  Elemental gallium can exist in several phases under ambient conditions. The
stable $\alpha$ phase has a superconducting transition temperature, $T_c$, of
0.9~K. By contrast, the $T_c$ of the metastable $\beta$ phase is around 6~K. To
understand the significant improvement in $T_c$ in the $\beta$ phase, we first
calculate the electronic structure, phonon dispersion, and the electron-phonon
coupling of gallium in the $\alpha$ and $\beta$ phase. Next, we solve the
Eliashberg equations to obtain the superconducting gaps and the transition
temperatures. Using these results, we relate the increased $T_c$ in the $\beta$
phase to structural differences between the phases that affect the electronic
and phonon properties. The structure motif of the $\alpha$ phase is Ga$_2$
dimers, which form strong covalent bonds leading to bonding and antibonding
states that suppress the density of states at the Fermi level. The $\beta$-Ga
structure consists of arrays of Ga chains that favor strong coupling between
the lattice vibrations and the electronic states near the Fermi level. The
increased density of states and strong coupling to the phonons for the
$\beta$-Ga chains compared to the $\alpha$ Ga$_2$ dimers enhance
superconductivity in the $\beta$-Ga phase.
