Negative thermal expansion of MgB$_{2}$ in the superconducting state and anomalous behavior of the bulk Gruneisen function

The thermal expansion coefficient $alpha$ of MgB$_2$ is revealed to change from positive to negative on cooling through the superconducting transition temperature $T_c$. The Gruneisen function also becomes negative at $T_c$ followed by a dramatic increase to large positive values at low temperature. The results suggest anomalous coupling between superconducting electrons and low-energy phonons.