Proton–proton chain reaction

From Wikipedia, the free encyclopedia

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

The proton–proton chain reaction is one of several fusion reactions by which stars convert hydrogen to helium, the primary alternative being the CNO cycle. The proton–proton chain dominates in stars the size of the Sun or smaller.^{[citation needed]}

In general, proton–proton fusion can occur only if the temperature (i.e. kinetic energy) of the protons is high enough to overcome their mutual electrostatic or Coulomb repulsion.^[1]

In the Sun, deuterium-producing events are so rare (diprotons, the much more common result of nuclear reactions within the star, immediately decay back into two protons) that a complete conversion of the star's hydrogen would take more than 10¹⁰ (ten billion) years at the prevailing conditions of its core.^[2] The fact that the Sun is still shining is due to the slow nature of this reaction; if it went more quickly, the Sun would have exhausted its hydrogen long ago.

History of the theory

The theory that proton–proton reactions were 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.^[3] This idea was part of the body of work in stellar nucleosynthesis for which Bethe won the 1967 Nobel Prize in Physics.

The proton–proton 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 1D

+

e+

+

ν e

with the overall formula:

1 1H

+

1 1H

→

2 1D

+

e+

+

ν e

+

0.42 MeV

This first step is extremely slow, because the beta-plus decay of the diproton to deuterium is extremely rare (the vast majority of the time, it decays back into hydrogen-1 through proton emission).

The positron immediately annihilates 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 this, 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

From here there are four possible paths to generate the helium isotope 4He. In pp I helium-4 comes from fusing two of the helium-3 nuclei produced; the pp II and pp III branches fuse 3He with a pre-existing 4He to make beryllium. In the Sun, branch pp I takes place 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. 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, while they may actually happen, do not show up because there are no stable isotopes of mass 5 or 8; the resulting products immediately decay into their initial reactants.

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 26.22 MeV^[4] The pp I branch is dominant at temperatures of 10 to 14 MK. Below 10 MK, the PP chain does not produce much 4He.^{[citation needed]}

The pp II branch

Proton–proton II chain reaction

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.

90% of the neutrinos produced in the reaction 7Be(e−,ν
e)7Li* carry an energy of 0.861 MeV, while the remaining 10% carry 0.383 MeV (depending on whether lithium-7 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 great 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 two protons and He4 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 do not help support the Sun against gravitational collapse. 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.^[5]

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.^[6]

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.^[7]

CNO cycle

From Wikipedia, the free encyclopedia

 

Overview of the CNO-I Cycle

The CNO cycle (for carbon–nitrogen–oxygen) is one of the two (known) sets of fusion reactions by which stars convert hydrogen to helium, the other being the proton–proton chain reaction. Unlike the latter, the CNO cycle is a catalytic cycle. Theoretical models^[which?] show that the CNO cycle is the dominant source of energy in stars more massive than about 1.3 times the mass of the Sun.^[1] The proton–proton chain is more important in stars the mass of the Sun or less. This difference stems from temperature dependency differences between the two reactions; pp-chain reactions start occurring at temperatures around 4×10⁶ K^[2] (4 megakelvins), making it the dominant energy source in smaller stars. A self-maintaining CNO chain starts occurring at approximately 15 MK, but its energy output rises much more rapidly with increasing temperatures.^[1] At approximately 17 MK, the CNO cycle starts becoming the dominant source of energy.^[3] The Sun has a core temperature of around 15.7 MK, and only 1.7% of 4He nuclei being produced in the Sun are born in the CNO cycle. The CNO-I process was independently proposed by Carl von Weizsäcker^[4] and Hans Bethe^[5] in 1938 and 1939, respectively.

In the CNO cycle, four protons fuse, using carbon, nitrogen and oxygen isotopes as a catalyst, to produce one alpha particle, two positrons and two electron neutrinos. Although there are various paths and catalysts involved in the CNO cycles, simply speaking all these cycles have the same net result:

4 1
1H  +  2 e−  →  4
2He  +  2 e+  +  2 e−  +  2 ν
e  +  3 γ  +  24.7 MeV  →  4
2He  +  2 ν
e  +  3 γ  +  26.7 MeV

The positrons will almost instantly annihilate with electrons, releasing energy in the form of gamma rays. The neutrinos escape from the star carrying away some energy. One nucleus goes to become carbon, nitrogen, and oxygen isotopes through a number of transformations in an endless loop.

Cold CNO cycles

Under typical conditions found in stellar plasmas, catalytic hydrogen burning by the CNO cycles is limited by proton captures. Specifically, the timescale for beta decay of radioactive nuclei produced is faster than the timescale for fusion. Because of the long timescales involved, the cold CNO cycles convert hydrogen to helium slowly, allowing them to power stars in quiescent equilibrium for many years.

