Beta decay: Difference between revisions

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 |last=Konya |first=J. |title=Nuclear and Radio-chemistry |last2=Nagy |first2=N. M. |publisher=[[Elsevier]] |year=2012 |isbn=978-0-12-391487-3 |pages=74–75}}</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 |last=Bijker |first=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 |last=Cottingham |first=W. N. |url=https://archive.org/details/introductiontonu0000cott/page/40 |title=An introduction to nuclear physics |last2=Greenwood |first2=D. A. |publisher=[[Cambridge University Press]] |year=1986 |isbn=978-0-521-31960-7 |page=[https://archive.org/details/introductiontonu0000cott/page/40 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.

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

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]].{{cn|date=July 2025}}

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.{{cn|date=July 2025}}

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 |last=Jensen |first=C. |url=https://www.springer.com/birkhauser/physics/book/978-3-7643-5313-1 |title=Controversy and Consensus: Nuclear Beta Decay 1911-1934 |publisher=[[Birkhäuser Verlag]] |year=2000 |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 |bibcode=1930ZPhy...60..143M |doi=10.1007/BF01339819 |issn=0044-3328 |url-access=subscription}}</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 |last=Brown |first=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.{{cn|date=July 2025}}

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.{{cn|date=July 2025}}

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 |last=Cowan |first=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.

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 |title=The Nobel Prize in Chemistry 1935 |url=https://www.nobelprize.org/nobel_prizes/chemistry/laureates/1935/ |access-date=2018-04-25 |website=www.nobelprize.org}}</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 ⁴⁸V.<ref name="k">{{Cite book |last=Segré |first=E. |title=Discovering Alvarez: Selected Works of Luis W. Alvarez |publisher=[[University of Chicago Press]] |year=1987 |isbn=978-0-226-81304-2 |editor-last=Trower |editor-first=P. W. |pages=[https://archive.org/details/discoveringalvar0000alva/page/11 11–12] |chapter=K-Electron Capture by Nuclei |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 |access-date=2009-10-07 |publisher=[[The Nobel Foundation]]}}</ref><ref>{{Cite journal |last=Alvarez |first=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 ⁶⁷Ga and other nuclides.<ref name="k" /><ref>{{Cite journal |last=Alvarez |first=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 |last=Alvarez |first=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>

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 |last=Lee |first=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 showed experimentally that an asymmetrical beta emission from [[cobalt-60|{{SimpleNuclide|cobalt|60}}]] proved that parity is not conserved in beta decay.<ref name="Wu1957">{{Cite journal |last=Wu |first=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 |last=Weinstock |first=Maia |title=Channeling Ada Lovelace: Chien-Shiung Wu, Courageous Hero of Physics |url=http://blogs.scientificamerican.com/guest-blog/2013/10/15/channeling-ada-lovelace-chien-shiung-wu-courageous-hero-of-physics/ |website=scientificamerican.com}}</ref><ref>{{Cite book |last=Laymon |first=R. |title=Case Studies in Experimental Physics |last2=Franklin |first2=A. |date=2022 |publisher=[[Springer International Publishing]] |isbn=978-3-031-12607-9 |location=Cham |pages=75–91 |doi=10.1007/978-3-031-12608-6_3}}</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 |title=The Nobel Prize in Physics 1957 |url=https://www.nobelprize.org/nobel_prizes/physics/laureates/1957/ |access-date=March 24, 2015 |publisher=[[The Nobel Foundation]]}}</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 |url=https://www.newscientist.com/people/chien-shiung-wu/ |access-date=February 18, 2025 |website=newscientist.com}}</ref>

[[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 |last=Ivanov |first=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 |arxiv=1706.08687 |bibcode=2017PhRvD..95k3006I |doi=10.1103/PhysRevD.95.113006 |issn=2470-0010 |s2cid=119103283}}</ref><ref>{{Cite journal |last=Ivanov |first=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 |arxiv=1805.09702 |bibcode=2018IJMPA..3350199I |doi=10.1142/S0217751X18501993 |issn=0217-751X |s2cid=119088802}}</ref>]]

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. |url=https://books.google.com/books?id=tEM9TCfAe7EC&pg=PA201 |title=Nuclear Principles in Engineering |publisher=[[Springer Science + Business Media]] |year=2009 |isbn=978-0-387-85608-7 |page=201}}</ref> If it comes from the L-shell, the process is called L-capture, etc.

