First Principles Study on the Physical Properties of Be-W Alloy

1. Introduction

Beryllium (Be) is one of the lightest structural metals, featuring an ultra-low density of 1.85 g/cm³, an extremely high elastic modulus of approximately 300 GPa, outstanding thermal stability, low thermal neutron absorption cross-section and favorable machinability. It is widely applied as lightweight structural materials and nuclear functional materials in aerospace, nuclear industry and precision instrument industries [1,2,3].Tungsten (W) boasts the highest melting point among all metals at 3422 °C, together with ultrahigh high-temperature strength, superior creep resistance, prominent irradiation damage tolerance and excellent corrosion resistance. It serves as a preferred candidate material for core components operating under extreme high-temperature conditions such as fusion reactors, aero-engines and high-temperature furnaces [4,5,6].Alloying beryllium with tungsten to form Be–W intermetallic compounds can effectively integrate the respective merits of the two constituent metals and realize synergistic properties including low density, high elastic modulus, high-temperature resistance, irradiation tolerance and corrosion resistance. Consequently, such intermetallics are promising candidate systems for next-generation plasma-facing components of fusion reactors, high-temperature load-bearing aerospace structures and control materials for nuclear reactors [7,8].

The key performance indicators of Be–W alloys, such as macroscopic mechanical properties, thermal stability and irradiation resistance, strongly depend on the phase composition, crystal structure, atomic arrangement and phase stability of their intrinsic intermetallic phases [9]. Multiple stoichiometric intermetallic compounds including Be₂W, Be₁₂W and Be₂₂W tend to generate during practical fabrication and service of Be–W alloys, and remarkable discrepancies in crystal structure, thermodynamic stability, mechanical performance and bonding characteristics among various phases dominantly determine the comprehensive service performance of the alloys [10]. Nevertheless, Be and W possess drastically different melting points (1287 °C for Be and 3422 °C for W). Problems including elemental segregation, phase decomposition and oxidative volatilization readily emerge during high-temperature alloying. It is experimentally difficult to synthesize high-purity single-phase Be–W intermetallics; furthermore, rigorous testing environments involving high temperature, high vacuum and irradiation exposure make it impossible to accurately acquire intrinsic thermodynamic parameters, electronic structures and inherent mechanical properties of these compounds. Conventional experimental investigations can only characterize the macroscopic properties of bulk alloys rather than uncover the underlying phase formation mechanisms and essential property origins at atomic and electronic scales, which greatly restricts the design and exploitation of Be–W-based materials.

Based on the fundamental principles of quantum mechanics, the first-principles calculation approach requires no empirical parameters and enables accurate prediction of intrinsic material properties including crystal structure, energetic parameters, electronic structure and elastic properties from the electronic scale. It has evolved into an indispensable core research tool for evaluating phase stability, bonding mechanism and property prediction of intermetallic compounds [11]. In recent years, first-principles calculations have been successfully implemented in numerous binary intermetallic systems, yielding abundant credible achievements concerning phase stability identification, bonding mechanism analysis and mechanical property forecasting. Existing first-principles investigations on intermetallics provide mature methodological references for analogous systems, covering the calculation of formation enthalpy and cohesive energy, density-of-states analysis, as well as the computation of elastic constants and the verification of mechanical stability criteria [12,13,14,15].

At present, systematic first-principles investigations on binary Be–W intermetallic compounds remain scarce. Most available studies focus on the physical property calculations of elemental beryllium and tungsten or preliminary exploration of simple alloy phases, while comprehensive comparative analyses of three typical intermetallics (Be₂W, Be₁₂W and Be₂₂W) are still lacking. The microscopic mechanisms governing the differences in their phase-forming ability, thermodynamic stability, mechanical stability, electronic bonding features and elastic performances have not yet been clarified. Accordingly, within the framework of density functional theory, the first-principles plane-wave pseudopotential method is employed in this work to systematically compute the lattice parameters, formation enthalpy, cohesive energy, band structure, density of states, elastic constants and macroscopic elastic moduli of the three representative Be–W intermetallics. Multi-dimensional analyses from thermodynamic, mechanical and electronic structure perspectives are performed to reveal the stability evolution and intrinsic origins of property discrepancies among various Be–W compounds, laying a scientific theoretical foundation for the rational design and experimental fabrication of high-temperature irradiation-resistant Be–W-based structural materials.

