Why does nuclear waste have to be stored until the constituent elements decay naturally? Fair warning, I have a bachelors in CS and have chemistry 211/212 under my belt. My understanding of the atom consists of a proton, neutron, and electron quasi-orbiting it in some sort of strange probability cloud. If everything I say past this paragraph is completely wrong, you can be comfortable in the fact that I will never actually get the chance to put this limited knowledge (mis)use.
So, I was reading about nuclear waste and vitrification, and a thought occurred to me, why don't we attempt to subject the unstable fission products to further fission until we "boil" them down to stable elements which don't undergo such quick nuclear decay? To my understanding, to force an atom to split you only need to bombard it with enough neutrons to add enough energy to overcome the binding energy of the neutron/proton nucleus.
I understand that there would be a law of diminishing returns because there is less binding energy in a smaller nucleus (In my mind this is analogous to there being less "binding" energy in the moon than there is the earth), but wouldn't that also mean that less energy would be required to force it to split?
 A: Apparently, the CANDU reactor can accept a variety of fuels, including what would be considered "waste" from other reactor types, although some amount of reprocessing is involved.
A: It might be possible, but it's uneconomical and probably also uses more electric energy than you get from the power plant.
Do you really mean fisson? Fission is splitting a nucleus in two approximately equal parts. This is done with certain isotopes e.g. uranium 235 which are only barely stable and sometimes fission spontaneously. Other isotopes such as 238-U are more stable can be fissioned with fast/energetic neutrons. The nuclear waste nuclei are much more stable and fissioning them requires very high energy impacts which could be achieved with particle accelerators, but this would be very slow, expensive and consume more energy than produced by the nuclear power plant.
OTOH you could just transmute long lived nuclear isotopes by irradiating them with neutrons into isotopes that undergo radioactive decay (alpha, beta, etc, which is different than fission) in a short time into stable isotopes. It would have to be researched which isotopes are best suited for trarnsmutation as there is only a limited amount of excess neutrons available in a nuclear reactor, as some neutrons are required to keep up the chain reaction.
Actually, the fission products are relatively short lived compared to actinides produced by neutron capture of reactor fuel without fissioning them. These could be bred to fissile material.
So, the quick decaying isotopes are less of a concern as they decay quick, though not always, into stable or short lived isotopes, but sometimes into long lived isotopes. And with neutron transmutation you usually get short lived isotopes anyways. Bigger atoms have a higher neutron/proton ratio, and even when some neutrons are released during fission the fission products have excess neutrons. You can't usually make them stable by adding neutrons.
