedron; (4) and finally non-bridging (O^∆ and O^□) oxygen atoms can exist in both polyhedra. Thus, there exist systematic differences between the individual bond lengths within a group according to the second Pauling’s rule: (a) for the bridging B–O bonds (<B^□–^□O^Δ> > <B^□–^□O^□> > <B^Δ–^ΔO^Δ> > <B^Δ–^ΔO^□>) and (b) for terminating B–O bonds (<B^Δ–^ΔO^Δ> > <B^Δ–O^Δ> and <B^□–^□O^Δ> > <B^□–O^□>). When the temperature increases, these trends remain [26].

Thermal variations of B–O bond lengths. Like silicates [73], the borates under study (see Section 4) show a weak contraction of the majority of individual and average strong and short B–O bond lengths when temperature changes, as it was observed in previously studied α-Na₂B₈O₁₃ [46], LiB₃O₅ (20–400) [47], α-CsB₅O₈ [48], and Li₂B₄O₇ [50,51]. The results of these studies were summarized in [26].

This contraction is an artefact of the X-ray diffraction method. It is caused by atomic thermal vibrations. When the bonds are strong in a group of atoms like TO₃ and TO₄ polyhedra, the entire group may undergo oscillation. This was called rigid body motion by Cruickshank [74]. The motion of each atom of the group is caused by the motion of the group. The effect of the thermal motion on bond lengths changes was first estimated from diffraction measurements by Busing and Levy [75]. Rigid body criteria were summarized by Downs ([73,76,77], and Refs therein). The atomic coordinates resulting from a single-crystal X-ray diffraction experiment are the maxima or the centroids of the electron density arising from the combined effects of atomic positions and thermal displacements. When temperature increases, the bond lengths change due to thermal atomic vibrations. The contraction can be described by the model of rigid-body motion [73,74,75,76,77]. Bond lengths are corrected according to the model of rigid-body motion using the formula given in [73,76]:

$$R_{corr}^2=R_{obs}^2+\frac{3}{8{\pi }^2}(B_{eq}(A)-B_{eq}(C)),$$

(1)

where R_corr and R_obs are corrected and observed B–O bond lengths, correspondingly; B_eq(A) and B_eq(C) are equivalent displacement parameters of the anion (oxygen) and the cation (boron, aluminium), correspondingly.

After applying this correction, the mean B–O bond lengths tend to increase slightly upon heating. This approach was described for borates in [26], and is applied here for the NLO borates under study (Figure 2). Individual B–O bond lengths are given as a function of temperature for β-BaB₂O₄ (a) and LiB₃O₅ (c). For K₂Al₂B₂O₇, Al–O bond lengths are shown in Figure 2b. In a temperature range of 23 to 1000 °C a mean thermal expansion for Al–O bond lengths in AlO₄ tetrahedra of zero was described by Hazen, Prewitt and Finger [66,67]. In the present study negative thermal expansion is found for the Al1–O1 and Al2–O2 bond lengths of AlO₄ tetrahedra in the range of 100–400 K (Figure 2b).

2.2.3. Thermal Invariability of Rigid B–O Groups in NLO Borates

The structures of the vast majority of existing NLO borates are based upon isolated BO₃ triangles. Those with single cyclic 3B-, double cyclic 4B- and 5B-groups of different dimensionality occur frequently in NLO borates; other groups occur occasionally. The main groups that occur in NLO borates are represented in Figure 1. Hence, at first we will consider thermal vibrations of the single BO₃ triangles. As an example, we discuss the Ca₄REEO(BO₃)₃ family [12] which is related to NLO borates based on isolated BO₃ triangles where REE = Gd (# 39716-ICSD) or Y, as well as K₂Al₂B₂O₇ [21] and BaBiBO₄ [24]. We have investigated K₂Al₂B₂O₇ (see Section 4.3) and BaBiBO₄ [58]. Nevertheless, the description of these borates is not uniquely defined: K₂Al₂B₂O₇ can be considered also as boroaluminate framework composed of non-rigid six-membered rings in which BO₃ triangles alternate with AlO₄ tetrahedra (see Section 4.3), and BaBiBO₄ can also be described as being built up from rigid borate-bismuthate chains. There are inconsistent data on the symmetry and NLO properties of BaBiBO₄: it was firstly determined as crystallizing in the non-centrosymmetric space group Pna2₁ with NLO properties (# 154105-ICSD, [24]), whilst later it was refined in space group Pnma (# 9424596-ICSD, [58]). In any case, in both borates, K₂Al₂B₂O₇ (see Section 4.3) and BaBiBO₄ [58], the BO₃ triangles exhibit thermal invariability (see Figure S1c).

There are four triborate rings that can be derived by permuting triangles for tetrahedra: (1) a ring of three triangles, 3B:<3∆>; (2) a ring of two triangles and a tetrahedron, 3B:<2∆□>, (3) a ring of a triangle and two tetrahedra, 3B:<∆2□>; and (4) a ring of three tetrahedra, 3B:<3□>. Two of them, <3∆> and especially <2∆□> (see Figure 1, 3B) occur frequently in NLO borates. All triborates of univalent metals except for the two modifications of α- and β-NaB₃O₅ are also built up from <2∆□> 3B-groups. These are the topologically identical three-dimensional frameworks of LiB₃O₅ (# 1585-, 66708-ICSD), CsB₃O₅ (# 2081-ICSD), CsLiB₆O₁₀ (# 80826-ICSD), TlB₃O₅ (#84855-ICSD), α- and β-RbB₃O₅ (# 91545, 87519-ICSD)-, and also another framework of KB₃O₅ (250224-ICSD). Most of them are borates with excellent NLO properties.

Among 4B-groups the 4B:<∆2□>=<∆2□> double ring (see Figure 1, 4B) is common in NLO borates (Li₂B₄O₇ (23876-, 34670-, 300010-, 65930-ICSD), LiNaB₄O₇ (186901-ICSD) etc. A unique four-fold cyclic 4B-group is formed by a condensation of four 3B-rings 4B:{<3B>=<3B>}≡{<3B>=<3B>} (see Figure 1, 4B:4□). It occurs only in the borosilicate Cs₂B₄SiO₉ (# 425583-ICSD) which shows NLO properties in the deep-ultraviolet range [70].

