The details would be lost on you, no offense. The point is it'll allow us to boost the reactor's power and get it running sooner than planned.”— Allie Filmore
The Beryllium agitator is a pre-War project that was being developed at the Mass Fusion facility in Boston. Although the Mass Fusion director, Karl Oslow, claimed that it was being developed for the "benefit of mankind", Oslow actually intended to have the agitator sent to Washington for military testing. When doctor Noel Chandrich, one of the scientists involved in the project, discovered this, he confronted Oslow about it, although Oslow stated that the agitator would be handed over to the military no matter what, which led to Chandrich being fired from the company.
Ultimately, the Great War occured on October 23rd, 2077, and Boston was hit by a nuclear device. The agitator was left unscathed in the Mass Fusion building's main reactor. It was left untouched for over 200 years until, in 2287, proof of its existence was discovered by the Institute and the Brotherhood of Steel.
The beryllium agitator is able to generate a considerable amount of energy. For that reason, the Institute planned to acquire it in order to power up the Institute's main generator, whereas the Brotherhood of Steel needed it in order to power up Liberty Prime and get it ready to fight the Institute.
The Beryllium agitator in Fallout 4 is based on the real use of beryllium as a way to start nuclear reactions.
If you finish the game with the Commonwealth Minutemen you may still collect the agitator after finding the Mass Fusion executive ID when relayed to the Mass Fusion executive suite. It is not considered a quest item after finishing this ending, meaning it can be dropped or sold.
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