2. Computational Methods and Crystal Models

All calculations in this work were performed based on density functional theory (DFT) using the CASTEP (Cambridge Serial Total Energy Package) quantum mechanics module embedded in Materials Studio. The Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) was adopted for the exchange-correlation potential. This functional exhibits high accuracy for intermetallic compounds composed of transition metals and light metals, and can reliably characterize orbital hybridization and energetic properties. The interaction between valence electrons and ionic cores was described by ultrasoft pseudopotentials, which can effectively reduce the plane-wave cutoff energy and improve computational efficiency [16,17,18].

The plane-wave cutoff energy was set to 450 eV to guarantee the convergence of total energy and the reliability of structural optimization. The Monkhorst-Pack grid scheme was used for Brillouin zone integration. Appropriate k-point meshes were selected according to the symmetry and unit cell size of each crystal structure to ensure convergence of self-consistent calculations. The convergence criteria for geometric optimization and energy calculations were defined as follows: the self-consistent field energy tolerance was 5×10⁻⁶ eV/atom, the interatomic force tolerance was 0.01 eV/nm, the unit cell stress tolerance was 0.02 GPa, and the atomic displacement tolerance was 5×10⁻⁴ Å. These settings ensure that all computational results meet the standard precision requirements of first-principles calculations.

Multiple stable intermetallic compounds exist in the binary Be–W system. Three stoichiometric phases with high formation tendency in experiments and prominent engineering application values were selected in this study, namely Be₂W, Be₁₂W and Be₂₂W. After establishing the initial crystal models, full geometric optimization was carried out on the lattice parameters, atomic coordinates and unit cell volume to release internal stress and obtain the ground-state equilibrium crystal structures. Subsequently, calculations of energy, electronic structure and elastic properties were implemented.

Figure 1. Crystal structures of Be-W: (a) Be₂W, (b) Be₁₂W and (c) Be₂₂W.

Figure 1. Crystal structures of Be-W: (a) Be₂W, (b) Be₁₂W and (c) Be₂₂W.

3. Calculation Results and Discussion

3.1. Crystal Structure and Thermodynamic Stability

The equilibrium crystal structure serves as the foundation for material property simulations and directly determines the accuracy and reliability of calculations. Therefore, geometric optimization was first performed on the crystal structures of Be₂W, Be₁₂W and Be₂₂W to obtain their stable configurations. The crystallographic parameters after geometric optimization are listed in Table 1.Cohesive energy (E_coh) refers to the energy released when isolated atoms assemble into a crystalline solid. It is commonly used to evaluate the structural stability of crystals and characterize the strength of atomic bonding. A negative cohesive energy value indicates a stable structure, and a smaller value corresponds to higher structural stability. Enthalpy of formation (ΔH) is defined as the energy absorbed or released during the formation of intermetallic compounds from solid elemental substances. It reflects the formation difficulty and alloying capacity of intermetallic compounds [19]. A negative enthalpy of formation means the compound tends to form readily, and a larger absolute value represents stronger alloying capability.

The cohesive energy and enthalpy of formation are calculated using Formula (1) and Formula (2), respectively [20]:

$$E_{coh}={\displaystyle \frac{1}{x+y}}(E_{total}^{AB}-x{E}_{atom}^A-y{E}_{atom}^B)$$

(1)

$$∆\mathrm{H}={\displaystyle \frac{1}{x+y}}(E_{total}^{AB}-x{E}_{solid}^A-y{E}_{solid}^B)$$

(2)

where $E_{total}^{AB}$ is the total energy of the compound; x and y represent the atomic numbers of element A and element B in the compound, respectively. $E_{atom}^A$ and $E_{atom}^B$ denote the energy of a single isolated atom of element A and element B, while $E_{solid}^A$ and $E_{solid}^B$ stand for the average energy per atom in the stable elemental solids of A and B. In this work, the cohesive energy and enthalpy of formation of Be₂W, Be₁₂W and Be₂₂W were calculated, and the results are summarized in Table 1.

The formation enthalpy can be used to judge the feasibility of a reaction. A negative formation enthalpy with a smaller absolute value indicates that the reaction occurs more readily. The cohesive energy is a key indicator for evaluating the stability of phase structures. According to the minimum energy principle, a lower cohesive energy corresponds to a more stable crystal structure. As shown in Table 1, Be₁₂W has the lowest formation enthalpy, meaning it is the easiest to form. In contrast, Be₂W exhibits the highest cohesive energy, which suggests that its crystal structure is the least stable among the three phases.

