# Thermodynamics and crystallography of hard light-element boron-rich compounds by in situ X-ray diffraction under high-pressure, high-temperature conditions

Alexandre Courac, Vladimir Turkevich, Yann Le Godec

<figure><img src="image_1.png" /></figure>

## To cite this version:

Alexandre Courac, Vladimir Turkevich, Yann Le Godec. Thermodynamics and crystallography of hard light-element boron-rich compounds by in situ X-ray diffraction under high-pressure, high-temperature conditions. Solid State Sciences, 2025, 163, pp.107909. 10.1016/j.solidstatesciences.2025.107909. hal-05064506

HAL Id: hal-05064506

https://hal.sorbonne-universite.fr/hal-05064506v1

Submitted on 14 Nov 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
<figure><img src="image_3.png" /></figure>

# Thermodynamics and crystallography of hard light-element boron-rich compounds by in situ X-ray diffraction under high-pressure, high-temperature conditions

Alexandre Courac,¹,² Vladimir Turkevich,³ Yann Le Godec¹

1. Institut de Minéralogie, de Physique des Matériaux et de Cosmochimie (IMPMC), UMR CNRS 7590, Muséum National d'Histoire Naturelle, IRD UMR 206, Sorbonne Université, 75005 Paris, France

2. Institut Universitaire de France (IUF), 75005 Paris, France

3. National Academy of Sciences of Ukraine, Institute for Superhard Materials (ISM), 04074 Kiev, Ukraine

## Abstract

Here we review the available experimental and theoretical data on phase transformations and thermodynamical properties of boron compounds that can be obtained by in situ methods at high-pressure and high-temperature (HPHT) conditions. The thermodynamic analysis allowed to establish the optimized sets of thermodynamic values and mechanical properties for boron allotropes and compounds such as boron suboxide $B_6O$, boron subnitride $B_{13}N_2$, and boron carbide.
# 1. In situ crystallography of boron-rich solids at HPHT

Recovery HPHT experiments are the base of industrial production of diamond and cubic boron nitride, discovery of new boron-rich solids [1], as well as sintered bindered superhard materials and semiconductors produced by volatile elements (e.g. nitrides like GaN). Generally, the best mechanical and other properties can be achieved in equilibrium synthesis, as it is well known for diamond. Boron carbide produced at HPHT conditions provides another example. However, it is not always clear whether HP phases form at HPHT conditions, or during cooling and/or decompression, how it happens in Si according to in situ studies [2, 3]. In situ methods allow observing phase transformations in solids under HPHT conditions, thereby removing the ambiguity that can arise when analyzing the recovered samples [4, 5].

In situ X-ray diffraction (XRD) of light element compounds [6] is a challenging task, particularly given the low atomic number of first group of elements. To date, only a limited number of light-element compounds with relatively simple crystal structures have been subjected to such analysis. Observation of X-ray diffraction and scattering of light-element solids and liquids remain a challenging axis of future developments of synchrotron beamlines for geophysics and materials science.

Following the diamond that can be obtained by high pressure, high temperature conditions, boron-rich solids synthesis at HPHT conditions attracted much attention in the past decades. B-C-N-O concentration tetrahedron (Fig. 1) contains a number of superhard materials that have been extensively studied, including boron and its compounds. However, the in situ HPHT thermodynamic data of boron and its compounds remains very limited as compared to graphite-to-diamond transformation in various systems (C-Ni-Fe [7], Mg-C [8-11], etc.).

In these proceedings we will get together all available thermodynamic data for boron-rich solids, with a particular focus on p-V-T equation of state and enthalpy of formation of HP phases that are lacking in conventional thermodynamic sources.

<figure><img src="image_4.png" /><figcaption>Fig. 1 Boron binary compounds of B-C-N-O concentration tetrahedron.</figcaption></figure>
## 2. Thermodynamic calculations at HPHT

The Gibbs energy at HPHT conditions can be estimated using p-V-T EOS according to

$$G(p, T) = G^{\circ}(T) + \int_{0}^{p} V(p', T) dp' \qquad (1)$$

Anderson-Grüneisen equation [12] takes into account the pressure dependence of thermal expansion through the volume change, i.e.

$$\alpha(p, T) = \alpha(0, T) \left[ \frac{V(p, T)}{V(0, T)} \right]^{\delta_T}, \qquad (2)$$

And can be integrated [13] (under the assumption that $\delta_T$ is constant) to

$$V(p, T) = [V(0, T)^{-\delta_T} + V(p, 300)^{-\delta_T} - V(0, 300)^{-\delta_T}]^{-1/\delta_T}, \qquad (3)$$

where thermal expansion (i.e. V(0,T) at 0.1 MPa) and isothermal compression (i.e. V(p,300) at 300 K) can be presented in any analytical form, e.g. polynomial

$$V(0,T) = V(0,300) [1+a (T-300)+b (T-300)^2] \qquad (4)$$

and Murnaghan [14] (or, in principle, any other) equation of state

$$V(p, 300) = V(0, 300) (1 + B'_{0} p / B_{0})^{-1/B_{0}}. \qquad (5)$$

The parameters describing the p-V-T EOS of phase are, therefore, $V_0 \equiv V(1 \text{ MPa}, 300 \text{ K}) \equiv M / \rho_0$, $B_0$, $B'_0$, a, b and $\delta_T$ (usually ~5-6 [15]). Such form of the EOS is useful to approximate the V(p,T) in the p-T domain of its formation and stability, which is often the principal domain of interest in the terms of HPHT thermodynamics. The values selected in this review were principally estimated in situ using X-ray diffraction (300-K EOS may be measured on decompression and recovery).

In some other cases, the knowledge of phase equilibrium curves may allow evaluating the HPHT EOS of unknown phase through the known one. For example, we succeeded to fit the experimental melting curves (p-T coordinates) of $\alpha$- and $\beta$-B$_2$O$_3$ and to find the parameters determining the p-V-T EOS for liquid B$_2$O$_3$ [16]. The melting was observed experimentally, while the bulk modulus and thermal expansion parameters were adjusted so that the theoretical curve fits the experimental one.
### 3. Boron and boron-rich compounds at HPHT conditions.

#### Boron

At present, at least five allotropes can be recovered from high pressure experiments at ambient conditions. Generally recognized icosahedral allotropes of boron are α-B₁₂ (or α-B, also may be denoted as α-rh B), β-B₁₀₉₋ₓ (or β-B), γ-B₂₈ (or γ-B [17, 18]), t-B₅₂₋ₓ (δ-B, [19]), and ε-B₁₅ (ε-B, [19]). For β-B and δ-B the number of atom per cell may change. δ-B is presumed to be metastable at any p-T condition by some authors [20], or to be stable at high temperature and high pressure by others [21]. At present time all icosahedral allotropes can be obtained in single crystal form by high pressure, high temperature synthesis. α-Ga structure of B become stable at high pressures, it has been predicted [17] and observed in situ [22], however it cannot be recovered at ambient conditions.

