View source for Nuclear fusion

{{Short description|Process of combining atomic nuclei}}
{{hatnote group|
{{About|the nuclear reaction|its use in producing energy|Fusion power|the journal|Nuclear Fusion (journal){{!}}''Nuclear Fusion'' (journal)|the song|Nuclear Fusion (song)}}
{{Distinguish|Nuclear fission}}
}}
{{Use dmy dates|date=December 2022}}
{{Use American English|date=December 2022}}
{{multiple image
| align             = right
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| image1            = The Sun in white light.jpg
| caption1          = The [[Sun]] is a [[main-sequence star]], and thus releases its [[energy]] by nuclear fusion of [[hydrogen]] nuclei into [[helium]]. In its core, the Sun fuses about 600 billion kilograms of hydrogen each second.<ref>{{Cite web |year=2012 |title=Ask Us: Sun |url=https://helios.gsfc.nasa.gov/qa_sun.html |url-status=dead |archive-url=https://web.archive.org/web/20180903223810/https://helios.gsfc.nasa.gov/qa_sun.html |archive-date=3 September 2018 |access-date=13 July 2017 |website=Cosmicopia |publisher=NASA}}</ref>
| image2            = ITER_Tokamak_and_Plant_Systems_(2016)_(41783636452).jpg
| caption2          = [[ITER]] is a collaboration of 33 countries researching fusion for eventual electricity generation. It is expected to complete assembly in 2033.
}}
[[File:Binding energy curve - common isotopes.svg|thumb|400 px|The [[nuclear binding energy]] curve. The formation of nuclei with masses up to [[iron-56]] releases energy, as illustrated above. ]]
{{Nuclear physics}}

'''Nuclear fusion''' is a [[nuclear reaction|reaction]] in which two or more [[atomic nuclei]], usually [[deuterium]] and [[tritium]] (hydrogen [[isotopes]]), combine to form one or more different atomic nuclei and [[subatomic particles]] ([[neutrons]] or [[protons]]). The difference in mass between the reactants and products is manifested as either the release or [[absorption (electromagnetic radiation)|absorption]] of [[energy]]. This difference in mass arises due to the difference in [[nuclear binding energy]] between the atomic nuclei before and after the reaction. Nuclear fusion is the process that powers active or [[main-sequence star]]s and other [[Absolute magnitude|high-magnitude]] stars, where [[Proton–proton chain reaction|large amounts of energy are released]].

A nuclear fusion process that produces atomic nuclei lighter than [[iron-56]] or [[nickel-62]] will generally release energy. These elements have a relatively small mass and a relatively large [[binding energy]] per [[nucleon]]. Fusion of nuclei lighter than these releases energy (an [[exothermic]] process), while the fusion of heavier nuclei results in energy retained by the product nucleons, and the resulting reaction is [[endothermic]]. The opposite is true for the reverse process, called [[nuclear fission]]. Nuclear fusion uses lighter elements, such as hydrogen and [[helium]], which are in general more fusible; while the heavier elements, such as [[uranium]], [[thorium]] and [[plutonium]], are more fissionable. The extreme [[astrophysical]] event of a [[supernova]] can produce enough energy to fuse nuclei into [[Chemical element|elements]] heavier than iron.

==History==
{{main|Timeline of nuclear fusion}}
American chemist [[William Draper Harkins]] was the first to propose the concept of nuclear fusion in 1915.<ref name=":0">{{Cite journal
|author=Robert S. Mulliken
|author-link=Robert S. Mulliken
|year=1975
|url=http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/harkins-william-d.pdf
|title=William Draper Harkins 1873 - 1951
|journal=Biographical Memoirs
|volume=46
|pages=47–80
|publisher=National Academy of Sciences
|access-date=23 August 2023
|archive-date=10 May 2017
|archive-url=https://web.archive.org/web/20170510115525/http://www.nasonline.org/publications/biographical-memoirs/memoir-pdfs/harkins-william-d.pdf
|url-status=live
}}</ref> Then in 1921, [[Arthur Eddington]] suggested [[Proton–proton chain|hydrogen–helium fusion]] could be the primary source of stellar energy.<ref>{{cite journal |last1=Eddington |first1=A.S. |title=The internal constitution of the stars |journal=Nature |date=2 September 1920 |volume=106 |issue=2653 |pages=14–20 |doi=10.1038/106014a0 |bibcode=1920Natur.106...14E |s2cid=36422819 |url=https://zenodo.org/record/1429642 |doi-access=free |access-date=25 March 2020 |archive-date=17 July 2022 |archive-url=https://web.archive.org/web/20220717115105/https://zenodo.org/record/1429642 |url-status=live }} 
* Reprinted in: {{cite journal |last1=Eddington |first1=A.S. |title=The internal constitution of the stars |journal=The Scientific Monthly |date=October 1920 |volume=11 |issue=4 |pages=297–303 |bibcode=1920SciMo..11..297E |url=https://babel.hathitrust.org/cgi/pt?id=hvd.32044102928264&view=1up&seq=313&skin=2021 |access-date=17 August 2022 |archive-date=17 August 2022 |archive-url=https://web.archive.org/web/20220817051553/https://babel.hathitrust.org/cgi/pt?id=hvd.32044102928264&view=1up&seq=313&skin=2021 |url-status=live }} 
* Reprinted in: {{cite journal |last1=Eddington |first1=A.S. |title=The internal constitution of the stars |journal=The Observatory |date=October 1920 |volume=43 |issue=557 |pages=341–358 |bibcode=1920Obs....43..341E |url=https://babel.hathitrust.org/cgi/pt?id=mdp.39015011432153&view=1up&seq=347&skin=2021 |access-date=17 August 2022 |archive-date=17 August 2022 |archive-url=https://web.archive.org/web/20220817060446/https://babel.hathitrust.org/cgi/pt?id=mdp.39015011432153&view=1up&seq=347&skin=2021 |url-status=live }}</ref> [[Quantum tunneling]] was discovered by [[Friedrich Hund]] in 1927,<ref>{{cite journal |last1=Hund |first1=F. |title=Zur Deutung der Molekelspektren. I. |journal=Zeitschrift für Physik |date=October 1927 |volume=40 |issue=10 |pages=742–764 |doi=10.1007/BF01400234 |bibcode=1927ZPhy...40..742H |s2cid=186239503 |trans-title=On the explanation of molecular spectra I. |language=German}}</ref><ref>Tunnelling was independently observed by Soviet scientists [[Grigory Samuilovich Landsberg]] and [[Leonid Isaakovich Mandelstam]]. See: 
* {{cite journal |last1=Ландсберг |first1=Г.С. |last2=Мандельштам |first2=Л.И. |title=Новое явление в рассеянии света (предварительный отчет) |journal=Журнал Русского физико-химического общества, Раздел физики [Journal of the Russian Physico-Chemical Society, Physics Section] |year=1928 |volume=60 |page=335 |trans-title=A new phenomenon in the scattering of light (preliminary report) |language=Russian}}
* {{cite journal |last1=Landsberg |first1=G. |last2=Mandelstam |first2=L. |title=Eine neue Erscheinung bei der Lichtzerstreuung in Krystallen |journal=Die Naturwissenschaften |year=1928 |volume=16 |issue=28 |pages=557–558 |doi=10.1007/BF01506807 |bibcode=1928NW.....16..557. |s2cid=22492141 |trans-title=A new phenomenon in the case of the scattering of light in crystals |language=German}} 
* {{cite journal |last1=Landsberg |first1=G.S. |last2=Mandelstam |first2=L.I. |title=Über die Lichtzerstreuung in Kristallen |journal=Zeitschrift für Physik |year=1928 |volume=50 |issue=11–12 |pages=769–780 |doi=10.1007/BF01339412 |bibcode=1928ZPhy...50..769L |s2cid=119357805 |trans-title=On the scattering of light in crystals |language=German}}</ref> and shortly afterwards [[Robert d'Escourt Atkinson|Robert Atkinson]] and [[Fritz Houtermans]] used the measured [[Atomic mass|masses]] of light elements to demonstrate that large amounts of energy could be released by fusing small nuclei.<ref>{{cite journal |last1=Atkinson |first1=R. d'E. |last2=Houtermans |first2=F. G. |title=Zur Frage der Aufbaumöglichkeit der Elemente in Sternen |journal=Zeitschrift für Physik |year=1929 |volume=54 |issue=9–10 |pages=656–665 |doi=10.1007/BF01341595 |bibcode=1929ZPhy...54..656A |s2cid=123658609 |trans-title=On the question of the possibility of forming elements in stars |language=German}}</ref> Building on the early experiments in artificial [[nuclear transmutation]] by [[Patrick Blackett]], laboratory fusion of [[Isotopes of hydrogen|hydrogen isotopes]] was accomplished by [[Mark Oliphant]] in 1932.<ref>{{cite journal |last1=Oliphant |first1=M.L.E. |last2=Harteck |first2=P. |last3=Rutherford |first3=E. |title=100 kilovolt discharges in deuterium plasmas |journal=Proceedings of the Royal Society A |year=1934 |volume=144 |pages=692–714}}</ref> In the remainder of that decade, the theory of the main cycle of nuclear fusion in stars was worked out by [[Hans Bethe]]. 

Research into [[Thermonuclear weapon|fusion for military purposes]] began in the early 1940s as part of the [[Manhattan Project]]. The first artificial thermonuclear fusion reaction occurred during the 1951 [[Greenhouse Item]] test of the first [[boosted fission weapon]], which uses a small amount of [[Deuterium–tritium fusion|deuterium–tritium]] gas to enhance the fission yield. The first [[thermonuclear weapon]] detonation, where the vast majority of the yield comes from fusion, was the 1952 [[Ivy Mike]] test of a [[Liquid hydrogen|liquid]] [[Deuterium fusion|deuterium-fusing]] device.

While fusion bomb detonations were [[Project PACER#Development|loosely considered for energy production]], the possibility of controlled and sustained reactions remained the scientific focus for peaceful fusion power. Research into developing controlled fusion inside [[fusion reactors]] has been ongoing since the 1930s, with [[Los Alamos National Laboratory]]'s Scylla I device producing the first laboratory thermonuclear fusion in 1958, but the technology is still in its developmental phase.<ref>{{Cite news |last=Videmšek |first=Boštjan |date=30 May 2022 |title=Nuclear fusion could give the world a limitless source of clean energy. We're closer than ever to it |publisher=CNN |url=https://www.cnn.com/interactive/2022/05/world/iter-nuclear-fusion-climate-intl-cnnphotos/ |access-date=13 December 2022 |archive-date=13 December 2022 |archive-url=https://web.archive.org/web/20221213004108/https://www.cnn.com/interactive/2022/05/world/iter-nuclear-fusion-climate-intl-cnnphotos/ |url-status=live }}</ref>

The US [[National Ignition Facility]], which uses laser-driven [[inertial confinement fusion]], was designed with a goal of [[break-even]] fusion; the first large-scale laser target experiments were performed in June 2009 and ignition experiments began in early 2011.<ref name="programsNIF">{{cite journal |author=Moses, E. I. |year=2009 |title=The National Ignition Facility: Ushering in a new age for high energy density science |url=https://zenodo.org/record/1232045 |journal=Physics of Plasmas |volume=16 |issue=4 |pages=041006 |bibcode=2009PhPl...16d1006M |doi=10.1063/1.3116505 |access-date=25 March 2020 |archive-date=12 August 2020 |archive-url=https://web.archive.org/web/20200812160458/https://zenodo.org/record/1232045 |url-status=live }}</ref><ref>{{cite journal |author=Kramer, David |date=March 2011 |title=DOE looks again at inertial fusion as potential clean-energy source |journal=Physics Today |volume=64 |issue=3 |pages=26–28 |bibcode=2011PhT....64c..26K |doi=10.1063/1.3563814}}</ref>  On 13 December 2022, the [[United States Department of Energy]] announced that on 5 December 2022, they had successfully accomplished break-even fusion, "delivering&nbsp;2.05&nbsp;megajoules (MJ) of energy to the target, resulting in&nbsp;3.15 MJ of fusion energy output."<ref>{{cite web |title=DOE National Laboratory Makes History by Achieving Fusion Ignition |url=https://www.energy.gov/articles/doe-national-laboratory-makes-history-achieving-fusion-ignition |access-date=13 December 2022 |archive-date=19 February 2023 |archive-url=https://web.archive.org/web/20230219060607/https://www.energy.gov/articles/doe-national-laboratory-makes-history-achieving-fusion-ignition |url-status=live }}</ref>

