Radioactive decay constant

The rate of radioactive transformation, or the activity, of a source equals the number A of identical radioactive atoms present in the source, multiplied by their characteristic radioactive decay constant λ. Thus Eq. (1) holds,

(1)

where the decay constant λ has dimensions of s⁻¹. The numerical value of λ expresses the statistical probability of decay of each radioactive atom in a group of identical atoms, per time. For example, if λ = 0.01 s⁻¹ for a particular radioactive species, then each atom has a chance of 0.01 (1%) of decaying in 1 s, and a chance of 0.99 (99%) of not decaying in any given 1-s interval. The constant λ is one of the most important characteristics of each radioactive nuclide: λ is essentially independent of all physical and chemical conditions such as temperature, pressure, concentration, chemical combination, or age of the radioactive atoms. There are a few cases where very small measurable effects are observed for different chemical combinations and pressure. However, in the Sun and in space a nucleus may be stripped of its atomic electrons, and then electron capture decay rates can change markedly; for example, the half-life of ⁷Be is 70 d in the Sun and 53 d on Earth, and the half-life of ⁵⁴Mn is estimated to be (1–2) × 10⁶ y in cosmic rays compared to 312 d on Earth because on average they have essentially no orbital electrons for electron captures to occur.

The identification of some radioactive samples can be made simply by measuring λ, which then serves as an equivalent of qualitative chemical analysis. For the most common radioactive nuclides, the range of λ extends from 3 × 10⁶ s⁻¹ (for thorium C′) to 1.6 × 10⁻¹⁸ s⁻¹ (for thorium).

Dual decay

Many radioactive nuclides have two or more independent and alternative modes of decay. For example, ²³⁸U can decay either by alpha-particle emission or by spontaneous fission. A single atom of ⁶⁴Cu can decay in any of three competing independent ways: negatron beta-particle emission, positron beta-particle emission, or electron capture. When two or more independent modes of decay are possible, the nuclide is said to exhibit dual decay.

The competing modes of decay of any nuclide have independent partial decay constants given by the probabilities λ₁, λ₂, λ₃ … per second, and the total probability of decay is represented by the total decay constant λ, defined by Eq. (2).

(2)

If there are A identical atoms present, the partial activities, as measured by the different modes of decay, are Aλ₁, Aλ₂, Aλ₃, …, and the total activity Aλ is given by Eq. (3).

(3)

The partial activities, Aλ₁, …, such as positron beta particles from ⁶⁴Cu, are proportional to the total activity, Aλ, at all times.

The branching ratio is the fraction of the decaying atoms which follow a particular mode of decay, and equals Aλ₁/Aλ or λ₁/λ. For example, in the case of ⁶⁴Cu the measured branching ratios are λ₁/λ = 0.40 for negatron beta decay, λ₂/λ = 0.20 for positron beta decay, and λ₃/λ = 0.40 for electron capture. The sum of all the branching ratios for a particular nuclide is unity.

Exponential decay law

The total activity, Aλ, equals the rate of decrease −dA/dt in the number of radioactive atoms A present. Because λ is independent of the age t of an atom, integration of the differential equation of radioactive decay, −dA/dt = Aλ, gives Eq. (4),

(4)

where ln represents the natural logarithm to the base e, and A atoms remain at time t if there were A₀ atoms initially present at time t₀. If t₀ = 0, then Eq. (4) can be rewritten as the exponential law of radioactive decay in its most common form, Eq. (5).

(5)

The initial activity at t = 0 was A₀λ, and the activity at t, when only A atoms remain untransformed, is Aλ. Because λ is a constant, the fractional activity Aλ/A₀λ at time t and the fractional amount of radioactive atoms A A₀ are given by Eq. (6).

(6)

In cases of dual decay, the partial activities Aλ₁, Aλ₂, … also decrease with time as e^{− λ t}₁, not as e^{− λ t}₁…, because Aλ₁/A₀λ₁ = A/A₀ = e^{−λ t} where λ is the total decay constant. This is because the decrease of each partial activity with time is due to the depletion of the total stock of atoms A, and this depletion is accomplished by the combined action of all the competing modes of decay.

Mean life

The actual life of any particular atom can have any value between zero and infinity. The average or mean life of a large number of identical radioactive atoms is, however, a definite and important quantity.

If there are A₀ atoms present initially at t = 0, then the number remaining undecayed at a later time t is A = A₀e^{− λt}, by Eq. (5). Each of these A atoms has a life longer than t. In an additional infinitesimally short time interval dt, between time t and t + dt, the absolute number of atoms which will decay on the average is Aλdt, and these atoms had a life-span t. The total L of the life-spans of all the A₀ atoms is the sum or integral of tAλ dt from t = 0 to t = ∞, which is given by Eq. (7).

(7)

Then the average lifetime L/A₀, which is called the mean life τ, is given by Eq. (8),

(8)

where λ is the total radioactive decay constant of Eq. (2).

Substitution of t = τ = 1/λ into Eq. (6) shows that the mean life is the time required for the number of atoms, or their activity, to fall to e⁻¹ = 0.368 of any initial value.

Half-period (half-life)

The time interval over which the chance of survival of a particular radioactive atom is exactly one-half is called the half-period T (also called half-life, written T₁/2. From Eq. (4), Eq. (9) is obtained.

(9)

Then the half-period T is related to the total radioactive decay constant λ, and to the mean life τ, by Eq. (10).

(10)

For mnemonic reasons, the half-period T is much more frequently employed than the total decay constant λ or the mean life τ. For example, it is more common to speak of ²³²Th as having a half-period of 1.4 × 10¹⁰ years than to speak of its mean life of 2.0 × 10¹⁰ years or its total decay constant of 1.6 × 10⁻¹⁸ s⁻¹, although all three are equivalent statements of the average longevity of ²³²Th atoms.

Any initial activity A₀λ is reduced to ½ in one half-period T, to 1/e in one mean life τ, to ¼ in two half-periods 2T, and so on (Fig. 2). The slope of the activity curve, or rate of decrease of activity, is d(Aλ)/dt = λdA/dt = −λ(Aλ). Thus the initial slope is −λ(A₀λ) = −(A₀λ)/τ. The area under the activity curve, if integrated to t = ∞, is simply A₀, the total initial number of radioactive atoms. Also, the initial activity A₀λ, if it could continue at a constant value for one mean life τ, would exactly destroy all the atoms because (A₀λ)τ = A₀. See also: Half-life

Fig. 2 Graphical representation of relationships in decay of a single radioactive nuclide.