Double 5B-groups <3B>–<3B> composed of two 3B-rings via a common tetrahedron (see Figure 1, 5B) occur quite frequently for instance 5B:4∆1□:<2∆□>–<2∆□> in K[B₅O₆(OH)₄]·2H₂O (18211-ICSD), 5B:3∆2□:<2∆□>–<2∆□> in K₂SrVB₅O₁₂ (185934-ICSD), 5B:2∆3□:<2∆□>–<2∆□> in La₂CaB₁₀O₁₉ (92866-ICSD).

The following groups occasionally occur in NLO borates: 2B:2∆:∆∆ in Pb₄O(BO₃)₂ (261420-ICSD), 6B:3∆3□:[O]<∆2□>|<∆2□>|<∆2□>| in K₃B₆O₁₀Cl (262005-ICSD), 7B:5∆2□:<2∆□>–<∆2□>–<2∆□> in Li₄Cs₃B₇O₁₄ (261420-ICSD) (see Figure 1, 7B). FBBs composed of two rigid groups are rare but do occur, for instance in LiKB₄O₇ (93601-ICSD) and LiRbB₄O₇ (93602-ICSD).

Thermal variations of B–O–B angles. As defined in Section 2.1 a rigid group is devoid of internal degrees of freedom whilst the external degrees of freedom of a rigid group change as impact on temperature. In other words, angles between polyhedra have to be almost invariable within the group and changeable between groups with temperature varying. This will now be discussed for different rigid groups:

In isolated triborate <3∆>3B-groups (β-BaB₂O₄) the average change in the B–O–B angles equals 0.6° in the range of 98 to 693 K (Section 4.1).

Tetraborate groups (Li₂B₄O₇ [50,51]) have B–O–B angles in the triangles and tetrahedra forming the rigid group that change by 0.1° whilst the angles between the rigid groups change by 1.6° on average.

The pentaborate group (α-CsB₅O₈ [48] and α-Na₂B₈O₁₃ [46]) contains six independent B–O–B angles within the group. Among them, the mean changes of the values of the angles do not exceed 0.5°. However B–O–B angles between the groups increase by 1.9°–2.3°.

Summary. The conclusion can be drawn that—at least in the temperature range of 20 to 500 °C—the variations of B–O–B angles within rigid groups are usually equal or less than 0.5°. This value is comparable with the error bars. For the same temperatures, the angles between rigid groups vary over a wider range (about 2°). The B–O bond lengths are almost invariable within accuracy of experiment (Section 2.2.2). Thus the thermal evolution of rigid groups in NLO borates shows that they are thermally invariable ([26,34]; see Section 4.1, Section 4.2 and Section 4.3).

2.2.4. Non-Rigid Cyclic Groups in NLO Borates

As discussed in Section 4.3 when analysing K₂Al₂B₂O₇, AlO₄ tetrahedra are located in a flexible six-fold ring that is composed of alternating BO₃ triangles and AlO₄ tetrahedra (see Figure S1d). The T–O bond lengths (in this case T = Al) practically do not change with temperature both in rigid and non-rigid groups because the TO₄ tetrahedron is a trivial rigid group. As for O–O–O angles, they change both inside the six-fold ring and outside of it (see Section 3.6), and the changes can reach 2°.

Another flexible ring consisting from eight alternating BO₃ triangles and BO₄ tetrahedra is present in the layered polyanion of the NLO borate α-BiB₃O₆ [72] (see Figure 1, 8B). Here, the angles change by more than 0.5° in a relatively small temperature range of 200 K (100–295 K) (see Section 3.6), and it reaches a more considerable value of 1.5° after increasing the range to 400°.

Interatomic O–O distances in BO₃ triangles and BO₄ tetrahedra of the eight-fold ring practically do not change with temperature. A value of zero given for the O–O edge of the polyhedron means that in the mentioned range of temperatures the changes do not exceed the measurement error of 0.5°.

Summary. One can conclude that, unlike in the rigid groups, the angles within the non-rigid B–O groups change with temperature. This holds for angles between polyhedra both inside the ring and outside. Changes of the interatomic O–O distances in the polyhedra are within the limits of 0.02 and 0.04 Å for both rigid and non-rigid groups.

2.3. Anisotropic Thermal Expansion of Borates from Powder HTXRD Data

In parallel to temperature-dependent structural studies the thermal expansion of more than 70 borates including many NLO borates has been examined by high-temperature X-ray powder diffraction (in air). The results have been published before in [26,34,42,43], etc. and are summarized in Table 2. They are also supplemented by more recent data obtained after 2013, also given in Table 2. Here, except for the formula of the compound, its symmetry, the studied temperature range (∆t) and the reference of the source, the tensor of the thermal expansion coefficients α₁₁, α₂₂, α₃₃ is given, the expansion coefficient α_V of the volume and the difference ∆ = α_max − α_min. The borates are assigned to groups according to the cation. There are more than two dozen non-centrosymmetric borates that are marked in bold in Table 2, and nearly half of them generate a second harmonic (marked by asterisks). A more detailed data analysis of Table 2 is given later on (Section 3).

3. Discussion. Desirable Crystal Chemical Criteria for NLO Borates

Here we consider desirable crystal chemical criteria for NLO crystals based on experimental data revealed in the present study. Besides the manifestation of the anisotropy of the optical properties (strong birefringence), we propose to analyse a few more possible conditions.

As discussed by us for the first time [47], “nonlinear optical effects might be caused by the anharmonicity of atomic thermal displacements that can be seen in thermal expansion tensor and in deviations from Gaussian shape of the probability density function of atomic thermal displacement factors”. Particularly, the thermal expansion tensor coefficients analysis (see Table 2) of two dozens of non-centrosymmetric borates versus centrosymmetric ones showed that most of the NLO borates expand strongly anisotropic.

