View source for Beta decay

{{short description|Type of radioactive decay}}
[[Image:Beta-minus Decay.svg|thumb|240px|{{SubatomicParticle|Beta-}}&nbsp;decay in an [[atomic nucleus]] (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron. Neither of these depictions shows the intermediate [[Virtual particle|virtual]] {{SubatomicParticle|W boson-|link=yes}} boson.]]
{{Nuclear physics}} 
In [[nuclear physics]], '''beta decay''' (β-decay) is a type of [[radioactive decay]] in which an [[atomic nucleus]] emits a [[beta particle]] (fast energetic [[electron]] or [[positron]]), transforming into an [[isobar (nuclide)|isobar]] of that nuclide. For example, beta decay of a [[neutron]] transforms it into a [[proton]] by the emission of an electron accompanied by an [[antineutrino]]; or, conversely a proton is converted into a neutron by the emission of a positron with a [[neutrino]] in what is called ''[[positron emission]]''. Neither the beta particle nor its associated (anti-)neutrino exist within the nucleus prior to beta decay, but are created in the decay process. By this process, unstable atoms obtain a more stable [[proton–neutron ratio|ratio of protons to neutrons]]. The probability of a nuclide decaying due to beta and other forms of decay is determined by its [[nuclear binding energy]]. The binding energies of all existing nuclides form what is called the nuclear band or [[valley of stability]].<ref name="konya74">
{{cite book
 |last1=Konya |first1=J.
 |last2=Nagy |first2=N. M.
 |year=2012
 |title=Nuclear and Radio-chemistry
 |pages=74–75
 |publisher=[[Elsevier]]
 |isbn=978-0-12-391487-3
}}</ref> For either electron or positron emission to be energetically possible, the energy release ([[#Energy release|see below]]) or [[Q value (nuclear science)|''Q'' value]] must be positive.

Beta decay is a consequence of the [[Weak interaction|weak force]], which is characterized by relatively long decay times. Nucleons are composed of [[up quark]]s and [[down quark]]s,<ref>{{cite journal
 |last1=Bijker |first1=R.
 |last2=Santopinto |first2=E.
 |year=2015
 |title=Valence and sea quarks in the nucleon
 |journal=[[Journal of Physics: Conference Series]]
 |volume=578 |issue=1
 |page=012015
 |arxiv= 1412.5559
 |bibcode=2015JPhCS.578a2015B
 |doi=10.1088/1742-6596/578/1/012015
| s2cid=118499855 }}</ref> and the weak force allows a [[quark]] to change its [[flavour (particle physics)|flavour]] by means of a virtual [[W boson]] leading to creation of an electron/antineutrino or positron/neutrino pair. For example, a neutron, composed of two down quarks and an up quark, decays to a proton composed of a down quark and two up quarks.

[[Electron capture]] is sometimes included as a type of beta decay,<ref>{{Cite book
 |last1=Cottingham
 |first1=W. N.
 |last2=Greenwood
 |first2=D. A.
 |year=1986
 |title=An introduction to nuclear physics
 |page=[https://archive.org/details/introductiontonu0000cott/page/40 40]
 |publisher=[[Cambridge University Press]]
 |isbn=978-0-521-31960-7
 |url=https://archive.org/details/introductiontonu0000cott/page/40 }}</ref> because the basic nuclear process, mediated by the weak force, is the same. In electron capture, an inner atomic electron is captured by a proton in the nucleus, transforming it into a neutron, and an [[electron neutrino]] is released.

==Description==
The two types of beta decay are known as ''beta minus'' and ''beta plus''. In beta minus (β<sup>−</sup>) decay, a neutron is converted to a proton, and the process creates an electron and an [[electron antineutrino]]; while in beta plus (β<sup>+</sup>) decay, a proton is converted to a neutron and the process creates a positron and an electron neutrino. β<sup>+</sup> decay is also known as [[positron emission]].<ref>{{Cite book
 |last1=Basdevant |first1=J.-L.
 |last2=Rich |first2=J.
 |last3=Spiro |first3=M.
 |year=2005
 |title=Fundamentals in Nuclear Physics: From Nuclear Structure to Cosmology
 |publisher=[[Springer (publisher)|Springer]]
 |isbn=978-0-387-01672-6 }}</ref>

Beta decay conserves a quantum number known as the [[lepton number]], or the number of electrons and their associated neutrinos (other leptons are the [[muon]] and [[Tau (particle)|tau]] particles). These particles have lepton number +1, while their antiparticles have lepton number −1. Since a proton or neutron has lepton number zero, β<sup>+</sup> decay (a positron, or antielectron) must be accompanied with an electron neutrino, while β<sup>−</sup> decay (an electron) must be accompanied by an electron antineutrino.

An example of electron emission (β<sup>−</sup> decay) is the decay of [[carbon-14]] into [[nitrogen-14]] with a [[half-life]] of about 5,730 years:
:{{nuclide|carbon|14}} → {{nuclide|nitrogen|14}} + {{subatomic particle|electron}} + {{math|{{subatomic particle|electron antineutrino}}}}

In this form of decay, the original element becomes a new chemical element in a process known as [[nuclear transmutation]]. This new element has an unchanged [[mass number]] {{mvar|A}}, but an [[atomic number]] {{mvar|Z}} that is increased by one. As in all nuclear decays, the decaying element (in this case {{nuclide|carbon|14}}) is known as the ''parent nuclide'' while the resulting element (in this case {{nuclide|nitrogen|14}}) is known as the ''daughter nuclide''.

Another example is the decay of hydrogen-3 ([[tritium]]) into [[helium-3]] with a half-life of about 12.3 years:
:{{nuclide|hydrogen|3}} → {{nuclide|helium|3}} + {{subatomic particle|electron}} + {{math|{{subatomic particle|electron antineutrino}}}}

An example of positron emission (β<sup>+</sup> decay) is the decay of [[magnesium-23]] into [[sodium-23]] with a half-life of about 11.3 s:

:{{nuclide|Magnesium|23}} → {{nuclide|Sodium|23}} + {{subatomic particle|positron}} + {{math|{{subatomic particle|electron neutrino}}}}
β<sup>+</sup> decay also results in nuclear transmutation, with the daughter element having an atomic number that is decreased by one.

[[File:RaE1.jpg|thumb|A beta spectrum, showing a typical division of energy between electron and antineutrino]]
The beta spectrum, or distribution of energy values for the beta particles, is continuous. The total energy of the decay process is divided between the electron, the antineutrino, and the recoiling nuclide. In the figure to the right, an example of an electron with 0.40 MeV energy from the beta decay of <sup>210</sup>Bi is shown. In this example, the total decay energy is 1.16 MeV, so the antineutrino has the remaining energy: {{nowrap|1=1.16 MeV − 0.40 MeV = 0.76 MeV}}. An electron at the far right of the curve would have the maximum possible kinetic energy, leaving the energy of the neutrino to be only its small rest mass.

==History==

===Discovery and initial characterization===
Radioactivity was discovered in 1896 by [[Henri Becquerel]] in [[uranium]], and subsequently observed by [[Marie Curie|Marie]] and [[Pierre Curie]] in [[thorium]] and in the newly discovered elements [[polonium]] and [[radium]]. In 1899, [[Ernest Rutherford]] separated radioactive emissions into two types: alpha and beta (now beta minus), based on penetration of objects and ability to cause ionization. Alpha rays could be stopped by thin sheets of paper or aluminium, whereas beta rays could penetrate several millimetres of aluminium. In 1900, [[Paul Villard]] identified a still more penetrating type of radiation, which Rutherford identified as a fundamentally new type in 1903 and termed [[gamma ray]]s. Alpha, beta, and gamma are the first three letters of the [[Greek alphabet]].

In 1900, Becquerel measured the [[mass-to-charge ratio]] ({{math|''m''/''e''}}) for beta particles by the method of [[J.J.&nbsp;Thomson]] used to study cathode rays and identify the electron. He found that {{math|''m''/''e''}} for a beta particle is the same as for Thomson's electron, and therefore suggested that the beta particle is in fact an electron.<ref name="Handbook">{{cite book|last1=L'Annunziata|first1=Michael|title=Handbook of Radioactivity Analysis|date=2012|publisher=Elsevier Inc.|page=3|edition=Third|url=https://books.google.com/books?id=S4FrejzJy0cC&pg=PA3|access-date=4 October 2017|isbn=978-0-12-384874-1}}</ref>

In 1901, Rutherford and [[Frederick Soddy]] showed that alpha and beta radioactivity involves the [[Nuclear transmutation|transmutation]] of atoms into atoms of other chemical elements. In 1913, after the products of more radioactive decays were known, Soddy and [[Kazimierz Fajans]] independently proposed their [[Radioactive displacement law of Fajans and Soddy|radioactive displacement law]], which states that beta (i.e., {{SubatomicParticle|Beta-}})&nbsp;emission from one element produces another element one place to the right in the [[periodic table]], while alpha emission produces an element two places to the left.