3.2. Electronic Structure

After geometric optimization, the band structures of Be₂ W, Be₁₂ W and Be₂₂ W along the high-symmetry directions of the Brillouin zone are presented in Figure 2. The total density of states and partial density of states of the three alloys are shown in Figure 3, where the horizontal dashed line at 0 eV marks the Fermi level. It can be observed from Figure 2 that all three alloys possess electrical conductivity, as no obvious band gap is detected. Electrons can readily gain energy and jump into the conduction band to conduct electricity.

The band structure diagrams reveal that the valence band of Be2 W ranges from −9 eV to 0 eV and its conduction band from 0 eV to 23 eV (Figure 2(a)). For Be₁₂ W, the valence band spans −11 eV to 0 eV and the conduction band 0 eV to 24 eV (Figure 2(b)). As for Be₂₂ W, its valence band is located between −9 eV and 0 eV, and the conduction band between 0 eV and 22 eV (Figure 2(c)). The variation in band width and energy level crossing of each phase can also be clearly seen in Figure 2.Be12 W features a relatively broad band structure with drastic fluctuations, indicating that electrons within the bands have a small effective mass and high delocalization, and the atomic orbitals constructing these bands exhibit strong extensibility. In contrast, Be₂₂ W has narrow bands, which means the eigenstates corresponding to these bands are mainly composed of atomic orbitals localized at specific lattice sites. Electrons in these bands are highly localized and have a relatively large effective mass. Be₂ W shows considerable overlap between adjacent orbitals, suggesting strong bonding characteristics. Since the melting point and hardness of intermetallic compounds are positively correlated with the type and strength of chemical bonds, the elastic modulus decreases gradually while brittleness increases correspondingly. Consequently, Be₂ W exhibits relatively high brittleness.

Density of states (DOS) describes the number of allowed electrons per unit energy interval, namely the electron distribution within a certain energy range [21]. Since atomic orbitals are primarily classified by energy levels, DOS can reflect the electron distribution in various orbitals, the interactions between atoms, and the characteristics of chemical bonds.The total density of states (DOS) and partial density of states (PDOS) of three Be-W intermetallic compounds, namely Be₂ W, Be₁₂ W and Be₂₂ W, are presented in Figure 3(a) to 3(c). The vertical dashed line at 0 eV denotes the Fermi level (EF ) [22].

As illustrated in the figures, the density of states at the Fermi level is greater than zero for all three Be-W intermetallic compounds, demonstrating their metallic nature. Near the Fermi level, the electron orbitals of Be and W exhibit similar sharp peaks at identical energy positions. The total density of states is mainly contributed by the Be p-orbitals and W d-orbitals, indicating the formation of p-d hybridization between Be and W atoms.It can also be seen from the total and partial density of states that partial peak overlap occurs between the Be p-orbitals and W s-orbitals around 8 eV above the Fermi level, which verifies the formation of covalent bonding between Be and W in this system.

It can be seen from the density of states profile of Be₂ W that the valence band region spanning 7 eV around 0 eV is mainly contributed by the d-orbitals of W, while the conduction band ranging from 5 eV to 12 eV is dominated by the p-orbitals of Be (Figure 3(a)). The density of states profiles of Be₁₂ W and Be₂₂ W (Figure 3(b) and 3(c)) show similar characteristics: their valence bands are primarily contributed by W d-orbitals and conduction bands by Be p-orbitals.

Further analysis of Figure 3 reveals that the density of states at the Fermi level EF mainly originates from the p-orbitals of Be and W. Specifically, the peak value of Be₂ W is 3.2 eV, that of Be₁₂ W is 4.3 eV, and that of Be₂₂ W reaches 26.4 eV. In general, the position of the Fermi level and the density of states at the Fermi level N(EF ) determine the structural stability of materials. A lower N(EF ) value corresponds to a more stable structure. Since the intermetallic compound Be₂₂ W has a relatively high N(EF ), it possesses inferior structural stability [23].