High pressure synthesis of single crystals of α-boron is feasible when employing a Pt/B or Cu/B contact [23]; HPHT crystallization of α-B can be achieved in the presence of carbon [24]. α-B has a simple unit cell of rhombohedral symmetry with twelve boron atoms organized as icosahedra with so-called 3-center electron-deficient bonds where the electron charge is accumulated at the center of a triangle formed by three adjacent atoms.

γ-B₂₈ is an established discovered high-pressure phase of boron. Its structure consists of icosahedral B₁₂ clusters and B₂ dumbbells in a NaCl-type arrangement (B₂)⁶⁺(B₁₂)⁶⁻ and displays a significant charge transfer δ~0.5-0.6. The discovery of this phase proved essential for the understanding and construction of the phase diagram of boron [17]. This phase has highest bulk modulus and hardness [25]. Also, it is a transparent phase, which is not typical for boron-rich solids. At the same time, in situ synthesis and phase transformations of this phase has not been successful so far.

Thermodynamic stability and fundamental state of boron were previously discussed quite a lot, however, no consensus can be made due to the lack of in situ data. Only direct irreversible transformation of alpha boron into beta has been observed in situ. The heat capacity data indicate that at high temperature beta phase is surely stable, however the low-temperature part lack the enthalpy of formation measurements between alpha and beta phases. The β-rhombohedral allotrope, β-B, proved to be a significant challenge in terms of understanding its crystal structure [26] and stability. It was observed that the stability of the crystal structure can be altered by changes in temperature or pressure, which in turn affects the occupancy of the crystal sites [20]. It is generally accepted that β-B is a fundamental state at ambient conditions [27]. α-B becomes stable at high pressure, but only at low and moderate temperatures, since its heat capacity increase stronger than that of β-B (Fig. 2). Experimental pressure-temperature phase diagram of boron has been also studied in the presence of Pt and Cu metals, and the single crystal growth data [28] is in reasonable agreement with theoretical diagrams [21].

Tab. 1 present the full data set available (experimental data and estimates) available for boron allotropes. The experimental melting curve allowed us obtaining estimate of bulk modulus of liquid boron, a value to be confirmed from future in situ studies of liquid boron structure.
<figure><img src="image_5.png" /><figcaption>Fig. 2 Heat capacity of α-B and β-B and relative stability of allotropes from ref. [29].</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Fig. 3 (a) In situ XRD patterns collected at 5 GPa and temperatures up to boron melting [21]. α-B passes into β-B above 2000 K and melts above 2400 K. (b) Melting curve of β-B.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Fig. 4 Bulk moduli of boron allotropes [30] and compounds of α-B structural family [31].</figcaption></figure>

<table><caption>Table 1. Parameters of p-V-T equation of state data for boron-rich solids</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs.</th></tr></thead><tbody><tr><td>γ-B<sub>28</sub></td><td>2.544</td><td>237</td><td>2.7</td><td>6</td><td>0</td><td>5.5</td><td>[30]</td></tr><tr><td>t-B<sub>52</sub></td><td>2.493</td><td>232</td><td>2.7</td><td>6</td><td>0</td><td>5.5</td><td>[13]</td></tr><tr><td>α-B<sub>12</sub></td><td>2.447</td><td>224</td><td>3.0</td><td>18.3</td><td>0</td><td>5.5</td><td>[32, 33]</td></tr><tr><td>B<sub>6</sub>O</td><td>2.601</td><td>180</td><td>6</td><td>14</td><td>5</td><td>6</td><td>[34, 35]</td></tr><tr><td>B<sub>13</sub>N<sub>2</sub></td><td>2.666</td><td>200</td><td>4</td><td>21.3</td><td>0</td><td>1.5</td><td>[36-38]</td></tr><tr><td>HP-B<sub>50</sub>N<sub>2</sub></td><td>2.408</td><td>167</td><td>4</td><td>15.7</td><td>0</td><td>5.5</td><td>[37, 38]</td></tr><tr><td>β-B</td><td>2.365</td><td>210</td><td>2.23</td><td>19.2</td><td>0</td><td>5.5</td><td>[39-41]</td></tr><tr><td>B<sub>4</sub>C</td><td></td><td>221</td><td>3.3</td><td>17.2</td><td>0</td><td>5.5</td><td>[42]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless.

Experimental phase diagrams have been proposed by different groups [16, 28], at the same time the crystallization of metastable phases cannot be precluded, as well as the crystal growth conditions cannot be unambiguously attributed to the highest pressure and temperature values
achieved in recovery experiments. The best estimate of up-to-date experimental phase equilibria can be presented as a self-consistent set of equations that agree with available thermodynamic data (Tab. 2):

$$T_{L/\beta} = 2360 + 15.5 p \qquad (6)$$

$$T_{\gamma/\beta} = -4534 + 707.8 p \qquad (7)$$

$$T_{t'/\gamma} = 2176 + 5.1 p \qquad (8)$$

$$T_{t'/\beta} = 3756 - 160.2 p \qquad (9)$$

In general case, one can consider such simplified equations as a form of thermodynamic equation

$$T_{eq} = \Delta H/\Delta S + p \Delta V/\Delta S \qquad (10)$$

with constant ΔH, ΔS and ΔV.

<table><caption>Table 2. Thermodynamic data on phase transformations in boron at HPHT.</caption><thead><tr><th>Y/X</th><th>ΔH<sub>Y/X</sub> (kJ/mol),<br>ΔS<sub>Y/X</sub> (kJ/mol K)<br>or ΔV<sub>Y/X</sub> (cm<sup>3</sup>/mol)</th><th>In situ / ex situ data</th><th>Reference data</th></tr></thead><tbody><tr><td>L/β</td><td>ΔH = 50</td><td>Experimental</td><td>ΔH<sub>melt</sub> = 50.2 -48.93</td></tr><tr><td></td><td>ΔV = 0.32</td><td>In situ HPHT data on melting</td><td>ΔV = 0.22 to 0.46</td></tr><tr><td></td><td>ΔS = 2.1·10<sup>-2</sup></td><td>Experimental</td><td></td></tr><tr><td>γ/β</td><td>ΔH = 2.5</td><td>Ab initio</td><td>ΔH<sub>0K</sub> = 2.5</td></tr><tr><td></td><td>ΔV = -0.35</td><td>Experimental from ex situ<br>data</td><td>ΔV<sub>300K, 1MPa</sub> = -0.32</td></tr><tr><td></td><td>ΔS = -5.5·10<sup>-4</sup></td><td>Thermodynamic consistency</td><td></td></tr><tr><td>t'/β</td><td>ΔH = 8.8</td><td>Fitted</td><td></td></tr><tr><td></td><td>ΔV = -0.38</td><td>Experimental, ex situ</td><td>ΔV<sub>300K, 1MPa</sub> = -0.30</td></tr><tr><td></td><td>ΔS = 2.3·10<sup>-3</sup></td><td></td><td></td></tr><tr><td>t'/γ</td><td>ΔH = 6.3</td><td>Thermodynamic consistency</td><td></td></tr></tbody></table>
<table><tr><td></td><td>ΔV = 0.01</td><td>Experimental, ex situ<br>Thermodynamic consistency</td><td>ΔV<sub>300K, 1MPa</sub> = 0.02</td></tr><tr><td></td><td>ΔS = 2.9·10<sup>-3</sup></td><td></td><td></td></tr><tr><td>L/t'</td><td>ΔH = 41.2</td><td>Thermodynamic consistency</td><td></td></tr><tr><td></td><td>ΔV = 0.7</td><td>Thermodynamic consistency</td><td></td></tr><tr><td></td><td>ΔS = 1.9·10<sup>-2</sup></td><td></td><td></td></tr></table>

