Monazite-structure orthophosphates, including ${\mathrm{LaPO}}_{4},$ ${\mathrm{PrPO}}_{4},$ ${\mathrm{NdPO}}_{4},$ ${\mathrm{SmPO}}_{4},$ ${\mathrm{EuPO}}_{4},$ ${\mathrm{GdPO}}_{4},$ and natural monazite, and their zircon-structure analogs, including ${\mathrm{ScPO}}_{4},$ ${\mathrm{YPO}}_{4},$ ${\mathrm{TbPO}}_{4},$ ${\mathrm{TmPO}}_{4},$ ${\mathrm{YbPO}}_{4},$ and ${\mathrm{LuPO}}_{4},$ were irradiated by $800{\mathrm{keV}\mathrm{}\mathrm{Kr}}^{2+}$ ions in the temperature range of 20 to 600 K. The critical amorphization dose was determined in situ as a function of temperature using selected-area electron diffraction. Amorphization doses were in the range of ${10}^{14}$ to ${10}^{16}{\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}/\mathrm{c}\mathrm{m}}^{2},$ depending on the temperature. Materials with the zircon structure were amorphized at higher temperatures than those with the monazite structure. The critical amorphization temperature ranged from 350 to 485 K for orthophosphates with the monazite structure and from 480 to 580 K for those with the zircon structure. However, natural zircon $({\mathrm{ZrSiO}}_{4})$ can be amorphized at over 1000 K. Within each structure type, the critical temperature of amorphization increased with the atomic number of the lanthanide cation. Structural topology models are consistent with the observed differences between the two structure types, but do not predict the relative amorphization doses for different compositions. The ratio of electronic-to-nuclear stopping correlates well with the observed sequence of susceptibility to amorphization within each structure type, consistent with previous results that electronic-energy losses enhance defect recombination in the orthophosphates.