View source for Pycnonuclear fusion

{{Short description|Type of nuclear fusion that occurs at high densities & low temperatures}}
{{distinguish|cold fusion}}
'''Pycnonuclear fusion''' ({{Langx|grc|πυκνός|links=yes|lit=dense, compact, thick|translit=pyknós}}) is a type of [[Nuclear fusion|nuclear fusion reaction]] which occurs due to [[Zero-point energy|zero-point oscillations]] of nuclei around their [[equilibrium point]] bound in their [[Crystal structure|crystal lattice]].<ref name=":0">{{Cite web |last1=Afanasjev |first1=A.V. |last2=Gasques |first2=L.R. |last3=Frauendorf |first3=S. |last4=Wiescher |first4=M. |title=Pycnonuclear Reactions |url=https://www.hzdr.de/FWK/jb03/PDF/s14.pdf |access-date=2022-08-06}}</ref><ref name=":6">{{Cite journal |last1=Afanasjev |first1=A. V. |last2=Beard |first2=M |last3=Chugunov |first3=A. I. |date=May 2012 |title=Large collection of astrophysical S factors and their compact representation |journal=Physical Review C |volume=85 |issue=5 |page=054615 |arxiv=1204.3174 |doi=10.1103/PhysRevC.85.054615 |bibcode=2012PhRvC..85e4615A |s2cid=119159326 }}</ref> In [[Quantum mechanics|quantum physics]], the phenomenon can be interpreted as overlap of the [[wave function]]s of neighboring [[ion]]s, and is proportional to the overlapping amplitude.<ref name=":1">{{cite journal |last1=Son |first1=S. |last2=Fisch |first2=N.J. |title=Pycnonuclear reaction and possible chain reactions in an ultra-dense DT plasma |journal=Physics Letters A |date=April 2005 |volume=337 |issue=4–6 |pages=397–407 |doi=10.1016/j.physleta.2005.01.084 }}</ref> Under the conditions of [[above-threshold ionization]], the reactions of neutronization and pycnonuclear fusion can lead to the creation of [[Lyapunov stability|absolutely stable]] environments in superdense substances.<ref>{{Cite book |last=Саакян |first=Г. С. |title=Равновесные конфигурации вырожденных газовых масс |publisher=М.: Наука |year=1972 |pages=344 |language=Ru}}</ref>

The term "'''pycnonuclear'''" was coined by A.G.W. Cameron in 1959, but research showing the possibility of nuclear fusion in extremely dense and cold compositions was published by W. A. Wildhack in 1940.<ref>{{Cite journal |last=Wildhack |first=W. A. |date=15 January 1940 |title=The Proton-Deuteron Transformation As a Source of Energy in Dense Stars |journal=Physical Review|publisher=American Physical Society |volume=57 |issue=2 |pages=81–86 |doi=10.1103/PhysRev.57.81 |bibcode=1940PhRv...57...81W }}</ref><ref name=":2">{{Cite journal |last=Cameron |first=A.G.W. |date=November 1959 |title=Pycnonuclear Relations and Nova Explosions |journal=Astrophysical Journal |volume=130 |pages=916 |doi=10.1086/146782|bibcode=1959ApJ...130..916C }}</ref>

== Astrophysics ==
'''Pycnonuclear reactions''' can occur anywhere and in any matter, but under standard conditions, the speed of the reaction is exceedingly low, and thus, have no significant role outside of [[Dense|extremely dense]] systems, neutron-rich and free electron-rich environments, such as the inner crust of a [[neutron star]].<ref name=":6" /><ref name=":4">{{Cite journal |last=Zeldovich |first=Y. B. |date=1957 |title=О ядерных реакциях в сверхплотном холодном водороде |url=http://elib.biblioatom.ru/text/zeldovich_izbannye-trudy_t2_1985/go,198/ |journal=Журнал экспериментальной и теоретической физики |language=Ru |volume=33 |pages=991}}</ref> A feature of pycnonuclear reactions is that the rate of the reaction is directly proportional to the density of the space that the reaction is occurring in, but is almost fully independent of the temperature of the environment.<ref name=":1" />

