Beta-decay stable isobars are the set of nuclides which cannot undergo beta decay, that is, the transformation of a neutron to a proton or a proton to a neutron within the nucleus. A subset of these nuclides are also stable with regards to double beta decay or theoretically higher simultaneous beta decay, as they have the lowest energy of all isobars with the same mass number.
This set of nuclides is also known as the line of beta stability, a term already in common use in 1965.[1][2] This line lies along the bottom of the nuclear valley of stability.
Introduction
The line of beta stability can be defined mathematically by finding the nuclide with the greatest binding energy for a given mass number, by a model such as the classical semi-empirical mass formula developed by C. F. Weizsäcker. These nuclides are local maxima in terms of binding energy for a given mass number.
From 36 to 72, only eight (36, 40, 46, 50, 54, 58, 64, 70) have two, and the remaining 11 have one.
From 74 to 122, three (88, 90, 118) have one, and the remaining 22 have two.
From 124 to 154, only one (140) has one, five have three, and the remaining 10 have two.
From 156 to 262, only eighteen have one, and the remaining 36 have two, though there may also exist some undiscovered ones.
All primordial nuclides are beta decay stable, with the exception of 40K, 50V, 87Rb, 113Cd, 115In, 138La, 176Lu, and 187Re. In addition, 123Te and 180mTa have not been observed to decay, but are believed to undergo beta decay with an extremely long half-life (over 1015 years). (123Te can only undergo electron capture to 123Sb, whereas 180mTa can decay in both directions, to 180Hf or 180W.) Among non-primordial nuclides, there are some other cases of theoretically possible but never-observed beta decay, notably including 222Rn[3] and 247Cm (the most stable isotopes of their elements considering all decay modes). Finally, 48Ca and 96Zr have not been observed to undergo beta decay (theoretically possible for both) which is extremely suppressed, but double beta decay is known for both. Similar suppression of single beta decay occurs also for 148Gd, a rather short-lived alpha emitter.
All elements up to and including nobelium, except technetium, promethium, and mendelevium, are known to have at least one beta-stable isotope. It is known that technetium and promethium have no beta-stable isotopes; current measurement uncertainties are not enough to say whether mendelevium has them or not.
346 nuclides (including 260Fm whose discovery is unconfirmed) have been definitively identified as beta-stable.[4][5] Theoretically predicted or experimentally observed double beta decay is shown by arrows, i.e. arrows point toward the lightest-mass isobar. This is sometimes dominated by alpha decay or spontaneous fission, especially for the heavy elements. Observed decay modes are listed as α for alpha decay, SF for spontaneous fission, and n for neutron emission in the special case of 5He. For mass 5 there are no bound isobars at all; mass 8 has bound isobars, but the beta-stable 8Be is unbound.[6]
Two beta-decay stable nuclides exist for odd neutron numbers 1 (2H and 3He), 3 (5He and 6Li – the former has an extremely short half-life), 5 (9Be and 10B), 7 (13C and 14N), 55 (97Mo and 99Ru), and 85 (145Nd and 147Sm); the first four cases involve very light nuclides where odd-odd nuclides are more stable than their surrounding even-even isobars, and the last two surround the proton numbers 43 and 61 which have no beta-stable isotopes. Also, two beta-decay stable nuclides exist for odd proton numbers 1, 3, 5, 7, 17, 19, 29, 31, 35, 47, 51, 63, 77, 81, and 95; the first four cases involve very light nuclides where odd-odd nuclides are more stable than their surrounding even-even isobars, and the other numbers surround the neutron numbers 19, 21, 35, 39, 45, 61, 71, 89, 115, 123, 147 which have no beta-stable isotopes. (For N = 21 the long-lived primordial 40K exists, and for N = 71 there is 123Te whose electron capture has not yet been observed, but neither are beta-stable.)
All even proton numbers 2 ≤ Z ≤ 102 have at least two beta-decay stable nuclides, with exactly two for Z = 4 (8Be and 9Be – the former having an extremely short half-life) and 6 (12C and 13C). Also, the only even neutron numbers with only one beta-decay stable nuclide are 0 (1H) and 2 (4He); at least two beta-decay stable nuclides exist for even neutron numbers in the range 4 ≤ N ≤ 160, with exactly two for N = 4 (7Li and 8Be), 6 (11B and 12C), 8 (15N and 16O), 66 (114Cd and 116Sn, noting also primordial but not beta-stable 115In), 120 (198Pt and 200Hg), and 128 (212Po and 214Rn – both very unstable to alpha decay). Seven beta-decay stable nuclides exist for the magic N = 82 (136Xe, 138Ba, 139La, 140Ce, 141Pr, 142Nd, and 144Sm) and five for N = 20 (36S, 37Cl, 38Ar, 39K, and 40Ca), 50 (86Kr, 88Sr, 89Y, 90Zr, and 92Mo, noting also primordial but not beta-stable 87Rb), 58 (100Mo, 102Ru, 103Rh, 104Pd, and 106Cd), 74 (124Sn, 126Te, 127I, 128Xe, and 130Ba), 78 (130Te, 132Xe, 133Cs, 134Ba, and 136Ce), 88 (148Nd, 150Sm, 151Eu, 152Gd, and 154Dy – the last not primordial), and 90 (150Nd, 152Sm, 153Eu, 154Gd, and 156Dy).
