# Are He-3 and Ne-21 stable isotopes?

The short answer is yes. However, a physicist could say that they are not stable. There are some reasons not to consider that these nuclides will not last forever. For example, they both contain protons. In theory, protons decay in 1031 to 1036 years. That is more than sextillion times the age of the universe! Why should we care abut this? Well, we shouldn’t. Practically, 3He and 21Ne are stable for the purpose of cosmonuclide applications. Therefore, their concentration C in a mineral exposed to cosmic radiation is defined by $C=P \cdot t$

where P is the production rate and t is the exposure time. On the other hand, when we define the accumulation of a cosmogenic radionuclide (e.g., 10Be), we use the formula: $C=\frac{P}{\lambda}\cdot(1-e^{-\lambda \cdot t})$

where λ is the decay constant, that is $=\ln{(2)}/t_{1/2}$, being $t_{1/2}$ the half-life of the nuclide. In the case of 10Be, $t_{1/2}$ = 1.386 My.

This means that, when programming, we should use the first formula when simulating the accumulation of stable nuclides, and the second one for radioactive nuclides. In pseudocode:

if λ>0
C=P/λ*(1-exp(-λ*t))
else
C=P*t
end


However, introducing an “if” statement in a code that is used many times in a simulation implies increasing computing time substantially.

That is why I normally consider ridiculously slow decay rates for these isotopes in my code (e.g., 100 times the age of the Earth in NUNAIT code, lines 28-36). By doing this, we can always use the non-stable formulas, avoiding 1/0 errors.

Even if we want to use these models to simulate very long histories, like accumulation in meteorites, we can always assign the theoretical proton decay constant to these “stable” isotopes with practically no numeric impact!