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Revealing the generation mechanism of oxygen-vacancy defect centers ${E}_{\ensuremath{\gamma}}^{\ensuremath{'}}$ in $a$-SiO${}_{2}$ is at the heart of hardening silicon electronics against radiation damage in extreme applications. The nature of these defect centers is not fully understood, though. This study addresses the problem using state-of-the-art first-principles calculations, and proposes a ``nonradiative carrier capture--structural relaxation'' mechanism and a fractional-power-law model to predict defect generation and evolution dynamics. The calculated defect behaviors align with experiments over wide dose and temperature ranges, and can support a range of oxygen-vacancy-related reliability and functionality studies.