===Neutrinos===

The study of beta decay provided the first physical evidence for the existence of the [[neutrino]]. In both alpha and gamma decay, the resulting alpha or gamma particle has a narrow energy [[Frequency distribution|distribution]], since the particle carries the energy from the difference between the initial and final nuclear states. However, the kinetic energy distribution, or spectrum, of beta particles measured by [[Lise Meitner]] and [[Otto Hahn]] in 1911 and by [[Jean Danysz]] in 1913 showed multiple lines on a diffuse background. These measurements offered the first hint that beta particles have a continuous spectrum.<ref name="Jensen">{{cite book
 |last1=Jensen |first1=C.
 |year=2000
 |title=Controversy and Consensus: Nuclear Beta Decay 1911-1934
 |url=https://www.springer.com/birkhauser/physics/book/978-3-7643-5313-1
 |publisher=[[Birkhäuser Verlag]]
 |isbn=978-3-7643-5313-1
}}</ref> In 1914, [[James Chadwick]] used a magnetic [[spectrometer]] with one of [[Hans Geiger|Hans Geiger's]] new [[Geiger counter|counters]] to make more accurate measurements which showed that the spectrum was continuous.<ref name="Jensen" /><ref>{{cite journal |last=Chadwick |first=J. |year=1914 |title=Intensitätsverteilung im magnetischen Spektren der β-Strahlen von Radium B + C |journal=[[Verhandlungen der Deutschen Physikalischen Gesellschaft]] |language=de |volume=16 |pages=383–391}}</ref> The results, which appeared to be in contradiction to the [[law of conservation of energy]], were validated by means of calorimetric measurements in 1929 by [[Lise Meitner]] and [[Wilhelm Orthmann]].<ref>{{Cite journal |last=Meitner |first=Lise |last2=Orthmann |first2=Wilhelm |date=1930-03-01 |title=Über eine absolute Bestimmung der Energie der primären β-Strahlen von Radium E |url=https://link.springer.com/article/10.1007/BF01339819 |journal=Zeitschrift für Physik |language=de |volume=60 |issue=3 |pages=143–155 |doi=10.1007/BF01339819 |issn=0044-3328}}</ref> If beta decay were simply electron emission as assumed at the time, then the energy of the emitted electron should have a particular, well-defined value.<ref name=Brown>{{cite journal
 |last1=Brown |first1=L. M.
 |year=1978
 |title=The idea of the neutrino
 |journal=[[Physics Today]]
 |volume=31 |issue=9 |pages=23–8
 |bibcode=1978PhT....31i..23B
 |doi=10.1063/1.2995181
}}</ref> For beta decay, however, the observed broad distribution of energies suggested that energy is lost in the beta decay process. This spectrum was puzzling for many years.

A second problem is related to the [[conservation of angular momentum]]. Molecular band spectra showed that the [[nuclear spin]] of [[nitrogen-14]] is 1 (i.e., equal to the [[reduced Planck constant]]) and more generally that the spin is integral for nuclei of even [[mass number]] and half-integral for nuclei of odd mass number. This was later explained by the [[Discovery of the neutron#Proton–neutron model of the nucleus|proton-neutron model of the nucleus]].<ref name=Brown/> Beta decay leaves the mass number unchanged, so the change of nuclear spin must be an integer. However, the electron spin is 1/2, hence angular momentum would not be conserved if beta decay were simply electron emission.

From 1920 to 1927, [[Charles Drummond Ellis]] (along with Chadwick and colleagues) further established that the beta decay spectrum is continuous. In 1933, Ellis and [[Nevill Mott]] obtained strong evidence that the beta spectrum has an effective upper bound in energy. [[Niels Bohr]] had suggested that the beta spectrum could be explained if [[conservation of energy]] was true only in a statistical sense, thus this [[Laws of science|principle]] might be violated in any given decay.<ref name=Brown/>{{rp|27}} However, the upper bound in beta energies determined by Ellis and Mott ruled out that notion. Now, the problem of how to account for the variability of energy in known beta decay products, as well as for conservation of momentum and angular momentum in the process, became acute.

In a [[Electron neutrino#Pauli's letter|famous letter]] written in 1930, [[Wolfgang Pauli]] attempted to resolve the beta-particle energy conundrum by suggesting that, in addition to electrons and protons, atomic nuclei also contained an extremely light neutral particle, which he called the neutron. He suggested that this "neutron" was also emitted during beta decay (thus accounting for the known missing energy, momentum, and angular momentum), but it had simply not yet been observed. In 1931, [[Enrico Fermi]] renamed Pauli's "neutron" the "neutrino" ('little neutral one' in Italian). In 1933, Fermi published his landmark [[Fermi's interaction|theory for beta decay]], where he applied the principles of quantum mechanics to matter particles, supposing that they can be created and annihilated, just as the light quanta in atomic transitions. Thus, according to Fermi, neutrinos are created in the beta-decay process, rather than contained in the nucleus; the same happens to electrons. The neutrino interaction with matter was so weak that detecting it proved a severe experimental challenge. Further indirect evidence of the existence of the neutrino was obtained by observing the recoil of nuclei that emitted such a particle after absorbing an electron. Neutrinos were finally detected directly in 1956 by the American physicists [[Clyde Cowan]] and [[Frederick Reines]] in the [[Cowan–Reines neutrino experiment]].<ref>{{cite journal
 |last1=Cowan |first1=C. L. Jr.
 |last2=Reines |first2=F.
 |last3=Harrison |first3=F. B.
 |last4=Kruse |first4=H. W.
 |last5=McGuire |first5=A. D.
 |year=1956
 |title=Detection of the Free Neutrino: a Confirmation
 |journal=[[Science (journal)|Science]]
 |volume=124 |issue=3212 |pages=103–104
 |bibcode=1956Sci...124..103C
 |doi=10.1126/science.124.3212.103
 |pmid=17796274
}}</ref> The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.

==={{SubatomicParticle|Beta+}}&nbsp;decay and electron capture===
In 1934, [[Frédéric Joliot-Curie|Frédéric]] and [[Irène Joliot-Curie]] bombarded aluminium with alpha particles to effect the nuclear reaction {{nuclide|Helium|4}}&nbsp;+&nbsp;{{nuclide|Aluminium|27}}&nbsp;→ {{nuclide|Phosphorus|30}}&nbsp;+&nbsp;{{nuclide|neutronium|1}}, and observed that the product isotope {{nuclide|Phosphorus|30}} emits a positron identical to those found in cosmic rays (discovered by [[Carl David Anderson]] in 1932). This was the first example of {{SubatomicParticle|Beta+}}&nbsp;decay ([[positron emission]]), which they termed [[artificial radioactivity]] since {{nuclide|Phosphorus|30}} is a short-lived nuclide which does not exist in nature. In recognition of their discovery, the couple were awarded the [[Nobel Prize in Chemistry]] in 1935.<ref>{{Cite web|url=https://www.nobelprize.org/nobel_prizes/chemistry/laureates/1935/|title=The Nobel Prize in Chemistry 1935|website=www.nobelprize.org|access-date=2018-04-25}}</ref>

The theory of [[electron capture]] was first discussed by [[Gian-Carlo Wick]] in a 1934 paper, and then developed by [[Hideki Yukawa]] and others. K-electron capture was first observed in 1937 by [[Luis Walter Alvarez|Luis Alvarez]], in the nuclide <sup>48</sup>V.<ref name=k>{{cite book
 |last=Segré
 |first=E.
 |year=1987
 |chapter=K-Electron Capture by Nuclei
 |editor1-last=Trower
 |editor1-first=P. W.
 |title=Discovering Alvarez: Selected Works of Luis W. Alvarez
 |pages=[https://archive.org/details/discoveringalvar0000alva/page/11 11–12]
 |publisher=[[University of Chicago Press]]
 |isbn=978-0-226-81304-2
 |chapter-url=https://archive.org/details/discoveringalvar0000alva/page/11
 }}</ref><ref>{{cite web |title=The Nobel Prize in Physics 1968: Luis Alvarez |url=http://nobelprize.org/nobel_prizes/physics/laureates/1968/alvarez-bio.html |publisher=[[The Nobel Foundation]] |access-date=2009-10-07 }}</ref><ref>{{cite journal
 |last1=Alvarez |first1=L. W.
 |year=1937
 |title=Nuclear K Electron Capture
 |journal=[[Physical Review]]
 |volume=52 |issue=2 |pages=134–135
 |bibcode=1937PhRv...52..134A
 |doi=10.1103/PhysRev.52.134
}}</ref> Alvarez went on to study electron capture in <sup>67</sup>Ga and other nuclides.<ref name=k /><ref>{{cite journal
 |last1=Alvarez |first1=L. W.
 |year=1938
 |title=Electron Capture and Internal Conversion in Gallium 67
 |journal=[[Physical Review]]
 |volume=53 |issue=7 |page=606
 |bibcode=1938PhRv...53..606A
 |doi=10.1103/PhysRev.53.606
}}</ref><ref>{{cite journal
 |last1=Alvarez|first1=L. W.
 |year=1938
 |title=The Capture of Orbital Electrons by Nuclei
 |journal=[[Physical Review]]
 |volume=54 |issue=7 |pages=486–497
 |bibcode=1938PhRv...54..486A
 |doi=10.1103/PhysRev.54.486
}}</ref>

