# DFT-based thermodynamic stabilities of the ternary compounds and calculation of the system U-B-C

J. Vrestal, R. Podloucky, H. Noel, P. F. Rogl

► To cite this version:

J. Vrestal, R. Podloucky, H. Noel, P. F. Rogl. DFT-based thermodynamic stabilities of the ternary compounds and calculation of the system U-B-C. Journal of Alloys and Compounds, 2025, 1028, pp.180592. 10.1016/j.jallcom.2025.180592. hal-05108378

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Research Article

# DFT-based thermodynamic stabilities of the ternary compounds and calculation of the system U-B-C

J. Vrestal<sup>a</sup>, R. Podloucky<sup>b</sup>, H. Noël<sup>c</sup>, P.F. Rogl<sup>b,*</sup>

<figure><img src="image_3.png" /></figure>

ARTICLE INFO

Keywords:

Uranium boron carbides
First-principles calculation
Phase diagram calculation
Electronic density of states

ABSTRACT

Based on experimental data for (a) the isothermal section at 1600°C, (b) the melting points of UBC and UB₂C as well as (c) for the energy of formation of the aforementioned two ternary compounds, which were calculated from density functional theory (DFT), the thermodynamic phase equilibria in the ternary system have been calculated i.e. phase relations have been defined for the isothermal sections at 1600°C, 1800°C, 2000°C, 2200°C, 2400°C and for four isopleths B - UC, U - B₄C, UB₂ - UC₂, and B₄C - UC. The equilibrium solidification path of alloys is outlined in a liquidus surface and a Schultz-Scheil diagram. The thermodynamic modelling perfectly complies with the phase equilibria observed experimentally, with the relevant homogeneity regions of the binary and ternary phases, as well as with the regions of primary crystallization of the ternary solid phases, UB₁ₓC₁+x and UB₂C. Due to the two-phase equilibria formed between the ternary compounds and between binary borides and carbides, we neither observe compatibility between uranium metal and B₄C, nor between B₄C and UC. Applying the Vienna Ab initio Simulation Package VASP for density-functional-theory (DFT) based first-principles calculations, the compounds UBC, UB₂C (high and low temperature modifications) as well as the hypothetical compounds "U₂B₂C₃" and "U₃B₂C₃", which both adopt the structure of the Th-homologues, were studied. In all cases the crystal structures were fully relaxed by optimizing total energy and strain. Because of the heavy element U, spin-orbit coupling (SOC) was included in the DFT calculations. The resulting total energies serve as basis for deriving the enthalpy of formation at zero Kelvin. Electron densities of states (eDOS) are presented as well as results for magnetic moments, which are in the range of ~0.2 to ~1.7 µB.

1. Introduction

As part of the so-called Dragon Project [1] in the fifties of the last century, the ternary system uranium - boron - carbon was at the focus of a high temperature reactor with actinoid-metal carbide fuel. Recent generation IV fission plants reconsidered dense carbide and carbide/-oxide fuels [2-4]. These activities demand a detailed knowledge and understanding of the physical and chemical behaviour of the nuclear fuel during fabrication processes, during normal operation as well as in accident situations. In demand of a general understanding of the phenomena occurring during a severe accident in light water reactors (LWRs with UO₂-fuel in Zircaloy cladding), implying partial or complete core melting, thermodynamic models are generated to predict the behaviour of the melts formed from the degradation of the core materials. Therefore, thermodynamic calculations tried to establish a

database (MASCA project) of the molten corium stratification extending the base system U-Zr-O-Fe also to B₄C absorber rods [5-8]. In this context the ternary system uranium - boron - carbon has drawn increased interest. A thorough description of the constitution of the ternary phase diagram U-B-C was given by Rogl [9] comprising the early stages of phase diagram work [10], the reinvestigation of the system at 1600°C [11] and the refinement of the crystal structures of the three ternary compounds UB₁ₓC₁+x (0 ≤ x ≤ 0.22 at 1600°C [12]), UB₂C with a high and a low temperature modification [13] (diffusion-controlled transformation at 1675 ± 25°C) and U₅B₂C₇ (detailed structure still unknown). Details on the physical properties of the compounds are available for UB₂C, which in the high temperature modification is an itinerant ferromagnet below 75 K exhibiting strong electron correlations (Kondo interactions) [14-17], whereas the low temperature modification behaves as a temperature independent paramagnet [12]. The

<footnote><sup>a</sup> Department of Chemistry, Faculty of Science, Masaryk University, Kotlarska 2, Brno 61137, Czech Republic</footnote>

<footnote><sup>b</sup> Institute of Materials Chemistry, Universität Wien, Währingerstrasse 42, Wien A-1090, Austria</footnote>

<footnote><sup>c</sup> Laboratoire de Chimie du Solide et Inorganique Moléculaire, UMR-CNRS 6511, Université de Rennes I, Avenue du Général Leclerc, Rennes Cedex F-35042, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: peter.franz.rogl@univie.ac.at (P.F. Rogl).</footnote>
<table><caption>**Table 1**

