There’s two types, broadly speaking, of atomic bombs; those that use fission (splitting atoms) and those that use fusion (combining atoms). All modern atomic bombs use fusion, but you’re asking about fission, which was used in the bombs dropped by the USA during WWII.
Radioactive materials like uranium and plutonium are constantly going through what’s called spontaneous fission. These elements (particularly versions with extra neutrons attached) are really unstable and prone to just falling apart, turning one atom into two and kicking off 2 or 3 neutrons in the process.
Those neutrons can hit *other* uranium or plutonium atoms, turning them into an extra-neutron extra-unstable version and causing them to split, releasing 2 or 3 *more* neutrons.
If you have a certain amount of the radioactive element in a close enough space, you’ve achieved what’s called *criticality* where enough neutrons hit other atoms to keep the chain reaction going, and in fact *increasing* in rate until all the radioactive atoms are split. This out-of-control chain reaction generates a huge amount of heat and explosive force – i.e. it is a bomb.
Early fission bombs basically had two pieces of uranium in them. Each piece was small enough that it was not critical. They were slowly decaying, but each decay didn’t manage to trigger 2 new decays so the reaction didn’t spiral out of control.
Once the bomb hit the ground, a conventional explosion would force the two chunks of uranium together, forming a combined mass that *was* critical. A chain reaction very quickly took place, splitting all those atoms and releasing tremendous amounts of heat.
Later designs would use a *single* piece of radioactive material, then squish it from all directions with explosions around it. By squishing it inwards it was made more dense, and thus critical.
Other designs involve shooting neutrons at the radioactive material to trigger the reaction. And again, this does not cover *fusion* bombs, which were first constructed in the 1950s and are likely the only type constructed today.
Latest Answers