===Non-conservation of parity===
In 1956, [[Tsung-Dao Lee]] and [[Chen Ning Yang]] noticed that there was no evidence that [[parity (physics)|parity]] was conserved in weak interactions, and so they postulated that this symmetry may not be preserved by the weak force. They sketched the design for an experiment for testing conservation of parity in the laboratory.<ref>{{cite journal
 |last1=Lee |first1=T. D.
 |last2=Yang |first2=C. N.
 |year=1956
 |title=Question of Parity Conservation in Weak Interactions
 |journal=[[Physical Review]]
 |volume=104 |issue=1
 |pages=254–258
 |bibcode=1956PhRv..104..254L
 |doi=10.1103/PhysRev.104.254
|doi-access=free
 }}</ref> Later that year, [[Chien-Shiung Wu]] and coworkers conducted the [[Wu experiment]] showing an asymmetrical beta decay of [[cobalt-60|{{SimpleNuclide|cobalt|60}}]] at cold temperatures that proved that parity is not conserved in beta decay.<ref name="Wu1957" >{{cite journal
 |last1=Wu |first1=C.-S.
 |last2=Ambler |first2=E.
 |last3=Hayward |first3=R. W.
 |last4=Hoppes |first4=D. D.
 |last5=Hudson |first5=R. P.
 |year=1957
 |title=Experimental Test of Parity Conservation in Beta Decay
 |journal=[[Physical Review]]
 |volume=105 |issue=4
 |pages=1413–1415
 |bibcode=1957PhRv..105.1413W
 |doi=10.1103/PhysRev.105.1413
|doi-access=free
 }}</ref><ref>{{cite web|url=http://blogs.scientificamerican.com/guest-blog/2013/10/15/channeling-ada-lovelace-chien-shiung-wu-courageous-hero-of-physics/|title=Channeling Ada Lovelace: Chien-Shiung Wu, Courageous Hero of Physics| first=Maia|last=Weinstock | website=scientificamerican.com}}</ref> This surprising result overturned long-held assumptions about parity and the weak force. In recognition of their theoretical work, Lee and Yang were awarded the [[Nobel Prize for Physics]] in 1957.<ref>{{cite web |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1957/ | title=The Nobel Prize in Physics 1957 |publisher=[[The Nobel Foundation]] |access-date=March 24, 2015}}</ref> However Wu, who was female, was not awarded the Nobel prize.<ref>{{cite web|last=Webb| first=Richard| title=Chien-Shiung Wu {{!}} Particle physicist denied a Nobel prize| website=newscientist.com| url=https://www.newscientist.com/people/chien-shiung-wu/| access-date=February 18, 2025}}</ref>

==β<sup>−</sup> decay <span class="anchor" id="beta minus decay"></span>==
[[Image:Beta Negative Decay.svg|thumb|right|The leading-order [[Feynman diagram]] for {{SubatomicParticle|Beta-}}&nbsp;decay of a [[neutron]] into a [[proton]], [[electron]], and [[electron antineutrino]] via a virtual [[W boson|{{SubatomicParticle|W boson-}} boson]]. For higher-order diagrams see <ref>{{Cite journal|last1=Ivanov|first1=A. N.|last2=Höllwieser|first2=R.| last3=Troitskaya|first3=N. I.|last4=Wellenzohn|first4=M.|last5=Berdnikov|first5=Ya. A.|date=2017-06-26|title=Precision theoretical analysis of neutron radiative beta decay to order O ( α 2 / π 2 )|journal=Physical Review D|language=en|volume=95| issue=11| page=113006|doi=10.1103/PhysRevD.95.113006|issn=2470-0010|arxiv=1706.08687|bibcode=2017PhRvD..95k3006I| s2cid=119103283}}</ref><ref>{{Cite journal|last1=Ivanov|first1=A. N.|last2=Höllwieser|first2=R.|last3=Troitskaya|first3=N. I.| last4=Wellenzohn|first4=M.|last5=Berdnikov|first5=Ya. A.|date=2018-11-30|title=Gauge properties of hadronic structure of nucleon in neutron radiative beta decay to order O(α/π) in standard ''V – A'' effective theory with QED and linear sigma model of strong low-energy interactions|journal=International Journal of Modern Physics A|language=en|volume=33|issue=33|page=1850199| doi=10.1142/S0217751X18501993| issn=0217-751X|arxiv=1805.09702|bibcode=2018IJMPA..3350199I |s2cid=119088802}}</ref>]]
In {{SubatomicParticle|Beta-}}&nbsp;decay, the [[weak interaction]] converts an [[atomic nucleus]] into a nucleus with [[atomic number]] increased by one, while emitting an electron ({{SubatomicParticle|Electron|link=yes}}) and an electron [[Neutrino#Antineutrinos|antineutrino]] ({{SubatomicParticle|Electron antineutrino}}). {{SubatomicParticle|Beta-}}&nbsp;decay generally occurs in neutron-rich nuclei.<ref name=Loveland>{{cite book
 |last=Loveland |first=W. D.
 |year=2005
 |title=Modern Nuclear Chemistry
 |url=https://books.google.com/books?id=ZAHJkrJlwbYC&pg=PA233
 |page=232
 |publisher=[[John Wiley & Sons|Wiley]]
 |isbn=978-0-471-11532-8
}}</ref> The generic equation is:

: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''+1}}|X′}} + {{SubatomicParticle|Electron}} + {{math|{{SubatomicParticle|Electron Antineutrino}}}}<ref name="konya74"/>
where {{mvar|A}} and {{mvar|Z}} are the [[mass number]] and [[atomic number]]<!-- "charge" is linked from another place. "mass number" is paired with "atomic number" --> of the decaying nucleus, and X and X′ are the initial and final elements, respectively.

Another example is when the [[free neutron]] ({{nuclide|neutronium|1}}) decays by {{SubatomicParticle|Beta-}}&nbsp;decay into a proton ({{SubatomicParticle|Proton}}):
: {{SubatomicParticle|Neutron}} → {{SubatomicParticle|Proton}} + {{SubatomicParticle|Electron}} + {{math|{{SubatomicParticle|Electron Antineutrino}}}}.
<!-- never use <math> for it, because symbols of particles may not be italicized! -->

At the [[fundamental particle|fundamental]] level (as depicted in the [[Feynman diagram]] on the right), this is caused by the conversion of the negatively charged ({{math|−{{sfrac|1|3}} [[elementary charge|e]]}}) down quark to the positively charged ({{math|+{{sfrac|2|3}} e}}) up quark, which is promoted by a virtual [[W boson|{{SubatomicParticle|W boson-}} boson]]; the {{SubatomicParticle|W boson-}} boson subsequently decays into an electron and an electron antineutrino:
:{{Subatomic particle|down quark}} → {{Subatomic particle|up quark}} + {{SubatomicParticle|Electron}} + {{math|{{SubatomicParticle|Electron Antineutrino}}}}.