Thermodynamic stabilities (DFT-data) of the ternary U-B-C compounds (tw = this work).</caption><thead><tr><th rowspan="2">Compound</th><th rowspan="2">Structure Type</th><th rowspan="2">Space Group</th><th colspan="3">Lattice Parameters (in nm)</th><th rowspan="2">ΔH<sub>f</sub> (0 K)<br>kJ/gatom</th><th rowspan="2">Refs.</th></tr><tr><th>a</th><th>b</th><th>c</th></tr></thead><tbody><tr><td>UBC<sup>§</sup></td><td>ThBC</td><td>P4<sub>1</sub>2<sub>1</sub></td><td>0.34612</td><td>0.34612</td><td>2.39481</td><td><b>-46.422</b></td><td>DFT, tw</td></tr><tr><td>UBB<sup>*</sup></td><td>UBC</td><td>Cmcm</td><td>0.38942</td><td>1.23928</td><td>0.30824</td><td><b>-15.960</b></td><td>DFT, tw</td></tr><tr><td rowspan="2">UBC</td><td rowspan="2">UBC</td><td rowspan="2">Cmcm</td><td>0.35732</td><td>1.18584</td><td>0.33881</td><td>-</td><td>[11]</td></tr><tr><td>0.36254</td><td>1.19504</td><td>0.32895</td><td><b>-49.025</b></td><td>DFT, tw</td></tr><tr><td>UB<sub>0.78</sub>C<sub>1.22</sub></td><td>UBC</td><td>Cmcm</td><td>0.35908</td><td>1.19957</td><td>0.33470</td><td>-</td><td>[11]</td></tr><tr><td>UB<sub>0.75</sub>C<sub>1.25</sub></td><td>UBC</td><td>Cmcm</td><td>0.36503</td><td>1.18304</td><td>0.32344</td><td><b>-41.726</b></td><td>DFT, tw</td></tr><tr><td>UCC<sup>*</sup></td><td>UBC</td><td>Cmcm</td><td>0.36046</td><td>1.14295</td><td>0.33208</td><td><b>-5.687</b></td><td>DFT, tw</td></tr><tr><td rowspan="2">ℓT-UB<sub>2</sub>C<sup>#</sup></td><td rowspan="2">UB<sub>2</sub>C</td><td rowspan="2">Pmma</td><td>0.60338</td><td>0.35177</td><td>0.41067</td><td>-</td><td>[13]</td></tr><tr><td>0.60257</td><td>0.35045</td><td>0.41026</td><td><b>-43.687</b></td><td>DFT, tw</td></tr><tr><td rowspan="2">hT-UB<sub>2</sub>C<sup>#</sup></td><td rowspan="2">ThB<sub>2</sub>C</td><td rowspan="2">R-3m</td><td>0.65224</td><td>0.65224</td><td>1.07909</td><td>-</td><td>[12]</td></tr><tr><td>0.65413</td><td>0.65413</td><td>1.07232</td><td><b>-41.743</b></td><td>DFT, tw</td></tr><tr><td>U<sub>5</sub>B<sub>2</sub>C<sub>7</sub></td><td>Ho<sub>5</sub>B<sub>2</sub>C<sub>6-7</sub></td><td>P4/ncc</td><td>0.784</td><td>0.784</td><td>2.358</td><td><b>-44.3</b><sup>§</sup></td><td>[11]</td></tr><tr><td>U<sub>2</sub>B<sub>2</sub>C<sub>3</sub><sup>§</sup></td><td>Th<sub>2</sub>B<sub>2</sub>C<sub>3</sub></td><td>Pnnm</td><td>1.23642</td><td>0.39046</td><td>0.33214</td><td><b>-33.452</b></td><td>DFT, tw</td></tr><tr><td>U<sub>3</sub>B<sub>2</sub>C<sub>3</sub><sup>§</sup></td><td>Th<sub>3</sub>B<sub>2</sub>C<sub>3</sub></td><td>P2/m</td><td>0.33087</td><td>0.36042</td><td>0.85949</td><td><b>-40.503</b></td><td>DFT, tw</td></tr></tbody><tfoot><tr><td colspan="8" style="text-align: center;">β= 100.06°</td></tr></tfoot></table>

<figure><img src="image_4.png" /><figcaption>**Fig. 1.** DFT calculated energies of formation (at 0 K) for ternary compounds in the system U-B-C along the section B - UC.</figcaption></figure>

paramagnetic solid solution UB<sub>1-x</sub>C<sub>1+x</sub> reveals typical signs of a valence instability of the U-atoms [12]. Earlier studies with quantum chemical approaches were made for rare-earth B-C compounds with crystal structures similar to the present U-B-C compounds focussing on the B-C substructures [18–23].