Pycnonuclear reactions are observed in [[neutron star]]s or [[white dwarf]]s, with evidence present of them occurring in lab-generated [[deuterium-tritium]] plasma.<ref name=":1" /><ref name=":2" /> Some speculations also relate the fact that [[Jupiter]] emits more [[radiation]] than it receives from the [[Sun]] with pycnonuclear reactions or [[cold fusion]].<ref name=":1" /><ref>{{Cite journal |last=Horowitz |first=C. J. |date=20 January 1991 |title=Cold nuclear fusion in dense metallic hydrogen |journal=Astrophysical Journal |volume=367 |pages=288–295 |doi=10.1086/169627 |bibcode=1991ApJ...367..288H |doi-access=free }}</ref>

=== Black dwarfs ===

=== White dwarfs ===
In [[white dwarf]]s, the core of the star is cold, under which conditions, so, if treated ''classically,'' the nuclei that arrange themselves into a [[crystal lattice]] are in their [[ground state]]. The [[Zero-point energy|zero-point oscillations]] of nuclei in the [[Crystal structure|crystal lattice]] with energy at the energy <math>E_0</math> at [[Gamow peak|Gamow's peak]] equal to <math>E_0 \thicksim \hbar w</math> can overcome the [[Coulomb barrier]], actuating pycnonuclear reactions. A semi-analytical model indicates that in white dwarfs, a [[Thermonuclear fusion|thermonuclear runaway]] can occur at much earlier ages than that of the [[Age of the universe|universe]], as the pycnonuclear reactions in the cores of white dwarfs exceed the luminosity of the white dwarfs, allowing [[Carbon-burning process|C-burning]] to occur, which catalyzes the formation of [[Supernova|type Ia supernovas]] in accreting white dwarfs, whose mass is equal to the [[Chandrasekhar mass]].<ref name=":0" /><ref>{{cite journal |last1=Chiosi |first1=E. |last2=Chiosi |first2=C. |last3=Trevisan |first3=P. |last4=Piovan |first4=L. |last5=Ovio |first5=M. |date=26 February 2015 |title=Exploring an alternative channel of evolution towards SNa&nbsp;Ia explosion |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=448 |issue=3 |pages=2100–2125 |doi=10.1093/mnras/stv084 |doi-access=free |arxiv=1409.1104 }}</ref><ref name=":3">{{cite journal |last1=Salpeter |first1=E. E. |last2=van Horn |first2=M.M. |date=January 1969 |title=Nuclear reaction rates at high densities |journal=Astrophysical Journal |volume=155 |pages=183 |doi=10.1086/149858 |bibcode=1969ApJ...155..183S }}</ref><ref name=":5">{{cite book |last1=Shapiro |first1=Stuart L. |last2=Teukolsky |first2=Saul A. |year=1983 |title=Black Holes, White Dwarfs, and Neutron Stars: The physics of compact objects |publisher=A Wiley-Interscience Publication |isbn=978-0-471-87316-7 |edition=1st }}{{pn|date=July 2024}}</ref>

Some studies indicate that the contribution of pycnonuclear reactions towards instability of white dwarfs is only significant in [[carbon]] [[white dwarfs]], while in [[oxygen]] white dwarfs, such instability is caused mostly due to [[electron capture]].<ref>{{cite journal |last1=Malheiro |first1=Manuel |last2=Otoniel |first2=Edson |last3=Coelho |first3=Jaziel G. |title=Relevance of Dynamical Nuclear Processes in Quantum Complex Systems of Massive White Dwarfs |journal=Brazilian Journal of Physics |date=April 2021 |volume=51 |issue=2 |pages=223–230 |doi=10.1007/s13538-020-00840-0 |bibcode=2021BrJPh..51..223M |s2cid=234330353 }}</ref> Although other authors disagree that the pycnonuclear reactions can act as major long-term heating sources for ''massive'' (1.25&nbsp;{{solar mass}}) white dwarfs, as their density would not suffice for a high rate of pycnonuclear reactions.<ref>{{cite journal |last=Horowitz |first=C.J. |date=2020-10-27 |title=Nuclear and dark matter heating in massive white dwarf stars |journal=Physical Review D |volume=102 |issue=8 |pages=083031 |doi=10.1103/PhysRevD.102.083031 |arxiv=2008.03291 |bibcode=2020PhRvD.102h3031H |s2cid=221083150 }}</ref>

