Proton–proton chain reaction

The proton–proton chain reaction dominates in stars the size of the Sun or smaller.

The proton–proton chain reaction is one of two nuclear fusion reactions, along with the CNO cycle, by which stars convert hydrogen to helium and which dominates in stars the size of the Sun or smaller.[1]

In general, proton–proton fusion can occur only if the kinetic energy (i.e. temperature) of the protons is high enough to overcome their mutual electrostatic or Coulomb repulsion.[2]

In the Sun, deuterium-producing events are rare as diprotons, the much more common result of nuclear reactions within the star, immediately decay back into two protons. A complete conversion of the hydrogen in the solar core is calculated to take more than 1010 (ten billion) years.[3]

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 Stanley Eddington in the 1920s. At the time, the temperature of the Sun was considered 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 immediately dissociates back into a pair of protons. In 1939, Hans Bethe proposed that one of the protons could beta decay into a neutron via the weak interaction during the brief moment of fusion, making deuterium the initial product in the chain.[4] This idea was part of the body of work in stellar nucleosynthesis for which Bethe won the 1967 Nobel Prize in Physics.

The pp chain reaction

The first step involves the fusion of two 1H 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:

1
1H 		+ 1
1H 		→ 2
2He 		+ γ

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

2
2He 		→ 2
1H 		+ e+ + ν
e

with the overall formula:

1
1H 		+ 1
1H 		→ 2
1H 		+ e+ + ν
e 		+ 0.42 MeV

This first step is extremely slow because the beta-plus decay of the diproton to deuterium has a negative Q value and so is extremely rare (the vast majority of the time, the diproton decays back into hydrogen-1 through proton emission). The half-life of a proton in the core of the Sun before it is involved in a successful p-p fusion is estimated to be a billion years, even at the extreme pressure and temperatures found there.

The positron emitted by the beta-decay is almost immediately annihilated with an electron, and their mass energy, as well as their kinetic energy, is carried off by two gamma ray photons.

e− + e+ → 2 γ + 1.02 MeV

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

2
1D 		+ 1
1H 		→ 3
2He 		+ γ + 5.49 MeV

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, a newly created deuterium nucleus exists for only about 4 seconds before it is converted to He-3.

From here there are four possible paths to generate 4He. In pp I, helium-4 is produced by fusing two helium-3 nuclei; the pp II and pp III branches fuse 3He with pre-existing 4He 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.[5]

In the Sun, 4He synthesis via branch pp I occurs with a frequency of 86%, pp II with 14% and pp III with 0.11%. There is also an extremely rare pp IV branch. Additionally, other even less frequent reactions may occur; however, 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 mass 5 or 8; the resulting products immediately decay into their initial reactants.

The overall reaction is:

4p → 4He + e+ + νe

The pp I branch

3
2He 		+ 3
2He 		→ 4
2He 		+ 2 1
1H 		+ 12.86 MeV

The complete pp I chain reaction releases a net energy of 26.732 MeV.[6] Two percent of this energy is lost to the neutrinos that are produced.[7] The pp I branch is dominant at temperatures of 10 to 14 MK. Below 10 MK, the PP chain does not produce much 4He.

The pp II branch

Proton–proton II chain reaction
See also: lithium burning
3
2He 		+ 4
2He 		→ 7
4Be 		+ γ
7
4Be 		+ e− → 7
3Li 		+ ν
e 		+ 0.861 MeV / 0.383 MeV
7
3Li 		+ 1
1H 		→ 2 4
2He

The pp II branch is dominant at temperatures of 14 to 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% of the neutrinos produced in the reaction of 7Be to 7Li carry an energy of 0.861 MeV, while the remaining 10% carry 0.383 MeV. The difference is whether the lithium-7 produced is in the ground state or an excited state, respectively.

The pp III branch

Proton–proton III chain reaction
3
2He 		+ 4
2He 		→ 7
4Be 		+ γ
7
4Be 		+ 1
1H 		→ 8
5B 		+ γ
8
5B 		  → 8
4Be 		+ e+ + ν
e 
8
4Be 		  → 2 4
2He

The pp III chain is dominant if the temperature exceeds 23 MK.

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

The pp IV (Hep) branch

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

3
2He 		+ 1
1H 		→ 4
2He 		+ e+ + ν
e 		+ 18.8 MeV

Energy release

Comparing the mass of the final helium-4 atom with the masses of the four protons reveals that 0.007 or 0.7% 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 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 pp-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 ppI, ppII and ppIII chains carry away 2.0%, 4.0%, and 28.3% of the energy in those reactions, respectively.[8]

The pep reaction

Proton–proton and electron-capture chain reactions in a star

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

1
1H 		+ e− + 1
1H 		→ 2
1D 		+ ν
e

In the Sun, the frequency ratio of the pep reaction versus the pp 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 pp reaction range in energy up to 0.42 MeV, the pep reaction produces sharp-energy-line neutrinos of 1.44 MeV. Detection of solar neutrinos from this reaction were reported by the Borexino collaboration in 2012.[9]

Both the pep and pp 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 anti-electron. This is represented in the figure of proton–proton and electron-capture chain reactions in a star, available at the NDM'06 web site.[10]

See also

Wikimedia Commons has media related to Proton-proton chain reaction.

References

  1. ↑ The Proton-Proton Chain
  2. ↑ Ishfaq Ahmad, The Nucleus, 1:42,59, (1971), The Proton type-nuclear fission reaction
  3. ↑ Kenneth S. Krane, Introductory Nuclear Physics , Wiley , 1987, p. 537.
  4. ↑ Hans A. Bethe, Physical Review 55:103, 434 (1939); cited in Donald D. Clayton, Principles of Stellar Evolution and Nucleosynthesis, The University of Chicago Press, 1983, p. 366.
  5. ↑ This time and the two other times above come from: Byrne, J. Neutrons, Nuclei, and Matter, Dover Publications, Mineola, New York, 2011, ISBN 0486482383, p 8.
  6. ↑ LeBlanc, Francis. An Introduction to Stellar Astrophysics.
  7. ↑ Burbidge, E.; Burbidge, G.; Fowler, William; Hoyle, F. (1 October 1957). "Synthesis of the Elements in Stars". Reviews of Modern Physics 29 (4): 547–650. Bibcode:1957RvMP...29..547B. doi:10.1103/RevModPhys.29.547. This value excludes the 2% neutrino energy loss.
  8. ↑ Claus E. Rolfs and William S. Rodney, Cauldrons in the Cosmos, The University of Chicago Press, 1988, p. 354.
  9. ↑ "First Evidence of pep Solar Neutrinos by Direct Detection in Borexino" (preprint on arXiv): Phys. Rev. Lett. 108, (5), 051302 (2012)
  10. ↑ Int'l Conference on Neutrino and Dark Matter, Thursday 07 Sept 2006, http://indico.lal.in2p3.fr/getFile.py/access?contribId=s16t1&sessionId=s16&resId=1&materialId=0&confId=a05162 Session 14.
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