Anomalous thermal expansion in α-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 similar to 170 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.