==β<sup>+</sup> decay <span class="anchor" id="beta plus decay"></span>==
{{main|Positron emission}}
[[File:Electron Capture Decay.svg|thumb|The leading-order [[Feynman diagram]] for ''{{SubatomicParticle|Beta+}}''&nbsp;decay of a [[proton]] into a [[neutron]], [[positron]], and [[electron neutrino]] via an intermediate virtual [[W boson|{{SubatomicParticle|W boson+}} boson]]]]

In {{SubatomicParticle|Beta+}}&nbsp;decay, or positron emission, the weak interaction converts an atomic nucleus into a nucleus with atomic number decreased by one, while emitting a positron ({{SubatomicParticle|Positron}}) and an [[electron neutrino]] ({{SubatomicParticle|Electron neutrino}}). ''{{SubatomicParticle|Beta+}}''&nbsp;decay generally occurs in proton-rich nuclei. The generic equation is:

: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{SubatomicParticle|Positron}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}<ref name="konya74"/>
This may be considered as the decay of a proton inside the nucleus to a neutron:
:p → n + {{SubatomicParticle|Positron}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}<ref name="konya74"/>

However, {{SubatomicParticle|Beta+}}&nbsp;decay cannot occur in an isolated proton because it requires energy, due to the [[mass]] of the neutron being greater than the mass of the proton. {{SubatomicParticle|Beta+}}&nbsp;decay can only happen inside nuclei when the daughter nucleus has a greater [[nuclear binding energy|binding energy]] (and therefore a lower total energy) than the mother nucleus. The difference between these energies goes into the reaction of converting a proton into a neutron, a positron, and a neutrino and into the kinetic energy of these particles. This process is opposite to negative beta decay, in that the weak interaction converts a proton into a neutron by converting an up quark into a down quark resulting in the emission of a {{SubatomicParticle|W boson+}} or the absorption of a {{SubatomicParticle|W boson-}}. When a {{SubatomicParticle|W boson+}} boson is emitted, it decays into a [[positron]] and an [[electron neutrino]]:
:{{Subatomic particle|up quark}} → {{Subatomic particle|down quark}} + {{SubatomicParticle|Positron}} + {{math|{{SubatomicParticle|Electron neutrino}}}}.

==Electron capture (K-capture/L-capture)==
{{main|Electron capture}}
[[File:Electron-capture.svg|alt=Leading-order EC Feynman diagrams|thumb|309x309px|The leading-order [[Feynman diagram]]s for [[electron capture]] decay. An [[electron]] interacts with an [[up quark]] in the nucleus via a [[W and Z bosons|W boson]] to create a [[down quark]] and [[electron neutrino]]. Two diagrams comprise the leading (second) order, though as a [[virtual particle]], the type (and charge) of the W-boson is indistinguishable.]]
In all cases where {{SubatomicParticle|Beta+}}&nbsp;decay (positron emission) of a nucleus is allowed energetically, so too is [[electron capture]] allowed. This is a process during which a nucleus captures one of its atomic electrons, resulting in the emission of a neutrino:

: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} + {{SubatomicParticle|Electron}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}

An example of electron capture is one of the decay modes of [[krypton-81]] into [[bromine-81]]:
:{{nuclide|Krypton|81}} + {{subatomic particle|electron}} → {{nuclide|Bromine|81}} + {{math|{{subatomic particle|electron neutrino}}}}

All emitted neutrinos are of the same energy. In proton-rich nuclei where the energy difference between the initial and final states is less than 2{{math|''m''<sub>e</sub>''c''<sup>2</sup>}}, {{SubatomicParticle|Beta+}}&nbsp;decay is not energetically possible, and electron capture is the sole decay mode.<ref name="Zuber" />

If the captured electron comes from the innermost shell of the atom, the [[Electron shell|K-shell]], which has the highest probability to interact with the nucleus, the process is called K-capture.<ref name="Jevremovic2009">{{cite book
 |last=Jevremovic |first=T.
 |year=2009
 |title=Nuclear Principles in Engineering
 |url=https://books.google.com/books?id=tEM9TCfAe7EC&pg=PA201
 |page=201
 |publisher=[[Springer Science + Business Media]]
 |isbn=978-0-387-85608-7
}}</ref> If it comes from the L-shell, the process is called L-capture, etc.

Electron capture is a competing (simultaneous) decay process for all nuclei that can undergo β<sup>+</sup> decay. The converse, however, is not true: electron capture is the ''only'' type of decay that is allowed in proton-rich nuclides that do not have sufficient energy to emit a positron and neutrino.<ref name="Zuber">{{cite book
 |last=Zuber |first=K.
 |year=2011
 |title=Neutrino Physics
 |page=466 |edition=2nd
 |publisher=[[CRC Press]]
 |isbn=978-1-4200-6471-1
}}</ref>

==Nuclear transmutation==
[[File:Table isotopes en.svg|250px|right|thumb|Graph of isotopes by type of nuclear decay. Orange and blue nuclides are unstable, with the black squares between these regions representing stable nuclides. The unbroken line passing below many of the nuclides represents the theoretical position on the graph of nuclides for which proton number is the same as neutron number. The graph shows that elements with more than 20 protons must have more neutrons than protons, in order to be stable.]]
{{See also|Nuclear drip line}}
If the proton and neutron are part of an [[atomic nucleus]], the above described decay processes [[Nuclear transmutation|transmute]] one chemical element into another. For example:

<!-- Autogenerated using Phykiformulae 0.11 by SkyLined]]
Cs-137 _ _ -> Ba-137 + e- + !ve (beta_minus_decay)
Na-22 _ _ -> Ne-22 + e+ + ve (beta_plus_decay)
Na-22 + e- -> Ne-22 + ve _ _ (electron_capture)
-->:{|border="0"
|- style="height:2em;"
|{{nuclide|link=yes|caesium|137}}&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|barium|137}}&nbsp;||+&nbsp;||{{SubatomicParticle|link=yes|Electron}}&nbsp;||+&nbsp;||{{math|{{SubatomicParticle|link=yes|Electron Antineutrino}}}}&nbsp;||(beta minus decay)
|- style="height:2em;"
|{{nuclide|link=yes|sodium|22}}&nbsp;||&nbsp;||&nbsp;||→&nbsp;||{{nuclide|link=yes|neon|22}}&nbsp;||+&nbsp;||{{math|{{SubatomicParticle|link=yes|Positron}}}}&nbsp;||+&nbsp;||{{math|{{SubatomicParticle|link=yes|Electron Neutrino}}}}&nbsp;||(beta plus decay)
|- style="height:2em;"
|{{nuclide|link=yes|sodium|22}}&nbsp;||+&nbsp;||{{SubatomicParticle|link=yes|Electron}}&nbsp;||→&nbsp;||{{nuclide|link=yes|neon|22}}&nbsp;||+&nbsp;||{{math|{{SubatomicParticle|link=yes|Electron Neutrino}}}}&nbsp;||&nbsp;||&nbsp;||(electron capture)
|}

Beta decay does not change the number&nbsp;({{mvar|A}}) of [[nucleon]]s in the nucleus, but changes only its [[electric charge|charge]]&nbsp;{{mvar|Z}}. Thus the set of all [[nuclide]]s with the same&nbsp;{{mvar|A}} can be introduced; these [[isobar (nuclide)|''isobaric'' nuclides]] may turn into each other via beta decay. For a given {{mvar|A}} there is one that is most stable. It is said to be beta stable, because it presents a local minimum of the [[mass excess]]: if such a nucleus has {{math|(''A'', ''Z'')}} numbers, the neighbour nuclei {{math|(''A'', ''Z''−1)}} and {{math|(''A'', ''Z''+1)}} have higher mass excess and can beta decay into {{math|(''A'', ''Z'')}}, but not vice versa. For all odd mass numbers {{mvar|A}}, there is only one known beta-stable isobar. For even&nbsp;{{mvar|A}}, there are up to three different beta-stable isobars experimentally known; for example, {{nuclide|tin|124}}, {{nuclide|tellurium|124}}, and {{nuclide|xenon|124}} are all beta-stable. There are about 350 known [[beta-decay stable isobars|beta-decay stable nuclides]].<ref name="nndc_Inte">{{Cite web | title=Interactive Chart of Nuclides | url=http://www.nndc.bnl.gov/chart/ | publisher=National Nuclear Data Center, Brookhaven National Laboratory | access-date=2014-09-18 | archive-date=2018-10-10 | archive-url=https://web.archive.org/web/20181010070007/http://www.nndc.bnl.gov/chart/  }}</ref>

===Competition of beta decay types===
Usually unstable nuclides are clearly either "neutron rich" or "proton rich", with the former undergoing beta decay and the latter undergoing electron capture (or more rarely, due to the higher energy requirements, positron decay). However, in a few cases of odd-proton, odd-neutron radionuclides, it may be energetically favorable for the radionuclide to decay to an even-proton, even-neutron isobar either by undergoing beta-positive or beta-negative decay. An often-cited example is the single isotope {{nuclide|copper|64|link=yes}} (29 protons, 35 neutrons), which illustrates three types of beta decay in competition. Copper-64 has a half-life of about 12.7 hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay. This particular nuclide (though not all nuclides in this situation) is almost equally likely to decay through proton decay by [[positron emission]] ({{val|18|u=%}}) or [[electron capture]] ({{val|43|u=%}}) to {{nuclide|nickel|64}}, as it is through neutron decay by electron emission ({{val|39|u=%}}) to {{nuclide|zinc|64}}.<ref>{{Cite web | title=WWW Table of Radioactive Isotopes, Copper 64 | publisher=Lawrence Berkeley National Laboratory | work=LBNL Isotopes Project | access-date=2014-09-18 | url=http://ie.lbl.gov/toi/nuclide.asp?iZA=290064  | archive-url=https://web.archive.org/web/20131214161059/http://ie.lbl.gov/toi/nuclide.asp?iZA=290064 | archive-date=2013-12-14}}</ref>