Radioactive equilibrium

In Fig. 4, the ratio Bλ_B/Aλ_A of the activities of the parent A and the daughter product B change with time. The activity Bλ_B is zero initially and also after a very long time, when all the atoms have decayed. Thus Bλ_B passes through a maximum value, and it can be shown that this occurs at a time t_m given by Eq. (15).

(15)

The situation in which the activities Aλ_A and Bλ_B are exactly equal to each other is called ideal equilibrium, and exists only at the moment t_m .

If the parent A is longer-lived than the daughter B, as occurs in many cases, then at a time which is long compared with the mean life τ_B of B, the activity ratio approaches a constant value given by Eq. (16),

(16)

where T_A and T_B are the half-periods of A and of B. When the activity ratio Bλ_B/Aλ_A is constant, a particular type of radioactive equilibrium exists. This is spoken of as secular equilibrium if the activity ratio is experimentally indistinguishable from unity, as occurs when T_A is very much greater than T_B .

Equilibrium concepts are applied also between a long-lived parent and any of its decay products in a long series. For example, in a sufficiently old uranium ore, radium (T = 1620 years) is in secular equilibrium with its ultimate parent uranium (T = 4.5 × 10⁹ years) although there are four intermediate radioactive substances intervening in the series between uranium and radium. Here, secular equilibrium shows that activities of radium and uranium continue to be equal to each other even though the activity of the parent uranium is decreasing with time.

When T_B is comparable with T_A, Eq. (16) shows that the equilibrium ratio will clearly exceed unity; this situation is spoken of as transient equilibrium. For example, in fission-product decay series (17)

(17)

the half-period of ¹⁴⁰Ba is 307 h and that of ¹⁴⁰La is 40 h. In an initially pure source of ¹⁴⁰Ba the activity of ¹⁴⁰La starts at zero, rises to a maximum at t_m = 135 h [Eq. (15)], then decreases, and after a few hundred hours is in transient equilibrium with its parent, when the ¹⁴⁰La activity [by Eq. (16)] is 307/(307 – 40) = 1.15 times the activity of its parent ¹⁴⁰Ba.

Radioactivity in the Earth

A number of isotopes of elements found in the Earth are radioactive (Table 2). All known or theoretically predicted isotopes of elements above lead are radioactive. As a general rule, one can detect the presence of a radioactive substance for about 10 half-lives. Therefore, activities with T ≲ 0.3 × 10⁹ years should not be found in the Earth. For example, present-day uranium is an isotopic mixture containing 99.3% ²³⁸U, whose half-period is 4.5 × 10⁹ years, and only 0.7% of the shorter-lived uranium isotope ²³⁵U, whose half-period is 0.7 × 10⁹ years, whereas these isotopes presumably were produced in roughly equal amounts in the Earth a few billion years ago. Geophysical evidence indicates that originally some ²³⁶U was present also, but none is found in nature now as expected with its half-period of 0.02 × 10⁹ years. The elements technetium (Z = 43) and promethium (Z = 61) are not found in the Earth's crust because all their isotopes are radioactive with much shorter half-periods (their longest-lived are T = 2.6 × 10⁶ years for ⁹⁷Tc and T = 17.7 years for ¹⁴⁵Pm).

Uranium-238 decays through a long series of 14 radioactive decay products before ending as a stable isotope of lead, ²⁰⁶Pb. Some of these members of the ²³⁸U decay chain have very short half-periods, so their existence in nature is entirely dependent on the presence of their long-lived parent, and thus is a genealogical accident. For example, radium occurs in nature only in the minerals of its parent, uranium. The decay series of ²³⁵U supports 14, and the decay series of ²³²Th supports 10, short-lived radioactive substances found in nature.

A few of the common elements contain long-lived, naturally radioactive isotopes. For example, all terrestrial potassium contains 0.012% of the radioactive isotope ⁴⁰K, which has a half-period of 1.3 × 10⁹ years, and emits negatron or positron beta particles (plus decay via electron capture) and gamma rays in a dual decay to stable ⁴⁰Ca and ⁴⁰Ar. This isotope is the principal source of radioactivity in a normal human being; each human contains about 0.1 microcurie (3.7 × 10³ becquerels) of the radioactive potassium isotope ⁴⁰K.

Geological age measurements are based on the accumulation of decay products of long-lived isotopes, especially in the cases of ⁴⁰K, ⁸⁷Rb, ²³²Th, ²³⁵U, and ²³⁸U.

Laboratory-produced radioactive nuclei

With particle accelerators and nuclear reactors, of the order of 2500 radioactive isotopes not found in detectable quantities in the Earth's crust have been produced in the laboratory since 1935, including those of 26 new chemical elements up to element 118. Earlier titles of induced or artificial radioactivities for these isotopes are misnomers. Many of these now have been identified in meteorites and in stars, and others are produced in the atmosphere by cosmic rays. There are over 5000 isotopes theoretically predicted to exist. As one approaches the place where a proton or neutron is no longer bound in a nucleus of an element (the limits of the existence of that element), the half-periods become extremely short. See also: Transuranium elements

For example, carbon-14 is a negatron beta-particle emitter, with a half-period of about 5600 years, which can be produced in the laboratory as the product of a variety of different nuclear transmutation experiments. Nuclear bombardment of ¹¹B nuclei by alpha particles (helium nuclei) can produce excited compound nuclei of ¹⁵N which promptly emit a proton (hydrogen nucleus), leaving ¹⁴C as the end product of the transmutation. The same end-product ¹⁴C can be produced by bombarding ¹⁴N with neutrons, resulting in nuclear reaction (18).

