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

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.


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 Crystals 2017, 7, 93 2 of 32 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 KB 5 O 8 •4H 2 O 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: [16], LiB 3 O 5 [17], CsB 3 O 5 [18], Sr 2 Be 2 B 2 O 7 [19], CsLiB 6 O 10 [20], and K 2 Al 2 B 2 O 7 [21,22].Later on, α-BiB 3 O 6 [23], BaBiBO 4 [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 BO 3 triangles or/and the BO 4 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 BO 4 tetrahedra, thus forming multiple cyclic rigid groups.BO 4 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 BO 3 triangles, BO 4 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-BaB 2 O 4 , Crystals 2017, 7, 93 3 of 32 LiB 3 O 5 and K 2 Al 2 B 2 O 7 , measured within the intervals 98-693, 89-298 and 98-348 K, respectively.Finally, for the first time structural data on α-BaB 2 O 4 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.

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 BO 3 group and the tetrahedral BO 4 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.

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 BO 3 and BO 4 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 BO 3 triangle or a BO 4 tetrahedron like the TO 4 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 B 2 O 3 are mainly represented by isolated BO 3 triangles and BO 4 tetrahedra or non-cyclic diortho groups (dimers), e.g., two corner-sharing triangles in [B 2 O 5 ] and two corner-sharing tetrahedra in [B 2 O 7 ].Isolated BO 3 and BO 4 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 Bϕ 3 triangles and isolated Bϕ 4 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.
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 x O y :B 2 O 3 = 1 (modifying agent to boron oxide ratio).This means that they are forming from approximately the centre of the M x O y -B 2 O 3 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 BO 4 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 BO 4 tetrahedra forming a large B 4 O 10 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 B 4 O 10 tetrahedron was experimentally found for the first time by Wu et  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   [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  [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 SiO 4 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 BO 3 and BO 4 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 B 3+ -O bonds in BO 3 triangles (1 v. u. per each B-O bond in average) and BO 4 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 BO 3 triangles and BO 4 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 K 2 Al 2 B 2 O 7 [21,22] and the 8B-ring in α-BiB 3 O 6 [72], as described in the following paragraph.

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 BO 3 triangles and finite groups (0D) to double-framework (3D) were studied (Table 1).We also refined the structures of two NLO borates (LiB 3 O 5 and Li 2 B 4 O 7 ) 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-BaB 2 O 4 polymorph in comparison to centrosymmetric α-BaB 2 O 4 , recorded over a wide temperature range, and data for K 2 Al 2 B 2 O 7 and LiB 3 O 5 at low temperatures.Moreover, for the first time the crystal structure of α-BaB 2 O 4 is refined in anisotropic approximation in addition to the isotropic one [60].Structures of β-BaB 2 O 4 , K 2 Al 2 B 2 O 7 and LiB 3 O 5 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 Sections 3.4 and 3.5 (see Table 2 [46] and α-CsB 5 O 8 [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).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 SiO 4 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 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-Na 2 B 8 O 13 [46], LiB 3 O 5   [47], α-CsB 5 O 8 [48], and Li 2 B 4 O 7 [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 TO 3 and TO 4 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]: 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).[66,67].In the present study negative thermal expansion is found for the Al1-O1 and Al2-O2 bond lengths of AlO 4 tetrahedra in the range of 100-400 K (Figure 2b).
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]: where Rcorr and Robs are corrected and observed B-O bond lengths, correspondingly; Beq(A) and Beq(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).

Thermal Invariability of Rigid B-O Groups in NLO Borates
The structures of the vast majority of existing NLO borates are based upon isolated BO 3 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 BO 3 triangles.As an example, we discuss the Ca 4 REEO(BO 3 ) 3 family [12] which is related to NLO borates based on isolated BO 3 triangles where REE = Gd (# 39716-ICSD) or Y, as well as K 2 Al 2 B 2 O 7 [21] and BaBiBO 4 [24].We have investigated K 2 Al 2 B 2 O 7 (see Section 4.3) and BaBiBO 4 [58].Nevertheless, the description of these borates is not uniquely defined: K 2 Al 2 B 2 O 7 can be considered also as boroaluminate framework composed of non-rigid six-membered rings in which BO 3 triangles alternate with AlO 4 tetrahedra (see Section 4.3), and BaBiBO 4 can also be described as being built up from rigid borate-bismuthate chains.There are inconsistent data on the symmetry and NLO properties of BaBiBO 4 : it was firstly determined as crystallizing in the non-centrosymmetric space group Pna2 1 with NLO properties (# 154105-ICSD, [24]), whilst later it was refined in space group Pnma (# 9424596-ICSD, [58]).In any case, in both borates, K 2 Al 2 B 2 O 7 (see Section 4.3) and BaBiBO 4 [58], the BO 3 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-NaB 3 O 5 are also built up from <2∆ > 3B-groups.These are the topologically identical three-dimensional frameworks of LiB 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-BaB 2 O 4 ) 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 (Li 2 B 4 O 7 [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-CsB 5 O 8 [48] and α-Na 2 B 8 O 13 [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 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 Sections 4.1-4.3).