Ab initio phase diagrams are presented at Fig. 5. The latest simulations, taking into account different possibilities of interstitial site occupancies indicate the instability of any tetragonal phase, either α-B or β-B, at high temperature. The actual extreme-condition challenges are (i) in situ melting of gamma boron and (ii) in situ observation, nature and melting of HPHT tetragonal phase, if exists and is stable. It should be noted here that t-B<sub>190+x</sub> phase, known only by chemical route synthesis, was never observed at HPHT conditions. Some ab initio simulations suggest its HPHT thermodynamic stability [17] and others not [20]. Various t-B<sub>52</sub> species of pure boron was observed experimentally at such HPHT conditions [21, 43, 44].

<figure><img src="image_8.png" /><figcaption>Fig. 5 Ab initio phase diagrams of boron with tetragonal phase (t-B<sub>190+x</sub>) [17] and without any tetragonal phase [20].</figcaption></figure>
## Boron-rich families of boron allotropes

Most allotropes of boron, except gamma phase, give rise to a family of compounds with similar structures, sometimes called as related to α-rh B (B₆O, B₁₃N₂ and B₄C, Fig. 6); β-rh B (see ref. [27]), α-T B or δ-B (B₅₀C₂, B₅₀N₂ [45]) and β-T B or I-T (α-AlB₁₂ [46]). Crystal structures were studied in the past, and are characterized by high degree of structural disorder for some allotropes (e.g. β-rh B [26]). At the same time, it was previously reported and seems to be correct that high pressure synthesis allows obtaining equilibrium structures with highest order in atomic distribution.

<figure><img src="image_9.png" /><figcaption>Fig. 6 Lattice parameters (a) and simplified crystal structure model (b) of alpha-boron type compounds and solid solutions. Oxygen can be replaced with nitrogen. Carbon occupancy may be zero, in the case of suboxide and subnitride (N replace O). In the case of X = Y = B, the structure becomes ε-B₁₅.</figcaption></figure>

Melting of some boron-rich compounds has been studied experimentally, in situ by electrical measurements or/and XRD. The pressure dependence of melting curves of B₄C [47], B₁₃N₂ [48] and B₆O [49] have the following analytical expressions in the 2.7 -7.7 GPa pressure range:

$$ \text{B}_4\text{C (19.5 at\% B): } T_m = 2720(50) - 13(6)\times p \qquad (11) $$

$$ \text{B}_{13}\text{N}_{2}: T_{m} = 2430(20) + 31(3) \times p \qquad (12) $$

$$ \text{B}_6\text{O: } T_m = 2350 + 60(10)\times p \qquad (13) $$

Only B₆O [49] melts congruently in the mentioned pressure range with positive slope; while melting temperature of B₄C decrease with pressure. B₁₃N₂ melts with decomposition.
<figure><img src="image_10.png" /><figcaption>Fig. 7 Lattice parameters (a) of various phases with structure related to tetragonal boron t-B$_{52}$ and best quality reported ex situ DRX (b).</figcaption></figure>

During past years, a number of reports have appeared on high-pressure synthesis of tetragonal allotrope of boron, t-B$_{52}$ (δ-B) phase. Although detailed characterization of the crystal structure has been performed in some simple cases, there remains remarkable variation of the a/c lattice-parameter ratio that raises strong doubts about its uniqueness. The Rietveld refinement (Fig. 7b) or single crystal study remain unique. For example, crystal structure of the high pressure – high temperature boron phase pc-B$_{52}$ synthesized by direct solid-state transformation of rhombohedral β-B$_{106}$ at 20 GPa and 2500 K belongs to the t-B$_{52}$ type, and its structure can be considered as pseudo-cubic with a/c ratio of √2 [44].

At Fig. 7 solid circle (●) represents pc-B$_{52}$ phase synthesized by us, open circles (○) show the lattice parameters of phases obtained at ambient or low pressure (compounds B$_{50}$C$_{2}$ and B$_{50}$B$_{2}$) [50], triangles correspond to t-B$_{52}$ phases synthesized at high pressures from β-B$_{106}$ (V) [50], and borane (Δ) [51].
## Boron nitride

Known allotropes of boron nitride with parameters that allows calculate pressure term in Gibbs energy are listed in Tab. 3 [52]. Their crystal structures are relatively simple and they are presented here just for full thermodynamic consideration of the B-BN system.

Phase diagram to quite high pressure (~25 GPa) contains only 3 phases, hBN, cBN and liquid [53]. Some thermodynamic data were obtained *in situ*, includes p-V-T EOS [53] and some phase equilibria [54]. Experimental phase diagram typical in situ XRD sequence are shown at Fig. 8.

<figure><img src="image_11.png" /><figcaption>Figure 8. Phase diagram of cBN [53] with in situ observation of crystallization [54].</figcaption></figure>

<table><caption>Table 3. Parameters of p-V-T equation of state data for some boron compounds</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs</th></tr></thead><tbody><tr><td>cBN</td><td>3.489</td><td>390</td><td>3.35</td><td>1</td><td>5</td><td>3</td><td>[52]</td></tr><tr><td>wBN</td><td>3.524</td><td>375</td><td>2.3</td><td>1</td><td>5</td><td>5.5</td><td>[52]</td></tr><tr><td>hBN</td><td>2.279</td><td>36.7</td><td>5.6</td><td>41</td><td>0</td><td>2.5</td><td>[52]</td></tr><tr><td>tBN</td><td>2.212</td><td>17.2</td><td>11.4</td><td>41</td><td>0</td><td>2.5</td><td>[52]</td></tr><tr><td>Nano cBN</td><td>3.489</td><td>375</td><td>2.3</td><td>1</td><td>5</td><td>3</td><td>[55]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δₜ is dimensionless.
## Boron oxide

Boron oxide is not a boron-rich compound, but it participates into HPHT equilibria important for synthesis of superhard boron suboxide. Boron oxide exists in two allotropes, namely α-B₂O₃ and β-B₂O₃ high-pressure phase. Crystal structures were previously resolved [56] and can be easily recognized by in situ XRD diffraction.

High pressure thermodynamics in situ have been studied in our previous work [16], including such important points as reversibility of transformation, to avoid equilibrium pressure overestimation due to the hysteresis. Boron oxide is a glass-forming material, and its crystalline forms (including ambient-pressure form) can be obtained principally by HP techniques. It is also important to note that at 0.1 MPa metastable melting of high pressure β-B₂O₃ phase occurs at higher temperature than that of α-B₂O₃. The set of thermodynamic parameters have been established (Tab. 4) that calculated phase equilibria match experimental phase diagram (Fig. 9a).

