Material Properties

In physics of nuclear decays, a radioactive equilibrium exists when a radioactive nuclide is decaying at the same rate at which it is being produced. The disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The daughter nucleus can either be stable or radioactive. If it is radioactive, then it decays into a daughter nucleus and so on. Thus, each radioactive parent nucleus can initiate a series of decays, with each decay-product having its own characteristic decay constant.

Concentration of daughter nuclei in the radioactive equilibrium depends primarily on proportions of half-lives (or decay constants) of parent and daughter nuclei. Since the production rate and decay rate are equal, the number of atoms present remains constant over time. In any case, a radioactive equilibrium is not established immediately, but it only takes place after a transition period. This period is of the order of few half-lifes of the longest-lived nucleus in the decay chain. In case of radioactive decay chains, a radioactive equilibrium may be established between each member of the decay chain.

As was written, proportionality of half-lives is a key parameter, which determines type of radioactive equilibrium:

• Radioactive equilibrium is not established when a half-life of the parent nucleus is shorter than a half-life of the daughter nucleus. In this case the production rate and decay rate of certain member of decay chain cannot be equal.
• Secular radioactive equilibrium exists when the parent nucleus has an extremely long half-life. This type of equilibrium is particularly important in nature. Over the 4.5 billion years of the Earth’s history, especially uranium 238, uranium 235 and thorium 232 and members of their decay chains have reached radioactive equilibria between the parent nucleus and the various descendants.
• Transient radioactive equilibrium exists when a half-life of the parent nucleus is longer than a half-life of the daughter nucleus. In this case, the parent nuclide and the daughter nuclide decay at essentially the same rate.

Secular Equilibrium

Secular radioactive equilibrium exists when the parent nucleus has an extremely long half-life. Secular equilibrium is typical for natural radioactive series, such as the thorium series or the uranium series. For the uranium series with uranium-238 (with a half-life of 4.47 billion years), where all of the elements in the chain are in secular equilibrium, each of the descendants has built up to an equilibrium amount and all decay at the rate set by the original parent. The only exception is the final stable element (lead-206) on the end of the chain. Its number of atoms is constantly increasing. In any case, a radioactive equilibrium is not established immediately, but it only takes place after a transition period. This period is of the order of few half-lifes of the longest-lived nucleus in the decay chain (²³⁴U for the uranium series; ²³¹Pa for the actinium series). In case of radioactive decay chains, a radioactive equilibrium may be established between each member of the decay chain.

As can be seen,  the secular equilibrium is particularly important in nature. Over the 4.5 billion years of the Earth’s history, especially uranium 238, uranium 235 and thorium 232 and members of their decay chains have reached radioactive equilibria between the parent nucleus and the various descendants. The half-lives of all their descendants are all extremely variable, and it is difficult to represent a range of timescales going from individual seconds to millions of years. Since daughter radioisotopes have different half-lives then secular equilibrium is reached after some time. In the long decay chain for a naturally radioactive element, such as uranium-238, where all of the elements in the chain are in secular equilibrium, each of the descendants has built up to an equilibrium amount and all decay at the rate set by the original parent. If and when equilibrium is achieved, each successive daughter isotope is present in direct proportion to its half-life (or to its decay constant).

Since its activity is inversely proportional to its half-life, each nuclide in the decay chain finally contributes as many individual transformations as the head of the chain. Radioactive cascade in the secular equilibrium significantly influences radioactivity (disintegrations per second) of natural samples and natural materials. All the descendants are present, at least transiently, in any natural sample, whether metal, compound, or mineral. For example, pure uranium-238 is weakly radioactive (proportional to its long half-life), but a uranium ore is about 13 times more radioactive (approximately 13 descendants in the decay chain) than the pure uranium-238 metal because of its daughter isotopes (e.g. radon, radium etc.) it contains. Not only are unstable radium isotopes significant radioactivity emitters, but as the next stage in the decay chain they also generate radon, a heavy, inert, naturally occurring radioactive gas. Moreover the decay heat of uranium and its decay products (e.g. radon, radium etc.) contributes to heating of Earth’s core. Together with thorium and potassium-40 in the Earth’s mantle is thought that these elements are the main source of heat that keeps the Earth’s core liquid.