(18)

This reaction is easily carried out by using neutrons from nuclear accelerators or a nuclear reactor. This particular transmutation reaction is one which occurs in nature also, because the nitrogen in the Earth's atmosphere is continually bombarded by neutrons which are produced by cosmic rays, thus producing radioactive ¹⁴C. Mixing of ¹⁴C with stable carbon provides the basis for radiocarbon dating of systems that absorb carbon for times up to about 50,000 years ago (10 half-lives). See also: Cosmic ray; Nuclear reaction; Nuclear reactor; Particle accelerator; Radiocarbon dating

Radioactive hydrogen, ³H, is also formed in the atmosphere from the ¹⁴N + neutron →¹²C + ³H reaction. Also, ³H is produced in the Sun, and the Earth's water as well as satellites show an additional concentration of ³H from the Sun. Over two dozen radioactive products, ranging in half-life from a few days to millions of years, have been identified in meteorites that have fallen to Earth. The carbon and hydrogen burning cycles that produce energy for stars produce radioactive ¹³N, ¹⁵O, ³H. At higher temperatures the radioactivities ⁷Be and even ⁸Be (T ≈ 10⁻¹⁶ s) help burn hydrogen and helium. In addition to the production of radioactive as well as stable isotopes prior to the formation of the solar system, nucleosynthesis continues to go on in stars with the production of many short-lived radioactive atoms by different processes. See also: Carbon-nitrogen-oxygen cycles (astrophysics); Nucleosynthesis; Proton-proton chain

The yield of any radioactivity produced in the laboratory is the initial rate of the activity under the particular conditions of nuclear bombardment. When a target material A is bombarded to produce a radioactive product B whose radioactive decay constant is λ_B, the number of atoms B which are present after a bombardment of duration t, and their activity Bλ_B, are given by Eq. (19),

(19)

where the yield Y has dimensions equivalent to curies of activity produced per second of bombardment. The yield Y depends on the number of atoms A present in the target, the intensity of the beam of bombarding particles, and the cross section, or probability of the reaction per bombarding particle under the conditions of bombardment.

Radioactive transformation series

As noted in Eqs. (12)–(14), many radioactive substances have decay products which are also radioactive. Accordingly, many long chains or series of radioactive transformations are known. The three naturally occurring transformation series are headed by ²³²Th, ²³⁵U, and ²³⁸U (Fig. 4 and Table 3).

Fig. 4 Main line of decay of uranium series, or 4n + 2 series, of heavy radioactive nuclides, headed in nature by uranium-238. Each member has a mass number given by 4n + 2, where n is an integer. Note: For historical purposes, the early names for certain elements are given in this figure, referenced in the following Table 3. For example, RaA is actually a polonium isotope.

Each of the naturally occurring radioactive isotopes in these transformation series has two synonymous names. For example, the commercially important radioisotope whose classical name is mesothorium-1 is known to be an isotope of radium with mass number of 228 and is designated as radium-228 (²²⁸Ra). Table 3 summarizes the names, symbols, and some radioactive properties of these three transformation series. However, these chains are not the only ones. Their uniqueness or importance as chains is an accident of the very long half-lives of ²³²Th, ²³⁵U, and ²³⁸U. For example, element 105 of mass 260 has a succession of seven alpha decays and one electron capture and positron decay to ²³²Th. The special importance of the chains in Table 3 is related to the fact that they were essentially the only early sources of radioactive materials, and they also play a role in nuclear power. See also: Nuclear power

Transformation series are now known for every element in the periodic table except hydrogen. Chains of neutron-rich isotopes have been produced and studied among the products of nuclear fission. Heavy-ion-induced reactions and high-flux reactors have been used to extend knowledge of the elements beyond uranium. The elements from number 93 (neptunium) to 118 (oganesson), which have so far not been found on Earth, were made in the laboratory. Both proton- and heavy-ion-induced reactions have extended knowledge of chains and neutron-deficient isotopes of the stable elements.

Geiger-Nuttall rule

Among all the known alpha-particle emitters, most alpha-particle energy spectra lie in the domain of 4–6 MeV, although a few extend as low as 2 MeV (¹⁴⁷₆₂Sm) and as high as 10 MeV (²¹²₈₄Po or ThC′). There is a systematic relationship between the kinetic energy of the emitted alpha particles and the half-period of the alpha emitter. The highest-energy alpha particles are emitted by short-lived nuclides, and the lowest-energy alpha particles are emitted by the very long lived alpha-particle emitters. The German physicist Hans Geiger and the English physicist John Mitchell Nuttall showed that there is a linear relationship between log λ and the energy of the alpha particle.

The Geiger-Nuttall rule is inexplicable by classical physics but emerges clearly from quantum, or wave, mechanics. In 1928, the hypothesis of transmission through nuclear potential barriers, as introduced by Russian-born U.S. physicist George Gamow and independently by British theoretical physicist Ronald Wilfred Gurney and U.S. nuclear physicist Edward Condon, was shown to give a satisfactory account of the alpha-decay data, and it has been altered subsequently only in details. The form of the barrier-penetration equations is such that correlation plots of log λ against 1/E give nearly straight lines. See also: Quantum mechanics

Nuclear potential barrier

At distances r which are large compared with the nuclear radius, the potential energy of an alpha particle, whose charge is 2e, in the field of a residual nucleus whose charge is (Z − 2)e, is 2(Z − 2)e²/r. At very close distances, this electrostatic repulsion is opposed and overcome by short-range, nuclear, attractive forces. The net potential energy U as a function of the separation r between the alpha particle and its residual nucleus is the nuclear potential barrier.

One of several operating definitions of the nuclear radius R is the distance r = R at which the attractive nuclear forces just balance the repulsive electrostatic forces. At this distance, called the top of the nuclear barrier, the potential energy is about 25–30 MeV for typical cases of heavy, alpha-emitting nuclei (Fig. 6).

Fig. 6 Schematic of nuclear potential barrier, illustrating emission of an alpha particle as a wave which can be transmitted through the barrier.

Inside the nucleus, the alpha particle is represented as a de Broglie matter wave. According to wave mechanics, this wave has a very small but finite probability of being transmitted through the nuclear potential energy barrier and thus of emerging as an alpha particle emitted from the nucleus. The transmission of a particle through such an energy barrier is completely forbidden in classical electrodynamics but is possible according to wave mechanics. This transmission of a matter wave through an energy barrier is analogous to the familiar case of the transmission of ordinary visible light through an opaque metal such as gold: if the gold is thin enough, some light does get through, as in the case of the thin gold leaf which is sometimes used for lettering signs on store windows. See also: Quantum mechanics

The wave-mechanical probability of the transmission of an alpha particle through the nuclear potential barrier is very strongly dependent upon the energy of the emitted alpha particle. Analytically, the probability of transmission T depends exponentially upon a barrier transmission exponent γ according to Eq. (20).

