Beta-decay stable isobars

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 nuclides 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.

β decay stable / even A
βDS	One	Two	Three
2-34	17
36-58	6	6
60-72	5	2
74-116	2	20
118-154	2	12	5
156-192	5	14
194-210	6	3
212-262	7	19
Total	50	75	6

All odd mass numbers have only one beta decay stable nuclide.

Among even mass number, five (124, 130, 136, 150, 154) have three beta-stable nuclides. None have more than three; all others have either one or two.

From 2 to 34, all have only one.
From 36 to 72, only eight (36, 40, 46, 50, 54, 58, 64, 70) have two, and the remaining 12 have one.
From 74 to 122, three (88, 90, 118) have one, and the remaining 23 have two.
From 124 to 154, only one (140) has one, six 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). Non-primordial ²⁴⁷Cm should undergo beta decay to ²⁴⁷Bk, but has also never been observed to do so. Finally, ⁴⁸Ca and ⁹⁶Zr have not been observed to undergo beta decay (which is theoretically possible for both), but double beta decay is known for both. All elements up to and including nobelium, except technetium and promethium, are known to have at least one beta-stable isotope.

List of known beta-decay stable isobars

350 beta-decay stable nuclides are currently known.[3][4] Theoretically predicted or experimentally observed double beta-decay is shown by arrows, i.e. arrows point towards the lightest-mass isobar. (This is sometimes dominated by alpha decay or spontaneous fission, especially for the heavy elements.)

No beta-decay stable nuclide has proton number 43 or 61 and no beta-decay stable nuclide has neutron number 19, 21, 35, 39, 45, 61, 71, 89, 115, 123, or 147.