Non-Rigid Cyclic Groups in NLO Borates
As discussed in Section 4.3 when analysing K 2 Al 2 B 2 O 7 , AlO 4 tetrahedra are located in a flexible six-fold ring that is composed of alternating BO 3 triangles and AlO 4 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 TO 4 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 BO 3 triangles and BO 4 tetrahedra is present in the layered polyanion of the NLO borate α-BiB 3 O 6 [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 BO 3 triangles and BO 4 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.

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 α 11 , α 22 , α 33 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).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).

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.

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., LiB 3 O 5 [47], Li 2 B 4 O 7 [50,54] and β-BaB 2 O 4 (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 LiB 3 O 5 .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 Li 2 B 4 O 7 (Figure 3b), and the vibrations of the Ba atoms in β-BaB 2 O 4 are the weakest (Figure 3c).It is remarkable that the NLO properties are also expressed weaker in Li 2 B 4 O 7 than in LiB 3 O 5 [12,15], etc.This can be caused by the stronger anharmonicity of the Li atoms' vibrations.The anharmonicity increases sharply with temperature in LiB 3 O 5 [47] and in Li 2 B 4 O 7 [50,54], especially close to the melting point, as found by Senyshyn and co-authors [54].

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 3с).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 Li 2 B 4 O 7 borate, preferably oscillates along this channel.It gives another possible structural criterion of NLO materials-irregular coordination of the atoms.

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 (Figures 3a and 4a).Thus, Li atoms are coordinated by four oxygen atoms with the Li-O bond lengths being 1.979-2.180Å [47]; the next oxygen O5 iii atom is at a distance of 2.684 Å from Li.The distorted LiO 4 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-O5 iii 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 (Figures 3a and 4a).
Crystals 2017, 7, 93 14 of 32 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.

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 (Figures 3a and 4a).Thus, Li atoms are coordinated by four oxygen atoms with the Li-O bond lengths being 1.979-2.180Å [47]; the next oxygen O5 iii 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-O5 iii 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 (Figures 3a and 4a).A similar situation occurs in Li2B4O7: the LiO4 tetrahedron at low temperatures [47] becomes an LiO3 triangle at high temperatures (Figures 3b and 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 Ba 2+ 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 Beq 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 (Figures 3a and 5a).Li + cation thermal vibrations in Li2B4O7 rapidly increase on heating to 773 K, the isotropic parameter Beq(Li) increases by a factor of 2.5 [50,51].The isotropic displacements Ueq 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).A similar situation occurs in Li 2 B 4 O 7 : the LiO 4 tetrahedron at low temperatures [47] becomes an LiO 3 triangle at high temperatures (Figures 3b and 4b), although screw chains of LiO 5 polyhedra remain at a wide range of temperatures (Figure 5b).In the case of heavier and higher charges of Ba 2+ in β-BaB 2 O 4 , 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 (Figures 3a and 5a).Li + cation thermal vibrations in Li 2 B 4 O 7 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-BaB 2 O 4 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 LiB 3 O 5 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-BaB 2 O 4 , the heavier Ba atom is moved towards screw 3 1 axis less significantly (see Section 4.1).
In LiB 3 O 5 LiO 5 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) = 1.968-2.003Å).The fourth O5 iii oxygen that contributes to the tetrahedron at 20 • C moves away from Li (at a distance of 2.381 Å) and the fifth O5 i oxygen is 2.6 Å away from Li (Figures 3a and 4a).At 650 K Li is clearly coordinated by three oxygen atoms, forming a planar LiO 3 triangle in the first coordination sphere, with two additional oxygen atoms on both sides of this triangle (second coordination sphere).
In 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 Li 2 B 4 O 7 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 Li 2 B 4 O 7 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 LiB 3 O 5 , the real reason for thermal deformations of the double interpenetrating boron-oxygen framework in Li 2 B 4 O 7 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 LiO 4 into LiO 3 triangle [47,51].It is notable that the screw chains of the LiO-polyhedra remain existent through the transformation of the LiO 4 tetrahedron into a LiO 3 triangle, as it is shown in the anharmonic approximation in LiB 3 O 5 [47,51] and in Li 2 B 4 O 7 [51,54] (Figure 5).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) = 1.968-2.003Å).The fourth O5 iii oxygen that contributes to the tetrahedron at 20 °C moves away from Li (at a distance of 2.381 Å) and the fifth O5 i oxygen is 2.6 Å away from Li (Figures 3a and 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 Ǻ 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: αa = 101, αb = 31, αc = −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 (αV = 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 (αa = 16 × 10 −6 K −1 ).Along the main axis, the c parameter shows compression (αc < 0) in negative and low temperature ranges (−180-250 °C) and expansion (αc > 0) at intermediate and high temperatures (250-750 °C) with the minimum (αc = 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 LiB 3 O 5 , CsLiB 6 O 10 , and CsB 3 O 5 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 [B 3 O 7 ] 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 [B 3 O 7 ] 5− anion.

