Anomalous Thermal Expansion in $\alpha$-Titanium

We provide a complete quantitative explanation for the anisotropic thermal expansion of hcp Ti at low temperature. The observed negative thermal expansion along the $c$ axis is reproduced theoretically by means of a parameter free theory which involves both the electron and phonon contributions to the free energy. The thermal expansion of titanium is calculated and found to be negative along the $c$ axis for temperatures below $\sim 170\mathrm{K}$, in good agreement with observations. We have identified a saddle point van Hove singularity near the Fermi level as the main reason for the anisotropic thermal expansion in $\alpha$-titanium.

Figure 1

The phonon dispersion of hcp Ti at room temperature and ambient pressure. The solid curve is the calculated frequencies from GGA and the dashed curve represents calculated frequencies from LDA. Both calculations are done at the experimental volume $V_0=15.91{\AA }^3$. The filled circles are the experimental data of Stassis et al. [22].

Figure 2

Linear thermal expansion for hcp Ti at ambient pressure. The solid lines are the theoretical calculation using GGA and the dashed line is from an LDA calculation. The filled circles are the experimental data of Nizhankovskii et al. [5]. The filled squares are the experimental volume expansion coefficient of Mal’ko et al. [24]. The filled triangle and the open triangle are the experimental data of ${\alpha }_a$ and ${\alpha }_c$, respectively, of Pawar and Deshpande [25]. The open circles are the experimental data of Nizhankovskii et al. scaled to give a volume expansion coefficient in agreement with Ref. 24.

Figure 3

Calculated Fermi surface of hcp Ti at $T=0$ equilibrium volume for three different lattice constants $c$, at $c=0.988c_0$ (a), at $c=c_0$ (b) and at $c=1.012c_0$ (c). Here $c_0$ corresponds to the ($T=0$) equilibrium lattice constant.