	Even N	Odd N
Even Z	Even A	Odd A
Odd Z	Odd A	Even A

All known beta-decay stable isobars sorted by mass number
Odd A	Even A	Odd A	Even A	Odd A	Even A	Odd A	Even A
¹H	²H	³He	⁴He	⁵He (n)	⁶Li	⁷Li	⁸Be (α)
⁹Be	¹⁰B	¹¹B	¹²C	¹³C	¹⁴N	¹⁵N	¹⁶O
¹⁷O	¹⁸O	¹⁹F	²⁰Ne	²¹Ne	²²Ne	²³Na	²⁴Mg
²⁵Mg	²⁶Mg	²⁷Al	²⁸Si	²⁹Si	³⁰Si	³¹P	³²S
³³S	³⁴S	³⁵Cl	³⁶S ← ³⁶Ar	³⁷Cl	³⁸Ar	³⁹K	⁴⁰Ar ← ⁴⁰Ca
⁴¹K	⁴²Ca	⁴³Ca	⁴⁴Ca	⁴⁵Sc	⁴⁶Ca → ⁴⁶Ti	⁴⁷Ti	⁴⁸Ti[lower-alpha 1]
⁴⁹Ti	⁵⁰Ti ← ⁵⁰Cr	⁵¹V	⁵²Cr	⁵³Cr	⁵⁴Cr ← ⁵⁴Fe	⁵⁵Mn	⁵⁶Fe
⁵⁷Fe	⁵⁸Fe ← ⁵⁸Ni	⁵⁹Co	⁶⁰Ni	⁶¹Ni	⁶²Ni	⁶³Cu	⁶⁴Ni ← ⁶⁴Zn
⁶⁵Cu	⁶⁶Zn	⁶⁷Zn	⁶⁸Zn	⁶⁹Ga	⁷⁰Zn → ⁷⁰Ge	⁷¹Ga	⁷²Ge
⁷³Ge	⁷⁴Ge ← ⁷⁴Se	⁷⁵As	⁷⁶Ge → ⁷⁶Se	⁷⁷Se	⁷⁸Se ← ⁷⁸Kr	⁷⁹Br	⁸⁰Se → ⁸⁰Kr
⁸¹Br	⁸²Se → ⁸²Kr	⁸³Kr	⁸⁴Kr ← ⁸⁴Sr	⁸⁵Rb	⁸⁶Kr → ⁸⁶Sr	⁸⁷Sr	⁸⁸Sr
⁸⁹Y	⁹⁰Zr	⁹¹Zr	⁹²Zr ← ⁹²Mo	⁹³Nb	⁹⁴Zr → ⁹⁴Mo	⁹⁵Mo	⁹⁶Mo ← ⁹⁶Ru[lower-alpha 2]
⁹⁷Mo	⁹⁸Mo → ⁹⁸Ru	⁹⁹Ru	¹⁰⁰Mo → ¹⁰⁰Ru	¹⁰¹Ru	¹⁰²Ru ← ¹⁰²Pd	¹⁰³Rh	¹⁰⁴Ru → ¹⁰⁴Pd
¹⁰⁵Pd	¹⁰⁶Pd ← ¹⁰⁶Cd	¹⁰⁷Ag	¹⁰⁸Pd ← ¹⁰⁸Cd	¹⁰⁹Ag	¹¹⁰Pd → ¹¹⁰Cd	¹¹¹Cd	¹¹²Cd ← ¹¹²Sn
¹¹³In	¹¹⁴Cd → ¹¹⁴Sn	¹¹⁵Sn	¹¹⁶Cd → ¹¹⁶Sn	¹¹⁷Sn	¹¹⁸Sn	¹¹⁹Sn	¹²⁰Sn ← ¹²⁰Te
¹²¹Sb	¹²²Sn → ¹²²Te	¹²³Sb	¹²⁴Sn → ¹²⁴Te ← ¹²⁴Xe	¹²⁵Te	¹²⁶Te ← ¹²⁶Xe	¹²⁷I	¹²⁸Te → ¹²⁸Xe
¹²⁹Xe	¹³⁰Te → ¹³⁰Xe ← ¹³⁰Ba	¹³¹Xe	¹³²Xe ← ¹³²Ba	¹³³Cs	¹³⁴Xe → ¹³⁴Ba	¹³⁵Ba	¹³⁶Xe → ¹³⁶Ba ← ¹³⁶Ce
¹³⁷Ba	¹³⁸Ba ← ¹³⁸Ce	¹³⁹La	¹⁴⁰Ce	¹⁴¹Pr	¹⁴²Ce → ¹⁴²Nd	¹⁴³Nd	¹⁴⁴Nd (α) ← ¹⁴⁴Sm
¹⁴⁵Nd	¹⁴⁶Nd → ¹⁴⁶Sm (α)	¹⁴⁷Sm (α)	¹⁴⁸Nd → ¹⁴⁸Sm (α)	¹⁴⁹Sm	¹⁵⁰Nd → ¹⁵⁰Sm ← ¹⁵⁰Gd (α)	¹⁵¹Eu (α)	¹⁵²Sm ← ¹⁵²Gd