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:

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/ |archive-url=https://web.archive.org/web/20181010070007/http://www.nndc.bnl.gov/chart/ |archive-date=2018-10-10 |access-date=2014-09-18 |publisher=National Nuclear Data Center, Brookhaven National Laboratory}}</ref>

Three types of beta decay in competition are illustrated by the single isotope {{nuclide|copper|64|link=yes}} (29 protons, 35 neutrons), which has a half-life of about 12.7 hours.<ref name="Cu-64">[http://www.lnhb.fr/nuclides/Cu-64_tables.pdf Atomic and Nuclear Data: Chapter 12 Cu-64 ] {{Webarchive|url=https://web.archive.org/web/20240502181344/http://www.lnhb.fr/nuclides/Cu-64_tables.pdf |date=2024-05-02 }} Laboratoire National Henri Becquerel, 2011. Retrieved on 2024-05-01.</ref> This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay.<ref name="Copper-64">{{Cite web |last=Gilbert |first=Thomas R. |title=Problem 20: Copper-64 is an unusual radionuclide |url=https://www.vaia.com/en-us/textbooks/chemistry/chemistry-the-science-in-context-5-edition/chapter-19/problem-20-copper-64-is-an-unusual-radionuclide-in-that-it-m/ |url-status=live |archive-url=https://web.archive.org/web/20240502182148/https://www.vaia.com/en-us/textbooks/chemistry/chemistry-the-science-in-context-5-edition/chapter-19/problem-20-copper-64-is-an-unusual-radionuclide-in-that-it-m/ |archive-date=2 May 2024 |access-date=2 May 2024 |website=Chemistry The Science in Context |publisher=Vaia}}</ref> This particular nuclide is almost equally likely to undergo proton decay (by [[positron emission]], 18% or by [[electron capture]], 43%; both forming [[Isotopes of nickel|{{SimpleNuclide|Nickel|64}}]]) or neutron decay (by electron emission, 39%; forming [[Isotopes of zinc|{{SimpleNuclide|Zinc|64}}]]).<ref name="Cu-64" /><ref name="Copper-64" />

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 |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 |access-date=2014-09-18 |website=LBNL Isotopes Project |publisher=Lawrence Berkeley National Laboratory}}</ref>

| [[fluorine-20|^{{0}}20F]] || 5390.86 ||style="text-align:center;"| β⁻

|[[Bismuth-210|²¹⁰Bi]] || 1162.2 ||style="text-align:center;"| β⁻

Tritium β⁻ decay being used in the [[KATRIN]] experimental search for [[sterile neutrinos]].<ref>{{Cite journal |last=Mertens |first=Susanne |date=2015-01-01 |title=Status of the KATRIN Experiment and Prospects to Search for keV-mass Sterile Neutrinos in Tritium β-decay |journal=Physics Procedia |series=13th International Conference on Topics in Astroparticle and Underground Physics, TAUP 2013 |volume=61 |pages=267–273 |bibcode=2015PhPro..61..267M |doi=10.1016/j.phpro.2014.12.043 |issn=1875-3892 |doi-access=free}}</ref>

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 |access-date=2013-03-09 |website=[[HyperPhysics]]}}</ref>

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

Other approximations can be found in the literature.<ref>{{Cite journal |last=Venkataramaiah |first=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 |last=Schenter |first=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 |bibcode=1983NSE....83..393S |doi=10.13182/NSE83-A17574 |osti=5307377}}</ref>

