{\it Cmcm} post-perovskite: a new alumina polymorph

Alumina, Al$_2$O$_3$, is a model ceramic material with important applications in high pressure science, particularly as the ruby pressure scale. It is isoelectronic with MgSiO$_3$, the major Earth forming mineral. Here we show by first principles that the newly found post-perovskite polymorph of MgSiO$_3$, CaIrO$_3$ type structure with {\it Cmcm} symmetry, is also a stable high pressure phase of Al$_2$O$_3$ and should be stabilized in the pressure range in which the ruby scale has been calibrated. The sequence of polymorphs under pressure in these minerals is therefore analog: corundum/ilmenite $\rightarrow$ {\it Pbnm}-perovskite for MgSiO3 and Rh$_2$O$_3$(II) type for Al$_2 $O$_3$ $\rightarrow$ {\it Cmcm} post-perovskite. The reason for the greater stability of {\it Pbnm}-perovskite in MgSiO$_3$ versus Rh$_2$O$_3$(II)-type in Al$_2$O$_3$ is the difference in cation polyhedral types and volumes in the former, that favors for ABX$_3$-type composition.