Truly one of the most outstanding works of non-fiction of the past 25 years. The first paragraph of the first chapter sets the tone of this compelling book:
The anecdote concerns Szilard's flash of insight concerning the recently observed naturally occurring fission of uranium, with neutrons (themselved only recently identified) as byproducts.
A single atom splitting released an incredible amount of energy for its mass. It was estimated that a single proton's mass, if completely converted into energy according to Einstein's famous equation, would release enough energy to make a very small mass such as a pollen grain jump enough to be visible under an ordinary microscope.
But a neutron, he realized, could easily wander into another nucleus and cause it to split. And if a typical split released two or more neutrons (the number was not yet known), a chain reaction could occur. And if this reaction was rapid, as Szilard had every reason to think it would be, an unprecedentedly energetic explosion might result. In other words, an "atomic bomb."
I also recommend Rhodes' Dark Sun, which carries the story forward into the thermonuclear era, and tells the story of the parallel Soviet effort, for which he gained access to former Soviet archives and surviving scientists.