Physics:Double beta decay

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Short description: Type of radioactive decay
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In nuclear physics, double beta decay is a type of radioactive decay in which two neutrons are simultaneously transformed into two protons, or vice versa, inside an atomic nucleus. As in single beta decay, this process allows the atom to move closer to the optimal ratio of protons and neutrons. As a result of this transformation, the nucleus emits two detectable beta particles, which are electrons or positrons.

The literature distinguishes between two types of double beta decay: ordinary double beta decay and neutrinoless double beta decay. In ordinary double beta decay, which has been observed in several isotopes, two electrons and two electron antineutrinos are emitted from the decaying nucleus. In neutrinoless double beta decay, a hypothesized process that has never been observed, only electrons would be emitted.

History

The idea of double beta decay was first proposed by Maria Goeppert Mayer in 1935.[1][2] In 1937, Ettore Majorana demonstrated that all results of beta decay theory remain unchanged if the neutrino were its own antiparticle, now known as a Majorana particle.[3] In 1939, Wendell H. Furry proposed that if neutrinos are Majorana particles, then double beta decay can proceed without the emission of any neutrinos, via the process now called neutrinoless double beta decay.[4] It is not yet known whether the neutrino is a Majorana particle, and, relatedly, whether neutrinoless double beta decay exists in nature.[5]

As parity violation in weak interactions would not be discovered until 1956, earlier calculations showed that neutrinoless double beta decay should be much more likely to occur than ordinary double beta decay, if neutrinos were Majorana particles. The predicted half-lives were on the order of 1015~1016 years.[5] Efforts to observe the process in laboratory date back to at least 1948 when E.L. Fireman made the first attempt to directly measure the half-life of the 124Sn isotope with a Geiger counter.[6] Radiometric experiments through about 1960 produced negative results or false positives, not confirmed by later experiments. In 1950, for the first time the double beta decay half-life of 130Te was measured by geochemical methods to be 1.4×1021 years,[7] reasonably close to the modern value. This involved detecting the concentration in minerals of the xenon produced by the decay.

In 1956, after the V − A nature of weak interactions was established, it became clear that the half-life of neutrinoless double beta decay would significantly exceed that of ordinary double beta decay. Despite significant progress in experimental techniques in 1960–1970s, double beta decay was not observed in a laboratory until the 1980s. Experiments had only been able to establish the lower bound for the half-life – about 1021 years. At the same time, geochemical experiments detected the double beta decay of 82Se and 128Te.[5]

Double beta decay was first observed in a laboratory in 1987 by the group of Michael Moe at UC Irvine in 82Se.[8] Since then, many experiments have observed ordinary double beta decay in other isotopes. None of those experiments have produced positive results for the neutrinoless process, raising the half-life lower bound to approximately 1025 years. Geochemical experiments continued through the 1990s, producing positive results for several isotopes.[5] Double beta decay is the rarest known kind of radioactive decay; as of 2019 it has been observed in only 14 isotopes (including double electron capture in 130Ba observed in 2001, 78Kr observed in 2013, and 124Xe observed in 2019), and all have a mean lifetime over 1018 yr (table below).[5]

Ordinary double beta decay

In a typical double beta decay, two neutrons in the nucleus are converted to protons, and two electrons and two electron antineutrinos are emitted. The process can be thought as two simultaneous beta minus decays. In order for (double) beta decay to be possible, the final nucleus must have a larger binding energy than the original nucleus. For some nuclei, such as germanium-76, the isobar one atomic number higher (arsenic-76) has a smaller binding energy, preventing single beta decay. However, the isobar with atomic number two higher, selenium-76, has a larger binding energy, so double beta decay is allowed.

The emission spectrum of the two electrons can be computed in a similar way to beta emission spectrum using Fermi's golden rule. The differential rate is given by

[math]\displaystyle{ \frac{dN(T_1, T_2, \cos\theta)}{dT_1 dT_2 d\cos\theta} = F(Z,T_1) F(Z,T_2) w_1 p_1 w_2 p_2 (Q-T_1-T_2)^5 (1 - v_1 v_2 \cos \theta) }[/math]

where the subscripts refer to each electron, T is kinetic energy, w is total energy, F(Z, T) is the Fermi function with Z the charge of the final-state nucleus, p is momentum, v is velocity in units of c, [math]\displaystyle{ \cos\theta }[/math] is the angle between the electrons, and Q is the Q value of the decay.

