β− decay in anatomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron. Neither of these depictions shows the intermediatevirtualW− boson.
Innuclear physics,beta decay (β-decay) is a type ofradioactive decay in which anatomic nucleus emits abeta particle (fast energeticelectron orpositron), transforming into anisobar of that nuclide. For example, beta decay of aneutron transforms it into aproton by the emission of an electron accompanied by anantineutrino; or, conversely a proton is converted into a neutron by the emission of a positron with aneutrino in what is calledpositron 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 stableratio of protons to neutrons. The probability of a nuclide decaying due to beta and other forms of decay is determined by itsnuclear binding energy. The binding energies of all existing nuclides form what is called the nuclear band orvalley of stability.[1] For either electron or positron emission to be energetically possible, the energy release (see below) orQ value must be positive.
Beta decay is a consequence of theweak force, which is characterized by relatively long decay times. Nucleons are composed ofup quarks anddown quarks,[2] and the weak force allows aquark to change itsflavour by means of a virtualW 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,[3] 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 anelectron neutrino is released.
The two types of beta decay are known asbeta minus andbeta plus. In beta minus (β−) decay, a neutron is converted to a proton, and the process creates an electron and anelectron 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 aspositron emission.[4]
Beta decay conserves a quantum number known as thelepton number, or the number of electrons and their associated neutrinos (other leptons are themuon andtau particles). These particles have lepton number +1, while their antiparticles have lepton number −1. Since a proton or neutron has lepton number zero, β+ decay (a positron, or antielectron) must be accompanied with an electron neutrino, while β− decay (an electron) must be accompanied by an electron antineutrino.
An example of electron emission (β− decay) is the decay ofcarbon-14 intonitrogen-14 with ahalf-life of about 5,700 years:
14 6C →14 7N +e− +ν e
In this form of decay, the original element becomes a new chemical element in a process known asnuclear transmutation. This new element has an unchangedmass numberA, but anatomic numberZ that is increased by one. As in all nuclear decays, the decaying element (in this case14 6C) is known as theparent nuclide while the resulting element (in this case14 7N) is known as thedaughter nuclide.
Another example is the decay of hydrogen-3 (tritium) intohelium-3 with a half-life of about 12.3 years:
3 1H →3 2He +e− +ν e
An example of positron emission (β+ decay) is the decay ofmagnesium-23 intosodium-23 with a half-life of about 11.3 s:
23 12Mg →23 11Na +e+ +ν e
β+ decay also results in nuclear transmutation, with the daughter element having an atomic number that is decreased by one.
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 of210Bi is shown. In this example, the total decay energy is 1.16 MeV, so the antineutrino has the remaining energy: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.
Radioactivity was discovered in 1896 byHenri Becquerel inuranium, and subsequently observed byMarie andPierre Curie inthorium and in the newly discovered elementspolonium andradium. 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 termedgamma rays. Alpha, beta, and gamma are the first three letters of theGreek alphabet.
In 1900, Becquerel measured themass-to-charge ratio (m/e) for beta particles by the method ofJ.J. Thomson used to study cathode rays and identify the electron. He found thatm/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.[5]
In 1901, Rutherford andFrederick Soddy showed that alpha and beta radioactivity involves thetransmutation of atoms into atoms of other chemical elements. In 1913, after the products of more radioactive decays were known, Soddy andKazimierz Fajans independently proposed theirradioactive displacement law, which states that beta (i.e.,β− ) emission from one element produces another element one place to the right in theperiodic table, while alpha emission produces an element two places to the left.
The study of beta decay provided the first physical evidence for the existence of theneutrino. In both alpha and gamma decay, the resulting alpha or gamma particle has a narrow energydistribution, 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 byLise Meitner andOtto Hahn in 1911 and byJean Danysz in 1913 showed multiple lines on a diffuse background. These measurements offered the first hint that beta particles have a continuous spectrum.[6] In 1914,James Chadwick used a magneticspectrometer with one ofHans Geiger's newcounters to make more accurate measurements which showed that the spectrum was continuous.[6][7] The results, which appeared to be in contradiction to thelaw of conservation of energy, were validated by means of calorimetric measurements in 1929 byLise Meitner andWilhelm Orthmann.[8] 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.[9] 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 theconservation of angular momentum. Molecular band spectra showed that thenuclear spin ofnitrogen-14 is 1 (i.e., equal to thereduced Planck constant) and more generally that the spin is integral for nuclei of evenmass number and half-integral for nuclei of odd mass number. This was later explained by theproton-neutron model of the nucleus.[9] 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 andNevill 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 ifconservation of energy was true only in a statistical sense, thus thisprinciple might be violated in any given decay.[9]: 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 afamous 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 landmarktheory 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 physicistsClyde Cowan andFrederick Reines in theCowan–Reines neutrino experiment.[10] The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.