Although phase relations were used to estimate the Gibbs free energy of formation for UBC, UB<sub>2</sub>C and U<sub>5</sub>B<sub>2</sub>C<sub>7</sub> [11], no attempts were made yet to perform a thermodynamic calculation of the phase constitution of the entire ternary system including the liquid. For comprehensive knowledge on a multi-component fuel-control system a detailed description of the Gibbs energy of all the phases of a system is a necessity. Therefore, the present paper will provide (i) a thorough thermodynamic description of the stability of all ternary compounds involved via DFT calculations and will build on it (ii) a critical thermodynamic CALPHAD assessment of the constitution of the entire U-B-C phase diagram based on the available experimental phase relations and melting temperatures as given in [11]; (iii) DFT calculations of the electronic structure of the ternary compounds will furthermore provide insight into the structure property relations.

## 2. Energetics

### 2.1. DFT calculations

Density functional theory (DFT) calculations for the uranium boron carbides studied in this work were performed by applying the Vienna ab initio simulation package (VASP) [24,25] for which pseudo-potentials were constructed according to the projector-augmented-wave method [26]. The corresponding atomic configuration of the U pseudo-potential consisted of the 14 valence states 6s<sup>2</sup>, 6p<sup>6</sup>, 6d<sup>2</sup>, 5f<sup>2</sup>, 7s<sup>2</sup>, whereas for C the 4 valence states 2s<sup>2</sup>, 2p<sup>2</sup> and for B the 3 valence states 2s<sup>2</sup>, 2p<sup>1</sup> were included. For the exchange-correlation functional the generalized gradient approximation according to Perdew et al. (PBE) [27] was chosen. The VASP calculations were done with high precision according to the energy cutoff at 600 eV. Because of the heavy element U, relativistic effects were included in terms of spin orbit coupling (SOC) in the Hamiltonian. The $\vec{k}$ space grids for integrating over the Brillouin zones were derived for the linear tetrahedron integration method [28] according to the parameter KSPACING = 0.1 in the INCAR file of VASP. This choice resulted for example in a 23 × 23 × 23 grid of $\vec{k}$ points in the Brillouin zone of UBC. The crystal structures - atomic positions as well as cell shapes and parameters - were fully relaxed by minimizing forces and components of stress tensors.

<footnote>§ hypothetical Th-analogues; * binary end-members of UB<sub>1-x</sub>C<sub>1+x</sub>; & space group C2/m: a= 1.19013, b= 0.35835, c= 0.33692 nm, β= 90.0°; # transition temperature 1675 ± 25°C; † derived from ternary phase relations [11].</footnote>
<table><caption>**Table 2**

Assessed parameters for the B-C-U system (binary parameters from [36,38,39]).</caption><thead><tr><th colspan="2">BCU</th></tr></thead><tbody><tr><td colspan="2">3 SUBLATTICES SITES 1:1:1</td></tr><tr><td colspan="2">CONSTITUENTS: B,C:B,C:U:</td></tr><tr><td>G<sup>BCU</sup><sub>B:C:U</sub></td><td>= 298.15 &lt; T &lt; 1873 -147070.-137.22*T<br/>1873 &lt; T &lt; 2373 -147070.-137.22*T-(T-1873)*121.41<br/>2373 &lt; T &lt; 2573 -147070.-137.22*T-(T-1873)*121.41-(T-2373)*55.14<br/>2573 &lt; T &lt; 6000 -147070.-137.22*T-(T-1873)*121.41-(T-2373)*55.14+(T-2573)*105.04</td></tr><tr><td>G<sup>BCU</sup><sub>C:C:U</sub></td><td>= 298.15 &lt; T &lt; 6000 -17060.-175.*T</td></tr><tr><td>G<sup>BCU</sup><sub>B:B:U</sub></td><td>= 298.15 &lt; T &lt; 6000 -47880.-140.*T</td></tr></tbody></table>

<table><thead><tr><th colspan="2">BCCU</th></tr></thead><tbody><tr><td colspan="2">3 SUBLATTICES SITES 1.25: 0.75: 1</td></tr><tr><td colspan="2">CONSTITUENTS: B,C:B,C:U:</td></tr><tr><td>G<sup>BCCU</sup><sub>C:B:U</sub></td><td>= 298.15 &lt; T &lt; 6000 -125176.-145.*T</td></tr></tbody></table>

<table><thead><tr><th colspan="2">B2CU</th></tr></thead><tbody><tr><td colspan="2">3 SUBLATTICES SITES 2: 1: 1</td></tr><tr><td colspan="2">CONSTITUENTS: B:C:U:</td></tr><tr><td>G<sup>B2CU</sup><sub>B:C:U</sub></td><td>= 298.15 &lt; T &lt; 1873 -174740.-163.0*T<br/>1873 &lt; T &lt; 2373 -174740.-163.0*T-(T-1873)*168.25<br/>2373 &lt; T &lt; 6000 -174740.-163.0*T-(T-1873)*168.25-(T-2373)*50.00</td></tr></tbody></table>