3.3. Mechanical Properties

Elastic constants are fundamental parameters characterizing the mechanical response of crystals and can be calculated from the stress-strain relationship. The number of independent elastic constants varies among different crystal systems. Specifically, hexagonal Be₂ W has five independent elastic constants, tetragonal Be₁₂ W has six, and cubic Be₂₂ W has three. Material stability covers both energetic stability and mechanical stability. The Born-Huang stability criteria are adopted to evaluate the mechanical stability of crystals [24].

A hexagonal crystal system has five independent elastic constants, namely C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄ . The criteria for mechanical stability of hexagonal crystals are as follows: C₁₁ >0; C₁₁ −C₁₂ >0; C₄₄ >0; (C₁₁ +C₁₂ )C₃₃ >2C₁₃² [25].

A tetragonal crystal system possesses six independent elastic constants: C₁₁, C₁₂, C₁₃, C₃₃, C₄₄ and C₆₆ . Its mechanical stability conditions are: C₁₁ >0, C₃₃ >0, C₄₄ >0, C₆₆ >0, C₁₁ −C₁₂ >0, C₁₁ +C₃₃ −2C₁₃ >0, 2(C₁₁ +C₁₂ )+C₃₃ +4C₁₃ >0 [26,27].

For a cubic crystal system, there are three independent elastic constants (C₁₁, C₁₂, C₄₄ ). The corresponding stability criteria are: C₁₁ >0, C₄₄ >0, C₁₁ −C₁₂ >0 and C₁₁ +2C₁₂ >0 [19].

It can be seen from Table 2 that Be₂ W satisfies all mechanical stability criteria: C₁₁ >0, C₁₁ −C₁₂ >0, C₄₄ >0 and (C₁₁ +C₁₂ )C₃₃ >2C₁₃² . Therefore, Be₂ W is mechanically stable. Similarly, Be₁₂ W and Be₂₂ W are also verified to be mechanically stable according to the corresponding stability constraints.

Elastic modulus describes the tensile and compressive properties of materials within the elastic limit, which is generally characterized by bulk modulus (B), shear modulus (G), Young’s modulus (Y) and Poisson’s ratio (ν).The elastic moduli of polycrystalline materials can be derived from the elastic constants of single crystals via reasonable approximations. The Voigt method assumes uniform stress distribution throughout the sample, and the bulk modulus and shear modulus are further expressed in terms of elastic constants C_ij .

$$B_V={\displaystyle \frac{C_{11}{+C}_{22}{+C}_{33}{+2(C}_{12}{+C}_{23}{+C}_{13})}{9}}$$

$$G_V={\displaystyle \frac{C_{11}{+C}_{22}{+C}_{33}{-C}_{12}{-C}_{23}{-C}_{13})}{15}}+{\displaystyle \frac{C_{44}{+C}_{55}{+C}_{66}}{5}}$$

The Reuss method adopts the assumption of uniform strain across the sample, and the bulk modulus and shear modulus are derived in terms of elastic compliance constants S_ij .

$$B_R={\displaystyle \frac{1}{S_{11}{+S}_{22}{+S}_{33}{+2(S}_{12}{+S}_{23}{+S}_{13})}}$$

$$G_R={\displaystyle \frac{15}{4\left(S_{11}{+S}_{22}{+S}_{33}{-S}_{12}{-S}_{23}{-S}_{13}\right)+3\left(S_{44}{+S}_{55}{+S}_{66}\right)}}$$

Theoretically, Hill proved that the bulk modulus and shear modulus obtained by the Voigt method and the Reuss method correspond to the maximum and minimum values for polycrystals, respectively. He proposed that the actual elastic moduli of polycrystals can be expressed as the arithmetic mean of the values calculated by the Voigt and Reuss methods:

$$B_H={\displaystyle \frac{(B_V+B_R)}{2}}$$

$$G_H={\displaystyle \frac{(G_V+G_R)}{2}}$$

In addition, Young’s modulus E and Poisson’s ratio ν can be calculated by the following formulas:

$$\mathrm{E}={\displaystyle \frac{9BG}{3B+G}}$$

$$\mathsf{\nu }={\displaystyle \frac{3B-2G}{2(3B+G)}}$$

The elastic properties including average bulk modulus B, Young’s modulus E, shear modulus G and Poisson’s ratio ν of Be₂ W, Be₁₂ W and Be₂₂ W alloys calculated from elastic constants are listed in Table 3.