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 BO 3 triangular radicals and mainly rigid groups containing BO 3 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 BO 3 triangles, flat 3B-and other B-O rigid groups containing BO 3 triangles.
In the case of a BO 3 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 BO 3 triangle, thus B and O atoms maximally vibrate preferably perpendicular to the plane of an isolated BO 3 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 BO 3 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 BO 3 groups exhibit excellent NLO properties this has motivated in finding new NLO materials with other rigid TO 3 groups, such as carbonates and nitrates.In the most structures based upon isolated TO 3 triangles (T = B, C, N), TO 3 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 BO 3 triangles in comparison to the pole figure of thermal expansion of the crystal structure of β-BaB 2 O 4 (Figure 6c), a crystal structure that is based on isolated triborate groups of three triangles 3B:<3∆> (see Figure 1, n = 3).For β-BaB 2 O 4 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-BaB 2 O 4 consisting of <3∆> groups.
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.

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.

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 ∆ = 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.(∆ = 37 × 10 −6 K −1 )-lower anisotropy.Among caesium borates the nonlinear-optical compound CsB 3 O 5 shows relatively low anisotropy of the thermal expansion (∆ = 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-RbB 3 O 5 (∆ = (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 RbB 3 O 5 ((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 SrB 4 O 7 borate expands almost isotropically.As noted by Becker [35], the birefringence of this borate and the isostructural PbB 4 O 7 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.

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 BO 3 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 SiO 4 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 (TO 3 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.
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, BO 3 and BO 4 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]  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".

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-BaB 2 O 4 , LiB 3 O 5 as well as K 2 Al 2 B 2 O 7 , which were obtained from single crystal X-ray diffraction data for a wide range of temperatures.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-BaB 2 O 4 was described in [127].Afterwards it was refined at different temperatures At the same time, the crystal structure of HT modification of α-BaB 2 O 4 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-BaB 2 O 4 and R-3c for α-BaB 2 O 4 .The borate anion of both polymorphs is characterized by isolated cyclic 3B-groups of three BO 3 , 3B:3∆:<3∆> (see Figure 1).A review on single-crystal growth, properties and crystal structures of the polymorphs of BaB 2 O 4 was given in [128].Previously [104], we examined the thermal expansion of β-BaB 2 O 4 by powder HTXRD methods, and found it to be strongly anisotropic, similar to the thermal expansion of α-BaB 2 O 4 briefly presented in [26,38,43].Here we present data on the crystal structures of β-BaB 2 O 4 and α-BaB 2 O 4 , 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-BaB 2 O 4 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 (Figures 8 and 9) shows, that the nonlinear-optical polymorph expands more anisotropically (α 11 = α a = 3, α 33 = α c = 45 × 10 −6 K −1 ) than the centrosymmetric phase, α-BaB 2 O 4 (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-BaB 2 O 4 .
The examination of the crystal structures of both modifications, β-and α-BaB 2 O 4 , 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-BaB 2 O 4 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       Thermal deformations of the crystal structure of β-BaB 2 O 4 are considerably more anisotropic compared to those of α-BaB 2 O 4 , 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 BaO 8 polyhedron in β-BaB 2 O 4 by a considerable value 0.035 Å along the direction of the 3 1 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-BaB 2 O 4 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-BaB 2 O 4 and 1.42(2) Å for α-BaB 2 O 4 at 298 K) are longer than for the apical ones (1.329 (10), 1.323 (11) for β-BaB 2 O 4 and 1.39(1) Å for α-BaB 2 O 4 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.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.(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 (αa = 101, αb = 31, αc = −71 × 10 −6 K −1 ), with significant values of the coefficients of anharmonic approximation for the Li atoms (see Sections 3.1 and 3.2), high mobility and dramatic shifts of Li of 0.26 Å [47].

LiB3O5
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)   (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 Pna2 1 .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 <2∆ > 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 Sections 3.1 and 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 Crystals 2017, 7, 93 23 of 32 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 LiB 3 O 5 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 2 1 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.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: αa = 8, αc = 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.The crystal structure of K 2 Al 2 B 2 O 7 is trigonal, with space group P321 [21,[132][133][134].The structure contains a framework built of corner-sharing tetrahedra of AlO 4 and triangles of BO 3 .Triangles and tetrahedra form six-fold rings of three AlO 4 tetrahedra parallel to the ab plane.They alternate with three BO 3 triangles, resulting in the general formula Al 3 B 3 O 12 (Figure 12a).Planes of triangles are almost parallel to the ab plane, while two AlO 4 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 BO 3 triangles and AlO 4 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 BO 3 triangles are parallel to the ab plane.A decrease of the angle between the BO 3 triangles and the ab plane can be the reason for the strong expansion along the c axis.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.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.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.

