Beta decay

In nuclear physics, beta decay (β-decay) is a type of radioactive decay in which a beta ray (fast energetic electron or positron) is emitted from an atomic nucleus. 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 (positron emission) with a neutrino, thus changing the nuclide type. 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.[1] For either electron or positron emission to be energetically possible, the energy release (see below) or Q value must be positive.

Beta decay is a consequence of the weak force, which is characterized by relatively lengthy decay times. Nucleons are composed of up quarks and down quarks,[2] and the weak force allows a quark to change type by the exchange of a W boson and the 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. Decay times for many nuclides that are subject to beta decay can be thousands of years.

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 an electron neutrino is released.

Beta-minus Decay

β
 decay in an atomic nucleus (the accompanying antineutrino is omitted). The inset shows beta decay of a free neutron. In both processes, an intermediate virtual
W
boson is not shown.

Description

The two types of beta decay are known as beta minus and beta plus. In beta minus (β) decay, a neutron is converted to a proton, and the process creates an electron and an electron 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 as positron emission.[4]

Beta decay conserves a quantum number known as the lepton number, or the number of electrons and their associated neutrinos (other leptons are the muon and tau 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 of carbon-14 into nitrogen-14 with a half-life of about 5,730 years:

14
6
C
14
7
N
+
e
+
ν
e

In this form of decay, the original element becomes a new chemical element in a process known as nuclear transmutation. This new element has an unchanged mass number A, but an atomic number Z that is increased by one. As in all nuclear decays, the decaying element (in this case 14
6
C
) is known as the parent nuclide while the resulting element (in this case 14
7
N
) is known as the daughter nuclide.

Another example is the decay of hydrogen-3 (tritium) into helium-3 with a half-life of about 12.3 years:

3
1
H
3
2
He
+
e
+
ν
e

An example of positron emission (β+ decay) is the decay of magnesium-23 into sodium-23 with a half-life of about 11.3 s:

23
12
Mg
23
11
Na
+
e+
+
ν
e

β+ decay also results in nuclear transmutation, with the resulting element having an atomic number that is decreased by one.

RaE1
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 of 210Bi is shown. In this example, the total decay energy is 1.16 MeV, so the antineutrino has the remaining energy: 1.16-0.40=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.

History

Discovery and initial characterization

Radioactivity was discovered in 1896 by Henri Becquerel in uranium, and subsequently observed by Marie and Pierre Curie in thorium and in the new elements polonium and radium. 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 termed gamma rays. Alpha, beta, and gamma are the first three letters of the Greek alphabet.

In 1900, Becquerel measured the mass-to-charge ratio (m/e) for beta particles by the method of J.J. Thomson used to study cathode rays and identify the electron. He found that m/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 and Frederick Soddy showed that alpha and beta radioactivity involves the transmutation of atoms into atoms of other chemical elements. In 1913, after the products of more radioactive decays were known, Soddy and Kazimierz Fajans independently proposed their radioactive displacement law, which states that beta (i.e.,
β
) emission from one element produces another element one place to the right in the periodic table, while alpha emission produces an element two places to the left.

Neutrinos

The study of beta decay provided the first physical evidence for the existence of the neutrino. In both alpha and gamma decay, the resulting particle has a narrow energy distribution, 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 by Lise Meitner and Otto Hahn in 1911 and by Jean 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 magnetic spectrometer with one of Hans Geiger's new counters to make more accurate measurements which showed that the spectrum was continuous.[6][7] The distribution of beta particle energies was in apparent contradiction to the law of conservation of energy. 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.[8] 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 the conservation of angular momentum. Molecular band spectra showed that the nuclear spin of nitrogen-14 is 1 (i.e. equal to the reduced Planck constant) and more generally that the spin is integral for nuclei of even mass number and half-integral for nuclei of odd mass number. This was later explained by the proton-neutron model of the nucleus.[8] 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 1920–1927, Charles Drummond Ellis (along with Chadwick and colleagues) further established that the beta decay spectrum is continuous. In 1933, Ellis and Nevill 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 if conservation of energy was true only in a statistical sense, thus this principle might be violated in any given decay.[8]: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 a famous 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" (roughly 'little neutral one' in Italian). In 1934, Fermi published his landmark theory 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 Clyde Cowan and Frederick Reines in the Cowan–Reines neutrino experiment.[9] The properties of neutrinos were (with a few minor modifications) as predicted by Pauli and Fermi.


β+
 decay and electron capture

In 1934, Frédéric and Irène Joliot-Curie bombarded aluminium with alpha particles to effect the nuclear reaction 4
2
He
 + 27
13
Al
 → 30
15
P
 + 1
0
n
, and observed that the product isotope 30
15
P
emits a positron identical to those found in cosmic rays (discovered by Carl David Anderson in 1932). This was the first example of
β+
 decay (positron emission), which they termed artificial radioactivity since 30
15
P
is a short-lived nuclide which does not exist in nature. In recognition of their discovery the couple were awarded the Nobel Prize in Chemistry in 1935[10].

