CNO cycle#CNO-I

Under typical conditions found in stars, catalytic hydrogen burning by the CNO cycles is limited by proton captures. Specifically, the timescale for beta decay of the 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.

The first proposed catalytic cycle for the conversion of hydrogen into helium was initially called the carbon–nitrogen cycle (CN-cycle), also referred to as the Bethe–Weizsäcker cycle in honor of the independent work of Carl Friedrich von Weizsäcker in 1937–38 and Hans Bethe. Bethe's 1939 papers on the CN-cycle{{cite journal

|last = Bethe |first = Hans A. |author-link = Hans Bethe

|year = 1939

|title = Energy production in stars

|journal = Physical Review

|volume = 55 |issue = 5 |pages = 434–456

|doi = 10.1103/PhysRev.55.434 |doi-access = free

|pmid = 17835673 |bibcode = 1939PhRv...55..434B

}} drew on three earlier papers written in collaboration with Robert Bacher and Milton Stanley Livingston{{cite journal

|last1 = Bethe |first1 = Hans A. |author1-link = Hans Bethe

|last2 = Bacher |first2 = Robert |author2-link = Robert Bacher

|year = 1936

|title = Nuclear Physics, A: Stationary states of nuclei

|journal = Reviews of Modern Physics

|volume = 8 |issue = 2 |pages = 82–229

|doi = 10.1103/RevModPhys.8.82

|bibcode = 1936RvMP....8...82B

|url = https://authors.library.caltech.edu/51288/1/RevModPhys.8.82.pdf

}}{{cite journal

|last = Bethe |first = Hans A. |author-link = Hans Bethe

|year = 1937

|title = Nuclear Physics, B: Nuclear dynamics, theoretical

|journal = Reviews of Modern Physics

|volume = 9 |issue = 2 |pages = 69–244

|doi= 10.1103/RevModPhys.9.69 |bibcode = 1937RvMP....9...69B

}}{{cite journal

|last1 = Bethe |first1 = Hans A. |author1-link = Hans Bethe

|last2 = Livingston |first2 = Milton S. |author2-link = Milton Stanley Livingston

|year = 1937

|title = Nuclear Physics, C: Nuclear Dynamics, Experimental

|journal = Reviews of Modern Physics

|volume = 9 |issue = 2 |pages = 245–390

|doi = 10.1103/RevModPhys.9.245 |bibcode = 1937RvMP....9..245L

}} and which came to be known informally as Bethe's Bible. It was considered the standard work on nuclear physics for many years and was a significant factor in his being awarded the 1967 Nobel Prize in Physics.{{cite magazine

|first = Jason Socrates |last = Bardi

|date = 23 January 2008

|title = Landmarks: What makes the stars shine?

|magazine = Physical Review Focus

|volume = 21 |issue = 3

|doi = 10.1103/physrevfocus.21.3

|url = https://physics.aps.org/story/v21/st3

|access-date = 26 November 2018

}} Bethe's original calculations suggested the CN-cycle was the Sun's primary source of energy. This conclusion arose from a belief that is now known to be mistaken, that the abundance of nitrogen in the sun is approximately 10%; it is actually less than half a percent. The CN-cycle, named as it contains no stable isotope of oxygen, involves the following cycle of transformations:

{{Cite book

|last=Krane |first=Kenneth S.

|year=1988

|title=Introductory Nuclear Physics

|page=[https://archive.org/details/introductorynucl00kran/page/n559 537]

|publisher=John Wiley & Sons

|isbn=0-471-80553-X

|url=https://archive.org/details/introductorynucl00kran

|url-access=limited

}}

: {{nuclide|carbon|12|link=yes}}   →  {{nuclide|nitrogen|13|link=yes}}   →  {{nuclide|carbon|13|link=yes}}   →   {{nuclide|nitrogen|14|link=yes}}   →   {{nuclide|oxygen|15|link=yes}}   →   {{nuclide|nitrogen|15|link=yes}}   →   {{nuclide|carbon|12}}

This cycle is now understood as being the first part of a larger process, the CNO-cycle, and the main reactions in this part of the cycle (CNO-I) are:

:

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 {{val|2.04|u=MeV}}, the total energy released in one cycle is 26.73 MeV; 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.{{cite web

|last1=Wapstra

|first1=Aaldert

|last2=Audi

|first2=Georges

|date=18 November 2003

|title=The 2003 Atomic Mass Evaluation

|publisher=Atomic Mass Data Center

|url=http://amdc.in2p3.fr/web/masseval.html

|access-date=25 October 2011

|archive-date=28 September 2011

|archive-url=https://web.archive.org/web/20110928003450/http://amdc.in2p3.fr/web/masseval.html

|url-status=dead

}}

The limiting (slowest) reaction in the CNO-I cycle is the proton capture on {{nuclide|nitrogen|14}}. In 2006 it was experimentally measured down to stellar energies, revising the calculated age of globular clusters by around 1 billion years.