Materials and Methods
Experimental approaches and techniques.Single crystals of β-, α-BaB 2 O 4 and K 2 Al 2 B 2 O 7 were grown by the TSSG method described in [135][136][137], while the crystals of LiB 3 O 5 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-BaB 2 O 4 (98,123,173,223,295, 323 K), LiB 3 O 5 (98,123,148,173,198,223,248,273, 298 K) and K 2 Al 2 B 2 O 7 (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-BaB 2 O 4 (298, 693 K) and α-BaB 2 O 4 (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-BaB 2 O 4 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 K 2 Al 2 B 2 O 7 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 LiB 3 O 5 [47] and K 2 2 B 2 O 7 [134].Crystal structures of β-and α-BaB 2 O 4 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-BaB 2 O 4 , α-BaB 2 O 4 , LiB 3 O 5 and K 2 Al 2 B 2 O 7 are given in Tables S1-S8, S9-S14, S15-S21 and S22-S28, respectively.
Thermal expansion of α-BaB 2 O 4 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].

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 BO 3 triangles, followed by single 3B-rings and double cyclic 4B-and 5B-groups, composed of 3B-rings; other groups are scarce.
The BO 3 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 BO 3 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-BaB 2 O 4 (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.

Figure 1 .
Figure 1.Main B-O groups occurring in NLO borates.

2. 2 .
Review of Temperature-Dependent Structural Studies of NLO Borates from Single-Crystal LTXRD and HTXRD Data Individual B-O bond lengths are given as a function of temperature for β-BaB 2 O 4 (a) and LiB 3 O 5 (c).For K 2 Al 2 B 2 O 7 , 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 AlO 4 tetrahedra of zero was described by Hazen, Prewitt and Finger
Li 2 B 4 O 7 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 LiO 4 tetrahedron at high temperatures, almost a triangle like in LiB 3 O 5 .Structural mechanism of thermal expansion.Lithium triborate, LiB 3 O 5 , 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 .
For example, among Ba borates β-BaB 2 O 4 (∆ = 42 × 10 −6 K −1 ) is featured with the most anisotropy, among bismuth borates-α-BiB 3 O 6 (∆ = 82 × 10 −6 K −1 ), and among lithium borates-LiB 3 O 5 (∆ = 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-KB 5 O 8 (∆ = 63 × 10 −6 K −1 ) shows the highest anisotropy whilst NLO K[B 5 O 6 (OH) 4 ]•2H 2 O 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 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).Crystals 2017, 7, 93 19 of 32 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.

Figure 7 .
Figure 7. Changes of the O-O-O angles of boron-oxygen groups in the structures of (a) β-BaB2O4; (b) LiB3O5; (c) Li2B4O7; (d) α-CsB5O8; (e) α-NaB8O13; (f) BiB3O6; (g) K2Al2B2O7.Inner angles change less than 0.5° in rigid groups (a-e); they are shown by a bold line with no numbers.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

Figure 8 .
Figure 8. Temperature dependence of lattice parameters and the volumes of the unit cells of β-BaB2O4 (a) and α-BaB2O4 (b).

Figure 8 .
Figure 8. Temperature dependence of lattice parameters and the volumes of the unit cells of β-BaB 2 O 4 (a) and α-BaB 2 O 4 (b).

Figure 11 .
Figure 11.Crystal structure of LiB3O5 versus the pole figure of the coefficients of the thermal expansion (a).The sharp anisotropy of the thermal expansion is caused by the contraction of the chain of 3B-groups.Upon heating from 98 up to 650 K the angle between planes of adjacent 3B-groups is changed for more than 3° (b).

Figure 11 .
Figure 11.Crystal structure of LiB 3 O 5 versus the pole figure of the coefficients of the thermal expansion (a).The sharp anisotropy of the thermal expansion is caused by the contraction of the chain of 3B-groups.Upon heating from 98 up to 650 K the angle between planes of adjacent 3B-groups is changed for more than 3 • (b).

Figure 12 .
Figure 12.Projections of the crystal structure of K 2 Al 2 B 2 O 7 onto (a) ab and (b) bc planes versus the pole figures of thermal expansion coefficients.Changes of O-O-O angles in between polyhedra are shown on (a).Next to the K-O bonds in (b) are shown their coefficients of thermal expansion (×10 −6 K −1 ).

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

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