3.1. Anharmonicity of Atomic Vibrations in NLO Materials

Up to date, there are only a few structures of nonlinear borates refined in the anharmonic approximation, i.e., LiB₃O₅ [47], Li₂B₄O₇ [50,54] and β-BaB₂O₄ (Table S4). It has been found that the atomic vibrations of Li atoms are asymmetrical more strongly than those of Ba atoms (Figure 3). At the same time, the anharmonic parameters for B and O atoms are of the same order as their estimated standard deviations (ESDs). Anharmonic parameters for Ba atoms exceed their error bars whilst B and O atoms vibrate practically anisotropically. The strongest anharmonicity of vibrations of Li atoms is found in LiB₃O₅. The thermal vibrations figure of the Li atom is oliform (Figure 3a), narrower on one side and wider on the other. The anharmonic vibrations of the Li atom are weaker in Li₂B₄O₇ (Figure 3b), and the vibrations of the Ba atoms in β-BaB₂O₄ are the weakest (Figure 3c). It is remarkable that the NLO properties are also expressed weaker in Li₂B₄O₇ than in LiB₃O₅ [12,15], etc. This can be caused by the stronger anharmonicity of the Li atoms’ vibrations. The anharmonicity increases sharply with temperature in LiB₃O₅ [47] and in Li₂B₄O₇ [50,54], especially close to the melting point, as found by Senyshyn and co-authors [54].

In general, the character of thermal vibrations of the atom is largely determined by its environment. Anharmonic vibrations may indicate a sharp asymmetry of the atom environment. According to [51,54], the Li atom, being in the channel of the double boron–oxygen framework of Li₂B₄O₇ borate, preferably oscillates along this channel. It gives another possible structural criterion of NLO materials—irregular coordination of the atoms.

3.2. High Mobility of Cations in NLO Borates

Thermal decreasing of cation coordination. The Li coordination at room temperature is described in [124] as a distorted tetrahedron where the Li atom is displaced from the centre towards the O2O3O4 plane (Figure 3a and Figure 4a). Thus, Li atoms are coordinated by four oxygen atoms with the Li–O bond lengths being 1.979–2.180 Å [47]; the next oxygen O5ⁱⁱⁱ atom is at a distance of 2.684 Å from Li. The distorted LiO₄ tetrahedron (Figure 3a) at low (see Section 4.2) and room temperature transforms into a triangle at 377 °C [47]. Three bond lengths (Li–O2, Li–O3 and Li–O4) are very close to each other (~2.0 Å) and remain practically the same (Figure 4a). The fourth Li–O5 bond length increases by 0.3 Å with increasing temperature and the fifth Li–O5ⁱⁱⁱ bond decreases by 0.25 Å and becomes almost to the same as the fourth one. Hence, the Li atom could be considered to be coordinated by three oxygen atoms at high temperature (Figure 3a and Figure 4a).

A similar situation occurs in Li₂B₄O₇: the LiO₄ tetrahedron at low temperatures [47] becomes an LiO₃ triangle at high temperatures (Figure 3b and Figure 4b), although screw chains of LiO₅ polyhedra remain at a wide range of temperatures (Figure 5b). In the case of heavier and higher charges of Ba²⁺ in β-BaB₂O₄, the trend of decreasing of cation coordination number remains, although it is less distinctive (see Section 4.1).

Cations vibrations increasing. The isotropic displacements B_eq of the Li atoms increase more than four times upon heating to 377 °C. The anisotropy of Li thermal vibrations also increases. The elongation of the ellipsoid of thermal vibrations and the anharmonic figure lie near the ac plane and approximately under angles equal to a and c axis (Figure 3a and Figure 5a). Li⁺ cation thermal vibrations in Li₂B₄O₇ rapidly increase on heating to 773 K, the isotropic parameter B_eq(Li) increases by a factor of 2.5 [50,51]. The isotropic displacements U_eq of the Ba atom in β-BaB₂O₄ from 98 K to 650 K increases 5-fold (Table S16).

Thermal shifting of cations. As coordination number decreases under heating shifting of cations is observed [47,50,51]. For instance, in LiB₃O₅ the vibrational-active Li atom is displaced by 0.31 Å in the direction of the O2–O3–O4 plane upon heating from low temperatures to 377 °C (other atomic shifts are 0.04 Å on the average). In β-BaB₂O₄, the heavier Ba atom is moved towards screw 3₁ axis less significantly (see Section 4.1).

In LiB₃O₅ LiO₅ polyhedra linked by O5 atoms form chains within the channels, alternating with the chains of the triborate groups (see Section 4.2). At 500 K, Li moves 0.25 Å towards the O2O3O4 plane, and its coordination becomes triangular (d(Li–O) = 1.968–2.003 Å). The fourth O5ⁱⁱⁱ oxygen that contributes to the tetrahedron at 20 °C moves away from Li (at a distance of 2.381 Å) and the fifth O5ⁱ oxygen is 2.6 Å away from Li (Figure 3a and Figure 4a). At 650 K Li is clearly coordinated by three oxygen atoms, forming a planar LiO₃ triangle in the first coordination sphere, with two additional oxygen atoms on both sides of this triangle (second coordination sphere).

In Li₂B₄O₇ the Li atoms shift maximally (0.007 Å) along the c axis. The elongation of Li–O1 bonds by 0.115 Å leads to a removal of O1 from the first coordination sphere, whereas other bonds elongate by 0.006, 0.022, and 0.032 Å as shown by arrows in Figure 3b. The shift of the O1 atom leads to a highly irregular LiO₄ tetrahedron at high temperatures, almost a triangle like in LiB₃O₅.

Structural mechanism of thermal expansion. Lithium triborate, LiB₃O₅, demonstrates the maximum anisotropy of thermal expansion among all of the borates: α_a = 101, α_b = 31, α_c = −71 × 10⁻⁶ K⁻¹. It can be seen that the difference of maximum and minimum values of the thermal expansion coefficients is 172 units [47]. At the same time, the thermal expansion coefficient of the volume (α_V = 60 × 10⁻⁶ K⁻¹) equals the average for alkali metal borates.