Prior to this breakthrough, controlled fusion reactions had been unable to produce break-even (self-sustaining) controlled fusion.<ref>{{cite web |title=Progress in Fusion |url=http://www.iter.org/sci/beyonditer |access-date=15 February 2010 |publisher=[[ITER]] |archive-date=1 June 2010 |archive-url=https://web.archive.org/web/20100601070234/http://www.iter.org/sci/BeyondITER |url-status=live }}</ref> The two most advanced approaches for it are [[Magnetic confinement fusion|magnetic confinement]] (toroid designs) and inertial confinement (laser designs). Workable designs for a toroidal reactor that theoretically will deliver ten times more fusion energy than the amount needed to heat plasma to the required temperatures are in development (see [[ITER]]). The ITER facility is expected to finish its construction phase in 2025. It will start commissioning the reactor that same year and initiate plasma experiments in 2025, but is not expected to begin full deuterium–tritium fusion until 2035.<ref>{{cite web |year=2014 |title=ITER – the way to new energy |url=http://www.iter.org/proj/iterandbeyond |url-status=dead |archive-url=https://web.archive.org/web/20120922162049/http://www.iter.org/proj/iterandbeyond |archive-date=22 September 2012 |website=ITER}}</ref>

Private companies pursuing the commercialization of nuclear fusion received $2.6 billion in private funding in 2021 alone, going to many notable startups including but not limited to [[Commonwealth Fusion Systems]], [[Helion Energy|Helion Energy Inc]]., [[General Fusion]], [[TAE Technologies]] Inc. and [[Zap Energy]] Inc.<ref>{{Cite news |date=2022-12-14 |title=Nuclear Fusion Breakthrough Set to Send Billions of Dollars Flowing to Atomic Startups |language=en |work=Bloomberg.com |url=https://www.bloomberg.com/news/articles/2022-12-14/fusion-milestone-draws-billions-to-replicate-power-of-the-stars |access-date=2023-01-10 |archive-date=31 January 2023 |archive-url=https://web.archive.org/web/20230131083723/https://www.bloomberg.com/news/articles/2022-12-14/fusion-milestone-draws-billions-to-replicate-power-of-the-stars |url-status=live |author-last1=Wade|author-first1=Will}}</ref>

One of the most recent breakthroughs to date in maintaining a sustained fusion reaction occurred in France's WEST fusion reactor. It maintained a 90 million degree plasma for a record time of six minutes. This is a tokamak style reactor which is the same style as the upcoming ITER reactor.<ref>{{Cite web |last=McGrath |first=Jenny |date=2024-05-07 |title=Fusion Breakthrough: 6 Minutes of Plasma Sets New Reactor Record |url=https://www.sciencealert.com/fusion-breakthrough-6-minutes-of-plasma-sets-new-reactor-record |access-date=2024-09-27 |website=ScienceAlert |language=en-US}}</ref>

==Process==
[[File:Deuterium-tritium fusion.svg|thumb|Fusion of [[deuterium]] with [[tritium]] creating [[helium-4]], freeing a neutron, and releasing 17.59 [[Electronvolt|MeV]] as [[kinetic energy]] of the products while a corresponding amount of [[Mass–energy equivalence|mass disappears]], in agreement with ''kinetic E'' = ∆''mc''<sup>2</sup>, where Δ''m'' is the decrease in the total rest mass of particles<ref name=Shultis>
{{cite book
 |author1=Shultis, J.K.  |author2=Faw, R.E. 
  |name-list-style=amp |year=2002
 |title=Fundamentals of nuclear science and engineering
 |url=https://books.google.com/books?id=SO4Lmw8XoEMC&pg=PA151
 |page=151
 |publisher=[[CRC Press]]
 |isbn=978-0-8247-0834-4
}}</ref>]]
The release of energy with the fusion of light elements is due to the interplay of two opposing forces: the [[nuclear force]], a manifestation of the [[strong interaction]], which holds protons and neutrons tightly together in the [[atomic nucleus]]; and the [[Coulomb's law|Coulomb force]], which causes positively [[Electric charge|charged]] [[proton]]s in the nucleus to repel each other.<ref>[http://www.ck12.org/flexbook/chapter/1903 Physics Flexbook] {{webarchive|url=https://web.archive.org/web/20111228011150/http://www.ck12.org/flexbook/chapter/1903 |date=28 December 2011 }}. Ck12.org. Retrieved 19 December 2012.</ref> Lighter nuclei (nuclei smaller than iron and nickel) are sufficiently small and proton-poor to allow the nuclear force to overcome the Coulomb force. This is because the nucleus is sufficiently small that all nucleons feel the short-range attractive force at least as strongly as they feel the infinite-range Coulomb repulsion. Building up nuclei from lighter nuclei by fusion releases the extra energy from the net attraction of particles. [[iron peak|For larger nuclei]], however, no energy is released, because the nuclear force is short-range and cannot act across larger nuclei.

Fusion powers [[star]]s and produces virtually all elements in a process called [[nucleosynthesis]]. The Sun is a main-sequence star, and, as such, generates its energy by nuclear fusion of hydrogen nuclei into helium. In its core, the Sun fuses 620&nbsp;million metric tons of hydrogen and makes 616&nbsp;million metric tons of helium each second.<!-- Meant generally; please do not insert info about [[stellar nucleosynthesis]], [[supernova nucleosynthesis]] and other types of specific nucleosynthesis here. --> The fusion of lighter elements in stars releases energy and the mass that always accompanies it. For example, in the fusion of two hydrogen nuclei to form helium, 0.645% of the mass is carried away in the form of kinetic energy of an [[alpha particle]] or other forms of energy, such as electromagnetic radiation.<ref name="bulletin1950">{{cite journal |last=Bethe |first=Hans A. |url=https://books.google.com/books?id=Mg4AAAAAMBAJ&pg=PA99 |title=The Hydrogen Bomb |journal=Bulletin of the Atomic Scientists |date=April 1950 |volume=6 |issue=4 |pages=99–104, 125– |doi=10.1080/00963402.1950.11461231 |bibcode=1950BuAtS...6d..99B |access-date=14 September 2018 |archive-date=14 January 2023 |archive-url=https://web.archive.org/web/20230114063514/https://books.google.com/books?id=Mg4AAAAAMBAJ&pg=PA99 |url-status=live }}</ref>

It takes considerable energy to force nuclei to fuse, even those of the lightest element, [[hydrogen]]. When accelerated to high enough speeds, nuclei can overcome this electrostatic repulsion and be brought close enough such that the attractive [[nuclear force]] is greater than the repulsive Coulomb force. The [[strong force]] grows rapidly once the nuclei are close enough, and the fusing nucleons can essentially "fall" into each other and the result is fusion; this is an [[exothermic reaction|exothermic process]].<ref>{{cite book |last1=Smith |first1=Peter F. |title=Building for a Changing Climate: The Challenge for Construction, Planning and Energy |date=2009 |publisher=Earthscan |isbn=978-1-84977-439-0 |page=129 |url=https://books.google.com/books?id=Dx-ZkpLS4wMC |language=en |access-date=20 June 2023 |archive-date=5 November 2023 |archive-url=https://web.archive.org/web/20231105221740/https://books.google.com/books?id=Dx-ZkpLS4wMC |url-status=live }}</ref>

Energy released in most [[nuclear reaction]]s is much larger than in [[chemical reaction]]s, because the [[binding energy]] that holds a nucleus together is greater than the energy that holds [[electron]]s to a nucleus. For example, the [[ionization energy]] gained by adding an electron to a hydrogen nucleus is {{val|13.6|ul=eV}}—less than one-millionth of the {{val|17.6|ul=MeV}} released in the [[deuterium]]–[[tritium]] (D–T) reaction shown in the adjacent diagram. Fusion reactions have an [[energy density]] many times greater than [[nuclear fission]]; the reactions produce far greater energy per unit of mass even though ''individual'' fission reactions are generally much more energetic than ''individual'' fusion ones, which are themselves millions of times more energetic than chemical reactions. Only direct conversion of [[Mass–energy equivalence|mass into energy]], such as that caused by the [[annihilation|annihilatory]] collision of [[matter]] and [[antimatter]], is more energetic per unit of mass than nuclear fusion. (The complete conversion of one gram of matter would release {{val|9|e=13|u=joules}} of energy.)

==In stars==
[[File:Fusion in the Sun.svg|thumb|250px|right|The [[proton–proton chain]] reaction, branch&nbsp;I, dominates in stars the size of the Sun or smaller.]]
[[File:CNO Cycle.svg|thumb|250px|right|The [[CNO cycle]] dominates in stars heavier than the Sun.]]

An important fusion process is the [[stellar nucleosynthesis]] that powers [[star]]s, including the Sun. In the 20th century, it was recognized that the energy released from nuclear fusion reactions accounts for the longevity of stellar heat and light. The fusion of nuclei in a star, starting from its initial hydrogen and helium abundance, provides that energy and synthesizes new nuclei. Different reaction chains are involved, depending on the mass of the star (and therefore the pressure and temperature in its core).

Around 1920, [[Arthur Eddington]] anticipated the discovery and mechanism of nuclear fusion processes in stars, in his paper ''The Internal Constitution of the Stars''.<ref name=eddington>{{cite journal |title=The Internal Constitution of the Stars |first=A. S. |last=Eddington |journal=The Scientific Monthly |volume=11 |issue=4 |pages=297–303 |date=October 1920 |doi=10.1126/science.52.1341.233 |jstor=6491 |pmid=17747682 |bibcode=1920Sci....52..233E |url=https://zenodo.org/record/1429642 |access-date=25 March 2020 |archive-date=17 July 2022 |archive-url=https://web.archive.org/web/20220717115105/https://zenodo.org/record/1429642 |url-status=live }}</ref><ref name=eddington2>{{cite journal|bibcode=1916MNRAS..77...16E|title=On the radiative equilibrium of the stars|journal=Monthly Notices of the Royal Astronomical Society|volume=77|pages=16–35|last1=Eddington|first1=A. S.|year=1916|doi=10.1093/mnras/77.1.16|doi-access=free}}</ref> At that time, the source of stellar energy was unknown; Eddington correctly speculated that the source was fusion of hydrogen into helium, liberating enormous energy according to [[Mass–energy equivalence|Einstein's equation]] {{math|''E'' {{=}} ''mc''<sup>2</sup>}}. This was a particularly remarkable development since at that time fusion and thermonuclear energy had not yet been discovered, nor even that stars are largely composed of [[hydrogen]] (see [[metallicity]]). Eddington's paper reasoned that:

# The leading theory of stellar energy, the contraction hypothesis, should cause the rotation of a star to visibly speed up due to [[conservation of angular momentum]]. But observations of [[Cepheid]] variable stars showed this was not happening.
# The only other known plausible source of energy was conversion of matter to energy; Einstein had shown some years earlier that a small amount of matter was equivalent to a large amount of energy. 
# [[Francis William Aston|Francis Aston]] had also recently shown that the mass of a helium atom was about 0.8% less than the mass of the four hydrogen atoms which would, combined, form a helium atom (according to the then-prevailing theory of atomic structure which held atomic weight to be the distinguishing property between elements; work by [[Henry Moseley]] and [[Antonius van den Broek]] would later show that nucleic charge was the distinguishing property and that a helium nucleus, therefore, consisted of two hydrogen nuclei plus additional mass). This suggested that if such a combination could happen, it would release considerable energy as a byproduct.
# If a star contained just 5% of fusible hydrogen, it would suffice to explain how stars got their energy. (It is now known that most 'ordinary' stars are usually made of around 70% to 75% hydrogen)
# Further elements might also be fused, and other scientists had speculated that stars were the "crucible" in which light elements combined to create heavy elements, but without more accurate measurements of their [[atomic mass]]es nothing more could be said at the time.

All of these speculations were proven correct in the following decades.

The primary source of solar energy, and that of similar size stars, is the fusion of hydrogen to form helium (the [[proton–proton chain]] reaction), which occurs at a solar-core temperature of 14&nbsp;million kelvin. The net result is the fusion of four [[proton]]s into one [[alpha particle]], with the release of two [[positron]]s and two [[neutrino]]s (which changes two of the protons into neutrons), and energy. In heavier stars, the [[CNO cycle]] and other processes are more important. As a star uses up a substantial fraction of its hydrogen, it begins to synthesize heavier elements. The heaviest elements are synthesized by fusion that occurs when a more massive star undergoes a violent [[supernova]] at the end of its life, a process known as [[supernova nucleosynthesis]].

==Requirements==
{{more citations|date=October 2024}}
A substantial energy barrier of electrostatic forces must be overcome before fusion can occur. At large distances, two naked nuclei repel one another because of the repulsive [[electrostatic force]] between their [[electric charge|positively charged]] protons. If two nuclei can be brought close enough together, however, the electrostatic repulsion can be overcome by the quantum effect in which nuclei can [[quantum tunnelling#Nuclear fusion|tunnel]] through coulomb forces.

When a [[nucleon]] such as a [[proton]] or [[neutron]] is added to a nucleus, the nuclear force attracts it to all the other nucleons of the nucleus (if the atom is small enough), but primarily to its immediate neighbors due to the short range of the force. The nucleons in the interior of a nucleus have more neighboring nucleons than those on the surface. Since smaller nuclei have a larger surface-area-to-volume ratio, the binding energy per nucleon due to the [[nuclear force]] generally increases with the size of the nucleus but approaches a limiting value corresponding to that of a nucleus with a diameter of about four nucleons. It is important to keep in mind that nucleons are [[Quantum physics|quantum objects]]. So, for example, since two neutrons in a nucleus are identical to each other, the goal of distinguishing one from the other, such as which one is in the interior and which is on the surface, is in fact meaningless, and the inclusion of quantum mechanics is therefore necessary for proper calculations.