<table><thead><tr><th colspan="2">U5B2C7</th></tr></thead><tbody><tr><td colspan="2">3 SUBLATTICES SITES 5: 2: 7</td></tr><tr><td colspan="2">CONSTITUENTS: U:B:C:</td></tr><tr><td>G<sup>USB2C7</sup><sub>U:B:C</sub></td><td>= 298.15 &lt; T &lt; 1873 -630000.-661.0*T<br/>1873 &lt; T &lt; 6000 -1868053.-1119.0*(T-1873)</td></tr></tbody></table>

<table><thead><tr><th colspan="2">LIQUID</th></tr></thead><tbody><tr><td colspan="2">1 SUBLATTICE 1</td></tr><tr><td colspan="2">CONSTITUENTS: B,C,U:</td></tr><tr><td><sup>0</sup>L<sup>LIQUID</sup><sub>B:C,U</sub></td><td>= 298.15 &lt; T &lt; 6000 -205312.33</td></tr></tbody></table>

## 2.2. Thermodynamic modelling

Thermodynamic modelling of the B-C-U system was performed by the CALPHAD method [29] calculating equilibrium phase compositions of the system by determining the minimum of its total Gibbs energy. Searching of this minimum was done by the PANDAT software package [30], which was used also for phase diagram calculation. The thermodynamic description of phase φ is based on the thermodynamic description (Gibbs energy) of elements i in the phase φ in the commonly used polynomial form:

$$G_i^{\Phi} = a + bT + cT \ln T + \Sigma d_n T^n \quad (1)$$

where a, b, c, d_n and n (n typically equal to 2, 3, and -1) are constants characteristic for the particular structure of the element i in a given temperature interval [31].

In the solution phases, Gibbs energy can be expressed in the Compound Energy Formalism (CEF) as the sum of proper contributions beside the reference value: Gibbs energy of ideal mixing (G_id), Gibbs energy of excess contribution by mutual interactions (G_ex) [32] and further possible contributions (magnetic, pressure etc.):

$$G^{\Phi} = \Sigma_{i} x_{i} G_{i}^{\Phi} + G_{id} + G_{ex} \quad (2)$$

The ideal mixing contribution to the total Gibbs energy is given by:

$$G_{id} = RT \Sigma_{i} x_{i} \ln x_{i} \quad (3)$$

Excess Gibbs energy contribution from mutual interaction of components is expressed by the Redlich-Kister polynomial [33]:

$$\exp G_m^{\Phi} = \sum_{i=1}^{n} (-1)^{\sum_{j=1}^{n} x_j x_{ij} L_{ijk}}, \quad \text{and} \quad \exp G_m^{\Phi} = \sum_{i=1}^{n-2} \sum_{j=1}^{n-1} \sum_{k=j+1}^{n} x_i x_j x_k L_{ijk}, \quad (4)$$

where x_i, x_j, x_k are molar fractions of components and n (n = 0, ..., n) is the summation index for parameters L, which may be temperature dependent:

$$L_{12} = A_{12} + B_{12}T, \quad \text{and} \quad L_{123} = A_{123} + B_{123}T \quad (5)$$

For compounds, the Gibbs energy of formation of the phase φ is expressed by the equation:

$$G^{\Phi} = \Delta H^{\Phi} - T \Delta S^{\Phi} \quad (6)$$

where ΔH^Φ is the enthalpy of formation at 0 K, calculated by the DFT procedure and the temperature coefficient is optimized on phase equilibrium data [34,35].

# 3. Results and discussion

## 3.1. DFT data for the ternary compounds and solid solutions

The DFT total energies for the fully relaxed crystal structures of the compounds E_DFT (U_kB_mC_n) were exploited for deriving the DFT formation enthalpies:

$$\Delta H_f(U_kB_mC_n, 0 K) = E_{\text{DFT}}(U_kB_mC_n) - (kE_{\text{DFT}}(U) + m(E_{\text{DFT}}(B) + nE_{\text{DFT}}(C))) \quad (7)$$

There, E_DFT(U), E_DFT(B), E_DFT(C) are the total energies for the compounds and the elemental solids. For calculating E_DFT(C) the diamond phase of C was chosen. It should be noted that total energies of DFT calculations are thermodynamically strictly valid only at T = 0 K.

The DFT calculated total energies, including also the unstable compounds isotypic with the homologous Th-system, are listed in Table 1 and those within the isopleth B - UC are depicted in Fig. 1.

## 3.2. Thermodynamic assessment of the U-B-C ternary phase diagram

Phase diagrams of the binary boundary systems pertinent to the U-B-
<figure><img src="image_5.png" /></figure>

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>**Fig. 2.** Isothermal sections in the system U-B-C. Upper left: experimental section at 1600°C. Rest of the figures: sections calculated at 1600°C, 1800°C, 2000°C, 2200°C, 2400°C as indicated in the graphs.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 3. Calculated isopleths of the system U-B-C. Section B-UC (top); section UB₂-UC₂ (second from top); section B₄C-UC (second from bottom) and section B₄C-U (bottom).</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>**Fig. 4.** Liquidus projection (calculated) and temperatures of isothermal reactions (in °C). Isotherms every 100°C. Red circles denote eutectic maxima on the monovariant lines.</figcaption></figure>

C ternary are available in the versions of (i) B-C as given in a comprehensive experimental and thermodynamic assessment [36], (ii) U-B and UC, both presented in the Pauling file [37] as well as by thermodynamic calculations in the data base ThermoAlloysData [38,39].