(20)

To a good approximation, Eq. (21) holds,

(21)

where h = 6.626 × 10⁻³⁴ joule-second is Planck's constant, and M is the so-called reduced mass of the alpha particle. For the alpha decay of ²²⁶Ra, the numerical value of γ is about 71: hence T = e⁻⁷¹ = 10⁻³¹. The first term on the right side of Eq. (21) is about 154 and is therefore the dominant term. When this term is taken alone, e^{−(4π2/h)(Z−2)2e2/V} is called the Gamow factor for barrier penetration. See also: Planck's constant

Inspection of Eq. (21) shows that the barrier transmission decreases with increasing nuclear charge (Z − 2)e, increases with increasing velocity V of emission of the alpha particle, and increases with increasing radius R of the nucleus. When the experimentally known values of alpha-decay energy are substituted into Eq. (21), with R about 10⁻¹² cm and Z about 90, the transmission coefficient T = e^−λ is found to extend over a domain of about 10⁻²⁰ to 10⁻⁴⁰. This range of about 10²⁰ is just what is needed to relate the alpha-decay energy to the broad domain of known alpha-decay half-periods. Equation (21) thus explains the Geiger-Nuttall rule very successfully (Fig. 7). From Eq. (20), one can derive the relationship between the mean life τ in seconds and the alpha-particle energy E_α in MeV, ln τ = AE^−1/2_α + B, where A and B are constants for different parent nuclei. From Fig. 7 it may be noted that, if the alpha decay energies of ²³²Th and ^235,238U had been 0.2 to 0.5 MeV higher, their half-lives would have been too short for them to still be present in the Earth's crust. If that had been the case, then radioactivity and the nucleus of the atom might never have been discovered.

Fig. 7 Systematics of the broad range of half-periods for alpha-particle decay and their strong dependence on alpha-decay energy and weaker dependence on nuclear charge. Numbers beside experimental points are mass numbers of parent alpha-particle emitters. Lines connect parent isotopes and are drawn using wave-mechanical theory of alpha-particle transmission through nuclear potential barriers.

Knowledge of alpha-emitting isotopes was greatly enlarged through the identification of many isotopes' far off stability in the region just above tin and in the broad region from neodymium all the way to uranium. For example, fusion reactions between 290-MeV ⁵⁸Ni ions and ⁵⁸Ni and ⁶³Cu targets have been used to produce and study very neutron deficient radioactive isotopes, including 12 alpha emitters between tin and cesium. These results provide important data on the atomic masses of nuclei far from the stable ones in nature. These data test understanding of nuclear mass formulas and their validity in new regions of the periodic table.

Neutrinos

The continuous spectrum of beta-particle energies (Fig. 8) implies the simultaneous emission of a second particle besides the beta particle, in order to conserve energy and angular momentum for each decaying nucleus. This particle is the neutrino. The sum of the kinetic energy of the neutrino and the beta particle equals E_max for the particular transition involved except in the rare cases where internal bremsstrahlung or shake-off electrons are emitted along with the beta particle and neutrino. The neutrino has zero charge and nearly zero rest mass, travels at essentially the same speed as light (3 × 10⁸ m/s or 1.86 × 10⁵ mi/s), and is emitted as a companion particle with each beta particle. See also: Neutrino

Two forms of neutrinos are distinguished in beta decay. In positron beta decay, a proton p in the nucleus transforms into a neutron n in the nucleus, thus reducing the nuclear charge by 1 unit. At the time of this transition, two particles, the positron β⁺ and the neutrino ν, are created and emitted. The emitted β⁺ and ν together carry away the energy E_max of the transition and provide for conservation of energy, momentum, angular momentum, charge, and statistics. Thus positron beta decay is represented by decay (22).

(22)

Negatron beta decay is a closely related process, except that a neutron n changes to a proton p in the nucleus, and a negatron beta particle β⁻ and its characteristic companion particle, the antineutrino, are emitted, as in decay (23).

(23)

The antineutrino is the antiparticle of the neutrino as the β⁺ is the antiparticle of the β⁻. The ν and $\overline{\text{ν}}$ have the same properties of zero charge and essentially zero rest mass, and differ only with respect to the direction of alignment of their intrinsic spin along their direction of motion. In most beta-decay contexts, the term “neutrino” includes both its forms, neutrino and antineutrino. See also: Antimatter

There are, in fact, three classes of neutrinos. The neutrinos emitted in the two types of beta decay (the decays in equations 22 and 23) are called electron neutrinos. In addition, there is a neutrino and antineutrino associated with the mu meson (μ±) and neutrinos and antineutrinos associated with the tau (τ±). These three particles, electron, mu, and tau, together with their neutrinos, ν_e, ν_μ, and ν_τ, and their respective antiparticles, make up a class of particles called leptons. The number of leptons in a decay or reaction is considered to be conserved; this rule is called lepton number conservation. There are separate conservation numbers for e, μ, and τ. There must be the same net number of each type of lepton on each side of a decay or reaction. For example, because there are no leptons on the left sides of the decays in equations 22 and 23, there must be no net leptons on the right sides. Thus the decay in equation 22 has on the right side an antielectron (positron) with lepton number L = −1 and a neutrino with lepton number L = 1. The decay in equation 23 has an electron with L = 1 and an antineutrino with L = −1, and so the lepton number is L = 1 − 1 = 0 leptons on the right side as well. The decays in equations 22 and 23 also conserve nucleon number. A nonzero rest mass of neutrinos opens up the possibility of neutrons oscillating from ν_e to ν_μ or vice versa, as well as from ν_μ to $\overline{\text{ν}{}_{\text{μ}}}$ and other types, as required in grand unified theories. See also: Lepton

Because the neutron rest mass is greater than the proton rest mass, free neutrons can undergo beta decay (in decay in equation 23), but protons must use part of the nuclear binding energy available inside a nucleus to make up the rest mass difference in the decay in equation 22.

