We systematically calculate the structure, formation enthalpy, formation free energy, elastic constants and electronic structure of Ti$_{0.98}$X$_{0.02}$ system by density functional theory (DFT) simulations to explore the effect of transition metal X (X=Ag, Cd, Co, Cr, Cu, Fe, Mn, Mo, Nb, Ni, Pd, Rh, Ru, Tc, and Zn) on the stability mechanism of $beta$-titanium. Based on our calculations, the results of formation enthalpy and free energy show that adding trace X is beneficial to the thermodynamic stability of $beta$-titanium. This behavior is well explained by the density of state (DOS). However, the tetragonal shear moduli of Ti$_{0.98}$X$_{0.02}$ systems are negative, indicating that $beta$-titanium doping with a low concentration of X is still elastically unstable at 0 K. Therefore, we theoretically explain that $beta$-titanium doping with trace transition metal X is unstable in the ground state.
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