What triggers a radioactive nucleus to decay spontaneously?

[Recycler]

what conditions must occur randomly to allow a radioactive nucleus to decay spontaneously?

 

[rootone]

A radioactive decay of a specific kind of atom occurs statistically on average.
We call it half-life,there are not specific conditions which lead to fission of a particular atom.
It is possible to make conditions different, like saturating the material containing the atoms with high energy neutrons,
That's how breeder reactors make Plutonium.

 

[jtbell]

There is no "condition" that triggers a nucleus to decay at a specific moment, as far as we know. Each nuclide (type of nucleus) has a certain probability of decaying during the next short time interval: the decay constant $\lambda$ which has units of probabiility per second. It's related to the half-life by $t_\rm{1/2} = (\ln 2) / \lambda$. The decay constant can be predicted (at least in principle) by applying quantum mechanics or quantum field theory to the system in question.

If $\lambda = 0.1$ per second, then the nucleus has a one-in-ten chance of decaying during the next second. But we can't force it to happen at any particular point in time during that second, and we don't know what (if anything) immediately triggers it.