The interaction of neutrinos with matter is exceedingly feeble. A neutrino can pass all the way through the Sun with little chance of collision. The thickness of lead required to attenuate neutrinos by the factor ¹/₂ is about 10¹⁸ m (10¹⁵ mi), or 100 light-years of lead. See also: Light-year

Average beta energy

Charged particles, such as beta particles or alpha particles, are easily absorbed in matter, and their kinetic energy is thereby converted into heat. In beta decay, the average energy E_av of the beta particles is far less than the maximum energy E_max of the particular beta-particle spectrum. The detailed shape of beta-particle spectra and hence the exact value of the ratio E_av /E_max varies somewhat with Z, E_max, the degree of forbiddenness of the transition, and the sign of charge of the emitted beta particle. A rough rule of thumb which covers many practical cases is E_av = (0.40 ± 0.05)E_max, with slightly higher values for positron beta-particle spectra than for negatron beta-particle spectra. The remaining decay energy is emitted as kinetic energy of neutrinos and is not recoverable in finite absorbers.

There are other processes that carry off part of the energy of beta decay, including internal bremsstrahlung (gamma rays) and shake-off electrons (atomic electrons). The total probabilities for these additional two processes are on the order of 1% or much less per beta decay, and the probability of their emission decreases rapidly with increasing energy, so they are mainly low-energy (less than about 50 keV) radiations. In internal bremsstrahlung, through an interaction of the beta particle and the emitting nucleus, part of the decay energy is emitted as a gamma ray. In the shake-off process, part of the beta-decay energy is given to one of the atomic electrons. The gamma rays are not absorbed in matter as easily as the beta particles. In addition, if one tries to absorb the beta particles in matter, the beta particles can interact with the atoms and give off external bremsstrahlung (gamma rays). The number of these gamma rays again is a strongly decreasing function of energy, but their emission extends up to the maximum energies of the beta particles. See also: Bremsstrahlung

Fermi theory

By postulating the simultaneous emission of a beta particle and a neutrino, as in reaction (22), Italian-born U.S. physicist Enrico Fermi developed in 1934 a quantum-mechanical theory which satisfactorily gives the shape of beta-particle spectra (Fig. 8), and the relative half-periods of beta-particle emitters for allowed beta decays. The energy distribution of beta particles in allowed transitions is then given by Eq. (24).

(24)

Here N(W) dW = number of beta particles in energy range W to W + dW; W = 1 + E/(m₀ c²) = total energy of beta particle in units of rest energy m₀ c² = 0.51 MeV for an electron (m₀ = electron mass, c = velocity of light); W₀ = 1 + E_max/(m₀ c²) = maximum energy of the beta-particle spectrum; |P|² = squared matrix element for the transition, and is of the order of unity for allowed transitions; τ₀ = time constant ≈ 7000 s; F(Z,W) = complex, dimensionless function involving the nuclear radius, nuclear charge, beta-particle energy, and whether the decay is β⁻ or β⁺.

Physically, this distribution function involves the product of the energy W and momentum (W²− 1)^1/2 of the beta particle times the energy (W₀ − W) and the momentum (W₀ − W)/c of the neutrino. The product of these factors gives a “statistical” distribution for the number of beta particles as a function of energy (Fig. 8). The observed spectra show an excess of low-energy β⁻ and a deficiency of low-energy β⁺ particles. This arises because of the Coulomb attraction and repulsion of the nucleus for β⁻ and β⁺. The statistical spectrum is corrected by the Fermi function, F(Z,W), and the new distribution agrees with experiments (Fig. 8).

Equation (24) essentially matches the energy spectra of allowed beta-particle transitions and therefore furnishes one type of experimental verification of the properties of neutrinos. Its counterpart in terms of the beta-particle momentum spectrum is often used for the analysis of spectra, and is given by Eq. (25).

(25)

The momentum distribution is much more nearly symmetric than its corresponding energy spectrum. Here N(η)dη = number of beta particles in the momentum interval from η to η + dη; η = (W²− 1)^1/2 = momentum of the beta particle in units of m₀ c; and F(Z,η) = F(Z,W) of Eq. (24).

Konopinski-Uhlenbeck theory

After the work of Fermi which explained allowed decay, Polish-born U.S. nuclear physicist Emil John Konopinski and Dutch-born U.S. theoretical physicist George Uhlenbeck in 1941 developed the theory of forbidden beta decay. Allowed decays occur between nuclear states which differ in spin by 0 or 1 unit and which have the same parity. Konopinski and Uhlenbeck developed a theory to describe beta decays where energy is available for decay but the allowed selection rules on spin or parity or both are violated. These beta transitions occur at a slower rate and are called forbidden transitions. In 1949, the theory of forbidden beta decay was confirmed by U.S. nuclear physicist Lawrence Langer and U.S physicist H. Clay Price Jr. The orders of forbiddenness, which retard the rate of decay, are the following: once-forbidden decay when the change in nuclear spin ΔJ is again 0 or 1 as in allowed decay, but a parity change Δπ occurs; once-forbidden unique decay when Δπ changes and ΔJ = 2; n-times forbidden decay when ΔJ = n, Δπ = (−)ⁿ, where Δπ = − indicates a parity change; and n-times forbidden unique decay when ΔJ = n + 1, Δπ = (−)ⁿ [Table 4]. In forbidden decays, the first-order allowed matrix elements of the Fermi theory in Eq. (24) vanish because of the selection rules on angular momentum and spin. Then the much smaller higher-order matrix elements that can be neglected compared to the large allowed matrix elements come into play. See also: Parity (quantum mechanics); Selection rules (physics)

Comparative half-lives, fT

The half-period T of beta decay can be derived from Eq. (24) because the radioactive decay constant λ = 0.693/T is simply the total probability of decay, or N(W) dW integrated over all possible values of the beta-particle energy from W = 1 to W = W₀.

For allowed decays, the matrix elements are not functions of the beta energy and can be factored out of Eq. (24), so Eq. (26) is valid,

(26)

where f is given by Eq. (27),

(27)

and the constants include |P|² of Eq. (24). Equation (26) can be rearranged as Eq. (28).

