All of the organizations mentioned are under the Department of Atomic Energy. Therefore, the correct option is: I, II, III, and IV.

I. Atomic Minerals Directorate for Research and Exploration (AMD): It is responsible for the exploration, research, and development of atomic minerals in India. It conducts surveys and assessments to identify and evaluate mineral resources, including uranium and thorium.

II. Heavy Water Board (HWB): It is responsible for the production of heavy water (deuterium oxide), which is used as a moderator and coolant in nuclear reactors. HWB operates several heavy water plants in India.

III. Indian Rare Earths Limited (IREL): It is a public sector undertaking under the Department of Atomic Energy. IREL is involved in the mining, processing, and refining of rare earth elements, thorium, and other valuable minerals.

IV. Uranium Corporation of India (UCIL): It is responsible for the mining and processing of uranium ore in India. UCIL operates several mines and ore processing plants to meet the country`s uranium requirements for nuclear power generation.

All of these organizations play critical roles in India`s nuclear energy sector and are part of the Department of Atomic Energy, which is responsible for promoting and developing nuclear energy technologies in the country.