While most studies indicate that at the end of their lifecycle, white dwarfs slowly decay into [[black dwarfs]], where pycnonuclear reactions slowly turn their cores into <chem>^56Fe</chem>, according to some versions, a collapse of [[black dwarfs]] is possible: M.E.&nbsp;Caplan (2020) theorizes that in the most massive black dwarfs (1.25&nbsp;{{solar mass}}), due to their declining electron fraction resulting from <chem>^56Fe</chem> production, they will exceed the [[Chandrasekhar limit]] in the very far future, speculating that their lifetime and delay time can stretch to up to {{10^|1100
}}&nbsp;years.<ref>{{cite journal |last=Caplan |first=M.E. |date=7 August 2020 |title=Black dwarf supernova in the far future |journal=[[Monthly Notices of the Royal Astronomical Society]] |volume=497 |issue=4 |pages=4357–4362 |doi=10.1093/mnras/staa2262 |doi-access=free |arxiv=2008.02296 }}</ref>

=== Neutron stars ===
{{Expand section|need expansion on energy release per nucleon reaction in magnetars. It requires an explanation of the arbitrary unit B* that the study has built upon|date=August 2022|small=yes}}
As the [[neutron stars]] undergo [[Accretion (astrophysics)|accretion]], the density in the crust increases, passing the [[electron capture]] threshold. As the electron capture threshold (<math>\rho = 1.455 * 10^{12}</math> g cm<sup>−3</sup>) is exceeded, it allows for the formation of light nuclei from the process of [[double electron capture]] (<chem>^40Mg + 2e -> ^34Ne + 6n + </chem>{{math|{{SubatomicParticle|Electron antineutrino}}}}), forming the light [[neon]] nuclei and [[free neutron]]s, which further increases the density of the crust.  As the density increases, the crystal lattices of neutron-rich nuclei are forced closer together due to [[Gravitational|gravitational collapse]] of [[Accretion (astrophysics)|accreting material]], and at a point where the nuclei are pushed so close together that their zero-point oscillations allow them to break through the [[Coulomb barrier]], [[Nuclear fusion|fusion]] occurs. While the main site of pycnonuclear fusion within neutron stars is the [[Crust (geology)|inner crust]], pycnonuclear reactions between light nuclei can occur even in the [[Plasma (physics)|plasma ocean]].<ref name=":9">{{Cite journal |last1=Haensel |first1=P |last2=Zdunik |first2=J. L. |date=January 1990 |title=Non-equilibrium processes in the crust of an accreting neutron star |journal=Astronomy and Astrophysics |volume=227 |issue=2 |pages=431–436 |bibcode=1990A&A...227..431H }}</ref><ref>{{Cite journal |last1=Horowitz |first1=C. J. |last2=Dussan |first2=H. |last3=Berry |first3=D. K. |date=29 April 2008 |title=Fusion of neutron-rich oxygen isotopes in the crust of accreting neutron stars |journal=Physical Review C |volume=77 |issue=4 |pages=045807 |doi=10.1103/PhysRevC.77.045807 |arxiv=0710.5714 |bibcode=2008PhRvC..77d5807H |s2cid=118639621 }}</ref> Since the core of neutron stars was approximated to be <math>3*10^{14}</math> g cm<sup>−3</sup>, at such extreme densities, pycnonuclear reactions play a large role as demonstrated by Haensel & Zdunik, who showed that at densities of <math>\rho = 10^{12} - 10^{13}</math> g cm<sup>−3</sup>, they serve as a major heat source.<ref>{{Cite journal |last1=Baym |first1=Gordon |last2=Hans A. |first2=Bethe |last3=Christopher J. |first3=Pethick |date=8 November 1971 |title=Neutron star matter |journal=Nuclear Physics A |volume=175 |issue=2 |pages=225–271 |doi=10.1016/0375-9474(71)90281-8 |bibcode=1971NuPhA.175..225B }}</ref><ref>{{Cite journal |last=Katsuhiko |first=Sato |date=February 1975 |title=Neutrino Degeneracy in Supernova Cores and Neutral Current of Weak Interaction |journal=Progress of Theoretical Physics |volume=53 |issue=2 |pages=595–597 |doi=10.1143/PTP.53.595 |bibcode=1975PThPh..53..595S |doi-access=free }}</ref><ref name=":10">{{Cite journal |last1=Haensel |first1=P. |last2=Zdunik |first2=J. L. |date=9 January 2008 |title=Models of crustal heating in accreting neutron stars |journal=Astronomy & Astrophysics |volume=480 |issue=2 |pages=459–464 |arxiv=0708.3996 |doi=10.1051/0004-6361:20078578 |bibcode=2008A&A...480..459H |s2cid=12594425 }}</ref> In the fusion processes of the inner crust, the burning of neutron-rich nuclei (<chem>^{34}Ne + ^{34}Ne -> ^68Ca</chem>)<ref name=":3" /><ref name=":9" /> releases a lot of heat, allowing pycnonuclear fusion to perform as a major energy source, possibly even acting as an energy basin for [[gamma-ray burst]]s.<ref name=":0" /><ref name=":6" />

