  To investigate the structural and magnetic properties of Cr-doped M-type
strontium hexaferrite (SrFe$_{12}$O$_{19}$) with x = (0.0, 0.5, 1.0), we
perform first-principles total-energy calculations relied on density functional
theory. Based on the calculation of the substitution energy of Cr in strontium
hexaferrite and formation probability analysis, we conclude that the doped Cr
atoms prefer to occupy the 2a, 12k, and 4f$_{2}$ sites which is in good
agreement with the experimental findings. Due to Cr$^{3+}$ ion moment, 3
{$\mu_B$}, smaller than that of Fe$^{3+}$ ion, 5 {$\mu_B$}, saturation
magnetization (M$_{s}$) reduce rapidly as the concentration of Cr increases in
strontium hexaferrite. The magnetic anisotropic field $\left(H_{a}\right)$
rises with an increasing fraction of Cr despite a significant reduction of
magnetization and a slight increase of magnetocrystalline anisotropy
$\left(K_{1}\right)$.The cause for the rise in magnetic anisotropy field
$\left(H_{a}\right)$ with an increasing fraction of Cr is further emphasized by
our formation probability study. Cr$^{3+}$ ions prefer to occupy the 2a sites
at lower temperatures, but as the temperature rises, it is more likely that
they will occupy the 12k site. Cr$^{3+}$ ions are more likely to occupy the 12k
site than the 2a site at a specific annealing temperature (>700{\deg}C).