(28)

For different beta decays, T varies over a range greater than 10¹⁸ and inversely depends on the beta-decay energy in analogy to the Geiger-Nuttall rule for alpha decay. However, Eq. (28) says that the comparative half-life should be a constant. Indeed, it is found experimentally that different classes of beta decay do have very similar fT values. It is generally easier to give the log₁₀ fT for comparison. The groups (Table 4) include, in addition to the forbidden decays, three classes of allowed decays: the favored or superallowed decays of nuclei whose structures are very similar so that the matrix element in the denominator of Eq. (28) is large and log fT is small; normal allowed; and allowed I-forbidden where the total angular momentum selection rule holds, but the individual particle that is undergoing beta decay has a change of 2 units of orbital angular momentum. The matrix elements for each degree of forbiddenness get progressively smaller and so log fT values increase sharply with each degree of forbiddenness. The ranges of these fT values for each degree of forbiddenness are in general so well established that measurements of fT values can be used to establish changes in spins and parities between nuclear states in beta decay.

Kurie plots

For allowed transitions, the transition matrix element |P|² is independent of the momentum η. Then Eq. (25) can be put in the form of Eq. (29).

(29)

Therefore, a straight line results when the quantity (N/η² F)^1/2 is plotted against beta-particle energy, either as W or as E, on a linear scale. Such graphs are called Kurie plots, Fermi plots, or Fermi-Kurie plots. These are especially useful for revealing deviations from the allowed theory and for obtaining the upper energy limit E_max as the extrapolated intercept of N/η² F on the energy axis. Practically all of the results on the shapes of beta-particle spectra are published as Kurie plots, rather than as actual momentum or energy spectra.

When spectral data give a straight line, then N(η) is in agreement with the Fermi momentum distribution, Eq. (25); and the intercept of this straight line, on the energy axis, gives the decay energy E_max. For β⁺ decay the total decay energy is E_max (β⁺) plus 1.022 MeV. In a, theoretical curves are given for various values of the neutrino rest mass, and the data points, which are experimental values, lie on the curve corresponding to zero mass.

In addition to allowed decays, all but one known once-forbidden decays have Kurie plots that are essentially linear in energy (b). The once-forbidden unique decays have a pronounced characteristic energy dependence for their matrix elements, and thus the conventional Kurie plot has a characteristic shape that differs from a straight line (Fig. 9). When the data are corrected by the unique shape factor, given by Konopinski and Uhlenbeck, a linear Kurie plot is again obtained. This unique shape was the key to the discovery of forbidden beta decay by Langer and Price (Fig. 9). The higher-order forbidden spectra each show different strong energy dependencies in their Kurie plots, each characteristic of their degree of forbiddenness.

Fig. 9 Once-forbidden spectrum of ⁹¹Y: conventional Kurie plot and Kurie plot corrected by the unique shape factor, a₁ = W² − 1 + (W₀ − W)², given by Konopinski and Uhlenbeck, which linearizes the data. The quantity W is the total beta-particle energy in units of the electron rest energy; other quantities are defined in text.

Double-beta decay

When the ground state of a nucleus differing by two units of charge from nucleus A has lower energy than A, then it is theoretically possible for A to emit two beta particles, either β⁺β⁺ or β⁻β⁻ as the case may be, and two neutrinos or antineutrinos, and go from Z to Z ± 2. Here, two protons decay into two neutrons, or vice versa. This is a second-order process and so should go much slower than beta decay. There are a number of cases where such decays should occur, but their half-lives are of the order of 10²⁰ years or greater. Such decay processes are obviously very difficult to detect.

The first direct evidence for two-neutrino double-beta-minus decay of selenium, reaction (30),

(30)

was found in 1987, some 40 years after the first attempt to observe this rare process. The trajectories of the electrons from this decay were detected in a time-projection chamber with a half-period of 0.92 ± 0.07 × 10²⁰ y obtained. Previous geochemical and cosmochemical evidence of double-beta decay had been only indirect because it consisted of observations of buildup of the noble gases ⁸²₃₂Kr and ¹³⁰₅₄Xe in the ⁸⁴₃₂Se and ¹³⁰₅₂Te samples, respectively. Now there are 14 known isotopes that are observed to undergo double-beta decay with half-lives that range from 8.5 × 10¹⁸ to 2.2 × 10²⁴ y. See also: Time-projection chamber

Some unifications of the electroweak and strong forces suggest that the electron neutrino and antineutrino are identical (ν_e ≡ $\overline{\text{ν}}$_e), that is, that the neutrino is a Majorana particle. There is no experimental evidence at present for this. If true, this allows double-beta decay to proceed without the emission of neutrinos. In this neutrinoless double-beta decay, the first neutrino absorbs the second one so the decay is decay (31).

(31)

Extensive searches to discover such decays have shown that the half-period of this decay is greater than 0.9 × 10²⁶ y. Lepton number is not conserved in the decay in equation 31. This process is predicted to occur in some grand unified theories. See also: Double-beta decay; Grand unification theories

Electron-capture transitions

Whenever it is energetically allowed by the mass difference between neighboring isobars, a nucleus Z may capture one of its own atomic electrons and transform to the isobar of atomic number Z − 1 (Table 1). Usually the electron-capture (EC) transition involves an electron from the K shell of atomic electrons, because these innermost electrons have the greatest probability density of being in or near the nucleus. See also: Electron capture

In EC transitions, a proton p bound in the parent nucleus absorbs an electron e⁻ and changes to a bound neutron n. The decay energy is carried away by an emitted neutrino ν as in transition (32).

(32)

The residual nucleus may be left either in its ground level or in an excited level from which gamma-ray emission follows. EC transitions compete with all cases of positron beta-particle decay. EC has an energetic advantage over β⁺ decay equivalent to the mass of two electrons, or 1.02 MeV, because in transition in equation 32, one electron mass e⁻ enters on the left and is available, whereas in the decay in equation 22, one electron mass β⁺ must be produced as a product of the positron beta-particle decay. For example, in the radioactive decay of ⁶⁴₂₉Cu, twice as many transitions go by EC to ⁶⁴₂₈Ni as go by positron beta decay to the same decay product. In the heavy, high-Z elements, EC is greatly favored over the competing β⁺ decay, and examples of measurable β⁺ decay are practically unknown for Z greater than 80, although there are a large number of examples of electron capture. As the energy for decay increases beyond 1.02 MeV, the probability of β⁺ decay increases relative to EC and dominates at several megaelectronvolts of energy.