The secular equilibrium can occasionally be disrupted when one of the intermediary nuclei leaves the sample (e.g. radon released from soil) where its ancestors are confined. These local disruptions are important to consider in the use of dating techniques. Disruption of the secular equilibrium also influences radioactivity of fresh nuclear fuel. Most of PWRs use the uranium fuel, which is in the form of uranium dioxide. But this uranium dioxide must be chemically purified and most of its decay chain is not present in the fresh nuclear fuel.

Radioactive Equilibrium of Uranium-234

Isotope of uranium-234 is a member of the uranium series. This isotope has the half-life of only 2.46×10⁵ years and therefore it do not belong to primordial nuclides (unlike ²³⁵U and ²³⁸U). On the other hand, this isotope is still present in the Earth’s crust, but this is due to the fact ²³⁴U is an indirect decay product of ²³⁸U. ²³⁸U decays via alpha decay into ²³⁴U. ²³⁴U decays via alpha decay into 230Th, except very small fraction (on the order of ppm) of nuclei which decays by spontaneous fission.

In a natural sample of uranium, these nuclei are present in the unalterable proportions of the radioactive equilibrium of the ²³⁸U filiation at a ratio of one atom of ²³⁴U for about 18 500 nuclei of ²³⁸U. As a result of this equilibrium these two isotopes (²³⁸U and ²³⁴U) contribute equally to the radioactivity of natural uranium.

Transient Radioactive Equilibrium

The transient radioactive equilibrium exists when a half-life of the parent nucleus is longer than a half-life of the daughter nucleus, but the concentration of parent nuclei significantly decreases in time. In this case, the parent nuclide and the daughter nuclide may decay at essentially the same rate, but both concentrations of nuclides decreases as the concentration of parent nuclei decreases. Contrary to secular equilibrium, the half-life of the daughter nuclei is not negligible compared to parent’s half-life.

An example of this type of compound decay process is a Technetium-99m generator producing technetium-99m for nuclear medicine diagnostic procedures from molybdenum-99. Technetium-99m’s short half-life of 6 hours makes storage impossible and would make transport very expensive. Therefere, for medical purposes, molybdenum-99 is used to produce technetium-99m. These two isotopes are in the transient equilibrium. The decay constant for molybdenum-99 is considerably smaller than the decay constant for technetium-99m. Although the decay constant for molybdenum-99 is smaller, the actual rate of decay is initially larger than that of molybdenum-99 because of the great difference in their initial concentrations. As the concentration of the daughter increases, the rate of decay of the daughter will approach and eventually match the decay rate of the parent. When this occurs, they are said to be in the transient equilibrium. In case of  Technetium-99m generator, transient equilibrium occurs after about four half-lives. Today, technetium-99m is the most utilized element in nuclear medicine and is employed in a wide variety of nuclear medicine imaging studies.

Also the transient equilibrium can occasionally be disrupted when one of the intermediary nuclei leaves the sample where its ancestors are confined.

Transient Radioactive Equilibrium with Source – Example

A special example of radioactive equilibrium are concentrations of iodine-135 and xenon-135 in a nuclear reactor, but in this case, the xenon burnup must be taken into account. Note that, in this special case, the half-life of the parent nucleus is shorter than the half-life of the daughter nucleus. The production and removal of xenon can be characterized by the following differential equations:

When the rate of production of iodine equals the rate of removal of iodine, equilibrium exists. This equilibrium also known as “xenon 135 reservoir”, since all of this iodine must undergo a decay to xenon. In equilibrium, the iodine concentration remains constant and is designated N_I(eq). The following equation for the iodine equilibrium concentration can be determined from the preceding equation by setting the dN_I/dt =0. Since the equilibrium iodine concentration is proportional to the fission reaction rate, it is also proportional to reactor power level.