CNO-I

The first proposed catalytic cycle for the conversion of hydrogen into helium was at first simply called the carbon–nitrogen cycle (CN cycle), also honorarily referred to as the Bethe–Weizsäcker cycle, because it does not involve a stable isotope of oxygen. Bethe's original calculations suggested the CN-cycle was the Sun's primary source of energy, owing to the belief at the time that the Sun's composition was 10% nitrogen;^[5] the solar abundance of nitrogen is now known to be less than half a percent. This cycle is now recognized as the first part of the larger CNO nuclear burning network.

The main reactions of the CNO-I cycle are 12
6C→13
7N→13
6C→14
7N→15
8O→15
7N→12
6C:^[6]

12 6C	+	1 1H	→	13 7N	+	γ			+	1.95 MeV
13 7N			→	13 6C	+	e+	+	ν e	+	1.20 MeV	(half-life of 9.965 minutes[7])
13 6C	+	1 1H	→	14 7N	+	γ			+	7.54 MeV
14 7N	+	1 1H	→	15 8O	+	γ			+	7.35 MeV
15 8O			→	15 7N	+	e+	+	ν e	+	1.73 MeV	(half-life of 122.24 seconds[7])
15 7N	+	1 1H	→	12 6C	+	4 2He			+	4.96 MeV

where the Carbon-12 nucleus used in the first reaction is regenerated in the last reaction. After the two positrons emitted annihilate with two ambient electrons producing an additional 2.04 MeV, the total energy released in one cycle is 26.73 MeV; it should be noted that in some texts, authors are erroneously including the positron annihilation energy in with the beta-decay Q-value and then neglecting the equal amount of energy released by annihilation, leading to possible confusion. All values are calculated with reference to the Atomic Mass Evaluation 2003.^[8]

The limiting (slowest) reaction in the CNO-I cycle is the proton capture on 14
7N; it was recently experimentally measured down to stellar energies, revising the calculated age of globular clusters by around 1 billion years.^[9]

The neutrinos emitted in beta decay will have a spectrum of energy ranges, because although momentum is conserved, the momentum can be shared in any way between the positron and neutrino, with either being emitted at rest and the other taking away the full energy, or anything in between, so long as all the energy from the Q-value is used. All momentum which get the electron and the neutrino together is not great enough to cause a significant recoil of the much heavier daughter nucleus and hence, its contribution to kinetic energy of the products, for the precision of values given here, can be neglected. Thus the neutrino emitted during the decay of nitrogen-13 can have an energy from zero up to 1.20 MeV, and the neutrino emitted during the decay of oxygen-15 can have an energy from zero up to 1.73 MeV. On average, about 1.7 MeV of the total energy output is taken away by neutrinos for each loop of the cycle, leaving about 25 MeV available for producing luminosity.^[10]

CNO-II

In a minor branch of the reaction, occurring in the Sun's inner part, the core, just 0.04% of the time, the final reaction shown above does not produce carbon-12 and an alpha particle, but instead produces oxygen-16 and a photon and continues 15
7N→16
8O→17
9F→17
8O→14
7N→15
8O→15
7N:

15 7N	+	1 1H	→	16 8O	+	γ			+	12.13 MeV
16 8O	+	1 1H	→	17 9F	+	γ			+	0.60 MeV
17 9F			→	17 8O	+	e+	+	ν e	+	2.76 MeV	(half-life of 64.49 seconds)
17 8O	+	1 1H	→	14 7N	+	4 2He			+	1.19 MeV
14 7N	+	1 1H	→	15 8O	+	γ			+	7.35 MeV
15 8O			→	15 7N	+	e+	+	ν e	+	2.75 MeV	(half-life of 122.24 seconds)

Like the carbon, nitrogen, and oxygen involved in the main branch, the fluorine produced in the minor branch is merely an intermediate product and at steady state, does not accumulate in the star.

CNO-III

This subdominant branch is significant only for massive stars. The reactions are started when one of the reactions in CNO-II results in fluorine-18 and gamma instead of nitrogen-14 and alpha, and continues 17
8O→18
9F→18
8O→15
7N→16
8O→17
9F→17
8O:

17 8O	+	1 1H	→	18 9F	+	γ			+	5.61 MeV
18 9F			→	18 8O	+	e+	+	ν e	+	1.656 MeV	(half-life of 109.771 minutes)
18 8O	+	1 1H	→	15 7N	+	4 2He			+	3.98 MeV
15 7N	+	1 1H	→	16 8O	+	γ			+	12.13 MeV
16 8O	+	1 1H	→	17 9F	+	γ			+	0.60 MeV
17 9F			→	17 8O	+	e+	+	ν e	+	2.76 MeV	(half-life of 64.49 seconds)

CNO-IV

A proton reacts with a nucleus causing release of an alpha particle.

