  We have investigated the phonon, elastic and thermodynamic properties of
L1$_{2}$ phase Al$_{3}$Ta by density functional theory approach combining with
quasi-harmonic approximation model. The results of phonon band structure shows
that L1$_{2}$ phase Al$_{3}$Ta possesses dynamical stability in the pressure
range from 0 to 80 GPa due to the absence of imaginary frequencies. The
pressure dependences of the elastic constants $C_{ij}$, bulk modulus $B$, shear
modulus $G$, Young's modulus $Y$, $B/G$ and Poisson's ratio $\nu$ have been
analysed. The elastic constants are satisfied with mechanical stability
criteria up to the external pressure of 80 GPa. The results of the elastic
properties studies show that Al$_{3}$Ta compound possesses a higher hardness,
improved ductility and plasticity under higher pressures. Further, we
systematically investigate the thermodynamic properties, such as the Debye
temperature $\Theta$, heat capacity $C_{p}$, and thermal expansion coefficient
$\alpha$, and provide the relationships between thermal parameters and
pressure.

Materials Science