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β−
decay in an atomic nucleus. The intermediate emission of a virtual W−
boson is omitted.

In nuclear physics, beta decay is a type of radioactive decay in which a beta particle (an electron or a positron) is emitted from an atom. There are two types of beta decay: beta minus and beta plus. In the case of beta decay that produces an electron emission, it is referred to as beta minus (β−
), while in the case of a positron emission as beta plus (β+
). In electron emission, an electron antineutrino is also emitted, while positron emission is accompanied by an electron neutrino. Beta decay is mediated by the weak force.

Emitted beta particles have a continuous kinetic energy spectrum, ranging from 0 to the maximal available energy (Q), which depends on the parent and daughter nuclear states that participate in the decay. A typical Q is around 1 MeV, but it can range from a few keV to a few tens of MeV. Since the rest mass energy of the electron is 511 keV, the most energetic beta particles are ultrarelativistic, with speeds very close to the speed of light.

Sometimes electron capture decay is included as a type of beta decay (and is referred to as "inverse beta decay"), because the basic process, mediated by the weak force is the same. However, no beta particle is emitted, but only an electron neutrino. Instead of beta-plus emission, an inner atomic electron is captured by a proton in the nucleus. This type of decay is therefore analogous to positron emission (and also happens, as an alternative decay route, in all positron-emitters). However, the route of 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). These may still reach a lower energy state, by the equivalent process of electron-capture and neutrino-emission.

[edit] β−
decay

The 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 a nucleus into its next-higher neighbor on the periodic table while emitting an electron (e−
) and an electron antineutrino (ν
e):

Failed to parse (Missing texvc executable; please see math/README to configure.): ^A_ZN \rightarrow ~ ^{~~~A}_{Z+1}N' + e^- + \bar{\nu_e}.

where Failed to parse (Missing texvc executable; please see math/README to configure.): A

and Failed to parse (Missing texvc executable; please see math/README to configure.): Z
are the mass number and charge of the decaying nucleus. For example, the neutron (n) decays by β−
 decay into a proton (p):

Failed to parse (Missing texvc executable; please see math/README to configure.): n \rightarrow p + e^- + \bar{\nu_e}.

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

Failed to parse (Missing texvc executable; please see math/README to configure.): d \rightarrow u + e^- + \bar{\nu_e}.

β−
decay generally occurs in neutron-rich nuclei.

[edit] β+
decay

In β+
decay, or "positron emission", the weak interaction converts a nucleus into its next-lower neighbor on the periodic table while emitting an positron (e+
) and an electron neutrino (ν
e):

Failed to parse (Missing texvc executable; please see math/README to configure.): ^A_ZN \rightarrow ~ ^{~~~A}_{Z-1}N' + e^+ + \nu_e.

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 value of the binding energy of the mother nucleus is less than that of the daughter 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. In an opposite process to the Beta- decay, the weak interaction converts a proton into a neutron by converting an up quark into a down quark by having it emit a W+
or absorb a W−
.

Electron capture (K-capture)[link]

In all the cases where β+
decay is allowed energetically, the electron capture process, when an atomic electron is captured by a nucleus with the emission of a neutrino, is also allowed:

Failed to parse (Missing texvc executable; please see math/README to configure.): ^A_ZN + e^- \rightarrow ~ ^{~~~A}_{Z-1}N' + {\nu_e}

The emitted neutrino is mono-energetic. In proton-rich nuclei where the energy difference between initial and final states is less than 2m_ec², β+
decay is not energetically possible, and electron capture is the sole decay mode.

This decay is also called K-capture because the inner most electron of an atom belongs to the K-shell of the electronic configuration of the atom, and this has the highest probability to interact with the nucleus.

There is an analogous process possible in theory in antimatter: antiproton-rich antimatter radioisotopes might decay via an analogous process of positron capture, but in practice no such complex antimatter nuclides have either been discovered or artificially constructed.