===Stability of naturally occurring nuclides===
Most naturally occurring nuclides on earth are beta stable. Nuclides that are not beta stable have [[half-life|half-lives]] ranging from under a second to periods of time significantly greater than the [[age of the universe]]. One common example of a long-lived isotope is the odd-proton odd-neutron nuclide {{nuclide|link=yes|Potassium|40}}, which undergoes all three types of beta decay ({{SubatomicParticle|Beta-}}, {{SubatomicParticle|Beta+}} and electron capture) with a half-life of {{val|1.277|e=9|u=years}}.<ref>
{{Cite web
 |title=WWW Table of Radioactive Isotopes, Potassium 40
 |publisher=Lawrence Berkeley National Laboratory
 |work=LBNL Isotopes Project
 |access-date=2014-09-18
 |url=http://ie.lbl.gov/toi/nuclide.asp?iZA=190040
 
 |archive-url=https://web.archive.org/web/20131009123338/http://ie.lbl.gov/toi/nuclide.asp?iZA=190040
 |archive-date=2013-10-09
}}</ref>

==Conservation rules for beta decay==

===Baryon number is conserved===

<math display="block">B=\frac{n_q - n_{\bar{q}}}{3}</math>
where
* <math>n_q</math> is the number of constituent quarks, and
* <math>n_{\overline{q}}</math> is the number of constituent antiquarks.

Beta decay just changes [[neutron]] to [[proton]] or, in the case of positive beta decay ([[electron capture]]) [[proton]] to [[neutron]] so the number of individual [[quarks]] doesn't change. It is only the baryon flavor that changes, here labelled as the [[isospin]].

Up and down [[quarks]] have total isospin <math display="inline">I=\frac{1}{2}</math> and isospin projections
<math display="block">I_\text{z}=\begin{cases}
\frac{1}{2} & \text{up quark} \\
-\frac{1}{2} & \text{down quark}
\end{cases}</math>

All other quarks have {{math|1=''I'' = 0}}.

In general<!-- antiquarks are not expected! -->
<math display="block">I_\text{z}=\frac{1}{2} (n_\text{u} - n_\text{d})</math>

===Lepton number is conserved===

<math display="block">L \equiv n_{\ell} - n_{\bar{\ell}}</math>

so all leptons have assigned a value of +1, antileptons &minus;1, and non-leptonic particles 0. 
<math display="block">\begin{matrix}
 & \text{n} & \rightarrow & \text{p} & + & \text{e}^- & + & \bar{\nu}_\text{e} \\
L: & 0 &=& 0 & + & 1 & - & 1
\end{matrix}</math>

===Angular momentum===
For allowed decays, the net orbital angular momentum is zero, hence only spin quantum numbers are considered.

The electron and antineutrino are [[fermions]], spin-1/2 objects, therefore they may couple to total <math>S=1</math> (parallel) or <math>S=0</math> (anti-parallel).

For forbidden decays, orbital angular momentum must also be taken into consideration.

==Energy release==
The [[Q value (nuclear science)|{{mvar|Q}} value]] is defined as the total energy released in a given nuclear decay. In beta decay, {{mvar|Q}} is therefore also the sum of the kinetic energies of the emitted beta particle, neutrino, and recoiling nucleus. (Because of the large mass of the nucleus compared to that of the beta particle and neutrino, the kinetic energy of the recoiling nucleus can generally be neglected.) Beta particles can therefore be emitted with any [[kinetic energy]] ranging from 0 to {{mvar|Q}}.<ref name="konya74"/> A typical {{mvar|Q}} is around 1&nbsp;[[MeV]], but can range from a few [[keV]] to a few tens of MeV.

Since the [[Mass–energy equivalence|rest mass]] of the electron is 511&nbsp;keV, the most energetic beta particles are [[Ultrarelativistic limit|ultrarelativistic]], with speeds very close to the [[speed of light]].
In the case of {{sup|187}}Re, the maximum speed of the beta particle is only 9.8% of the speed of light.

The following table gives some examples:

{| class="wikitable zebra"
|+ Examples of beta decay energies
|-
! Isotope || Energy<br />([[Electronvolt|keV]])|| Decay mode || Comments
|-
| free<br />[[Neutron]] || {{0}}782.33 ||style="text-align:center;"| β<sup>−</sup> ||
|-
| <sup>{{0|00}}3</sup>H<br /><small>(Tritium)</small> || {{0|00}}18.59 || align="center"| β<sup>−</sup> || Second lowest known β<sup>−</sup> energy, being used in the [[KATRIN]] experiment.
|-
| <sup>{{0}}11</sup>C || {{0}}960.4<br />1982.4 ||style="text-align:center;"| β<sup>+</sup><br /><span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> || <!--beides möglich, sucht noch jemand die Übergangswahrscheinlichkeiten raus?-->
|-
| [[Carbon-14|<sup>{{0}}14</sup>C]] || {{0}}156.475 ||style="text-align:center;"| β<sup>−</sup> ||
|-
| <sup>{{0}}20</sup>F || 5390.86 ||style="text-align:center;"| β<sup>−</sup> ||
|-
| <sup>{{0}}37</sup>K || 5125.48<br />6147.48 ||style="text-align:center;"| β<sup>+</sup><br /><span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> || <!--beides möglich, sucht noch jemand die Übergangswahrscheinlichkeiten raus?-->
|-
| <sup>163</sup>Ho || {{0|000}}2.555 ||style="text-align:center;"| <span style="font-size:larger;">ε{{0|<sup>+</sup>}}</span> || 
|-
| <sup>187</sup>Re || {{0|000}}2.467 ||style="text-align:center;"| β<sup>−</sup> || Lowest known β<sup>−</sup> energy, being used in the [[Microcalorimeter Arrays for a Rhenium Experiment]] experiment
|-
| <sup>210</sup>Bi || 1162.2 ||style="text-align:center;"| β<sup>−</sup> ||
|}

===β<sup>−</sup> decay===
Consider the generic equation for beta decay
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''+1}}|X′}} + {{SubatomicParticle|Electron}} + {{math|{{SubatomicParticle|Electron Antineutrino}}}}.
The {{mvar|Q}} value for this decay is
:<math chem>Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m_N\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)-m_e-m_{\overline\nu_e}\right]c^2</math>,
where <math chem>m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)</math> is the mass of the nucleus of the {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} atom, <math chem>m_e</math> is the mass of the electron, and <math chem>m_{\overline\nu_e}</math> is the mass of the electron antineutrino. In other words, the total energy released is the mass energy of the initial nucleus, minus the mass energy of the final nucleus, electron, and antineutrino. The mass of the nucleus {{mvar|m<sub>N</sub>}} is related to the standard [[atomic mass]] {{mvar|m}} by
<math chem display="block">m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)c^2=m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)c^2 + Z m_e c^2-\sum_{i=1}^Z B_i.</math>
That is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of all ''electron'' binding energies {{mvar|B<sub>i</sub>}} for the atom. This equation is rearranged to find <math chem>m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right)</math>, and <math chem>m_N\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)</math> is found similarly. Substituting these nuclear masses into the {{math|Q}}-value equation, while neglecting the nearly-zero antineutrino mass and the difference in electron binding energies, which is very small for high-{{mvar|Z}} atoms, we have
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)-m\left(\ce{^\mathit{A}_{\mathit{Z}+1}X'}\right)\right]c^2</math>
This energy is carried away as kinetic energy by the electron and antineutrino.

