Trans. Nonferrous Met. Soc. China 24(2014) 2920-2929

Mechanical properties and electronic structures of MgCu₂, Mg₂Ca and MgZn₂ Laves phases by first principles calculations

Ping-li MAO, Bo YU, Zheng LIU, Feng WANG, Yang JU

School of Materials Science and Engineering, Shenyang University of Technology, Shenyang 110870, China

Received 8 October 2013; accepted 14 January 2014

Abstract: Mechanical properties and electronic structure of MgCu₂, Mg₂Ca and MgZn₂ phases were investigated by means of first principles calculations from CASTEP program based on density functional theory (DFT). The calculated lattice parameters are in good agreement with the experimental and literature values. The calculated heat of formation and cohesive energies showed that MgCu₂ has the strongest alloying ability and structural stability. Elastic constants of MgCu₂, Mg₂Ca and MgZn₂were calculated, and the bulk moduli, shear moduli, elastic moduli and Poisson ratio were derived. The calculated results show that MgCu₂, Mg₂Ca and MgZn₂ are all ductile phases. Among the three phases, MgCu₂ has the strongest stiffness and the plasticity of MgZn₂ phase is the best. Melting points of the three phases were predicted using cohesive energy and elastic constants. Density of states (DOS), Mulliken population, electron occupation number and charge density difference were discussed. Finally, Debye temperature was calculated and discussed.

Key words: magnesium alloy; MgCu₂; Mg₂Ca; MgZn₂; Laves phases; electronic structure; mechanical property; Debye temperature

1 Introduction

Magnesium alloys have many advantages, such as low density, high specific strength and stiffness, good damping capacity and machinability, easy recycling, and have been widely used in aerospace industry, automotive industry, 3C products, national defense and other fields. Magnesium is the lightest metal structural material, known as the “green” engineering material [1,2]. However, due to the low heat resistance, low strength and ductility, magnesium alloys are limited to the extensive use on the heat resistant structure parts. Zinc element is a commonly used alloying element in magnesium alloy, it has the same electrovalence and crystal structure with Mg and can form a large concentration solid solution with Mg [3].

In recent years, high-performance wrought magnesium alloys, especially Mg-Zn based alloys, have been received increasing attention. Mg-Zn binary alloys contain MgZn, Mg₂Zn₃, MgZn₂ and Mg₂Zn₁₁ phases, of which the most important strengthening phase is MgZn₂[4,5]. The study shows that grain refinement can significantly improve the strength and ductility of magnesium and its alloys [6]. In the Mg-Zn binary alloy, the grains can be significantly refined by adding light and inexpensive elements Ca, and also Ca element can increase the density of aging precipitates MgZn₂phases, thereby improving the mechanical properties of the alloy [7]. Due to a potential performance of high temperature alloy, Mg-Zn-Ca alloys have been gained widespread attention [8,9]. Alloying is an effective method to improve the microstructure and mechanical properties of magnesium alloys. LUO and PEKGUCERYUZ [10] showed that by adding Cu, the strength and plasticity of magnesium alloys can be improved by forming MgCu₂ phase. QIU et al [11] studied the amorphous forming ability and mechanical properties in Mg-Zn-Ca alloy by adding Cu based on the experimental method. SENKOV and SCOTT [12] studied the formation and thermal stability of Ca-Mg-Zn and Ca-Mg-Zn-Cu bulk metallic glasses. LIU et al [13] investigated the structural and electronic properties of MgCu₂ Laves phase under pressure by first principles calculations. However, no systematic theoretical investigations have been reported on electronic structure, mechanical properties and Debye temperature of MgCu₂, Mg₂Ca and MgZn₂ Laves phases in Cu alloyed Mg-Zn-Ca alloy by first principles calculations. The elastic constants of compounds have been used to determine the mechanical properties of the compounds, and due to the experimental inconvenience, and the elastic constants of metals have been investigated by quantum mechanics method based on density functional theory, giving some satisfactory results for the calculated bulk modulus, shear modulus and other elastic constants in recent years [14]. Therefore, using the theory method to study elastic constants (C_ij) of MgCu₂, Mg₂Ca and MgZn₂ is feasible.

In the present work, the first principles calculations are used to investigate the structural, elastic and electronic properties of the binary MgCu₂, Mg₂Ca and MgZn₂ Laves phases. The melting points and Debye temperatures are also calculated. The obtained results are compared with the available experimental and theoretical values.