The electrostatic force, on the other hand, is an [[inverse square law|inverse-square force]], so a proton added to a nucleus will feel an electrostatic repulsion from ''all'' the other protons in the nucleus. The electrostatic energy per nucleon due to the electrostatic force thus increases without limit as nuclei atomic number grows.

[[File:Nuclear fusion forces diagram.svg|left|350px|thumb|The [[electrostatic force]] between the positively charged nuclei is repulsive, but when the separation is small enough, the quantum effect will tunnel through the wall. Therefore, the prerequisite for fusion is that the two nuclei be brought close enough together for a long enough time for quantum tunneling to act.]]

The net result of the opposing electrostatic and strong nuclear forces is that the binding energy per nucleon generally increases with increasing size, up to the elements [[iron]] and [[nickel]], and then decreases for heavier nuclei. Eventually, the [[binding energy]] becomes negative and very heavy nuclei (all with more than 208 nucleons, corresponding to a diameter of about 6 nucleons) are not stable. The four most tightly bound nuclei, in decreasing order of [[binding energy]] per nucleon, are {{SimpleNuclide|link=yes|Nickel|62}}, {{SimpleNuclide|link=yes|Iron|58}}, {{SimpleNuclide|link=yes|Iron|56}}, and {{SimpleNuclide|link=yes|Nickel|60}}.<ref>[http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nucbin2.html#c1 The Most Tightly Bound Nuclei] {{Webarchive|url=https://web.archive.org/web/20110514050922/http://hyperphysics.phy-astr.gsu.edu/hbase/nucene/nucbin2.html#c1 |date=14 May 2011 }}. Hyperphysics.phy-astr.gsu.edu. Retrieved 17 August 2011.</ref> Even though the [[isotopes of nickel|nickel isotope]], {{SimpleNuclide|link=yes|Nickel|62}}, is more stable, the [[isotopes of iron|iron isotope]] {{SimpleNuclide|link=yes|Iron|56}} is an [[order of magnitude]] more common. This is due to the fact that there is no easy way for stars to create {{SimpleNuclide|link=yes|Nickel|62}} through the [[alpha process]].

An exception to this general trend is the [[helium-4]] nucleus, whose binding energy is higher than that of [[lithium]], the next heavier element. This is because protons and neutrons are [[fermion]]s, which according to the [[Pauli exclusion principle]] cannot exist in the same nucleus in exactly the same state. Each proton or neutron's energy state in a nucleus can accommodate both a spin up particle and a spin down particle. Helium-4 has an anomalously large binding energy because its nucleus consists of two protons and two neutrons (it is a [[doubly magic]] nucleus), so all four of its nucleons can be in the ground state. Any additional nucleons would have to go into higher energy states. Indeed, the helium-4 nucleus is so tightly bound that it is commonly treated as a single quantum mechanical particle in nuclear physics, namely, the [[alpha particle]].

The situation is similar if two nuclei are brought together. As they approach each other, all the protons in one nucleus repel all the protons in the other. Not until the two nuclei actually come close enough for long enough so the strong attractive [[nuclear force]] can take over and overcome the repulsive electrostatic force. This can also be described as the nuclei overcoming the so-called [[Coulomb barrier]]. The kinetic energy to achieve this can be lower than the barrier itself because of quantum tunneling.

The [[Coulomb barrier]] is smallest for isotopes of hydrogen, as their nuclei contain only a single positive charge. A [[diproton]] is not stable, so neutrons must also be involved, ideally in such a way that a helium nucleus, with its extremely tight binding, is one of the products.

Using [[Tritium#Deuterium|deuterium–tritium]] fuel, the resulting energy barrier is about 0.1&nbsp;MeV. In comparison, the energy needed to remove an [[electron]] from [[hydrogen]] is 13.6&nbsp;eV. The (intermediate) result of the fusion is an unstable <sup>5</sup>He nucleus, which immediately ejects a neutron with 14.1&nbsp;MeV. The recoil energy of the remaining <sup>4</sup>He nucleus is 3.5&nbsp;MeV, so the total energy liberated is 17.6&nbsp;MeV. This is many times more than what was needed to overcome the energy barrier.

[[File:fusion rxnrate.svg|right|300px|thumb|The fusion reaction rate increases rapidly with temperature until it maximizes and then gradually drops off. The DT rate peaks at a lower temperature (about 70&nbsp;keV, or 800 million kelvin) and at a higher value than other reactions commonly considered for fusion energy.]]

The reaction [[cross section (physics)|cross section]] (σ) is a measure of the probability of a fusion reaction as a function of the relative velocity of the two reactant nuclei. If the reactants have a distribution of velocities, e.g. a thermal distribution, then it is useful to perform an average over the distributions of the product of cross-section and velocity. This average is called the 'reactivity', denoted {{math|{{angbr|''σv''}}}}. The reaction rate (fusions per volume per time) is {{math|{{angbr|''σv''}}}} times the product of the reactant number densities:

:<math>f = n_1 n_2 \langle \sigma v \rangle.</math>

If a species of nuclei is reacting with a nucleus like itself, such as the DD reaction, then the product <math>n_1n_2</math> must be replaced by <math>n^2/2</math>.

<math>\langle \sigma v \rangle</math> increases from virtually zero at room temperatures up to meaningful magnitudes at temperatures of [[1 E-15 J|10]]–[[1 E-14 J|100]]&nbsp;keV. At these temperatures, well above typical [[ion]]ization energies (13.6&nbsp;eV in the hydrogen case), the fusion reactants exist in a [[Plasma physics|plasma]] state.

The significance of <math>\langle \sigma v \rangle</math> as a function of temperature in a device with a particular energy [[confinement time]] is found by considering the [[Lawson criterion]]. This is an extremely challenging barrier to overcome on Earth, which explains why fusion research has taken many years to reach the current advanced technical state.<ref name=lawson>{{Cite web|url=https://www.scienceworldreport.com/articles/5763/20130323/lawson-criteria-make-fusion-power-viable-iter.htm|title=What Is The Lawson Criteria, Or How to Make Fusion Power Viable|first=Science World|last=Report|date=23 March 2013|website=Science World Report|access-date=14 March 2021|archive-date=3 August 2021|archive-url=https://web.archive.org/web/20210803184652/https://www.scienceworldreport.com/articles/5763/20130323/lawson-criteria-make-fusion-power-viable-iter.htm|url-status=live}}</ref>

== Artificial fusion ==
{{main article|Fusion power}}

=== Thermonuclear fusion ===
{{unreferenced section|date=August 2023}}
Thermonuclear fusion is the process of atomic nuclei combining or "fusing" using high temperatures to drive them close enough together for this to become possible. Such temperatures cause the matter to become a [[plasma physics|plasma]] and, if confined, fusion reactions may occur due to collisions with extreme thermal kinetic energies of the particles. There are two forms of thermonuclear fusion: ''uncontrolled'', in which the resulting energy is released in an uncontrolled manner, as it is in [[thermonuclear weapon]]s ("hydrogen bombs") and in most [[star]]s; and ''controlled'', where the fusion reactions take place in an environment allowing some or all of the energy released to be harnessed for constructive purposes.

Temperature is a measure of the average [[kinetic energy]] of particles, so by heating the material it will gain energy. After reaching sufficient temperature, given by the [[Lawson criterion]], the energy of accidental collisions within the [[plasma (physics)|plasma]] is high enough to overcome the [[Coulomb barrier]] and the particles may fuse together.

In a [[Deuterium–tritium fusion|deuterium–tritium fusion reaction]], for example, the energy necessary to overcome the [[Coulomb barrier]] is 0.1&nbsp;[[Electronvolt|MeV]]. Converting between energy and temperature shows that the 0.1 MeV barrier would be overcome at a temperature [[Orders of magnitude (temperature)|in excess of 1.2 billion]] [[kelvin]].

There are two effects that are needed to lower the actual temperature. One is the fact that [[temperature]] is the ''average'' kinetic energy, implying that some nuclei at this temperature would actually have much higher energy than 0.1&nbsp;MeV, while others would be much lower. It is the nuclei in the high-energy tail of the [[Distribution function (physics)|velocity distribution]] that account for most of the fusion reactions. The other effect is [[quantum tunnelling]]. The nuclei do not actually have to have enough energy to overcome the Coulomb barrier completely. If they have nearly enough energy, they can tunnel through the remaining barrier. For these reasons fuel at lower temperatures will still undergo fusion events, at a lower rate.

''Thermonuclear'' fusion is one of the methods being researched in the attempts to produce [[fusion power]]. If thermonuclear fusion becomes favorable to use, it would significantly reduce the world's [[carbon footprint]].

===Beam–beam or beam–target fusion===
{{main|Colliding beam fusion}}

Accelerator-based light-ion fusion is a technique using [[particle accelerator]]s to achieve particle kinetic energies sufficient to induce light-ion fusion reactions.<ref>{{Cite book |url=https://link.springer.com/book/10.1007/978-3-030-62308-1 |title=Accelerator Technology |series=Particle Acceleration and Detection |year=2020 |language=en |doi=10.1007/978-3-030-62308-1 |isbn=978-3-030-62307-4 |s2cid=229610872 |last1=Möller |first1=Sören |access-date=20 September 2022 |archive-date=23 September 2022 |archive-url=https://web.archive.org/web/20220923225830/https://link.springer.com/book/10.1007/978-3-030-62308-1 |url-status=live }}</ref>

Accelerating light ions is relatively easy, and can be done in an efficient manner—requiring only a vacuum tube, a pair of electrodes, and a high-voltage transformer; fusion can be observed with as little as 10 kV between the electrodes.{{citation needed|date=August 2023}} The system can be arranged to accelerate ions into a static fuel-infused target, known as ''beam–target'' fusion, or by accelerating two streams of ions towards each other, ''beam–beam'' fusion.{{citation needed|date=August 2023}} The key problem with accelerator-based fusion (and with cold targets in general) is that fusion cross sections are many orders of magnitude lower than Coulomb interaction cross-sections. Therefore, the vast majority of ions expend their energy emitting [[bremsstrahlung]] radiation and the ionization of atoms of the target. Devices referred to as sealed-tube [[neutron generator]]s are particularly relevant to this discussion. These small devices are miniature particle accelerators filled with deuterium and tritium gas in an arrangement that allows ions of those nuclei to be accelerated against hydride targets, also containing deuterium and tritium, where fusion takes place, releasing a flux of neutrons. Hundreds of neutron generators are produced annually for use in the petroleum industry where they are used in measurement equipment for locating and mapping oil reserves.{{citation needed|date=August 2023}}

A number of attempts to recirculate the ions that "miss" collisions have been made over the years. One of the better-known attempts in the 1970s was [[Migma]], which used a unique particle [[storage ring]] to capture ions into circular orbits and return them to the reaction area. Theoretical calculations made during funding reviews pointed out that the system would have significant difficulty scaling up to contain enough fusion fuel to be relevant as a power source. In the 1990s, a new arrangement using a [[field-reversed configuration]] (FRC) as the storage system was proposed by [[Norman Rostoker]] and continues to be studied by [[TAE Technologies]] {{as of|2021|lc=yes}}. A closely related approach is to merge two FRC's rotating in opposite directions,<ref>J. Slough, G. Votroubek, and C. Pihl, "Creation of a high-temperature plasma through merging and compression of supersonic field reversed configuration plasmoids" Nucl. Fusion 51,053008 (2011).</ref> which is being actively studied by [[Helion Energy]]. Because these approaches all have ion energies well beyond the [[Coulomb barrier]], they often suggest the use of alternative fuel cycles like p-[[Boron#Depleted boron (boron-11)|<sup>11</sup>B]] that are too difficult to attempt using conventional approaches.<ref>A. Asle Zaeem et al "Aneutronic Fusion in Collision of Oppositely Directed Plasmoids" Plasma Physics Reports, Vol. 44, No. 3, pp. 378–386 (2018).</ref>

===Muon-catalyzed fusion===
[[Muon-catalyzed fusion]] is a fusion process that occurs at ordinary temperatures. It was studied in detail by [[Steven E. Jones|Steven Jones]] in the early 1980s. Net energy production from this reaction has been unsuccessful because of the high energy required to create [[muon]]s, their short 2.2&nbsp;μs [[half-life]], and the high chance that a muon will bind to the new [[alpha particle]] and thus stop catalyzing fusion.<ref>{{cite journal |author=Jones, S.E. |title=Muon-Catalysed Fusion Revisited |journal=Nature |volume=321 |pages=127–133 |year=1986 |doi=10.1038/321127a0|bibcode = 1986Natur.321..127J |issue=6066|s2cid=39819102 }}</ref>

===Other principles===
[[File:TCV vue gen.jpg|thumb|The ''[[Tokamak à configuration variable]]'', research fusion reactor, at the [[École Polytechnique Fédérale de Lausanne]] (Switzerland)]]

Some other confinement principles have been investigated.