<figure><img src="image_12.png" /><figcaption>Fig. 9 (a) Phase diagram of B₂O₃ with experimental in situ data [16]; and (b) bulk moduli of crystalline and liquid forms of boron oxide B₂O₃.</figcaption></figure>

<table><caption>Table 4. Parameters of p-V-T equation of state data for boron oxide phases</caption><thead><tr><th></th><th>ρ<sub>0</sub></th><th>B<sub>0</sub></th><th>B'<sub>0</sub></th><th>a</th><th>b</th><th>δ<sub>T</sub></th><th>Refs.</th></tr></thead><tbody><tr><td>α-B<sub>2</sub>O<sub>3</sub></td><td>2.554</td><td>40</td><td>5.5</td><td>30</td><td>80</td><td>6</td><td>[16]</td></tr><tr><td>β-B<sub>2</sub>O<sub>3</sub></td><td>3.11</td><td>170</td><td>5</td><td>15</td><td>11</td><td>5.5</td><td>[16, 35]</td></tr><tr><td>L-B<sub>2</sub>O<sub>3</sub></td><td>2.00 (extr)</td><td>16</td><td>8</td><td>620</td><td>-400*</td><td>7.5</td><td>[16]</td></tr></tbody></table>

Units: ρ₀ in g/cm³; B₀ in GPa; B₀ is dimensionless; a in 10⁶ K⁻¹; b in 10⁹ K⁻²; δT is dimensionless.
## 4. HPHT thermodynamics of binary and ternary systems of B-C-N-O tetrahedron.

### Boron suboxide and B-B₂O₃ system

Boron suboxide [57] is usually believed to be of B₆O stoichiometry with typical oxygen deficiency when produced without high pressure [58]. High pressure allows getting highly stoichiometric species [59], including single crystals. Crystal structures reported are typical for B₆O (Fig. 6b) and similar to other chalcogenides. At higher pressure, although, higher boron content of unit cell is suggested, and the a-c lattice parameter evolution follows similar tendency as in the case of boron carbide with high boron content (B-doped B₄C from thermodynamic point of view).

Theoretical phase diagrams can be speculated taking into account the predicted stability of β-B₆O polymorph [60]. This phase has not been observed experimentally, to the best of our knowledge.

Thermodynamics of B₆O has been extensively studied in situ [34, 49, 61] using energy-dispersive diffraction. Thermoelastic equation of state has been reported based on numerous HPHT data obtained during study of phase transformations.

Experimental melting curve of B₆O is shown at Fig. 10a [49]. At low pressures, up to several GPa, B₆O decomposes into boron-rich and oxygen-rich liquids, and fast quench allows recovery of decomposition products. At the same time at higher pressure the melting is congruent and one can recover well crystallized B₆O ingots.

In situ data up to 5 GPa allowed to get general features of the B-B₂O₃ phase diagram [61], like melting temperature at given concentration, the corresponding phase diagram has been calculated using available thermodynamic data for simple substances and compounds at 0.1 MPa, p-V-T EOS and adjusted parameters for subregulars solution model that allows describing the interaction in liquid phase.

<figure><img src="image_13.png" /><figcaption>Fig. 10 Melting curve of B₆O (a), in situ XRD patterns in the course of B + B₂O₃ = B₆O reaction (b,c), and phase diagram of B-B₂O₃ system (c) with typical in situ experimental points (solid, solid/liquid and liquid states by solid, crossed and open circles respectively).</figcaption></figure>

The HPHT transformations and chemical reactions generally take place in the presence of liquid phase, and under pressure the diffusion process is quite fast to reach the thermodynamic equilibrium. Formation of liquid phase, in some cases, may be explained by metastable eutectic
melting and is often useful at the phase diagram (dashed lines). It has not been observed in the B-B₂O₃ system at 5 GPa.

Phase diagram of B-B₂O₃ system has been calculated up to 20 GPa in recent works using Thermocalc software [62]. Two diagrams at ambient pressure and at 8 GPa are shown at Fig. 11.

<figure><img src="image_14.png" /><figcaption>Fig. 11 Simulated B-B₂O₃ phase diagrams at 0.1 MPa and 8 GPa using ThermoCalc database [62].</figcaption></figure>
## Boron subnitrides and B-BN system

Long debates on the stoichiometry and crystal structure of boron subnitrides [36, 63, 64] let to general admittance of the B₁₃N₂ structural model (Fig. 6b) which is different from both B₆O and B₄C structured, and can be considered as an ordered B₁₃C₂-like structural type. Theoretical calculations also predict thermodynamic stability of B₁₃N₂ (B₁₂.₆₇N₂ exact stoichiometry) [65] rather than B₆N structure. At Fig. 12 we show the thermal expansion of boron subnitride data [37] combined with compressibility data [31] and two experimental points of Fig. 13. The dimensionless Anderson Grüneisen parameter δT is thus expected to be close to 1.5.

<figure><img src="image_15.png" /><figcaption>Fig. 12 The p-V-T equation of state of B₁₃N₂. Experimental points were obtained at HASYLAB (multianvil press MAX80, resistive heating).</figcaption></figure>

The chemical interaction and phase relations in the B-BN system have been in situ studied at 5 GPa and temperatures up to 2800 K using X-ray diffraction with synchrotron radiation [66]. Thus, the thermodynamic data obtained in situ were obtained only for the lowest temperature part of phase diagram. The thermodynamic analysis of the B-BN system based on experimental data allowed us to construct equilibrium and metastable phase diagrams of the system at 5 GPa. The only thermodynamically stable boron subnitride, B₁₃N₂, melts incongruently at 2600 K and forms eutectic equilibrium with boron at 2300 K and 4 at% of nitrogen.

Later studies at high pressure indicated the existence of another thermodynamically stable compounds, B₅₀N₂ [67], and the phase diagram of B-BN system has been constructed up to 20 GPa (Fig. 14).
<figure><img src="image_16.png" /><figcaption>Fig. 13 Typical in situ XRD patterns collected at MAX80: observation of chemical reaction B + BN = B₁₃N₂ at 5 GPa (a) and phase diagram of N-BN system (b).</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14 Simulated B-BN phase diagrams at 0.1 MPa and 8 GPa using ThermoCalc database from ref. [67].</figcaption></figure>
B-N-O system: boron-rich solid solutions

Crystal structures of B~6~(O,N,B) solid solutions reported so far [64] well agree with proposed above simplified model (Fig. 6b).

Typical combined in situ and ex situ data for thermodynamic analysis of B-N-O system [68] is shown at Fig. 15. At first step, the concentration of eutectic in the B~6~O-BN binary section was established ex situ in quench and recovery experiments at 5 GPa, by optical observation of microstructure of polished samples (sample 2). After that, the melting temperature of preformed eutectic alloy has been in situ measured at MAX80 at the same pressure. The adjustment of parameters of subregular liquid solution model was made in order to match this experimental melting temperature; and the ternary phase diagram was constructed using this model.