For some nuclei, the process occurs as conversion of two protons to neutrons, emitting two electron neutrinos and absorbing two orbital electrons (double electron capture). If the mass difference between the parent and daughter atoms is more than 1.022 MeV/c2 (two electron masses), another decay is accessible, capture of one orbital electron and emission of one positron. When the mass difference is more than 2.044 MeV/c2 (four electron masses), emission of two positrons is possible. These theoretical decay branches have not been observed.

Known double beta decay isotopes

There are 35 naturally occurring isotopes capable of double beta decay.[9] In practice, the decay can be observed when the single beta decay is forbidden by energy conservation. This happens for elements with an even atomic number and even neutron number, which are more stable due to spin-coupling. When single beta decay or alpha decay also occur, the double beta decay rate is generally too low to observe. However, the double beta decay of 238U (also an alpha emitter) has been measured radiochemically. Two other nuclides in which double beta decay has been observed, 48Ca and 96Zr, can also theoretically single beta decay, but this decay is extremely suppressed and has never been observed. Similar suppression of energetically barely possible single beta decay occurs for 148Gd and 222Rn, but both these nuclides are rather short-lived alpha emitters.

Fourteen isotopes have been experimentally observed undergoing two-neutrino double beta decay (β–β–) or double electron capture (εε).[10] The table below contains nuclides with the latest experimentally measured half-lives, as of December 2016, except for 124Xe (for which double electron capture was first observed in 2019). Where two uncertainties are specified, the first one is statistical uncertainty and the second is systematic.

Nuclide Half-life, 1021 years Mode Transition Method Experiment
48Ca 0.064+0.007−0.006 ± +0.012−0.009 β–β– direct NEMO-3[11]
76Ge 1.926 ±0.094 β–β– direct GERDA[10]
78Kr 9.2 +5.5−2.6 ±1.3 εε direct BAKSAN[10]
82Se 0.096 ± 0.003 ± 0.010 β–β– direct NEMO-3[10]
96Zr 0.0235 ± 0.0014 ± 0.0016 β–β– direct NEMO-3[10]
100Mo 0.00693 ± 0.00004 β–β– direct NEMO-3[10]
0.69+0.10−0.08 ± 0.07 β–β– 0+→ 0+1 Ge coincidence[10]
116Cd 0.028 ± 0.001 ± 0.003
0.026+0.009−0.005 		β–β– direct NEMO-3[10]
ELEGANT IV[10]
128Te 7200 ± 400
1800 ± 700 		β–β– geochemical [10]
130Te 0.82 ± 0.02 ± 0.06 β–β– direct CUORE-0[12]
124Xe 18 ± 5 ± 1 εε direct XENON1T[13]
136Xe 2.165 ± 0.016 ± 0.059 β–β– direct EXO-200[10]
130Ba (0.5 – 2.7) εε geochemical [14][15]
150Nd 0.00911+0.00025−0.00022 ± 0.00063 β–β– direct NEMO-3[10]
0.107+0.046−0.026 β–β– 0+→ 0+1 Ge coincidence[10]
238U 2.0 ± 0.6 β–β– radiochemical [10]

Searches for double beta decay in isotopes that present significantly greater experimental challenges are ongoing. One such isotope is 134Xe.[16]

The following known beta-stable (or almost beta-stable in the cases 48Ca, 96Zr, and 222Rn)[17] nuclides with A ≤ 260 are theoretically capable of double beta decay, where red are isotopes that have a double-beta rate measured experimentally and black have yet to be measured experimentally: 46Ca, 48Ca, 70Zn, 76Ge, 80Se, 82Se, 86Kr, 94Zr, 96Zr, 98Mo, 100Mo, 104Ru, 110Pd, 114Cd, 116Cd, 122Sn, 124Sn, 128Te, 130Te, 134Xe, 136Xe, 142Ce, 146Nd, 148Nd, 150Nd, 154Sm, 160Gd, 170Er, 176Yb, 186W, 192Os, 198Pt, 204Hg, 216Po, 220Rn, 222Rn, 226Ra, 232Th, 238U, 244Pu, 248Cm, 254Cf, 256Cf, and 260Fm.[9]

The following known beta-stable (or almost beta-stable in the case 148Gd) nuclides with A ≤ 260 are theoretically capable of double electron capture, where red are isotopes that have a double-electron capture rate measured and black have yet to be measured experimentally: 36Ar, 40Ca, 50Cr, 54Fe, 58Ni, 64Zn, 74Se, 78Kr, 84Sr, 92Mo, 96Ru, 102Pd, 106Cd, 108Cd, 112Sn, 120Te, 124Xe, 126Xe, 130Ba, 132Ba, 136Ce, 138Ce, 144Sm, 148Gd, 150Gd, 152Gd, 154Dy, 156Dy, 158Dy, 162Er, 164Er, 168Yb, 174Hf, 180W, 184Os, 190Pt, 196Hg, 212Rn, 214Rn, 218Ra, 224Th, 230U, 236Pu, 242Cm, 252Fm, and 258No.[9]

In particular, 36Ar is the lightest observationally stable nuclide whose decay is energetically possible.