The theory of electron capture was first discussed by Gian-Carlo Wick in a 1934 paper, and then developed by Hideki Yukawa and others. K-electron capture was first observed in 1937 by Luis Alvarez, in the nuclide 48V.[11][12][13] Alvarez went on to study electron capture in 67Ga and other nuclides.[11][14][15]

Non-conservation of parity

In 1956, Tsung-Dao Lee and Chen Ning Yang noticed that there was no evidence that parity 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.[16] Later that year, Chien-Shiung Wu and coworkers conducted the Wu experiment showing an asymmetrical beta decay of cobalt-60 at cold temperatures that proved that parity is not conserved in beta decay.[17][18] This surprising result overturned long-held assumptions about parity and the weak force. In recognition of their theoretical work, Lee and Yang were awarded the Nobel Prize for Physics in 1957.[19]

β decay

Beta Negative Decay
The leading-order Feynman diagram for
β
 decay of a neutron into a proton, electron, and electron antineutrino via an intermediate
W
boson

In
β
 decay, the weak interaction converts an atomic nucleus into a nucleus with atomic number increased by one, while emitting an electron (
e
) and an electron antineutrino (
ν
e
).
β
 decay generally occurs in neutron-rich nuclei.[20] The generic equation is:

A
Z
X
A
Z+1
X'
+
e
+
ν
e
[1]

where A and Z are the mass number and atomic number of the decaying nucleus, and X and X' are the initial and final elements, respectively.

Another example is when the free neutron (1
0
n
) decays by
β
 decay into a proton (
p
):


n

p
+
e
+
ν
e
.

At the fundamental level (as depicted in the Feynman diagram on the right), this is caused by the conversion of the negatively charged (−1/3 e) down quark to the positively charged (+2/3 e) up quark by emission of a
W
boson
; the
W
boson subsequently decays into an electron and an electron antineutrino:


d

u
+
e
+
ν
e
.

β+ decay

Electron Capture Decay
The leading-order Feynman diagram for
β+
 decay of a proton into a neutron, positron, and electron neutrino via an intermediate
W+
boson
.

In
β+
 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 an electron neutrino (
ν
e
).
β+
 decay generally occurs in proton-rich nuclei. The generic equation is:

A
Z
X
A
Z−1
X’
+
e+
+
ν
e
[1]

This may be considered as the decay of a proton inside the nucleus to a neutron

p → n +
e+
+
ν
e
[1]

However,
β+
 decay cannot occur in an isolated proton because it requires energy, due to the mass of the neutron being greater than the mass of the proton.
β+
 decay can only happen inside nuclei when the daughter nucleus has a greater binding 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 a
W+
or the absorption of a
W
.

Electron capture (K-capture)

In all cases where
β+
 decay (positron emission) of a nucleus is allowed energetically, so too is electron capture allowed. This is a process during which a nucleus captures one of its atomic electrons, resulting in the emission of a neutrino:

A
Z
X
+
e
A
Z−1
X’
+
ν
e

An example of electron capture is one of the decay modes of krypton-81 into bromine-81:

81
36
Kr
+
e
81
35
Br
+
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.[21]

If the captured electron comes from the innermost shell of the atom, the K-shell, which has the highest probability to interact with the nucleus, the process is called K-capture.[22] 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 the only type of decay that is allowed in proton-rich nuclides that do not have sufficient energy to emit a positron and neutrino.[21]

Nuclear transmutation

Table isotopes en

If the proton and neutron are part of an atomic nucleus, the above described decay processes transmute one chemical element into another. For example:

137
55
Cs
 
    →  137
56
Ba
 

e
 

ν
e
 
(beta minus decay)
22
11
Na
 
    →  22
10
Ne
 

e+
 

ν
e
 
(beta plus decay)
22
11
Na
 

e
 
→  22
10
Ne
 

ν
e
 
    (electron capture)

Beta decay does not change the number (A) of nucleons in the nucleus, but changes only its charge Z. Thus the set of all nuclides with the same A can be introduced; these isobaric nuclides may turn into each other via beta decay. For a given A there is one that is most stable. It is said to be beta stable, because it presents a local minima of the mass 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 numbers A, there is only one known beta-stable isobar. For even A, there are up to three different beta-stable isobars experimentally known; for example, 96
40
Zr
, 96
42
Mo
, and 96
44
Ru
are all beta-stable. There are about 355 known beta-decay stable nuclides.[23]

Competition of beta decay types

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. An often-cited example is the single isotope 64
29
Cu
(29 protons, 35 neutrons), which illustrates three types of beta decay in competition. Copper-64 has a half-life of about 12.7 hours. This isotope has one unpaired proton and one unpaired neutron, so either the proton or the neutron can decay. This particular nuclide (though not all nuclides in this situation) is almost equally likely to decay through proton decay by positron emission (18%) or electron capture (43%) to 64
28
Ni
, as it is through neutron decay by electron emission (39%) to 64
30
Zn
.[24]

