Investigating the β-Mg17Al12 Alloy under Pressure Using First-Principles Methods: Structure, Elastic Properties, and Mechanical Properties

Abstract

Calculations of first principles were employed to explore the elastic constants of the $\beta$-Mg₁₇Al₁₂ intermetallic complex under pressure, along with several related physical parameters, including the bulk modulus, the shear modulus, Young’s modulus, Poisson’s ratio, and the anisotropy index. The volume of the $\beta$-Mg${}_{17}$Al${}_{12}$ crystal in the ground state was V₀ = 1180.353 Å${}^3$, and the lattice parameter was 10.57 Å. This is in agreement with the available results in the literature, which indicate that the calculations were correct. The three independent elastic constants, C₁₁, C₁₂, and C₄₄, increased with increasing pressure. The bulk modulus B, shear modulus G, and Young’s modulus E increased with increasing pressure, indicating that the bulk deformation resistance, shear deformation resistance, and stiffness of the $\beta$-Mg${}_{17}$Al${}_{12}$ phase increased with increasing pressure. The phase had a $B/G$ > 1.75 and a Poisson’s ratio of $\nu$ > 0.26 and increased with pressure, indicating that the $\beta$-Mg${}_{17}$Al${}_{12}$ crystals were ductile and that the ductility increased with pressure. The Cauchy pressure C${}_{12}$–C${}_{44}$ increased with increasing pressure. The anisotropy coefficients A${}_{\left(100\right)}$ and A${}_{\left(110\right)}$ deviated further from 1, and the anisotropy increased. The electronic structure calculations showed that the total density of the states (TDOS) was achieved mainly by the Mg-3p and Al-3p states, and the total density of states moved toward the higher energy regions under pressure, with enhanced interatomic bonding, leading to an increase in the elastic constants and ultimately to an increase in each physical property with increasing pressure.

1. Introduction

Demand for lighter cars has led to the increasing popularity of lightweight alloys. Magnesium has become more desirable for use in modern industries due to its low quality, maximum strength-to-weight ratio, outstanding ductility, superb castability, and high specific strength [1,2,3]. However, during high temperatures, strength, fatigue, and creep resistance are reduced; hence, the applications for this material are still limited [4]. Thus, Mg alloys have been studied extensively in order to enhance their mechanical properties. There is evidence that the alloying effect can greatly optimize the mechanical properties for magnesium alloys [5]. The incorporation of aluminum particles into the alloying process leads to a magnesium–aluminum alloy, which has the benefits of low density, a light weight, stiffness, a strong strength-to-weight ratio, good castability, and ductility; so, it can be used in the transport, aircraft, and aeronautics industries [6,7]. The oldest and most commonly used magnesium alloy for casting is the binary Mg–Al alloy [8]. Recently, countries have been paying more attention to environmental and energy issues. Multiple transportation companies have reduced the weight of various transportation vehicles in an effort to conserve energy and reduce emissions through the continuous improvement of lightweight alloy materials. As a consequence, Mg–Al alloys are growing in popularity in various industries around the world, most notably the automotive industry. Alloys made from magnesium and aluminum demonstrate a high degree of stiffness, creep resistance, and fatigue resistance; on the other hand, they have restricted physicochemical properties, particularly under high heat conditions, severely limiting their application [9,10].

Generally, the organization of Mg–Al alloys in the as-cast state consists of an $\alpha$-Mg matrix and a dissociated eutectic $\beta$-Mg${}_{17}$Al${}_{12}$ phase due to nonequilibrium crystallization. $\beta$-Mg${}_{17}$Al${}_{12}$ is a key phase in Mg–Al alloys that is required for crystal border stabilization and crystal unfolding in heat. In addition, the bulk of the $\beta$-Mg${}_{17}$Al${}_{12}$ phase is dispersed as a coarse lattice along the grain limit, which could contribute to fracture initiation and stress accumulation [11]. Many researchers have studied how to improve the performance of the magnesium–aluminum alloy to meet the demand for various work environments and overcome its deficiencies. $\beta$-Mg${}_{17}$Al${}_{12}$ is an intermediate phase that may block the dislocation motion at grain boundaries and play an essential role in fortification. This can significantly boost the grain boundary strength of the magnesium–aluminum alloy. First-principles calculations make it possible to provide the theoretical support and data to calculate and predict the microstructure of materials with the help of condensed matter theory, and these have been widely used in the fields of materials design, biology, geophysics, and chemistry. Some investigators are currently employing density functional theory to cases studying the intermediate phase of $\beta$-Mg${}_{17}$Al${}_{12}$ in the magnesium–aluminum alloy. The structure, formation energy, and resilient performance of the $\beta$-Mg${}_{17}$Al${}_{12}$ and Mg${}_{24}$Y${}_5$ phases were examined by Wang et al. using density functional theory, and the outcome revealed that the $\beta$-Mg${}_{17}$Al${}_{12}$ phase had a slightly higher stiffness and improved ductility compared with Mg${}_{24}$24Y${}_5$ [12]. It is well-known that applying pressure to a material can have a significant impact on its chemical and physical properties and phase transition. Furthermore, it may change the phase structure of the material and its electron interactions with other atoms in its microstructure. In addition, in the wake of advances in high-pressure methods, some researchers are pursuing the use of stress to influence the properties and structure of products and materials. Fei et al. examined the optical properties, phase transitions, and electronic structures of Mg${}_2$Si using density functional theory applied to high pressure [13]. It was demonstrated that magnesium silicate experienced two stress-induced phase changes at 8.38 GPa and 28.84 GPa, respectively, and became a metallic substance under high pressure. According to Wang et al. [14], when the alloy material consolidated under high pressure, advanced equiaxed dendrites formed. A pressure increase resulted in smaller and smaller volumes of the Mg${}_{32}$(Al, Zn)${}_{49}$ phases, and their quantities decreased as well. Furthermore, Mg and Zn are much more robustly soluble in a-Al at elevated pressures than they are at lower pressures. It is therefore evident that pressure dramatically and significantly influences the way in which materials’ structures and properties are controlled.

A thorough study of the pressure-related characteristics of the $\beta$-Mg${}_{17}$Al${}_{12}$ intermediate phase is lacking. To optimize the mechanical performances of magnesium–aluminum alloy, it is critical to understand the electrical construction and mechanical performances of this interphase. In this study, the mechanics, electronic density, and crystal structure of the $\beta$-Mg${}_{17}$Al${}_{12}$ interphase were calculated under a variety of pressures. The stability of the mechanical performance as well as the anisotropy were derived from a first-principles study using the generalized density function theory. This work provides information on a theoretical framework for examining how the presence of the $\beta$-Mg${}_{17}$Al${}_{12}$ intermediate phase affects the properties of the Mg–Al alloy. In addition, a theoretical foundation is developed for future investigation of how the intermediate $\beta$-Mg${}_{17}$Al${}_{12}$ phase impacts the properties of Mg–Al alloys.

2. Calculation Details

First-principles simulations were undertaken using the Cambridge Serial Total Energy Package (CASTEP) procedure [15,16,17], which is based on plane wave pseudopotential DFT [18,19]. To illustrate the concrete mathematics of exchange-related functions, we used the generalized gradient approximation (GGA) for Perdewe–Burkee–Eruzerhof (PBE) [20]. It is hypothesized that charged particles exist in the electronic states of Mg(2p63s2) and Al(3s23p1), and the ion–electron interactions are calculated using ultrasoft pseudopotentials (USPP) [21] of the Vanderbilt type. Brillouin zone sampling was performed with a k-point mesh of 6 × 6 × 6 Monkhorst Pack points [22] in all the electronic calculations of $\beta$-Mg${}_{17}$Al${}_{12}$. The cutoff energy of the fundamental set of plane waves was 380 eV. The cubic crystalline structure of $\beta$-Mg${}_{17}$Al${}_{12}$ belongs to the space group $I-43m$ (see Figure 1), and the design of the structure was optimized using the BFGS method [23], similar to the hydrostatic pressure ranging between 0 GPa and 80 GPa. The parameters of the self-consistent calculation were set so that the total energy per atom was 1.0 × ${10}^{-5}$ eV/atom, while the most intense force that acted on it was below 3.0 × ${10}^{-2}$ eV. At the same time, the converged threshold’s maximum stress was limited to 5.0 × ${10}^{-2}$ GPa. In order to obtain a more accurate resolution of the fundamental wave equation, the self-consistent field sensitivity was tuned to 1.0 × ${10}^{-6}$ eV/atom. In order to perform electron removal, the concentration mixing technique was implemented in the self-consistent computation. All the electronic structure simulations in this work used these parameters.

3. Results and Discussion

3.1. Structural Properties and Stability

For the purpose of demonstrating the most consistent crystal structure possible of $\beta$-Mg${}_{17}$Al${}_{12}$, an equation of state (EOS) based on the Birch–Murnaghan expressions [24] of state was used for the calculation of the total energy data for volume V as a function of convention unit volume $V_0$. Figure 2 displays the relationship plot between the energy and volume ($E-V$). The bulk modulus $B_0$ was calculated by matching the expected values of E to V of the Birch–Murnaghan EOS, and the pressure volume relation was extended to Equation (1) [24].

(1)$E\left(V\right)=E_0+\frac{9}{16}{V}_0{B}_0\left\{{\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]}^3{B}_0^{\prime }+{\left[{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}-1\right]}^2\left[6-4{\left(\frac{V_0}{V}\right)}^{\frac{2}{3}}\right]\right\}$

$B_0$ refers to the bulk modulus, while $B_0^{\prime }$ refers to the pressure derivative of $B_0$, which can be represented as $B_0^{\prime }=d{B}_0/dP$. $E\left(V\right)$ and $E_0$ signify the total energy at the deformed (V) and reference ($V_0$) volumes, respectively.

As can be seen in Figure 2, the volume V was comparable to volume $V_0$ ($V_0$ = 1180.35 Å${}^3$), the total energy ($E_t$ = −34,470.85 eV) reached its lowest value, and the lattice parameter was 10.57. The lattice parameter of $\beta$-Mg${}_{17}$Al${}_{12}$ ranges from 10.52 Å to 10.57 Å, and the volume of the single-element cell ranges from 1168.7 ${\AA }^3$ to 1181.48 ${\AA }^3$, based on the known comparison of experimental and theoretical results [12,25,26,27,28]. Our computational results corresponded well with the values in the literature, demonstrating that the procedure was valid. In equilibrium, $a_0$ and $V_0$ are the balance of lattice constants and initial unit volumes, respectively, at pressure $P=0$ and temperature $T=0$.

To study how the pressure affected the crystalline architecture constant and the single-element volume of the $\beta$-Mg${}_{17}$Al${}_{12}$ cell, the equilibrium lattice constant was calculated for various pressures. In Figure 3, the correspondence to the dimensionless ratios $a/a_0$ and $V/V_0$ were plotted as a function of pressure. The dimensionless ratios $a/a_0$ and $V/V_0$ decreased as the applied pressure increased, as shown in Figure 3. On the other hand, under pressure, the volume compression ratio was substantially larger than the lattice constant. This suggests that there was a reduction in the interatomic distances and an increase in the atomic electron interactions. In Equation (2), the relation between the no dimensions ratios a to $a_0$ and V to $V_0$ as a functionality is expressed by a third-order polynomial.

(2)$\begin{array}{c}a/a_0=0.99815-4.99\times {10}^{-3}P+6.43666\times {10}^{-5}{P}^2-3.50886\times {10}^{-7}{P}^3\hfill \\ V/V_0=0.9938-0.01414\times {10}^{-2}+2.00933\times {10}^{-4}{P}^2-1.1242\times {10}^{-6}{P}^3\hfill \end{array}$

It is imperative to understand that elastic constants play an important role in defining an anisotropic material’s structural stability, as well as its ability to withstand the applied forces. The coefficients $C_{11}$, $C_{12}$, and $C_{44}$ are used to express the elastic constants in the square-shaped crystal. According to the cubic stability requirement of the crystal [29], it is necessary for these elastic constants to satisfy the expressions of the generalization in Equation (3).

(3)$\left(C_{11}-C_{12}\right)>0,C_{11}>0,C_{44}>0,\left(C_{11}+2C_{12}\right)>0$

Computational values for the additional bulk modulus B, Young’s modulus E, and shear modulus G calculated for $T=0$ and $P=0$ are shown in Table 1, which also includes the calculated values of the elastic constants. Our calculated conclusions agreed with those published in the literature, based on the comparisons in Table 1 [12,26,27,28,30,31].

In addition, the estimated elastic constants’ dependencies for the pressure exerted are shown in Figure 4. The three independent elasticity constants, $C_{11}$, $C_{12}$, and $C_{44}$, increased as the pressure increased over the pressure region from 0 to 80 GPa, and the $C_{11}$ value increased most rapidly and was always larger than the $C_{12}$ value, while the values of $C_{44}$ were positive, and the diameters of the elastic constants in the pressure range studied were consistent with the mechanical stability criterion in Equation (3), indicating that $\beta$-Mg${}_{17}$Al${}_{12}$ may exist stably within a pressure range of 80 GPa.

3.2. Mechanical Properties

As is well-known, alloy products exhibit varied mechanical performance, which includes high strength, tenacity, and abrasion resistance. These mechanical properties are typically influenced by the material’s moduli. A material’s bulk modulus B, the Young’s modulus E, and the shear modulus G make up its mechanical properties. A higher number indicates greater resistance to deformation of the material. For the cubic system, the following Equation (4) can be used to calculate the bulk modulus and shear modulus parameters, which are based on the Voigt–Reuss–Hill (VRH) approximation, and relate to the material elastic constants.

(4)$B=B_V=B_R=\frac{1}{3}\left(C_{11}+2C_{12}\right)G=\frac{1}{2}\left(G_V+G_R\right)$

Assuming the VRH approximation, the Voigt shear modulus $G_V$ and the Reuss shear modulus $G_R$ denote the max and min data, respectively, and the formulae are supplied in Equation (5).

(5)$G_V=\frac{1}{3}\left(C_{11}-C_{12}+3C_{44}\right)G_R=\frac{5\left(C_{11}-C_{12}\right)C_{44}}{4C_{44}+3\left(C_{11}-C_{12}\right)}$

The Young’s modulus and Poisson’s ratio can be calculated with Equation (6), once the bulk modulus B and the shear modulus G have been determined.

(6)$E=\frac{9BG}{3B+G}v=\frac{3B-2G}{6B+2G}$

As a function of the pressure exerted, the bulk modulus, Young’s modulus, and shear modulus are displayed as fluctuations in Figure 5. A material’s shear modulus represents its tensile shear strain; a material’s bulk modulus measures its resistance to the strain under applied stress. By measuring the Young’s modulus, we can determine the stiffness of a material. The higher the value, the more rigid the material is. In the range of projected pressures from 0 to 80 GPa, the bulk modulus B, shear modulus G, and Young’s modulus E increased uniformly with increasing applied pressure. The results show that this alloy material became harder and more resistant to volume strain and shear strain with the increase in pressure.

Exploring the phase changes in materials requires consideration of the ductile-to-brittle transition. According to Pugh [32], the ductile/brittle quality of pure polycrystalline metals is determined by the ratio of the bulk moduli to the shear moduli. In general, 1.75 is the critical $B/G$ ratio for determining ductile from brittle materials. Material that has a $B/G$ greater than 1.75 is said to be ductile, while material that has a $B/G$ less than 1.75 is said to be brittle. Furthermore, a higher $B/G$ ratio results in a ductile material; otherwise, it is a brittle one. It is crucial for Poisson’s ratio ($\nu$) to be 0.26 in order to be able to determine the stability of the crystal’s shear stress. Upon exceeding a Poisson’s ratio of 0.26, it becomes ductile. As $\nu$ increases, the ductility of the material increases [33]. For the $\beta$-Mg${}_{17}$Al${}_{12}$, Figure 6 shows the $B/G$ values and Poisson’s ratio plotted as a function of the pressure. As shown in Figure 6a, the $B/G$ value was always greater than 1.75 over the studied pressure range of 0 to 80 GPa, indicating an increase in ductility with pressure. With increasing pressure, $\beta$-Mg${}_{17}$Al${}_{12}$’s Poisson’s ratio also increased, along with its ductility over the pressure range, as shown in Figure 6b.

The Cauchy pressure $C_{12}-C_{44}$ is believed to be a critical specification for describing the angular nature of the combination of atoms in the metal and chemical compound, as it reveals the properties of bonds at the level of atoms [34]. The atomic bonding is characterized by the metallic bonding when the globular atoms are enveloped by a homogeneous electron gas and do not have any regionality or directionality in the distribution of electrons when the positive Cauchy pressure is present. When the Cauchy pressure is negative, however, a directional bonding pattern with an angular character is observed. As the directional feature becomes stronger, the Cauchy pressure increases [35,36]. As the Cauchy pressures were always positive within the expected pressure range and rose rapidly with increasing applied pressure, Figure 7 shows that the metal bonding was the most common mode of bonding and became stronger under pressure.

3.3. Anisotropy

To study the mechanical properties of materials, it is necessary to fully understand the elastic anisotropy. A is an anisotropy factor that can be used to study the flexible anisotropy of the materials. For isotropic materials, the anisotropy factor A is unity (A = 1), while other anisotropy factor values denote varied material anisotropy levels. Material anisotropy is present when the anisotropy coefficient of a material is considerably higher or significantly lower than the unit [37,38]. Yoo [39] presented the cross-slip–rotation mechanism, in which a large value of A can enhance the driving force (or tangential force) acting on the screw dislocation and thus accelerate the cross-slip–rotation process; this implies that the cross-slip of the screws’ displacement can indeed be enhanced with a significant increase in the anisotropy coefficient.

In this case, the anisotropy factors for $\beta$-Mg${}_{17}$Al${}_{12}$ with cubic crystals were calculated by Equation (7), which is associated with the planar and symmetrical axes in terms of the three elastic constants [40,41]:

(7)$A_{\left(100\right)}=\frac{2C_{44}}{C_{11}-C_{12}}{A}_{\left(110\right)}=\frac{C_{44}\left(C^{\prime }+2C_{12}+C_{11}\right)}{C_{11}{C}^{\prime }-C_{12}^2}$

where $C^{\prime }=C_{44}+\frac{C_{11}+C_{12}}{2}$.

$A_{\left(100\right)}$ and $A_{\left(110\right)}$ are the anisotropy factors in the (100) direction and the (110) direction, respectively. Figure 8 shows that the above formulations of both anisotropy factors can be used to offer data relevant to the study of the mechanical performance of the $\beta$-Mg${}_{17}$Al${}_{12}$ alloy. The $\beta$-Mg${}_{17}$Al${}_{12}$ alloy showed anisotropy at $P=0$ due to $A_{\left(100\right)}=0.64827$ and $A_{\left(110\right)}=0.7143$, and these two anisotropy coefficients rapidly decreased with the increase in the pressure applied, making the material more anisotropic. By applying pressure to the screw dislocations, the cross-slip can be effectively promoted.

Another important factor in characterizing a material’s mechanical properties is its Poisson’s ratio, which normally ranges from 0.5 to 1. A higher Poisson’s ratio suggests that the material is more malleable. The kind of interatomic bonding is the main factor determining the Poisson’s ratio when considering the main atomic scale parameters [41]. According to Reed et al. [42], the bottom and upper bounds of the central force solids are given by $\nu$ = 0.25 and $\nu$ = 0.5, respectively. In the current investigation, the parameters ${\nu }_{\left[001\right]}$ and ${\nu }_{\left[111\right]}$ referred to the Poisson’s ratios in the single-axis directions [001] and [111], which were appropriately formulated by Equation (8) [41,43], and the relevant functions were plotted in Figure 9 in relation to the pressure used.

(8)${\nu }_{\left[001\right]}=\frac{C_{12}}{C_{11}+C_{12}}{\nu }_{\left[111\right]}=\frac{C_{11}+2C_{12}-2C_{44}}{2\left(C_{11}+2C_{12}+C_{44}\right)}$

At $P=0$, we obtained $V_{\left[111\right]}=0.31$, as shown in Figure 9. These results imply that the core force was the most prevalent form of interatomic bonding $\beta$-Mg${}_{17}$Al${}_{12}$ in the uniaxial [111] direction, and there was an increase in the value of ${\nu }_{\left[111\right]}$ with increasing pressure; therefore, a larger pressure may provide a greater centering power and better agility in the uniaxial [111] orientation.

3.4. Electronic Properties

In metals or alloys, the electronic structures between atoms frequently reveal the binding process. They were used in this study to characterize how the structural stabilization of the $\beta$-Mg${}_{17}$Al${}_{12}$ alloy varied as a function of the pressure. Figure 10 depicts the total density of states (TDOS) as well as the partial density of states (PDOS) for the alloy $\beta$-Mg${}_{17}$Al${}_{12}$ at 0 GPa. The Fermi level is indicated by the dotted line at 0 eV. The TDOS at the Fermi energy (EF) was not zero, which indicates that the $\beta$-Mg${}_{17}$Al${}_{12}$ alloys possessed metallic properties, which was in good agreement with the outcomes shown in Figure 7. The PDOS of the $\beta$-Mg${}_{17}$Al${}_{12}$ alloy showed that the Mg-3p and Al-3p states had the most significant contributions to the valency belt at EF, while the Mg-s and Al-s states had a smaller contribution.

To ascertain the impact of the high pressure on the TDOS of the $\beta$-Mg${}_{17}$Al${}_{12}$ alloy, we depicted the TDOS change profiles at 0, 40, and 80 GPa pressures near EF, as shown in Figure 11. As the pressure increased, the conduction band energy decreased slightly, while the valence band energy increased. This allowed more electrons inside the crystal to jump from the valence band to the conduction band, which resulted in stronger bonding between atoms and an increase in the elastic constant. All other physical properties were calculated based on the elastic constants. The calculation of the density of states reflects the physical nature of each macroscopic property as a function of pressure.

4. Conclusions

The structural, elastic, and mechanical properties of $\beta$-Mg${}_{17}$Al${}_{12}$ were investigated in the context of the density generalized function theory using the CASTEP program at pressures of 0–80 GPa. The crystal lattice particles at 0 GPa pressure were in agreement with the values in the literature, indicating that the calculation method was correct. The volume of $\beta$-Mg${}_{17}$Al${}_{12}$ crystals in the ground state was V${}_0$ = 1180.35 Å${}^3$, and the lattice parameter was 10.57 Å. This is consistent with the available results in the literature and indicates that the calculations were correct. With greater pressure, the three separate elastic constants $C_{11}$, $C_{12}$, and $C_{44}$ increased. The bulk modulus B, shear modulus G, and Young’s modulus E all rose with the pressure, which demonstrated that the bulk deformation resistance, shear deformation resistance, and stiffness of the $\beta$-Mg${}_{17}$Al${}_{12}$ phase all increased as the pressure increased. The $B/G$ > 1.75 and Poisson’s ratio $\nu$ > 0.26 increased with increasing pressure, indicating that the $\beta$-Mg${}_{17}$Al${}_{12}$ crystals were ductile, and the ductility increased with increasing pressure. Furthermore, the Cauchy pressure C${}_{12}$–C${}_{44}$ increased with increasing pressure; the anisotropy factor A${}_{\left(100\right)}$ and A${}_{\left(110\right)}$ deviated further from 1, and the anisotropy was enhanced. The electronic structure calculations showed that the total density of states TDOS was achieved by the Mg-3p and Al-3p states and that the total density of states movement to the higher energy area under pressure and the interatomic bonding was enhanced, which led to an increase in the elastic constant and, ultimately, to an increase in each physical property with increasing pressure.

Author Contributions

Conceptualization, C.X. and X.Z.; methodology, C.X.; software, Y.W.; validation, C.X. and X.Z.; formal analysis, Z.T.; investigation, Z.T. and Z.L.; data curation, L.L. and Z.L.; writing—original draft preparation, C.X. and X.Z.; writing—review and editing, Z.T., Z.L. and Y.W.; visualization, C.X. and L.L.; project administration, C.X., L.L. and Y.W.; funding acquisition, C.X., L.L. and Y.W. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.

Conflicts of Interest

The authors declare no conflict of interest.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Figures and Table

Figure 1. The cubic crystal structure of the [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.] alloy.

Figure 2. Conventional [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.] cell volume as a function of the predicted total energy fluctuation curve.

Figure 3. Based on the applied pressure, the plots depict the variation in [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] for the [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.] alloys.

Figure 4. The elastic constants versus pressure for the [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.] phase.

Figure 5. Plot of the variation in the bulk modulus B, Young’s modulus E, and shear modulus G, as a function of the applied pressure for [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 6. G/B (a,b) versus the applied pressure for [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 7. Cauchy pressures [Forumla omitted. See PDF.] as a function of pressure for [Forumla omitted. See PDF.] -Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 8. Plots of the anisotropy factor variation as a function of applied pressure for [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 9. The Poisson’s ratio as a function of applied pressure for [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 10. Density of electronic states of [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.].

Figure 11. The density of states for [Forumla omitted. See PDF.]-Mg[Forumla omitted. See PDF.]Al[Forumla omitted. See PDF.] under different pressures.

Table 1

The elastic constants, modulus, and Poisson’s ratio of the $\beta$-Mg${}_{17}$Al${}_{12}$.

	${\mathit{C}}_{11}$	${\mathit{C}}_{12}$	${\mathit{C}}_{44}$	B	G	E	$\mathit{\nu }$
Present	92.52	26.24	21.48	48.33	25.57	65.21	0.27
Literature [28]	116.90	18.01	35.15	50.97	40.31	95.69	0.18
Literature [26]	98.00	28.00	31.00	51.55	32.54	80.60	0.24
Literature [27]	90.50	26.70	29.70	48.02	30.56	76.30	0.24
Literature [12]	87.00	29.00	20.00	48.33	23.22	60.04	0.29
Literature [30]	88.38	23.54	26.95	45.15	29.14	71.94	0.23
Literature [31]	89.86	27.21	33.32	48.09	32.49	79.62	0.23

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