2 Computational method

Cambridge sequential total energy package (CASTEP), a first principles plane wave pseudo- potentials method based on density function theory (DFT) [15], is used for the calculations. Generalized gradient approximation (GGA) of Perdew–Burke–Ernzerhof (PBE) [16] is used to describe the exchange-correlation energy function. The ultrasoft pseudo-potential [17] is used to describe the interaction between ion core and valence electron. The outermost electron configuration for Mg is 3s², and others are described by 4s², 3d¹⁰ for Ca, 4s² for Zn, 3d¹⁰, 4s¹ for Cu, respectively. The parameters that affect the calculation accuracy are kinetic energy cutoff and the number of k–points network in brillouin zone; the cut-off energy of plane wave is set to 380 eV; the Monkhost–Pack scheme with a k–points separation for Mg₂Ca and MgZn₂ are 6×6×4, for MgCu₂ is 6×6×6, respectively. Geometry optimization is carried out under the electron relaxation until the total energy convergence value is 5.0×10^-8 eV/atom; the maximum Hellmann– Feynaman force is within 0.01 eV/ ; the maximum displacement is within 5×10^-4 ; and the maximum stress is within 0.02 GPa.

The elastic constants of MgCu₂, Mg₂Ca and MgZn₂are calculated by the stress–strain method. The maximum strain amplitude is set to 0.003. The total energy convergence value is 1.0×10^-6 eV/atom, the maximum force is within 0.002 eV/ , and the maximum displacement is within 1.0×10^-4 .

3 Results and discussion

3.1 Crystal structure and lattice constant

The crystal structures of MgCu₂, Mg₂Ca and MgZn₂ are shown in Fig. 1. The structure parameters and lattice constants are listed in Tables 1 and 2. The calculated lattice constants are in good agreement with the experimental and other theoretical values, and the error is only 2%. The fairly good agreement between the theoretical and experimental values shows that the present calculations are highly reliable.

Fig. 1 Crystal structures of MgCu₂ (a), Mg₂Ca (b) and MgZn₂(c)

Table 1 Structure parameters of MgCu₂, Mg₂Ca and MgZn₂

Table 2 Equilibrium crystal parameters (a, c), unit cell volume (V₀) and density (ρ) of MgCu₂, Mg₂Ca and MgZn₂

3.2 Heat of formation and cohesive energies

Heat of formation of MgCu₂, Mg₂Ca and MgZn₂ is calculated by formula as follows:

               (1)

where ΔH is the heat of formation;  is the total energy of AB₂ type intermetallic compounds;  and  are the energy per atom of pure elements A and B, respectively;  and  are the numbers of A and B atoms in unit cell, respectively. The calculated energies of Mg, Cu, Ca and Zn in solid states are -973.9652, -1476.5411, -1001.5392 and -1709.8504 eV/atom, respectively. Based on the results of ,  and , the calculated heat of formation of MgCu₂, Mg₂Ca and MgZn₂ are listed in Table 3.

The structural stability of MgCu₂, Mg₂Ca and MgZn₂ is investigated by means of cohesive energy. Generally, the cohesive energy is defined as the energy needed to decompose the compound into a single atom. Hence, the larger the calculated value is, the more stable the crystal structure is [25]. Cohesive energies (E_coh) of MgCu₂, Mg₂Ca and MgZn₂ are calculated by expression as follows:

              (2)

where  and  are the energies of A and B atoms in the free state, respectively. The calculated energies of Mg, Cu, Ca and Zn in free states are -972.4847, -1472.8555, -999.6320 and -1708.6826 eV/atom, respectively. Based on the results of ,  and , the obtained results are also listed in Table 3.

As can be seen from Table 3, the calculated values in this work are in good agreement with the experimental and theoretical values, proving the reliability of the calculation methods. The negative heat values of formation of MgCu₂, Mg₂Ca and MgZn₂ show that these phases can exist stably. The lower the heat of formation is, the stronger the forming ability is. From Table 3, it can be concluded that MgCu₂ phase has the strongest forming ability, then MgZn₂, finally Mg₂Ca.

From Table 3, we can know that MgCu₂ has the largest value of cohesive energy, and far larger than those of Mg₂Ca and MgZn₂, indicating that MgCu₂ has the most stable structure, next Mg₂Ca, finally MgZn₂. Further analysis found that adding Cu into Mg–Zn–Ca alloy can improve the structural stability through forming MgCu₂ phase.