For thermodynamic modelling of the ternary system U-B-C we used data for the respective binary systems B-C [36], U-B [38] and U-C [39]. Modelling of the ternary compound phases was based on first principles data (DFT) of their energy of formation, listed in Table 1, followed by optimising their Gibbs energies on phase equilibrium data (i.e. isothermal section at 1600 °C, congruent melting temperatures of UBC and UB₂C). During optimization, ternary parameters for the liquid phase had to be added and Gibbs energy data for phases UCₓ (FCC_B1), UCₓ (BCT) and U₂C₃ in the U-C system [39], needed to be destabilized above 3000 K. In order to model the solid solution phase UB₁ₓC₁+x (0 ≤ x ≤ 0.22 at 1600°C) Gibbs energy data have been provided from ab-initio calculations for the isostructural binary end-members UBB and UCC as well as for the end points of the solid solution phase at x = 0, 0.25 (see values listed in Table 1). The assessed thermodynamic parameters are presented in Table 2.

The results of the thermodynamic modelling are summarized in a series of isothermal sections at 1600°C, 1800°C, 2000°C, 2200°C, 2400°C in Fig. 2 as well as in a series of isopleths B - UC, UB₂ - UC₂, B₄C - UC and B₄C - U in Fig. 3. The liquidus surface is presented in Fig. 4 displaying also the thermodynamically derived temperatures of the isothermal reactions. The thermodynamic database (tdb-file) is added in section "Supplementary Material". The modelling perfectly describes not only the phase equilibria observed experimentally in the isothermal section at 1600°C as well as the relevant homogeneity regions of the binary and ternary phases, but also the regions of primary crystallisation of the ternary compounds UB₁ₓC₁+x and UB₂C (high temperature modification). The liquid phase starts at low-melting near γU (T_m = 1135°C) and at 2000°C covers already most of the central part of the ternary diagram.

The invariant three- and four-phase equilibria involving the liquid, listed in Table 3, served to draw a Schultz-Scheil diagram in Fig. 5, providing deeper insight into the solidification processes under equilibrium conditions. As far as the formation of U₅B₂C₇ is concerned, we obtain a peritectoid formation at 1784 °C: UC + U₂C₃ + UB₁ₓC₁+x ↔ U₅B₂C₇. Furthermore, phase relations involving B₄C undergo a transition reaction U6 in the solid state at 1602°C: UB₂C + B₄C ↔ UB₄ + C.

### 3.3. Electronic structure and bonding

For the DOS of SOC the VASP calculations information on the individual spinor components is available for the site projected density of states, which are decomposed in s-, p-, d- and f-like components. DOS results are shown in Fig. 6a to 6e for selected compounds such as U₄B₄C₄ (UBC), U₂B₄C₂ (ℓT-UB₂C), U₉B₁₈C₉ (hT-UB₂C), U₄B₄C₆ (hypothetical U₂B₂C₃ with Th₂B₂C₃-type) and U₃B₂C₃ (hypothetical U₃B₂C₃ with Th₃B₂C₃-type). The DOS figures are presented for all atoms in the unit cell (chemical composition in brackets), which in general results in quite different scales for the DOS values depending on the atomic composition. Embedded in the figures is a sketch of essential structural elements with U atoms in red, and B and C atoms in green and black, respectively. For U, the d-like (black dashes) and f-like (red line) contributions are shown summed over all U atoms. For B and C the total local DOS (sum over all ℓ-like contributions and atoms) are drawn multiplied with the shown factors for better visibility. The total DOS is colored blue.

Starting with UBC (Fig. 6a) a significant U-like DOS begins at about -5 eV below E<sub>F</sub>, which is of mixed d- and f-character. Above ~ -2 eV the f-character dominates culminating in two large f-peaks around and above E<sub>F</sub>, which are split due to the relativistic effect of spin-orbit coupling. The peaks are rather narrow due to the high symmetry of the crystal structure and a gap opens up. If symmetry is reduced such as in ℓT-UB₂C and hT-UB₂C (see Figs. 6b, 6c) these peaks are broadened and the gap above E<sub>F</sub>, which is distinctive for UBC, disappears. For UBC
<caption>**Table 3**</caption>

Invariant equilibria as calculated (only involving the liquid phase). Among the participating phases of a reaction only the compositions for the liquid, L, and for the solid solution phases are given.