¹⁵³Eu	¹⁵⁴Sm → ¹⁵⁴Gd ← ¹⁵⁴Dy (α)	¹⁵⁵Gd	¹⁵⁶Gd ← ¹⁵⁶Dy	¹⁵⁷Gd	¹⁵⁸Gd ← ¹⁵⁸Dy	¹⁵⁹Tb	¹⁶⁰Gd → ¹⁶⁰Dy
¹⁶¹Dy	¹⁶²Dy ← ¹⁶²Er	¹⁶³Dy	¹⁶⁴Dy ← ¹⁶⁴Er	¹⁶⁵Ho	¹⁶⁶Er	¹⁶⁷Er	¹⁶⁸Er ← ¹⁶⁸Yb
¹⁶⁹Tm	¹⁷⁰Er → ¹⁷⁰Yb	¹⁷¹Yb	¹⁷²Yb	¹⁷³Yb	¹⁷⁴Yb ← ¹⁷⁴Hf (α)	¹⁷⁵Lu	¹⁷⁶Yb → ¹⁷⁶Hf
¹⁷⁷Hf	¹⁷⁸Hf	¹⁷⁹Hf	¹⁸⁰Hf ← ¹⁸⁰W (α)	¹⁸¹Ta	¹⁸²W	¹⁸³W	¹⁸⁴W ← ¹⁸⁴Os (α)
¹⁸⁵Re	¹⁸⁶W → ¹⁸⁶Os (α)	¹⁸⁷Os	¹⁸⁸Os	¹⁸⁹Os	¹⁹⁰Os ← ¹⁹⁰Pt (α)	¹⁹¹Ir	¹⁹²Os → ¹⁹²Pt
¹⁹³Ir	¹⁹⁴Pt	¹⁹⁵Pt	¹⁹⁶Pt ← ¹⁹⁶Hg	¹⁹⁷Au	¹⁹⁸Pt → ¹⁹⁸Hg	¹⁹⁹Hg	²⁰⁰Hg
²⁰¹Hg	²⁰²Hg	²⁰³Tl	²⁰⁴Hg → ²⁰⁴Pb	²⁰⁵Tl	²⁰⁶Pb	²⁰⁷Pb	²⁰⁸Pb
²⁰⁹Bi (α)	²¹⁰Po (α)	²¹¹Po (α)	²¹²Po (α) ← ²¹²Rn (α)	²¹³Po (α)	²¹⁴Po (α) ← ²¹⁴Rn (α)	²¹⁵At (α)	²¹⁶Po (α) → ²¹⁶Rn (α)
²¹⁷Rn (α)	²¹⁸Rn (α) ← ²¹⁸Ra (α)	²¹⁹Fr (α)	²²⁰Rn (α) → ²²⁰Ra (α)	²²¹Ra (α)	²²²Ra[lower-alpha 3] (α)	²²³Ra (α)	²²⁴Ra (α) ← ²²⁴Th (α)
²²⁵Ac (α)	²²⁶Ra (α) → ²²⁶Th (α)	²²⁷Th (α)	²²⁸Th (α)	²²⁹Th (α)	²³⁰Th (α) ← ²³⁰U (α)	²³¹Pa (α)	²³²Th (α) → ²³²U (α)
²³³U (α)	²³⁴U (α)	²³⁵U (α)	²³⁶U (α) ← ²³⁶Pu (α)	²³⁷Np (α)	²³⁸U (α) → ²³⁸Pu (α)	²³⁹Pu (α)	²⁴⁰Pu (α)
²⁴¹Am (α)	²⁴²Pu (α) ← ²⁴²Cm (α)	²⁴³Am (α)	²⁴⁴Pu (α) → ²⁴⁴Cm (α)	²⁴⁵Cm (α)	²⁴⁶Cm (α)	²⁴⁷Bk (α)	²⁴⁸Cm (α) → ²⁴⁸Cf (α)
²⁴⁹Cf (α)	²⁵⁰Cf (α)	²⁵¹Cf (α)	²⁵²Cf (α) ← ²⁵²Fm (α)	²⁵³Es (α)	²⁵⁴Cf (SF) → ²⁵⁴Fm (α)	²⁵⁵Fm (α)	²⁵⁶Cf (SF) → ²⁵⁶Fm (SF)
²⁵⁷Fm (α)	²⁵⁸Fm (SF) ← ²⁵⁸No (SF)	²⁵⁹Md (SF)	²⁶⁰Fm (SF) → ²⁶⁰No (SF)		²⁶²No (SF)