* [[Antimatter catalyzed nuclear pulse propulsion|Antimatter-initialized fusion]] uses small amounts of [[antimatter]] to trigger a tiny fusion explosion. This has been studied primarily in the context of making [[nuclear pulse propulsion]], and [[pure fusion bomb]]s feasible. This is not near becoming a practical power source, due to the cost of manufacturing antimatter alone.
* [[Pyroelectric fusion]] was reported in April 2005 by a team at [[University of California, Los Angeles|UCLA]]. The scientists used a [[pyroelectricity|pyroelectric]] crystal heated from {{convert|-34|to|7|C|F}}, combined with a [[tungsten]] needle to produce an [[electric field]] of about 25&nbsp;gigavolts per meter to ionize and accelerate [[deuterium]] nuclei into an [[erbium]] deuteride target. At the estimated energy levels,<ref>[http://www.nature.com/nature/journal/v434/n7037/suppinfo/nature03575.html Supplementary methods for "Observation of nuclear fusion driven by a pyroelectric crystal"] {{Webarchive|url=https://web.archive.org/web/20170204193327/http://www.nature.com/nature/journal/v434/n7037/suppinfo/nature03575.html |date=4 February 2017 }}. Main article {{cite journal|doi=10.1038/nature03575|title=Observation of nuclear fusion driven by a pyroelectric crystal|year=2005|last1=Naranjo|first1=B.|last2=Gimzewski|first2=J.K.|last3=Putterman|first3=S.|journal=Nature|volume=434|issue=7037|pages=1115–1117|pmid=15858570|bibcode = 2005Natur.434.1115N |s2cid=4407334}}</ref> the D–D fusion reaction may occur, producing [[helium-3]] and a 2.45&nbsp;MeV [[neutron]]. Although it makes a useful neutron generator, the apparatus is not intended for power generation since it requires far more energy than it produces.<ref>[http://rodan.physics.ucla.edu/pyrofusion/ UCLA Crystal Fusion]. Rodan.physics.ucla.edu. Retrieved 17 August 2011. {{webarchive |url=https://web.archive.org/web/20150608184332/http://rodan.physics.ucla.edu/pyrofusion/ |date=8 June 2015 }}</ref><ref>{{cite journal |url=http://www.aip.org/pnu/2005/split/729-1.html |title=Pyrofusion: A Room-Temperature, Palm-Sized Nuclear Fusion Device |author1=Schewe, Phil |author2=Stein, Ben |name-list-style=amp |journal=Physics News Update |volume=729 |issue=1 |year=2005 |url-status=dead |archive-url=https://web.archive.org/web/20131112155702/http://www.aip.org/pnu/2005/split/729-1.html |archive-date=12 November 2013 |accessdate=3 May 2006 }}</ref><ref>[http://www.csmonitor.com/2005/0606/p25s01-stss.html Coming in out of the cold: nuclear fusion, for real] {{Webarchive|url=https://web.archive.org/web/20120122115832/http://www.csmonitor.com/2005/0606/p25s01-stss.html |date=22 January 2012 }}. ''The Christian Science Monitor''. (6 June 2005). Retrieved 17 August 2011.</ref><ref>[http://www.nbcnews.com/id/7654627 Nuclear fusion on the desktop ... really!] {{Webarchive|url=https://web.archive.org/web/20160904023602/http://www.nbcnews.com/id/7654627/ |date=4 September 2016 }}. MSNBC (27 April 2005). Retrieved 17 August 2011.</ref> D–T fusion reactions have been observed with a tritiated erbium target.<ref>{{cite journal|doi=10.1016/j.nima.2010.08.003|title=Pyroelectric fusion using a tritiated target|journal=Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment|volume=632|issue=1|pages=43–46|year=2011|last1=Naranjo|first1=B.|last2=Putterman|first2=S.|last3=Venhaus|first3=T.|bibcode=2011NIMPA.632...43N}}</ref>
* [[Nuclear fusion–fission hybrid]] (hybrid nuclear power) is a proposed means of generating [[Electrical power industry|power]] by use of a combination of nuclear fusion and [[Nuclear fission|fission]] processes. The concept dates to the 1950s, and was briefly advocated by [[Hans Bethe]] during the 1970s, but largely remained unexplored until a revival of interest in 2009, due to the delays in the realization of pure fusion.<ref name="hybrid">{{cite journal | author = Gerstner, E. | title = Nuclear energy: The hybrid returns | year = 2009 | journal = [[Nature (journal)|Nature]] | volume = 460 | issue = 7251| pages = 25–28 | pmid = 19571861|doi=10.1038/460025a| doi-access = free }}</ref>
* [[Project PACER]], carried out at [[Los Alamos National Laboratory]] (LANL) in the mid-1970s, explored the possibility of a fusion power system that would involve exploding small [[H-bomb|hydrogen bomb]]s (fusion bombs) inside an underground cavity. As an energy source, the system is the only fusion power system that could be demonstrated to work using existing technology. However, it would also require a large, continuous supply of nuclear bombs, making the economics of such a system rather questionable.
* [[Bubble fusion]] also called '''sonofusion''' was a proposed mechanism for achieving fusion via [[sonic cavitation]] which rose to prominence in the early 2000s. Subsequent attempts at replication failed and the principal investigator, [[Rusi Taleyarkhan]], was judged guilty of [[research misconduct]] in 2008.<ref>{{cite news |last1=Maugh II |first1=Thomas |title=Physicist is found guilty of misconduct |url=https://www.latimes.com/archives/la-xpm-2008-jul-19-sci-misconduct19-story.html |access-date=17 April 2019 |work=Los Angeles Times |archive-date=17 April 2019 |archive-url=https://web.archive.org/web/20190417184641/https://www.latimes.com/archives/la-xpm-2008-jul-19-sci-misconduct19-story.html |url-status=live }}</ref>

==Confinement in thermonuclear fusion==
The key problem in achieving thermonuclear fusion is how to confine the hot plasma. Due to the high temperature, the plasma cannot be in direct contact with any solid material, so it has to be located in a [[vacuum]]. Also, high temperatures imply high pressures. The plasma tends to expand immediately and some force is necessary to act against it. This force can take one of three forms: gravitation in stars, magnetic forces in magnetic confinement fusion reactors, or [[inertia]]l as the fusion reaction may occur before the plasma starts to expand, so the plasma's inertia is keeping the material together.

=== Gravitational confinement ===
{{main|Stellar nucleosynthesis}}{{unreferenced section|date=August 2023}}
One force capable of confining the fuel well enough to satisfy the [[Lawson criterion]] is [[gravity]]. The mass needed, however, is so great that gravitational confinement is only found in [[star]]s—the least massive stars capable of sustained fusion are [[red dwarf]]s, while [[brown dwarf]]s are able to fuse [[deuterium]] and [[lithium]] if they are of sufficient mass. In stars [[Asymptotic giant branch|heavy enough]], after the supply of hydrogen is exhausted in their cores, their cores (or a shell around the core) start fusing [[triple-alpha process|helium to carbon]]. In the most massive stars (at least 8–11 [[solar mass]]es), the process is continued until some of their energy is produced by [[Silicon burning process|fusing lighter elements to iron]]. As iron has one of the highest [[binding energy#Nuclear binding energy curve|binding energies]], reactions producing heavier elements are generally [[endothermic]]. Therefore, significant amounts of heavier elements are not formed during stable periods of massive star evolution, but are formed in [[R-process|supernova explosions]]. [[s-process|Some lighter stars]] also form these elements in the outer parts of the stars over long periods of time, by absorbing energy from fusion in the inside of the star, by absorbing neutrons that are emitted from the fusion process.

All of the elements heavier than iron have some potential energy to release, in theory. At the extremely heavy end of element production, these heavier elements can [[exothermic|produce energy]] in the process of being split again back toward the size of iron, in the process of [[nuclear fission]]. Nuclear fission thus releases energy that has been stored, sometimes billions of years before, during stellar [[nucleosynthesis]].

=== Magnetic confinement ===
{{main|Magnetic confinement fusion}}
Electrically charged particles (such as fuel ions) will follow [[magnetic field]] lines (see [[Guiding center#Gyration|Guiding centre]]). The fusion fuel can therefore be trapped using a strong magnetic field. A variety of magnetic configurations exist, including the toroidal geometries of [[tokamak]]s and [[stellarator]]s and open-ended mirror confinement systems.

=== Inertial confinement ===
{{main|Inertial confinement fusion}}
A third confinement principle is to apply a rapid pulse of energy to a large part of the surface of a pellet of fusion fuel, causing it to simultaneously "implode" and heat to very high pressure and temperature. If the fuel is dense enough and hot enough, the fusion reaction rate will be high enough to burn a significant fraction of the fuel before it has dissipated. To achieve these extreme conditions, the initially cold fuel must be explosively compressed. Inertial confinement is used in the [[hydrogen bomb]], where the driver is [[x-rays]] created by a fission bomb. Inertial confinement is also attempted in "controlled" nuclear fusion, where the driver is a [[laser]], [[ion]], or [[electron]] beam, or a [[Z-pinch]]. Another method is to use conventional high [[explosive material]] to compress a fuel to fusion conditions.<ref>F. Winterberg "[https://arxiv.org/abs/0802.3408 Conjectured Metastable Super-Explosives formed under High Pressure for Thermonuclear Ignition] {{Webarchive|url=https://web.archive.org/web/20230303045020/https://arxiv.org/abs/0802.3408 |date=3 March 2023 }}"</ref><ref>Zhang, Fan; Murray, Stephen Burke; Higgins, Andrew (2005) "[http://www.wipo.int/pctdb/images4/PCT-PAGES/2006/102006/06024137/06024137.pdf Super compressed detonation method and device to effect such detonation]{{dead link|date=January 2012}}"</ref> The UTIAS explosive-driven-implosion facility was used to produce stable, centred and focused hemispherical implosions<ref>I.I. Glass and J.C. Poinssot "[https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700009349_1970009349.pdf IMPLOSION DRIVEN SHOCK TUBE] {{Webarchive|url=https://web.archive.org/web/20230402100741/https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/19700009349_1970009349.pdf |date=2 April 2023 }}". NASA</ref> to generate [[neutron]]s from D-D reactions. The simplest and most direct method proved to be in a predetonated stoichiometric mixture of [[deuterium]]-[[oxygen]]. The other successful method was using a miniature [[Voitenko compressor]],<ref>D.Sagie and I.I. Glass (1982) "[https://web.archive.org/web/20110522030502/http://handle.dtic.mil/100.2/ADA121652 Explosive-driven hemispherical implosions for generating fusion plasmas]"</ref> where a plane diaphragm was driven by the implosion wave into a secondary small spherical cavity that contained pure [[deuterium]] gas at one atmosphere.<ref>T. Saito, A. K. Kudian and I. I. Glass "[http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADP000254&Location=U2&doc=GetTRDoc.pdf Temperature Measurements Of An Implosion Focus] {{Webarchive|url=https://web.archive.org/web/20120720174842/http://www.dtic.mil/cgi-bin/GetTRDoc?AD=ADP000254&Location=U2&doc=GetTRDoc.pdf |date=2012-07-20 }}"</ref>

===Electrostatic confinement ===
{{main|Inertial electrostatic confinement}}
There are also [[Inertial electrostatic confinement|electrostatic confinement fusion]] devices. These devices confine [[ion]]s using electrostatic fields. The best known is the [[fusor]]. This device has a cathode inside an anode wire cage. Positive ions fly towards the negative inner cage, and are heated by the electric field in the process. If they miss the inner cage they can collide and fuse. Ions typically hit the cathode, however, creating prohibitory high [[conduction (heat)|conduction]] losses. Also, fusion rates in [[fusor]]s are very low due to competing physical effects, such as energy loss in the form of light radiation.<ref>Ion Flow and Fusion Reactivity, Characterization of a Spherically convergent ion Focus. PhD Thesis, Dr. Timothy A Thorson, Wisconsin-Madison 1996.</ref> Designs have been proposed to avoid the problems associated with the cage, by generating the field using a non-neutral cloud. These include a plasma oscillating device,<ref>"Stable, thermal equilibrium, large-amplitude, spherical plasma oscillations in electrostatic confinement devices", DC Barnes and Rick Nebel, PHYSICS OF PLASMAS VOLUME 5, NUMBER 7 JULY 1998</ref> a [[Penning trap]] and the [[polywell]].<ref>Carr, M.; Khachan, J. (2013). "A biased probe analysis of potential well formation in an electron only, low beta Polywell magnetic field". Physics of Plasmas 20 (5): 052504. {{Bibcode|2013PhPl...20e2504C}}. {{doi|10.1063/1.4804279}}</ref> The technology is relatively immature, however, and many scientific and engineering questions remain.

