Proton–proton chain reaction

History of the theory

The theory that proton–proton reactions are the basic principle by which the Sun and other stars burn was advocated by Arthur Eddington in the 1920s. At the time, the temperature of the Sun was considered to be too low to overcome the Coulomb barrier. After the development of quantum mechanics, it was discovered that tunneling of the wavefunctions of the protons through the repulsive barrier allows for fusion at a lower temperature than the classical prediction.

Even so, it was unclear how proton–proton fusion might proceed, because the most obvious product, helium-2 (diproton), is unstable and almost instantly dissociates back into two protons. In 1939, Hans Bethe proposed that one of the protons could decay by beta emission into a neutron via the weak interaction during the brief moment of fusion, making deuterium a vital product in the chain. This idea was part of the body of work in stellar nucleosynthesis for which Bethe won the Nobel Prize in Physics in 1967.

The proton-proton chain reaction

The first step involves the fusion of two 1
H
nuclei (protons) into deuterium, releasing a positron and a neutrino as one proton changes into a neutron. It is a two-stage process; first, two protons fuse to form a diproton:

followed by the beta-plus decay of the diproton to deuterium:

with the overall formula:

This first step is extremely slow because the positron emission of the diproton to deuterium is extremely rare (the vast majority of the time, the diproton decays back into hydrogen-1 through proton emission). This is because the emission of the positron is brought about by the weak nuclear force, which is immensely weaker than the strong nuclear force and the electromagnetic force.

The half-life of a proton in the core of the Sun before it is involved in a successful proton-proton fusion is estimated to be about one billion years, even at the extreme pressures and temperatures found there.

The positron emitted by the decay by beta emission usually annihilates immediately with an electron. Their mass-energies plus their kinetic energy is carried off by two gamma rays (photons) with the mass-energy of 511 KeV (thousand electron-volts) apiece.

After it is formed, the deuterium produced in the first stage can fuse with another proton to produce the light isotope of helium, 3
He
:

This process, mediated by the strong nuclear force rather than the weak force, is extremely fast by comparison to the first step. It is estimated that, under the conditions in the Sun's core, each newly created deuterium nucleus exists for only about four seconds before it is converted to He-3.

From here there are four possible paths to generate 4
He
. In p-p I, helium-4 is produced by fusing two helium-3 nuclei; the p-p II and p-p III branches fuse 3
He
with pre-existing 4
He
to form beryllium-7, which undergoes further reactions to produce two helium-4 nuclei. In the Sun, the helium-3 produced in these reactions exists for only about 400 years before it is converted into helium-4.

In the Sun, 4
He
synthesis via branch p-p I occurs with a frequency of 83.30 percent, p-p II with 16.68 percent, and p-p III with 0.02 percent.

There is also the extremely rare p-p IV branch. Other even-rarer reactions may occur. The rate of these reactions is very low due to very small cross-sections, or because the number of reacting particles is so low that any reactions that might happen are statistically insignificant. This is partly why no mass-5 or mass-8 elements are seen. While the reactions that would produce them, such as a proton + helium-4 producing lithium-5, or two helium-4 nuclei coming together to form beryllium-8, may actually happen, these elements are not detected because there are no stable isotopes of atomic masses 5 or 8; the resulting products immediately decay into their initial reactants.

The overall reaction is:

4p → 4
He
+ 2e⁺ + 2ν_e

The complete p-p I chain reaction releases a net energy of 6988428293818504840♠26.732 MeV. Two percent of this energy is lost to the neutrinos that are produced. The p-p I branch is dominant at temperatures of 10 to 7007140000000000000♠14 MK. Below 7007100000000000000♠10 MK, the P-P chain does not produce much 4
He
.

The P-P II branch is dominant at temperatures of 14 to 7007230000000000000♠23 MK.

Note that the energies in the equation above are not the energy released by the reaction. Rather, they are the energies of the neutrinos that are produced by the reaction. 90 percent of the neutrinos produced in the reaction of 7
Be
to 7
Li
carry an energy of 6987137947395530700♠0.861 MeV, while the remaining 10 percent carry 6986613633594520999♠0.383 MeV. The difference is whether the lithium-7 produced is in the ground state or an excited (metastable) state, respectively.

The P-P III chain is dominant if the temperature exceeds 7007230000000000000♠23 MK.

The p-p III chain is not a major source of energy in the Sun (only 0.11 percent), but it was very important in the solar neutrino problem because it generates very high energy neutrinos (up to 6988225266014072200♠14.06 MeV).

The P-P IV (Hep) branch

This reaction is predicted theoretically, but it has never been observed due to its rarity (about 6993300000000000000♠0.3 ppm in the Sun). In this reaction, helium-3 captures a proton directly to give helium-4, with an even higher possible neutrino energy (up to 18.8 MeV).

Energy release

Comparing the mass of the final helium-4 atom with the masses of the four protons reveals that 0.7 percent of the mass of the original protons has been lost. This mass has been converted into energy, in the form of gamma rays and neutrinos released during each of the individual reactions. The total energy yield of one whole chain is 6988428261774975100♠26.73 MeV.

Energy released as gamma rays will interact with electrons and protons and heat the interior of the Sun. Also kinetic energy of fusion products (e.g. of the two protons and the 4
2He
from the p-p I reaction) increases the temperature of plasma in the Sun. This heating supports the Sun and prevents it from collapsing under its own weight.

Neutrinos do not interact significantly with matter and therefore do not help support the Sun against gravitational collapse. Their energy is lost: the neutrinos in the p-p I, p-p II, and p-p III chains carry away 2.0%, 4.0%, and 28.3% of the energy in those reactions, respectively.

The PEP reaction

Deuterium can also be produced by the rare pep (proton–electron–proton) reaction (electron capture):

In the Sun, the frequency ratio of the pep reaction versus the p-p reaction is 1:400. However, the neutrinos released by the pep reaction are far more energetic: while neutrinos produced in the first step of the p-p reaction range in energy up to 6986672914124540000♠0.42 MeV, the pep reaction produces sharp-energy-line neutrinos of 6987230713414127999♠1.44 MeV. Detection of solar neutrinos from this reaction were reported by the Borexino collaboration in 2012.

Both the pep and p-p reactions can be seen as two different Feynman representations of the same basic interaction, where the electron passes to the right side of the reaction as an antielectron. This is represented in the figure of proton–proton and electron-capture chain reactions in a star, available at the NDM'06 web site.