Dark current spectroscopy is used to directly investigate deep-level traps induced by alpha particle irradiation in charge-coupled devices, at trap concentrations as low as ~2 times 107 cm-3. The irradiated devices have long dark current histogram tails that are quantized at long imager integration times. This quantization is due to the discrete nature of the distribution of the radiation-induced traps within the device pixels. Four distinct traps, with different dark current generation rates, are found and characterized in terms of their activation energies and capture cross-sections. Two of the traps, identified as the divacancy and the E-center (phosphorus-vacancy), have the highest concentrations and generation rates and are the main constituents of the irradiation tails. The origin of these irradiation tails is explained as a superposition of peaks formed by pixels that contain from a single deep-level trap to multiple traps of the same or different types. Divacancy annealing is shown to result in the formation of four additional traps, which are tentatively attributed to vacancy-oxygen, vacancy-hydrogen, and high-order silicon-vacancy complexes.