Stability of naturally occurring nuclides

Most naturally occurring nuclides on earth are beta stable. Those that are not have half-lives ranging from under a second to periods of time significantly greater than the age of the universe. One common example of a long-lived isotope is the odd-proton odd-neutron nuclide 40
19
K
, which undergoes all three types of beta decay (
β
,
β+
and electron capture) with a half-life of 1.277×109 years.[25]

Conservation rules for beta decay

Baryon number is conserved

where

is the number of constituent quarks, and
is the number of constituent antiquarks.

Beta decay just changes neutron to proton or, in the case of positive beta decay (electron capture) proton to neutron so the number of individual quarks doesn't change. It is only the baryon flavor that changes, here labelled as the isospin.

Up and down quarks have total isospin and isospin projections

All other quarks have I = 0.

In general

Lepton number is conserved

so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0.

Angular momentum

For allowed decays, the net orbital angular momentum is zero, hence only spin quantum numbers are considered.

The electron and antineutrino are fermions, spin-1/2 objects, therefore they may couple to total (parallel) or (anti-parallel).

For forbidden decays, orbital angular momentum must also be taken into consideration.

Energy release

The Q 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 any kinetic energy ranging from 0 to Q.[1] A typical Q is around 1 MeV, but can range from a few keV to a few tens of MeV.

Since the rest mass of the electron is 511 keV, the most energetic beta particles are ultrarelativistic, with speeds very close to the speed of light.

β decay

Consider the generic equation for beta decay

A
Z
X
A
Z+1
X’
+
e
+
ν
e
.

The Q value for this decay is

,

where is the mass of the nucleus of the A
Z
X
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 nucleus mN is related to the standard atomic mass m by

.

That is, the total atomic mass is the mass of the nucleus, plus the mass of the electrons, minus the sum of all electron binding energies Bi for the atom. This equation is rearranged to find , and is found similarly. Substituting these nuclear masses into the Q-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 have

This energy is carried away as kinetic energy by the electron and neutrino.

Because the reaction will proceed only when the Q-value is positive, β decay can occur when the mass of atom A
Z
X
is greater than the mass of atom A
Z+1
X’
.[26]

β+ decay

The equations for β+ decay are similar, with the generic equation

A
Z
X
A
Z−1
X’
+
e+
+
ν
e

giving

.

However, in this equation, the electron masses do not cancel, and we are left with

Because the reaction will proceed only when the Q-value is positive, β+ decay can occur when the mass of atom A
Z
X
exceeds that of A
Z-1
X’
by at least twice the mass of the electron.[26]

Electron capture

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
Z
X
+
e
A
Z−1
X’
+
ν
e

we have

,

which simplifies to

,

where Bn 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.[26]

Beta emission spectrum

Beta spectrum of RaE
Beta spectrum of 210Bi. Emax=Q=1.16 MeV is the maximum energy

Beta decay can be considered as a perturbation as described in quantum mechanics, and thus Fermi's Golden Rule can be applied. This leads to an expression for the kinetic energy spectrum N(T) of emitted betas as follows:[27]

where T 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) with Z the charge of the final-state nucleus, E=T + mc2 is the total energy, p=(E/c)2 − (mc)2 is the momentum, and Q is the Q value of the decay. The kinetic energy of the emitted neutrino is given approximately by Q minus the kinetic energy of the beta.

As an example, the beta decay spectrum of 210Bi (originally called RaE) is shown to the right.

Fermi function

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:[28]

where S=1 − α2 Z2 (α is the fine-structure constant), η=± αZE/pc (+ for electrons, − for positrons), ρ=rN/ℏ (rN is the radius of the final state nucleus), and Γ is the Gamma function.

For non-relativistic betas (Qmec2), this expression can be approximated by:[29]

Other approximations can be found in the literature.[30][31]

Kurie plot

A Kurie plot (also known as a Fermi–Kurie plot) is a graph used in studying beta decay developed by Franz N. D. Kurie, in which the square root of the number of beta particles whose momenta (or energy) lie within a certain narrow range, divided by the Fermi function, is plotted against beta-particle energy.[32][33] 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's Q-value). With a Kurie plot one can find the limit on the effective mass of a neutrino.[34]

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 negative helicity, i.e., they move, naively speaking, like left-handed screws driven into a material (they have negative longitudinal polarization).[35] Conversely, positrons have mostly positive helicity, i.e., they move like right-handed screws. Neutrinos (emitted in positron decay) have positive helicity, while antineutrinos (emitted in electron decay) have negative helicity.[36]

The higher the energy of the particles, the higher their polarization.

Types of beta decay transitions

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.