Neutrinoless double beta decay

Main page: Physics:Neutrinoless double beta decay
Feynman diagram of neutrinoless double beta decay, with two neutrons decaying to two protons. The only emitted products in this process are two electrons, which can occur if the neutrino and antineutrino are the same particle (i.e. Majorana neutrinos) so the same neutrino can be emitted and absorbed within the nucleus. In conventional double beta decay, two antineutrinos — one arising from each W vertex — are emitted from the nucleus, in addition to the two electrons. The detection of neutrinoless double beta decay is thus a sensitive test of whether neutrinos are Majorana particles.

If the neutrino is a Majorana particle (i.e., the antineutrino and the neutrino are actually the same particle), and at least one type of neutrino has non-zero mass (which has been established by the neutrino oscillation experiments), then it is possible for neutrinoless double beta decay to occur. Neutrinoless double beta decay is a lepton number violating process. In the simplest theoretical treatment, known as light neutrino exchange, a nucleon absorbs the neutrino emitted by another nucleon. The exchanged neutrinos are virtual particles.

With only two electrons in the final state, the electrons' total kinetic energy would be approximately the binding energy difference of the initial and final nuclei, with the nuclear recoil accounting for the rest. Because of momentum conservation, electrons are generally emitted back-to-back. The decay rate for this process is given by [math]\displaystyle{ \Gamma = G |M|^2 |m_{\beta \beta}|^2, }[/math] where G is the two-body phase-space factor, M is the nuclear matrix element, and mββ is the effective Majorana mass of the electron neutrino. In the context of light Majorana neutrino exchange, mββ is given by

[math]\displaystyle{ m_{\beta \beta} = \sum_{i=1}^3 m_i U^2_{ei}, }[/math]

where mi are the neutrino masses and the Uei are elements of the Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix. Therefore, observing neutrinoless double beta decay, in addition to confirming the Majorana neutrino nature, can give information on the absolute neutrino mass scale and Majorana phases in the PMNS matrix, subject to interpretation through theoretical models of the nucleus, which determine the nuclear matrix elements, and models of the decay.[18][19]

The observation of neutrinoless double beta decay would require that at least one neutrino is a Majorana particle, irrespective of whether the process is engendered by neutrino exchange.[20]

Experiments

Numerous experiments have searched for neutrinoless double beta decay. The best-performing experiments have a high mass of the decaying isotope and low backgrounds, with some experiments able to perform particle discrimination and electron tracking. In order to remove backgrounds from cosmic rays, most experiments are located in underground laboratories around the world.

Recent and proposed experiments include:

  • Completed experiments:
    • Gotthard TPC
    • Heidelberg-Moscow, 76Ge detectors (1997–2001)
    • IGEX, 76Ge detectors (1999–2002)[21]
    • NEMO, various isotopes using tracking calorimeters (2003–2011)
    • Cuoricino, 130Te in ultracold TeO2 crystals (2003–2008)[22]
  • Experiments taking data as of November 2017:
    • COBRA, 116Cd in room temperature CdZnTe crystals
    • CUORE, 130Te in ultracold TeO2 crystals
    • EXO, a 136Xe and 134Xe search
    • GERDA, a 76Ge detector
    • KamLAND-Zen, a 136Xe search. Data collection from 2011.[22]
    • Majorana, using high purity 76Ge p-type point-contact detectors.[23]
    • XMASS using liquid Xe
  • Proposed/future experiments:
    • CUPID, neutrinoless double-beta decay of 100Mo
    • CANDLES, 48Ca in CaF2, at Kamioka Observatory
    • MOON, developing 100Mo detectors
    • AMoRE, 100Mo enriched CaMoO4 crystals at YangYang underground laboratory[24]
    • nEXO, using liquid 136Xe in a time projection chamber [25]
    • LEGEND, Neutrinoless Double-beta Decay of 76Ge.
    • LUMINEU, exploring 100Mo enriched ZnMoO4 crystals at LSM, France.
    • NEXT, a Xenon TPC. NEXT-DEMO ran and NEXT-100 will run in 2016.
    • SNO+, a liquid scintillator, will study 130Te
    • SuperNEMO, a NEMO upgrade, will study 82Se
    • TIN.TIN, a 124Sn detector at INO
    • PandaX-III, an experiment with 200 kg to 1000 kg of 90% enriched 136Xe
    • DUNE, a TPC filled with liquid Argon doped with 136Xe.