Further studies have established that most [[magnetar]]s are found at densities of <math>\rho = 10^{10}-10^{11}</math>g cm<sup>−3</sup>, indicating that pycnonuclear reactions along with subsequent electron capture reactions could serve as major heat sources.<ref>{{Cite journal |last1=Chamel |first1=Nicolas |last2=Fantina |first2=Anthea Francesca |last3=Suleiman |first3=Lami |last4=Zdunik |first4=Julian-Leszek |last5=Haensel |first5=Pawel |date=June 2021 |title=Heating in Magnetar Crusts from Electron Captures |journal=Universe |volume=7 |issue=6 |pages=193 |doi=10.3390/universe7060193 |arxiv=2106.05040 |bibcode=2021Univ....7..193C |doi-access=free }}</ref>

=== Triple-alpha reaction ===
{{Expand section|date=August 2022|talksection=Gasques, L. R.; Afanasjev, A. V.; et al. (29 August 2005) & D. G. Yakovlev, M. Beard, L. R. Gasques, and M. Wiescher, Phys. Rev. C 82, 044609 (2010)|small=yes}}

In [[Wolf–Rayet star]]s, the triple-alpha reaction is accommodated by the low-energy of <chem>^8Be</chem> [[resonance]]. However, in neutron stars the temperature in the core is so low that the triple-alpha reactions can occur via the pycnonuclear pathway.<ref name=":7">{{cite journal |last1=Schramm |first1=S. |last2=Koonin |first2=S. E. |title=Pycnonuclear fusion rates |journal=The Astrophysical Journal |date=December 1990 |volume=365 |pages=296 |bibcode=1990ApJ...365..296S |doi=10.1086/169482 }}</ref>

== Mathematical model ==

{{Expand section|date=August 2022|talksection=Gasques, L. R.; Afanasjev, A. V.; et al. (29 August 2005) & D. G. Yakovlev, M. Beard, L. R. Gasques, and M. Wiescher, Phys. Rev. C 82, 044609 (2010)|small=yes|The article should expand on the ideas of the rate of pycnonuclear fusion in MCP in various conditions which depend on the temperature, screening and also the reaction of the lattice to tunneling}}

As the density increases, the [[Gamow peak]] increases in height and shifts towards lower energy, while the potential barriers are depressed. If the potential barriers are depressed by the amount of <math>E_0</math>, the Gamow peak is shifted across the origin, making the reactions density-dependent, as the Gamow peak energy is much larger than the thermal energy. The material becomes a [[degenerate gas]] at such densities. Harrison proposed that models fully independent of temperature be called ''cryonuclear''.<ref>{{Cite journal |last=Harrison |first=E. R. |date=August 1964 |title=Thermonuclear and pycnonuclear reactions |journal=Proceedings of the Physical Society |volume=84 |issue=213 |pages=213–229 |doi=10.1088/0370-1328/84/2/304 |bibcode=1964PPS....84..213H }}</ref>

Pycnonuclear reactions can proceed in two ways: direct (<chem>^{34}Ne + ^{34}Ne</chem> or <chem>^{40}Mg + ^{40}Mg</chem>) or through chain of electron capture reactions (<chem>^25N + ^40Mg</chem>).<ref>{{cite journal |last1=Lau |first1=R. |last2=Beard |first2=M. |last3=Gupta |first3=S. S. |last4=Schatz |first4=H. |last5=Afanasjev |first5=A. V. |last6=Brown |first6=E. F. |last7=Deibel |first7=A. |last8=Gasques |first8=L. R. |last9=Hitt |first9=G. W. |last10=Hix |first10=W. R. |last11=Keek |first11=L. |last12=Möller |first12=P. |last13=Shternin |first13=P. S. |last14=Steiner |first14=A. W. |last15=Wiescher |first15=M. |last16=Xu |first16=Y. |title=Nuclear Reactions in the Crusts of Accreting Neutron Stars |journal=The Astrophysical Journal |date=20 May 2018 |volume=859 |issue=1 |pages=62 |doi=10.3847/1538-4357/aabfe0 |arxiv=1803.03818 |bibcode=2018ApJ...859...62L |s2cid=56316978 |doi-access=free }}</ref>