In 1934,Frédéric andIrène Joliot-Curie bombarded aluminium with alpha particles to effect the nuclear reaction4 2He + 27 13Al →30 15P + 1 0n, and observed that the product isotope30 15P emits a positron identical to those found in cosmic rays (discovered byCarl David Anderson in 1932). This was the first example ofβ+ decay (positron emission), which they termedartificial radioactivity since30 15P is a short-lived nuclide which does not exist in nature. In recognition of their discovery, the couple were awarded theNobel Prize in Chemistry in 1935.[11]
In 1956,Tsung-Dao Lee andChen Ning Yang noticed that there was no evidence thatparity 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.[17] Later that year,Chien-Shiung Wu and coworkers showed experimentally that an asymmetrical beta emission from60 Co proved that parity is not conserved in beta decay.[18][19][20] This surprising result overturned long-held assumptions about parity and the weak force. In recognition of their theoretical work, Lee and Yang were awarded theNobel Prize for Physics in 1957.[21] However Wu, who was female, was not awarded the Nobel prize.[22]
Inβ− decay, theweak interaction converts anatomic nucleus into a nucleus withatomic number increased by one, while emitting an electron (e− ) and an electronantineutrino (ν e).β− decay generally occurs in neutron-rich nuclei.[25] The generic equation is:
whereA andZ are themass number andatomic number of the decaying nucleus, and X and X′ are the initial and final elements, respectively.
Another example is when thefree neutron (1 0n) decays byβ− decay into a proton (p):
n →p +e− +ν e.
At thefundamental level (as depicted in theFeynman diagram on the right), this is caused by the conversion of the negatively charged (−1/3e) down quark to the positively charged (+2/3e) up quark, which is promoted by a virtualW− boson; theW− boson subsequently decays into an electron and an electron antineutrino:
Inβ+ decay, or positron emission, the weak interaction converts an atomic nucleus into a nucleus with atomic number decreased by one, while emitting a positron (e+ ) and anelectron neutrino (ν e).β+ decay generally occurs in proton-rich nuclei. The generic equation is:
However,β+ decay cannot occur in an isolated proton because it requires energy, due to themass of the neutron being greater than the mass of the proton.β+ decay can only happen inside nuclei when the daughter nucleus has a greaterbinding 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 aW+ or the absorption of aW− . When aW+ boson is emitted, it decays into apositron and anelectron neutrino:
In all cases whereβ+ decay (positron emission) of a nucleus is allowed energetically, so too iselectron capture allowed. This is a process during which a nucleus captures one of its atomic electrons, resulting in the emission of a neutrino:
A ZX +e− →A Z−1X′ +ν e
An example of electron capture is one of the decay modes ofkrypton-81 intobromine-81:
81 36Kr +e− →81 35Br +ν e
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 2mec2,β+ decay is not energetically possible, and electron capture is the sole decay mode.[26]
If the captured electron comes from the innermost shell of the atom, theK-shell, which has the highest probability to interact with the nucleus, the process is called K-capture.[27] 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+ decay. The converse, however, is not true: electron capture is theonly type of decay that is allowed in proton-rich nuclides that do not have sufficient energy to emit a positron and neutrino.[26]
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 diagonal line is where proton number is the same as neutron number.
Beta decay does not change the number (A) ofnucleons in the nucleus, but changes only itschargeZ. Thus the set of allnuclides with the same A can be introduced; theseisobaric nuclides may turn into each other via beta decay. For a givenA there is one that is most stable. It is said to be beta stable, because it presents a local minimum of themass excess: if such a nucleus has(A,Z) numbers, the neighbour nuclei(A,Z−1) and(A,Z+1) have higher mass excess and can beta decay into(A,Z), but not vice versa. For all odd mass numbersA, there is only one known beta-stable isobar. For even A, there are up to three different beta-stable isobars experimentally known; for example,124 50Sn,124 52Te, and124 54Xe are all beta-stable. There are about 350 knownbeta-decay stable nuclides.[28]
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.