Status

While some experiments have claimed a discovery of neutrinoless double beta decay, modern searches have found no evidence for the decay.

Heidelberg-Moscow Controversy

Some members of the Heidelberg-Moscow collaboration claimed a detection of neutrinoless beta decay in 76Ge in 2001.[26] This claim was criticized by outside physicists[1][27][28][29] as well as other members of the collaboration.[30] In 2006, a refined estimate by the same authors stated the half-life was 2.3×1025 years.[31] This half-life has been excluded at high confidence by other experiments, including in 76Ge by GERDA.[32]

Current results

As of 2017, the strongest limits on neutrinoless double beta decay have come from GERDA in 76Ge, CUORE in 130Te, and EXO-200 and KamLAND-Zen in 136Xe.

Higher order simultaneous beta decay

For mass numbers with more than two beta-stable isobars, quadruple beta decay and its inverse, quadruple electron capture, have been proposed as alternatives to double beta decay in the isobars with the greatest energy excess. These decays are energetically possible in eight nuclei, though partial half-lives compared to single or double beta decay are predicted to be very long; hence, quadruple beta decay is unlikely to be observed. The seven candidate nuclei for quadruple beta decay include 96Zr, 136Xe, and 150Nd capable of quadruple beta-minus decay, and 124Xe, 130Ba, 148Gd, and 154Dy capable of quadruple beta-plus decay or electron capture (though 148Gd and 154Dy are non-primordial alpha-emitters with geologically short half-lives). In theory, quadruple beta decay may be experimentally observable in three of these nuclei – 96Zr, 136Xe, and 150Nd – with the most promising candidate being 150Nd. Triple beta decay is also possible for 48Ca, 96Zr, and 150Nd.[33]

Moreover, such a decay mode could also be neutrinoless in physics beyond the standard model.[34] Neutrinoless quadruple beta decay would violate lepton number in 4 units, as opposed to a lepton number breaking of two units in the case of neutrinoless double beta decay. Therefore, there is no 'black-box theorem' and neutrinos could be Dirac particles while allowing these type of processes. In particular, if neutrinoless quadruple beta decay is found before neutrinoless double beta decay then the expectation is that neutrinos will be Dirac particles.[35]

So far, searches for triple and quadruple beta decay in 150Nd have remained unsuccessful.[33]