{{cite journal

| collaboration=LUNA Collaboration

| last1=Lemut | first1=A. | last2=Bemmerer | first2=D.

| last3=Confortola | first3=F. | last4=Bonetti | first4=R.

| last5=Broggini |first5=C. | last6=Corvisiero | first6=P.

| last7=Costantini |first7=H. | last8=Cruz | first8=J.

| last9=Formicola |first9=A. | last10=Fülöp | first10=Zs.

| last11=Gervino | first11=G. | last12=Guglielmetti | first12=A.

| last13=Gustavino | first13=C. | last14=Gyürky | first14=Gy.

| last15=Imbriani | first15=G. | last16=Jesus A. | first16=P.

| last17=Junker | first17=M. | last18=Limata | first18=B.

| last19=Menegazzo | first19=R. | last20=Prati | first20=P.

| last21=Roca | first21=V. | last22=Rogalla | first22=D.

| last23=Rolfs | first23=C. | last24=Romano | first24=M.

| last25=Rossi Alvarez | first25=C. | last26=Schümann | first26=F.

| last27=Somorjai | first27=E. | last28=Straniero | first28=O.

| last29=Strieder | first29=F. | last30=Terrasi | first30=F.

| last31=Trautvetter | first31=H.P. | display-authors=6

| year=2006

| title=First measurement of the {{sup|14}}N(p,γ){{sup|15}}O cross section down to 70 keV

| journal=Physics Letters B

| volume=634 | issue=5–6 | pages=483–487

| bibcode=2006PhLB..634..483L | arxiv = nucl-ex/0602012

| s2cid=16875233 | doi=10.1016/j.physletb.2006.02.021

}}

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 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. The total momentum received by the positron and the neutrino is not great enough to cause a significant recoil of the much heavier daughter nucleus{{efn| Note: It is not important how invariant masses of e and ν are small, because they are already small enough to become relativistic. What is important is that the daughter nucleus is heavy compared to {{frac|p|c}} .}} 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 {{val|1.20|u=MeV}}, and the neutrino emitted during the decay of oxygen-15 can have an energy from zero up to {{val|1.73|u=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 {{val|25|u=MeV}} available for producing luminosity.

{{cite book

| last1 = Scheffler | first1 = Helmut

| last2 = Elsässer | first2 = Hans

| year = 1990

| title = Die Physik der Sterne und der Sonne

| trans-title = The Physics of the Stars and the Sun

| publisher = Bibliographisches Institut (Mannheim, Wien, Zürich)

| isbn = 3-411-14172-7

}}

In a minor branch of the above reaction, occurring in the Sun's core 0.04% of the time, the final reaction involving {{nuclide|nitrogen|15|link=yes}} shown above does not produce carbon-12 and an alpha particle, but instead produces oxygen-16 and a photon and continues

:{{Separated entries

|{{nuclide|nitrogen|15|link=yes}}|{{nuclide|oxygen|16|link=yes}}|{{nuclide|fluorine|17|link=yes}}|{{nuclide|oxygen|17|link=yes}}|{{nuclide|nitrogen|14|link=yes}}|{{nuclide|oxygen|15|link=yes}}|{{nuclide|nitrogen|15}}|separator= → }}

In detail:

:

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

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 a photon instead of nitrogen-14 and an alpha particle, and continues

: {{nuclide|oxygen|17|link=yes}} → {{nuclide|fluorine|18|link=yes}} → {{nuclide|oxygen|18|link=yes}} → {{nuclide|nitrogen|15|link=yes}} → {{nuclide|oxygen|16|link=yes}} → {{nuclide|fluorine|17|link=yes}} → {{nuclide|oxygen|17}}

In detail:

:

Image:NuclearReaction.svg

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 a photon instead of nitrogen-15 and an alpha particle, and continues

:{{Separated entries|

{{nuclide|oxygen|18|link=yes}}|{{nuclide|fluorine|19|link=yes}}|{{nuclide|oxygen|16|link=yes}}|{{nuclide|fluorine|17|link=yes}}|{{nuclide|oxygen|17|link=yes}}|{{nuclide|fluorine|18|link=yes}}|{{nuclide|oxygen|18|link=yes}}|separator= → }}

In detail:

:

In some instances {{nuclide|fluorine|18}} can combine with a helium nucleus to start a neon-sodium cycle, in which:{{cite web |url = https://core.ac.uk/download/pdf/31144835.pdf |title = The neon-sodium cycle: Study of the 22Ne(p, γ)23Na reaction at astrophysical energies |first = Rosanna |last = Depalo |access-date = 16 April 2020 |archive-date = 28 July 2020 |archive-url = https://web.archive.org/web/20200728182717/https://core.ac.uk/download/pdf/31144835.pdf |url-status = dead }}{{Cite journal |last=Depalo |first=R |last2=LUNA collaboration |date=2016-01-05 |title=Towards a study of 22 Ne(p γ ) 23 Na at LUNA |url=https://iopscience.iop.org/article/10.1088/1742-6596/665/1/012017 |journal=Journal of Physics: Conference Series |volume=665 |pages=012017 |doi=10.1088/1742-6596/665/1/012017 |issn=1742-6588|hdl=2434/931202 |hdl-access=free }}

{{Separated entries|

{{nuclide|neon|20|link=yes}}|{{nuclide|sodium|21|link=yes}}|{{nuclide|neon|21|link=yes}}|{{nuclide|sodium|22|link=yes}}|{{nuclide|neon|22|link=yes}}|{{nuclide|sodium|23|link=yes}}|{{nuclide|neon|20|link=yes}}|separator= → }}

The sodium-23 can also turn into magesium-24 after proton bombardment, initiating the magnesium-aluminum cycle.

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 before 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.{{clarify|date=August 2015}}

The difference between the CNO-I cycle and the HCNO-I cycle is that {{nuclide|nitrogen|13|link=yes}} captures a proton instead of decaying, leading to the total sequence

:{{nuclide|carbon|12|link=yes}}→{{nuclide|nitrogen|13|link=yes}}→{{nuclide|oxygen|14|link=yes}}→{{nuclide|nitrogen|14|link=yes}}→{{nuclide|oxygen|15|link=yes}}→{{nuclide|nitrogen|15|link=yes}}→{{nuclide|carbon|12}}

In detail:

:

The notable difference between the CNO-II cycle and the HCNO-II cycle is that {{nuclide|fluorine|17|link=yes}} captures a proton instead of decaying, and neon is produced in a subsequent reaction on {{nuclide|fluorine|18|link=yes}}, leading to the total sequence

:{{nuclide|nitrogen|15|link=yes}}→{{nuclide|oxygen|16|link=yes}}→{{nuclide|fluorine|17|link=yes}}→{{nuclide|neon|18|link=yes}}→{{nuclide|fluorine|18|link=yes}}→{{nuclide|oxygen|15|link=yes}}→{{nuclide|nitrogen|15}}

In detail:

:

An alternative to the HCNO-II cycle is that {{nuclide|fluorine|18|link=yes}} captures a proton moving towards higher mass and using the same helium production mechanism as the CNO-IV cycle as

:{{nuclide|fluorine|18}}→{{nuclide|neon|19|link=yes}}→{{nuclide|fluorine|19|link=yes}}→{{nuclide|oxygen|16|link=yes}}→{{nuclide|fluorine|17|link=yes}}→{{nuclide|neon|18|link=yes}}→{{nuclide|fluorine|18}}

In detail:

:

While the total number of "catalytic" 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 moves material, within 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 do main sequence stars, which is considered to be convincing evidence for the operation of the CNO cycle.{{cite journal |last1=Marks and Sarna |title=The chemical evolution of the secondary stars in close binaries, arising from common-envelope evolution and nova outbursts |journal=Monthly Notices of the Royal Astronomical Society |date=December 1998 |volume=301 |issue=3 |pages=699–720 |doi=10.1046/j.1365-8711.1998.02039.x |doi-access=free |bibcode=1998MNRAS.301..699M }}

• Aneutronic fusion
• Cold fusion
• Fusion power
• Nuclear fusion
• Proton–proton chain, as found in stars like the Sun
• Stellar nucleosynthesis, the whole topic
• Triple-alpha process, how {{SimpleNuclide|carbon|12}} is produced from lighter nuclei

{{notelist}}

{{reflist}}

• {{cite journal

|last=Bethe |first=H. A.

|year=1939

|title=Energy Production in Stars

|journal=Physical Review

|volume=55 |issue=5 |pages=434–56

|bibcode=1939PhRv...55..434B

|doi= 10.1103/PhysRev.55.434

|pmid=17835673

|doi-access=free

}}

• {{cite journal

|last=Iben |first=I.

|year=1967

|title=Stellar Evolution Within and off the Main Sequence

|journal=Annual Review of Astronomy and Astrophysics

|volume=5 |pages=571–626

|bibcode=1967ARA&A...5..571I

|doi=10.1146/annurev.aa.05.090167.003035

}}

{{Nuclear processes}}

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Category:Nuclear fusion reactions

Category:Carbon

Category:Nitrogen

Category:Oxygen

Category:Fluorine

Category:Neon

Category:Nucleosynthesis

Category:Hans Bethe