The thermal expansion of the NLO borate Li₂B₄O₇ that contains tetraborate groups (see Table 2) is maximal in the ab plane (α_a = 16 × 10⁻⁶ K⁻¹). Along the main axis, the c parameter shows compression (α_c < 0) in negative and low temperature ranges (−180–250 °C) and expansion (α_c > 0) at intermediate and high temperatures (250–750 °C) with the minimum (α_c = 0) at about 250 °C. Here, we will try to explain the main features of this enigmatic thermal expansion. Structural studies [50,51] of Li₂B₄O₇ in a temperature range of −150 to 500 °C revealed considerable changes of the B1–O1–B2 angle between 4B-groups. They amount to about +2°, in contrast to the stability (0.1°) of the angles within the rigid 4B-groups.

As in the case of the unique framework of the triborate LiB₃O₅, the real reason for thermal deformations of the double interpenetrating boron–oxygen framework in Li₂B₄O₇ is the character of the thermal vibrations of the Li⁺ cation. It is notable that in NLO Li borates the screw chains of LiO-polyhedra remain after conversion of LiO₄ into LiO₃ triangle [47,51]. It is notable that the screw chains of the LiO-polyhedra remain existent through the transformation of the LiO₄ tetrahedron into a LiO₃ triangle, as it is shown in the anharmonic approximation in LiB₃O₅ [47,51] and in Li₂B₄O₇ [51,54] (Figure 5).

The second-order NLO properties of LiB₃O₅, CsLiB₆O₁₀, and CsB₃O₅ borates, which all contain a topologically identical framework composed of 3B-groups, have been quantitatively studied from the viewpoint of the chemical bond [125]. It is noted that (i) differences in the NLO properties within this group of borates arise from the contributions of the different cations due to a changing coordination environment, i.e., the different interaction between the cation and the rigid anionic group; (ii) the [B₃O₇]⁵⁻ group is very important for borate crystals. Heavier cations decrease the covalence values for the bonds in the anionic group and increase the susceptibility of the cation–oxygen bond. The latter, especially, leads to an increase in the resulting macroscopic susceptibility induced by the heavier cation. The present work shows that heavier cations strengthen the interaction between the cation and the [B₃O₇]⁵⁻ anion.

3.3. Self-Assembly of Rigid Groups into NLO Borates

Anisotropic thermal vibrations of atoms could play a key role in the anisotropy of thermal expansion and in the self-assembly of rigid groups into crystal structure [26,43,45]. Here, we describe the self-assembly of BO₃ triangular radicals and mainly rigid groups containing BO₃ triangles into ordered crystal structures from the viewpoint of atomic thermal motion. The organizing force of self-assembly in borates is strong anisotropy of the thermal atomic vibrations in the BO₃ triangles, flat 3B- and other B–O rigid groups containing BO₃ triangles.

In the case of a BO₃ triangle (see Figure 1, 1B), O and B atoms oscillate mainly perpendicular to the strong B–O bond. There are three B–O bonds in the BO₃ triangle, thus B and O atoms maximally vibrate preferably perpendicular to the plane of an isolated BO₃ triangle (see Figure 1, 1B). The triangle as a whole has to vibrate to the direction perpendicular plane of the triangle. These triangles and rigid groups, due to the sharp anisotropy of thermal vibrations of B and O atoms, tend to be arranged in the parallel (or preferable) orientation. As a general trend [26,43,45], for NLO borates the self-assembly of rigid B–O groups containing triborate rings is characteristic in the process of crystal growth. Self-assembly is manifested in a parallel or preferred orientation of BO₃ triangles and 3B-rings. This is the cause of the high anisotropic thermal expansion of borate, which may contribute to the appearance of NLO properties in borates (see Table 2). Since most borates based upon asymmetric rigid BO₃ groups exhibit excellent NLO properties this has motivated in finding new NLO materials with other rigid TO₃ groups, such as carbonates and nitrates. In the most structures based upon isolated TO₃ triangles (T = B, C, N), TO₃ groups are represented in preferred orientation parallel to each other and the most of them show strong anisotropic thermal expansion [45]. Recently, for this reason carbonates have been widely studied, and a series of the materials have been discovered with proper NLO properties ([126], and Refs therein).

Most impressively, it is manifested in the case of a 3B-group composed of three triangles: internal B and O atoms vibrate maximally perpendicular to the plane of the triborate group (Figure 6a), and the whole group vibrates in the same direction. Hence a structure with isolated triborate groups has to expand considerably perpendicular to the plane of a group and expand weakly in parallel to the plane of a group (Figure 6c).

As an example, we can refer to the triborate group composed of three BO₃ triangles in comparison to the pole figure of thermal expansion of the crystal structure of β-BaB₂O₄ (Figure 6c), a crystal structure that is based on isolated triborate groups of three triangles 3B:<3∆> (see Figure 1, n = 3). For β-BaB₂O₄ we also investigated the refractive indices (Figure 6b): N_g and N_p directions coincide to α_min and α_max [104], respectively. As seen from Figure 6 and other examples in [20], isolated triborate groups are usually arranged in parallel to each other to form 0D-3B structures, as seen in α- and β-BaB₂O₄ consisting of <3∆> groups.

3.4. Strong Anisotropy of the Thermal Expansion in NLO Borates

For most borates a high anisotropy of the thermal expansion is characteristic, as has been found in [26,34,42]. It can be analysed to serve as an additional criterion for NLO borates, like high values of n_g − n_p birefringence [12,15], etc. Similarly, we evaluated the degree of the anisotropy of thermal expansion by a value Δ = α_max − α_min (Table 2). Non-centrosymmetric as well as NLO borates often show stronger anisotropy of thermal expansion in comparison to centrosymmetric borates of the same group. For example, among Ba borates β-BaB₂O₄ (Δ = 42 × 10⁻⁶ K⁻¹) is featured with the most anisotropy, among bismuth borates—α-BiB₃O₆ (Δ = 82 × 10⁻⁶ K⁻¹), and among lithium borates—LiB₃O₅ (Δ = 172 × 10⁻⁶ K⁻¹) is the evident record-breaker. However, there are exceptions as for any semi-empirical rule. So, among K-borates the centrosymmetric polymorph β-KB₅O₈ (Δ = 63 × 10⁻⁶ K⁻¹) shows the highest anisotropy whilst NLO K[B₅O₆(OH)₄]·2H₂O (Δ = 37 × 10⁻⁶ K⁻¹)—lower anisotropy. Among caesium borates the nonlinear-optical compound CsB₃O₅ shows relatively low anisotropy of the thermal expansion (Δ = 25 × 10⁻⁶ K⁻¹), while other caesium borates expand more anisotropically. It should be noted that among rubidium borates the non-centrosymmetric polymorphs α- and β-RbB₃O₅ (Δ = (91–100) × 10⁻⁶ K⁻¹) show the highest anisotropy of expansion; however, their NLO properties are not known. However, a record anisotropy of thermal expansion among the rubidium borates of both non-centrosymmetric (α- and β-) polymorphs RbB₃O₅ ((91–100) × 10⁻⁶ K⁻¹) leads us to expect that at least one of them generates a significant second harmonic.

