  We use first-principles total-energy calculations based on density functional
theory to study the site occupancy and magnetic properties of Al-substituted
$M$-type strontium hexaferrite SrFe$_{12-x}$Al$_{x}$O$_{19}$ with $x=0.5$ and
$x=1.0$. We find that the non-magnetic Al$^{3+}$ ions preferentially replace
Fe$^{3+}$ ions at two of the majority spin sites, $2a$ and $12k$, eliminating
their positive contribution to the total magnetization causing the saturation
magnetization $M_s$ to be reduced as Al concentration $x$ is increased. Our
formation probability analysis further provides the explanation for increased
magnetic anisotropy field when the fraction of Al is increased. Although
Al$^{3+}$ ions preferentially occupy the $2a$ sites at a low temperature, the
occupation probability of the $12k$ site increases with the rise of the
temperature. At a typical annealing temperature ($> 700\,^{\circ}{\rm C}$)
Al$^{3+}$ ions are much more likely to occupy the $12k$ site than the $2a$
site. Although this causes the magnetocrystalline anisotropy $K_1$ to be
reduced slightly, the reduction in $M_s$ is much more significant. Their
combined effect causes the anisotropy field $H_a$ to increase as the fraction
of Al is increased, consistent with recent experimental measurements.
