Negative thermal expansion of MgB$_{2}$ below T$_{c}$ - violation of the Ehrenfest relation

The thermal expansion coefficient $\alpha$ of MgB$_{2}$ is observed to change from positive to negative upon cooling through the superconducting transition temperature $T_{c}$. The bulk Gr\"uneisen parameter, which is proportional to $\alpha$, changes sign at $T_{c}$ as well. The jumps in $\alpha$ and the heat capacity at $T_c$ are used to estimate its pressure dependence, $dT_{c}$/$dP$; comparison to the measured value reveals violation of the Ehrenfest relation by 42\%. The results suggest anomalous coupling between the lattice and superconducting electrons via low-energy phonons.