The atomic matter piles were energy-production components utilized in Federation starship propulsion systems.
In 2254, Starfleet officer Benjamin Finney, while serving aboard the USS Republic, accidentally left a circuit open to the atomic matter piles, a mistake which could have resulted in the destruction of the ship. This oversight was logged by Finney's old friend, Ensign James T. Kirk, who subsequently closed the switch, and Finney was moved to the bottom of the promotion list as a result. This led to a rift in the friendship between Kirk and Finney. (TOS: "Court Martial")
When this episode was written in the mid-1960's, an atomic matter pile was defined in the real world as a …type of nuclear reactor whose core consisted of layers of graphite block interspersed with uranium, designed to create a sustained fission reaction. [1]. An example of a nuclear pile reactor was the Chicago Pile-1 which, in 1942, produced the first self-sustaining nuclear reaction.