For A ≤ 209, the only beta-decay stable nuclides that are not primordial nuclides are 5He, 8Be, 146Sm, 150Gd, and 154Dy. (146Sm has a half-life long enough that it should barely survive as a primordial nuclide, but it has never been experimentally confirmed as such.) All beta-decay stable nuclides with A ≥ 209 are known to undergo alpha decay, though for some, spontaneous fission is the dominant decay mode. Cluster decay is sometimes also possible, but in all known cases it is a minor branch compared to alpha decay or spontaneous fission. Alpha decay is energetically possible for all beta-stable nuclides with A ≥ 165 with the single exception of 204Hg, but in most cases the Q-value is small enough that such decay has never been seen.[7]
With the exception of 262No, no nuclides with A > 260 are currently known to be beta-stable. Moreover, the known beta-stable nuclei for individual masses A = 222, A = 256, and A ≥ 258 (corresponding to proton numbers Z = 86 and Z ≥ 98, or to neutron numbers N = 136 and N ≥ 158) may not represent the complete set.[8][9]
One chart of known and predicted nuclides up to Z = 149, N = 256. Black denotes the predicted beta-stability line, which is in good agreement with experimental data, though it fails to predict that Tc and Pm have no beta-stable isotope (the mass differences causing these anomalies are small). Islands of stability are predicted to center near Ds and 126, beyond which the model appears to deviate from several rules of the semi-empirical mass formula.
The general patterns of beta-stability are expected to continue into the region of superheavy elements, though the exact location of the center of the valley of stability is model dependent. It is widely believed that an island of stability exists along the beta-stability line for isotopes of elements around copernicium that are stabilized by shell closures in the region; such isotopes would decay primarily through alpha decay or spontaneous fission.[13] Beyond the island of stability, various models that correctly predict many known beta-stable isotopes also predict anomalies in the beta-stability line that are unobserved in any known nuclides, such as the existence of two beta-stable nuclides with the same odd mass number.[8][14] This is a consequence of the fact that a semi-empirical mass formula must consider shell correction and nuclear deformation, which become far more pronounced for heavy nuclides.[14][15]
The beta-stable fully ionized nuclei (with all electrons stripped) are somewhat different. Firstly, if a proton-rich nuclide can only decay by electron capture (because the energy difference between the parent and daughter is less than 1.022MeV, the amount of decay energy needed for positron emission), then full ionization makes decay impossible. This happens for example for 7Be.[16] Moreover, sometimes the energy difference is such that while β− decay violates conservation of energy for a neutral atom, bound-state β− decay (in which the decay electron remains bound to the daughter in an atomic orbital) is possible for the corresponding bare nucleus. Within the range 2 ≤ A ≤ 270, this means that 163Dy, 193Ir, 205Tl, 215At, and 243Am among beta-stable neutral nuclides cease to be beta-stable as bare nuclides, and are replaced by their daughters 163Ho, 193Pt, 205Pb, 215Rn, and 243Cm.[17]
Beta decay toward minimum mass
Beta decay generally causes nuclides to decay toward the isobar with the lowest mass (which is often, but not always, the one with highest binding energy) with the same mass number. Those with lower atomic number and higher neutron number than the minimum-mass isobar undergo beta-minus decay, while those with higher atomic number and lower neutron number undergo beta-plus decay or electron capture.
However, there are a few odd-odd nuclides between two beta-stable even-even isobars, that predominantly decay to the higher-mass of the two beta-stable isobars. For example, 40K could either undergo electron capture or positron emission to 40Ar, or undergo beta minus decay to 40Ca: both possible products are beta-stable. The former process would produce the lighter of the two beta-stable isobars, yet the latter is more common.
↑ 48Ca is theoretically capable of beta decay to 48Sc, thus making it not a beta-stable nuclide. However, such a process has never been observed, having a partial half-life greater than 1.1+0.8 −0.6×1021years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[10]
↑ 96Zr is theoretically capable of beta decay to 96Nb, thus making it not a beta-stable nuclide. However, such a process has never been observed, having a partial half-life greater than 2.4×1019years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[11]
↑ 148Gd was previously thought to be a third beta-stable isobar for mass 148,[6] but according to current mass determinations it has a higher mass than 148Eu and can undergo electron capture. Nevertheless, the mass difference is very small (27.0keV, even lower than likewise unseen electron capture of 123Te), and only alpha decay has been observed experimentally for 148Gd.