Fermi transitions

A Fermi 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 by

with the weak vector coupling constant, the isospin raising and lowering operators, and running over all protons and neutrons in the nucleus.

Gamow-Teller transitions

A Gamow-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 by

with the weak axial-vector coupling constant, and the spin Pauli matrices, which can produce a spin-flip in the decaying nucleon.

Forbidden transitions

When L > 0, the decay is referred to as "forbidden". Nuclear selection rules require high L-values to be accompanied by changes in nuclear spin (J) and parity (π). The selection rules for the Lth forbidden transitions are:

where Δπ=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:

Forbiddenness ΔJ Δπ
Superallowed 0 no
Allowed 0, 1 no
First forbidden 0, 1, 2 yes
Second forbidden 1, 2, 3 no
Third forbidden 2, 3, 4 yes

Rare decay modes

Bound-state β decay

A very small minority of free neutron decays (about four per million) are so-called "two-body decays", in which 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 neutral hydrogen atom.[37] In this type of beta decay, in essence all of the neutron decay 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 by Daudel, Jean, and Lecoin in 1947,[38] and the phenomenon in fully ionized atoms was first observed for 163Dy66+ in 1992 by Jung et al. of the Darmstadt Heavy-Ion Research group. Although neutral 163Dy is a stable isotope, the fully ionized 163Dy66+ undergoes β decay into the K and L shells with a half-life of 47 days.[39]

Another possibility is that a fully ionized atom undergoes greatly accelerated β decay, as observed for 187Re by Bosch et al., also at Darmstadt. Neutral 187Re does undergo β decay with a half-life of 42 × 109 years, but for fully ionized 187Re75+ this is shortened by a factor of 109 to only 32.9 years.[40] For comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1%.

Double beta decay

Some nuclei can undergo double beta decay (ββ decay) where the charge of the nucleus changes by two units. Double beta decay is difficult to study, as the process has an extremely long half-life. In nuclei for which both β decay and ββ decay are possible, the rarer ββ decay process is effectively impossible to observe. However, in nuclei where β decay is forbidden but ββ decay is allowed, the process can be seen and a half-life measured.[41] Thus, ββ decay is usually studied only for beta stable nuclei. Like single beta decay, double beta decay does not change A; thus, at least one of the nuclides with some given A has to be stable with regard to both single and double beta decay.

"Ordinary" double beta decay results in the emission of two electrons and two antineutrinos. If neutrinos are Majorana particles (i.e., they are their own antiparticles), then a decay known as neutrinoless double beta decay will occur. Most neutrino physicists believe that neutrinoless double beta decay has never been observed.[41]

See also

References

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Bibliography

External links

  • Beta decay simulation [1]
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. This line lies along the bottom of the nuclear valley of stability.

Beta (finance)

In finance, the beta (β or beta coefficient) of an investment indicates whether the investment is more or less volatile than the market as a whole.

Beta is a measure of the risk arising from exposure to general market movements as opposed to idiosyncratic factors. The market portfolio of all investable assets has a beta of exactly 1. A beta below 1 can indicate either an investment with lower volatility than the market, or a volatile investment whose price movements are not highly correlated with the market. An example of the first is a treasury bill: the price does not go up or down a lot, so it has a low beta. An example of the second is gold. The price of gold does go up and down a lot, but not in the same direction or at the same time as the market.A beta greater than 1 generally means that the asset both is volatile and tends to move up and down with the market. An example is a stock in a big technology company. Negative betas are possible for investments that tend to go down when the market goes up, and vice versa. There are few fundamental investments with consistent and significant negative betas, but some derivatives like put options can have large negative betas.Beta is important because it measures the risk of an investment that cannot be reduced by diversification. It does not measure the risk of an investment held on a stand-alone basis, but the amount of risk the investment adds to an already-diversified portfolio. In the Capital Asset Pricing Model (CAPM), beta risk is the only kind of risk for which investors should receive an expected return higher than the risk-free rate of interest.The definition above covers only theoretical beta. The term is used in many related ways in finance. For example, the betas commonly quoted in mutual fund analyses generally measure the risk of the fund arising from exposure to a benchmark for the fund, rather than from exposure to the entire market portfolio. Thus they measure the amount of risk the fund adds to a diversified portfolio of funds of the same type, rather than to a portfolio diversified among all fund types.Beta decay refers to the tendency for a company with a high beta coefficient (β > 1) to have its beta coefficient decline to the market beta. It is an example of regression toward the mean.

Beta decay transition

A Fermi transition or a Gamow–Teller transition are types of nuclear beta decay determined by changes in angular momentum or spin. In the Fermi transition, the spins of the emitted particles are antiparallel, coupling to , so the angular momentum of the initial and final angular momentum states of the nucleus are unchanged (). This is in contrast to a Gamow-Teller transition, where the spins of the emitted electron (positron) and antineutrino (neutrino) couple to total spin , leading to an angular momentum change between the initial and final angular momentum states of the nucleus.

