  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 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.