See also

References

  1. ↑ 1.0 1.1 Giuliani, A.; Poves, A. (2012). "Neutrinoless double-beta decay". Advances in High Energy Physics 2012: 1–38. doi:10.1155/2012/857016. http://downloads.hindawi.com/journals/ahep/2012/857016.pdf. 
  2. ↑ "Double beta-disintegration". Physical Review 48 (6): 512–516. 1935. doi:10.1103/PhysRev.48.512. Bibcode: 1935PhRv...48..512G. 
  3. ↑ "Teoria simmetrica dell'elettrone e del positrone" (in it). Il Nuovo Cimento 14 (4): 171–184. 1937. doi:10.1007/BF02961314. Bibcode: 1937NCim...14..171M. 
  4. ↑ "On Transition Probabilities in Double Beta-Disintegration". Physical Review 56 (12): 1184–1193. 1939. doi:10.1103/PhysRev.56.1184. Bibcode: 1939PhRv...56.1184F. 
  5. ↑ 5.0 5.1 5.2 5.3 5.4 Barabash, A.S. (2011). "Experiment double beta decay: Historical review of 75 years of research". Physics of Atomic Nuclei 74 (4): 603–613. doi:10.1134/S1063778811030070. Bibcode: 2011PAN....74..603B. 
  6. ↑ "Double beta decay". Physical Review 74 (9): 1201–1253. 1948. doi:10.1103/PhysRev.74.1201. Bibcode: 1948PhRv...74.1201.. 
  7. ↑ Inghram, M.G.; Reynolds, J.H. (1950). "Double Beta-Decay of 130Te". Physical Review 78 (6): 822–823. doi:10.1103/PhysRev.78.822.2. Bibcode: 1950PhRv...78..822I. 
  8. ↑ Elliott, S. R.; Hahn, A. A.; Moe; M. K. (1987). "Direct evidence for two-neutrino double-beta decay in 82Se". Physical Review Letters 59 (18): 2020–2023. doi:10.1103/PhysRevLett.59.2020. PMID 10035397. Bibcode: 1987PhRvL..59.2020E. 
  9. ↑ 9.0 9.1 9.2 Tretyak, V.I.; Zdesenko, Yu.G. (2002). "Tables of Double Beta Decay Data — An Update". At. Data Nucl. Data Tables 80 (1): 83–116. doi:10.1006/adnd.2001.0873. Bibcode: 2002ADNDT..80...83T. 
  10. ↑ 10.00 10.01 10.02 10.03 10.04 10.05 10.06 10.07 10.08 10.09 10.10 10.11 10.12 10.13 Patrignani, C. (2016). "Review of Particle Physics". Chinese Physics C 40 (10): 100001. doi:10.1088/1674-1137/40/10/100001. Bibcode: 2016ChPhC..40j0001P. http://inspirehep.net/record/1489868/files/openaccess_cpc_40_10_100001.pdf.  See p. 768
  11. ↑ Arnold, R. (2016). "Measurement of the double-beta decay half-life and search for the neutrinoless double-beta decay of 48Ca with the NEMO-3 detector". Physical Review D 93 (11): 112008. doi:10.1103/PhysRevD.93.112008. Bibcode: 2016PhRvD..93k2008A. 
  12. ↑ Alduino, C. (2016). "Measurement of the Two-Neutrino Double Beta Decay Half-life of 130Te with the CUORE-0 Experiment". The European Physical Journal C 77 (1): 13. doi:10.1140/epjc/s10052-016-4498-6. Bibcode: 2017EPJC...77...13A. 
  13. ↑ Aprile, E. et al. (2019). "Observation of two-neutrino double electron capture in 124Xe with XENON1T". Nature 568 (7753): 532–535. doi:10.1038/s41586-019-1124-4. PMID 31019319. Bibcode: 2019Natur.568..532X. 
  14. ↑ A. P. Meshik; C. M. Hohenberg; O. V. Pravdivtseva; Ya. S. Kapusta (2001). "Weak decay of 130Ba and 132Ba: Geochemical measurements". Physical Review C 64 (3): 035205 [6 pages]. doi:10.1103/PhysRevC.64.035205. Bibcode: 2001PhRvC..64c5205M. https://zenodo.org/record/1063702. 
  15. ↑ M. Pujol; B. Marty; P. Burnard; P. Philippot (2009). "Xenon in Archean barite: Weak decay of 130Ba, mass-dependent isotopic fractionation and implication for barite formation". Geochimica et Cosmochimica Acta 73 (22): 6834–6846. doi:10.1016/j.gca.2009.08.002. Bibcode: 2009GeCoA..73.6834P. 
  16. ↑ Albert, J. B. (3 November 2017). "Searches for Double Beta Decay of 134Xe with EXO-200". Physical Review D 96 (9): 092001. doi:10.1103/PhysRevD.96.092001. Bibcode: 2017PhRvD..96i2001A. 
  17. ↑ Belli, P.; Bernabei, R.; Danevich, F. A. et al. (2019). "Experimental searches for rare alpha and beta decays". European Physical Journal A 55 (8): 140–1–140–7. doi:10.1140/epja/i2019-12823-2. ISSN 1434-601X. Bibcode: 2019EPJA...55..140B. 
  18. ↑ Grotz, K.; Klapdor, H. V. (1990). The Weak Interaction in Nuclear, Particle and Astrophysics. CRC Press. ISBN 978-0-85274-313-3. https://archive.org/details/weakinteractioni0000grot. 