An analysis of the Voigt, Reuss and Hill average elastic parameters of the three Be-W intermetallic compounds (Be₂ W, Be₁₂ W and Be₂₂ W) in Table 3 shows that the Voigt and Reuss models correspond to the theoretical upper and lower limits of the elastic modulus of polycrystalline materials, respectively. The bulk modulus, shear modulus, Young’s modulus and Poisson’s ratio derived from the Hill average, which is calculated as the arithmetic mean of the above two models, can more accurately characterize the macroscopic elastic mechanical behavior of polycrystalline compounds.

According to the Hill average results, the bulk modulus follows the order: Be₂ W>Be₁₂ W>Be₂₂ W. This indicates that Be₂ W has stronger resistance to hydrostatic pressure and volume deformation, while Be₂₂ W exhibits the weakest resistance to volumetric compression. Both the shear modulus and Young’s modulus decrease in the sequence: Be₁₂ W>Be₂ W>Be₂₂ W, demonstrating that Be₁₂ W possesses higher elastic stiffness, better resistance to shear and plastic deformation, and the optimal comprehensive load-bearing performance. In terms of Poisson’s ratio, the values rank as: Be₂₂ W>Be₂ W>Be₁₂ W. A lower Poisson’s ratio implies more remarkable brittleness. Be₁₂ W has the minimum Poisson’s ratio and thus relatively higher brittleness. Be₂ W and Be₂₂ W show similar Poisson’s ratios, presenting better coordination of transverse deformation under loading and superior overall ductility. In addition, for all three compounds, the Voigt elastic modulus is higher than the Hill average, and the Hill average is larger than the Reuss elastic modulus. This distribution is highly consistent with the classic Voigt-Reuss-Hill theory, which further verifies the rationality and reliability of the calculated elastic constants and fitted moduli in this work.

4. Conclusions

Based on density functional theory (DFT), this work systematically investigates the crystal structures, thermodynamic stability, electronic structures and mechanical properties of three Be-W intermetallic compounds, namely Be₂ W, Be₁₂ W and Be₂₂ W, by adopting the first-principles plane-wave pseudopotential method. The main conclusions are drawn as follows:

(1) Crystal structure and thermodynamic stability

The lattice constants obtained via geometric optimization are in good agreement with experimental values, which verifies the reliability of the established models. The calculated formation enthalpies indicate that Be₁₂ W is the easiest to form and has the strongest alloying capacity. According to the cohesive energy and density of states at the Fermi level, Be₂ W possesses the poorest structural stability, while Be₁₂ W and Be₂₂ W exhibit better stability.

(2) Electronic structure and bonding characteristics

No obvious band gaps are observed for the three compounds, which demonstrates their metallic conductivity. The total density of states is mainly contributed by Be p-orbitals and W d-orbitals, and prominent p-d orbital hybridization occurs, with covalent bonds acting as the dominant bonding form. Be₁₂ W features highly extended orbitals and favorable bonding behavior, whereas Be₂ W has relatively weak bonding strength.

(3) Mechanical stability and elastic properties

Be₂ W (hexagonal), Be₁₂ W (tetragonal) and Be₂₂ W (cubic) all satisfy the Born criteria for mechanical stability of their corresponding crystal systems, proving that all three phases are mechanically stable. The bulk modulus follows the order: Be₂ W>Be₁₂ W>Be₂₂ W, suggesting a gradual decline in the resistance to volume deformation. Both shear modulus and Young’s modulus rank as Be₁₂ W>Be₂ W>Be₂₂ W, meaning Be₁₂ W has the highest stiffness and deformation resistance. Poisson’s ratio results reveal that Be₁₂ W shows relatively higher brittleness, while Be₂ W and Be₂₂ W have slightly better ductility.

In summary, Be₁₂W stands out with superior formability, structural stability and comprehensive mechanical properties, and thus is a promising candidate phase for Be-W based high-temperature radiation-resistant structural materials. This study supplements the fundamental physical property data of binary Be-W intermetallic compounds and clarifies the correlation mechanism between microscopic bonding behavior and macroscopic performances. It provides theoretical basis and data support for the theoretical design, experimental preparation and engineering application of materials in this system.

This work was supported by the Science and Technology Department of Ningxia [grant numbers 2025AAC031060, 2025AAC031057].