Nuclear transmutation[link]

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

137 55Cs			→	137 56Ba	+	e−	+	ν e	(beta minus decay)
22 11Na			→	22 10Ne	+	e+	+	ν e	(beta plus decay)
22 11Na	+	e−	→	22 10Ne	+	ν e			(electron capture)

Beta decay does not change the number of nucleons, A, 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. Among them, several nuclides (at least one) are beta stable, because they present 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 the global minimum is also the unique local minimum. For even A, there are up to three different beta-stable isobars experimentally known; for example, 96
40Zr, 96
42Mo, and 96
44Ru are all beta-stable, though the first one can undergo a very rare double beta decay (see below). There are about 355 known beta-decay stable nuclides total.

A beta-stable nucleus may undergo other kinds of radioactive decay (alpha decay, for example). In nature, most isotopes are beta stable, but a few exceptions exist with half-lives so long that they have not had enough time to decay since the moment of their nucleosynthesis. One example is 40
19K, which undergoes all three types of beta decay (β−
, β+
and electron capture) with a half-life of 1.277×10⁹ years.

Double beta decay[link]

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 in most practically interesting cases, because both β decay and ββ decay are possible, with probability favoring β decay; the rarer ββ decay process is masked by these events. 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.

[edit] Bound-state β^- decay

For fully ionized atoms (bare nuclei), it is possible for electrons to be emitted from the nucleus into low-lying atomic bound states (orbitals). This can not occur for neutral atoms whose low-lying bound states are already filled.

The phenomenon 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.^[1]

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 x 10⁹ years, but for fully ionized ¹⁸⁷Re⁷⁵⁺ this is shortened by a factor of 10⁹ to only 32.9 years.^[2] For comparison the variation of decay rates of other nuclear processes due to chemical environment is less than 1%. (See Radioactive decay#Changing decay rates)

Forbidden transitions[link]

Beta decays can be classified according to the L-value of the emitted radiation. 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 L^th forbidden transitions are:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Delta J = L-1, L, L+1; \Delta \pi = (-1)^L,

where Δπ = 1 or -1 corresponds to no parity change or parity change, respectively. The special case of a 0⁺ → 0⁺ transition is referred to as "superallowed". The following table lists the ΔJ and Δπ values for the first few values of L:

Beta emission spectrum[link]

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^[3]:

Failed to parse (Missing texvc executable; please see math/README to configure.): N(T) = C_L(T) F(Z,T) p E (Q-T)^2

where T is the kinetic energy, C_L(T) 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 + m c² is the total energy, p = [(E/c)² − (mc)²]^1/2 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.

Fermi function[link]

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^[4]:

Failed to parse (Missing texvc executable; please see math/README to configure.): F(Z,T) = \frac{2 (1+S)}{\Gamma(1+2S)^2} (2 p \rho)^{2S-2} e^{\pi \eta} |\Gamma(S+i \eta)|^2,

where S = [1 − α² Z²]^1/2 (α is the fine-structure constant), η = ± α Z E / p (+ for electrons, - for positrons), ρ = r_N / ℏ (r_N is the radius of the final state nucleus), and Γ is the Gamma function.

For non-relativistic betas (Q << m_ec²), this expression can be approximated by^[5]:

Failed to parse (Missing texvc executable; please see math/README to configure.): F(Z,T) \approx \frac{2 \pi \eta}{1 - e^{- 2 \pi \eta}}.

Other approximations can be found in the literature^[6]

Kurie plot[link]

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. 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).

History[link]

[edit] Discovery and characterization of β−
decay

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 aluminum, whereas beta rays could penetrate several millimetres of aluminum. (In 1900 Paul Villard identified gamma rays as a third, still more penetrating type of radiation.)

In 1900 Becquerel measured the ratio of electric charge to mass (e/m) for beta particles by the method of J.J. Thomson used to study cathode rays and identify the electron. He found that e/m for a beta particle is the same as for Thomson’s electron, and therefore suggested that the beta particle is in fact an electron.

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 in beta decay[link]

Historically, the study of beta decay provided the first physical evidence of the neutrino. In 1911 Lise Meitner and Otto Hahn performed an experiment that showed that the energies of electrons emitted by beta decay had a continuous rather than discrete spectrum. This was in apparent contradiction to the law of conservation of energy, as it appeared that energy was lost in the beta decay process. A second problem was that the spin of the Nitrogen-14 atom was 1, in contradiction to the Rutherford prediction of ½.