  19. ↑ Klapdor-Kleingrothaus, H. V.; Staudt, A. (1998). Non-accelerator Particle Physics (Reprint ed.). IOP Publishing. ISBN 978-0-7503-0305-7. http://web-docs.gsi.de/~gkalicy/books/NonAccelerator%20particle%20physics.pdf. 
  20. ↑ Schechter, J.; Valle, J. W. F. (1982). "Neutrinoless double-β decay in SU(2)×U(1) theories". Physical Review D 25 (11): 2951–2954. doi:10.1103/PhysRevD.25.2951. Bibcode: 1982PhRvD..25.2951S. 
  21. ↑ Aalseth, C. E. (2000). "Recent Results of the IGEX 76Ge Double-Beta Decay Experiment". Physics of Atomic Nuclei 63 (7): 1225–1228. doi:10.1134/1.855774. Bibcode: 2000PAN....63.1225A. 
  22. ↑ 22.0 22.1 Schwingenheuer, B. (2013). "Status and prospects of searches for neutrinoless double beta decay". Annalen der Physik 525 (4): 269–280. doi:10.1002/andp.201200222. Bibcode: 2013AnP...525..269S. 
  23. ↑ Xu, W. (2015). "The Majorana Demonstrator: A Search for Neutrinoless Double-beta Decay of 76Ge". Journal of Physics 606 (1): 012004. doi:10.1088/1742-6596/606/1/012004. Bibcode: 2015JPhCS.606a2004X. 
  24. ↑ Khanbekov, N. D. (2013). "AMoRE: Collaboration for searches for the neutrinoless double-beta decay of the isotope of 100Mo with the aid of 40Ca100MoO4 as a cryogenic scintillation detector". Physics of Atomic Nuclei 76 (9): 1086–1089. doi:10.1134/S1063778813090093. Bibcode: 2013PAN....76.1086K. 
  25. ↑ Albert, J. B. (2018). "Sensitivity and Discovery Potential of nEXO to Neutrinoless Double Beta Decay". Physical Review C 97 (6): 065503. doi:10.1103/PhysRevC.97.065503. Bibcode: 2018PhRvC..97f5503A. 
  26. ↑ Klapdor-Kleingrothaus, H. V.; Dietz, A.; Harney, H. L.; Krivosheina, I. V. (2001). "Evidence for Neutrinoless Double Beta Decay". Modern Physics Letters A 16 (37): 2409–2420. doi:10.1142/S0217732301005825. Bibcode: 2001MPLA...16.2409K. 
  27. ↑ Feruglio, F.; Strumia, A.; Vissani, F. (2002). "Neutrino oscillations and signals in beta and 0nu2beta experiments". Nuclear Physics 637 (1): 345–377. doi:10.1016/S0550-3213(02)00345-0. Bibcode: 2002NuPhB.637..345F. 
  28. ↑ Aalseth, C. E. (2002). "Comment on "evidence for Neutrinoless Double Beta Decay"". Modern Physics Letters A 17 (22): 1475–1478. doi:10.1142/S0217732302007715. Bibcode: 2002MPLA...17.1475A. 
  29. ↑ Zdesenko, Y. G.; Danevich, F. A.; Tretyak, V. I. (2002). "Has neutrinoless double β decay of 76Ge been really observed?". Physics Letters B 546 (3–4): 206. doi:10.1016/S0370-2693(02)02705-3. Bibcode: 2002PhLB..546..206Z. 
  30. ↑ Bakalyarov, A. M.; Balysh, A. Y.; Belyaev, S. T.; Lebedev, V. I.; Zhukov, S. V. (2005). "Results of the experiment on investigation of Germanium-76 double beta decay". Physics of Particles and Nuclei Letters 2 (2005): 77–81. Bibcode: 2003hep.ex....9016B. 
  31. ↑ Klapdor-Kleingrothaus, H. V.; Krivosheina, I. V. (2006). "The Evidence for the Observation of 0νββ Decay: The Identification of 0νββ Events from the Full Spectra". Modern Physics Letters A 21 (20): 1547. doi:10.1142/S0217732306020937. Bibcode: 2006MPLA...21.1547K. 
  32. ↑ Agostini, M. (2017). "Background-free search for neutrinoless double-β decay of 76Ge with GERDA". Nature 544 (7648): 47–52. doi:10.1038/nature21717. PMID 28382980. Bibcode: 2017Natur.544...47A. 
  33. ↑ 33.0 33.1 Barabash, A. S.; Hubert, Ph.; Nachab, A.; Umatov, V. I. (2019). "Search for triple and quadruple β decay of Nd150". Physical Review C 100 (4): 045502. doi:10.1103/PhysRevC.100.045502. 
  34. ↑ Heeck, J.; Rodejohann, W. (2013). "Neutrinoless Quadruple Beta Decay". Europhysics Letters 103 (3): 32001. doi:10.1209/0295-5075/103/32001. Bibcode: 2013EL....10332001H. 
  35. ↑ Hirsch, M.; Srivastava, R.; Valle, JWF. (2018). "Can one ever prove that neutrinos are Dirac particles?". Physics Letters B 781: 302–305. doi:10.1016/j.physletb.2018.03.073. Bibcode: 2018PhLB..781..302H. 

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