<figure><img src="image_18.png" /></figure>

**Figure 15.** Typical combined in situ and ex situ data for thermodynamic analysis of B-N-O system: (a1,2,3) concentration of eutectic (optical microscopy ex situ), (b) binary section B~6~O-BN with melting temperature of eutectic (in situ at MAX80) and (c) B-B~2~O~3~-BN phase diagram (liquidus surface BY adjustment of subregular liquid solution model to match experimental melting temperature).
## B-C system at HPHT

From thermodynamic point B₄C is a compound that can dissolve boron in it, decreasing the carbon concentration from ~20 at% down to 9 at% [69]. It can be produced by various methods (e.g. CVD), including high pressure synthesis from the elements [70, 71]. Crystal structure(s) has been long time debated and till now the agreement have not been achieved [72-75]. Various models can be efficiently used for analysis of experimental data. Simplified model for crystal structure refinement is shown at Fig. 6b and seems to be most suitable for HPHT samples, in which the structural disorder is typically less pronounced [76]. Similar models have been developed for thermodynamic simulations (e.g. using CALPHAD method) [69]. At the same time, the predictions of such models should be taken with precaution for HPHT conditions. As for high pressure, it is generally believed to stabilize the regular atomic distribution and all crystallographic sites to be occupied (no vacations noted as V at Fig. 6b) [76]. A number of reports for ‘molecular-like’ [77], ‘inversed molecular-like’ [78] or neither [79] solid behavior can be found in the EOS studies at room temperature. This point, however, is not relevant to thermodynamic stability of various carbides and will not be discussed in this review.

Theoretical phase stability in B-C system was studied by Nathaly Vast's group [73] allows alternative insight into the thermodynamic stability and convex hull of both boron-rich and carbon-like structures. Four atom chains between icosahedra were predicted to be more stable.

Thermodynamic in situ data is very limited [47], and the actual experimental phase diagrams represent principally the data of recovery experiments at ambient conditions. In situ data up to 5 GPa allowed to get general features of the B-C phase diagram, like melting temperature of B₄C [47] that decrease with pressure (in contrast with most other known compounds), the corresponding phase diagram has been calculated using available thermodynamic data for simple substances and compounds at 0.1 MPa, p-V-T EOS and adjusted parameters for subregulars solution model that allows describing the interaction in liquid phase. Phase diagram of B-C system can calculated up to 20 GPa using ThermoCalc software and database reported in ref. [80] (Fig. 16).

<figure><img src="image_19.png" /><figcaption>Figure 16. Phase diagrams of B-C system at 0.1 MPa and 8 GPa simulated using ThermoCalc [80].</figcaption></figure>
### B-C-O system: boron-rich solutions

A number of boron-rich B-C-O solid solutions has been reported in previous studies by McMillan group [64]. Crystal structures of reported so far B-C-O boron-rich solids of alfa-boron structural family can be understood in the framework of described above simplified model (Fig. 6b).

This ternary system has not been studied in situ so far, and only thermodynamic CALPHAD simulations using interaction parameters of binary systems [62, 80] can be made. High-temperature isothermal sections of this diagram are shown at Fig. 17, which is the preliminary estimate the phase stability domains prior to getting reliable experimental data at ambient and high pressures.

<figure><img src="image_20.png" /><figcaption>Figure 17. Tentative isothermal sections of the B-C-B₂O₃ phase diagram at 0.1 MPa and 8 GPa at 2400 K.</figcaption></figure>
### B-C phases with carbon structures

The B-C graphite-like and diamond-like phases are not formally boron rich compounds with crystal structures produced by boron clusters, but boron concentration can reach 50 at% [81]. They are generally considered as metastable, and often decomposes into boron (rich) carbides and boron-doped diamond at high enough pressure. Strictly speaking, they are not electronic analogues of carbon, but can be considered similarly to some extent in both theoretical [82] and experimental [83] research.

The 300-K equations of state of graphite-like B–C phases with boron content from 0 to 50 at% have been studied using powder X-ray diffraction with synchrotron radiation up to 35 GPa [84-86]. The compression axis moduli show the non-monotone dependence on boron content and strongly correlate with lattice parameters. The values of bulk moduli $B_0$ vary from 17.5 GPa (t-BC₄) to 30.7 GPa (t-BC₁.₅); while for turbostratic graphite $B_0$ = 29.5 GPa.

<figure><img src="image_21.png" /><figcaption>**Figure 18.** The p-V-T equation of state of c-BC₅. Experimental points [83] were obtained in situ at ESRF (diamond anvil cell, laser heating).</figcaption></figure>

For diamond-like c-BC₅ [83, 87] the parameters for calculation of high-pressure term in Gibbs energy are: $\rho_0$ = 3.267 gcm⁻³, $B_0$ = 335 GPa, $B_0'$ = 4.5, $a$ = 13 K⁻¹, $b$ = 0 K⁻² and $\delta_T$ = 1.5 (Fig. 18). The last value has been refined using the only HPHT observation in DAC. The parameters of the 300 K EOS of graphite-like phases [84-86] can be found in Tab. 10.
<table><caption>Table 10. 300 K EOS data of turbostratic heavily boron-doped graphitic carbons and turbostratic graphite</caption><thead><tr><th>Phase</th><th>Lattice parameters, Å</th><th>EOS parameters: ρ<sub>0</sub> (g/cm<sup>3</sup>), B<sub>0</sub> (GPa), B<sub>0</sub>'</th></tr></thead><tbody><tr><td>t-BC</td><td>a = 2.450<br>c = 3.405</td><td>ρ<sub>0</sub> = 2.141, B<sub>0</sub>=23, B<sub>0</sub>'=8.0</td></tr><tr><td>t-BC<sub>1.5</sub></td><td>a = 2.459<br>c = 3.449</td><td>ρ<sub>0</sub> = 2.120, B<sub>0</sub>=30.7, B<sub>0</sub>'=8.6</td></tr><tr><td>t-BC<sub>3</sub></td><td>a = 2.478<br>c = 3.442</td><td>ρ<sub>0</sub> = 2.125, B<sub>0</sub>=24.8, B<sub>0</sub>'=12.5</td></tr><tr><td>t-BC<sub>4</sub></td><td>a = 2.469<br>c = 3.430</td><td>ρ<sub>0</sub> = 2.159, B<sub>0</sub>=17.5, B<sub>0</sub>'=13.9</td></tr><tr><td>t-C</td><td>a = 2.440<br>c = 3.440</td><td>ρ<sub>0</sub> = 2.249, B<sub>0</sub>=29.5, B<sub>0</sub>'=9.4</td></tr></tbody></table>
**5. Hardness of boron-rich compounds of B-C-N-O tetrahedron.**

Hardness is a complex property that reflect the ability of solid to resist the penetration or scratching. It is a non-trivial function of thermodynamic parameters like composition (for solid solutions), temperature and pressure. Two letters are important, for example, for high speed cutting tools and deep earth and planetary science. Theoretical models of hardness often refer to thermodynamic properties of solids and their crystal chemistry characteristics. Such models can be purely empirical [88], semiempirical ab initio [89] or purely ab initio taking into account stress-strain relationships in different crystallographic directions [90] or dislocation propagation energy [91]. Generally, all of them provide the 10% accuracy, both at ambient conditions and at high temperatures (Fig. 19a).

