Posted: Apr 05, 2010 7:53 am
I can think of one reason why we can find isotopes with shorter half-lives than 100 million years. They're the decay products of some of those longer-lived isotopes. That's why we find radium in coal dust - it's a decay product of uranium. Any reasonably ancient piece of rock containing uranium will also contain radium in small but measurable quantities. Since uranium percolates through geological water flows into coal seams and becomes resident in the coal in measurable quantities, radium appears in the coal as well. It's one of the reasons that ash from coal burning is a quantifiable radiological hazard. Find out more about this from this scientific paper among others. 
Then, some of those longer-lived isotopes induce other elements alongside them to become radioactive. Neutron activation is a well-known phenomenon arising from nuclear reactors. My old favourite resource, Kaye & Laby, has a nice long article on neutron activation cross sections. Basically, what happens is this. Long-lived alpha-decay emitters such as uranium and thorium emit alpha particles (helium nuclei). In the case of 232Th, these have an energy of around 35.4 MeV, whilst those for 238U have an energy of 47.3 MeV. These can interact with other nuclei via collisions, producing an effect known as spallation, where the collision results in the liberation of a neutron or other fragment from the target nucleus. That neutron, in turn, can be absorbed by a stable nucleus, and transformed into a radioactive nucleus. For example, a stable 58Ni nucleus can absorb a spallation neutron, becoming 59Ni, which is radioactive, decaying via electron capture to the stable 59Co, with a half-life of 75,400 years.
Of course, which of the surrounding stable nuclei absorb the spallation neutron is dependent upon several factors, but this mechanism produces elements such as technetium in tiny quantities in appropriate geological strata. No isotope of technetium has a half life greater than 4.2 million years (this is an element with no stable isotopes), so its detection in a natural rock sample indicates that it was produced by neutron activation of molybdenum followed by beta decay (typically neutron activation of 98Mo to become 99Mo, which beta-decays with a half-life of 65 hours to produce 99Tc, which has a half-life of 211,000 years. Other technetium isotopes are produced by spontaneous fission of uranium, such as that occurring in the Oklo natural nuclear reactor.
Incidentally, a tiny fraction of 238U nuclei can undergo spontaneous fission, not only producing elements such as technetium directly, but producing other elements via neutron activation (three neutrons are released by the fissioning nucleus). However, a comparison of the interaction cross sections shows that this is a very infrequently occurring process - the cross section for 238U fission is around 4 × 10-6 barns (which means it doesn't happen very often). Compare this with the fission cross section for 235U, which is 583 barns, or 239Pu, which is 748 barns (both are fissile materials capable of being utilised in a nuclear reactor or a nuclear weapon as a result).
Then, some of those longer-lived isotopes induce other elements alongside them to become radioactive. Neutron activation is a well-known phenomenon arising from nuclear reactors. My old favourite resource, Kaye & Laby, has a nice long article on neutron activation cross sections. Basically, what happens is this. Long-lived alpha-decay emitters such as uranium and thorium emit alpha particles (helium nuclei). In the case of 232Th, these have an energy of around 35.4 MeV, whilst those for 238U have an energy of 47.3 MeV. These can interact with other nuclei via collisions, producing an effect known as spallation, where the collision results in the liberation of a neutron or other fragment from the target nucleus. That neutron, in turn, can be absorbed by a stable nucleus, and transformed into a radioactive nucleus. For example, a stable 58Ni nucleus can absorb a spallation neutron, becoming 59Ni, which is radioactive, decaying via electron capture to the stable 59Co, with a half-life of 75,400 years.
Of course, which of the surrounding stable nuclei absorb the spallation neutron is dependent upon several factors, but this mechanism produces elements such as technetium in tiny quantities in appropriate geological strata. No isotope of technetium has a half life greater than 4.2 million years (this is an element with no stable isotopes), so its detection in a natural rock sample indicates that it was produced by neutron activation of molybdenum followed by beta decay (typically neutron activation of 98Mo to become 99Mo, which beta-decays with a half-life of 65 hours to produce 99Tc, which has a half-life of 211,000 years. Other technetium isotopes are produced by spontaneous fission of uranium, such as that occurring in the Oklo natural nuclear reactor.
Incidentally, a tiny fraction of 238U nuclei can undergo spontaneous fission, not only producing elements such as technetium directly, but producing other elements via neutron activation (three neutrons are released by the fissioning nucleus). However, a comparison of the interaction cross sections shows that this is a very infrequently occurring process - the cross section for 238U fission is around 4 × 10-6 barns (which means it doesn't happen very often). Compare this with the fission cross section for 235U, which is 583 barns, or 239Pu, which is 748 barns (both are fissile materials capable of being utilised in a nuclear reactor or a nuclear weapon as a result).