In 1920-1927, Charles Drummond Ellis (along with James Chadwick and colleagues) established clearly that the beta decay spectrum is really continuous, ending all controversies.

In a famous letter written in 1930 Wolfgang Pauli suggested that in addition to electrons and protons atoms also contained an extremely light neutral particle which he called the neutron. He suggested that this "neutron" was also emitted during beta decay and had simply not yet been observed. In 1931 Enrico Fermi renamed Pauli's "neutron" to neutrino, and in 1934 Fermi published a very successful model of beta decay in which neutrinos were produced.

Discovery of other types of beta decay[link]

In 1934 Frédéric and Irène Joliot-Curie bombarded aluminum with alpha particles to effect the nuclear reaction ⁴He + ²⁷Al → ³⁰P + n, and observed that the product isotope ³⁰P emits a positron identical to those found in cosmic rays by Carl David Anderson in 1932. This was the first example of β+
decay, which they termed artificial radioactivity since ³⁰P is a short-lived nuclide which does not exist in nature.

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.^[7][8][9] Alvarez went on to study electron capture in ⁶⁷Ga and other nuclides.^[7][10][11]

See also[link]

• Double beta decay
• Electron capture
• Neutrino
• Alpha decay
• Betavoltaics
• Particle radiation
• Radioactive isotope
• Tritium illumination, a form of fluorescent lighting powered by beta decay

References[link]

• Franz N. D. Kurie, J. R. Richardson, H. C. Paxton (March 1936). "The Radiations Emitted from Artificially Produced Radioactive Substances. I. The Upper Limits and Shapes of the β-Ray Spectra from Several Elements". Physical Review 49 (5): 368–381. Bibcode 1936PhRv...49..368K. DOI:10.1103/PhysRev.49.368. 
• F. N. D. Kurie (May 1948). "On the Use of the Kurie Plot". Physical Review 73 (10): 1207. Bibcode 1948PhRv...73.1207K. DOI:10.1103/PhysRev.73.1207. 
• Jagdish K. Tuli, Nuclear Wallet Cards, 7th edition, April 2005, Brookhaven National Laboratory, US National Nuclear Data Center

1. ^ M. Jung et al., Phys. Rev. Lett. 69, 2164 (1992) First observation of bound-state beta minus decay.
2. ^ F. Bosch et al., Phys. Rev. Lett. 77, 5190 (1996) Observation of bound-state beta minus decay of fully ionized ¹⁸⁷Re: ¹⁸⁷Re-¹⁸⁷Os Cosmochronometry
3. ^ Energy and Momentum Spectra for Beta Decay, accessed on line March 9, 2012.
4. ^ E. Fermi, Z. Phys. 88, 161 (1934).
5. ^ N. F. Mott and H. S. W. Massey, The Theory of Atomic Collisions, Clarendon, Oxford (1933).
6. ^ P. Venkataramaiah et al., J. Phys. G: Nucl. Phys. 11, 359 (1985); G. K. Schenter and P. Vogel, Nucl. Sci. Eng. 83, 393 (1983).
7. ^ ^{a b} pp. 11–12, K-Electron Capture by Nuclei, Emilio Segré, chapter 3 in Discovering Alvarez: selected works of Luis W. Alvarez, with commentary by his students and colleagues, Luis W. Alvarez and W. Peter Trower, University of Chicago Press, 1987, ISBN 0-226-81304-5.
8. ^ Luis Alvarez, The Nobel Prize in Physics 1968, biography, nobelprize.org. Accessed on line October 7, 2009.
9. ^ Nuclear K Electron Capture, Luis W. Alvarez, Physical Review 52 (1937), pp. 134–135, doi:10.1103/PhysRev.52.134 .
10. ^ Electron Capture and Internal Conversion in Gallium 67, Luis W. Alvarez, Physical Review 53 (1937), p. 606, doi:10.1103/PhysRev.53.606.
11. ^ The Capture of Orbital Electrons by Nuclei, Luis W. Alvarez, Physical Review 54 (October 1, 1938), pp. 486–497, doi:10.1103/PhysRev.54.486.

External links[link]

• The Live Chart of Nuclides - IAEA with filter on decay type