Theoretical hardness scale for boron-rich solids has been proposed [92]. Hardness as a function of thermodynamic parameters can be also predicted using so-called “thermodynamic” model of hardness [93] and well agrees with experimental data, at least for concentration and temperature dependences (Fig. 19b,c). A number of ab initio predictions have been made to highlight how mechanical properties can be improved via crystallographic and thermodynamic consideration. Experimentally, the best mechanical properties of boron carbide ceramics require high pressure sintering.

<figure><img src="image_22.png" /><figcaption>**Figure 19.** (a) Theoretical harness scaled vs experimental Vickers hardness. Inset shows B₁₃N₂ single crystal. (b,c) Temperature and concentration dependence of Vickers hardness of boron</figcaption></figure>
carbide: experimental values (symbols) vs predictions of thermodynamic model of hardness (curves).

At Fig. 20 theoretical (o) and experimental (x) values of Vickers hardness of boron-containing binary compounds are presented as a function of the atomic fraction of second element. The approximate boundaries of hard ($H_v$ > 10 GPa), superhard ($H_v$ > 40 GPa) and boron-rich (boron content ≥ 80 at%) phases are given by horizontal and vertical lines. The highest values of hardness can be observed only for pure elements, i.e. HV = 60 GPa in the case of γ-B, around 60% of diamond, the hardest known phase.

<figure><img src="image_23.png" /><figcaption>Figure 20. Vickers hardness of boron-rich compounds.</figcaption></figure>
4. Conclusions

In situ XRD studies of elemental boron and its compounds with light elements are very limited, and only α-B with a relatively simple crystal structure could be clearly observed in some experiments, including its transformation to β-B and melting. This paper combines for the first time available in situ, ex situ and ab initio data that is relevant to calculation of phase equilibria at high pressure, high temperature conditions involving boron and its compounds.
Graphical abstract

<figure><img src="image_24.png" /></figure>
**Supporting information**

N/A

**Author statement**

The paper have been written by the contribution of all coauthors.

**Declaration of competing interest**

No competing interest to declare.

**Data availability**

Data will be available on request.

**Acknowledgements**

AC and YLG acknowledge the ANR – FRANCE (French National Research Agency) for its financial support of the BCSi project number ANR-21-CE08-0018. VZT work was supported by project No. 2023.05/0007 financed by the National Research Fund of Ukraine. Also, VZT thanks for financial support from INP CNRS.

**References**

1. Kurakevych, O.O., Superhard phases of simple substances and binary compounds of the B-C-N-O system: from diamond to the latest results (a Review). Journal of Superhard Materials, 2009. **31**(3): p. 139-157.

2. Kurakevych, O.O., et al., Synthesis of Bulk BC8 Silicon Allotrope by Direct Transformation and Reduced-Pressure Chemical Pathways (v6ol 55, pg 8943, 2016). Inorganic Chemistry, 2016. **55**(19): p. 9949-9949.

3. Kurakevych, O.O., et al., Synthesis of Bulk BC8 Silicon Allotrope by Direct Transformation and Reduced-Pressure Chemical Pathways. Inorganic Chemistry, 2016. **55**(17): p. 8943-8950.

4. Shimomura, O., Current Activity of MAX80 at the Photon Factory. Physica B+C, 1986. **139-140**: p. 292-300.

5. Guignard, J. and W.A. Crichton, The large volume press facility at ID06 beamline of the European synchrotron radiation facility as a High Pressure-High Temperature deformation apparatus. Rev. Sci. Instrum., 2015. **86**(8): p. 085112.

6. Le Godec, Y. and A. Courac, In Situ High-Pressure Synthesis of New Outstanding Light-Element Materials under Industrial P-T Range. Materials, 2021. **14**(15): p. 4245.

7. Solozhenko, V.L., et al., Kinetics of diamond crystallization from the melt of the Fe-Ni-C system. Journal of Physical Chemistry B, 2002. **106**(26): p. 6634-6637.

8. Turkevich, V.Z., et al., Mg–C System up to 20 GPa: Its Phase Diagram and Stable Magnesium Carbides. The Journal of Physical Chemistry C, 2023. **127**(4): p. 1965-1972.

9. Strobel, T.A., et al., Synthesis of beta-Mg2C3: A Monoclinic High-Pressure Polymorph of Magnesium Sesquicarbide. Inorganic Chemistry, 2014. **53**(13): p. 7020-7027.

10. Kurakevych, O.O., et al., High-Pressure and High-Temperature Stability of Antifluorite Mg2C by in Situ X-ray Diffraction and ab Initio Calculations. Journal of Physical Chemistry C, 2014. **118**(15): p. 8128-8133.

11. Kurakevych, O.O., et al., Synthesis of Mg2C: A Magnesium Methanide. Angewandte Chemie-International Edition, 2013. **52**(34): p. 8930-8933.
12. Anderson, O.L. and D.G. Isaak, *The dependence of the Anderson-Grueneisen parameter $\delta_T$ upon compression at extreme conditions*. J. Phys. Chem. Solids, 1993. **54**(2): p. 221-227.

13. Kurakevych, O.O., Y. Le Godec, and V.L. Solozhenko, *Integrated form of the Anderson-Gruneisen equation of state for p-V-T data fit: Application to the compounds of boron, carbon, silicon and some metals*. Journal of Physics - Conference Series, 2017. **950**: p. 042023.

14. Murnaghan, F.D., *The compressibility of media under extreme pressures*. Proc. Nation. Acad. Sci. USA, 1944. **30**(9): p. 244-247.

15. Chopelas, A. and R. Boehler, *Thermal expansion measurements at very high pressure, systematics, and a case for a chemically homogeneous mantle*. Geophysical Research Letters, 1989. **16**(11): p. 1347-1350.

16. Solozhenko, V.L., et al., *Thermodynamically Consistent p-T Phase Diagram of Boron Oxide $B_2O_3$ by in Situ Probing and Thermodynamic Analysis*. Journal of Physical Chemistry C, 2015. **119**(35): p. 20600-20605.

17. Oganov, A.R., et al., *Ionic high-pressure form of elemental boron*. Nature, 2009. **457**(7231): p. 863-867.

18. Oganov, A.R., et al., *Ionic high-pressure form of elemental boron (vol 457, pg 863, 2009)*. Nature, 2009. **460**(7252): p. 292-292.

19. Parakhonskiy, G., et al., *High pressure synthesis and investigation of single crystals of metastable boron phases*. High Pressure Research, 2013. **33**(3): p. 673-683.

20. Hayami, W., et al., *Thermodynamic stability of elemental boron allotropes with varying numbers of interstitial atoms*. Journal of Solid State Chemistry, 2024. **329**: p. 124407.

21. Solozhenko, V.L. and O.O. Kurakevych, *Equilibrium p-T Phase Diagram of Boron: Experimental Study and Thermodynamic Analysis*. SCIENTIFIC REPORTS, 2013. **3**.

22. Chuvashova, I., et al., *Nonicosahedral boron allotrope synthesized at high pressure and high temperature*. Physical Review B, 2017. **95**(18): p. 180102.

23. Parakhonskiy, G., et al., *High pressure synthesis of single crystals of $\alpha$-boron*. Journal of Crystal Growth, 2011. **321**(1): p. 162-166.

