Phase Stability, Elastic Modulus and Elastic Anisotropy of X Doped (X = Zn, Zr and Ag) Al₃Li: Insight from First-Principles Calculations

Abstract

In present work, the effects of alloying elements X (X = Zn, Zr and Ag) doping on the phase stability, elastic properties, anisotropy and Debye temperature of Al₃Li were studied by the first-principles method. Results showed that pure and doped Al₃Li can exist and be stable at 0 K. Zn and Ag elements preferentially occupy the Al sites and Zr elements tend to occupy the Li sites. All the C_ij obey the mechanical stability criteria, indicating the mechanical stability of these compounds. The overall anisotropy decreases in the following order: Al₂₃Li₈Ag > Al₃Li > Al₂₃Li₈Zn > Al₂₄Li₇Zr, which shows that the addition of Zn and Zr has a positive effect on reducing the anisotropy of Al₃Li. The shear anisotropic factors for Zn and Zr doped Al₃Li are very close to one, meaning that elastic moduli do not strongly depend on different shear planes. For pure and doped Al₃Li phase, the transverse sound velocities ν_t1 and ν_t2 among the three directions are smaller than the longitudinal sound velocity ν_l. Moreover, only the addition of Zn is beneficial to increasing the Θ_D of Al₃Li among the three elements.

1. Introduction

Al-Li alloys have gained widespread attention for their use as lightweight structural materials in the aerospace field due to their low density, high elastic modulus and specific stiffness compared with other alloys [1,2,3,4]. The mechanical behaviors of Al-Li alloys depend to a large extent on the structure and properties of precipitates [5,6]. Al₃Li (δ′) phase plays a vital strengthening role in high lithium aluminum alloys, especially in binary Al-Li alloys [7,8]. However, Al-Li alloys invariably possess poor ductility due to the planar slip caused by the shear of δ′ precipitates [8,9]. Given this situation, microalloying can improve the mechanical properties of Al-Li alloys.

The microalloying elements may affect the structure and properties of precipitate [10,11], which can be researched by different methods [12,13,14]. For example, the existence of Mg in Al-Li-Mg alloys increases the lattice parameters of the matrix and δ′, indicating that Mg has been incorporated into δ′ [15]. Gault et al. have demonstrated that the Mg atoms are partitioned to Li sublattices in δ‘precipitation [16,17]. Therefore, other alloying elements may also occupy the sublattice of either Al or Li in δ′ precipitate. Hirosawa and Sato researched the atomistic behavior of various microalloying elements in Al-Li alloys at 273 K, which revealed that Ag, Pb and Pd are preferentially occupying Al sites and Mg, Zn and Cu tend to occupy the Li sites [18]. In addition, the influence of microalloying elements on the precipitation process of δ′ phase has been systematically studied through experimental analysis [19,20]. Our previous work studied the influence of alloying elements on the elastic properties of δ′ phase at higher doping concentrations [21]. However, there are few reports concerning the influence of microalloying elements on the elastic anisotropy of doped δ′ phase at the doping concentration of 3.125 at. %.

In the present study, the enthalpy of formation and transfer energy were employed to predict the relative stability of doped Al₃Li and the site preferences of alloying elements X (X = Zn, Zr and Ag), respectively. We adopted the first-principles method to reveal the effects of alloying elements X on the elastic modulus, anisotropy and Debye temperature of Al₃Li phase.

2. Computational Studies

Al₃Li phase is a cubic structure with a = b = c = 4.010 Å and a 2 × 2 × 2 supercell with 32 atoms which was employed to simulate the influence of doping elements, as shown in Figure 1. Only one alloying atom was selected to substitute either one Al atom or one Li atom, and the doping concentration of alloying element is 3.125 at. %. Hence the chemical formulas of doped Al₃Li phase can be labeled as Al₂₄Li₇X and Al₂₃Li₈X (X = Zn, Zr and Ag), respectively.

All the calculations in this work were carried out using the Cambridge Sequential Total Energy Package based on the density functional theory (DFT) [22]. The generalized gradient approximation (GGA) with the Perdew–Burke–Ernzerhof (PBE) function was employed [23]. The Broyden–Fletcher–Goldfarb–Shanno (BFGS) was selected to optimize the structure of pure and doped Al₃Li, and the convergence thresholds for maximum displacement were set to 5.0 × 10⁻⁴ Å. For all calculations, the plane-wave cut-off energy was set to 500.0 eV, and a Monkhorst–Pack mesh with 26 × 26 × 26 and 9 × 9 × 9 k-points was chosen to sample the Brillouin zone of pure Al₃Li and alloying elements X doped Al₃Li, respectively.

3. Results and Discussion

3.1. Phase Stability and Site Preference

In this work, the enthalpy of formation (ΔH_f) is used to predict the relative stability of alloying elements X doped Al₃Li [24] and the ΔH_f can be calculated using the following formula [24]:

$$\Delta H_f=\frac{1}{\left(a+b+1\right)}(E_{tot}-a{E}_{solid}^{Al}-b{E}_{solid}^{Li}-E_{solid}^X)$$

(1)

where a and b correspond to the number of Al and Li atoms, E_tot is the total energy of alloying elements X doped Al₃Li phase, and $E_{solid}^{Al}$, $E_{solid}^{Li}$ and $E_{solid}^X$ represent the energies per atom of Al, Li, X in solid states, respectively. As shown in

Table 1. Predicted enthalpy of formation ΔH_f (eV), normalized transfer energy ${\tilde{E}}_{Li\to Al}^X$ and site preference of doped Al₃Li at 0 GPa.

Element X.	ΔHf		${\tilde{\mathit{E}}}_{\mathit{L}\mathit{i}\to \mathit{A}\mathit{l}}^{\mathit{X}}$	Site Preference
	Al24Li7X	Al23Li8X
Zn	−0.073	−0.097	−0.09	Al
Zr	−0.139	−0.096	3.15	Li
Ag	−0.080	−0.106	−2.27	Al

, the enthalpies of formation for doped samples are negative, which implies that they can exist and be stable [25,26]. Moreover, a phase with lower ΔH_f is much easier to form than others. The ΔH_f of Al₂₄Li₇X (X = Zn and Ag) are larger than those of Al₂₃Li₈X (X = Zn and Ag), which indicates that Al₂₃Li₈X (X = Zn and Ag) are easier to form. For Zr doped Al₃Li phase, the ΔH_f of Al₂₄Li₇Zr is smaller than Al₂₃Li₈Zr, meaning that Al₂₃Li₈Zr is harder to form. Al₂₄Li₇Zr phase has the smallest value of ΔH_f, suggesting that Al₂₄Li₇Zr is more stable than other doped samples.

The site preference of a microalloying element in Al₃Li phase was investigated by normalized transfer energy (${\tilde{E}}_{Li\to Al}^X$) and the formula of ${\tilde{E}}_{Li\to Al}^X$ is defined as follows [27,28]:

$$E_{Al\to Li}^X=E^X\left(\mathrm{AL}\right)+E^{AL}\left(\mathrm{Li}\right)-E^X\left(\mathrm{Li}\right)-E_{\mathrm{Al}3\mathrm{Li}}$$

(2)

$$E_{Al\to Li}^X=E^X\left(\mathrm{Li}\right)+E^{Li}\left(\mathrm{Al}\right)-E^X\left(\mathrm{Al}\right)-E_{\mathrm{Al}3\mathrm{Li}}$$

(3)

$$E_X=E_{Al\to Li}^X+E_{Li\to Al}^X$$

(4)

$${\tilde{E}}_{Li\to Al}^X=E_{Li\to Al}^X/E_X$$

(5)

Here, $E_{Li\to Al}^X$ and $E_{Al\to Li}^X$ represent the transfer energy of moving an alloying element X atom from a Li site to an Al site and from an Al site to a Li site, respectively. $E^X\left(\mathrm{Al}\right)$ and $E^X\left(\mathrm{Li}\right)$ stand for the energies of Al₃Li with an alloying element X atom in an Al site and a Li site, respectively. $E^{Al}\left(\mathrm{Li}\right)$ and $E^{Li}\left(\mathrm{Al}\right)$ represent the energies of an Al antisite and a Li antisite of Al₃Li. $E_{\mathrm{Al}3\mathrm{Li}}$ stands for the total energy of Al₃Li. As listed in Table 1, the values of ${\tilde{E}}_{Li\to Al}^X$ for Zn and Ag doped Al₃Li are negative, which means that Zn and Ag show strong Al site preference [27,28]. The value of ${\tilde{E}}_{Li\to Al}^X$ for Zr doped Al₃Li is bigger than one, indicating that Zr elements tend to occupy the Li site. This result may be related to atomic radius difference between these elements. The atomic radius of Zn (1.39 Å) and Ag (1.44 Å) are closer to the atomic radius of Al (1.43 Å) and they tend to occupy the Al site. On the other hand, the atomic radius difference between Li (1.54 Å) and Zr (1.60 Å) is smaller than that of Al and Zr, which could be used to explain the site preference of Zr.

3.2. Elastic Properties

The elastic behavior, which when considered as a basic physical property, can be described by elastic constants (C_ij). Pure and Zr doped Al₃Li have three independent elastic constants, i.e., C₁₁, C₁₂ and C₄₄. However, with the addition of alloying elements X (X = Zn and Ag), the lattice symmetry of Al₃Li decreases, thus, increasing the independent C_ij [29]. Hence, the Al₂₃Li₈X (X = Zn and Ag) phases have a tetragonal structure, and the average elastic constants were not considered. As shown in

Table 2. Computed elastic constants C_ij (GPa), bulk modulus B (GPa), shear modulus G (GPa), Young’s modulus E (GPa) and B/G ratio for pure and doped Al₃Li compounds.

Phase	C11	C33	C44	C66	C12	C13	B	G	E	B/G
Al3Li	125.3	125.3	40.8	40.8	31.7	31.7	62.9	43.1	105.2	1.46
Al23Li8Zn	127.7	125.7	45.6	47.6	31.1	30.7	62.9	47.0	112.9	1.34
Al24Li7Zr	127.9	127.9	43.1	43.1	38.1	38.1	68.0	43.8	108.2	1.55
Al23Li8Ag	128.8	127.6	40.7	44.2	32.3	31.2	63.9	44.3	108.0	1.44

, the predicted C_ij of Al₃Li are in good agreement with other reported values [30,31]. In addition, all the obtained C_ij obey the mechanical stability criteria [32,33], implying the mechanical stability of pure and doped Al₃Li.

The bulk modulus B, shear modulus G and Young’s modulus E of pure and doped Al₃Li phase were calculated by adopting Voigt–Reuss–Hill approximation [34]. As displayed in Table 2, the value of B for Al₂₄Li₇Zr is larger than that of other doped Al₃Li, indicating that Al₂₄Li₇Zr has a stronger resistance to volume change. Moreover, Zr and Ag elements can improve the B of Al₃Li, while the value of B between pure and Zn doped is similar. The values of G can be sorted in the order of Al₂₃Li₈Zn > Al₂₃Li₈Ag > Al₂₄Li₇Zr > Al₃Li, which illustrate that the Zn, Ag and Zr can improve the shear deformation resistance. Compared to pure Al₃Li, the values of G for Al₂₃Li₈Zn, Al₂₃Li₈Ag and Al₂₄Li₇Zr increase by 9.1%, 2.9% and 1.6%, respectively. The values of E follow the order of Al₂₃Li₈Zn > Al₂₄Li₇Zr > Al₂₃Li₈Ag > Al₃Li, indicating that Al₂₃Li₈Zn has the highest stiffness among these phases. Furthermore, the B/G ratios for all considered compounds are less than 1.75, implying that these compounds tend to exhibit brittle behavior based on the Pugh formulation [35]. The B/G ratios of Zr doped Al₃Li is larger than others, which suggests that Zr slightly decreases the brittleness of Al₃Li. As a whole, the Zn element plays a prominent role in enhancing the stiffness and shear deformation resistance compared with Ag and Zr elements. These three alloying elements have different effects on the elastic properties of Al₃Li, which may be due to the different strength of chemical bonds between different alloying elements and Al or Li.

3.3. Elastic Anisotropy

In order to comprehensively investigate the elastic anisotropy performance of Al₃Li with doping elements, a variety of different elastic anisotropy indices were selected. The universal anisotropy index A^U is applied to estimate the overall anisotropy of compounds and the calculation formula is as follows [36]:

$$A^U=5\frac{G_V}{G_R}+\frac{B_V}{B_R}-6$$

(6)

where G_V (B_V) and G_R (B_R) represent the Voigt and the Reuss shear (bulk) modulus, respectively. When the value of A^U is zero, the crystal tends to be isotropic. Significant deviations from zero indicate high anisotropic characteristics. As displayed in

Table 3. Predicted universal anisotropy index (A^U) and shear anisotropic factors (A₁, A₂ and A₃) of pure and doped Al₃Li.

Phase	A1	A2	A3	AU
Al3Li	0.871	0.871	0.871	0.023
Al23Li8Zn	0.951	0.954	0.986	0.003
Al24Li7Zr	0.959	0.959	0.959	0.002
Al23Li8Ag	0.839	0.857	0.915	0.030

, the A^U decreases in the following order: Al₂₃Li₈Ag > Al₃Li > Al₂₃Li₈Zn > Al₂₄Li₇Zr. The obtained values of A^U are in the range of 0.002 to 0.030, indicating that pure and doped Al₃Li have relatively weak anisotropy. Consequently, Al₂₃Li₈Ag has the strongest anisotropic behavior. In contrast, the existence of Zn and Zr has a positive effect on reducing the anisotropy of Al₃Li. Furthermore, the A^U for Al₂₄Li₇Zr and Al₂₃Li₈Zn is close to zero, implying that the elastic modulus of Al₂₄Li₇Zr andAl₂₃Li₈Zn shows similar elastic properties in all directions.

The shear anisotropic factors (A₁, A₂ and A₃) can be used to describe the shear anisotropies of crystal and are defined as follows [37]:

For cubic symmetry:

$$A_1=A_2=A_3=\frac{2C_{44}}{C_{11}-C_{12}}\mathrm{for}\mathrm{the}\{100\}\mathrm{plane}$$

(7)

For tetragonal symmetry:

$$A_1=\frac{C_{44}\left(C_{11}+2C_{13}+C_{33}\right)}{C_{11}{C}_{33}-C_{13}^2}\mathrm{for}\mathrm{the}(010)\mathrm{or}(100)\mathrm{plane}$$

(8)

$$A_2=\frac{C_{44}\left(C_L+2C_{13}+C_{33}\right)}{C_L{C}_{33}-C_{13}^2}\mathrm{for}\mathrm{the}(1\overline{1}0)\mathrm{plane}$$

(9)

$$C_L=C_{44}+\frac{\left(C_{11}+C_{12}\right)}{2}$$

(10)

$$A_3=\frac{2C_{66}}{C_{11}-C_{12}}\mathrm{for}\mathrm{the}(001)\mathrm{plane}$$

(11)

For an isotropic crystal, the values of A₁, A₂ and A₃ are equal to one, otherwise, the crystal presented anisotropy. As listed in Table 3, the predicted A₁, A₂ and A₃ are less than one, which reveals that they show anisotropic characteristics in different shear planes. The shear anisotropy of Al₂₃Li₈Ag along the (010), (100) and (1$\overline{1}$0) plane is much higher than others, while the degree of anisotropy of Al₂₃Li₈Zn is the smallest in the (001) plane. Compared with the other two microalloying elements, Ag slightly increased the shear anisotropy of Al₃Li in the (010), (100) and (1$\overline{1}$0) plane. In addition, the shear anisotropic factors for Zn and Zr doped Al₃Li are close to one, meaning that their elastic moduli do not strongly depend on the (010), (100), (001) and (1$\overline{1}$0) planes.

The three-dimensional (3D) surface construction of Young’s modulus has become an effective method to display the elastic anisotropy of compounds visually and can be determined by the following expressions [38]:

For cubic symmetry:

$$\frac{1}{E}=S_{11}-2\left(S_{11}-S_{12}-\frac{S_{44}}{2}\right)\left(l_1^2{l}_2^2+l_2^2{l}_3^2+l_1^2{l}_3^2\right)$$

(12)

For tetragonal symmetry:

$$\frac{1}{E}=S_{11}\left(l_1^4+l_2^4\right)+\left(2S_{13}+S_{44}\right)\left(l_1^2{l}_3^2+l_2^2{l}_3^2\right)+S_{33}{l}_3^4+\left(2S_{12}+S_{66}\right)l_1^2{l}_2^2$$

(13)

where, S_ij represents the elastic compliance coefficients of pure and doped Al₃Li, and l₁, l₂ and l₃ stand for the directional cosines corresponding to the x, y and z axes. If the 3D surface diagram is spherical, it indicates that the phase shows isotropic behavior. Otherwise, the phase exhibits anisotropic behavior. As plotted in Figure 2, the 3D surface diagrams of pure and doped Al₃Li deviate from the sphere, which match well with the results of A^U. The 3D surface diagram of Al₂₄Li₇Zr and Al₂₃Li₈Zn are close to the sphere, especially Al₂₄Li₇Zr.

3.4. Debye Temperature and Anisotropic Sound Velocities

The thermodynamic properties of crystals, such as specific heat and melting point, can be described by Debye temperature (Θ_D). The Θ_D can be calculated by the following expression [39]:

$${\Theta }_D=\frac{h}{k_B}{[\frac{3n}{4\pi }(\frac{N_A\rho }{M})]}^{\frac{1}{3}}{v}_m$$

(14)

$$v_m={[\frac{1}{3}(\frac{2}{v_s{}^3}+\frac{1}{v_l{}^3})]}^{-1/3}$$

(15)

$$v_t=\sqrt{\frac{G}{\rho }}$$

(16)

$$v_l=\sqrt{\frac{3B+4G}{3\rho }}$$

(17)

Here, h, k_B, n, N_A, ρ, M and v_m represent Planck’s constant, Boltzmann’s constant, total number of atoms, Avogadro’s number, density, molecular weight and average wave velocity, respectively. v_t and v_l are the transverse and longitudinal sound velocities of materials, respectively.

The sound velocities, density and Debye temperature of pure and doped Al₃Li phase are summarized in

Table 4. The density ρ, transverse sound velocity v_t, longitudinal sound velocity v_l, mean sound velocity v_m and Debye temperature Θ_D of pure and doped Al₃Li phase.

Phase	ρ (g/cm3)	vt (m/s)	vl (m/s)	vm (m/s)	ΘD (K)
Al3Li	2.22	4403.2	7358.2	4872.2	570.9
Al23Li8Zn	2.36	4466.8	7301.2	4931.8	578.8
Al24Li7Zr	2.47	4213.4	7159.1	4669.6	545.5
Al23Li8Ag	2.48	4225.0	7036.3	4673.6	548.0

. The obtained Θ_D of Al₃Li is 570.9 K, which is consistent with the reported value in the literature [40]. The value of Θ_D from high to low is as follows: Al₂₃Li₈Zn > Al₃Li > Al₂₃Li₈Ag > Al₂₄Li₇Zr, which indicates that the Θ_D of Al₂₄Li₇Zr is smaller than that of the others. Among the three elements, only the addition of Zn is beneficial to increase the Θ_D of Al₃Li at the doping concentration of 3.125 at. %.

According to the Debye model, the sound velocity is one of the critical parameters for obtaining Debye temperature, which is closely related to the elastic properties of crystals. Longitudinal sound waves are related to the compressibility and density of crystals. The transverse sound velocity is considered to be the origin of shear deformation, which can be obtained by shear modulus and density [41]. In addition, the direction of sound propagation has an impact on the sound velocity of crystal. Thus, the sound velocity anisotropy of doped Al₃Li phases were systemically studied and the formulas are as follows [41]:

For tetragonal crystals:

$$[100]:\left[100\right] {\nu }_l=\left[010\right] {\nu }_l=\sqrt{\frac{C_{11}}{\rho }};\left[001\right] {\nu }_{t1}=\sqrt{\frac{C_{44}}{\rho }};\left[010\right] {\nu }_{t2}=\sqrt{\frac{C_{66}}{\rho }}$$

(18)

$$[001]:\left[001\right] {\nu }_l=\sqrt{\frac{C_{33}}{\rho }};\left[100\right] {\nu }_{t1}=\left[010\right] {\nu }_{t2}=\sqrt{\frac{C_{66}}{\rho }}$$

(19)

$$[110]:\left[110\right] {\nu }_l=\sqrt{\frac{\left(C_{11}+C_{12}+2C_{66}\right)}{2\rho }};\left[001\right] {\nu }_{tl}=\sqrt{\frac{C_{44}}{\rho }};\left[1\overline{1}0\right] {\nu }_{t2}=\sqrt{\frac{\left(C_{11}-C_{12}\right)}{2\rho }}$$

(20)

For cubic crystals:

$$[100]:\left[100\right] {\nu }_l=\sqrt{\frac{C_{11}}{\rho }};\left[010\right] {\nu }_{t1}=\left[001\right] {\nu }_{t2}=\sqrt{\frac{C_{44}}{\rho }}$$

(21)

$$[110]:\left[110\right] {\nu }_l=\sqrt{\frac{\left(C_{11}+C_{12}+2C_{44}\right)}{2\rho }};\left[1\overline{1}0\right] {\nu }_{t1}=\sqrt{\frac{\left(C_{11}-C_{12}\right)}{\rho }}\left[001\right] {\nu }_{t2}=\sqrt{\frac{C_{44}}{\rho }}$$

(22)

$$[111]:\left[111\right] {\nu }_l=\sqrt{\frac{\left(C_{11}+2C_{12}+4C_{44}\right)}{3\rho }};\left[11\overline{2}\right] {\nu }_{t1}={\nu }_{t2}=\sqrt{\frac{\left(C_{11}-C_{12}+C_{44}\right)}{3\rho }}$$

(23)

where v_t1 and v_t2 stand for the first and second mode transverse sound velocity. The directional sound velocities for pure and doped Al₃Li phases are listed in

Table 5. The calculated anisotropic sound velocities (m/s) of pure and doped Al₃Li phase.

Cubic Crystals	[100]			[110]			[111]
	[100]	[010] νt1	[001] νt2	[110] νl	[$\mathbf{1}\overline{\mathbf{1}}$0] νt1	[001] νt2	[111] νl	[$\mathbf{11}\overline{\mathbf{2}}$] νt1	[$\mathbf{11}\overline{\mathbf{2}}$] νt2
Al3Li	7596.8	4411.3	4411.3	7429.4	6629.3	4411.3	7372.8	4597.3	4597.3
Al24Li7Zr	7201.2	4178.1	4178.1	7148.9	6034.3	4178.1	7131.4	4237.5	4237.5
Tetragonal Crystals	[100]			[001]			[110]
	[100] νl	[001] νt1	[010] νt2	[001] νl	[100] νt1	[010] νt2	[110] νl	[001] νt1	[$\mathbf{1}\overline{\mathbf{1}}$0] νt2
Al23Li8Zn	7363.3	4401.4	4495.6	6993.0	3951.6	3951.6	7343.2	4401.4	4528.5
Al23Li8Ag	7201.1	4047.6	4216.1	7367.0	4534.9	4534.9	7085.2	4047.6	4408.0

. The values of sound velocities are different in different directions, indicating the anisotropy of sound velocity. The v_t1 and v_t2 for pure and Zr doped Al₃Li are equal in the [100] and [111] directions, while the v_t1 and v_t2 of Al₂₃Li₈X (X = Zn, Ag) have the same value along the [001] directions. However, the values between v_t1 and v_t2 are different for Al₃Li and Al₂₃Li₈Zr in the [110] direction. For Al₃Li, the longitudinal sound velocity decreases in the [100] direction after element doping. The transverse sound velocities ν_t1 and ν_t2 among the three directions are smaller than the longitudinal sound velocity ν_l. For Al₃Li, Al₂₃Li₈Zn and Al₂₄Li₇Zr, the value of ν_l in the [100] direction is the largest, while Al₂₃Li₈Ag has the largest ν_l along the [001] direction.

4. Conclusions

We investigated the effects of alloying elements X (X = Zn, Zr and Ag) on the phase stability, elastic properties, anisotropy and Debye temperature of pure and doped Al₃Li phase. The following conclusions are reached:

(1) The enthalpies of formation for doped samples are negative, indicating that they can exist and be stable at 0 K. The Zn and Ag elements preferentially occupy the Al sites and Zr elements tend to occupy the Li sites.

(2) Pure and doped Al₃Li are mechanically stable and tend to exhibit brittle behavior. The addition of Zr and Ag elements can improve the B. Moreover, the Zn element plays an obvious role in enhancing the stiffness and shear deformation resistance.

(3) The A^U decreases in the following order: Al₂₃Li₈Ag > Al₃Li > Al₂₃Li₈Zn > Al₂₄Li₇Zr, implying that Zn and Zr can reduce the anisotropy of Al₃Li. Pure and doped Al₃Li show anisotropy in different shear planes. The addition of Ag slightly increased the anisotropy of Al₃Li in the (010), (100) and (1$\overline{1}$0) plane. The longitudinal sound velocity of Al₃Li decreased in the [100] direction after element doping. In addition, the transverse sound velocities ν_t1 and ν_t2 among the three directions are smaller than the longitudinal sound velocity ν_l.

(4) The value of Θ_D from high to low is as follows: Al₂₃Li₈Zn > Al₃Li > Al₂₃Li₈Ag > Al₂₄Li₇Zr, which shows that the addition of Zn is beneficial for increasing the Θ_D.