Three types of beta decay in competition are illustrated by the single isotope64 29Cu (29 protons, 35 neutrons), which has a half-life of about 12.7 hours.[29] This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay.[30] This particular nuclide is almost equally likely to undergo proton decay (bypositron emission, 18% or byelectron capture, 43%; both forming64 Ni) or neutron decay (by electron emission, 39%; forming64 Zn).[29][30]
Most naturally occurring nuclides on earth are beta stable. Nuclides that are not beta stable havehalf-lives ranging from under a second to periods of time significantly greater than theage of the universe. One common example of a long-lived isotope is the odd-proton odd-neutron nuclide40 19K, which undergoes all three types of beta decay (β− ,β+ and electron capture) with a half-life of1.277×109 years.[31]
Beta decay just changesneutron toproton or, in the case of positive beta decay (electron capture)proton toneutron so the number of individualquarks doesn't change. It is only the baryon flavor that changes, here labelled as theisospin.
Up and downquarks have total isospin and isospin projections
TheQ value is defined as the total energy released in a given nuclear decay. In beta decay,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 anykinetic energy ranging from 0 toQ.[1] A typicalQ is around 1 MeV, but can range from a fewkeV to a few tens of MeV.
Since therest mass of the electron is 511 keV, the most energetic beta particles areultrarelativistic, with speeds very close to thespeed of light.In the case of187Re, the maximum speed of the beta particle is only 9.8% of the speed of light.
where is the mass of the nucleus of theA ZX atom, is the mass of the electron, and 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 nucleusmN is related to the standardatomic massm byThat is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of allelectron binding energiesBi for the atom. This equation is rearranged to find, and is found similarly. Substituting these nuclear masses into theQ-value equation, while neglecting the nearly zero antineutrino mass and the difference in electron binding energies, which is very small for high-Z atoms, we haveThis energy is carried away as kinetic energy by the electron and antineutrino.
Because the reaction will proceed only when theQ value is positive, β− decay can occur when the mass of atomA ZX is greater than the mass of atomA Z+1X′.[33]
The equations for β+ decay are similar, with the generic equation
A ZX →A Z−1X′ +e+ +ν e
givingHowever, in this equation, the electron masses do not cancel, and we are left with
Because the reaction will proceed only when theQ value is positive, β+ decay can occur when the mass of atomA ZX exceeds that ofA Z−1X′ by at least twice the mass of the electron.[33]
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
A ZX +e− →A Z−1X′ +ν e
we havewhich simplifies towhereBn 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+ decay can always also undergo electron capture, but the reverse is not true.[33]
Beta spectrum of210Bi.Emax =Q = 1.16 MeV is the maximum energy
Beta decay can be considered as aperturbation as described in quantum mechanics, and thusFermi's Golden Rule can be applied. This leads to an expression for the kinetic energy spectrumN(T) of emitted betas as follows:[34]whereT is the kinetic energy,CL is a shape function that depends on the forbiddenness of the decay (it is constant for allowed decays),F(Z,T) is the Fermi Function (see below) withZ the charge of the final-state nucleus,E =T +mc2 is the total energy, is the momentum, andQ is theQ value of the decay. The kinetic energy of the emitted neutrino is given approximately byQ minus the kinetic energy of the beta.
As an example, the beta decay spectrum of210Bi (originally called RaE) is shown to the right.
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:[35]wherep is the final momentum, Γ theGamma function, and (ifα is thefine-structure constant andrN the radius of the final state nucleus), (+ for electrons, − for positrons), and.
For non-relativistic betas (Q ≪mec2), this expression can be approximated by:[36]
Other approximations can be found in the literature.[37][38]
AKurie plot (also known as aFermi–Kurie plot) is a graph used in studying beta decay developed byFranz 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.[39][40] 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'sQ value). With a Kurie plot one can find the limit on the effective mass of a neutrino.[41]
Helicity (polarization) of neutrinos, electrons and positrons emitted in beta decay
After the discovery of parity non-conservation (see§ History), it was found that, in beta decay, electrons are emitted mostly with negativehelicity, i.e., they move, naively speaking, like left-handed screws driven into a material (they have negative longitudinalpolarization).[42] 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.[43]
The higher the energy of the particles, the higher their polarization.
Beta decays can be classified according to the angular momentum (L value) and total spin (S 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 (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.
AFermi transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin, leading to an angular momentum change 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 bywith the weak vector coupling constant, theisospinraising and lowering operators, and running over all protons and neutrons in the nucleus.
