Structural phase transformations in Ti$_{3}$Al$_{2}$Nb system, a first-principles approach

First-principles method is employed to determine the vibrational entropy and Gibbs free energy as a function of temperature of the homogenous Ti$_{3}$Al$_{2}$Nb system. Calculated energies at T=0 K show instabilities in ternary B2 Ti$_{3}$Al$_{2}$Nb alloy against the $\omega $ and B8$_{2}$ structures. We show that at high temperatures the B2 phase is stabilized by the vibrational entropy. The transition temperatures for B2 $\to \quad \omega $ and B2 $\to $ B8$_{2}$ have been calculated and are in excellent agreement with experiment.