It is remarkable that the non-centrosymmetric borate SrB₄O₇ borate expands almost isotropically. As noted by Becker [35], the birefringence of this borate and the isostructural PbB₄O₇ are inconsiderable, and as a consequence phase matching is not possible. Reasons for the occurrence of strong anisotropy are discussed in [44]. The reasons for a manifestation of high anisotropy of NLO borates in comparison to centrosymmetric borates are discussed in the next Section 3.5.

3.5. Reasons for Strong Anisotropy of the Thermal Expansion of Borates

Borates exhibit thermal expansion that are strongly anisotropic and appear to be maximal for those compounds that contain isolated BO₃ triangles or rigid boron–oxygen groups and unfixed lattice angles (monoclinic and triclinic crystals).

The B and O atoms in planar, triangular 3B-groups vibrate maximally along the normal to the plane of the triangle, and minimally in the plane itself, generating maximal and minimal expansion, respectively.

As it was shown in Section 2.2.3, rigid groups almost do not change their size and configuration with temperature and adjust themselves to variable temperatures only by mutual rotation around shared oxygen atoms. This large hinge mechanism is strongly anisotropic by virtue of its nature.

The question can arise, why such anisotropy of structural (in particular thermal) deformations is characteristic for borates but not to the same extent for silicates, for example? The answer is simple—there are almost no unalterable atomic groups in silicates, such as rigid boron–oxygen groups; only the SiO₄ tetrahedra are unalterable with temperature, but larger polyanions (pyroxene chains, amphibole ribbons, tetrahedral nets, etc.) are not rigid, i.e., they have internal degrees of freedom for reconstruction to reach an energy minimum. However, silicates exhibit a strong anisotropy of thermal expansion by another mechanism—for oblique angle (monoclinic and triclinic) crystals by way of shears ([43], and Refs therein) connected to changes of angular lattice parameters. Shears are definitely characteristic of borates, too.

Thus, the manifestation of a strong anisotropy of the thermal deformation of crystals has at least three main reasons [44]: (1) shears, or shear deformations, which are characterised by changes of angular lattice parameters; (2) strong anisotropy of atomic thermal vibrations in planar anionic groups (TO₃ triangles); (3) hinges, or deformations of an assembly of corner-sharing rigid groups, having no own (internal) degrees of freedom to adjust themselves to varying thermodynamic conditions (T, p, X). The first of the reasons listed above can be realized in silicates, the first and the second in carbonates, and all the three reasons are realized in borates. That is why borates most often demonstrate a thermal expansion that is strongly anisotropic. It can be assumed that this is characteristic not only for thermal deformations, but also for pressure, composition (chemical) and other types of crystal structure deformations. In all cases, rigid boron–oxygen groups make a decisive contribution to the sharp anisotropy of the structural deformations of borates. For this reason, we will clarify this concept.

3.6. “Rigidity” of Rigid Groups

At ambient conditions BO₃ triangles and BO₄ tetrahedra demonstrate systematic variations of bond lengths and angles, i.e., (a) the average <B––O> bond length is 1.37 Å for triangles and 1.47 Å for tetrahedra; (b) approximately the same values are characteristic of terminal <B––O> bonds; (c) regular variations of the bridging B––O bonds <B^□–^□O^∆> > <B^□–^□O^□> > <B^∆–^∆O^∆> > <B^∆–^∆O^□> and so on are caused by various environments of O and B atoms [34]. Under high pressure, the number of BO₄ tetrahedra and edge-sharing (O–O) tetrahedra increases [63,64,65].

An earlier review [26] the results on single crystal HTXRD and LTXRD studies were summarized for more than ten borate structures as well as high-temperature powder X-ray diffraction data for about 70 borates. These studies allow to formulate the following basic principles of high-temperature borate crystal chemistry. On heating, BO₃ and BO₄ polyhedra and rigid groups consisting of these polyhedra, maintain their configuration and size (accurate to the vibrations amplitudes), i.e., they are thermally rigid. But rigid groups are able to rotate like hinges exhibiting thermal expansion that is highly anisotropical, including negative linear expansion.

As it has been noted in [26], the thermal invariability of groups means that changes in the rigid groups do not exceed the error bars of 0.003 Å for average bond lengths and of 0.5° for B–O–B bond angles. This holds for the temperature range of 20 to 500 °C.

Angles between rigid groups can vary over a wider range from about 2° to 4° for the same temperature range. Within groups considered as non-rigid ones, the changes in bond angles are similar to those in between rigid groups.

Thus, in [26] the temperature dependence analysis of B–O bond lengths and B–O–B bond angles was presented for a characteristic thermal behaviour of rigid boron–oxygen groups; however, changes of the O–O–O angles with temperature were left without attention.

Now, the experimental data from [26] were used for additional calculations on the O–O–O angles at different temperatures. The results of these calculations are represented in Figure 7 for different types of B–O groups. Values of the changes of the O–O–O angles are given for each group in a specified temperature range. If such changes do not exceed the measurement error (0.5°), then the value of the change of the angle is not specified in the figure.

It is seen from Figure 7 that the O–O–O inner angle (inside the rigid group) almost does not change with temperature. Changes of the outer angles (along the perimeter of a rigid group) exceed the measurement errors and can reach a few degrees.

