In this paper spectroscopic investigations of $Gd_3Ga_5O_{12}$ (GGG) polycrystals, containing $Cr^{3+}$ and intentionally doped with $Pr^{3+}$ of concentrations 0.5, 1 and 1.5 mol% are presented. We have measured the steady state luminescence and luminescence excitation spectra, as well as the time resolved spectra and luminescence kinetics. The main goal was to investigate the excitation energy transfer from lattice to impurity and between impurities. We found that relative intensity of $Cr^{3+}$ and GGG lattice luminescence decreased when material was doped with $Pr^{3+}$. On the other hand, time resolved spectroscopy and luminescence decay measurements showed that the $Cr^{3+}$ and GGG lattice luminescence decays were independent of $Pr^{3+}$ content. The lifetime of $Pr^{3+}$ luminescence related to $\text{}^{1}D_{2} \rightarrow \text{}^{3}H_{4}$ and $\text{}^{3}P_{0} \rightarrow \text{}^{3}H_{4}$ transitions decreased with concentration of $Pr^{3+}$, which was attributed to the concentration luminescence quenching. No energy transfer between GGG lattice defects and $Cr^{3+}$, and $Pr^{3+}$ ions was observed. We proposed the model of radiative recombination of electron and hole, which took place through three independent pathways: by GGG host emission that peaked at 12750 $cm^{-1}$, by $Cr^{3+}$ luminescence that peaked at 15400 $cm^{-1}$, and by $Pr^{3+}$ luminescence.
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