Because the reaction will proceed only when the {{mvar|Q}}&nbsp;value is positive, β<sup>−</sup> decay can occur when the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} is greater than the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''+1}}|X′}}.<ref name="Krane1987">{{cite book|author=Kenneth S. Krane|title=Introductory Nuclear Physics| url=https://books.google.com/books?id=ConwAAAAMAAJ|date=5 November 1987|publisher=Wiley|isbn=978-0-471-80553-3}}</ref>

===β<sup>+</sup> decay===
The equations for β<sup>+</sup> decay are similar, with the generic equation
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{SubatomicParticle|Positron}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}
giving
<math chem display="block">Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m_N\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-m_e-m_{\nu_e}\right]c^2.</math>
However, in this equation, the electron masses do not cancel, and we are left with
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right)-m\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-2m_e\right]c^2.</math>

Because the reaction will proceed only when the {{mvar|Q}}&nbsp;value is positive, β<sup>+</sup> decay can occur when the mass of atom {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} exceeds that of {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''-1}}|X′}} by at least twice the mass of the electron.<ref name="Krane1987" />

===Electron capture===
The analogous calculation for electron capture must take into account the binding energy of the electrons. This is because the atom will be left in an excited state after capturing the electron, and the binding energy of the captured innermost electron is significant. Using the generic equation for electron capture
: {{Physics particle|TL={{mvar|A}}|BL={{mvar|Z}}|X}} + {{SubatomicParticle|Electron}} → {{Physics particle|TL={{mvar|A}}|BL={{math|''Z''−1}}|X′}} + {{math|{{SubatomicParticle|Electron Neutrino}}}}
we have
<math chem display="block">Q=\left[m_N\left(\ce{^\mathit{A}_\mathit{Z}X}\right) + m_e - m_N\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)-m_{\nu_e}\right]c^2,</math>
which simplifies to
<math chem display="block">Q=\left[m\left(\ce{^\mathit{A}_\mathit{Z}X}\right) - m\left(\ce{^\mathit{A}_{\mathit{Z}-1}X'}\right)\right]c^2-B_n,</math>
where {{mvar|B<sub>n</sub>}} is the binding energy of the captured electron.

Because the binding energy of the electron is much less than the mass of the electron, nuclei that can undergo β<sup>+</sup> decay can always also undergo electron capture, but the reverse is not true.<ref name="Krane1987" />

==Beta emission spectrum==
[[File:Beta spectrum of RaE.jpg|thumb|Beta spectrum of <sup>210</sup>Bi. ''E''<sub>max</sub> = ''Q'' = 1.16 MeV is the maximum energy]]
Beta decay can be considered as a [[Perturbation theory (quantum mechanics)|perturbation]] as described in quantum mechanics, and thus [[Fermi's Golden Rule]] can be applied. This leads to an expression for the kinetic energy spectrum {{math|''N''(''T'')}} of emitted betas as follows:<ref>{{cite web | last=Nave | first=C. R. |title=Energy and Momentum Spectra for Beta Decay | url=http://hyperphysics.phy-astr.gsu.edu/hbase/nuclear/beta2.html |work=[[HyperPhysics]] |access-date=2013-03-09}}</ref>

<math display="block">N(T) = C_L(T) F(Z,T) p E (Q-T)^2</math>

where {{mvar|T}} is the kinetic energy, {{mvar|C<sub>L</sub>}} is a shape function that depends on the forbiddenness of the decay (it is constant for allowed decays), {{math|''F''(''Z'', ''T'')}} is the Fermi Function (see below) with ''Z'' the charge of the final-state nucleus, {{math|1=''E'' = ''T'' + ''mc''<sup>2</sup>}} is the total energy, <math> p = \sqrt{(E/c)^2 - (mc)^2}</math> is the momentum, and {{mvar|Q}} is the [[Q value (nuclear science)|Q value]] of the decay. The kinetic energy of the emitted neutrino is given approximately by {{mvar|Q}} minus the kinetic energy of the beta.

As an example, the beta decay spectrum of <sup>210</sup>Bi (originally called RaE) is shown to the right.

===Fermi function===

The Fermi function that appears in the beta spectrum formula accounts for the Coulomb attraction / repulsion between the emitted beta and the final state nucleus. Approximating the associated wavefunctions to be spherically symmetric, the Fermi function can be analytically calculated to be:<ref>{{cite journal |last1=Fermi |first1=E. |year=1934 |title=Versuch einer Theorie der &beta;-Strahlen. I |journal=[[Zeitschrift für Physik]] |volume=88 |issue=3–4 |pages=161–177 |bibcode=1934ZPhy...88..161F |doi=10.1007/BF01351864|s2cid=125763380 }}</ref>

<math display="block">F(Z,T)=\frac{2 (1+S)}{\Gamma(1+2S)^2} (2 p \rho)^{2S-2} e^{\pi \eta} |\Gamma(S+i \eta)|^2,</math>

where {{mvar|p}} is the final momentum, Γ the [[Gamma function]], and (if {{mvar|α}} is the [[fine-structure constant]] and {{mvar|r<sub>N</sub>}} the radius of the final state nucleus) <math>S = \sqrt{1 - \alpha^2 Z^2}</math>, <math>\eta = \pm Ze^2E/(\hbar cp)</math> (+ for electrons, − for positrons), and <math>\rho = r_N/\hbar </math>.

For non-relativistic betas ({{math|''Q'' ≪ ''m''<sub>e</sub>''c''<sup>2</sup>}}), this expression can be approximated by:<ref>{{cite book
 |last1=Mott |first1=N. F.
 |last2=Massey |first2=H. S. W.
 |year=1933
 |title=The Theory of Atomic Collisions
 |publisher=[[Clarendon Press]]
 |lccn=34001940
}}</ref>

<math display="block">F(Z,T) \approx \frac{2 \pi \eta}{1 - e^{- 2 \pi \eta}}.</math>

Other approximations can be found in the literature.<ref>{{cite journal
 |last1=Venkataramaiah |first1=P.
 |last2=Gopala |first2=K.
 |last3=Basavaraju |first3=A.
 |last4=Suryanarayana |first4=S. S.
 |last5=Sanjeeviah |first5=H.
 |year=1985
 |title=A simple relation for the Fermi function
 |journal=[[Journal of Physics G]]
 |volume=11 |issue=3 |pages=359–364
 |bibcode=1985JPhG...11..359V
 |doi=10.1088/0305-4616/11/3/014
|s2cid=250803189
 }}</ref><ref>{{cite journal
 |last1=Schenter |first1=G. K.
 |last2=Vogel |first2=P.
 |year=1983
 |title=A simple approximation of the fermi function in nuclear beta decay
 |journal=[[Nuclear Science and Engineering]]
 |volume=83 |issue=3 |pages=393–396
 |doi=10.13182/NSE83-A17574
 |bibcode=1983NSE....83..393S
 |osti=5307377
}}</ref>

===Kurie plot===
<!-- Note: Kurie is correct (see References below); no relation to the Curies -->

A '''Kurie plot''' (also known as a '''Fermi–Kurie plot''') is a graph used in studying beta decay developed by [[Franz N. D. Kurie]], in which the square root of the number of beta particles whose momenta (or energy) lie within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy.<ref>{{cite journal
 |last1=Kurie |first1=F. N. D. |author-link=Franz N. D. Kurie
 |last2=Richardson |first2=J. R.
 |last3=Paxton |first3=H. C.
 |year=1936
 |title=The Radiations Emitted from Artificially Produced Radioactive Substances. I. The Upper Limits and Shapes of the β-Ray Spectra from Several Elements
 |journal=[[Physical Review]]
 |volume=49 |issue=5 |pages=368–381
 |bibcode=1936PhRv...49..368K
 |doi=10.1103/PhysRev.49.368
}}</ref><ref>{{cite journal
 |last1=Kurie |first1=F. N. D. |author-link=Franz N. D. Kurie
 |year=1948
 |title=On the Use of the Kurie Plot
 |journal=[[Physical Review]]
 |volume=73 |issue=10 |page=1207
 |bibcode=1948PhRv...73.1207K
 |doi=10.1103/PhysRev.73.1207
}}</ref> It is a straight line for allowed transitions and some forbidden transitions, in accord with the Fermi beta-decay theory. The energy-axis (x-axis) intercept of a Kurie plot corresponds to the maximum energy imparted to the electron/positron (the decay's {{mvar|Q}}&nbsp;value). With a Kurie plot one can find the limit on the effective mass of a neutrino.<ref>{{Cite journal
 |last=Rodejohann |first=W.
 |year=2012
 |title=Neutrinoless double beta decay and neutrino physics
 |journal=Journal of Physics G: Nuclear and Particle Physics
 |volume=39
 |issue=12
 |page=124008
 |arxiv=1206.2560
 |doi=10.1088/0954-3899/39/12/124008
 |bibcode=2012JPhG...39l4008R
 |s2cid=119158221 }}</ref>

==Helicity (polarization) of neutrinos, electrons and positrons emitted in beta decay==
After the discovery of parity non-conservation (see [[Beta decay#Non-conservation of parity|History]]), it was found that, in beta decay, electrons are emitted mostly with negative [[Chirality (physics)|helicity]], i.e., they move, naively speaking, like left-handed screws driven into a material (they have negative longitudinal [[Spin polarization|polarization]]).<ref>{{cite journal
 |last1=Frauenfelder |first1=H.
 |display-authors=etal.
 |year=1957
 |title=Parity and the Polarization of Electrons fromCo60
 |journal=[[Physical Review]]
 |volume=106 |issue= 2|pages=386–387
 |bibcode= 1957PhRv..106..386F
 |doi=10.1103/physrev.106.386
}}</ref> Conversely, positrons have mostly positive helicity, i.e., they move like right-handed screws. Neutrinos (emitted in positron decay) have negative helicity, while antineutrinos (emitted in electron decay) have positive helicity.<ref>{{cite book
 |last1=Konopinski |first1=E. J.
 |last2=Rose |first2=M. E.
 |year=1966
 |chapter=The Theory of nuclear Beta Decay
 |editor-last=Siegbhan |editor-first=K.
 |title=Alpha-, Beta- and Gamma-Ray Spectroscopy
 |volume=2
 |publisher=[[North-Holland Publishing Company]]
}}</ref>

The higher the energy of the particles, the higher their polarization.

