The effect of diamagnetic dilution of the Fe sublattice on the structural and magnetic properties of the double perovskite ${\mathrm{Sr}}_{2}{\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Ga}}_{x}\mathrm{Re}{\mathrm{O}}_{6}$ $(0<x<0.7)$ has been explored by means of x-ray structural analysis, magnetometry, M\"ossbauer spectroscopy, and band structure calculations. The end member of the solid solution series, ${\mathrm{Sr}}_{2}\mathrm{Ga}\mathrm{Re}{\mathrm{O}}_{6}$, was predicted to be a half-metallic ferromagnet based on ab initio band structure calculations using the WIEN2k software package. Because the Re-containing double perovskites, like the parent compound ${\mathrm{Sr}}_{2}\mathrm{Fe}\mathrm{Re}{\mathrm{O}}_{6}$, rarely exhibit antisite disorder, the appearance of increasing amounts of antisite disorder in ${\mathrm{Sr}}_{2}{\mathrm{Fe}}_{1\ensuremath{-}x}{\mathrm{Ga}}_{x}\mathrm{Re}{\mathrm{O}}_{6}$ $(0<x<0.7)$ is unexpected. Although the amount of antisite disorder increases with increasing Ga content, it also depends on the sample preparation history. For $0<x<0.4$ only a tetragonal, magnetically ordered phase was detected by x-ray structural analysis as well as M\"ossbauer analysis. A phase separation into a tetragonal and a cubic phase for $x\ensuremath{\geqslant}0.4$ is detected by x-ray structural analysis accompanied by the observation of a magnetically ordered and a paramagnetic phase in the corresponding M\"ossbauer spectra. Below 20% Ga content, Ga statistically dilutes the -Fe-O-Re-O-Fe- double-exchange pathways. Phase separation begins at 20% Ga substitution; between 20% and 40% of Ga content, the paramagnetic Ga-based phase does not contain any Fe. The Fe-containing, paramagnetic cubic phases which can be detected by M\"ossbauer spectroscopy first appear for $x=0.4$.
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