3.3 Mechanical properties

Elastic constants are often used to characterize the deformation resistant capacity to an externally applied stress. Cubic has three independent elastic constants as follows: C₁₁, C₁₂and C₄₄, the corresponding stability criteria [26] are: (C₁₁+2C₁₂)/3>0, C₁₁–C₁₂>0, C₄₄>0. The independent elastic constants for hexagonal are as follows: C₁₁, C₁₂, C₁₃, C₃₃ and C₄₄. The corresponding stability criteria [27] are: C₁₁>0, C₁₁–C₁₂>0, C₄₄>0, >0,.

The calculated elastic constants of MgCu₂, Mg₂Ca and MgZn₂ are listed in Table 4. As can be seen from Table 4, the calculated results in this work are in good agreement with the experimental and theoretical values. Further analysis shows that the calculation results satisfy the stability criteria.

Bulk moduli (B) and shear moduli (G) of MgCu₂ are deduced by the following formula [30]:

                          (3)

                     (4)

Bulk moduli (B) and shear moduli (G) of Mg₂Ca and MgZn₂ are also estimated by Voigt–Reuss–Hill (VRH) approximation [30]. The Voigt bounds of B and G are

              (5)

      (6)

and the Reuss bounds are

                  (7)

  (8)

Table 3 Heat of formation (ΔH) and cohesive energy (E_coh) of MgCu₂, Mg₂Ca and MgZn₂

Table 4 Elastic constants (C₁₁, C₁₂, C₁₃, C₃₃and C₄₄) of MgCu₂, Mg₂Ca and MgZn₂

Finally, the expressions of B and G are

                           (9)

                          (10)

where the subscripts V and R represent the Voigh and the Reuss, respectively. Further, elastic moduli (E), Poisson ratio (ν) and anisotropic coefficient (A) are obtained according to the following formula [30]:

                              (11)

                            (12)

                             (13)

The calculated results are listed in Table 5. The bulk moduli B, shear moduli G, elastic moduli E and Poisson ratio ν of MgCu₂, Mg₂Ca and MgZn₂ are shown in Fig. 2. From Table 5, it can be found that E and G are satisfied with the relation G = E/2 (1+ν). The bulk moduli are usually assumed to be a measure of deformation resistant capacity upon applied pressure [31]. The larger the value of bulk modulus is, the stronger the capacity of the resist deformation is. From Fig. 2(a), we can conclude that the ability to resist deformation from strong to weak is MgCu₂, MgZn₂ and Mg₂Ca, respectively. Similarly, the shear modulus is a measure of resist reversible deformation by shear stress [31]. The larger the value is, the stronger the capacity of the resist shear deformation is. The calculated results demonstrate that MgCu₂ has the largest value, followed by Mg₂Ca and MgZn₂. Hence, the deformation resistant capacity of MgCu₂ would be much stronger than that of Mg₂Ca or MgZn₂. Besides, Poisson ratio is also a measure of the stability of the crystal against shear, which usually ranges from -1 to 0.5. The larger the Poisson ratio is, the better the plasticity is [27]. In Fig. 2(b), the calculated results demonstrate that MgZn₂ has the best plasticity because of the largest value of Poisson ratio, next MgCu₂, finally Mg₂Ca. Furthermore, elastic modulus is defined as the ratio between stress and strain, and it also provides a measure of stiffness of the solid materials. The larger the value is, the stiffer the material is. The calculated results show that the stiffness of MgCu₂ is the largest, then Mg₂Ca, and the last MgZn₂.

The ratio of shear moduli to bulk moduli (G/B) of polycrystalline phases is used to predict the brittle and ductile behavior of materials [31]. A high (low) G/B value is associated with brittleness (ductility). The critical value, which is used to separate brittleness from ductility, is about 0.57. The G/B values of MgCu₂, Mg₂Ca and MgZn₂ are 0.33, 0.53 and 0.18, respectively. So, MgCu₂, Mg₂Ca and MgZn₂ are all ductile. The C₁₂-C₄₄ value can also define the ductility (brittleness) of crystal [32,33]. If the value is positive, polycrystalline phases are ductile; otherwise, it is brittle. From Table 5, it can be obtained that MgCu₂, Mg₂Ca and MgZn₂ are all ductile because the C₁₂–C₄₄values are all positive, which is in good agreement with the previous (G/B) results. Hence, with the addition of Cu element, the plasticity of Mg-Zn-Ca alloy will be improved by forming the ductile phase of MgCu₂.