==Types of beta decay transitions==
{{main|Beta decay transition}}
Beta decays can be classified according to the angular momentum ([[Angular momentum operator|{{mvar|L}}&nbsp;value]]) and total spin ([[Spin (physics)|{{mvar|S}}&nbsp;value]]) of the emitted radiation. Since total angular momentum must be conserved, including orbital and spin angular momentum, beta decay occurs by a variety of quantum state transitions to various nuclear angular momentum or spin states, known as "Fermi" or "Gamow–Teller" transitions. When beta decay particles carry no angular momentum ({{math|1=''L'' = 0}}), the decay is referred to as "allowed", otherwise it is "forbidden".

Other decay modes, which are rare, are known as bound state decay and double beta decay.

===Fermi transitions===
A '''Fermi transition''' is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin <math>S=0</math>, leading to an angular momentum change <math>\Delta J=0</math> between the initial and final states of the nucleus (assuming an allowed transition). In the non-relativistic limit, the nuclear part of the operator for a Fermi transition is given by
<math display="block"> \mathcal{O}_{F}=G_{V}\sum_{a} \hat{\tau}_{a\pm} </math>
with <math>G_V</math> the weak vector coupling constant, <math>\tau_{\pm}</math> the [[isospin]] [[Ladder operator|raising and lowering operators]], and <math>a</math> running over all protons and neutrons in the nucleus.

===Gamow–Teller transitions===
A '''Gamow–Teller transition''' is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin <math>S=1</math>, leading to an angular momentum change <math>\Delta J=0,\pm 1</math> between the initial and final states of the nucleus (assuming an allowed transition).
In this case, the nuclear part of the operator is given by
<math display="block"> \mathcal{O}_{GT}=G_{A}\sum_{a} \hat{\sigma}_{a}\hat{\tau}_{a\pm} </math>
with <math>G_{A}</math> the weak axial-vector coupling constant, and <math>\sigma</math> the [[Pauli matrices|spin Pauli matrices]], which can produce a spin-flip in the decaying nucleon.

===Forbidden transitions===
When {{math|''L'' > 0}}, the decay is referred to as "[[forbidden transition|forbidden]]". Nuclear [[selection rule]]s require high {{mvar|L}}&nbsp;values to be accompanied by changes in [[nuclear spin]]&nbsp;({{mvar|J}}) and [[Parity (physics)|parity]]&nbsp;({{mvar|π}}). The selection rules for the {{mvar|L}}th forbidden transitions are:
<math display="block">\Delta J=L-1, L, L+1; \Delta \pi=(-1)^L, </math>
where {{math|1=Δ''π'' = 1 or −1}} corresponds to no parity change or parity change, respectively. The special case of a transition between isobaric analogue states, where the structure of the final state is very similar to the structure of the initial state, is referred to as "superallowed" for beta decay, and proceeds very quickly. The following table lists the Δ{{var|J}} and Δ{{var|π}} values for the first few values of&nbsp;{{mvar|L}}:

{|class="wikitable"
|-
! Forbiddenness !! Δ{{var|J}} !! Δ{{var|π}}
|-
|Superallowed ||0 ||{{no}}
|-
|Allowed ||0, 1 ||{{no}}
|-
|First forbidden ||0, 1, 2 ||{{yes}}
|-
|Second forbidden ||1, 2, 3 ||{{no}}
|-
|Third forbidden ||2, 3, 4 ||{{yes}}
|}

==Rare decay modes==

===Bound-state β{{sup|−}} decay===
A very small minority of free neutron decays (about four per million) are "two-body decays": the proton, electron and antineutrino are produced, but the electron fails to gain the 13.6 eV energy necessary to escape the proton, and therefore simply remains bound to it, as a neutral [[hydrogen atom]].<ref>[http://www.physi.uni-heidelberg.de/Publications/ckm_byrne.pdf An Overview Of Neutron Decay] {{Webarchive|url=https://web.archive.org/web/20170919055418/http://www.physi.uni-heidelberg.de/Publications/ckm_byrne.pdf |date=2017-09-19 }} J. Byrne in Quark-Mixing, CKM Unitarity (H. Abele and D. Mund, 2002), see p.XV</ref> In this type of beta decay, in essence all of the neutron [[decay energy]] is carried off by the antineutrino.

For fully ionized atoms (bare nuclei), it is possible in likewise manner for electrons to fail to escape the atom, and to be emitted from the nucleus into low-lying atomic bound states (orbitals). This cannot occur for neutral atoms with low-lying bound states which are already filled by electrons.

Bound-state β{{sup|−}} decays were predicted by [[Raymond Daudel|Daudel]], Jean, and Lecoin in 1947,<ref>{{cite journal| last1=Daudel | first1=Raymond | first2=Maurice | last2=Jean | first3=Marcel | last3=Lecoin | year=1947 | title=Sur la possibilité d'existence d'un type particulier de radioactivité phénomène de création e | journal=J. Phys. Radium | volume=8 | issue=8 | pages=238–243 | doi=10.1051/jphysrad:0194700808023800| url=https://hal.archives-ouvertes.fr/jpa-00234057/document }}</ref> and the phenomenon in fully ionized atoms was first observed for [[isotopes of dysprosium|{{sup|163}}Dy{{sup|66+}}]] in 1992 by Jung et al. of the [[GSI Helmholtz Centre for Heavy Ion Research|Darmstadt Heavy-Ion Research Center]]. Though neutral {{sup|163}}Dy is stable, fully ionized {{sup|163}}Dy{{sup|66+}} undergoes β{{sup|−}} decay into the K and L shells with a half-life of 47 days.<ref>{{cite journal
 |last1=Jung |first1=M.
 |display-authors=etal
 |year=1992
 |title=First observation of bound-state β<sup>&minus;</sup> decay
 |journal=[[Physical Review Letters]]
 |volume=69 |issue=15 |pages=2164–2167
 |bibcode=1992PhRvL..69.2164J
 |doi=10.1103/PhysRevLett.69.2164
 |pmid=10046415
}}</ref> The resulting nucleus – [[isotopes of holmium|{{sup|163}}Ho{{sup|66+}}]] – is stable only in this almost fully ionized state and will decay via [[electron capture]] into {{sup|163}}Dy in the neutral state. Likewise, while being stable in the neutral state, the fully ionized [[Isotopes of thallium|{{sup|205}}Tl{{sup|81+}}]] undergoes bound-state β{{sup|−}} decay to [[isotopes of lead|{{sup|205}}Pb{{sup|81+}}]] with a half-life of {{val|291|33|27|}} days.<ref>{{cite web|url=http://www.ca.infn.it/~oldeman/resneu/p1522_1.pdf |title=Bound-state beta decay of highly ionized atoms|access-date=June 9, 2013 |url-status=dead |archive-url=https://web.archive.org/web/20131029205727/http://www.ca.infn.it/~oldeman/resneu/p1522_1.pdf |archive-date=October 29, 2013 }}</ref><ref>{{cite journal |last1=Bai |first1=M. |last2=Blaum |first2=K. |last3=Boev |first3=B. |last4=Bosch |first4=F. |last5=Brandau |first5=C. |last6=Cvetković |first6=V. |last7=Dickel |first7=T. |last8=Dillmann |first8=I. |last9=Dmytriiev |first9=D. |last10=Faestermann |first10=T. |last11=Forstner |first11=O. |last12=Franczak |first12=B. |last13=Geissel |first13=H. |last14=Gernhäuser |first14=R. |last15=Glorius |first15=J. |last16=Griffin |first16=C. J. |last17=Gumberidze |first17=A. |last18=Haettner |first18=E. |last19=Hillenbrand |first19=P.-M. |last20=Kienle |first20=P. |last21=Korten |first21=W. |last22=Kozhuharov |first22=Ch. |last23=Kuzminchuk |first23=N. |last24=Langanke |first24=K. |last25=Litvinov |first25=S. |last26=Menz |first26=E. |last27=Morgenroth |first27=T. |last28=Nociforo |first28=C. |last29=Nolden |first29=F. |last30=Pavićević |first30=M. K. |last31=Petridis |first31=N. |last32=Popp |first32=U. |last33=Purushothaman |first33=S. |last34=Reifarth |first34=R. |last35=Sanjari |first35=M. S. |last36=Scheidenberger |first36=C. |last37=Spillmann |first37=U. |last38=Steck |first38=M. |last39=Stöhlker |first39=Th. |last40=Tanaka |first40=Y. K. |last41=Trassinelli |first41=M. |last42=Trotsenko |first42=S. |last43=Varga |first43=L. |last44=Wang |first44=M. |last45=Weick |first45=H. |last46=Woods |first46=P. J. |last47=Yamaguchi |first47=T. |last48=Zhang |first48=Y. H. |last49=Zhao |first49=J. |last50=Zuber |first50=K. |title=Bound-State Beta Decay of {{sup|205}}Tl{{sup|81+}} Ions and the LOREX Project |collaboration=E121 Collaboration and LOREX Collaboration |journal=Physical Review Letters |date=2 December 2024 |volume=133 |issue=23 |pages=232701 |publisher=American Physical Society |doi=10.1103/PhysRevLett.133.232701 |url=https://link.aps.org/doi/10.1103/PhysRevLett.133.232701|arxiv=2501.06029 }}</ref> The half-lives of neutral {{sup|163}}Ho and {{sup|205}}Pb are respectively 4570 years and {{val|1.73|e=7}} years. In addition, it is estimated that β{{sup|−}} decay is energetically impossible for natural atoms but theoretically possible when fully ionized also for <sup>193</sup>Ir, <sup>194</sup>Au, <sup>202</sup>Tl, <sup>215</sup>At, <sup>243</sup>Am, and <sup>246</sup>Bk.<ref name="bs-prediction">{{cite journal |last1=Liu |first1=Shuo |last2=Gao |first2=Chao |last3=Xu |first3=Chang |date=2021 |title=Investigation of bound state β<sup>−</sup> decay half-lives of bare atoms |url= |journal=Physical Review C |volume=104 |issue=2 |pages=024304 |doi=10.1103/PhysRevC.104.024304 |access-date=}}</ref>