Let us return to the understanding of the “thermal rigidity” term of the rigid group. The authors of [26] understand it as follows: “Average B–O bond lengths change in triangles and tetrahedra in the temperature range 20–500 °C usually do not exceed the error bars 0.003 Å.” “Variations of B–O–B angles within rigid boron–oxygen groups usually are equal or less than 0.5° at least in the temperature range of 20–500 °C. This value is comparable with the error bars. Angles between rigid groups can vary in wider range (about 2°) at the same temperatures. Within groups considered as non-rigid ones, the changes in bond angles are similar to those in-between rigid groups”.

Now, after calculation of the O–O–O angles, it is possible to give a more generalized definition: the thermal invariability of the rigid group means that the lengths and bond angles of the rigid group remain almost unchanged when the temperature changes (angles O–O–O are not the bonds angles).

4. New Temperature-Dependent Structural Studies of NLO Borates from Single-Crystal LTXRD and HTXRD Data

As it was mentioned in the introduction, this review also contains new experimental data on NLO borates. Here we represent our findings on the crystal structures of β- and α-BaB₂O₄, LiB₃O₅ as well as K₂Al₂B₂O₇, which were obtained from single crystal X-ray diffraction data for a wide range of temperatures.

4.1. β-BaB₂O₄ (98, 123, 173, 223, 295, 323, 693 K) and α-BaB₂O₄ (295, 673 K)

This barium borate allows it to compare the crystal structures and thermal deformations of a NLO and a LO phase of the same chemical composition. Both modifications are based on the same isolated groups. For the first time, the crystal structure of β-BaB₂O₄ was described in [127]. Afterwards it was refined at different temperatures [49]. At the same time, the crystal structure of HT modification of α-BaB₂O₄ was published [60], with the parameters of the thermal displacement of atoms adjusted using an isotropic approximation. Both modifications crystallize in the trigonal system, the space groups are R3c for β-BaB₂O₄ and R-3c for α-BaB₂O₄. The borate anion of both polymorphs is characterized by isolated cyclic 3B-groups of three BO₃, 3B:3∆:<3∆> (see Figure 1). A review on single-crystal growth, properties and crystal structures of the polymorphs of BaB₂O₄ was given in [128]. Previously [104], we examined the thermal expansion of β-BaB₂O₄ by powder HTXRD methods, and found it to be strongly anisotropic, similar to the thermal expansion of α-BaB₂O₄ briefly presented in [26,38,43]. Here we present data on the crystal structures of β-BaB₂O₄ and α-BaB₂O₄, determined by single-crystal X-ray diffraction data in the temperature ranges of 98–673 K and 298–694 K, respectively.

The temperature dependence of the cell parameters (Figure 8) was described for β- and α-BaB₂O₄ by first-order polynomials, like in [38]. Figures of the tensor of the thermal expansion versus the structure are shown in Figure 9. The anisotropy of the expansion is dictated by the orientation of rigid 3B-groups in the structure, where expansion in the plane is minimal and perpendicular to the plane is maximal. The comparison between both modifications (Figure 8 and Figure 9) shows, that the nonlinear-optical polymorph expands more anisotropically (α₁₁ = α_a = 3, α₃₃ = α_c = 45 × 10⁻⁶ K⁻¹) than the centrosymmetric phase, α-BaB₂O₄ (α_a = 6, α_c = 28 × 10⁻⁶ K⁻¹), although their volumetric expansion is comparable: α_V = 40 and 51 × 10⁻⁶ K⁻¹ for β and α-BaB₂O₄.

The examination of the crystal structures of both modifications, β- and α-BaB₂O₄, over a wide range of temperatures (98–673 and 298–694 K, respectively) shows also the difference in thermal transformations of the structure. In the β-BaB₂O₄ structure, the Ba atom occupies a general position 6b, where it is coordinated by eight oxygen atoms at distances of 2.643(4), 2.708(4), 2.766(5), 2.781(5), 2.837(4), 2.828(6), 2.906(4) and 3.058(5) Å (at 298 K). Temperature dependencies of Ba–O bond lengths are shown in Figure 10a. The thermal expansion coefficients of these bond lengths are 12, −1, 21, 7, 40, −11, 6 and 35 × 10⁻⁶ K⁻¹, respectively.

In α-BaB₂O₄ the Ba atoms occupy two special positions, 6a and 12c, where they are coordinated by a regular trigonal prism (bond length 2.66(3) Å × 6 at 298 K) and by nine oxygen atoms (2.74(1) Å × 3, 2.83(3) Å × 3, 3.08(1) Å × 3). Temperature dependencies of Ba–O bond lengths are shown in Figure 10b. The thermal expansion coefficients of these Ba–O bond lengths in the BaO₉ polyhedron are 3, 28 and 23 × 10⁻⁶ K⁻¹, and in the BaO₆ polyhedron they are 19 × 10⁻⁶ K⁻¹.

Thermal deformations of the crystal structure of β-BaB₂O₄ are considerably more anisotropic compared to those of α-BaB₂O₄, and they show themselves mainly in the coordination sphere of barium. A thermal change that is strongly anisotropic leads to a shift of Ba in the BaO₈ polyhedron in β-BaB₂O₄ by a considerable value 0.035 Å along the direction of the 3₁ screw axis (Figure 10c). These changes in the coordination sphere are followed by an anharmonicity of the vibrations of Ba, although inconsiderable (see Figure 3c).

Lengths and angles of the bonds in the 3B-groups of β-BaB₂O₄ are almost stable upon temperature changes. In the 3B-groups, the B–O bond lengths of the bridging oxygen atoms (1.393(9)–1.415(7) Å for β-BaB₂O₄ and 1.42(2) Å for α-BaB₂O₄ at 298 K) are longer than for the apical ones (1.329(10), 1.323(11) for β-BaB₂O₄ and 1.39(1) Å for α-BaB₂O₄ at 298 K). This is related to the fact that the terminating O atom is linked with one B atom, while the bridging O atom is linked with two boron atoms. As a result, the related bridging B–O bond is weaker and longer than for the terminal B–O bonds, which is in accordance to analogous observations in other borates [26]. Individual B–O bond lengths as a function of temperature are given in Figure 2a.

