We revisit the magnetic properties of the hexagonal ferrite PbFe$_{12-x}$Ga$_x$O$_{19}$. Recent experiments have reported puzzling dependencies of the ordering temperature and the saturation magnetization on the Ga concentration $x$. To explain these observations, we perform large-scale Monte Carlo simulations, focusing on the effects of an unequal distribution of the Ga impurities over the five distinct Fe sublattices. Ab-initio density-functional calculations predict that the Ga ions preferably occupy the $12k$ sublattice and (to a lesser extent) the $2a$ sublattice. We incorporate this insight into a nonuniform model of the Ga distribution. Monte Carlo simulations using this model lead to an excellent agreement between the theoretical and experimental values of the ordering temperature and saturation magnetization, indicating that the unequal distribution of the Ga impurities is the main reason for the unusual magnetic properties of PbFe$_{12-x}$Ga$_x$O$_{19}$. We also compute the temperature and concentration dependencies of the sublattice magnetizations, and we study the character of the zero-temperature transition that takes place when the ordering temperature is tuned to zero.

11 pages, 12 figures, final version as published