Effect of pressure on the superconducting transition temperature of doped and neutron-damaged MgB$_2$

Measurements of the superconducting transition temperature for Al-doped, C-doped and neutron-damaged-annealed MgB$_2$ samples under pressure up to $\sim 8$ kbar are presented. The $dT_c/dP$ values change systematically with the decrease of the ambient pressure $T_c$ in a regular fashion for each group of the samples. The evolution of the pressure derivatives can be understood assuming that the change in phonon spectrum is a dominant contribution to $dT_c/dP$.