Can an element decay into an infinite loop? I've been working on a problem and have been wondering is there any isotope of any element that beta decays under certain conditions but then under differing conditions the daughter nucleus electron captures or positron decays. And what conditions would help catalyze this opposing positron decay or electron capture?
In addition, I am aware that external conditions can effect decay in light weight elements and complete ionization of heavier elements can have a similar effect. If any of you nuclear chemists out there or knowledgeable person on the subject could elaborate it would help a lot, thanks.
 A: The beryllium-7 nucleus is stable, but the beryllium-7 atom may decay by electron capture. This is because the reaction 
$$
\rm ^7_4Be^+ + e^- \to {}^7_3Li + \nu_e + 0.861\,MeV
$$
is energetically allowed. The equivalent reaction, with the electron on "before" side replaced with a positron on the "after" side
$$
\rm ^7_4Be \to {} {}^7_3Li^- + e^+ + \nu_e + (-0.161\,MeV)
$$
is not allowed, because the two extra leptons add 1.022 MeV to the final rest mass.
Bare nuclei are therefore stable against positron emission; this species appears in cosmic rays, as long-lived ions, but is unstable on Earth where it can collect negative charge.
There are several "odd-odd" nuclei (with odd proton number $Z$ and odd neutron number $N$) which are nearly stable, but sandwiched between more stable even-even nuclei with the same mass number $A=N+Z$.  These may decay either by beta or positron emission.  For instance, postassium-40 may beta-decay to calcium-40, or positron-decay/electron capture to argon-40.  The beta and positron decays are weak interaction effects.
I suppose you might consider some weak transitions induced by neutrino irradiation as a "catalyzed" decay.  That's how the neutrino was actually discovered by Reines and Cowan, using the reaction
$$
\rm \bar\nu_e + p^+ \to n + e^+
$$
Likewise solar neutrinos were first observed by Davis using
$$
\rm \nu_e + {}^{37}_{17}Cl \to {}^{37}_{18}Ar^+ + e^-
$$
In both of these you are taking a stable nucleus and transmuting it to an unstable nucleus by the flavor-changing weak interaction.  You could just as well do this starting with an unstable species, like
$$
\rm \bar\nu_e + {}^{37}Ar \to {}^{37}Cl + e^+
$$
However it'd be hard to tell if you had actually done this "catalysis," because neutrino cross sections are so feeble and beta emitters are so happy to decay on their own.
All spontaneous decays are exothermic; you cannot have an infinite loop of decays for thermodynamical reasons.  You can have loops of nuclear reactions if there is extra energy and extra particles about.  For instance in stars, carbon nuclei catalyze hydrogen fusion in what's called the CNO cycle.
A: In astrophysics, rates of beta decay and electron capture can be influenced by environment, specifically the ambient density of free electrons.
If the gas is dense enough, the Fermi energy of the electrons could be higher than the maximum possible beta decay electron energy. This would suppress beta decay and enhance electron capture.
Take the example of the dense gas at the centre of a white dwarf star, that might be composed largely of $^{12}$C.
This of course would usually be stable, but above densities of about $4\times 10^{13}$ kg/m$^{3}$ the electron Fermi energy rises above the threshold for electron capture and produces $^{12}$B.
The reverse process of beta decay is then blocked by the electron degeneracy, but then if you could somehow move the $^{12}$B somewhere where the electron density were lower, perhaps by convection, then it would beta decay back to Carbon.
Fortunately for us, this is unlikely. The electron capture results in a fall in degeneracy pressure, an increase in density and an instability that can ultimately result in type Ia supernovae producing a significant fraction of the iron in the universe.