Fermi and Gamow-Teller transitions correspond to two different forms of leading order behavior of the weak interaction Hamiltonian in the non-relativistic limit:

= isospin transition matrix which turn protons to neutrons and vice versa
= Pauli spin matrices, which lead to .
= identity operator in spin space, leaving unchanged.
= Weak vector coupling constant.
= Weak axial-vector coupling constant.

The theoretical work in describing these transitions was done between 1934 and 1936 by Nuclear Physicists George Gamow and Edward Teller at George Washington University.

Beta particle

A beta particle, also called beta ray or beta radiation (symbol β), is a high-energy, high-speed electron or positron emitted by the radioactive decay of an atomic nucleus during the process of beta decay. There are two forms of beta decay, β− decay and β+ decay, which produce electrons and positrons respectively.Beta particles with an energy of 0.5 MeV have a range of about one metre in air; the distance is dependent on the particle energy.

Beta particles are a type of ionizing radiation and for radiation protection purposes are regarded as being less ionising than alpha particles, but more ionising than gamma rays. The higher the ionising effect, the greater the damage to living tissue.

CUORE

The Cryogenic Underground Observatory for Rare Events (CUORE, pronounced [ˈkwɔːre]) is a particle physics facility located underground at the Laboratori Nazionali del Gran Sasso in Assergi, Italy. CUORE was designed primarily as a search for neutrinoless double beta decay in 130Te, a process that has never been observed. It uses tellurium dioxide (TeO2) crystals as both the source of the decay and as bolometers to detect the resulting electrons. CUORE searches for the characteristic signal of neutrinoless double beta decay, a small peak in the observed energy spectrum around the known decay energy; for 130Te, this is Q = 2527.518 ± 0.013 keV. CUORE can also search for signals from dark matter candidates, such as axions and WIMPs.An observation of neutrinoless double beta decay would conclusively show that neutrinos are Majorana fermions; that is, they are their own antiparticles. This is relevant to many topics in particle physics, including lepton number conservation, nuclear structure, and neutrino masses and properties.

The CUORE collaboration involves physicists from several countries, primarily from the United States and Italy. CUORE is funded by the Istituto Nazionale di Fisica Nucleare of Italy, the United States Department of Energy, and the National Science Foundation of the United States.

In September 2014, as part of the testing of the CUORE dilution refrigerator, scientists in the CUORE collaboration cooled a copper vessel with a volume of one cubic meter to 6 mK (0.006 K, −273.144 °C) for 15 days, setting a record for the lowest temperature in the universe over such a large contiguous volume.

Calcium-48

Calcium-48 is a scarce isotope of calcium containing 20 protons and 28 neutrons. It makes up 0.187% of natural calcium by mole fraction. Although it is unusually neutron-rich for such a light nucleus, its beta decay is extremely hindered, and so the only radioactive decay pathway that it has been observed to undergo is the extremely rare process of double beta decay. Its half-life is about 6.4×1019 years, so for all practical purposes it can be treated as stable. One factor contributing to this unusual stability is that 20 and 28 are both magic numbers, making 48Ca a "doubly magic" nucleus.

Since 48Ca is both practically stable and neutron-rich, it is a valuable starting material for the production of new nuclei in particle accelerators, both by fragmentation and by fusion reactions with other nuclei, for example in the discoveries of the heaviest five elements on the periodic table, from flerovium to oganesson. Heavier nuclei generally require a greater fraction of neutrons for maximum stability, so neutron-rich starting materials are necessary.

48Ca is the lightest nucleus known to undergo double beta decay and the only one simple enough to be analyzed with the sd nuclear shell model. It also releases more energy (4.27 MeV) than any other double beta decay candidate. These properties make it an interesting probe of nuclear structure models and a promising candidate in the ongoing search for neutrinoless double beta decay.

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.

Electron capture

Electron capture (K-electron capture, also K-capture, or L-electron capture, L-capture) is a process in which the proton-rich nucleus of an electrically neutral atom absorbs an inner atomic electron, usually from the K or L electron shell. This process thereby changes a nuclear proton to a neutron and simultaneously causes the emission of an electron neutrino.

p + e− → n + νeSince this single emitted neutrino carries the entire decay energy, it has this single characteristic energy. Similarly, the momentum of the neutrino emission causes the daughter atom to recoil with a single characteristic momentum.

The resulting daughter nuclide, if it is in an excited state, then transitions to its ground state. Usually, a gamma ray is emitted during this transition, but nuclear de-excitation may also take place by internal conversion.

Following capture of an inner electron from the atom, an outer electron replaces the electron that was captured and one or more characteristic X-ray photons is emitted in this process. Electron capture sometimes also results in the Auger effect, where an electron is ejected from the atom's electron shell due to interactions between the atom's electrons in the process of seeking a lower energy electron state.

