Nuclear Reactions

What are the nuclear reactions taking place in the interior of the stars, and what are their elementary steps?

There are many. In these reactions, light elements fuse into heavier elements. The more massive is a star and the higher the temperature of its interior, the higher the number of possible reactions available, which transform atoms into heavier and heavier elements in a process known as nucleosynthesis.

The elementary steps of nuclear reactions need to fulfill some conditions:

  • The electric charge needs to be conserved.
  • The number of leptons (electrons, positrons, neutrinos, and antineutrinos) needs to be conserved.
    Conditions: matter leptons – antimatter leptons = constant
    Note: electron + positron = 2 photons (γ) (matter and antimatter annihilate each other producing photons)

elementary-particles

Neutrinos are somewhat special in that they have a very small mass, and a very small cross section, therefore they have very long mean free paths (of the order of 10e9 solar radii!).

In the following, I represent nuclei by the symbol:

symbol-nucleum

One possible reaction is the conversion of hydrogen into helium. This happens via the proton-proton chain or via the CNO cycle, where carbon, nitrogen, and oxygen act as catalysts.

proton-proton-chain

The proton-proton chain consists on 3 branches: PP-I, PP-II, PP-III.

PP-I. Source.

PP-I. Source.

The slowest step in the PP-I chain is the first one, which involves the weak nuclear force.

PP-II. Source.

PP-II. Source.

PP-III. Source.

PP-III. Source.

In an environment similar to the center of the Sun, 69% of the time the PP-I chain takes place, while 31% of the time the PP-II chain occurs. In such environment, the occurrence of the PP-III chain is more rare, with a probability of 0.3%. These probabilities change depending on the temperature on the star.

In summary, the three branches of the PP-chain:

The next pic represents the CNO cycle.

CNO cycle. Source.

CNO cycle. Source.

The cycle above culminates with the production of carbon-12 and helium-4. Another possibility, less probable (0.04%) is the production of oxygen-16 and a photon instead.

The effect of fusing hydrogen into helium is the increase of the mean molecular weight (μ). The star begins to collapse, and this is compensated by a gradual increase of T and ρ.

Therefore, the PP-chain is dominant in low mass stars with low T, while the CNO cycle dominates at higher T and more massive stars.

In the triple alpha process, helium is converted into carbon. The “alpha” comes from the historical mysterious alpha particles, which turned out to be helium-4 nuclei. This process may be thought of as a 3-body interaction, and it has a very dramatic T dependence.

Triple alpha process. Source.

Triple alpha process. Source.

The power laws for the reactions above are follow. Simply note the temperature dependence of each reaction. Each needs larger T than the previous, but the T-dependence is much stronger for the latter (indicated by larger powers). Thus a given increase in T causes a larger and larger increase in the reaction rates:

pp-chain-rate

CNO-cycle-rate

triple-alpha-rate

At sufficiently high temperatures, many more reactions become available, consisting on the formation of heavier and heavier nuclei. Examples are carbon burning reactions, yielding oxygen, neon, sodium, and magnesium, and oxygen burning reactions, yielding magnesium, silicium, phosphorous, and sulphur.

If we plot the binding energy per nucleon (nucleons = neutrons and protons), Eb/A, we obtain a maximum at iron, and a series of stability peaks corresponding to particularly stable nuclei (e.g. helium, carbon, oxygen). These “enhanced stability peaks” are caused by the shell structure of atomic energy levels (similar to the shell structure of the electrons around a nucleus).

Generally speaking, fusion reactions of elements with atomic mass larger than 56, tend to consume energy, while fusion reactions resulting in iron or elements lighter than iron tend to liberate energy.

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