Why is the isotope of lead-206 stable against alpha decay? The mass of Lead-206 is larger than that of Mercury-202 + Helium-4. Why is then Lead-206 stable against alpha decay? I have heard that the beta-decay can stabilize a nucleus against alpha decay, and that this is the case of the Lead-206. How does it happen?
 A: Lead 206 is not stable against alpha decay. I was surprised to find out that even the doubly-magic nucleus 208Pb is unstable: the Q value (energy released in the decay) in units of amu is $+5.549(4)\times10^{-4}$.
Alpha decay half-lives approximately follow the rule that the log of the half-life varies linearly with $Q^{-1/2}$. The physics underlying this is that the nucleus has to tunnel out through the Coulomb barrier, and the barrier penetration probability depends  exponentially on the square root of the amount by which the energy falls short of the classically allowed energy. (It's counterintuitive but true that the electrical force, which is repulsive, hinders these decays.) The $Q$ values for the "stable" isotopes of lead are extremely small. Therefore their half-lives are expected to be extremely long.
I guess this sort of make sense with hindsight, if you think about how alpha decay rates depend on $Z$. The lower you go in atomic number, the longer the half-lives become, until you reach lead, which is where we start to get effectively "stable" nuclei. It makes sense that as you get to lead, which is the boundary, there should be nuclei that  have very long half-lives.
Keep in mind that lots of forms of matter can be theoretically unstable, but stable for practical purposes. For example, it's theoretically possible for 124Te to decay into two 62Ni nuclei plus four electrons and four antineutrinos, because the $Q$ value for such a decay is $+0.04$ amu. That doesn't mean that we ever expect to observe such a decay. Quantum mechanics says that whatever is not forbidden is compulsory, but that doesn't mean that it has a high probability.

I have heard that the beta-decay can stabilize a nucleus against alpha decay, and that this is the case of the Lead-206.

That doesn't make much sense to me. Could you point us to a source on this? I imagine it's stable against beta decay, since it's even-even and close to the line of stability. The two processes are in any case independent. Having an additional decay process available will lower the overall half-life, not lengthen it.
