Nuclear Fission

Back at Los Alamos, the remaining scientists grappled with much harder problems than Solitaire, like figuring out how neutrons behave inside a nuclear core. In a nuclear core, there are trillions and trillions of neutrons all interacting with their surroundings. So the number of possible outcomes is immense, and computing it directly seemed impossible. - [Derek] But when Ulam returned to work, he had a sudden revelation.

What if we could simulate these systems by generating lots of random outcomes like I did with Solitaire? He shared this idea with von Neumann, who immediately recognized its power, but also spotted a key problem. - See, in Solitaire, each game is independent. How the cards are dealt in one game have no effect on the next, but neutrons aren't like that. A neutron's behavior depends on where it is and what it has done before.

So you couldn't just sample random outcomes like in Solitaire. Instead, you needed to model a whole chain of events where each step influenced the next. What von Neumann realized is that you needed a Markov chain. So they made one and a much simplified version of it works something like this. Now, the starting state is just a neutron traveling through the core, and from there, three things can happen. It can scatter off an atom and keep traveling, so that gives you an arrow going back to itself.

It can leave the system or get absorbed by a non-fissile material, in which case it no longer takes part in the chain reaction, and so it ends its Markov chain, or it can strike another uranium-235 atom, triggering a fission event and releasing two or three more neutrons that then start their own chains. But in this chain, the transition probabilities aren't fixed, they depend on things like the neutron's position, velocity and energy, as well as the overall configuration and mass of uranium.

So a fast-moving neutron might have a 30% chance to scatter, a 50% chance to be absorbed or leave, and a 20% chance to cause fission. But a slower-moving neutron would have different probabilities. Next, they ran this chain on the world's first electronic computer, the ENIAC. The computer started by randomly generating a neutron starting conditions and stepped through the chain to keep track of how many neutrons were produced on average per run, known as the multiplication factor k. So if, on average, one neutron produces another two neutrons, then k is equal to two.

And if on average every two neutrons produce three neutrons, then k is equal to three over two, and so on. Then, after stepping through the full chain for a specified number of steps, we collect the average k-value and record that number in a histogram. This process was then repeated hundreds of times, and the results tallied up, giving you a statistical distribution of the outcome. If you find that in most cases, k is less than one, the reaction dies down.

If it's equal to one, there's a self-sustaining chain reaction, but it doesn't grow. And if k is larger than one, the reaction grows exponentially and you've got a bomb. - With it, von Neumann and Ulam had a statistical way to figure out how many neutrons were produced without having to do any exact calculations. In other words, they could approximate differential equations that were too hard to solve analytically. All that was needed was a name for the new method.