  We investigate native point defects and rare-earth (co)dopants in
SrAl$_2$O$_4$ using hybrid density-functional defect calculations. Europium
(Eu) and dysprosium (Dy) are found to be mixed valence and energetically most
favorable at the Sr lattice sites. However, unlike Eu where both Eu$^{2+}$ and
Eu$^{3+}$ can be realized in synthesis, Dy is stable predominantly as
Dy$^{3+}$, and the divalent Dy$^{2+}$ may only be photogenerated under
irradiation. On the basis of an analysis of Eu-related band-defect (including
charge-transfer) and interconfigurational $5d$-$4f$ optical transitions, we
assign the characteristic broad blue (445 nm) and green (520 nm) emission bands
in Eu$^{2+}$-doped SrAl$_2$O$_4$ to the $4f^65d^1$ $\rightarrow$ $4f^7$
transition in Eu$^{2+}$ incorporated at the Sr1 and Sr2 sites, respectively.
Strontium interstitials (not oxygen vacancies, in contrast to what is commonly
believed) and Dy$_{\rm Sr}$ can act as efficient electron traps for
room-temperature persistent luminescence. This work calls for a re-assessment
of certain assumptions regarding specific carrier trapping centers made in all
mechanisms previously proposed for the persistent luminescence in Eu- and
(Eu,Dy)-doped SrAl$_2$O$_4$. It also serves as a methodological template for
the understanding and design of rare-earth doped phosphors.