24. Chakraborti, A., et al., *When carbon impurities trigger the synthesis of $\alpha$-boron at high pressure and high temperature*. Acta Materialia, 2023. **249**: p. 118820.

25. Solozhenko, V.L., O.O. Kurakevych, and A.R. Oganov, *On the hardness of a new boron phase, orthorhombic $\gamma\alpha_2B-28$*. Journal of Superhard Materials, 2008. **30**(6): p. 428-429.

26. Ogitsu, T., E. Schwegler, and G. Galli, *$\beta$-Rhombohedral Boron: At the Crossroads of the Chemistry of Boron and the Physics of Frustration*. Chemical Reviews, 2013. **113**(5): p. 3425-3449.

27. Albert, B. and H. Hillebrecht, *Boron: Elementary Challenge for Experimenters and Theoreticians*. Angewandte Chemie International Edition, 2009. **48**(46): p. 8640-8668.

28. Parakhonskiy, G., et al., *Experimental pressure-temperature phase diagram of boron: resolving the long-standing enigma*. Scientific Reports, 2011. **1**(1): p. 96.

29. White, M.A., et al., *Determination of Phase Stability of Elemental Boron*. Angewandte Chemie International Edition, 2015. **54**(12): p. 3626-3629.

30. Le Godec, Y., et al., *Equation of state of orthorhombic boron, $\gamma\alpha_2B-28$. Solid State Communications, 2009. **149**(33-34): p. 1356-1358.

31. Kurakevych, O.O. and V.L. Solozhenko, *300-K equation of state of rhombohedral boron subnitride, $B_13N_2$. Solid State Communications, 2009. **149**(47-48): p. 2169-2171.

32. Chuvashova, I., et al., *High-pressure behavior of $\alpha$-boron studied on single crystals by X-ray diffraction, Raman and IR spectroscopy*. Journal of Solid State Chemistry, 2017. **245**: p. 50-60.

33. Cherednichenko, K.A. and V.L. Solozhenko, *Thermal expansion of $\alpha$-boron and some boron-rich pnictides*. Solid State Communications, 2019. **303-304**: p. 113735.
34. Kurakevych, O.O. and V.L. Solozhenko, *Thermoelastic equation of state of boron suboxide B6O up to 6 GPa and 2700 K: Simplified Anderson-Gruneisen model and thermodynamic consistency*. Journal of Superhard Materials, 2014, **36**(4): p. 270-278.

35. Nieto-Sanz, D., et al., *X-ray study of the synthesis of boron oxides at high pressure: phase diagram and equation of state*. Physical Review B, 2004. **70**: p. 214108 1-6.

36. Kurakevych, O.O. and V.L. Solozhenko, *Rhombohedral boron subnitride, B13N2, by X-ray powder diffraction*. Acta Crystallographica Section C-Crystal Structure Communications, 2007. **63**: p. I80-I82.

37. Cherednichenko, K.A., L. Gigli, and V.L. Solozhenko, *Thermal expansion of boron subnitrides*. SOLID STATE COMMUNICATIONS, 2018. **275**: p. 21-24.

38. Cherednichenko, K.A. and V.L. Solozhenko, *Structure and equation of state of tetragonal boron subnitride B50N2*. JOURNAL OF APPLIED PHYSICS, 2017. **122**(15).

39. Nelmes, R.J., et al., *Neutron- and x-ray-diffraction measurements of the bulk modulus of boron*. Physical Review B, 1993. **47**(13): p. 7668-7673.

40. Lundström, T., B. Lönnberg, and J. Bauer, *Thermal expansion of β-rhombohedral boron*. Journal of Alloys and Compounds, 1998. **267**(1): p. 54-58.

41. Sanz, D.N., P. Loubeyre, and M. Mezouar, *Equation of State and Pressure Induced Amorphization of β-Boron from X-Ray Measurements up to 100 GPa*. Physical Review Letters, 2002. **89**(24): p. 245501.

42. Somayazulu, M., et al., *P-V-T equation of state of boron carbide*. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2023. **381**(2258): p. 20220331.

43. Qin, J., et al., *Phase relations in boron at pressures up to 18 GPa and temperatures up to 2200 °C*. Physical Review B, 2012. **85**(1): p. 014107.

44. Kurakevych, O.O. and V.L. Solozhenko, *Crystal structure of dense pseudo-cubic boron allotrope, pc-B-52, by powder X-ray diffraction*. Journal of Superhard Materials, 2013. **35**(1): p. 60-63.

45. Ploog, K., et al., *B48B2C2 and B48B2N2, two non-metal borides with structure of so-called I-tetragonal boron*. Journal of Less-Common Metals, 1972. **29**: p. 161-169.

46. Vlasse, M., et al., *Crystal structure of tetragonal boron related to α-AlB12*. Journal of Solid State Chemistry, 1979. **28**(3): p. 289-301.

47. Mukhanov, V.A., P.S. Sokolov, and V.L. Solozhenko, *On melting of B4C boron carbide under pressure*. Journal of Superhard Materials, 2012. **34**(3): p. 211-213.

48. Solozhenko, V.L. and V.A. Mukhanov, *On melting of boron subnitride B13N2 under pressure*. Journal of Superhard Materials, 2017. **39**(2): p. 147-148.

49. Kurakevych, O.O. and V.L. Solozhenko, *Experimental Study and Critical Review of Structural, Thermodynamic and Mechanical Properties of Superhard Refractory Boron Suboxide B6O*. Journal of Superhard Materials, 2011. **33**(6): p. 421-428.

50. Kurakevych, O.O., et al., *Comparison of solid-state crystallization of boron polymorphs at ambient and high pressures*. High Pressure Research, 2012. **32**(1): p. 30-38.

51. Ekimov, E.A., I.P. Zibrov, and A.V. Zoteev, *Preparation of Boron Microcrystals via High-Pressure, High-Temperature Pyrolysis of Decaborane, B10H14*. Inorg. Mater., 2011. **47**(11): p. 1194-1198.

52. Kurakevych, O.O. and V.L. Solozhenko, *High-Pressure Design of Advanced BN-Based Materials*. Molecules, 2016. **21**(10).

53. Solozhenko, V.L., V.Z. Turkevich, and W.B. Holzapfel, *Refined Phase Diagram of Boron Nitride*. J. Phys. Chem. B, 1999. **103**(15): p. 2903-2905.

54. Solozhenko, V.L., et al., *In situ studies of boron nitride crystallization from BN solutions in supercritical N-H fluid at high pressures and temperatures*. Physical Chemistry Chemical Physics, 2002. **4**(21): p. 5386-5393.
55. Le Godec, Y., et al., *Effect of nanostructuration on compressibility of cubic BN*. Journal of Superhard Materials, 2012. **34**(5): p. 336-338.

56. Prewitt, C.T. and R.D. Shannon, *Crystal structure of a high-pressure form of B₂O₃*. Acta Crystallographica B, 1968. **24**: p. 869-874.

57. Rizzo, H.F., W.C. Simmons, and H.O. Bielstein, *The Existence and Formation of the Solid B₆O*. Journal of the Electrochemical Society, 1962. **109**(11): p. 1079-1082.