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. Islands of stability are predicted to center near ²⁹⁴Ds and ³⁵⁴126, beyond which the model appears to deviate from several rules of the semi-empirical mass formula.[8]

All beta-decay stable nuclides with A ≥ 209 were observed to decay by alpha decay except some where spontaneous fission dominates. With the exception of ²⁶²No, no nuclides with A > 260 have been definitively identified as beta-stable. ²⁶⁰Fm and ²⁶²No are unconfirmed.[4]

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.[9] Beyond the island of stability, various models that correctly predict the known beta-stable isotopes 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][10] 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.[10][11]

Beta decay toward minimum mass

Beta decay generally causes isotopes 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 not in italics in the table above. Thus, 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 four nuclides that are exceptions, in that the majority of their decays are in the opposite direction:

Chlorine-36	35.96830698	Potassium-40	39.96399848	Silver-108	107.905956	Promethium-146	145.914696
2% to Sulfur-36	35.96708076	11.2% to Argon-40	39.9623831225	3% to Palladium-108	107.903892	37% to Samarium-146	145.913041
98% to Argon-36	35.967545106	89% to Calcium-40	39.96259098	97% to Cadmium-108	107.904184	63% to Neodymium-146	145.9131169

Notes

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
_−0.6×10²¹ years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[5]
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 2.4×10¹⁹ years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[6]
While the AME2016 atomic mass evaluation gives ²²²Rn a lower mass than ²²²Fr,[4] implying beta stability, it is predicted that single beta decay of ²²²Rn is energetically possible (albeit with very low decay energy),[7] and it falls within the error margin given in AME2016.[4] Hence, ²²²Rn is probably not beta-stable, 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.[7]

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
Hansen, P. G. (1979). "Nuclei Far Away from the Line of Beta Stability: Studies by On-Line Mass Separation". Annual Review of Nuclear and Particle Science. 29: 69–119. Bibcode:1979ARNPS..29...69H. doi:10.1146/annurev.ns.29.120179.000441.
"Interactive Chart of Nuclides (Brookhaven National Laboratory)". Archived from the original on 2020-07-25. Retrieved 2009-06-19.
Audi, G.; Kondev, F. G.; Wang, M.; Huang, W. J.; Naimi, S. (2017). "The NUBASE2016 evaluation of nuclear properties" (PDF). Chinese Physics C. 41 (3): 030001. Bibcode:2017ChPhC..41c0001A. doi:10.1088/1674-1137/41/3/030001.
Aunola, M.; Suhonen, J.; Siiskonen, T. (1999). "Shell-model study of the highly forbidden beta decay ⁴⁸Ca → ⁴⁸Sc". EPL. 46 (5): 577. Bibcode:1999EL.....46..577A. doi:10.1209/epl/i1999-00301-2.
Finch, S.W.; Tornow, W. (2016). "Search for the β decay of ⁹⁶Zr". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 806: 70–74. Bibcode:2016NIMPA.806...70F. doi:10.1016/j.nima.2015.09.098.
Belli, P.; Bernabei, R.; Cappella, C.; Caracciolo, V.; Cerulli, R.; Danevich, F.A.; Di Marco, A.; Incicchitti, A.; Poda, D.V.; Polischuk, O.G.; Tretyak, V.I. (2014). "Investigation of rare nuclear decays with BaF₂ crystal scintillator contaminated by radium". European Physical Journal A. 50 (9): 134–143. arXiv:1407.5844. Bibcode:2014EPJA...50..134B. doi:10.1140/epja/i2014-14134-6. S2CID 118513731.
Koura, H. (2011). Decay modes and a limit of existence of nuclei in the superheavy mass region (PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 18 November 2018.
Zagrebaev, Valeriy; Karpov, Alexander; Greiner, Walter (2013). "Future of superheavy element research: Which nuclei could be synthesized within the next few years?" (PDF). Journal of Physics. 420 (1): 012001. arXiv:1207.5700. Bibcode:2013JPhCS.420a2001Z. doi:10.1088/1742-6596/420/1/012001. S2CID 55434734.
Möller, P.; Sierk, A.J.; Ichikawa, T.; Sagawa, H. (2016). "Nuclear ground-state masses and deformations: FRDM(2012)". Atomic Data and Nuclear Data Tables. 109–110: 1–204. arXiv:1508.06294. Bibcode:2016ADNDT.109....1M. doi:10.1016/j.adt.2015.10.002. S2CID 118707897.
Möller, P. (2016). "The limits of the nuclear chart set by fission and alpha decay" (PDF). EPJ Web of Conferences. 131: 03002:1–8. Bibcode:2016EPJWC.13103002M. doi:10.1051/epjconf/201613103002.

