Why did "tickling the dragons tail" by Louis Slotin not cause an explosion?
Your understanding is pretty much correct and your question quite a natural one.
The core did react: the release of energy heated the core and shells quickly, thus changing the neutron capture cross section for the plutonium in the core. A plutonium (or any fissionable) atom's ability to capture a neutron and undergo fission is weakly dependent on temperature: decreasing with increasing temperature. As the core heated, the lowered ability to capture neutrons meant that the core actually became subcritical pretty quickly, thus quenching the chain reaction. If Slotin would not have flipped the top shell off, the uncritical core would then begin to cool and become critical again, repeating the process.
Generally, and thankfully, big bangs are very hard to provoke with nuclear chain reactions. The immense release of energy in a small space means that the critical mass is going to heat up and quench the reaction very quickly. If you do get to explosion, the mass blows itself apart, quenching the reaction even more quickly, so unless there is very particular conditions, the bang is not going to be a big one: just enough to break apart the apparatus and drench every living thing nearby in a lethal dose of neutrons. Big nuclear explosions only happen when the process of assembling the critical mass is so fast and the process of crushing the critical mass and keeping it confined does so long enough that a huge amount of material has time to undergo fission before the core blows itself apart and puts an end to the whole process. In a plutonium bomb, this is done by crushing the hollow core with shaped explosives that produce a highly symmetrical, inward moving shock wave.
I can only answer qualitatively:
The experiment where the death occurred was on a subcritical mass of plutonium, and reflectors were being used to bring the number of neutrons to the ones required for criticallity.
The mean generation time, Λ, is the average time from a neutron emission to a capture that results in fission
and l is the the prompt neutron lifetime , the average time between the emission of neutrons and either their absorption in the system or their escape from the system
k = 1 (criticality): Every fission causes an average of one more fission, leading to a fission (and power) level that is constant. Nuclear power plants operate with k = 1 unless the power level is being increased or decreased.
k less then one is subcritical, and k larger than 1 is supercritical, the explosion stage of weapons.
They were experimenting on turning a subcritical lump to critical by using reflectors and increasing the number of neutrons in the sample.
Times in these experiments from subcritical to critical depend a lot on the boundary conditions, and in a sense the accident is an experiment. It tells us that subcritical can go to critical within times controllable by humans , i.e. seconds, ( though they do not survive). His reactions were fast enough to stop the chain reaction, though it is not clear whether the lump of plutonium would have gone to supercritical if the reflectors were not removed. The subsequent to the accidents precautions of using remote control and the observes at miles distances shows that the danger existed.
Anyway the experiment tells us that even though the prompt neutron lifetime is in microseconds the build up of the chain reaction for this particular geometry and mass is of the order of seconds, enough for a human to react.