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 momentum (or energy) lies within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy.<ref>{{Cite journal |last=Kurie |first=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 |last=Kurie |first=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 |bibcode=2012JPhG...39l4008R |doi=10.1088/0954-3899/39/12/124008 |s2cid=119158221}}</ref>

After the discovery of parity non-conservation (see ''[[#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 |last=Frauenfelder |first=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 |last=Konopinski |first=E. J. |title=Alpha-, Beta- and Gamma-Ray Spectroscopy |last2=Rose |first2=M. E. |publisher=[[North-Holland Publishing Company]] |year=1966 |editor-last=Siegbhan |editor-first=K. |volume=2 |chapter=The Theory of nuclear Beta Decay}}</ref>

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

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 |last=Daudel |first=Raymond |last2=Jean |first2=Maurice |last3=Lecoin |first3=Marcel |year=1947 |title=Sur la possibilité d'existence d'un type particulier de radioactivité phénomène de création e |url=https://hal.archives-ouvertes.fr/jpa-00234057/document |journal=J. Phys. Radium |volume=8 |issue=8 |pages=238–243 |doi=10.1051/jphysrad:0194700808023800}}</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 |last=Jung |first=M. |display-authors=etal |year=1992 |title=First observation of bound-state β⁻ 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 |title=Bound-state beta decay of highly ionized atoms |url=http://www.ca.infn.it/~oldeman/resneu/p1522_1.pdf |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 |access-date=June 9, 2013}}</ref><ref>{{Cite journal |last=Bai |first=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. |date=2 December 2024 |title=Bound-State Beta Decay of {{sup|205}}Tl{{sup|81+}} Ions and the LOREX Project |url=https://link.aps.org/doi/10.1103/PhysRevLett.133.232701 |journal=Physical Review Letters |publisher=American Physical Society |volume=133 |issue=23 |pages=232701 |arxiv=2501.06029 |doi=10.1103/PhysRevLett.133.232701 |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. |collaboration=E121 Collaboration and LOREX Collaboration}}</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 ¹⁹³Ir, ¹⁹⁴Au, ²⁰²Tl, ²¹⁵At, ²⁴³Am, and ²⁴⁶Bk.<ref name="bs-prediction">{{Cite journal |last=Liu |first=Shuo |last2=Gao |first2=Chao |last3=Xu |first3=Chang |date=2021 |title=Investigation of bound state β⁻ decay half-lives of bare atoms |journal=Physical Review C |volume=104 |issue=2 |pages=024304 |bibcode=2021PhRvC.104b4304L |doi=10.1103/PhysRevC.104.024304}}</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,{{NUBASE2020|ref}} but for fully ionized {{sup|187}}Re{{sup|75+}} this is shortened to only 32.9 years. 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>

<br />"Note also, that the decay of bare ¹⁸⁷Re is dominated by the nonunique transition to the first excited state of ¹⁸⁷Os, since the decay to the ground state has a much smaller matrix element."</ref> 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 |last=Takahashi |first=K. |last2=Boyd |first2=R. N. |last3=Mathews |first3=G. J. |last4=Yokoi |first4=K. |date=1 October 1987 |title=Bound-state beta decay of highly ionized atoms |url=https://www.researchgate.net/publication/13335547 |journal=Physical Review C |volume=36 |issue=4 |pages=1522–1528 |bibcode=1987PhRvC..36.1522T |doi=10.1103/PhysRevC.36.1522 |pmid=9954244}}</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 |last=Belli |first=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. |date=2014 |title=Investigation of rare nuclear decays with BaF₂ crystal scintillator contaminated by radium |journal=European Physical Journal A |volume=50 |issue=9 |pages=134–143 |arxiv=1407.5844 |bibcode=2014EPJA...50..134B |doi=10.1140/epja/i2014-14134-6 |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" />

== Bibliography ==

* '''Beta decay simulation''' [https://phet.colorado.edu/en/simulation/legacy/beta-decay]