Question

Are there any examples of common substances whose decay is not exponential?

We're used to thinking about radioactivity in terms of half-lives. This is a concept that makes sense only for a decay that is exponential. However, there are plenty of physics articles on the subject of non exponential decay. It seems to be theoretically ubiquitous. For example:

The decay of unstable quantum states is an ubiquitous process in virtually all fields of physics and energy ranges, from particle and nuclear physics to condensed matter, or atomic and molecular science. The exponential decay, by far the most common type, is surrounded by deviations at short and long times1,2. The short-time deviations have been much discussed, in particular in connection with the Zeno effect3,4,5 and the anti-Zeno effect6,7,8,9. Experimental observations of short10,11 and long12 time deviations are very recent. A difficulty for the experimental verification of long-time deviations has been the weakness of the decaying signal13, but also the measurement itself may be responsible, because of the suppression of the initial state reconstruction2,14.

1) L. A. Khalfin, Zurn. Eksp. Teor. Fiz. 33, 1371 (1957), English translation: Sov. Phys. JETP 6 1053 (1958).
2) L. Fonda and G. C. Ghirardi, Il Nuovo Cimento 7A, 180 (1972).

10.1103/PhysRevA.74.062102, F. Delgado, J. G. Muga, G. Garcia-Calderon
Suppression of Zeno effect for distant detectors

So are there any examples of deviations from long time decay? If not, then why not? Is the theory wrong or simply impractical? And is there a simple, intuitive explanation for why long decays should not be exponential?

Answer 1

For very long times, a decay process starts to compete with the inverse process. For instance, right now you are bathed in an ocean of matter and antimatter neutrinos with lots of different energies. For a given beta-decaying nucleus, some fraction of these background neutrinos will have enough energy to drive the inverse decay process, transforming the "daughter" nucleus into the "parent." Thus if you start off with a population of parent nuclei, you don't necessarily end up with zero parent nuclei and all daughter nuclei, as pure exponential decay would predict; instead you end up with a tiny fraction of the parent nuclei remaining in the sample. The size of this steady-state fraction depends on the local neutrino density and energy spectrum. You can make the same argument for other decay modes.