Like the CNO-III, this branch is also only significant in massive stars. The reactions are started when one of the reactions in CNO-III results in fluorine-19 and gamma instead of nitrogen-15 and alpha, and continues 19
9F→16
8O→17
9F→17
8O→18
9F→18
8O→19
9F:

19 9F	+	1 1H	→	16 8O	+	4 2He			+	8.114 MeV
16 8O	+	1 1H	→	17 9F	+	γ			+	0.60 MeV
17 9F			→	17 8O	+	e+	+	ν e	+	2.76 MeV	(half-life of 64.49 seconds)
17 8O	+	1 1H	→	18 9F	+	γ			+	5.61 MeV
18 9F			→	18 8O	+	e+	+	ν e	+	1.656 MeV	(half-life of 109.771 minutes)
18 8O	+	1 1H	→	19 9F	+	γ			+	7.994 MeV

Hot CNO cycles

Under conditions of higher temperature and pressure, such as those found in novae and x-ray bursts, the rate of proton captures exceeds the rate of beta-decay, pushing the burning to the proton drip line. The essential idea is that a radioactive species will capture a proton more quickly than it can beta decay, opening new nuclear burning pathways that are otherwise inaccessible. Because of the higher temperatures involved, these catalytic cycles are typically referred to as the hot CNO cycles; because the timescales are limited by beta decays instead of proton captures, they are also called the beta-limited CNO cycles.

HCNO-I

The difference between the CNO-I cycle and the HCNO-I cycle is that 13
7N captures a proton instead of decaying, leading to the total sequence 12
6C→13
7N→14
8O→14
7N→15
8O→15
7N→12
6C:

12 6C	+	1 1H	→	13 7N	+	γ			+	1.95 MeV
13 7N	+	1 1H	→	14 8O	+	γ			+	4.63 MeV
14 8O			→	14 7N	+	e+	+	ν e	+	5.14 MeV	(half-life of 70.641 seconds)
14 7N	+	1 1H	→	15 8O	+	γ			+	7.35 MeV
15 8O			→	15 7N	+	e+	+	ν e	+	2.75 MeV	(half-life of 122.24 seconds)
15 7N	+	1 1H	→	12 6C	+	4 2He			+	4.96 MeV

HCNO-II

The notable difference between the CNO-II cycle and the HCNO-II cycle is that 17
9F captures a proton instead of decaying, and helium is produced in a subsequent reaction on 18
9F, leading to the total sequence 15
7N→16
8O→17
9F→18
10Ne→18
9F→15
8O→15
7N:

15 7N	+	1 1H	→	16 8O	+	γ			+	12.13 MeV
16 8O	+	1 1H	→	17 9F	+	γ			+	0.60 MeV
17 9F	+	1 1H	→	18 10Ne	+	γ			+	3.92 MeV
18 10Ne			→	18 9F	+	e+	+	ν e	+	4.44 MeV	(half-life of 1.672 seconds)
18 9F	+	1 1H	→	15 8O	+	4 2He			+	2.88 MeV
15 8O			→	15 7N	+	e+	+	ν e	+	2.75 MeV	(half-life of 122.24 seconds)

HCNO-III

An alternative to the HCNO-II cycle is that 18
9F captures a proton moving towards higher mass and using the same helium production mechanism as the CNO-IV cycle as 18
9F→19
10Ne→19
9F→16
8O→17
9F→18
10Ne→18
9F:

18 9F	+	1 1H	→	19 10Ne	+	γ			+	6.41 MeV
19 10Ne			→	19 9F	+	e+	+	ν e	+	3.32 MeV	(half-life of 17.22 seconds)
19 9F	+	1 1H	→	16 8O	+	4 2He			+	8.11 MeV
16 8O	+	1 1H	→	17 9F	+	γ			+	0.60 MeV
17 9F	+	1 1H	→	18 10Ne	+	γ			+	3.92 MeV
18 10Ne			→	18 9F	+	e+	+	ν e	+	4.44 MeV	(half-life of 1.672 seconds)

Use in astronomy

While the total number of "catalytic" CNO nuclei are conserved in the cycle, in stellar evolution the relative proportions of the nuclei are altered. When the cycle is run to equilibrium, the ratio of the carbon-12/carbon-13 nuclei is driven to 3.5, and nitrogen-14 becomes the most numerous nucleus, regardless of initial composition. During a star's evolution, convective mixing episodes bring material in which the CNO cycle has operated from the star's interior to the surface, altering the observed composition of the star. Red giant stars are observed to have lower carbon-12/carbon-13 and carbon-12/nitrogen-14 ratios than main sequence stars, which is considered to be convincing evidence for the operation of the CNO cycle.^{[citation needed]}

The presence of the heavier elements carbon, nitrogen and oxygen places an upper bound of approximately 150 solar masses on the maximum size of massive stars.^{[citation needed]} It is thought^{[by whom?]} that the "metal-poor" early universe could have had stars, called Population III stars, up to 250 solar masses without interference from the CNO cycle at the beginning of their lifetime.