The most well known Inertial electrostatic confinement approach is the [[fusor]]. Starting in 1999, a number of amateurs have been able to do amateur fusion using these homemade devices.<ref>{{cite web |url=http://www.fusor.net/board/ |title=Fusor Forums • Index page |publisher=Fusor.net |access-date=24 August 2014 |archive-date=8 August 2014 |archive-url=https://web.archive.org/web/20140808074102/http://www.fusor.net/board/ |url-status=live }}</ref><ref>{{cite web |url=http://www.clhsonline.net/sciblog/index.php/2012/03/build-a-nuclear-fusion-reactor-no-problem/ |title=Build a Nuclear Fusion Reactor? No Problem |publisher=Clhsonline.net |date=23 March 2012 |access-date=24 August 2014 |url-status=dead |archive-url=https://web.archive.org/web/20141030210524/http://www.clhsonline.net/sciblog/index.php/2012/03/build-a-nuclear-fusion-reactor-no-problem/ |archive-date=30 October 2014 }}</ref><ref>{{cite news|url=https://www.bbc.co.uk/news/10385853|title=Extreme DIY: Building a homemade nuclear reactor in NYC|access-date=30 October 2014|date=23 June 2010|last1=Danzico|first1=Matthew|archive-date=16 May 2018|archive-url=https://web.archive.org/web/20180516133915/http://www.bbc.co.uk/news/10385853|url-status=live}}</ref><ref>{{cite web |last=Schechner |first=Sam |url=https://online.wsj.com/news/articles/SB121901740078248225 |title=Nuclear Ambitions: Amateur Scientists Get a Reaction From Fusion |work=The Wall Street Journal |date=18 August 2008 |access-date=24 August 2014 |archive-date=3 March 2014 |archive-url=https://web.archive.org/web/20140303102549/http://online.wsj.com/news/articles/SB121901740078248225 |url-status=live }}</ref> Other IEC devices include: the [[Polywell]], MIX POPS<ref>{{cite journal|bibcode=2005PhRvL..95a5003P|doi=10.1103/PhysRevLett.95.015003|title=Experimental Observation of a Periodically Oscillating Plasma Sphere in a Gridded Inertial Electrostatic Confinement Device|pmid=16090625|journal=Phys Rev Lett|year=2005|volume=95|issue=1|pages=015003|vauthors=Park J, Nebel RA, Stange S, Murali SK|url=https://zenodo.org/record/1233951|access-date=25 August 2020|archive-date=23 October 2020|archive-url=https://web.archive.org/web/20201023003428/https://zenodo.org/record/1233951|url-status=live}}</ref> and Marble concepts.<ref>"The Multiple Ambipolar Recirculating Beam Line Experiment" Poster presentation, 2011 US-Japan IEC conference, Dr. Alex Klein</ref>

==Important reactions==
<!-- This section is linked from [[Fusion power]] -->
{{unreferenced section|date=August 2023}}

===Stellar reaction chains===
At the temperatures and densities in stellar cores, the rates of fusion reactions are notoriously slow. For example, at solar core temperature (''T'' ≈ 15&nbsp;MK) and density (160&nbsp;g/cm<sup>3</sup>), the energy release rate is only 276&nbsp;μW/cm<sup>3</sup>—about a quarter of the volumetric rate at which a resting human body generates heat.<ref>[http://fusedweb.pppl.gov/CPEP/Chart_Pages/5.Plasmas/SunLayers.html FusEdWeb | Fusion Education]. Fusedweb.pppl.gov (9 November 1998). Retrieved 17 August 2011. {{Webarchive|url=https://web.archive.org/web/20071024012758/http://fusedweb.pppl.gov/CPEP/Chart_pages/5.Plasmas/SunLayers.html |date=24 October 2007 }}</ref> Thus, reproduction of stellar core conditions in a lab for nuclear fusion power production is completely impractical. Because nuclear reaction rates depend on density as well as temperature and most fusion schemes operate at relatively low densities, those methods are strongly dependent on higher temperatures. The fusion rate as a function of temperature (exp(−''E''/''kT'')), leads to the need to achieve temperatures in terrestrial reactors 10–100 times higher than in stellar interiors: ''T'' ≈ {{val|0.1|–|1.0|e=9|u=K}}.

===Criteria and candidates for terrestrial reactions===
{{main article|Fusion power#Fuels}}
In artificial fusion, the primary fuel is not constrained to be protons and higher temperatures can be used, so reactions with larger cross-sections are chosen. Another concern is the production of neutrons, which activate the reactor structure radiologically, but also have the advantages of allowing volumetric extraction of the fusion energy and [[tritium]] breeding. Reactions that release no neutrons are referred to as [[Aneutronic fusion|''aneutronic'']].

To be a useful energy source, a fusion reaction must satisfy several criteria. It must:

;Be [[exothermic]]: This limits the reactants to the low ''Z'' (number of protons) side of the [[Nuclear binding energy#Nuclear binding energy curve|curve of binding energy]]. It also makes helium {{SimpleNuclide|link=yes|Helium|4}} the most common product because of its extraordinarily tight binding, although {{SimpleNuclide|link=yes|Helium|3}} and {{SimpleNuclide|link=yes|Hydrogen|3}} also show up.
;Involve low atomic number (''Z'') nuclei: This is because the electrostatic repulsion that must be overcome before the nuclei are close enough to fuse ( [[Coulomb barrier]] ) is directly related to the number of protons it contains – its atomic number.
;Have two reactants: At anything less than stellar densities, three-body collisions are too improbable. In inertial confinement, both stellar densities and temperatures are exceeded to compensate for the shortcomings of the third parameter of the Lawson criterion, ICF's very short confinement time.
;Have two or more products: This allows simultaneous conservation of energy and momentum without relying on the electromagnetic force.
;Conserve both protons and neutrons: The cross sections for the weak interaction are too small.

Few reactions meet these criteria. The following are those with the largest cross sections:<ref>{{cite book
 |author1         = M. Kikuchi, K. Lackner
 |author2         = M. Q. Tran
 |name-list-style = amp
 |year            = 2012
 |title           = Fusion Physics
 |url             = http://www-pub.iaea.org/books/IAEABooks/8879/Fusion-Physics
 |page            = 22
 |publisher       = [[International Atomic Energy Agency]]
 |isbn            = 9789201304100
 |access-date     = 8 December 2015
 |archive-date    = 8 December 2015
 |archive-url     = https://web.archive.org/web/20151208122642/http://www-pub.iaea.org/books/IAEABooks/8879/Fusion-Physics
 |url-status      = live
}}</ref><ref>{{cite book
|author1=K. Miyamoto
|year=2005
|title=Plasma Physics and Controlled Nuclear Fusion
|publisher=[[Springer-Verlag]]
|isbn=3-540-24217-1
}}</ref>

<!-- Autogenerated using Phykiformulae 0.10 by [[User:SkyLined]]
(1)     D + T    → He   ( 3.52 MeV  ) + n   ( 14.06 MeV  )
(2i)    D + D    → T    ( 1.01 MeV ) + p    ( 3.02 MeV  ) _ _ _ _       _ _50%
(2ii)   _ _ _    → He-3 ( 0.82 MeV ) + n    ( 3.27 MeV  ) _ _ _ _       _ _50%
(3)     D + He-3 → He   ( 3.6 MeV  ) + p    ( 14.7 MeV  )
(4)     T + T    → He   _ _       _ + 2n   _ _        _ _ _ + 11.3MeV
(5)  He-3 + He-3 → He   _ _       _ + 2p   _ _        _ _ _ + 12.9MeV
(6i) He-3 + T    → He   _ _       _ + p    + n        _ _ _ + 12.1MeV _ _57%
(6ii)   _ _ _    → He   ( 4.8 MeV  ) + D    ( 9.5 MeV   ) _ _ _ _       _ _43%
(7i)    D + Li-6 → 2He  + 22.3 MeV
(7ii)   _ _ _    → He-3 + He      _ + n    _ _        _ _ _ + 1.8 MeV
(7iii)  _ _ _    → Li-7 ( 0.625 MeV ) + p (4.375 MeV)
(7iv)   _ _ _    → Be-7 ( 0.425 MeV )+ n ( 2.975 MeV )
(8)     p + Li-6 → He   ( 1.7 MeV  ) + He-3 (2.3 MeV)
(9)  He-3 + Li-6 → 2He  + p       _ _ _    _ _        _ _ _ + 16.9 MeV
(10)    p + B-11 → 3He  _ _       _ _ _    _ _        _ _ _ + 8.7 MeV

-->:{| border="0"
|- style="height:2em;"
|(1)&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|3.52|u=MeV}}||)&nbsp;||+&nbsp;||[[Neutron|n<sup>0</sup>]]&nbsp;||(&nbsp;||style="text-align:right;"|{{val|14.06|u=MeV}}||)
|- style="height:2em;"
|(2i)&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||→&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|1.01|u=MeV}}||)&nbsp;||+&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||(&nbsp;||style="text-align:right;"|{{val|3.02|u=MeV}}||)&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 50%
|- style="height:2em;"
|(2ii)&nbsp;||&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|0.82|u=MeV}}||)&nbsp;||+&nbsp;||[[Neutron|n<sup>0</sup>]]&nbsp;||(&nbsp;||style="text-align:right;"|{{val|2.45|u=MeV}}||)&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 50%
|- style="height:2em;"
|(3)&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|3.6|u=MeV}}||)&nbsp;||+&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||(&nbsp;||style="text-align:right;"|{{val|14.7|u=MeV}}||)
|- style="height:2em;"
|(4)&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||2&nbsp;[[Neutron|n<sup>0</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|11.3|u=MeV}}
|- style="height:2em;"
|(5)&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||2&nbsp;[[Proton|p<sup>+</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|12.9|u=MeV}}
|- style="height:2em;"
|(6i)&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||+&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||+&nbsp;||[[Neutron|n<sup>0</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|12.1|u=MeV}}||&nbsp;|| 57%
|- style="height:2em;"
|(6ii)&nbsp;||&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|4.8|u=MeV}}||)&nbsp;||+&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|9.5|u=MeV}}||)&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;|| 43%
|- style="height:2em;"
|(7i)&nbsp;||{{nuclide|link=yes|deuterium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|lithium|6}}&nbsp;||→&nbsp;||2&nbsp;{{nuclide|link=yes|helium|4}}&nbsp;||+&nbsp;||style="text-align:right;"|{{val|22.4|u=MeV}}
|- style="height:2em;"
|(7ii)&nbsp;||&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||&nbsp;||+&nbsp;||[[Neutron|n<sup>0</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|2.56|u=MeV}}
|- style="height:2em;"
|(7iii)&nbsp;||&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|lithium|7}}&nbsp;||+&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|5.0|u=MeV}}
|- style="height:2em;"
|(7iv)&nbsp;||&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|beryllium|7}}&nbsp;||+&nbsp;||[[Neutron|n<sup>0</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|3.4|u=MeV}}
|- style="height:2em;"
|(8)&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||+&nbsp;||{{nuclide|link=yes|lithium|6}}&nbsp;||→&nbsp;||{{nuclide|link=yes|helium|4}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|1.7|u=MeV}}||)&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||(&nbsp;||style="text-align:right;"|{{val|2.3|u=MeV}}||)
|- style="height:2em;"
|(9)&nbsp;||{{nuclide|link=yes|helium|3}}&nbsp;||+&nbsp;||{{nuclide|link=yes|lithium|6}}&nbsp;||→&nbsp;||2&nbsp;{{nuclide|link=yes|helium|4}}&nbsp;||+&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|16.9|u=MeV}}
|- style="height:2em;"
|(10)&nbsp;||[[Proton|p<sup>+</sup>]]&nbsp;||+&nbsp;||{{nuclide|link=yes|boron|11}}&nbsp;||→&nbsp;||3&nbsp;{{nuclide|link=yes|helium|4}}&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||&nbsp;||+&nbsp;||style="text-align:right;"|{{val|8.7|u=MeV}}
|}
{{Nucleosynthesis}}

For reactions with two products, the energy is divided between them in inverse proportion to their masses, as shown. In most reactions with three products, the distribution of energy varies. For reactions that can result in more than one set of products, the branching ratios are given.

Some reaction candidates can be eliminated at once. The D–<sup>6</sup>Li reaction has no advantage compared to [[Proton|p<sup>+</sup>]]–{{nuclide|link=yes|Boron|11}} because it is roughly as difficult to burn but produces substantially more neutrons through {{nuclide|link=yes|Deuterium}}–{{nuclide|link=yes|Deuterium}} side reactions. There is also a [[Proton|p<sup>+</sup>]]–{{nuclide|link=yes|Lithium|7}} reaction, but the cross section is far too low, except possibly when ''T''<sub>''i''</sub> > 1 MeV, but at such high temperatures an endothermic, direct neutron-producing reaction also becomes very significant. Finally there is also a [[Proton|p<sup>+</sup>]]–{{nuclide|link=yes|Beryllium|9}} reaction, which is not only difficult to burn, but {{nuclide|link=yes|Beryllium|9}} can be easily induced to split into two alpha particles and a neutron.