↑ While the AME2020 atomic mass evaluation gives 222Rn a lower mass than 222Fr (the β−-decay energy is given as (−6 ± 8) keV),[5] implying beta-stability, it is predicted that single beta decay of 222Rn is energetically possible (albeit with very low decay energy),[3] and it falls within the error margin given in AME2020. Hence, current mass determinations cannot decisively determine whether 222Rn is beta-stable or not, though only the alpha decay mode is experimentally known for that nuclide, and the search for beta decay yielded a lower partial half-life limit of 8 years.[3]
↑ While the AME2020 atomic mass evaluation gives 256Cf a lower mass than 256Es (the β−-decay energy is given as (−140# ± 330#) keV),[5] implying beta-stability, the error margin between them is larger than the mass difference. Hence, current mass determinations cannot decisively determine whether 256Cf is beta-stable or not.
↑ While the AME2020 atomic mass evaluation gives 259Md a lower mass than 259Fm (the β+-decay energy is given as (−140# ± 300#) keV),[5] implying beta-stability, the error margin between them is larger than the mass difference. Hence, current mass determinations cannot decisively determine which one of 259Fm and 259Md is beta-stable.
↑ There is no known beta-stable isobar for mass 261, although they are known for the surrounding masses 260 and 262. Various models suggest that one of the undiscovered 261Md and 261No should be beta-stable.[12][8]
↑ While the AME2020 atomic mass evaluation gives 262Rf a higher mass than 262Lr (the β+-decay energy is given as (290# ± 300#) keV),[5] implying non-beta-stability, the error margin between them is larger than the mass difference. Hence, current mass determinations cannot decisively determine whether 262Rf is beta-stable or not.
Related Research Articles
In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which an atomic nucleus emits a beta particle, transforming into an 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 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. For either electron or positron emission to be energetically possible, the energy release or Q value must be positive.
Stable nuclides are isotopes of a chemical element whose nucleons are in a configuration that does not permit them the surplus energy required to produce a radioactive emission. The nuclei of such isotopes are not radioactive and unlike radionuclides do not spontaneously undergo radioactive decay. When these nuclides are referred to in relation to specific elements they are usually called that element's stable isotopes.
In nuclear physics, the island of stability is a predicted set of isotopes of superheavy elements that may have considerably longer half-lives than known isotopes of these elements. It is predicted to appear as an "island" in the chart of nuclides, separated from known stable and long-lived primordial radionuclides. Its theoretical existence is attributed to stabilizing effects of predicted "magic numbers" of protons and neutrons in the superheavy mass region.
Nuclides are a class of atoms characterized by their number of protons, Z, their number of neutrons, N, and their nuclear energy state.
In nuclear engineering, fissile material is material that can undergo nuclear fission when struck by a neutron of low energy. A self-sustaining thermal chain reaction can only be achieved with fissile material. The predominant neutron energy in a system may be typified by either slow neutrons or fast neutrons. Fissile material can be used to fuel thermal-neutron reactors, fast-neutron reactors and nuclear explosives.
In nuclear science a decay chain refers to the predictable series of radioactive disintegrations undergone by the nuclei of certain unstable chemical elements.
Lead (82Pb) has four observationally stable isotopes: 204Pb, 206Pb, 207Pb, 208Pb. Lead-204 is entirely a primordial nuclide and is not a radiogenic nuclide. The three isotopes lead-206, lead-207, and lead-208 represent the ends of three decay chains: the uranium series, the actinium series, and the thorium series, respectively; a fourth decay chain, the neptunium series, terminates with the thallium isotope 205Tl. The three series terminating in lead represent the decay chain products of long-lived primordial 238U, 235U, and 232Th. Each isotope also occurs, to some extent, as primordial isotopes that were made in supernovae, rather than radiogenically as daughter products. The fixed ratio of lead-204 to the primordial amounts of the other lead isotopes may be used as the baseline to estimate the extra amounts of radiogenic lead present in rocks as a result of decay from uranium and thorium.
There are 39 known isotopes and 17 nuclear isomers of tellurium (52Te), with atomic masses that range from 104 to 142. These are listed in the table below.
Technetium (43Tc) is one of the two elements with Z < 83 that have no stable isotopes; the other such element is promethium. It is primarily artificial, with only trace quantities existing in nature produced by spontaneous fission or neutron capture by molybdenum. The first isotopes to be synthesized were 97Tc and 99Tc in 1936, the first artificial element to be produced. The most stable radioisotopes are 97Tc, 98Tc, and 99Tc.