Table 5 Bulk moduli (B), shear moduli (G), elastic moduli (E), elastic constants (C_ij), G/B, Poisson ratio (ν) and anisotropic coefficient (A) of MgCu₂, Mg₂Ca and MgZn₂

Fig. 2 Bulk moduli (B), shear moduli (G) (a) and elastic moduli (E) and Poisson ratio (ν) (b) of MgCu₂, Mg₂Ca and MgZn₂

The melting point of the materials can be predicted using the cohesive energy and elastic constants based on empirical formula. LI et al [23] found that the melting point (T_m) and cohesive energy exbibit a linear relationship in binary intermetallic compounds as follows:

T_m=3.84867E_coh                                     (14)

For metal materials, the melting point and elastic constants shows a linear relationship as follows [19]:

T_m/K=553+5.91C₁₁±300                       (15)

T_m/K=354+4.50[1/3(2C₁₁+C₃₃)]±300            (16)

The melting point values of MgCu₂, Mg₂Ca and MgZn₂ are shown in Fig. 3. It can be seen that the experimental values are in the predicting range of elastic constants (±300 K), but there is a certain deviation between the experimental value and the melting point predicted using the cohesive energy. We can know that the relative errors in melting points of MgCu₂, Mg₂Ca and MgZn₂ by adopting cohesive energy are 4.87%, 34.76% and 40.14%, respectively. The relative errors calculated using elastic constants are 11.89%, 13.44% and 26.57% (by Eq. (15)), 20.94%, 39.79% and 1.61% (by Eq. (16)), respectively. Hence, for MgCu₂, Mg₂Ca and MgZn₂, the melting point predicted using elastic constants is closer to the experimental values than using cohesive energy. There is a certain deviation between the predicted values and experimental values, but we can believe that the melting point of MgCu₂ is the highest among the three intermetallic compounds. The higher the melting point is, the better the structural stability at high temperature is. Hence, MgCu₂ has the best stability among the three intermetallic compounds.

Fig. 3 Melting temperatures of MgCu₂, Mg₂Ca and MgZn₂

3.4 Electronic structures

The electronic structure is calculated to understand the bonding characteristics of MgCu₂, Mg₂Ca and MgZn₂, and further reveal the mechanism about structural stability and elastic properties. Structural stability of intermetallic compound is associated with its bonding electron orbits. For covalent bond, it depends on the depth and width of band gap near Fermi level, while ionic bond is determined by the charge transfer between atoms. The calculated total (partial) densities of states (DOS) of the three intermetallic compounds are shown in Fig. 4. It is found that the main bonding peaks of MgCu₂, Mg₂Ca and MgZn₂ locate in the range from -10 eV to 0 eV, originating from the contribution of valance electron numbers of Mg s, Mg p, Cu s and Cu d orbits; Mg s, Mg p, Ca s and Ca p orbits; Mg s, Mg p, Zn s and Zn d orbits, respectively. From Fig. 4, it can be seen that for MgCu₂, orbits are mainly Mg s state and Cu s state hybrid; for Mg₂Ca, orbits are Mg s, p state and Cu s state hybrid; for MgZn₂, orbits are Mg s state and Zn s state hybrid. From the perspective of covalent bond, the stability of Mg₂Ca phase is stronger than that of MgCu₂ or MgZn₂, which are not entirely consistent with cohesive energy results. Therefore, it needs to consider the ionic bond feature of the three intermetallic compounds.

Fig. 4 Density of states (DOS) of MgCu₂ (a), Mg₂Ca (b) and MgZn₂(c)

The Mulliken population of MgCu₂, Mg₂Ca and MgZn₂ are tabulated in Table 6. Mg-Zn(1) represents the chemical bonds between Mg and the nearest neighbor Zn atom, Mg-Zn(2) represents the chemical bonds between Mg and the farthest Zn atom. If the chemical bond length is positive, the chemical bond in the compound can be built. From Table 6, we can know that the chemical bonds of these compounds are able to build, due to the fact that the bond lengths of these compounds are all positive. There are six types of chemical bonds in Mg₂Ca and the populations of Mg₂Ca are all positive. Hence, these data illustrate that the covalency of chemical bonds is strong in Mg₂Ca. In MgZn₂, the population of Mg-Zn(2) is -0.20. Comparing MgZn₂ with Mg₂Ca, it can be found that the covalency of chemical bonds in MgZn₂ is weaker than that in Mg₂Ca. There are three types of chemical bonds in MgCu₂ and the population of Mg-Mg, Mg-Cu and Cu-Cu bonds are -0.08, -0.13 and 0.53, respectively. It can be known that the metallicity of chemical bonds is significantly enhanced in MgCu₂.