Several examples are known of completely pure EC radioactivity in which there is insufficient nuclear energy to allow any positron beta-particle decay (total decay energy is less than 1.022 MeV). For example, ⁵⁵₂₆Fe emits no positron beta particles, but transforms with a half-period of 2.6 years entirely by EC to the ground level of ⁵⁵₂₅Mn. This radioactivity is detectable through the K-series x-rays which are emitted from ⁵⁵Mn when the atomic electron vacancy, produced by nuclear capture of a K electron, refills from the L shell of atomic electrons. Also, double-electron capture, analogous to double-beta decay, can occur, albeit rarely. Here two atomic electrons are captured and two neutrinos emitted.

Mean life for transitions

A reasonably successful approximate theory of the mean life for gamma-ray decay was developed by Austrian-born U.S. theoretical physicist Victor Weisskopf in 1951, using the single-particle shell model of nuclei (Fig. 10). An E2 transition of about 1 MeV is expected to take place with a mean life, τ_el, or mean delay in the upper level, of about 10⁻¹¹ s. Thus, most gamma-ray transitions are prompt transitions, in which the mean life of the excited level is too short to be measured easily. The mean life τ_mag for magnetic multipoles is of the order of 30 (for A = 20) to 150 (for A = 200) times longer than τ_el for electric multipole transitions (Fig. 10).

Fig. 10 Theoretical values of half-lives for decay of nuclear levels by emission of gamma rays and conversion electrons, for electric multipoles.

At low energies or high Z, or both, the internal conversion process becomes a very important additional mode of decay that markedly shortens the mean lives of the nuclear levels. In addition, in many cases the structure of the nucleus comes into play and alters the observed mean lives considerably compared to those predicted by the Weisskopf theory (Fig. 10). Electric dipole (E1) transitions are generally retarded (longer mean lives) by factors of 10⁶ over the Weisskopf estimates (Fig. 10). On the other hand, Danish physicist Aage Bohr and U.S.-born Danish nuclear physicist Benjamin Roy Mottelson developed a model of collective nuclear motions where E2 transitions are enhanced by factors of 100 or more (shorter τ) over the Weisskopf single-particle estimates, and these predictions are confirmed by experiments. The magnetic dipole (M1) transitions are also often hindered by factors of 100 or more. Measurements of the mean lives for gamma-ray decay provide important tests of nuclear models.

Internal conversion

An alternative type of deexcitation which always competes with gamma-ray emission is known as internal conversion. Instead of the emission of a gamma ray, the nuclear excitation energy can be transferred directly to a bound electron of the same atom. Then the nuclear energy difference is converted to energy of an atomic electron, which is ejected from the atom with a kinetic energy E_i given by Eq. (33).

(33)

Here B_i is the original atomic binding energy of the particular electron, which is ejected, and W is the nuclear transition energy which would otherwise have been emitted as a gamma-ray photon having energy hν = W.

The spectrum of internal conversion electrons is then a series of discrete energies, or “lines,” each corresponding to an individual value of B_i , for the K, L (L₁, L₂, L₃), M, … electrons in each shell and subshell of the atom. Thus conversion electron spectra are much more complex than gamma spectra (Fig. 11). From the spacing of the E_i values in this conversion electron spectrum, it is possible to assign definitely the atomic number Z of the atom in which the nuclear transition W took place. In this way it is known that the conversion electron and the competing gamma-ray emission are sequels and not antecedents of alpha decay, beta decay, and electron-capture transitions.

Fig. 11 Spectra from the decay of 30-s half-life ¹⁸⁶Tl far off stability (15 neutrons less than the lightest stable thallium isotope). Nuclear transition energies are given in kiloelectronvolts. (a) Internal conversion electrons. (b) Gamma rays. The strong 522-keV electron transition has no gamma ray associated with it. The strong 511-keV gamma ray is from annihilation of positrons and is not a nuclear transition.

The total internal conversion coefficient α_T is the ratio of the number of transitions proceeding by internal conversion, N_eT , to the number going by gamma-ray emission, N_γ, for any particular nuclear transformation from an excited level to a lower-lying level, as in Eq. (34).

(34)

The total internal conversion coefficient is a sum of the conversion coefficients for each shell [K, L(L₁ + L₂ + L₃), M(M₁ + · · ·), and so forth], and is given by Eq. (35),

(35)

where α_K = N_eK N_γ, α_L = N_eL /N_γ = α_L1 + α_L2 + α_L3 (α_L1 = N_eL1/N_γ, … ), and N_eK, N_eL (N_eL1, … ), and so forth are the numbers of electrons ejected from the K, L (L₁, L₂, L₃), … shells, respectively. In general, this probability of internal conversion relative to gamma-ray emission increases with increasing atomic number Z, with increasing multipole order 2^l, and with decreasing nuclear deexcitation energy W. In middle-weight elements, for W = 1 MeV, α is of the order of 10⁻² to 10⁻⁴; while for W = 0.2 MeV, α is of the order of 0.1 for electric l = 2 transitions, and 10 or larger for electric l = 5 transitions. When internal conversion electron decay occurs, this process is always followed by the emission of characteristic x-rays of the element and Auger electrons from outer shells when the inner shell vacancy is filled. This emission can include K x-rays, L x-rays and x-rays from higher shells, and K Auger electrons, L Auger electrons, and so forth. See also: Auger effect

Radiationless transitions

There are cases where gamma-ray emission is strictly forbidden and conversion electron emission allowed (Fig. 11). This occurs when both nuclear states have zero spin and the same parity. The conversion electrons are called electric monopole radiations, EO. These transitions occur because of the penetration of the atomic electrons into the nuclear volume where they interact directly with the nucleus. EO radiation can occur in principle whenever two states have the same spin and same parity, but in practice, EO decays are found to be very, very small in these cases. There are some exceptions in well-deformed nuclei and in nuclei which have states with quite different shapes. In these cases, EO decays can totally dominate the electron emission for transitions that have no change in spin and that involve decays between states with large differences in their nuclear shapes (Fig. 11).

The EO decays which arise because of the penetration of the atomic electrons into the nuclear volume are thus sensitive measures of changes in shape between two nuclear states, and have played important roles in establishing vibrations of the nuclear shape and the coexistence of states with quite different deformation in the same nucleus. There also are other circumstances where the penetration of the atomic electron into the nuclear volume gives rise to additional contributions to the conversion-electron decay. Again, these penetration effects probe details of the structure of the nucleus.