=== Uncertainties ===
The current consensus on the ''rate'' of pycnonuclear reactions is not coherent. There are currently a lot of uncertainties to consider when modelling the rate of pycnonuclear reactions, especially in spaces with high numbers of free particles. The primary focus of current research is on the effects of crystal lattice deformation and the presence of free neutrons on the reaction rate. Every time fusion occurs, nuclei are removed from the crystal lattice - creating a defect. The difficulty of approximating this model lies within the fact that the further changes occurring to the lattice and the effect of various deformations on the rate are thus far unknown. Since neighbouring lattices can affect the rate of reaction too, negligence of such deformations could lead to major discrepancies.<ref name=":3" /><ref name=":8">{{Cite book |last=Yakovlev |first=D. G. |url=http://www.femto.com.ua/articles/part_2/2828.html |title=Пикноядерные реакции. Физическая энциклопедия Т.5 |year=1992 |isbn=5-85270-034-7 |pages=672 |publisher=Советская энциклопедия |language=Ru}}</ref> Another confounding variable would be the presence of free neutrons in the crusts of [[neutron stars]]. The presence of free neutrons could potentially affect the [[Coulomb barrier]], making it either taller or thicker. A study published by D.G. Yakovlev in 2006 has shown that the rate calculation of the ''first'' pycnonuclear fusion of two <chem>^{34}Ne</chem> nuclei in the crust of a neutron star can have an uncertainty magnitude of up to ''seven''. In this study, Yakovlev also highlighted the uncertainty in the threshold of pycnonuclear fusion (e.g., at what density it starts), giving the approximate density required for the start of pycnonuclear fusion of <math>\rho_{pyc} \thickapprox 10^{12} - 10^{13} </math>g cm<sup>−3</sup>, arriving at a similar conclusion as Haesnel and Zdunik.<ref name=":3" /><ref name=":10" /><ref name=":11">{{Cite journal |last1=Yakovlev |first1=D. G. |last2=Gasques |first2=L. |last3=Wiescher |first3=M. |date=18 August 2006 |title=Pycnonuclear burning of 34Ne in accreting neutron stars |journal=Monthly Notices of the Royal Astronomical Society |volume=371 |issue=3 |pages=1322–1326 |doi=10.1111/j.1365-2966.2006.10753.x |bibcode=2006MNRAS.371.1322Y |s2cid=119393244 |doi-access=free }}</ref> According to Haesnel and Zdunik, additional uncertainty of rate calculations in neutron stars can also be due to uneven distribution of the crustal heating, which can impact the thermal states of neutron stars before and after [[Accretion (astrophysics)|accretion]].<ref name=":10" />

<!-- The Ukrainian research attempted to explain the results of Fleischmann M. and Pons S. in their 1989 electrolysis paper. This might be interesting for people filling on the Cold fusion article. -->In white dwarfs and neutron stars, the nuclear reaction rates can not only be affected by pycnonuclear reactions but also by the plasma screening of the Coulomb interaction.<ref name=":6" /><ref name=":3" /> A [[Ukraine|Ukrainian]] Electrodynamic Research Laboratory "Proton-21", established that by forming a thin [[Plasma (physics)|electron plasma]] layer on the surface of the target material, and, thus, forcing the self-compression of the target material at low temperatures, they could stimulate the process of pycnonuclear fusion. The startup of the process was due to the self-contracting [[Plasma (physics)|plasma]] "scanning" the entire volume of the target material, screening the [[Electric field|Coulomb field]].<ref>{{cite book |doi=10.1007/978-1-4020-5874-5 |title=Controlled Nucleosynthesis |date=2007 |isbn=978-1-4020-5873-8 |editor-last1=Adamenko |editor-last2=Selleri |editor-last3=Van Der Merwe |editor-first1=Stanislav |editor-first2=Franco |editor-first3=Alwyn }}{{pn|date=July 2024}}</ref>

=== Screening, Quantum Diffusion & Nuclear Fusion Regimes ===
Before delving into the mathematical model, it is important to understand that pycnonuclear fusion, in its essence, occurs due to two main events:

* A phenomenon of quantum nature called [[Quantum tunnelling|'''quantum diffusion''']]'''.'''
* Overlap of the wave functions of '''zero-point oscillations''' of the nuclei.