<table><thead><tr><th rowspan="2">Reaction</th><th rowspan="2">T (°C)</th><th rowspan="2">Type</th><th rowspan="2">Phase</th><th colspan="3">Composition (at%)</th></tr><tr><th>U</th><th>B</th><th>C</th></tr></thead><tbody><tr><td>L ↔ UB<sub>2</sub>C + UB<sub>4</sub></td><td>2186.8</td><td>e<sub>5</sub></td><td>L</td><td>23.3</td><td>58.3</td><td>18.4</td></tr><tr><td>L ↔ UB<sub>12</sub> + B<sub>4+δ</sub>C</td><td>2157.7</td><td>e<sub>7</sub></td><td>L</td><td>6.6</td><td>90.7</td><td>2.7</td></tr><tr><td rowspan="2">L ↔ UB<sub>4</sub> + B<sub>4+δ</sub>C</td><td rowspan="2">2130.6</td><td rowspan="2">e<sub>8</sub></td><td>B<sub>4+δ</sub>C</td><td>0.0</td><td>95.9</td><td>4.1</td></tr><tr><td>L</td><td>9.81</td><td>82.5</td><td>7.73</td></tr><tr><td>L ↔ UB<sub>4</sub> + UB<sub>12</sub> + B<sub>4+δ</sub>C</td><td>2126.1</td><td>E<sub>1</sub></td><td>L</td><td>0.0</td><td>90.8</td><td>9.2</td></tr><tr><td rowspan="2">L ↔ UB<sub>2</sub>C + UB<sub>1-x</sub>C<sub>1+x</sub></td><td rowspan="2">2095.9</td><td rowspan="2">e<sub>9</sub></td><td>L</td><td>9.1</td><td>85.6</td><td>5.3</td></tr><tr><td>B<sub>4+δ</sub>C</td><td>0.0</td><td>86.3</td><td>13.7</td></tr><tr><td rowspan="2">L ↔ UC + UB<sub>1-x</sub>C<sub>1+x</sub></td><td rowspan="2">2063.8</td><td rowspan="2">e<sub>10</sub></td><td>L</td><td>31.5</td><td>37.5</td><td>31.0</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>31.5</td><td>37.0</td><td>31.5</td></tr><tr><td rowspan="2">L ↔ UB<sub>12</sub> + βB + B<sub>4+δ</sub>C</td><td rowspan="2">2041.2</td><td rowspan="2">E<sub>2</sub></td><td>L</td><td>38.1</td><td>23.5</td><td>38.4</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>38.4</td><td>23.2</td><td>38.4</td></tr><tr><td rowspan="2">L + UB<sub>2</sub>C ↔ UB<sub>1-x</sub>C<sub>1+x</sub> + UB<sub>4</sub></td><td rowspan="2">2026.1</td><td rowspan="2">U<sub>1</sub></td><td>L</td><td>1.8</td><td>97.9</td><td>0.3</td></tr><tr><td>B<sub>4+δ</sub>C</td><td>0.0</td><td>87.1</td><td>12.9</td></tr><tr><td rowspan="2">L + UB<sub>4</sub> ↔ UB<sub>1-x</sub>C<sub>1+x</sub> + UB<sub>2</sub></td><td rowspan="2">1985.2</td><td rowspan="2">U<sub>2</sub></td><td>L</td><td>35.2</td><td>40.7</td><td>24.1</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>33.3</td><td>33.2</td><td>33.4</td></tr><tr><td rowspan="2">L + UB<sub>4</sub> ↔ UB<sub>2</sub>C + B<sub>4+δ</sub>C</td><td rowspan="2">1981.7</td><td rowspan="2">U<sub>3</sub></td><td>L</td><td>38.4</td><td>37.9</td><td>23.6</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>33.3</td><td>33.2</td><td>33.4</td></tr><tr><td rowspan="2">L ↔ UB<sub>2</sub>C + Cgraph</td><td rowspan="2">1956.8</td><td rowspan="2">e<sub>12</sub></td><td>L</td><td>15.1</td><td>62.1</td><td>22.8</td></tr><tr><td>B<sub>4+δ</sub>C</td><td>0.0</td><td>82.8</td><td>17.2</td></tr><tr><td rowspan="2">L + UC ↔ Cgraph + UB<sub>1-x</sub>C<sub>1+x</sub></td><td rowspan="2">1953.6</td><td rowspan="2">U<sub>4</sub></td><td>L</td><td>20.6</td><td>41.6</td><td>37.8</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>27.8</td><td>25.5</td><td>46.7</td></tr><tr><td rowspan="2">L ↔ UB<sub>2</sub>C + Cgraph + UB<sub>1-x</sub>C<sub>1+x</sub></td><td rowspan="2">1933.3</td><td rowspan="2">E<sub>3</sub></td><td>L</td><td>33.3</td><td>29.4</td><td>37.2</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>24.5</td><td>33.4</td><td>42.0</td></tr><tr><td rowspan="2">L ↔ UB<sub>2</sub>C + Cgraph + B<sub>4+δ</sub>C</td><td rowspan="2">1924.5</td><td rowspan="2">E<sub>4</sub></td><td>L</td><td>33.3</td><td>30.5</td><td>36.2</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>16.5</td><td>51.2</td><td>32.4</td></tr><tr><td rowspan="2">L + UB<sub>1-x</sub>C<sub>1+x</sub> ↔ UC + UB<sub>2</sub></td><td rowspan="2">1874.6</td><td rowspan="2">U<sub>5</sub></td><td>L</td><td>45.7</td><td>28.8</td><td>25.5</td></tr><tr><td>UB<sub>1-x</sub>C<sub>1+x</sub></td><td>33.3</td><td>33.2</td><td>33.5</td></tr><tr><td rowspan="2">L ↔ γU + UC + UB<sub>2</sub></td><td rowspan="2">1087.8</td><td rowspan="2">E<sub>5</sub></td><td>L</td><td>96.3</td><td>2.3</td><td>1.4</td></tr><tr><td>UC</td><td>50.1</td><td>0.0</td><td>49.9</td></tr></tbody></table>