Internal pair formation

When the energy between two states in the same nucleus exceeds 1.022 MeV, twice the rest mass energy of an electron, it is possible for the nucleus to give up its excess energy to an electron-positron pair—a pair creation process. This is a third alternative mode to gamma decay and conversion electron decay. This process becomes more important as the gamma-ray energy increases. It is relatively unimportant below 2–3 MeV of decay energy. See also: Electron-positron pair production and annihilation

Isomeric transitions

Measurably delayed radioactive transitions from an excited level of a nucleus are known as isomeric transitions. The measurably long-lived excited level is called an isomeric or metastable level or an isomer of the ground level. What constitutes an isomer is not well-defined. The terminology arose when it was difficult to measure mean lives shorter than 10⁻⁷ s. States with longer mean lives were isomers. Now mean lives down to 10⁻¹³ s can be measured for many transitions in different nuclei, but these are not generally called isomers. The break point is simply not defined.

Figure 12 (below) shows that if the excitation energy is small (say, 0.5 MeV or less) and the angular momentum difference l is large (say, l = 3 or more), then the mean life of an excited level for gamma-ray or conversion-electron emission can be of the order of 1 s up to several years.

Most of the long-lived isomers occur in nuclei which have odd mass number A. Then either the number of protons Z in the nucleus is odd, or the number of neutrons N in the nucleus is odd. The frequency distribution of odd-A isomeric pairs, excited level and ground level, displays so-called islands of isomerism in which the odd-proton or odd-neutron number is less than 50 or less than 82. The distribution is one of several lines of evidence for closed shells of identical nucleons at N or Z = 50 or 82 in nuclei, and it plays an important role in the so-called shell model of nuclei. See also: Nuclear isomerism; Nuclear shell model and magic numbers

Beta-delayed alpha radioactivity

The β⁻ decay of ²¹⁴Bi to ²¹⁴Po leaves the nucleus in such a high-energy excited state that it can emit an alpha particle and go to ²¹⁰Pb as an alternative to gamma-ray decay to lower levels in ²¹⁴Po. This is a two-step process with beta decay the first step. After beta decay, the nucleus is in such a highly excited state that it can emit either an alpha particle or gamma ray.

The β⁻ delayed alpha emission has been found relatively rarely, but in many cases beta (β⁺, EC) delayed alpha emission has been discovered. In proton-rich nuclei far from stability, the conditions are more favorable for beta (β⁺, EC) delayed alpha emission because of the excess of nuclear charge, and a number of such beta-delayed alpha emitters are now known.

Beta-delayed neutron radioactivity

In 1939, shortly after the discovery of nuclear fission, it was proposed that the delayed neutrons observed following fission were in fact beta-delayed neutrons. That is, after the nucleus fissioned, the beta decay of the neutron-rich fission fragments populated high-energy excited states that could promptly undergo dual decay, emitting either a gamma ray or neutron. The processes of beta-delayed two- and three-neutron emission were discovered in 1979 and 1980 in the decay of ¹¹Li, and β⁻2n to 4n decays were subsequently observed in other nuclei.

The process of beta-delayed neutron emission is essential for the control of nuclear fission reactors. When neutrons absorbed by ²³⁵U cause the ²³⁶U nucleus formed to fission, many of the fission products undergo beta-delayed neutron emission. These neutrons are important in producing more fission events. In a nuclear reactor, the rate of neutron-induced fission must be controlled to prevent the fission reactions from running away and destroying the reactor. The rate of fission depends on the number of neutrons available. The numbers of neutrons can be controlled by moving in and out of the reactor control rods, which contain material with very high neutron absorption rates. Because many of the neutrons emitted in fission are delayed by beta-decay half-lives, these half-lives allow time for the control rods to be mechanically inserted and removed to control the rate of fission. See also: Delayed neutron

Beta-delayed proton radioactivities

In addition to proton radioactivity, one can have beta-delayed proton and beta-delayed two-proton radioactivities, which again ultimately result in emission of protons from the nucleus. These latter processes also occur in quite proton-rich nuclei with very high decay energies; however, they are complex two-step decay modes whose fundamental first step is beta decay.

Dozens of nuclei ranging from ⁹₆C to ¹⁸³₈₀Hg have been identified to decay by the two-step mode of beta-delayed proton radioactivity. Typical is the decay of ³³₁₈Ar (Fig. 15), with a half-life of 173 ms; it was produced by the ³²₁₆S + ³₂He → ³³₁₈Ar + 2n reaction. This isotope decays by superallowed and allowed β⁺ decay to a number of levels in its daughter nucleus ³³₁₇Cl, which immediately (in less than 10⁻¹⁷ s) breaks up into ³²₁₆S and a proton. More than 30 proton groups arising from the decay of ³³₁₈Ar are observed, ranging in energy from 1 to approximately 6 MeV and varying in intensity over four orders of magnitude. Although it is normally very difficult to study many β-decay branches in the decay of a particular nuclide—because of the continuous nature of the energy spectrum of the emitted beta particles—it is possible to do so when investigating beta-delayed proton emitters. The observed proton group energies and intensities can be correlated with the levels fed in the preceding beta decay and their transition rates, thereby permitting sensitive tests via beta decay of nuclear wave functions arising from different models of the nucleus. β⁺-delayed two-proton decay has also been discovered, as well as β⁻-delayed deuteron and triton decay.

Fig. 15 Spectrum of β⁺-delayed protons from the decay of ³³₁₈Ar as observed in a counter telescope; the proton laboratory energy is indicated at the top. Proton groups are numbered 1 through 35.

Beta-delayed spontaneous fission

There are also observed beta-decay processes where the excited nucleus following beta decay has a probability of undergoing spontaneous fission rather than gamma-ray decay. This is the same process as in spontaneous or isomeric spontaneous fission. The excitation energy of the nuclear level provides the extra energy to make fission possible. The nucleus splits into two nearly equal fragments plus some neutrons. This process is like isomeric spontaneous fission except that the lifetime of the nuclear level is so short that the level would not normally be called an isomer.

Joseph H. Hamilton