Nuclear binding energy in experimental physics is the minimum energy that is required to disassemble the nucleus of an atom into its constituent protons and neutrons, known collectively as nucleons. The binding energy for stable nuclei is always a positive number, as the nucleus must gain energy for the nucleons to move apart from each other. Nucleons are attracted to each other by the strong nuclear force. In theoretical nuclear physics, the nuclear binding energy is considered a negative number. In this context it represents the energy of the nucleus relative to the energy of the constituent nucleons when they are infinitely far apart. Both the experimental and theoretical views are equivalent, with slightly different emphasis on what the binding energy means.
A table or chart of nuclides is a two-dimensional graph of isotopes of the elements, in which one axis represents the number of neutrons and the other represents the number of protons in the atomic nucleus. Each point plotted on the graph thus represents a nuclide of a known or hypothetical chemical element. This system of ordering nuclides can offer a greater insight into the characteristics of isotopes than the better-known periodic table, which shows only elements and not their isotopes. The chart of the nuclides is also known as the Segrè chart, after the Italian physicist Emilio Segrè.
In nuclear physics, the valley of stability is a characterization of the stability of nuclides to radioactivity based on their binding energy. Nuclides are composed of protons and neutrons. The shape of the valley refers to the profile of binding energy as a function of the numbers of neutrons and protons, with the lowest part of the valley corresponding to the region of most stable nuclei. The line of stable nuclides down the center of the valley of stability is known as the line of beta stability. The sides of the valley correspond to increasing instability to beta decay. The decay of a nuclide becomes more energetically favorable the further it is from the line of beta stability. The boundaries of the valley correspond to the nuclear drip lines, where nuclides become so unstable they emit single protons or single neutrons. Regions of instability within the valley at high atomic number also include radioactive decay by alpha radiation or spontaneous fission. The shape of the valley is roughly an elongated paraboloid corresponding to the nuclide binding energies as a function of neutron and atomic numbers.
The neutron number is the number of neutrons in a nuclide.
Isotopes are distinct nuclear species of the same chemical element. They have the same atomic number and position in the periodic table, but different nucleon numbers due to different numbers of neutrons in their nuclei. While all isotopes of a given element have similar chemical properties, they have different atomic masses and physical properties.
Isobars are atoms (nuclides) of different chemical elements that have the same number of nucleons. Correspondingly, isobars differ in atomic number but have the same mass number. An example of a series of isobars is 40S, 40Cl, 40Ar, 40K, and 40Ca. While the nuclei of these nuclides all contain 40 nucleons, they contain varying numbers of protons and neutrons.
The nuclear drip line is the boundary beyond which atomic nuclei are unbound with respect to the emission of a proton or neutron.
The Mattauch isobar rule, formulated by Josef Mattauch in 1934, states that if two adjacent elements on the periodic table have isotopes of the same mass number, one of the isotopes must be radioactive. Two nuclides that have the same mass number (isobars) can both be stable only if their atomic numbers differ by more than one. In fact, for currently observationally stable nuclides, the difference can only be 2 or 4, and in theory, two nuclides that have the same mass number cannot be both stable, but many such nuclides which are theoretically unstable to double beta decay have not been observed to decay, e.g. 134Xe. However, this rule cannot make predictions on the half-lives of these radioisotopes.
In nuclear physics, properties of a nucleus depend on evenness or oddness of its atomic number Z, neutron number N and, consequently, of their sum, the mass number A. Most importantly, oddness of both Z and N tends to lower the nuclear binding energy, making odd nuclei generally less stable. This effect is not only experimentally observed, but is included in the semi-empirical mass formula and explained by some other nuclear models, such as the nuclear shell model. This difference of nuclear binding energy between neighbouring nuclei, especially of odd-A isobars, has important consequences for beta decay.
The neutron–proton ratio of an atomic nucleus is the ratio of its number of neutrons to its number of protons. Among stable nuclei and naturally occurring nuclei, this ratio generally increases with increasing atomic number. This is because electrical repulsive forces between protons scale with distance differently than strong nuclear force attractions. In particular, most pairs of protons in large nuclei are not far enough apart, such that electrical repulsion dominates over the strong nuclear force, and thus proton density in stable larger nuclei must be lower than in stable smaller nuclei where more pairs of protons have appreciable short-range nuclear force attractions.
References
↑ Proc. Int. Symposium on Why and How should we investigate Nuclides Far Off the Stability Line", Lysekil, Sweden, August 1966, eds. W. Forsling, C.J. Herrlander and H. Ryde, Stockholm, Almqvist & Wiksell, 1967
↑ Liu, Shuo; Gao, Chao; Xu, Chang (2021). "Investigation of bound state β− decay half-lives of bare atoms". Physical Review C. 104 (2): 024304. doi:10.1103/PhysRevC.104.024304.
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