Helium Fusion and the Origin of Elements

Question

In the fifth of his Messenger Lectures Feynman mentions the fact that Carbon-12 has an energy level at 7.82 MeV. He then states that this is what enables the fusion of three Helium-4 ions into Carbon and ultimately enables the production of the other elements. Moreover one can apparently calculate that an energy level of 7.82 MeV is necessary in advance.

Now unfortunately I have almost no knowledge of nuclear physics and therefore do not know how fusion is modelled theoretically. And I do not know if it is hard or easy to estimate that energy level (i.e. it could simply be a matter of if energy conservation allows the process). So my question are:

What goes into this kind of calculation?

and more generally:

What are the current methods for a quantitative (qualitative) understanding of nuclear fusion?

Naively I would model the fusion process by an estimate for the transition amplitude between a $\lvert \text 3{He}\rangle$ state into a $\lvert C\rangle$ state maybe by Fermi's golden rule. This should give you approximately a delta function $\delta(E_f - E_i)$, where $E_f$ and $E_i$ are the energies of the initial and final state. Then you would need to estimate the energy levels of the $3\text{He}$ system and one of them would be approximately at 7.82 MeV.

You can assume in your answers that I took graduate level courses on quantum mechanics and quantum field theory, if that helps.

Answer 1

In the 1940s and 50s, physicists were trying to understand the formation of carbon-12 in stars. It was (correctly) proposed that two Helium-4 nuclei first fuse to produce beryllium-8, which then fuses with another Helium-4 to produce Carbon-12. This is known as the triple-alpha process. An apparent problem with this explanation was that the ground state of Carbon-12 had too low of an energy for this process to occur to the extent that it does. Fred Hoyle proposed in 1954 that there exists an excited state of C-12 just above the combined energy of He-4 and Be-8, meaning just more than 7.6 MeV above the ground state of C-12. Three years later, such an excited C-12 state was found 7.82 MeV above the ground state. So Fred Hoyle didn't really calculate the existance of the excited state, he reasoned that since carbon exists, there must be a way to form carbon and therefore such a state must exist. The excited state is now known as the Hoyle State.

Recently calculation of the Hoyle State from physics princples has been accomplished. See Carbon's Hoyle state calculated at long last.