at about −3 to −2 eV DOSs of all the atoms are sizeable indicating the hybridization between the corresponding electronic states. Around E<sub>F</sub>, however, the B and C DOSs are rather small pointing at localized U f-like states. Below −5 eV very distinctive DOS features due to the B-C interactions are seen. The isolated peak at −11 to −8 eV is made up by strongly hybridized B-s like and C-s like states whereby the C-like DOS is significantly larger than the B-like contribution. Other interacting B and C states produce the broad features above ~ −8 eV. Again, the shapes are rather similar but now the C-like DOS is distinctly smaller than the B-like contribution. These B-C features are due to the structure of zig-zag chains with each B bonded to a terminating C, as sketched in Fig. 6a. Other typical B-C atomic arrangements lead to similar and other distinctive features as observed for ℓT-UB<sub>2</sub>C and hT-UB<sub>2</sub>C with their rather ring-like B-C atomic arrangements (see Figs. 6b, 6c).

For U<sub>2</sub>B<sub>2</sub>C<sub>3</sub> (Fig. 6d) one observes an isolated B-C feature of the DOS centred at −8 eV, which is typical for interacting s-like electronic states of a linear chain. For this structure we refer to our investigation of Th<sub>2</sub>B<sub>2</sub>C<sub>3</sub> [40] with the same structure studying in some details DOS, electronic bands and charge density contours of the B-C structural element. In distinction to U, the Th atoms have no f-like valence states, but for the B-C interactions, which are well separated in energy from the Th- and U-like valence states, this makes no difference.

With C-branched boron zig-zag chains, the B-C like features of the DOS are broadened for U<sub>3</sub>B<sub>2</sub>C<sub>3</sub> (see Fig. 6e) and no such pronounced features as for U<sub>2</sub>B<sub>2</sub>C<sub>3</sub> arise.

In SOC calculations a direction dependent magnetization density with its components m(x), m(y), and m(z) is calculated. We present data for the average magnetic moment of $m_{av} = \sqrt{m_x^2 + m_y^2 + m_z^2}$ per U atom as listed in Table 4. The size of $m_{av}$ varies significantly from 0.18 µ<sub>B</sub> in U<sub>2</sub>B<sub>2</sub>C<sub>3</sub> to 1.67 µ<sub>B</sub> for the high temperature phase of UB<sub>2</sub>C. This is due to the B-C structural environment of U atoms as shown by the insets in the corresponding DOS figures. For hT-UB<sub>2</sub>C with its 9 U atoms in the unit cell, the local magnetic moments vary between 1.44 and 1.64 µ<sub>B</sub> in fine agreement with a neutron diffraction study (1.03–1.12 µ<sub>B</sub>/U-atom at 1.5 K [14]).

For U<sub>2</sub>B<sub>2</sub>C<sub>3</sub> the 4 local moments of U vary between 0.17 and 0.24 µ<sub>B</sub>. In a non-relativistic spin-polarized interpretation without SOC the magnetic ordering is ferromagnetic with magnetic moments as listed in Table 4. The smallest moments in Table 4 appear for the experimentally temperature independent paramagnetic compounds UBC and ℓT-UB<sub>2</sub>C [12].

The number of states at the Fermi level define the Sommerfeld coefficient, γ, of the specific heat (listed in Table 4) indicating metallic behaviour for all uranium boron carbides.