58. Olofsson, M. and T. Lundstrom, *Synthesis and structure of non-stoichiometric B₆O*. Journal of Alloys Compounds, 1997. **257**: p. 91-95.

59. Hubert, H., et al., *High-pressure, high-temperature synthesis and characterization of boron suboxide (B₆O)*. Chemistry of Materials, 1998. **10**(6): p. 1530-1537.

60. Dong, H., et al., *Prediction of a new ground state of superhard compound B₆O at ambient conditions*. Scientific Reports, 2016. **6**(1): p. 31288.

61. Solozhenko, V.L., et al., *Phase diagram of the B-B₂O₃ system at 5 GPa: Experimental and theoretical studies*. Journal of Physical Chemistry B, 2008. **112**(21): p. 6683-6687.

62. Turkevich, V.Z., D.V. Turkevich, and V.L. Solozhenko, *Phase diagram of the B–B₂O₃ system at pressures to 24 GPa*. Journal of Superhard Materials, 2016. **38**(3): p. 216-218.

63. Solozhenko, V.L., Y. Le Godec, and O.O. Kurakevych, *Solid-state synthesis of boron subnitride, B₆N: myth or reality?*. Comptes Rendus Chimie, 2006. **9**(11-12): p. 1472-1475.

64. Hubert, H., et al., *High-pressure, high-temperature syntheses in the B–C–N–O system. I. Preparation and Characterization*. Journal of Solid State Chemistry, 1997. **133**: p. 356-364.

65. Zhang, H., S. Yao, and M. Widom, *Predicted phase diagram of boron-carbon-nitrogen*. Physical Review B, 2016. **93**(14): p. 144107.

66. Solozhenko, V.L., et al., *Phase Diagram of the B-BN System at 5 GPa*. JOURNAL OF PHYSICAL CHEMISTRY B, 2010. **114**(17): p. 5819-5822.

67. Solozhenko, V.L. and V.Z. Turkevich, *Phase Diagram of the B-BN System at Pressures up to 24 GPa: Experimental Study and Thermodynamic Analysis*. JOURNAL OF PHYSICAL CHEMISTRY C, 2018. **122**(15): p. 8505-8509.

68. Solozhenko, V.L., et al., *Phase Equilibria in the B-BN-B₂O₃ System at 5 GPa*. Journal of Physical Chemistry C, 2013. **117**(36): p. 18642-18647.

69. Rogl, P.F., et al., *The B-rich side of the B–C phase diagram*. Calphad, 2014. **44**: p. 3-9.

70. Chakraborti, A., N. Vast, and Y. Le Godec, *Synthesis of boron carbide from its elements at high pressures and high temperatures*. Solid State Sciences, 2020. **104**: p. 106265.

71. Iwan, S., et al., *High-pressure high-temperature synthesis and characterization of B10C*. AIP Advances, 2024. **14**(6).

72. Shirai, K., K. Sakuma, and N. Uemura, *Theoretical study of the structure of boron carbide ${\mathrm{B}}_{2}{{\mathbf{B}}}_{6}$*. Physical Review B, 2014. **90**(6): p. 064109.

73. Jay, A., et al., *Theoretical phase diagram of boron carbide from ambient to high pressure and temperature*. Journal of Applied Physics, 2019. **125**(18).

74. Yang, X., W.A. Goddard, III, and Q. An, *Structure and Properties of Boron-Very-Rich Boron Carbides: B12 Icosahedra Linked through Bent CBB Chains*. The Journal of Physical Chemistry C, 2018. **122**(4): p. 2448-2453.

75. Clark, H.K. and J.L. Hoard, *The Crystal Structure of Boron Carbide*. Journal of the American Chemical Society, 1943. **65**(11): p. 2115-2119.

76. Mondal, S., et al., *Disorder and defects are not intrinsic to boron carbide*. Scientific Reports, 2016. **6**(1): p. 19330.

77. Lazzari, R., et al., *Atomic Structure and Vibrational Properties of Icosahedral ${\mathbf{B}}_{2}{{\mathbf{C}}}_{6}$*. Physical Review Letters, 1999. **83**(16): p. 3230-3233.
78. Nelmes, R.J., et al., *Observation of Inverted-Molecular Compression in Boron Carbide*. Physical Review Letters, 1995. **74**(12): p. 2268-2271.

79. Chuvashova, I., et al., *Structural stability and mechanism of compression of stoichiometric B13C2 up to 68GPa*. Scientific Reports, 2017. **7**(1): p. 8969.

80. Turkevich, V.Z. and V.L. Solozhenko, *Thermodynamic calculation of the B-C system at pressures to 24 GPa*. Journal of Superhard Materials, 2014. **36**(5): p. 358-360.

81. Shirasaki, T., et al., *Synthesis and characterization of boron-substituted carbons*. Carbon, 2000. **38**(10): p. 1461-1467.

82. Lowther, J.E., *Potential super-hard phases and the stability of diamond-like boron-carbon structures*. Journal of Physics: Condensed Matter, 2005. **17**(21): p. 3221.

83. Solozhenko, V.L., et al., *Ultimate Metastable Solubility of Boron in Diamond: Synthesis of Superhard Diamondlike BC5*. Physical Review Letters, 2009. **102**(1): p. 015506.

84. Kurakevych, O.O., T. Chauveau, and V.L. Solozhenko, *On Crystal Lattice Parameters of Graphite-Like Phases of the B-C System*. Journal of Superhard Materials, 2010. **32**(4): p. 231-235.

85. Kurakevych, O.O., E.G. Solozhenko, and V.L. Solozhenko, *High-pressure study of graphite-like B-C phases*. High Pressure Research, 2009. **29**(4): p. 605-611.

86. Solozhenko, V.L., et al., *Equation of state of graphite-like BC*. Solid State Communications, 2006. **137**(5): p. 268-271.

87. Solozhenko, V.L., et al., *Ultimate Metastable Solubility of Boron in Diamond: Synthesis of Superhard Diamondlike BC5 (vol 102, 015506, 2009)*. Physical Review Letters, 2009. **102**(17): p. 179901.

88. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *The interrelation between hardness and compressibility of substances and their structure and thermodynamic properties*. Journal of Superhard Materials, 2008. **30**(6): p. 368-378.

89. Mazhnik, E. and A.R. Oganov, *A model of hardness and fracture toughness of solids*. Journal of Applied Physics, 2019. **126**(12).

90. Veprek, S., R.F. Zhang, and A.S. Argon, *Mechanical properties and hardness of boron and boron-rich solids*. Journal of Superhard Materials, 2011. **33**(6): p. 409-420.

91. Gilman, J.J., *Why Silicon Is Hard*. Science, 1993. **261**(5127): p. 1436-1439.

92. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *Thermodynamic Model of Hardness: Particular Case of Boron-Rich Solids*. Journal of Superhard Materials, 2010. **32**(3): p. 167-176.

93. Mukhanov, V.A., O.O. Kurakevych, and V.L. Solozhenko, *Hardness of materials at high temperature and high pressure*. Philosophical Magazine, 2009. **89**(25): p. 2117-2127.
**Declaration of interests**

[x] The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

[ ] The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: