First-principles investigation of elastic, mechanical, electronic and thermodynamic properties of Al3Li compound under pressure

First-principles calculations are employed to study the elastic, mechanical, electronic and thermodynamic properties of Al3Li compound under pressure up to 25.9 GPa. Based on the elastic constant and phonon calculations, we found that Al3Li has mechanical and dynamical stability in the considered pressure range. The elastic constants and mechanical properties of Al3Li under pressure are calculated. The values of hardness, thermal conductivity and melting temperature under ambient pressure are compared with that of the pure Al and Al3Sc. The wave velocities, Debye temperatures, phonon vibrational frequencies positively increase with the increasing pressure. The results from the analysis of electronic density of states exhibit a metallic bonding behavior of Al3Li. Finally, the temperature dependent behaviors of thermodynamic properties of Al3Li as a function of temperature are determined within the quasi-harmonic approximation theory and compared with the available experimental and theoretical data.


Introduction
Al-Li alloys have attracted significant attention in cryogenic applications and aerospace industry because of the advantages of low density and high strength [1,2]. The strengthening of Al-Li alloy can be attributed to the spherical precipitations, d¢ phase of Al 3 Li with L1 2 -ordered structure which is formed in heat treatment [3,4]. The mechanical properties of Al 3 Li could be further improved by the addition of Sc, which can enhance the grain refinement of the alloys and improve its low-cycle fatigue properties [5]. On the other hand, the addition of Li in Al-Sc alloys can lead to higher hardness, which is introduced by the Al 3 X (X=Sc, Li) precipitates [6]. Recently, the metastable phase Al 3 Li is also experimentally observed in the Al-Cu-Li compounds [7].
Plenty of works have investigated the various properties of L1 2 -Al 3 Sc, under ambient pressure [4,8,9] and high pressure [10]. At the side of Al 3 Li compound with d¢ phase, the temperature dependence of elastic constants have been uncovered experimentally more than a half century ago [11]. After that, the mechanical, electronic and thermodynamic properties under ambient pressure have been extensively investigated. Noble et al [1] estimated the Young's moduli of Al 3 Li at room temperature, 96 GPa. Hu et al [4] performed a comparative study between Al 3 Sc and Al 3 Li compounds under 0 pressure and reported vast useful information, including energy, mechanical parameters as well as the properties based on elastic constants. Mao et al [12] studied the nucleation and stability of L1 2 -ordered precipitates in Al-Sc-Li alloys. Their study suggested that the contribution of strain energy to free energy is small due to the small lattice parameter difference between Al and Al 3 Li compound. Guo et al [13] calculated the elastic constants of Al 3 Li compound. The calculated Young's modulus showed a good agreement with the experimental observation when the experimental values are extrapolated to 0 K. The vibrational and thermal properties of Al 3 Li were studied by the first-principles pseudo-potential calculations [14].
Aluminides compounds are always used as structural materials, and their work environment may be under pressure. There are many reported investigations to exhibit that the pressure has significant impacts on the various properties in aluminides [10,[15][16][17][18][19]. The above statements indicate that a systematic investigation of aluminides under pressure is of great necessity and importance. However, to the best of our knowledge, properties of Al 3 Li under pressure are rarely studied. Therefore, in this paper, the first-principles calculations are performed to clarify and understand the elastic, mechanical, thermodynamic and electronic properties of Al 3 Li under high pressure up to 25.9 GPa.

Calculation methods
We used the Vienna Ab-initio Simulation Package (VASP) [20,21] to perform the first-principles calculations. Exchange correction function was treated by adopting the generalized gradient approximations of the Perdew-Burke-Ernzerhof form [22]. The Brillouin zone (BZ) of the unit cell was sampled by using the Monkhorst-Pack scheme and the BZ integration was performed by the Methfessel-Paxton method for stress calculations or by tetrahedron method for density of state calculations. The summation over the BZ is performed on a 0.1 Å −1 spacing k-point mesh with smearing of 0.08 eV. The kinetic cut-off energy of 500 eV has been tested to be sufficient for convergence. The structure relaxation convergence was set as´-1 10 6 eV. The Bader charge analysis code [23] was used to explore the volume of each atom [24]. The elastic constants were directly obtained based on the stress-strain relationship. The PHONOPY code [25] was employed to perform the calculations of phonon frequencies with a´2 2 2supercell for Al 3 Li. The thermal properties under different pressures were calculated by using the quasi-harmonic approximation theory [26].
There are two schemes to realize the pressurization on the Al 3 Li unit cell. The first method is that the destination hydrostatic pressure is set in the feed file of VASP code, i.e. setting PSTRESS tag in INCAR file. The second one is that the lattice constant of unit cell is changed to simulate the hydrostatic pressurization. These two schemes are all successfully employed in previous works [17]. Based on the consideration of computational cost, we picked the second scheme in this paper. Of cause, no matter which scheme is employed, the law induced by pressure should be the same.
The calculated pressure dependence of the relative volume, / V V , 0 (V 0 and V stand for the volume of cell under ambient and various pressures, respectively) of the unit cell for Al 3 Li is shown in figure 1(a). It reveals that the ratio of / V V 0 becomes more and more smaller with the increasing pressure, which indicates that the internal atomic distances in this compound should be more and more smaller. The obtained pressure-volume data are fitted to a third order Birch-Murnaghan equation of state as follows: The fitting parameter B 0 is the bulk modulus and ¢ B 0 is the first-order derivative of B . 0 The fitting is based on the least-squares fitting method, where the root-mean-square deviation s is estimated by The variables P , i V , and V fit are the pressure, volume of optimized geometry, and volume of fitting amount, respectively. The minimum s can be achieved with an excellent agreement between V and V .
fit Thus, the optimal value of B 0 can be obtained. A Fortran program has been developed to perform this fitting work [27,28].
The calculated values of B 0 and ¢ B 0 are 64.3 GPa and 4.2, with value of s less than 0.002, which agrees well with the experimental value of B , 0 63.1 GPa [11]. Based on the Bader charge analysis, the volume of Al and Li atoms in Al 3 Li unit cell could be obtained. Their volume ratios under different pressure are presented in figure 1(a). It is seen that Li atom has a smaller compressive trend than Al atom in Al 3 Li compound.
The cohesive energy of atom in compound can be directly obtained by dividing the system total energy by the number of atom in system. The additional cohesive energy is the energy introduced by the external pressure. It can be described by the difference between the energy of the unit cell under various pressures (E) and under ambient pressure (E 0 ). Figure 1(b) plots the additional cohesive energy introduced by external pressure. It is obvious that an external hydrostatic pressure up to 25.9 GPa results in an increase of about 0.22 eV/atom for the cohesive energy, which does not destroy the cubic stability of Al 3 Li.
The formation enthalpy per formula unit (f.u.) is defined as the total energy difference between the compound and its constituents in proportion to the composition, which can be obtained by the following equation: is the total energy of an Al 3 Li f.u. with different lattice parameters under various pressures, E Al and E Li are the cohesive energy per atom of pure element solids with their ground state, i.e. Al in FCC, Li in BCC under ambient pressure.

Elastic constant and mechanical behavior
As listed in table 1, the elastic constants at ambient pressure agree well with the experimental observations [11] and previous theoretical findings [4,9,[12][13][14]. Figure 2(a) presents these elastic constants for Al 3 Li under various pressures. All elastic constants positively increase with the increasing pressure, which is understandable because the interaction between atoms is enhanced by the external pressure. The mechanical stability criterion, Born criterion, for elastic constants is: C 11 -C 12 >0, C 44 >0, C 11 +2C 12 >0 [17,[30][31][32][33][34]. As seen in table 1, the elastic constants meet that criteria, suggesting that Al 3 Li is mechanically stable under pressure up to 25.9 GPa.
The bulk modulus B, shear modulus G, and Young's modulus E can be obtained with these elastic constants by the Voigt-Reuss-Hill method [35]. The calculation for Poisson ratio n also has been performed. For the cubic system, the formulas used are summarized as follows: , The calculated values of B and G at ambient pressure are 64.5 and 43.1 GPa, which agree well with the experimental values of 63.1 and 40.3 GPa [11]. The value of E is 105 GPa, which is 9.7 GPa larger than the experimental value obtained at 293 K [1], and it is in line with other theoretical results, as listed in table 1. It is obvious that the values of B, G and E increase with increasing pressure as shown in figure 2(b), which exhibits similar behavior to the elastic constants. This indicates that the elevated external pressure could improve the hardness of materials [36].
The ratio of B/G was proposed by Pugh to quantitatively describe the brittle or ductile behavior of materials [37]. A high B/G ratio is associated with ductility nature, vice versa. The critical value which separates ductile and brittle material is 1.75 [16]. The B/G ratio for Al 3 Li is 1.497 at zero pressure, meeting the experimental result 1.5 [11]. The B/G ratio positively increases with the increasing external pressure and reaches 1.570 at the pressure of 25.9 GPa. It illustrates that Al 3 Li is a brittle compound even if the pressure is up to 25.9 GPa.
The Cauchy pressure (C 12 -C 44 ) can also provide the brittle or ductile nature of materials [38]. Based on the values of C 12 and C 44 listed in table 1, the Cauchy pressures for Al 3 Li in the pressure range of 0-25.9 GPa are all negative, which suggests the brittle behavior of Al 3 Li under pressure.
Poisson ratio is employed to quantitatively describe the stability of crystals against shear deformation. The larger Poisson ratio (v) suggests the better plasticity in materials. The value of v of ductile material is larger than 0.26, while that value of brittle materials is less than 0.26 [39]. Poisson ratio of Al 3 Li increases from 0.227 to 0.242 when the pressure increases from 0 to 25.9 GPa, indicating Al 3 Li keeps brittle property in the considered pressure range. This is exactly consistent with the predictions from the B/G relationship and Cauchy pressure. Furthermore, Poisson ratio provides useful information about the characteristics of bonding forces in solid [40]. The values of n for the minimum and maximum limits for central force solids are 0.25 to 0.5, respectively. As    [11]. c GGA-PBE (VASP)calculations from [9]. d GGA-PAW (VASP) calculations from [12]. e FLAPW-LDA calculations from [13]. f LDA with pseudopotentials (VASP) calculations from [14]. g Experimental data at 293 K from [1].
this criterion and the values of n listed in table 1, the interatomic forces of Al 3 Li is not central force. Additionally, the values of Poisson ratio increase accordingly along with increasing pressure. The universal anisotropy index A U can be calculated by [41]: As seen in table 1, the values of A U approach 0, suggesting Al 3 Li exhibits isotropy property under the considered pressure. It is worth to notice that, the value of A U declines with the increasing pressure when the pressure is less than 11 GPa and increases again when the pressure is larger than 12.8 GPa. The Al 3 Li shows exact isotropy in the pressure range of 11.0-12.8 GPa.
In order to understand the elastic isotropy, we plot the directional dependences of the Young's modulus at different pressures in figure 3 [42][43][44][45]. We can note that this result is consistent with the analyses of A U under pressure. The difference between the upper and lower limits for Al 3 Li under the pressure of 11.0 and 12.8 GPa is only 2 GPa, which shows a good isotropic behavior as shown in figures 3(b) and (c). The decreasing and increasing behavior of the values of A U before and after 11 GPa is illustrated vividly in figure 3.

Anisotropic wave velocity and Debye temperature
The wave velocities in solid are related to their elastic constants. The wave velocities of both longitudinal and transverse along three different crystal directions, including [100], [110] and [111], were calculated for Al 3 Li. The equations used for these calculations are given by [46]: , 010 001 , The calculated wave velocities in different directions are listed in table 2, the results for Al at ambient pressure are also included. Our results for Al are in good agreement with the available experimental [10] and theoretical results [10,18,47]. Obviously, the wave velocities in different directions for Al 3 Li are larger than those of Al, and increase with the increasing pressure. Debye temperature (θ D ) is the highest temperature that can be achieved due to a single normal vibration for a compound [48,49]. It gives insight into several important physical properties, such as specific heat, ultrasonic wave velocity. The method for estimating the value of θ D is as [50]: A m 1 3 where h and k B are the Plank's constant and Boltzmann's constant. n and N A present the total number of atoms in the cell and Avogadro's number, r and M are the mass density and molecular weight of the compound. Moreover, v m can be approximately calculated from the longitudinal wave velocity v l and transverse wave velocity v t [50]. where B and G are bulk and shear modulus. The v , l v , t v m and θ D for Al 3 Li as a function of pressure are shown in figure 4. At ambient pressure, our results agree well with the values obtained under GGA-PBE scheme in [4]. However, the value of θ D from [13] is much higher than our result. According to the computational formulas of v , l v t and θ D , the value of θ D strongly depends on the bulk modulus and shear modulus. Those values in [13] are significantly higher than this paper, as listed in table 1. For Al 3 Li under other pressures, the wave velocity and Debye temperature all increase with the increasing external pressure up to 25.9 GPa, as shown in figure 4. The increasing Debye temperature corresponds to the increasing vibrational frequency with increasing pressure. Future experimental work could test our predication since there are no experimental data or theoretical results available in literature on these properties for Al 3 Li under pressure.

Thermal conductivity, hardness and melting temperature
The thermal conductivity is often used to quantify the thermal transportation behavior of materials. since the thermal conductivity will change with the external conditions, it is meaningful to determine the thermal conductivity of Al 3 Li under different pressures. Both Clark's model and Cahill's model were developed for studying on the minimum thermal conductivity (k min ) of crystals based on phonon-model but with different l v t and v m ) and Debye temperature (θ D ) for Al 3 Li. Theoretical results from [4] and [13] are also plotted. Table 2. Wave velocity along different directions for Al 3 Li under different pressures. The results for Al at 0 GPa are also included. Units of pressure, velocity and mass density are in GPa, Km/s and g/cm 3 . [100] [ 110] [ 111] approaches [51]. They work well for many materials and give an intuitive description of the phonon limit of thermal conductivities [52]. The formulas used in this work are: In Clark's model [53], , and in Cahill's model [35] , where k B is the Boltzmann's constants, M is the molar weight, n is the total number of atoms per f.u., E is the Young's modulus, r is the mass density, p is the density of number of atom per volume, and v l and v t are the longitudinal and transverse wave velocities, respectively. The k min values based on these two models are usually above the experimental observation because of the influence of temperature on phonon propagation. More information can be found in [51]. The values of k min obtained by Cahill's model are slightly larger than that of Clark's model at a given pressure throughout the whole pressurization process. At the pressure of 0 GPa, the value of k min for Al 3 Li is 1.28-1.40 Wm −1 K −1 , which is higher than that of pure Al (0.84-1.08 Wm −1 K −1 ), but slightly lower than Al 3 Sc, as listed in table 3. The minimum thermal conductivity is 40%-50% improved by alloying Li in Al to form Al 3 Li compound. The values of k min for Al 3 Li increase with the increasing pressure, as shown in figure 5(a), and it reaches 1.87-2.04 Wm −1 K −1 when the pressure is up to 25.9 GPa.
Moreover, the melting temperature (T m ) of Al 3 Li can be estimated by the empirical function [55]:  table 3. We can see the melting temperature of Al 3 Li is similar to that of pure Al and lower than that of Al 3 Sc. The calculated T m of Al 3 Li as a function of pressure is shown in figure 5(b). The T m calculated by T m 1 and T m 2 follows the same varying trend of pressure, that is the gradual increase with the increasing external pressure. It should note that the two empirical functions for T m might have potential risk in predicting the crystals melting point under pressure, one should be very cautious on the range of applicability of empirical function in the future study.
To obtain the hardness of Al 3 Li under pressure, we adopted the empirical scheme to evaluate their Vicker's hardness [57,58], which is determined by B and G as: V v 2 0.585 Table 3. Lattice constants (a 0 ), Elastic constants (C ij ), and the calculated hardness (H), lower limit thermal conductivity (k min ) and melting temperatures (T m ) of Al, Sl 3 Sc and Al 3 Li under the pressure of 0 GPa. a Experimental data from [14]. b Theoretical data (GGA-PAW, VASP) from [12]. c Experimental data from [54]. d Experimental data from [14] e Values are calculated based on the corresponding elastic constants.
Another relation for calculating microhardness (H m ) based on the Poisson ratio (n) and Young's modulus (E) is [59]: The calculated hardess for Al, Al 3 Sc and Al 3 Li under ambient pressure are listed in table 3. The values obtained by these two methods are close to each other for all these three systems. We can see that the hardness of Al 3 Sc, the other important precipitate in Al-Sc-Li alloy, is higher than Al 3 Li, and both of them are significantly higher than pure Al. Our result answers the peak hardness introduced by the precipitates of Al 3 Li and Al 3 Sc in Al-Li-Sc alloys [6]. From the last two columns in table 1, we find that the hardness H v (H m ) of Al 3 Li at ambient pressure is 8.271 (7.844) GPa, and it increases with the pressure, up to 12.265 (14.293) GPa at pressure 25.9 GPa. The reason is that the bond lengths gradually become shorter with the increasing pressure.

Electronic structure and chemical bonding
To analyze the bonding characteristics of the Al 3 Li compound, the electronic densities of states (DOS) were calculated under different pressures, which are presented in figure 6. At the ambient pressure, figure 6(a), Al 3 Li exhibits a metallic character because there is no energy gap near the Fermi level. The Li atom has one valence electron and Al atom has three. However, the electronic density of the Li atom is lower than 1/3 of Al, indicating that partial of Li electronic charge is transferred to Al atoms. This is consistent with the Born electronegative of Al and Li (Al: 1.61, Li: 0.98). The main bonding peaks of Al 3 Li are predominantly derived from Al_s with a little contribution from Li_sp states at the energy region below −3.5 eV. Al_p and Li_p states dominate the bonding peaks at the energy region of −3.5-0 eV. The increasing pressure hardly affects the bonding character of Al 3 Li in the considered pressure range, as shown in figures 6(b) and (c). The shape of DOS curves exhibits relatively few changes, and it is expanded in the energy scales introduced by the increasing of pressure.

Phonon dispersion and density of states
Lattice dynamics plays an important role in the understanding of physical properties of solids, such as phase stability. Phonon dispersions in materials are also interesting because of the anomalous electronic screening [60,61]. Therefore, the phonon dispersion has been calculated to investigate the lattice dynamic properties of Al 3 Li under pressure, as shown in figure 7. No imaginary frequency has been observed, suggesting that Al 3 Li is dynamically stable under pressure up to 25.9 GPa. There are 4 atoms in Al 3 Li cell, phonon spectra curves consists of 12 phonon branches, which contain 3 acoustic branches and 9 optical branches [18]. Generally, the frequency of optical models is higher than that of acoustic models [17]. The highest optical branches are separated clearly from the rest branches at 0 GPa because of the large mass difference between Al and Li elements. Four multi-band degenerations at the Gamma point are presented in the phonon dispersion curves, as shown in figure 7, and their frequencies are also printed. The type of degeneration does not change with pressure which reflects the stability of Al 3 Li under pressure. The frequencies of those degenerated bands are enhanced as the external pressure increase. This phenomenon can be explained by the shorter bond length when pressure increases. The shorter bond length can lead to the larger force constants, resulting in higher phonon frequencies [62]. At the meantime, the external pressure limits the vibration range of the atoms and increases the lattice vibration frequencies to consume the internal energy [18].
To further understand the lattice dynamic behavior of Al 3 Li under pressure, we calculated the phonon density of states (PDOS), presented in figure 8. We can see that the 3 separated optical branches are mainly derived from Li atoms and other branches are dominated by Al atoms at 0 GPa, as shown in figure 8(a). The increasing pressure lowers the absolute value of PDOS and increases their vibration frequency. With the increase of the pressure, the branches with low frequency, lower than the frequency at the pseudogap marked by an arrow, are always dominated by Al atoms with a constant contribution from Li while the branches with high frequency, higher than the frequency at the pseudogap, are dominated by Li atoms with a slightly increasing contribution from Al atoms, as shown in figures 8(b) and (c). Al atoms are more sensitive than Li to the effects introduced by external pressure.

Thermodynamic properties
The data of thermodynamic properties under high pressure and temperature can provide the valuable information for industrial applications of materials under extreme conditions [18]. The thermal properties of Al 3 Li as a function of temperature are calculated by the quasi-harmonic approximation theory [26].    3 Li along with the available experimental and theoretical data under the pressure of 0 GPa. We can see that our result is in good agreement with the results obtained in [14] with a constant difference about 4 GPa. This difference can be explained by the different value of C 12 , as listed in table 1, which leads to the 4 GPa difference of bulk modulus at 0 K. On the other hand, the limited experimental value of bulk modulus versus temperature apparently decreases faster than our calculations and [14]. It might be because Al 3 Li is a metastable phase in Al-Li alloy [1] and it is difficult to obtain reliable experimental results for such a phase. On the other hand, our result is obtained with the help of DFT calculation and quasi-harmonic approximation theory [26], which are widely employed in this field, wish the future solution could give new explanation on this mismatch. The thermal expansion coefficient of Al 3 Li at 0 GPa was shown in figure 9(b) along with the theoretical results from [14]. We can see the good agreement between them, especially when temperature is lower than 500 K.
The volume expansion coefficient, volume expansion and bulk modulus under pressure versus temperature are shown in figures 9(b)-(d), respectively. We can see that the thermal expansion coefficient increases exponentially with temperature below 200 K, and gradually approaches linear behavior beyond 300 K at ambient pressure. At a given temperature, the expansion coefficient decreases with the external pressure, as shown in figure 9(b). The similar phenomenon also presented in the volume expansion, figure 9(c). We can conclude that the external pressure weakens the thermal effect on volume expansion of Al 3 Li. The values of bulk modulus decrease accordingly with the increasing temperature, as shown in figure 9(d). At the same temperature, the bulk modulus increases along with the increasing external pressure, exhibiting the same trend as that of elastic constants.
The calculated free energies, entropies for Al 3 Li compound as a function of temperature under different pressures are displayed in figures 10(a) and (b). We can see that the free energy decreases gradually with increasing temperature under a certain pressure and increases with the increasing pressure at a given temperature. The entropies just possess the opposite varying trend of free energy on temperature, which the entropy increases with the increasing temperature under a given pressure and decreases with the increasing pressure at a certain temperature.
The variation of the isochoric heat capacity (C v ) with temperature is shown in figure 10(c). It can be seen that at low temperature, C v is proportional to third power of temperature. Whereas in the range of high temperature, C v approaches the Dulong-Petit limit of 99 J·mol −1 K −1 . Figure 10(d) illustrates the calculated isobaric heat  [11] and theoretical values from [14] are also presented. (b)Temperature dependence of thermal expansion coefficient, including theoretical values from [14]. (c) volume expansion, and (d) bulk modulus of Al 3 Li under various pressures. The unit of external pressure is GPa.
capacity (C p ) as a function of temperature under pressures up to 25.9 GPa. The results of C p for Al 3 Li under 0 GPa from [14] are also presented. It is obvious that our results agree well with that from [14]. The C p follows the similar behavior as C v except at the very high temperature. The values of C p at 800 K for Al 3 Li under the pressures of 0, 12.8 and 25.9 GPa are 111.6, 101.8 and 99.4 J·mol −1 K −1 , respectively.
One should note that there is electronic contribution to free energy, entropy, heat capacity and thermal conductivity of Al 3 Li as other metals.
The electronic contribution to free energy, F , el could be given by [63]: where n F is the DOS at Fermi level. From the above two equations we could get that a large DOS at the fermi energy implies a largely negative free energy contribution [63]. Obviously, the electronic contribution to entropy is proportional to the DOS at fermi level at a given temperature.
For heat capacity, the vibration contribution is shown in figure10(c), C v is proportional to the third power of temperature at low temperature, i.e.C T , v 3 the electronic contribution to C v could be given by [65]: Therefore, the influence of electronic contribution to heat capacity should be significantly at low temperature.
There is also electronic contribution to thermal conductivity due to the phonon and electron coupling. One can find more information from [66]. Figure 10. The temperature dependence of (a) free energy, (b) entropy, (c) isochoric heat capacity (C v ), (d) isobaric heat capacity (C p ), including the results at ambient pressure from [14], for Al 3 Li under different pressures.

Conclusion
In this study, we have comprehensively studied the elastic, mechanical, electronic and thermodynamic properties of Al 3 Li compound under pressure based on the first-principles calculations. The conclusions of this work may be summarized as following: (1) The calculated elastic constants for Al 3 Li show increase with the increasing pressure and satisfy the Born's criteria, indicating the mechanical stability is well even if the pressure is up to 25.9 GPa. There is no imaginary frequency presented in the calculation of the phonon dispersion spectrum also suggests the stability of Al 3 Li under pressure.
(2) The bulk modulus, Yang's modulus and shear modulus are 64.5, 105.7 and 43.1 GPa at 0 pressure and they increase accordingly to 133.3, 206.4 and 83.1 GPa as the pressure increases to 25.9 GPa. The Poisson ratio and Cauchy pressure both suggest that Al 3 Li sustains brittle nature in the whole considered pressure range. The universal elastic anisotropy index firstly declines with the increasing pressure when the pressure is less than 11 GPa, and it increases with the increasing pressure when the pressure is larger than 12.8 GPa. Based on the good agreement between the calculated wave velocities and experimental observation for pure Al, the wave velocities for Al 3 Li under pressure were calculated. The Debye temperature was also obtained further. They all accordingly increase along with the increasing pressure.
(3) According to empirical functions, the thermal conductivity, melting temperature and hardness were all predicted, and they show the same reaction to pressure as that of wave velocities. These quantities for Al, Al 3 Sc and Al 3 Li are compared when the pressure is 0. Al 3 Sc has the best hardness and the highest melting temperature, the Al 3 Li and Al present the similar melting temperature, Al has the worst thermal conductivity. The results from the analysis of electronic density of states exhibit a metallic bonding behavior of Al 3 Li.
(4) We gave a prediction of the bulk modulus, thermal expansion coefficient, isochoric heat capacity, isobaric heat capacity, free energy and entropy under various pressures for Al 3 Li as a function of temperature. The presented study would be helpful to future experimental and theoretical explorations.