The electron occupation numbers of MgCu₂, Mg₂Ca and MgZn₂are shown in Table 7. It is found that for MgCu₂, the valence electron configuration of Mg atom is 2p⁶3s²; the electron configuration of Mg is 2p^6.353s^0.4 obtained through calculation; the electronic number localized in Mg atom is 6.74; and the lost electron number of Mg is 0.75 e. The valence electron configuration of Cu is 3d¹⁰4s¹; after optimization, the electron configuration of Cu is 3d^9.754s^0.72; the electronic number localized in Cu is 11.63; the obtained electron number of Cu is 0.63 e. Similarly, for Mg₂Ca, the lost electron number of Ca is 0.9 e; the obtained electron number of Mg are 0.37 e and 0.48 e, respectively. For MgZn₂, the lost electron number of Mg is 1.03 e; the obtained electron number of Zn is 0.46 e and 0.53 e, respectively. For MgCu₂, the charge transfer from Mg atoms to Cu atoms and the transfer number is 5.04 (1.26×4). For Mg₂Ca, the charge transfers from Ca atoms to Mg atoms, and the transfer number is 3.60 (0.90×4). For MgZn₂, the charge transfers from Mg atoms to Zn atoms, and the transfer number is 4.12 (1.03×4). Hence, the ionic bonds of the three intermetallic compounds from strong to weak are: MgCu₂, MgZn₂and Mg₂Ca.

Table 6 Mulliken population analysis of MgCu₂, Mg₂Ca and MgZn₂

Table 7 Mulliken charge of MgCu₂, Mg₂Ca and MgZn₂

In this work, to further reveal the covalent and ionic bonding characteristics, the charge density difference is investigated. Charge density difference can directly reflect the bonding characteristics. The results are shown in Fig. 5. The contour lines are plotted from -0.2 e/ ³ to 0.2 e/ ³ with 0.1 e/ ³ interval. From Fig. 5(a), the bonding between Cu and its adjacent Cu atom is mainly covalent, the bonding between Cu and Mg is ionic and the bonding between Mg and Mg is metallic. In Fig. 5(b), the bonding between Mg and its nearest Mg atom is covalent, the bonding between Mg and Ca is ionic and the bonding between Ca and Ca is metallic. In Fig. 5(c), it is found that there are covalent Zn-Zn bonds, ionic Mg-Zn bonds and metallic Mg-Mg bonds. Generally, for AB₂ type Laves intermetallic compounds, there are mainly metallic bonding between A atoms, covalent bonding between B atoms and ionic bonding between A and B atoms. Based on the above discussion, the bonding characteristics of MgCu₂, Mg₂Ca and MgZn₂ are all covalent bonds, ionic bonds and metallic bonds, which lead to the structural stability.

The metallicity of MgCu₂, Mg₂Ca and MgZn₂ are calculated by [34]

                 (17)

where D_fis the DOS value at Fermi level; T is the temperature; k_Bis the Boltzmann constant; n_m and n_eare the densities of the thermal excited electrons and valence electron in the cell, respectively. n_eis calculated by n_e=N/V, where N is the total number of valence electrons and V is the cell volume. The relevant parameters and calculated results are listed in Table 8, from which we can obtain that metallic strength in ascending order is: MgZn₂< MgCu₂2Ca.

Fig. 5 Electron density difference of MgCu₂ (a), Mg₂Ca (b) and MgZn₂(c)

Table 8 Density of states at Fermi level D_f，total number of valence electrons N, cell volume V and metallicity parameter (f_m) of MgCu₂, Mg₂Ca and MgZn₂

3.5 Debye temperature

After the elastic constants and electronic structures of the alloys are got, Debye temperature (Θ_D) can be calculated at low temperature. Debye temperature of materials gives some insight into the thermodynamics deduced from elastic constants. It is known that the Debye temperature can be used to represent the strength of covalent bond in solids. The Debye temperatures of MgCu₂, Mg₂Ca and MgZn₂ are calculated using the average sound velocity (v_m) by the following equation [35,36]:

                  (18)

where h is the Planck constant; k_B is the Boltzmann constant; N_A is the Avogadro number; n is the total number of atoms per formula; ρ (=M/V) is the density, where M is the molecular mass per formula. The average wave velocity (v_m) in the polycrystalline material can be calculated by

                       (19)

where v_l and v_s are the longitudinal and shear sound velocities, respectively, which are gained from the values of Hill’s bulk moduli B and shear moduli G from Navier’s equation [37]:

,                (20)

The calculated results of longitudinal sound velocity v_l, shear sound velocity v_s, average sound velocity v_m and Debye temperature Θ_D are listed in Table 9, the calculated averaged elastic wave velocity of MgCu₂or Mg₂Ca is larger than that of MgZn₂, which is around 3000 m/s. A reasonable explanation is that these compounds have large bulk moduli (B) and shear moduli (G) values. The Debye temperature of MgCu₂ is estimated to be 315.57 K, which agrees well with the available value 332.5 K [38] derived by specific heat capacity measurements and 336.40 K [38], 339 K [39] obtained by elasticity measurements. The obtained result of MgZn₂ here is 207.62 K, which is close to the theoretical value of 259.46 K [40]. The little error is due to the smaller values of B and G calculated here. By now, we cannot find any other Debye temperature values of Mg₂Ca comparison, but due to the credibility values of MgCu₂and MgZn₂, we can say that the value of Mg₂Ca is available. The largest Θ_D is 328.70 K for Mg₂Ca. The larger the Debye temperature is, the stronger the covalent bond strength in solids is [41]. From Table 8, we can summarize that the covalent bond in Mg₂Ca is the strongest among these phases, which is consistent with the results of density of states (DOS) in section 3.4 that Mg₂Ca has the strongest covalent bond.

Table 9 Theoretically calculated thermal properties of MgCu₂, Mg₂Ca and MgZn₂ phases, including density (ρ), longitudinal sound velocity (v_l), shear velocity (v_s), average sound velocity (v_m) and Debye temperature (Θ_D)

4 Conclusions

1) The calculated lattice parameters are good consistent with the experimental and literature values. The calculated heat of formation and cohesive energies show that MgCu₂ has stronger alloying ability and structural stability than Mg₂Ca and MgZn₂.

2) The elastic constants of MgCu₂, Mg₂Ca and MgZn₂ phases are calculated. The results of bulk moduli B, shear moduli G, elastic moduli E and Poisson ratio ν show that MgCu₂has stronger deformation resistant capacity than Mg₂Ca and MgZn₂, the plasticity of MgZn₂ phase is the best, the stiffness of MgCu₂ is the largest and MgCu₂, Mg₂Ca and MgZn₂ are all ductile phases.

3) The melting points of MgCu₂,Mg₂Ca and MgZn₂ phases are predicted using the cohesive energy and elastic constants based on the empirical formula. The calculated results show that MgCu₂ has the highest melting point.

4) The bonding characteristics of MgCu₂, Mg₂Ca and MgZn₂ are all covalent, ionic and metallic, respectively.

5) The Debye temperatures obtained by the elastic constants are calculated and the results show that Mg₂Ca has the strongest covalent bond.

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MgCu₂, Mg₂Ca和MgZn₂ Laves相力学性质和电子结构的第一性原理计算

毛萍莉，于 波，刘 正，王 峰，鞠 阳

沈阳工业大学 材料科学与工程学院，沈阳 110870

摘  要：通过基于密度泛函理论的第一性原理计算方法，对MgCu₂, Mg₂Ca和MgZn₂的力学性能和电子结构进行计算，计算所得晶格参数与实验值和文献值相吻合。合金形成热和结合能的计算结果表明，MgCu₂ 具有最强的合金形成能力和结构稳定性。计算了MgCu₂, Mg₂Ca和MgZn₂ 的弹性常数，推导了体模量、剪切模量、弹性模量和泊松比。结果表明，MgCu₂、Mg₂Ca和MgZn₂ 均为延性相，MgCu₂的刚度最大，MgZn₂的塑性最好。通过对结合能和弹性常数的计算，预测了MgCu₂、Mg₂Ca和MgZn₂ 的熔点。通过对态密度(DOS)、Mulliken布居数、电子占据数和差分电荷密度的计算，分析了MgCu₂、Mg₂Ca和MgZn₂的结构稳定性和力学性能机制。最后，计算和讨论了3种金属间化合物的Debye温度。

关键词：镁合金；MgCu₂；Mg₂Ca；MgZn₂；Laves相；电子结构；力学性能；Debye温度

(Edited by Hua YANG)

Foundation item: Project (2013201018) supported by Scientific and Technological Project of Liaoning Province, China

Corresponding author: Ping-li MAO; Tel: +86-24-25497131; E-mail: pinglimao@yahoo.com

DOI: 10.1016/S1003-6326(14)63427-0