Following electron capture, the atomic number is reduced by one, the neutron number is increased by one, and there is no change in mass number. Simple electron capture by itself results in a neutral atom, since the loss of the electron in the electron shell is balanced by a loss of positive nuclear charge. However, a positive atomic ion may result from further Auger electron emission.

Electron capture is an example of weak interaction, one of the four fundamental forces.

Electron capture is the primary decay mode for isotopes with a relative superabundance of protons in the nucleus, but with insufficient energy difference between the isotope and its prospective daughter (the isobar with one less positive charge) for the nuclide to decay by emitting a positron. Electron capture is always an alternative decay mode for radioactive isotopes that do not have sufficient energy to decay by positron emission.

Electron capture is sometimes included as a type of beta decay, because the basic nuclear process, mediated by the weak force, is the same. In nuclear physics, beta decay is a type of radioactive decay in which a beta ray (fast energetic electron or positron) and a neutrino are emitted from an atomic nucleus.

Electron capture is sometimes called inverse beta decay, though this term usually refers to the interaction of an electron antineutrino with a proton.If the energy difference between the parent atom and the daughter atom is less than 1.022 MeV, positron emission is forbidden as not enough decay energy is available to allow it, and thus electron capture is the sole decay mode. For example, rubidium-83 (37 protons, 46 neutrons) will decay to krypton-83 (36 protons, 47 neutrons) solely by electron capture (the energy difference, or decay energy, is about 0.9 MeV).

Electron neutrino

The electron neutrino (νe) is a subatomic lepton elementary particle which has no net electric charge. Together with the electron it forms the first generation of leptons, hence the name electron neutrino. It was first hypothesized by Wolfgang Pauli in 1930, to account for missing momentum and missing energy in beta decay, and was discovered in 1956 by a team led by Clyde Cowan and Frederick Reines (see Cowan–Reines neutrino experiment).

Enriched Xenon Observatory

The Enriched Xenon Observatory (EXO) is a particle physics experiment searching for neutrinoless double beta decay of xenon-136 at WIPP near Carlsbad, New Mexico, U.S.

Neutrinoless double beta decay (0νββ) detection would prove the Majorana nature of neutrinos and impact the neutrino mass values and ordering. These are important open topics in particle physics.

EXO currently has a 200-kilogram xenon liquid time projection chamber (EXO-200) with R&D efforts on a ton-scale experiment (nEXO). Xenon double beta decay was detected and limits have been set for 0νββ.

Fermi's interaction

In particle physics, Fermi's interaction (also the Fermi theory of beta decay) is an explanation of the beta decay, proposed by Enrico Fermi in 1933. The theory posits four fermions directly interacting with one another (at one vertex of the associated Feynman diagram). This interaction explains beta decay of a neutron by direct coupling of a neutron with an electron, a neutrino (later determined to be an antineutrino) and a proton.Fermi first introduced this coupling in his description of beta decay in 1933. The Fermi interaction was the precursor to the theory for the weak interaction where the interaction between the proton–neutron and electron–antineutrino is mediated by a virtual W− boson.

Forbidden mechanism

In spectroscopy, a forbidden mechanism (forbidden transition or forbidden line) is a spectral line associated with absorption or emission of light by atomic nuclei, atoms, or molecules which undergo a transition that is not allowed by a particular selection rule but is allowed if the approximation associated with that rule is not made. For example, in a situation where, according to usual approximations (such as the electric-dipole approximation for the interaction with light), the process cannot happen, but at a higher level of approximation (e.g. magnetic dipole, or electric quadrupole) the process is allowed but at a much lower rate.

An example is phosphorescent glow in the dark materials, which absorb light and form an excited state whose decay involves a spin flip, and is therefore forbidden by electric dipole transitions. The result is emission of light slowly over minutes or hours.

Although the transitions are nominally forbidden, there is a small probability of their spontaneous occurrence, should an atomic nucleus, atom or molecule be raised to an excited state. More precisely, there is a certain probability that such an excited entity will make a forbidden transition to a lower energy state per unit time; by definition, this probability is much lower than that for any transition permitted or allowed by the selection rules. Therefore, if a state can de-excite via a permitted transition (or otherwise, e.g. via collisions) it will almost certainly do so before any transition occurs via a forbidden route. Nevertheless, most forbidden transitions are only relatively unlikely: states that can only decay in this way (so-called meta-stable states) usually have lifetimes on the order milliseconds to seconds, compared to less than a microsecond for decay via permitted transitions. In some radioactive decay systems, multiple levels of forbiddenness can stretch life times by many orders of magnitude for each additional unit by which the system changes beyond what is most allowed under the selection rules. Such excited states can last years, or even for many billions of years (too long to have been measured).