Both of these effects are heavily affected by screening. The term screening is generally used by nuclear physicists when referring to plasmas of particularly high density. In order for the pycnonuclear fusion to occur, the two particles must overcome the electrostatic repulsion between them - the energy required for this is called the Coulomb barrier. Due to the presence of other charged particles (mainly electrons) next to the reacting pair, they exert the effect of shielding - as the electrons create an [[electron cloud]] around the positively charged ions - effectively reducing the electrostatic repulsion between them, lowering the Coulomb barrier. This phenomenon of shielding is referred to as "'''screening'''", and in cases where it is particularly strong, it is called "'''strong screening'''". Consequently, in cases where the plasma has a strong screening effect, the rate of pycnonuclear fusion is substantially enhanced.<ref>{{cite journal |last1=Gasques |first1=L. R. |last2=Afanasjev |first2=A. V. |last3=Aguilera |first3=E. F. |last4=Beard |first4=M. |last5=Chamon |first5=L. C. |last6=Ring |first6=P. |last7=Wiescher |first7=M. |last8=Yakovlev |first8=D. G. |title=Nuclear fusion in dense matter: Reaction rate and carbon burning |journal=Physical Review C |date=29 August 2005 |volume=72 |issue=2 |page=025806 |doi=10.1103/PhysRevC.72.025806 |arxiv=astro-ph/0506386 |bibcode=2005PhRvC..72b5806G |s2cid=11119094 }}</ref>

[[Quantum tunnelling]] is the foundation of the quantum physical approach to pycnonuclear fusion. It is closely intertwined with the screening effect, as the ''transmission coefficient'' <math>T
</math> depends on the height of the [[potential barrier]], the mass of the particles, and their relative velocity (since the total energy of the system depends on the kinetic energy). From this follows that the transmission coefficient is very sensitive to the effects of screening. Thus, the effect of screening not only contributes to the reduction of the potential barrier that allows for "classical" fusion to occur via the overlap of the wave functions of the zero-point oscillations of the particles, but also to the increase of the transmission coefficient, both of which increase the rate of pycnonuclear fusion.<ref name=":5" /><ref name=":11" />

On top of the other various jargon related to pycnonuclear fusion, the papers also introduce various '''regimes''', that define the rate of pycnonuclear fusion. Specifically, they identify the '''zero-temperature''', '''intermediate''', and '''thermally-enhanced''' regimes as their main ones.<ref name=":3" />

==== One-Component Plasma (OCP) ====
The pioneers to the derivation of the rate of pycnonuclear fusion in one-component plasma (OCP) were Edwin Salpeter and David Van-Horn, with their article published in 1969. Their approach used a semiclassical method to solve the [[Schrödinger equation]] by using the [[WKB approximation|Wentzel-Kramers-Brillouin]] (WKB) approximation, and [[Wigner–Seitz radius|Wigner-Seitz]] (WS) spheres. Their model is heavily simplified, and whilst it is primitive, is required to understand other approaches which largely extrapolated on the works of Salpeter & Van Horn. They employed the WS spheres to simplify the OCP into regions containing one ion each, with the ions situated on the vertices of a [[Body centred cubic|BCC]] crystal lattice. Then, using the WKB approximation, they resolved the effect of quantum tunnelling on the fusing nuclei. Extrapolating this to the entire lattice allowed them to arrive at their formula for the rate of pycnonuclear fusion:<ref name=":3" />
<math display="block">P = {8 \over 2}{\rho \over \mu_AH} \langle p \rangle_{A_v}</math>
where <math>\rho</math> is the density of the plasma, <math>\mu_A</math> is the mean molecular weight per electron (atomic nucleus), <math>H</math> is a constant equal to <math>1.66044*10^{-24}</math> and serves as a conversion factor from atomic mass units to grams, and <math>\langle p \rangle_{A_v}</math> represents the thermal average of the pairwise reaction probability.

However, the big fault of the method proposed by Salpeter & Van-Horn is that they neglected the ''dynamic model'' of the lattice. This was improved upon by Schramm and Koonin in 1990. In their model, they found that the dynamic model cannot be neglected, but it is possible that the effects caused by the dynamicity can be cancelled out.<ref name=":3" /><ref name=":7" />

== See also ==
* [[Cold fusion]]
* [[Thermonuclear reaction]]
* [[Accretion (astrophysics)]]
* [[Plasma (physics)]]
* [[Quantum tunnelling]]

== References ==

{{reflist}}

[[Category:Nuclear fusion]]
[[Category:Neutron sources]]
[[Category:Astronomy]]