In addition to the fusion reactions, the following reactions with neutrons are important in order to "breed" tritium in "dry" fusion bombs and some proposed fusion reactors:
<!-- Autogenerated using Phykiformulae 0.10 by [[User:SkyLined]]
{{Nuclide2|neutron}} + {{Nuclide2|lithium|6}} -> T + He + 4.784 MeV
n + {{Nuclide2|lithium|7}} -> T + He + n − 2.467 MeV

-->:{| border="0"
|- style="height:2em;"
|[[Neutron|n<sup>0</sup>]]&nbsp;||+&nbsp;||{{nuclide|link=yes|lithium|6}}&nbsp;||→&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|4}} + 4.784 MeV
|- style="height:2em;"
|[[Neutron|n<sup>0</sup>]]&nbsp;||+&nbsp;||{{nuclide|link=yes|lithium|7}}&nbsp;||→&nbsp;||{{nuclide|link=yes|tritium}}&nbsp;||+&nbsp;||{{nuclide|link=yes|helium|4}} + [[Neutron|n<sup>0</sup>]] − 2.467 MeV
|}

The latter of the two equations was unknown when the U.S. conducted the [[Castle Bravo]] fusion bomb test in 1954. Being just the second fusion bomb ever tested (and the first to use lithium), the designers of the Castle Bravo "Shrimp" had understood the usefulness of <sup>6</sup>Li in tritium production, but had failed to recognize that <sup>7</sup>Li fission would greatly increase the yield of the bomb. While <sup>7</sup>Li has a small neutron cross-section for low neutron energies, it has a higher cross section above 5 MeV.<ref name=cross_section>[http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_7/4_7_4c.html Subsection 4.7.4c] {{Webarchive|url=https://web.archive.org/web/20180816195425/http://www.kayelaby.npl.co.uk/atomic_and_nuclear_physics/4_7/4_7_4c.html |date=16 August 2018 }}. Kayelaby.npl.co.uk. Retrieved 19 December 2012.</ref> The 15 Mt yield was 150% greater than the predicted 6 Mt and caused unexpected exposure to fallout.

To evaluate the usefulness of these reactions, in addition to the reactants, the products, and the energy released, one needs to know something about the [[nuclear cross section]]. Any given fusion device has a maximum plasma pressure it can sustain, and an economical device would always operate near this maximum. Given this pressure, the largest fusion output is obtained when the temperature is chosen so that {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}} is a maximum. This is also the temperature at which the value of the triple product {{mvar|nTτ}} required for [[Fusion energy gain factor#Ignition|ignition]] is a minimum, since that required value is inversely proportional to {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}} (see [[Lawson criterion]]). (A plasma is "ignited" if the fusion reactions produce enough power to maintain the temperature without external heating.) This optimum temperature and the value of {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}} at that temperature is given for a few of these reactions in the following table.

{| class="wikitable" style="margin:auto;"
|-
!fuel                                           !! ''T'' [keV] !! {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}} [m<sup>3</sup>/s/keV<sup>2</sup>]
|-
|{{nuclide|deuterium}}–{{nuclide|tritium}}      || 13.6        || {{val|1.24|e=-24}}
|-
|{{nuclide|deuterium}}–{{nuclide|deuterium}}    || 15          || {{val|1.28|e=-26}}
|-
|{{nuclide|deuterium}}–{{nuclide|helium|3}}     || 58          || {{val|2.24|e=-26}}
|-
|p<sup>+</sup>–{{nuclide|lithium|6}}            || 66          || {{val|1.46|e=-27}}
|-
|p<sup>+</sup>–{{nuclide|boron|11}}             || 123         || {{val|3.01|e=-27}}
|}

Note that many of the reactions form chains. For instance, a reactor fueled with {{nuclide|tritium}} and {{nuclide|helium|3}} creates some {{nuclide|deuterium}}, which is then possible to use in the {{nuclide|deuterium}}–{{nuclide|helium|3}} reaction if the energies are "right". An elegant idea is to combine the reactions (8) and (9). The {{nuclide|helium|3}} from reaction (8) can react with {{nuclide|lithium|6}} in reaction (9) before completely thermalizing. This produces an energetic proton, which in turn undergoes reaction (8) before thermalizing. Detailed analysis shows that this idea would not work well,{{Citation needed|date=April 2010}} but it is a good example of a case where the usual assumption of a [[Maxwell–Boltzmann distribution|Maxwellian]] plasma is not appropriate.

=== Abundance of the nuclear fusion fuels ===
{{See also|Abundance of the chemical elements|Abundance of elements in Earth's crust|Abundances of the elements (data page)|Aneutronic fusion|CNO cycle|Cold fusion}}
{| class="wikitable"
|+
!Nuclear Fusion Fuel Isotope
!Half-Life
!Abundance
|-
|{{Nuclide|hydrogen|1|link=yes}}<ref>[[CNO cycle#]]</ref><ref name=":5">[[CNO cycle#CNO-I]]</ref><ref name=":6">[[CNO cycle#CNO-II]]</ref><ref name=":7">[[CNO cycle#CNO-III]]</ref><ref name=":8">[[CNO cycle#CNO-IV]]</ref><ref name=":9">[[CNO cycle#HCNO-I]]</ref><ref name=":10">[[CNO cycle#HCNO-II]]</ref><ref name=":11">[[CNO cycle#HCNO-III]]</ref>
|Stable
|99.98%
|-
|{{Nuclide|deuterium|link=yes}}<ref name=":2">[[Nuclear fusion#Criteria and candidates for terrestrial reactions]]</ref><ref name=":3">[[Aneutronic fusion#Coulomb barrier]]</ref><ref name=":4">[[Aneutronic fusion#Candidate reactions]]</ref><ref name=":1">[[Cold fusion#Lack of expected reaction products]]</ref>
|Stable
|0.02%
|-
|{{Nuclide|tritium|link=yes}}<ref name=":2" /><ref name=":3" />
|12.32(2) y
|trace
|-
|{{Nuclide|Helium|3|link=yes}}<ref name=":2" /><ref name=":3" /><ref name=":4" />
|stable
|0.0002%
|-
|{{Nuclide|Helium|4|link=yes}}<ref name=":1" />
|stable
|99.9998%
|-
|{{Nuclide|lithium|6|link=yes}}<ref name=":2" /><ref name=":3" /><ref name=":4" />
|stable
|7.59%
|-
|{{Nuclide|lithium|7|link=yes}}<ref name=":2" /><ref name=":4" />
|stable
|92.41%
|-
|{{Nuclide|boron|11|link=yes}}<ref name=":2" /><ref name=":3" /><ref name=":4" /><ref>[[Aneutronic fusion#Residual radiation]]</ref>
|stable
|80%
|-
|{{Nuclide|carbon|12|link=yes}}<ref name=":5" /><ref name=":9" />
|stable
|98.9%
|-
|{{Nuclide|carbon|13|link=yes}}<ref name=":5" />
|stable
|1.1%
|-
|{{Nuclide|nitrogen|13|link=yes}}<ref name=":5" /><ref name=":9" />
|9.965(4) min
|syn
|-
|{{Nuclide|nitrogen|14|link=yes}}<ref name=":5" /><ref name=":6" /><ref name=":9" />
|stable
|99.6%
|-
|{{Nuclide|nitrogen|15|link=yes}}<ref name=":5" /><ref name=":6" /><ref name=":7" /><ref name=":9" /><ref name=":10" /><ref name=":4" />
|stable
|0.4%
|-
|{{Nuclide|oxygen|14|link=yes}}<ref name=":9" />
|70.621(11) s
|syn
|-
|{{Nuclide|oxygen|15|link=yes}}<ref name=":5" /><ref name=":6" /><ref name=":9" /><ref name=":10" />
|122.266(43) s
|syn
|-
|{{Nuclide|oxygen|16|link=yes}}<ref name=":6" /><ref name=":7" /><ref name=":8" /><ref name=":10" /><ref name=":11" />
|stable
|99.76%
|-
|{{Nuclide|oxygen|17|link=yes}}<ref name=":6" /><ref name=":7" /><ref name=":8" />
|stable
|0.04%
|-
|{{Nuclide|oxygen|18|link=yes}}<ref name=":7" /><ref name=":8" />
|stable
|0.20%
|-
|{{Nuclide|fluorine|17|link=yes}}<ref name=":6" /><ref name=":7" /><ref name=":8" /><ref name=":10" /><ref name=":11" />
|64.370(27) s
|syn
|-
|{{Nuclide|fluorine|18|link=yes}}<ref name=":7" /><ref name=":8" /><ref name=":10" /><ref name=":11" />
|109.734(8) min
|trace
|-
|{{Nuclide|fluorine|19|link=yes}}<ref name=":8" /><ref name=":11" />
|stable
|100%
|-
|{{Nuclide|Neon|18|link=yes}}<ref name=":10" /><ref name=":11" />
|1664.20(47) ms
|trace
|-
|{{Nuclide|Neon|19|link=yes}}<ref name=":11" />
|17.2569(19) s
|trace
|}

=== Neutronicity, confinement requirement, and power density ===
Any of the reactions above can in principle be the basis of [[fusion power]] production. In addition to the temperature and cross section discussed above, we must consider the total energy of the fusion products ''E''<sub>fus</sub>, the energy of the charged fusion products ''E''<sub>ch</sub>, and the atomic number ''Z'' of the non-hydrogenic reactant.

Specification of the {{nuclide|deuterium}}–{{nuclide|deuterium}} reaction entails some difficulties, though. To begin with, one must average over the two branches (2i) and (2ii). More difficult is to decide how to treat the {{nuclide|tritium}} and {{nuclide|helium|3}} products. {{nuclide|tritium}} burns so well in a deuterium plasma that it is almost impossible to extract from the plasma. The {{nuclide|deuterium}}–{{nuclide|helium|3}} reaction is optimized at a much higher temperature, so the burnup at the optimum {{nuclide|deuterium}}–{{nuclide|deuterium}} temperature may be low. Therefore, it seems reasonable to assume the {{nuclide|tritium}} but not the {{nuclide|helium|3}} gets burned up and adds its energy to the net reaction, which means the total reaction would be the sum of (2i), (2ii), and (1):

:5 {{nuclide|link=yes|deuterium}} → {{nuclide|link=yes|helium|4}} + 2 [[Neutron|n<sup>0</sup>]] + {{nuclide|link=yes|helium|3}} + [[Proton|p<sup>+</sup>]],  ''E''<sub>fus</sub> = 4.03 + 17.6 + 3.27 = 24.9&nbsp;MeV, ''E''<sub>ch</sub> = 4.03 + 3.5 + 0.82 = 8.35&nbsp;MeV.

For calculating the power of a reactor (in which the reaction rate is determined by the D–D step), we count the {{nuclide|deuterium}}–{{nuclide|deuterium}} fusion energy ''per D–D reaction'' as ''E''<sub>fus</sub> = (4.03&nbsp;MeV + 17.6&nbsp;MeV) × 50% + (3.27&nbsp;MeV) × 50% = 12.5&nbsp;MeV and the energy in charged particles as ''E''<sub>ch</sub> = (4.03&nbsp;MeV + 3.5&nbsp;MeV) × 50% + (0.82&nbsp;MeV) × 50% = 4.2&nbsp;MeV. (Note: if the tritium ion reacts with a deuteron while it still has a large kinetic energy, then the kinetic energy of the helium-4 produced may be quite different from 3.5 MeV,<ref>A momentum and energy balance shows that if the tritium has an energy of E<sub>T</sub> (and using relative masses of 1, 3, and 4 for the neutron, tritium, and helium) then the energy of the helium can be anything from [(12E<sub>T</sub>)<sup>1/2</sup>−(5×17.6MeV+2×E<sub>T</sub>)<sup>1/2</sup>]<sup>2</sup>/25 to [(12E<sub>T</sub>)<sup>1/2</sup>+(5×17.6MeV+2×E<sub>T</sub>)<sup>1/2</sup>]<sup>2</sup>/25. For E<sub>T</sub>=1.01&nbsp;MeV this gives a range from 1.44&nbsp;MeV to 6.73&nbsp;MeV.</ref> so this calculation of energy in charged particles is only an approximation of the average.) The amount of energy per deuteron consumed is 2/5 of this, or 5.0&nbsp;MeV (a [[specific energy]] of about 225&nbsp;million [[Megajoule|MJ]] per kilogram of deuterium).

Another unique aspect of the {{nuclide|deuterium}}–{{nuclide|deuterium}} reaction is that there is only one reactant, which must be taken into account when calculating the reaction rate.