Inverse beta decay

Inverse beta decay, commonly abbreviated to IBD, is a nuclear reaction involving electron antineutrino scattering off a proton, creating a positron and a neutron. This process is commonly used in the detection of electron antineutrinos in neutrino detectors, such as the first detection of antineutrinos in the Cowan–Reines neutrino experiment, or in neutrino experiments such as KamLAND and Borexino. It is an essential process to experiments involving low energy neutrinos (< 60 MeV) such as those studying neutrino oscillation, reactor neutrinos, sterile neutrinos, and geoneutrinos. The IBD reaction can only be used to detect antineutrinos (rather than normal matter neutrinos, such as from the sun) due to lepton conservation.

Isotopes of technetium

Technetium (43Tc) is the first of the two elements lighter than bismuth that have no stable isotopes; the other such element is promethium. It is primarily artificial, with only trace quantities (About 16 thousand metric tonnes) 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 (half-life of 4.21 million years), 98Tc (half-life: 4.2 million years) and 99Tc (half-life: 211,100 years).Thirty-three other radioisotopes have been characterized with atomic masses ranging from 85Tc to 120Tc. Most of these have half-lives that are less than an hour; the exceptions are 93Tc (half-life: 2.75 hours), 94Tc (half-life: 4.883 hours), 95Tc (half-life: 20 hours), and 96Tc (half-life: 4.28 days).Technetium also has numerous meta states. 97mTc is the most stable, with a half-life of 91.0 days (0.097 MeV). This is followed by 95mTc (half-life: 61 days, 0.038 MeV), and 99mTc (half-life: 6.04 hours, 0.143 MeV). 99mTc only emits gamma rays, subsequently decaying to 99Tc.For isotopes lighter than the most stable isotope, 98Tc, the primary decay mode is electron capture, giving molybdenum. For the heavier isotopes, the primary mode is beta emission, giving ruthenium, with the exception that 100Tc can decay both by beta emission and electron capture.Technetium-99 is the most common and most readily available isotope, as it is a major fission product from fission of actinides like uranium and plutonium with a fission product yield of 6% or more, and in fact the most significant long-lived fission product. Lighter isotopes of technetium are almost never produced in fission because the initial fission products normally have a higher neutron/proton ratio than is stable for their mass range, and therefore undergo beta decay until reaching the ultimate product. Beta decay of fission products of mass 95–98 stops at the stable isotopes of molybdenum of those masses and does not reach technetium. For mass 100 and greater, the technetium isotopes of those masses are very short-lived and quickly beta decay to isotopes of ruthenium. Therefore, the technetium in spent nuclear fuel is practically all 99Tc.

One gram of 99Tc produces 6.2×108 disintegrations a second (that is, 0.62 GBq/g).Technetium has no stable or nearly stable isotopes, and thus a standard atomic weight cannot be given.

Isotopes of zirconium

Naturally occurring zirconium (40Zr) is composed of four stable isotopes (of which one may in the future be found radioactive), and one very long-lived radioisotope (96Zr), a primordial nuclide that decays via double beta decay with an observed half-life of 2.0×1019 years; it can also undergo single beta decay, which is not yet observed, but the theoretically predicted value of t1/2 is 2.4×1020 years. The second most stable radioisotope is 93Zr, which has a half-life of 1.53 million years. Twenty-seven other radioisotopes have been observed. All have half-lives less than a day except for 95Zr (64.02 days), 88Zr (83.4 days), and 89Zr (78.41 hours). The primary decay mode is electron capture for isotopes lighter than 92Zr, and the primary mode for heavier isotopes is beta decay.

Neutrino

A neutrino ( or ) (denoted by the Greek letter ν) is a fermion (an elementary particle with half-integer spin) that interacts only via the weak subatomic force and gravity. The mass of the neutrino is much smaller than that of the other known elementary particles. Although only differences of squares of the three mass values are known as of 2016, cosmological observations imply that the sum of the three masses must be less than one millionth that of the electron. The neutrino is so named because it is electrically neutral and because its rest mass is so small (-ino) that it was long thought to be zero. The weak force has a very short range, the gravitational interaction is extremely weak, and neutrinos, as leptons, do not participate in the strong interaction. Thus, neutrinos typically pass through normal matter unimpeded and undetected.Weak interactions create neutrinos in one of three leptonic flavors: electron neutrinos (νe), muon neutrinos (νμ), or tau neutrinos (ντ), in association with the corresponding charged lepton. Although neutrinos were long believed to be massless, it is now known that there are three discrete neutrino masses with different tiny values, but they do not correspond uniquely to the three flavors. A neutrino created with a specific flavor is in an associated specific quantum superposition of all three mass states. As a result, neutrinos oscillate between different flavors in flight. For example, an electron neutrino produced in a beta decay reaction may interact in a distant detector as a muon or tau neutrino.For each neutrino, there also exists a corresponding antiparticle, called an antineutrino, which also has half-integer spin and no electric charge. They are distinguished from the neutrinos by having opposite signs of lepton number and chirality. To conserve total lepton number, in nuclear beta decay, electron neutrinos appear together with only positrons (anti-electrons) or electron-antineutrinos, and electron antineutrinos with electrons or electron neutrinos.Neutrinos are created by various radioactive decays, including in beta decay of atomic nuclei or hadrons, nuclear reactions such as those that take place in the core of a star or artificially in nuclear reactors, nuclear bombs or particle accelerators, during a supernova, in the spin-down of a neutron star, or when accelerated particle beams or cosmic rays strike atoms. The majority of neutrinos in the vicinity of the Earth are from nuclear reactions in the Sun. In the vicinity of the Earth, about 65 billion (6.5×1010) solar neutrinos per second pass through every square centimeter perpendicular to the direction of the Sun.For study, neutrinos can be created artificially with nuclear reactors and particle accelerators. There is intense research activity involving neutrinos, with goals that include the determination of the three neutrino mass values, the measurement of the degree of CP violation in the leptonic sector (leading to leptogenesis); and searches for evidence of physics beyond the Standard Model of particle physics, such as neutrinoless double beta decay, which would be evidence for violation of lepton number conservation. Neutrinos can also be used for tomography of the interior of the earth.

