Double beta decay

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 ¹²⁴
Sn
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 ¹³⁰
Te
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 ⁸²
Se
and ¹²⁸
Te
.^[5]

Double beta decay was first observed in a laboratory in 1987 by the group of Michael Moe at UC Irvine in ⁸²
Se
.^[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 ¹³⁰
Ba
observed in 2001, ⁷⁸
Kr
observed in 2013, and ¹²⁴
Xe
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

$${\displaystyle {\frac {dN(T_{1},T_{2},\cos \theta )}{dT_{1}dT_{2}d\cos \theta }}=F(Z,T_{1})F(Z,T_{2})w_{1}p_{1}w_{2}p_{2}(Q-T_{1}-T_{2})^{5}(1-v_{1}v_{2}\cos \theta )}$$

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, ${\displaystyle \cos \theta }$ 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 ²³⁸
U
(also an alpha emitter) has been measured radiochemically. Two other nuclides in which double beta decay has been observed, ⁴⁸
Ca
and ⁹⁶
Zr
, 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,^[10] 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).^[11] 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, 10^21 years	Mode	Transition	Method	Experiment
^48 Ca	0.064^+0.007 −0.006 ± ^+0.012 −0.009	β^–β^–		direct	NEMO-3[12]
^76 Ge	1.926±0.094	β^–β^–		direct	GERDA[11]
^78 Kr	9.2^+5.5 −2.6±1.3	εε		direct	BAKSAN[11]
^82 Se	0.096 ± 0.003 ± 0.010	β^–β^–		direct	NEMO-3[11]
^96 Zr	0.0235 ± 0.0014 ± 0.0016	β^–β^–		direct	NEMO-3[11]
^100 Mo	0.00693 ± 0.00004	β^–β^–		direct	NEMO-3[11]
	0.69^+0.10 −0.08 ± 0.07	β^–β^–	0^+→ 0^+1		Ge coincidence[11]
^116 Cd	0.028 ± 0.001 ± 0.003 0.026^+0.009 −0.005	β^–β^–		direct	NEMO-3[11] ELEGANT IV[11]
^128 Te	7200 ± 400 1800 ± 700	β^–β^–		geochemical	[11]
^130 Te	0.82 ± 0.02 ± 0.06	β^–β^–		direct	CUORE-0[13]
^124 Xe	18 ± 5 ± 1	εε		direct	XENON1T[14]
^136 Xe	2.165 ± 0.016 ± 0.059	β^–β^–		direct	EXO-200[11]
^130 Ba	(0.5 – 2.7)	εε		geochemical	[15][16]
^150 Nd	0.00911^+0.00025 −0.00022 ± 0.00063	β^–β^–		direct	NEMO-3[11]
	0.107^+0.046 −0.026	β^–β^–	0^+→ 0^+1		Ge coincidence[11]
^238 U	2.0 ± 0.6	β^–β^–		radiochemical	[11]

Searches for double beta decay in isotopes that present significantly greater experimental challenges are ongoing. One such isotope is ¹³⁴
Xe
.^[17]

The following known beta-stable (or almost beta-stable in the cases ⁴⁸Ca, ⁹⁶Zr, and ²²²Rn^[10])^[18] 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 article: 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 $${\displaystyle \Gamma =G|M|^{2}|m_{\beta \beta }|^{2},}$$ where G is the two-body phase-space factor, M is the nuclear matrix element, and m_ββ is the effective Majorana mass of the electron neutrino. In the context of light Majorana neutrino exchange, m_ββ is given by

$${\displaystyle m_{\beta \beta }=\sum _{i=1}^{3}m_{i}U_{ei}^{2},}$$

where m_i are the neutrino masses and the U_ei 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.^[19][20]

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.^[21]

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)^[22]
  • NEMO, various isotopes using tracking calorimeters (2003–2011)
  • Cuoricino, ¹³⁰Te in ultracold TeO₂ crystals (2003–2008)^[23]
• Experiments taking data as of November 2017:
  • AMoRE, ¹⁰⁰Mo enriched CaMoO₄ crystals at YangYang underground laboratory^[24][25]
  • 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.^[23]
  • Majorana, using high purity ⁷⁶Ge p-type point-contact detectors.^[26]
  • XMASS using liquid Xe
• Proposed/future experiments:
  • CUPID, neutrinoless double-beta decay of ¹⁰⁰Mo
  • CANDLES, ⁴⁸Ca in CaF_2, at Kamioka Observatory
  • MOON, developing ¹⁰⁰Mo detectors
  • nEXO, using liquid ¹³⁶Xe in a time projection chamber ^[27]
  • 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.
  • NuDoubt^++[28] will study double beta plus decays of ⁷⁸Kr in a pressurized hybrid-opaque liquid scintillation detector ^[29]

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.^[30] This claim was criticized by outside physicists^{[1][31][32][33]} as well as other members of the collaboration.^[34] In 2006, a refined estimate by the same authors stated the half-life was 2.3×10²⁵ years.^[35] This half-life has been excluded at high confidence by other experiments, including in ⁷⁶Ge by GERDA.^[36]

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-minus decay is also possible for ⁴⁸Ca, ⁹⁶Zr, and ¹⁵⁰Nd;^[37] triple beta-plus decay or electron capture is also possible for ¹⁴⁸Gd and ¹⁵⁴Dy.

Moreover, such a decay mode could also be neutrinoless in physics beyond the standard model.^[38] 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'^{[definition needed]} 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.^[39]

So far, searches for triple and quadruple beta decay in ¹⁵⁰Nd have remained unsuccessful.^[37]

See also

• Double electron capture
• Beta decay
• Neutrino
• Particle radiation
• Radioactive isotope

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External links

• Double beta decay on arxiv.org