AGamow–Teller transition is a beta decay in which the spins of the emitted electron (positron) and anti-neutrino (neutrino) couple to total spin, leading to an angular momentum change between the initial and final states of the nucleus (assuming an allowed transition).In this case, the nuclear part of the operator is given bywith the weak axial-vector coupling constant, and thespin Pauli matrices, which can produce a spin-flip in the decaying nucleon.
WhenL > 0, the decay is referred to as "forbidden". Nuclearselection rules require highL values to be accompanied by changes innuclear spin (J) andparity (π). The selection rules for theLth forbidden transitions are:whereΔπ = 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 ΔJ and Δπ values for the first few values of L:
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 neutralhydrogen atom.[44] In this type of beta decay, in essence all of the neutrondecay 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− decays were predicted byDaudel, Jean, and Lecoin in 1947,[45] and the phenomenon in fully ionized atoms was first observed for163Dy66+ in 1992 by Jung et al. of theDarmstadt Heavy-Ion Research Center. Though neutral163Dy is stable, fully ionized163Dy66+ undergoes β− decay into the K and L shells with a half-life of 47 days.[46] The resulting nucleus –163Ho66+ – is stable only in this almost fully ionized state and will decay viaelectron capture into163Dy in the neutral state. Likewise, while being stable in the neutral state, the fully ionized205Tl81+ undergoes bound-state β− decay to205Pb81+ with a half-life of291+33 −27 days.[47][48] The half-lives of neutral163Ho and205Pb are respectively 4570 years and1.73×107 years. In addition, it is estimated that β− decay is energetically impossible for natural atoms but theoretically possible when fully ionized also for193Ir,194Au,202Tl,215At,243Am, and246Bk.[49]
Another possibility is that a fully ionized atom undergoes greatly accelerated β decay, as observed for187Re by Bosch et al., also at Darmstadt. Neutral187Re does undergo β− decay, with half-life4.12×1010 years,[50] but for fully ionized187Re75+ this is shortened to only 32.9 years. This is because187Re75+ is energetically allowed to undergo β− decay to the first-excited state in187Os75+, a process energetically disallowed for natural187Re.[51] Similarly, neutral241Pu undergoes β− decay with a half-life of 14.3 years, but in its fully ionized state the beta-decay half-life of241Pu94+ decreases to 4.2 days.[52] For comparison, the variation of decay rates of other nuclear processes due to chemical environment isless than 1%. Moreover, current mass determinations cannot decisively determine whether222Rn is energetically possible to undergo β− decay (the decay energy given in AME2020 is (−6 ± 8) keV),[53][54] but in either case it is predicted that β− will be greatly accelerated for fully ionized222Rn86+.[49]
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 decay and 2β are possible, the rarer 2β process is effectively impossible to observe. However, in nuclei where β decay is forbidden but 2β is allowed, the process can be seen and a half-life measured.[55] Thus, 2β is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not changeA; thus, at least one of the nuclides with some givenA 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 areMajorana particles (i.e., they are their own antiparticles), then a decay known asneutrinoless double beta decay will occur. Most neutrino physicists believe that neutrinoless 2β has never been observed.[55]
^Ivanov, A. N.; Höllwieser, R.; Troitskaya, N. I.; Wellenzohn, M.; Berdnikov, Ya. A. (2018-11-30). "Gauge properties of hadronic structure of nucleon in neutron radiative beta decay to orderO(α/π) in standardV −A effective theory with QED and linear sigma model of strong low-energy interactions".International Journal of Modern Physics A.33 (33): 1850199.arXiv:1805.09702.Bibcode:2018IJMPA..3350199I.doi:10.1142/S0217751X18501993.ISSN0217-751X.S2CID119088802.
^Konopinski, E. J.; Rose, M. E. (1966). "The Theory of nuclear Beta Decay". In Siegbhan, K. (ed.).Alpha-, Beta- and Gamma-Ray Spectroscopy. Vol. 2.North-Holland Publishing Company.
^Bosch, F.; et al. (1996). "Observation of bound-state beta minus decay of fully ionized187Re:187Re–187Os Cosmochronometry".Physical Review Letters.77 (26):5190–5193.Bibcode:1996PhRvL..77.5190B.doi:10.1103/PhysRevLett.77.5190.PMID10062738. "Note also, that the decay of bare187Re is dominated by the nonunique transition to the first excited state of187Os, since the decay to the ground state has a much smaller matrix element."