4.2. LiB₃O₅ (98, 123, 148, 173, 198, 223, 248, 273, 298, 293, 500, 650 K)

Lithium triborate exhibits excellent nonlinear optical properties paralleled by a high threshold for laser damage, which is required for many applications. The compound crystallizes orthorhombically, space group Pna2₁. The crystal structure of this well-known NLO borate was solved at room temperature [128,129,130] and then refined, with special attention to the distribution of the electron density [124,131]. The structure consists of a framework of <2Δ□> 3B-rings consisting of two triangles and a tetrahedron (Figure 7b), and Li atoms located in the interspaces of this framework. Its peculiarity is a thermal expansion anisotropy that is extreme amongst borates (α_a = 101, α_b = 31, α_c = −71 × 10⁻⁶ K⁻¹), with significant values of the coefficients of anharmonic approximation for the Li atoms (see Section 3.1 and Section 3.2), high mobility and dramatic shifts of Li of 0.26 Å [47].

Upon heating from 98 to 650 K the Li atoms show an increase of the amplitudes of their vibrations, that is accompanied by anharmonicity of its vibrations, as shown by us [47]. Furthermore, the coordination number of lithium decreased from four to three. This was described in detail in Section 3.2. The thermal mobility of the lithium atoms is reflected in the changes of the boron–oxygen framework, which is compressed along the c axis and expands along the a and b axes. Within the boron–oxygen groups the B–O bonds lengths and angles remain (in the limit of error) almost unchanged (see Figure 2b). A slight contraction of the B–O bond lengths was corrected in the view of atomic thermal vibrations, so that the rigid 3B-groups in LiB₃O₅ did not change in configuration or size on heating, but they rotated relatively to each other. The plane of the 3B-group is considered to be the plane outlined through the boron atoms. In the rigid triborate the corner-shared groups are condensed to form screw chains around the 2₁ axis (Figure 11). Upon heating from 98 up to 650 K the angle between the planes of adjacent 3B-groups changed for more than 3°, and the correspondent B–O–B angle changed for 3.8°. Thus, the B–O screw chain is changed like a hinge: it contracts along the c-axis and expands in direction of the a-axis.

4.3. K₂Al₂B₂O₇ (98, 123, 173, 223, 298, 348 K)

The crystal structure of K₂Al₂B₂O₇ is trigonal, with space group P321 [21,132,133,134]. The structure contains a framework built of corner-sharing tetrahedra of AlO₄ and triangles of BO₃. Triangles and tetrahedra form six-fold rings of three AlO₄ tetrahedra parallel to the ab plane. They alternate with three BO₃ triangles, resulting in the general formula Al₃B₃O₁₂ (Figure 12a). Planes of triangles are almost parallel to the ab plane, while two AlO₄ tetrahedra of each ring are directed towards one side and one tetrahedron towards the opposite side. Layers of rings are polymerized to form a framework through the corners of aluminium–oxygen tetrahedra (Figure 12b).

We examined the crystal structure of this borate at 98, 123, 173, 223, 298, and 348 K. According to [85], the thermal expansion is less anisotropic compared to other borates: α_a = 8, α_c = 16 × 10⁻⁶ K⁻¹. Consisting of rigid BO₃ triangles and AlO₄ tetrahedra, the six-membered boron–aluminate ring is not rigid and—upon heating—is subject to considerable thermal rearrangements (see Section 3.6). B–O–Al angles inside the ring are 156° and 84°, however, between 98 and 348 K they change about +1.5°–1.9° and −1.2°–2.0°, respectively. Thus, the boroaluminate ring changes considerably in the ab plane; however, adjacent rings align their thermal deformations to the result, that the expansion in the ab plane becomes inconsiderable. With a rise in temperature the ring turns into a more regular shape, and the planes of the BO₃ triangles are parallel to the ab plane. A decrease of the angle between the BO₃ triangles and the ab plane can be the reason for the strong expansion along the c axis.

Considerable changes of the B–O–Al angles within the hexagonal ring are followed by significant transformations of the coordination sphere of potassium. Potassium atoms occupy two special positions K1 and K2, where they are coordinated by ten and nine atoms of oxygen at distances of 2.680(3) × 2, 2.954(1) × 2, 3.020(3) × 2, 3.198(2) × 2, 3.344(3) × 2 Å and 2.679(2) × 2, 2.751(3) × 2, 3.058(1), 3.212(3) × 2, 3.252(3) × 2 Å, respectively. Coefficients of the thermal expansion for these K–O bonds are 30 × 2, 38 × 2, 67 × 2, 22 × 2, −12 × 2 10⁻⁶ K⁻¹ and 62 × 2, 48 × 2, 37, 36 × 2, −4 × 2 10⁻⁶ K⁻¹, respectively. The temperature dependence of the K–O bonds lengths are shown in Figure 13. Analyzing the distribution of these bond lengths, it is easy to note that the thermal expansion coefficients are greater in one hemisphere (for example, 38 × 2 and 67 × 2 10⁻⁶ K⁻¹ for K1) than in the other (30 × 2 and −12 × 2 10⁻⁶ K⁻¹ for K1). This is connected to the temperature-induced displacements of the potassium cations relative to the boron–aluminate framework. Thus, rather considerable changes happen in the coordination sphere of potassium, and its coordination polyhedron varies anisotropically with temperature increasing.

5. Materials and Methods

Experimental approaches and techniques. Single crystals of β-, α-BaB₂O₄ and K₂Al₂B₂O₇ were grown by the TSSG method described in [135,136,137], while the crystals of LiB₃O₅ were obtained by cooling from a melt as described in [47]. Optically clear single crystals of suitable quality for X-ray diffraction were selected using a polarizing microscope, and mounted on a glass fibre. The crystals of β-BaB₂O₄ (98, 123, 173, 223, 295, 323 K), LiB₃O₅ (98, 123, 148, 173, 198, 223, 248, 273, 298 K) and K₂Al₂B₂O₇ (98, 123, 173, 223, 298, 348 K) were measured using STOE IPDS II diffractometer, graphite-monochromated MoKα-radiation, frame widths of 1° in ω. A hot/cold air blower (Oxford Cryosystems, Oxford, UK) was used for temperature control.