With this choice, we tabulate parameters for four of the most important reactions

{| class="wikitable" style="margin:auto; text-align:right;"
|-
!fuel                                          !!''Z''!!''E''<sub>fus</sub> [MeV]!!''E''<sub>ch</sub> [MeV]!!neutronicity
|-
|{{nuclide|deuterium}}–{{nuclide|tritium}}     ||  1  || 17.6 || 3.5 || 0.80
|-
|{{nuclide|deuterium}}–{{nuclide|deuterium}}   ||  1  || 12.5 || 4.2 || 0.66
|-
|{{nuclide|deuterium}}–{{nuclide|helium|3}}    ||  2  || 18.3 ||18.3 || ≈0.05
|-
|p<sup>+</sup>–{{nuclide|boron|11}}            ||  5  || 8.7  || 8.7 || ≈0.001
|}

The last column is the [[aneutronic fusion|neutronicity]] of the reaction, the fraction of the fusion energy released as neutrons. This is an important indicator of the magnitude of the problems associated with neutrons like radiation damage, biological shielding, remote handling, and safety. For the first two reactions it is calculated as {{nowrap|(''E''<sub>fus</sub> − ''E''<sub>ch</sub>)/''E''<sub>fus</sub>}}. For the last two reactions, where this calculation would give zero, the values quoted are rough estimates based on side reactions that produce neutrons in a plasma in thermal equilibrium.

Of course, the reactants should also be mixed in the optimal proportions. This is the case when each reactant ion plus its associated electrons accounts for half the pressure. Assuming that the total pressure is fixed, this means that particle density of the non-hydrogenic ion is smaller than that of the hydrogenic ion by a factor {{nowrap|2/(''Z'' + 1)}}. Therefore, the rate for these reactions is reduced by the same factor, on top of any differences in the values of {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}}. On the other hand, because the {{nuclide|deuterium}}–{{nuclide|deuterium}} reaction has only one reactant, its rate is twice as high as when the fuel is divided between two different hydrogenic species, thus creating a more efficient reaction.

Thus there is a "penalty" of {{nowrap|2/(''Z'' + 1)}} for non-hydrogenic fuels arising from the fact that they require more electrons, which take up pressure without participating in the fusion reaction. (It is usually a good assumption that the electron temperature will be nearly equal to the ion temperature. Some authors, however, discuss the possibility that the electrons could be maintained substantially colder than the ions. In such a case, known as a "hot ion mode", the "penalty" would not apply.) There is at the same time a "bonus" of a factor 2 for {{nuclide|deuterium}}–{{nuclide|deuterium}} because each ion can react with any of the other ions, not just a fraction of them.

We can now compare these reactions in the following table.

{| class="wikitable" style="margin:auto; text-align:right;"
|-
!fuel                                         !!{{math|{{angbr|''σv''}}/''T''<sup>2</sup>}}!!penalty/bonus !!inverse reactivity!!Lawson criterion!!power density [W/m<sup>3</sup>/kPa<sup>2</sup>]!!inverse ratio of power density
|-
|{{nuclide|deuterium}}–{{nuclide|tritium}}    || {{val|1.24|e=-24}} ||  1  ||    1 ||   1 || 34     ||    1
|-
|{{nuclide|deuterium}}–{{nuclide|deuterium}}  || {{val|1.28|e=-26}} ||  2  ||   48 ||  30 ||  0.5   ||   68
|-
|{{nuclide|deuterium}}–{{nuclide|helium|3}}   || {{val|2.24|e=-26}} || 2/3 ||   83 ||  16 ||  0.43  ||   80
|-
|p<sup>+</sup>–{{nuclide|lithium|6}}          || {{val|1.46|e=-27}} || 1/2 || 1700 ||     || 0.005  || 6800
|-
|p<sup>+</sup>–{{nuclide|boron|11}}           || {{val|3.01|e=-27}} || 1/3 || 1240 || 500 || 0.014  || 2500
|}

The maximum value of {{math|{{angbr|''σv''}}/''T''<sup>2</sup>}} is taken from a previous table. The "penalty/bonus" factor is that related to a non-hydrogenic reactant or a single-species reaction. The values in the column "inverse reactivity" are found by dividing {{val|1.24|e=-24}} by the product of the second and third columns. It indicates the factor by which the other reactions occur more slowly than the {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|tritium}} reaction under comparable conditions. The column "[[Lawson criterion]]" weights these results with ''E''<sub>ch</sub> and gives an indication of how much more difficult it is to achieve ignition with these reactions, relative to the difficulty for the {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|tritium}} reaction. The next-to-last column is labeled "power density" and weights the practical reactivity by ''E''<sub>fus</sub>. The final column indicates how much lower the fusion power density of the other reactions is compared to the {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|tritium}} reaction and can be considered a measure of the economic potential.

===Bremsstrahlung losses in quasineutral, isotropic plasmas===
{{unreferenced section|date=August 2023}}
The ions undergoing fusion in many systems will essentially never occur alone but will be mixed with [[electron]]s that in aggregate neutralize the ions' bulk [[electrical charge]] and form a [[Plasma (physics)|plasma]]. The electrons will generally have a temperature comparable to or greater than that of the ions, so they will collide with the ions and emit [[x-ray]] radiation of 10–30 keV energy, a process known as [[Bremsstrahlung]].

The huge size of the Sun and stars means that the x-rays produced in this process will not escape and will deposit their energy back into the plasma. They are said to be [[Opacity (optics)|opaque]] to x-rays. But any terrestrial fusion reactor will be [[Optical depth|optically thin]] for x-rays of this energy range. X-rays are difficult to reflect but they are effectively absorbed (and converted into heat) in less than mm thickness of stainless steel (which is part of a reactor's shield). This means the bremsstrahlung process is carrying energy out of the plasma, cooling it.
 
The ratio of fusion power produced to x-ray radiation lost to walls is an important figure of merit. This ratio is generally maximized at a much higher temperature than that which maximizes the power density (see the previous subsection). The following table shows estimates of the optimum temperature and the power ratio at that temperature for several reactions:

{| class="wikitable" style="margin:auto;"
|-
!fuel                                        !!''T''<sub>i</sub> [keV]!!''P''<sub>fusion</sub>/''P''<sub>Bremsstrahlung</sub>
|-
|{{nuclide|deuterium}}–{{nuclide|tritium}}   ||   50 || 140
|-
|{{nuclide|deuterium}}–{{nuclide|deuterium}} ||  500 ||   2.9
|-
|{{nuclide|deuterium}}–{{nuclide|helium|3}}  ||  100 ||   5.3
|-
|{{nuclide|helium|3}}–{{nuclide|helium|3}}   || 1000 ||   0.72
|-
|p<sup>+</sup>–{{nuclide|lithium|6}}         ||  800 ||   0.21
|-
|p<sup>+</sup>–{{nuclide|boron|11}}          ||  300 ||   0.57
|}

The actual ratios of fusion to Bremsstrahlung power will likely be significantly lower for several reasons. For one, the calculation assumes that the energy of the fusion products is transmitted completely to the fuel ions, which then lose energy to the electrons by collisions, which in turn lose energy by Bremsstrahlung. However, because the fusion products move much faster than the fuel ions, they will give up a significant fraction of their energy directly to the electrons. Secondly, the ions in the plasma are assumed to be purely fuel ions. In practice, there will be a significant proportion of impurity ions, which will then lower the ratio. In particular, the fusion products themselves ''must'' remain in the plasma until they have given up their energy, and ''will'' remain for some time after that in any proposed confinement scheme. Finally, all channels of energy loss other than Bremsstrahlung have been neglected. The last two factors are related. On theoretical and experimental grounds, particle and energy confinement seem to be closely related. In a confinement scheme that does a good job of retaining energy, fusion products will build up. If the fusion products are efficiently ejected, then energy confinement will be poor, too.

The temperatures maximizing the fusion power compared to the Bremsstrahlung are in every case higher than the temperature that maximizes the power density and minimizes the required value of the [[Lawson criterion|fusion triple product]]. This will not change the optimum operating point for {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|tritium}} very much because the Bremsstrahlung fraction is low, but it will push the other fuels into regimes where the power density relative to {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|tritium}} is even lower and the required confinement even more difficult to achieve. For {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|deuterium}} and {{nuclide|link=yes|deuterium}}–{{nuclide|link=yes|helium|3}}, Bremsstrahlung losses will be a serious, possibly prohibitive problem. For {{nuclide|link=yes|helium|3}}–{{nuclide|link=yes|helium|3}}, [[Proton|p<sup>+</sup>]]–{{nuclide|link=yes|lithium|6}} and [[Proton|p<sup>+</sup>]]–{{nuclide|link=yes|boron|11}} the Bremsstrahlung losses appear to make a fusion reactor using these fuels with a quasineutral, isotropic plasma impossible. Some ways out of this dilemma have been considered but rejected.<ref>{{cite journal|title=Fundamental Limitations on Plasma Fusion Systems not in Thermodynamic Equilibrium|journal=Dissertation Abstracts International|volume= 56-07 |issue=Section B|page= 3820|author=Rider, Todd Harrison|bibcode=1995PhDT........45R|year=1995}}</ref><ref>Rostoker, Norman; Binderbauer, Michl and Qerushi,
Artan. [https://web.archive.org/web/20051223133818/http://fusion.ps.uci.edu/artan/Posters/aps_poster_2.pdf Fundamental limitations on plasma fusion systems not in thermodynamic equilibrium]. fusion.ps.uci.edu</ref> This limitation does not apply to [[Non-neutral plasmas|non-neutral and anisotropic plasmas]]; however, these have their own challenges to contend with.

== Mathematical description of cross section ==

=== Fusion under classical physics ===
{{unreferenced section|date=August 2023}}
In a classical picture, nuclei can be understood as hard spheres that repel each other through the Coulomb force but fuse once the two spheres come close enough for contact. Estimating the radius of an atomic nuclei as about one femtometer, the energy needed for fusion of two hydrogen is:

:<math chem>E_{\ce{thresh}}= \frac{1}{4\pi \epsilon_0} \frac{Z_1 Z_2}{r} \ce{->[\text{2 protons}]} \frac{1}{4\pi \epsilon_0} \frac{e^2}{1\ \ce{fm}} \approx 1.4\ \ce{MeV}</math>

This would imply that for the core of the sun, which has a [[Boltzmann distribution]] with a temperature of around 1.4 keV, the probability hydrogen would reach the threshold is <math>10^{-290}</math>, that is, fusion would never occur. However, fusion in the sun does occur due to quantum mechanics.

=== Parameterization of cross section ===
{{unreferenced section|date=August 2023}}
The probability that fusion occurs is greatly increased compared to the classical picture, thanks to the smearing of the effective radius as the [[Matter wave|de Broglie wavelength]] as well as [[quantum tunneling]] through the potential barrier. To determine the rate of fusion reactions, the value of most interest is the [[Cross section (physics)|cross section]], which describes the probability that particles will fuse by giving a characteristic area of interaction. An estimation of the fusion cross-sectional area is often broken into three pieces:
:<math>\sigma \approx \sigma_\text{geometry} \times T \times R, </math>
where <math>\sigma_\text{geometry} </math> is the geometric cross section, {{mvar|T}} is the barrier transparency and {{mvar|R}} is the reaction characteristics of the reaction.

<math>\sigma_\text{geometry} </math> is of the order of the square of the de Broglie wavelength <math>\sigma_\text{geometry} \approx \lambda^2 = \bigg( \frac{\hbar}{m_r v} \bigg)^2 \propto \frac{1}{\epsilon} </math> where <math>m_r </math> is the reduced mass of the system and <math>\epsilon  </math> is the center of mass energy of the system.

{{mvar|T}} can be approximated by the Gamow transparency, which has the form: <math>T \approx e^ {- \sqrt{\epsilon_G /\epsilon} } </math> where <math>\epsilon_G = ( \pi \alpha Z_1 Z_2)^2 \times 2 m_r c^2 </math> is the [[Gamow factor]] and comes from estimating the quantum tunneling probability through the potential barrier.

{{mvar|R}} contains all the nuclear physics of the specific reaction and takes very different values depending on the nature of the interaction. However, for most reactions, the variation of <math>R(\epsilon)</math> is small compared to the variation from the Gamow factor and so is approximated by a function called the astrophysical [[S-factor]], <math>S(\epsilon)</math>, which is weakly varying in energy. Putting these dependencies together, one approximation for the fusion cross section as a function of energy takes the form:

:<math>\sigma(\epsilon) \approx \frac{S(\epsilon)}{\epsilon} e^{ - \sqrt{\epsilon_G / \epsilon}}</math>

More detailed forms of the cross-section can be derived through nuclear physics-based models and [[R-matrix]] theory.