Another possibility is that a fully ionized atom undergoes greatly accelerated β decay, as observed for [[Isotopes of rhenium|{{sup|187}}Re]] by Bosch et al., also at Darmstadt. Neutral {{sup|187}}Re does undergo β{{sup|−}} decay, with half-life {{val|4.12|e=10}} years,<ref>{{cite journal |doi=10.1126/science.271.5252.1099 |last1=Smoliar |first1=M.I. |last2=Walker |first2=R.J. |last3=Morgan |first3=J.W. |year=1996 |title=Re-Os ages of group IIA, IIIA, IVA, and IVB iron meteorites |journal=Science |volume=271 |pages=1099–1102 |bibcode=1996Sci...271.1099S |issue=5252|s2cid=96376008 }}</ref> but for fully ionized {{sup|187}}Re{{sup|75+}} this is shortened to only 32.9 years.<ref>{{cite journal
 |last1=Bosch |first1=F.
 |display-authors=etal
 |year=1996
 |title=Observation of bound-state beta minus decay of fully ionized {{sup|187}}Re: {{sup|187}}Re–{{sup|187}}Os Cosmochronometry
 |journal=[[Physical Review Letters]]
 |volume=77 |issue=26 |pages=5190–5193
 |bibcode=1996PhRvL..77.5190B
 |doi=10.1103/PhysRevLett.77.5190
 |pmid=10062738
}}</ref> This is because {{sup|187}}Re{{sup|75+}} is energetically allowed to undergo β{{sup|−}} decay to the first-excited state in {{sup|187}}Os{{sup|75+}}, a process energetically disallowed for natural {{sup|187}}Re.<ref name="bs-prediction" /> Similarly, neutral [[Plutonium-241|{{sup|241}}Pu]] undergoes β{{sup|−}} decay with a half-life of 14.3 years, but in its fully ionized state the beta-decay half-life of {{sup|241}}Pu{{sup|94+}} decreases to 4.2 days.<ref>{{cite journal |last1=Takahashi |first1=K. |last2=Boyd |first2=R. N. |last3=Mathews |first3=G. J. |last4=Yokoi |first4=K. |title=Bound-state beta decay of highly ionized atoms |journal=Physical Review C |date=1 October 1987 |volume=36 |issue=4 |pages=1522–1528 |doi=10.1103/PhysRevC.36.1522 |url=https://www.researchgate.net/publication/13335547_Bound-state_beta_decay_of_highly_ionized_atoms}}</ref> For comparison, the variation of decay rates of other nuclear processes due to chemical environment is [[radioactive decay#Changing rates|less than 1%]]. Moreover, current mass determinations cannot decisively determine whether {{sup|222}}Rn is energetically possible to undergo β{{sup|−}} decay (the decay energy given in AME2020 is (−6 ± 8) keV),{{AME2020 II|ref}}<ref>{{cite journal |last1=Belli |first1=P. |last2=Bernabei |first2=R. |last3=Cappella |first3=C. |last4=Caracciolo |first4=V. |last5=Cerulli |first5=R. |last6=Danevich |first6=F.A. |last7=Di Marco |first7=A. |last8=Incicchitti |first8=A. |last9=Poda |first9=D.V. |last10=Polischuk |first10=O.G. |last11=Tretyak |first11=V.I. |title=Investigation of rare nuclear decays with BaF<sub>2</sub> crystal scintillator contaminated by radium |date=2014 |journal=European Physical Journal A |volume=50 |issue=9 |pages=134–143 |doi=10.1140/epja/i2014-14134-6 |arxiv=1407.5844|bibcode=2014EPJA...50..134B |s2cid=118513731 }}</ref> but in either case it is predicted that β{{sup|−}} will be greatly accelerated for fully ionized {{sup|222}}Rn{{sup|86+}}.<ref name="bs-prediction" />

===Double beta decay===
{{main|Double beta decay}}
Some nuclei can undergo double beta decay (2β) where the charge of the nucleus changes by two units. Double beta decay is difficult to study, as it has an extremely long half-life. In nuclei for which both β&nbsp;decay and 2β are possible, the rarer 2β process is effectively impossible to observe. However, in nuclei where β&nbsp;decay is forbidden but 2β is allowed, the process can be seen and a half-life measured.<ref name="Bilenky">{{cite journal
 |last=Bilenky |first=S. M.
 |year=2010
 |title=Neutrinoless double beta-decay
 |volume=41 |issue=5 |pages=690–715
 |journal=[[Physics of Particles and Nuclei]]
 |arxiv=1001.1946
 |bibcode=2010PPN....41..690B
 |doi=10.1134/S1063779610050035
|hdl=10486/663891
 |s2cid=55217197
 }}</ref> Thus, 2β is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change {{mvar|A}}; thus, at least one of the nuclides with some given {{mvar|A}} has to be stable with regard to both single and double beta decay.

"Ordinary" 2β results in the emission of two electrons and two antineutrinos. If neutrinos are [[Majorana particle]]s (i.e., they are their own antiparticles), then a decay known as [[neutrinoless double beta decay]] will occur. Most neutrino physicists believe that neutrinoless 2β has never been observed.<ref name="Bilenky" />

==See also==
* [[Common beta emitters]]
*[[Neutrino]]
*[[Betavoltaics]]
*[[Particle radiation]]
*[[Radionuclide]]
*[[Tritium illumination]], a form of [[fluorescent lighting]] powered by beta decay
*[[Pandemonium effect]]
*[[Total absorption spectroscopy]]

==References==
{{reflist|30em}}

==Bibliography==
* {{cite book
 |last=Tomonaga |first=S.-I. |author-link=Sin-Itiro Tomonaga
 |year=1997
 |title=The Story of Spin
 |publisher=[[University of Chicago Press]]
}}
*{{cite book
 |last=Tuli |first=J. K.
 |title=Nuclear Wallet Cards
 |url=http://www.nndc.bnl.gov/wallet/wall35.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.nndc.bnl.gov/wallet/wall35.pdf |archive-date=2022-10-09 |url-status=live
 |edition=8th
 |year=2011
 |publisher=[[Brookhaven National Laboratory]]
}}

==External links==
* [[Image:Ndslivechart.png]] '''[http://www-nds.iaea.org/livechart The Live Chart of Nuclides - IAEA ]''' with filter on decay type
*'''Beta decay simulation''' [https://phet.colorado.edu/en/simulation/legacy/beta-decay]

{{Nuclear processes}}

{{Authority control}}

{{DEFAULTSORT:Beta Decay}}
[[Category:Nuclear physics]]
[[Category:Radioactivity]]