External links

Decay-Chains https://www-nds.iaea.org/relnsd/NdsEnsdf/masschain.html
(Russian) Beta-decay stable nuclides up to Z=118

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[6] 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
_−0.6×10²¹ years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[5]

[8] 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 2.4×10¹⁹ years, longer than its double beta decay half-life, meaning that double beta decay would usually occur first.[6]

[10] While the AME2016 atomic mass evaluation gives ²²²Rn a lower mass than ²²²Fr,[4] implying beta stability, it is predicted that single beta decay of ²²²Rn is energetically possible (albeit with very low decay energy),[7] and it falls within the error margin given in AME2016.[4] Hence, ²²²Rn is probably not beta-stable, 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.[7]

[1] 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

[2] Hansen, P. G. (1979). "Nuclei Far Away from the Line of Beta Stability: Studies by On-Line Mass Separation". Annual Review of Nuclear and Particle Science. 29: 69–119. Bibcode:1979ARNPS..29...69H. doi:10.1146/annurev.ns.29.120179.000441.

[3] "Interactive Chart of Nuclides (Brookhaven National Laboratory)". Archived from the original on 2020-07-25. Retrieved 2009-06-19.

[NUBASE2016-4] Audi, G.; Kondev, F. G.; Wang, M.; Huang, W. J.; Naimi, S. (2017). "The NUBASE2016 evaluation of nuclear properties" (PDF). Chinese Physics C. 41 (3): 030001. Bibcode:2017ChPhC..41c0001A. doi:10.1088/1674-1137/41/3/030001.

[5] Aunola, M.; Suhonen, J.; Siiskonen, T. (1999). "Shell-model study of the highly forbidden beta decay ⁴⁸Ca → ⁴⁸Sc". EPL. 46 (5): 577. Bibcode:1999EL.....46..577A. doi:10.1209/epl/i1999-00301-2.

[7] Finch, S.W.; Tornow, W. (2016). "Search for the β decay of ⁹⁶Zr". Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment. 806: 70–74. Bibcode:2016NIMPA.806...70F. doi:10.1016/j.nima.2015.09.098.

[222Rn01-9] Belli, P.; Bernabei, R.; Cappella, C.; Caracciolo, V.; Cerulli, R.; Danevich, F.A.; Di Marco, A.; Incicchitti, A.; Poda, D.V.; Polischuk, O.G.; Tretyak, V.I. (2014). "Investigation of rare nuclear decays with BaF₂ crystal scintillator contaminated by radium". European Physical Journal A. 50 (9): 134–143. arXiv:1407.5844. Bibcode:2014EPJA...50..134B. doi:10.1140/epja/i2014-14134-6. S2CID 118513731.

[SHlimit-11] Koura, H. (2011). Decay modes and a limit of existence of nuclei in the superheavy mass region (PDF). 4th International Conference on the Chemistry and Physics of the Transactinide Elements. Retrieved 18 November 2018.

[Zagrebaev-12] Zagrebaev, Valeriy; Karpov, Alexander; Greiner, Walter (2013). "Future of superheavy element research: Which nuclei could be synthesized within the next few years?" (PDF). Journal of Physics. 420 (1): 012001. arXiv:1207.5700. Bibcode:2013JPhCS.420a2001Z. doi:10.1088/1742-6596/420/1/012001. S2CID 55434734.

[frdm2012-13] Möller, P.; Sierk, A.J.; Ichikawa, T.; Sagawa, H. (2016). "Nuclear ground-state masses and deformations: FRDM(2012)". Atomic Data and Nuclear Data Tables. 109–110: 1–204. arXiv:1508.06294. Bibcode:2016ADNDT.109....1M. doi:10.1016/j.adt.2015.10.002. S2CID 118707897.

[liquiddrop-14] Möller, P. (2016). "The limits of the nuclear chart set by fission and alpha decay" (PDF). EPJ Web of Conferences. 131: 03002:1–8. Bibcode:2016EPJWC.13103002M. doi:10.1051/epjconf/201613103002.