=== Formulas of fusion cross sections ===
The Naval Research Lab's plasma physics formulary<ref>{{Cite web|url=https://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_13.pdf|title=NRL PLASMA FORMULARY|last=Huba|first=J.|year=2003|website=MIT Catalog|access-date=11 November 2018|archive-date=17 April 2018|archive-url=https://web.archive.org/web/20180417231213/http://library.psfc.mit.edu/catalog/online_pubs/NRL_FORMULARY_13.pdf|url-status=live}}</ref> gives the total cross section in [[Barn (unit)|barns]] as a function of the energy (in keV) of the incident particle towards a target ion at rest fit by the formula:

:<math>\sigma^\text{NRL}(\epsilon) = \frac{A_5 + \big( (A_4 - A_3 \epsilon)^2 + 1 \big)^{-1} A_2}{ \epsilon (e^{A_1 \epsilon^{-1/2} }-1)}</math> with the following coefficient values:
{| class="wikitable" style="text-align:right;"
|+NRL Formulary Cross Section Coefficients
!
!DT(1)
!DD(2i)
!DD(2ii)
!DHe<sup>3</sup>(3)
!TT(4)
!The<sup>3</sup>(6)
|-
|A1
|45.95
|46.097
|47.88
|89.27
|38.39
|123.1
|-
|A2
|50200
|372
|482
|25900
|448
|11250
|-
|A3
|{{val|1.368e-2}}
|{{val|4.36e-4}}
|{{val|3.08e-4}}
|{{val|3.98e-3}}
|{{val|1.02e-3}}
|0
|-
|A4
|1.076
|1.22
|1.177
|1.297
|2.09
|0
|-
|A5
|409
|0
|0
|647
|0
|0
|}
Bosch-Hale<ref>{{Cite journal|title=Improved formulas for fusion cross-sections and thermal reactivities|journal=Nuclear Fusion|volume=32|issue=4|pages=611–631|last=Bosch|first=H. S|year=1993|doi=10.1088/0029-5515/32/4/I07|s2cid=55303621}}</ref> also reports a R-matrix calculated cross sections fitting observation data with [[Padé approximant|Padé rational approximating coefficient]]s. With energy in units of keV and cross sections in units of millibarn, the factor has the form:

:<math>S^{\text{Bosch-Hale}}(\epsilon) = \frac{A_1 + \epsilon\bigg( A_2 + \epsilon \big(A_3 + \epsilon(A_4 + \epsilon A_5)\big)\bigg)}{1 + \epsilon\bigg(B_1 + \epsilon \big(B_2 + \epsilon (B_3 +\epsilon B_4)\big) \bigg)}</math>, with the coefficient values: 
{| class="wikitable" style="text-align:right;"
|+Bosch-Hale coefficients for the fusion cross section
!
!DT(1)
!DD(2ii)
!DHe<sup>3</sup>(3)
!The<sup>4</sup>
|-
|<math>\epsilon_G</math>
|31.3970
|68.7508
|31.3970
|34.3827
|-
|A1
|{{val|5.5576e4}}
|{{val|5.7501e6}}
|{{val|5.3701e4}}
|{{val|6.927e4}}
|-
|A2
|{{val|2.1054e2}}
|{{val|2.5226e3}}
|{{val|3.3027e2}}
|{{val|7.454e8}}
|-
|A3
| {{val|-3.2638e-2}}
| {{val|4.5566e1}}
| {{val|-1.2706e-1}}
|{{val|2.050e6}}
|-
|A4
|{{val|1.4987e-6}}
|0
|{{val|2.9327e-5}}
|{{val|5.2002e4}}
|-
|A5
|{{val|1.8181e-10}}
| 0
| {{val|-2.5151e-9}}
|0
|-
|B1
|0
| {{val|-3.1995e-3}}
|0
|{{val|6.38e1}}
|-
|B2
|0
| {{val|-8.5530e-6}}
| 0
| {{val|-9.95e-1}}
|-
|B3
|0
|{{val|5.9014e-8}}
|0
|{{val|6.981e-5}}
|-
|B4
|0
|0
|0
|{{val|1.728e-4}}
|-
|Applicable Energy Range [keV]
|0.5–5000 
|0.3–900
|0.5–4900
|0.5–550
|-
|<math>(\Delta S)_{\text{max}}\%</math>
|2.0
|2.2
|2.5
|1.9
|}
where <math>\sigma^{\text{Bosch-Hale}}(\epsilon) = \frac{S^{\text{Bosch-Hale}}(\epsilon)}{\epsilon \exp(\epsilon_G/\sqrt{\epsilon})}</math>

===Maxwell-averaged nuclear cross sections===
{{unreferenced section|date=August 2023}}
In fusion systems that are in thermal equilibrium, the particles are in a [[Maxwell–Boltzmann distribution]], meaning the particles have a range of energies centered around the plasma temperature. The sun, magnetically confined plasmas and inertial confinement fusion systems are well modeled to be in thermal equilibrium. In these cases, the value of interest is the fusion cross-section averaged across the Maxwell–Boltzmann distribution. The Naval Research Lab's plasma physics formulary tabulates Maxwell averaged fusion cross sections reactivities in <math>\mathrm{cm^3/s}</math>.
{| class="wikitable" style="text-align:right;"
|+NRL Formulary fusion reaction rates averaged over Maxwellian distributions
!Temperature [keV]
!DT(1)
!DD(2ii)
!DHe<sup>3</sup>(3)
!TT(4)
!The<sup>3</sup>(6)
|-
|1
|{{val|5.5e-21}}
|{{val|1.5e-22}}
|{{val|1.0e-26}}
|{{val|3.3e-22}}
|{{val|1.0e-28}}
|-
|2
|{{val|2.6e-19}}
|{{val|5.4e-21}}
|{{val|1.4e-23}}
|{{val|7.1e-21}}
|{{val|1.0e-25}}
|-
|5
|{{val|1.3e-17}}
|{{val|1.8e-19}}
|{{val|6.7e-21}}
|{{val|1.4e-19}}
|{{val|2.1e-22}}
|-
|10
| {{val|1.1e-16}}
| {{val|1.2e-18}}
| {{val|2.3e-19}}
|{{val|7.2e-19}}
|{{val|1.2e-20}}
|-
|20
|{{val|4.2e-16}}
|{{val|5.2e-18}}
|{{val|3.8e-18}}
|{{val|2.5e-18}}
|{{val|2.6e-19}}
|-
|50
|{{val|8.7e-16}}
| {{val|2.1e-17}}
| {{val|5.4e-17}}
|{{val|8.7e-18}}
|{{val|5.3e-18}}
|-
|100
|{{val|8.5e-16}}
| {{val|4.5e-17}}
|{{val|1.6e-16}}
|{{val|1.9e-17}}
|{{val|2.7e-17}}
|-
|200
|{{val|6.3e-16}}
| {{val|8.8e-17}}
| {{val|2.4e-16}}
| {{val|4.2e-17}}
|{{val|9.2e-17}}
|-
|500
|{{val|3.7e-16}}
|{{val|1.8e-16}}
|{{val|2.3e-16}}
|{{val|8.4e-17}}
|{{val|2.9e-16}}
|-
|1000
|{{val|2.7e-16}}
|{{val|2.2e-16}}
|{{val|1.8e-16}}
|{{val|8.0e-17}}
|{{val|5.2e-16}}
|}
For energies <math>T \le 25 \text{ keV}</math> the data can be represented by:

:<math>(\overline{\sigma v})_{DD} = 2.33\times 10^{-14}\cdot T^{-2/3} \cdot e^{-18.76\ T^{-1/3}} \mathrm{~{cm}^3/s}</math>

:<math>(\overline{\sigma v})_{DT} = 3.68\times 10^{-12}\cdot T^{-2/3} \cdot e^{-19.94\ T^{-1/3}} \mathrm{~{cm}^3/s}</math>
with {{mvar|T}} in units of keV.

==See also==
{{Columns-list|colwidth=22em|
*[[China Fusion Engineering Test Reactor]]
*[[Cold fusion]]
*[[Focus fusion]]
*[[Fusenet]]
*[[Fusion rocket]]
*[[Impulse generator]]
*[[Joint European Torus]]
*[[List of fusion experiments]]
*[[List of Fusor examples]]
*[[Neutron source]]
*[[Nuclear power|Nuclear energy]]
*[[Nuclear physics]]
*[[Nuclear reactor]]
*[[Periodic table]]
*[[Pulsed power]]
*[[Pure fusion weapon]]
*[[Teller–Ulam design]]
*[[Timeline of nuclear fusion]]
*[[Triple-alpha process]]
}}

==References==
{{Reflist}}

==Further reading==
*{{cite web
 |title        = What is Nuclear Fusion?
 |url          = http://www.nuclearfiles.org/menu/key-issues/nuclear-weapons/basics/what-is-fusion.htm
 |publisher    = NuclearFiles.org
 |access-date  = 12 January 2006
 |archive-url  = https://web.archive.org/web/20060928042248/http://www.nuclearfiles.org/menu/key-issues/nuclear-weapons/basics/what-is-fusion.htm
 |archive-date = 28 September 2006
 |url-status   = dead
 }}
*{{Cite book
 |last1         = Atzeni |first1 = Stefano
 |last2        = Meyer-ter-Vehn |first2 = Jürgen
 |title        = The physics of inertial fusion: beam plasma interaction, hydrodynamics, hot dense matter
 |publisher    = [[Clarendon Press]]
 |year         = 2004
 |isbn         = 978-0-19-856264-1
 |edition      = Repr
 |series       = International series of monographs on physics
 |location     = Oxford
 |chapter      = Nuclear fusion reactions
 |chapter-url  = http://www.oup.co.uk/pdf/0-19-856264-0.pdf
 |archive-url  = https://web.archive.org/web/20050124085615/http://www.oup.co.uk/pdf/0-19-856264-0.pdf
 |archive-date = 24 January 2005
 |url-status   = dead
}}
*{{Cite journal
 |last      = Brumfiel |first = Geoff
 |date      = 22 May 2006
 |title     = Chaos could keep fusion under control
 |journal   = [[Nature (journal)|Nature]]
 |doi       = 10.1038/news060522-2
 |issn      = 0028-0836
 |s2cid     = 62598131
}}
*{{Cite AV media
 |url       = https://www.youtube.com/watch?v=FhL5VO2NStU
 |title     = Should Google Go Nuclear? Clean, cheap, nuclear power...
 |date      = 8 October 2007
 |last      = Bussard |first = Robert |author-link = Robert Bussard
 |type      = Video
 |publisher = Google TechTalks
 |orig-date = 9 November 2006
 |via       = [[YouTube]]
}}
*{{Cite web
 |last1           = Wenisch |first1 = Antonia
 |last2          = Kromp |first2 = Richard
 |last3          = Reinberger |first3 = David
 |date           = November 2007
 |title          = Science or Fiction: Is there a Future for Nuclear?
 |url            = http://www.ecology.at/ecology/files/pr577_1.pdf
 |url-status     = dead
 |archive-url    = https://web.archive.org/web/20210126133058/http://www.ecology.at/ecology/files/pr577_1.pdf
 |archive-date   = 26 January 2021
 |access-date    = 8 October 2008
 |publisher      = Austrian Institute of Ecology
}}
*{{Cite book
 |last1            = Kikuchi |first1 = Mitsuru
 |last2           = Lackner |first2 = Karl
 |last3           = Tran |first3 = M. Q.
 |url             = http://www-pub.iaea.org/books/IAEABooks/8879/Fusion-Physics
 |title           = Fusion physics
 |publisher       = [[International Atomic Energy Agency]]
 |year            = 2012
 |isbn            = 978-92-0-130410-0
 |series          = Publication / Division of Scientific and Technical Information, International Atomic Energy Agency
 |location        = Vienna
 |page            = 22
 |access-date     = 8 December 2015
 |archive-url     = https://web.archive.org/web/20151208122642/http://www-pub.iaea.org/books/IAEABooks/8879/Fusion-Physics
 |archive-date    = 8 December 2015
 |url-status      = live
 |name-list-style = amp
}}
*{{Cite book
 |url             = http://link.springer.com/10.1007/978-1-4757-9319-2
 |title           = Atomic and Molecular Processes in Fusion Edge Plasmas
 |publisher       = [[Springer US]]
 |year            = 1995
 |isbn            = 978-1-4757-9321-5
 |editor-last     = Janev |editor-first = R. K.
 |location        = Boston, MA
 |doi             = 10.1007/978-1-4757-9319-2
 |access-date     = 16 January 2023
 |archive-url     = https://web.archive.org/web/20230116205120/https://link.springer.com/book/10.1007/978-1-4757-9319-2
 |archive-date    = 16 January 2023
 |url-status      = live
}}

==External links==
{{Commons|Nuclear fusion}}
*[https://web.archive.org/web/20161215103736/http://nuclearfiles.org/ NuclearFiles.org] – A repository of documents related to nuclear power.
*[https://web.archive.org/web/20160304054003/http://alsos.wlu.edu/qsearch.aspx?browse=science%2FFusion Annotated bibliography for nuclear fusion from the Alsos Digital Library for Nuclear Issues]
*[http://lasp.colorado.edu/~horanyi/5150/NRL_FORMULARY_07.pdf NRL Fusion Formulary] {{Webarchive|url=https://web.archive.org/web/20201026032602/http://lasp.colorado.edu/~horanyi/5150/NRL_FORMULARY_07.pdf |date=26 October 2020 }}
*{{cite web|url=https://wiki.fusion.ciemat.es/wiki/Main_Page|title=FusionWiki}}
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