## 4. Summary/conclusion

The thermodynamic description of the ternary system U-B-C is of significant value in monitoring the physical and chemical behaviour of uranium-based nuclear fuel in combination with B<sub>4</sub>C control material particularly in severe accident scenarios. In the present paper thermodynamic phase equilibria in the ternary system have successfully been modelled and consequently phase relations have been defined for the isothermal sections at 1600°C, 1800°C, 2000°C, 2200°C, 2400°C, for four isopleths B-UC, U-B<sub>4</sub>C, UB<sub>2</sub>-UC<sub>2</sub>, and B<sub>4</sub>C-UC as well as for the liquidus surface. These results have been achieved starting from the experimental isothermal sections at 1600°C, from the melting points of UBC and UB<sub>2</sub>C as well from the energy of formation data for the aforementioned two ternary compounds, which were calculated from ab-initio techniques (DFT). The equilibrium solidification path of alloys can now be followed precisely from the liquidus surface and a Schultz-Scheil diagram. The thermodynamic modelling perfectly describes (i) the experimental phase equilibria, (ii) the relevant homogeneity regions of the binary and ternary phases, and (iii) the regions of primary crystallization of the ternary phases, UB<sub>1-x</sub>C<sub>1+x</sub> and UB<sub>2</sub>C. It should be emphasized that the two-phase equilibria formed between the ternary compounds and binary borides and carbides, prevent compatibility
<figure><img src="image_18.png" /><figcaption>Fig. 5. Schultz-Scheil diagram for the system U – B – C, referring only to reactions with the liquid.</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>**Fig. 6.** a: Electronic density of states for UBC. Left insert portrays the crystal structure of UBC (unique type) as seen along the a-axis. Right insert is a view on the C-B-C groups connected via relatively long B-B bonds. b: Electronic density of states for IT-UB₂C. Left insert shows the crystal structure of IT-UB₂C (unique type) along the b-axis. Right insert reveals the B-C net. c: Electronic density of states for hT-UB₂C. Left insert shows the crystal structure of hT-UB₂C (ThB₂C type) in perspective view along the c-axis. Right insert presents the B-C net. d: Electronic density of states for (hypothetical) U₂B₂C₃. Left insert depicts the crystal structure (Th₂B₂C₃-type) along the b-axis. Right insert defines the C-branched -B-C-B-C- chain. e: Electronic density of states for (hypothetical) U₃B₂C₃. Left insert presents a view on the crystal structure (Th₃B₂C₃-type) and left inset shows the C-B-B-C groups connected via relatively long B-B bonds.</figcaption></figure>

between uranium metal and B₄C, as well as between B₄C and UC.

The DFT calculations provided total energies from which enthalpy of formations at 0 K were calculated for the structurally optimized compounds UB₁-xC₁+x (at x = 0 and 0.25), UB₂C (high and low temperature modification) as well as for the hypothetical compounds "U₂B₂C₃" and "U₃B₂C₃", which both are isostructural with their Th-homologues. The electronic DOS was discussed in particular with respect to the structural

arrangements of the B-C atoms. Because spin-orbit coupling was included in the DFT calculations the resulting magnetization density was used to define magnetic moments of the U-atoms varying between ~0.2 to ~1.7 µB depending on the chemical composition. Although localized U-f like states dominate the electronic structure at the Fermi energy the electronic conductivity is of metallic character for all the studied compounds.
**Table 4**

Averaged magnetic moments m<sub>av</sub> (in µ<sub>B</sub>) per U atom and Sommerfeld constant γ (in mJU<sup>-1</sup>K<sup>-2</sup>) for the listed U-B-C compounds. N<sub>U</sub> is the number of U atoms in the unit cell.

<table><thead><tr><th>Compound</th><th>N<sub>U</sub></th><th>m<sub>av</sub> [µ<sub>B</sub>]</th><th>γ [mJU<sup>-1</sup>K<sup>-2</sup>]</th></tr></thead><tbody><tr><td>UBC</td><td>4</td><td>0.41</td><td>10.6</td></tr><tr><td>εT-UB<sub>2</sub>C</td><td>2</td><td>1.01</td><td>15.3</td></tr><tr><td>hT-UB<sub>2</sub>C</td><td>9</td><td>1.67</td><td>6.8</td></tr><tr><td>U<sub>2</sub>B<sub>2</sub>C<sub>3</sub></td><td>4</td><td>0.18</td><td>11.8</td></tr><tr><td>U<sub>3</sub>B<sub>2</sub>C<sub>3</sub></td><td>3</td><td>0.39</td><td>9.4</td></tr></tbody></table>

**CRediT authorship contribution statement**

**Jan Vrestal:** Writing – review & editing, CALPHAD calculation.
**Raimund Podloucky:** Writing – review & editing, DFT calculations.
**Henri Noel:** Writing – review & editing, Validation, Investigation.
**Peter Rogl:** Writing – review & editing, Writing – original draft, Validation, Investigation, Conceptualization.

**Author statement**

All authors have contributed equally to the work.

**Declaration of Competing Interest**

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.

**Acknowledgements**

We thank Dr. Evelyn Fischer (ThermoAlloysData) and Dr. Christine Guéneau for the kind supply of the tdb files for the binary systems U-B and U-C. DFT calculations were done on the facilities of the Vienna Scientific Cluster VSC.

**Appendix A. Supporting information**

Supplementary data associated with this article can be found in the online version at doi:10.1016/j.jallcom.2025.180592.

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