Neutrino Ettore Majorana Observatory

The Neutrino Ettore Majorana Observatory (NEMO experiment) is an international collaboration of scientists searching for neutrinoless double beta decay (0νββ). The collaboration has been active since 1989. Observation of 0νββ would indicate neutrinos are Majorana particles and could be used to measure the neutrino mass. It is located in the Modane Underground Laboratory (LSM) in the Fréjus Road Tunnel. The experiment has (as of 2018) had 3 detectors, NEMO-1, NEMO-2, NEMO-3 (and a demonstrator module of SuperNEMO-detector) and is planning (as of 2018) to construct a new detector SuperNEMO. The NEMO-1 and NEMO-2 prototype detectors were used until 1997. Latest experiment NEMO-3 was under design and construction from 1994 onwards, took data from January 2003 to January 2011 and the final data analysis was published in 2018. The NEMO-2 and NEMO-3 detectors produced measurements for double neutrino decays and limits for neutrinoless double-beta decay for a number of elements, such as molybdenum-100 and selenium-82. These double beta decay times are important contributions to understanding the nucleus and are needed inputs for neutrinoless decay studies, which constrain neutrino mass.

The NEMO collaboration remains active and is constructing an improved SuperNEMO detector. Planning of SuperNEMO and commissioning of SuperNEMO demonstrator module is on-going as of 2019.

Positron emission

Positron emission or beta plus decay (β+ decay) is a subtype of radioactive decay called beta decay, in which a proton inside a radionuclide nucleus is converted into a neutron while releasing a positron and an electron neutrino (νe). Positron emission is mediated by the weak force. The positron is a type of beta particle (β+), the other beta particle being the electron (β−) emitted from the β− decay of a nucleus.

An example of positron emission (β+ decay) is shown with magnesium-23 decaying into sodium-23:

2312Mg → 2311Na + e+ + νeBecause positron emission decreases proton number relative to neutron number, positron decay happens typically in large "proton-rich" radionuclides. Positron decay results in nuclear transmutation, changing an atom of one chemical element into an atom of an element with an atomic number that is less by one unit.

Positron emission should not be confused with electron emission or beta minus decay (β− decay), which occurs when a neutron turns into a proton and the nucleus emits an electron and an antineutrino.

Positron emission is different from proton decay, the hypothetical decay of protons, not necessarily those bound with neutrons, not necessarily through the emission of a positron and not as part of nuclear physics, but rather of particle physics.

S-process

The slow neutron-capture process or s-process is a series of reactions in nuclear astrophysics that occur in stars, particularly AGB stars. The s-process is responsible for the creation (nucleosynthesis) of approximately half the atomic nuclei heavier than iron.

In the s-process, a seed nucleus undergoes neutron capture to form an isotope with one higher atomic mass. If the new isotope is stable, a series of increases in mass can occur, but if it is unstable, then beta decay will occur, producing an element of the next highest atomic number. The process is slow (hence the name) in the sense that there is sufficient time for this radioactive decay to occur before another neutron is captured. A series of these reactions produces stable isotopes by moving along the valley of beta-decay stable isobars in the chart of isotopes.

A range of elements and isotopes can be produced by the s-process, because of the intervention of alpha decay steps along the reaction chain. The relative abundances of elements and isotopes produced depends on the source of the neutrons and how their flux changes over time. Each branch of the s-process reaction chain eventually terminates at a cycle involving lead, bismuth, and polonium.

The s-process contrasts with the r-process, in which successive neutron captures are rapid: they happen more quickly than the beta decay can occur. The r-process dominates in environments with higher fluxes of free neutrons; it produces heavier elements and more neutron-rich isotopes than the s-process. Together the two processes account for most of the relative abundance of chemical elements heavier than iron.

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