High-temperature measurements on β-BaB₂O₄ (298, 693 K) and α-BaB₂O₄ (298, 673 K) were performed using an automatic three-circle diffractometer installed at the Institute of Silicate Chemistry of Russian Academy of Sciences (inventors—Prof. Yurij F. Shepelev and Prof. Yurij. I. Smolin; software development—Prof. Yurij F. Shepelev and Dr. Alexander. A. Levin) with the perpendicular beam scheme using graphite-monochromatised MoKα-radiation. The crystal placed on a glass capillary by a special high-temperature glue was blown with hot air while being measured. To study the anharmonicity of atomic vibration, the crystal of β-BaB₂O₄ was measured at room temperature with (sinθ/λ)_max = 1.03 using a Bruker Smart APEX II diffractometer (Bruker, Billerica, MA, USA).

The data were corrected for Lorentz, polarization, absorption, and background effects. The sample of K₂Al₂B₂O₇ was twinned with (−1, 0, 0, 0, −1, 0, 0, 0, 1) twinning matrix and a ~1:3 twin domains ratio. The structures were refined starting from the positional parameters of LiB₃O₅ [47] and K₂Al₂B₂O₇ [134]. Crystal structures of β- and α-BaB₂O₄ were solved using Shelx-97 software (Göttingen, Germany) [138] and refined with JANA2006 software [139]. The joint-probability density function was calculated from the inverse Fourier transform of the anharmonic ADPs approximated by the third- and fourth-order expansion of the Gram–Charlier series [140]. Anharmonic temperature factors (C_ijk) of the third order for oxygen and boron atoms appeared to be of the same order as their ESDs. C_ijk and D_ijkl coefficients were significant (more than 3σ) only for Ba atoms (see Table S4). Data visualization was performed with the Vesta software (Tsukuba, Japan) [141]. Experimental details, refinement results, final atomic positional and displacement parameters, selected bond lengths and angles for β-BaB₂O₄, α-BaB₂O₄, LiB₃O₅ and K₂Al₂B₂O₇ are given in Tables S1–S8, S9–S14, S15–S21 and S22–S28, respectively.

CCDC 1531519–1531543 contains the supplementary crystallographic data for this paper. These data can be obtained free of charge via http://www.ccdc.cam.ac.uk/conts/retrieving.html (or from the CCDC, 12 Union Road, Cambridge CB2 1EZ, UK; Fax: +44-1223-336033; E-mail: [email protected]).

Thermal expansion of α-BaB₂O₄ was studied in air by means of a Stoe Stadi P X-ray diffractometer with CuKα radiation (Darmstadt, Germany) with a high-temperature camera Büchler HDK S1 (Hechingen, Germany). The sample was prepared on a Pt–Rh plate using a suspension in heptane. The temperature step was 40 K in the range 293–373 K and 25 K in the range of 373–973 K. Unit-cell parameters of the compound at different temperatures were refined by the least-squares method. Main coefficients of the thermal expansion tensor were determined using a polynomial approximation of temperature dependencies for the unit-cell parameters by ThetaToTensor program (Institute of Silicate Chemistry of Russian Academy of Sciences, Saint-Petersburg, Russia) [142].

6. Conclusions

There is growing evidence that the NLO properties of a material are related to the particular features of its atomic molecular structure. In this context, there is a good reason to investigate the relationship between the crystal structure and the NLO properties of borates. The most common group in the NLO borates are isolated BO₃ triangles, followed by single 3B-rings and double cyclic 4B- and 5B-groups, composed of 3B-rings; other groups are scarce.

The BO₃ triangles and the cyclic 3B-groups are flat and asymmetrical in shape. Hence, we conclude that the asymmetrical flat shape of anionic groups is strongly preferred in NLO borates. A flat shape of the anionic groups leads to their preferable arrangement in parallel to each other, i.e., to the self-assembly of a drastically anisometric crystal structure of NLO borates.

The anisometric atomic molecular structure of NLO borates generally gives rise to a pronounced anisotropy of their physical properties, and thermal expansion follows this trend. It was noted that the sharp anisotropy of thermal expansion is common for borates that have a pseudo-layered structure. In this review, we show (see Table 2) that the borates generating the second harmonic exhibit stronger anisotropy of thermal expansion.

The sizable birefringence, mentioned in Section 4.3, is an example of a physical property that is governed by the “layered” structure of borates. Indeed, a pronounced anisotropy of physical properties ensues from the preferred orientation of BO₃ triangles and 3B-rings comprising such triangles. The “layers” are characterized by the highest density, the slowest propagation of light oscillating in the plane of the layer, thus the highest refractive index n_g. In contrast, the direction perpendicular to the layers has the minimal density, the fastest propagation of the waves oscillating in this plane, while the refractive index n_p reaches its minimal value. This gives rise to the sizable birefringence n_g − n_p.

Since the birefringence and the anisotropy of thermal expansion are functions of the same argument (pseudo-layered structure), the significant birefringence correlates with the high anisotropy of thermal expansion: n_g corresponds to α_min, n_p − α_max. The borate β-BaB₂O₄ (see Figure 6, n_g − n_p = 0.041) is a typical example. In this respect, high anisotropy can be considered a further criterion for NLO borates. In particular, by analysing the thermal expansion tensor of two dozen non-centrosymmetric borates versus their centrosymmetric counterparts (see Table 2), we demonstrated that most NLO borates exhibit a pronounced anisotropy of thermal expansion. In turn, the reason for the anisotropy of thermal expansion is the thermal vibrations of atoms.

As noted by Chen and co-authors, the anionic groups are primarily responsible for the second harmonic generation, although the cationic contributions cannot be neglected. We can conclude that the thermal mobility of cations rises significantly: their anharmonic components of thermal vibrations increase a few times to reach significant values, the cations shift, and, as a result, the coordination number of cations tends to decrease with temperature. The mobility of cations in centrosymmetric structures is less pronounced than in non-centrosymmetric ones.