The Flying Eyes, by J. Hunter Holly (epub/Kindle/pdf)

After that, Linc uses every resource of the Space Research Lab and the National Guard to destroy the Eyes. But nothing stops them. In desperation, Linc decides to capture an Eye. When he finally manages to communicate with it, he learns that the creatures need radiation to live. And they give him an ultimatum: Earth must explode a series of atom bombs to supply them with radiation...or they will turn the world's population into mindless robots! Given the choice between self-destruction from atomic fallout or annihilation via the sinister Flying Eyes -- can Linc find a third path that will save mankind?