When the rate of production of xenon 135 equals the rate of removal, equilibrium exists also for xenon. The xenon concentration remains constant and is designated N_Xe(eq). The following equation (1) for the xenon equilibrium concentration can be determined from the preceding equation by setting the dN_Xe/dt =0. For xenon 135 to be in equilibrium, iodine 135 must also be in equilibrium. Substituting the expression for equilibrium iodine 135 concentration into the equation for equilibrium xenon (1) results in the following (2).

From this equation it can be seen that the equilibrium value for xenon 135 increases as power increases, because the numerator is proportional to the fission reaction rate. But the thermal flux is also in the denominator. Therefore, as the thermal flux exceeds some value, the xenon burnup begins to dominate, and at approximately 10¹⁵ neutrons.cm^-2.s^-1, the xenon-135 concentration approaches a limiting value. The equilibrium iodine 135 and xenon 135 concentrations as a function of neutron flux are illustrated in the following figure.

Bateman Equations

In physics, the Bateman equations are a set of first-order differential equations, which describe the time evolution of nuclide concentrations undergoing serial or linear decay chain. The model was formulated by Ernest Rutherford in 1905 and the analytical solution for the case of radioactive decay in a linear chain was provided by Harry Bateman in 1910. This model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems.

For example, ORIGEN is a computer code system for calculating the buildup, decay, and processing of radioactive materials. ORIGEN uses a matrix exponential method to solve a large system of coupled, linear, first-order ordinary differential equations (similar to the Bateman equations) with constant coefficients.

The Bateman equations for radioactive decay case of n – nuclide series in linear chain describing nuclide concentrations are as follows:

Bateman Equations for Nuclear Transmutation

As was written, this model can be also used in nuclear depletion codes to solve nuclear transmutation and decay problems as well. In case of transmutation the decay constants that govern Bateman equations for a decay case are substituted by transmutation constants. By the transmutation constant λ_i,j we understand probability of the ith nuclide production per time unit from the jth nuclide destruction, as a result of nuclear interaction with the whole spectrum of interacting particles or due to the natural nuclear decay.

These equations are usually used for the exact evolution of isotopic changes in the nuclear fuel during fuel depletion. Fuel depletion is usually modelled mathematically as a set of differential equations known as evolution equations.

Special Reference: Jerzy Cetnar, General solution of Bateman equations for nuclear transmutations. Annals of Nuclear Energy 33 (2006). January 2006.

See also:

Radioactive Decay

In physics of nuclear decays, a radioactive equilibrium exists when a radioactive nuclide is decaying at the same rate at which it is being produced. The disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The daughter nucleus can either be stable or radioactive. If it is radioactive, then it decays into a daughter nucleus and so on. Thus, each radioactive parent nucleus can initiate a series of decays, with each decay-product having its own characteristic decay constant.

Concentration of daughter nuclei in the radioactive equilibrium depends primarily on proportions of half-lives (or decay constants) of parent and daughter nuclei. Since the production rate and decay rate are equal, the number of atoms present remains constant over time. In any case, a radioactive equilibrium is not established immediately, but it only takes place after a transition period. This period is of the order of few half-lifes of the longest-lived nucleus in the decay chain. In case of radioactive decay chains, a radioactive equilibrium may be established between each member of the decay chain.

As was written, proportionality of half-lives is a key parameter, which determines type of radioactive equilibrium:

• Radioactive equilibrium is not established when a half-life of the parent nucleus is shorter than a half-life of the daughter nucleus. In this case the production rate and decay rate of certain member of decay chain cannot be equal.
• Secular radioactive equilibrium exists when the parent nucleus has an extremely long half-life. This type of equilibrium is particularly important in nature. Over the 4.5 billion years of the Earth’s history, especially uranium 238, uranium 235 and thorium 232 and members of their decay chains have reached radioactive equilibria between the parent nucleus and the various descendants.
• Transient radioactive equilibrium exists when a half-life of the parent nucleus is longer than a half-life of the daughter nucleus. In this case, the parent nuclide and the daughter nuclide decay at essentially the same rate.

Secular Equilibrium

Secular radioactive equilibrium exists when the parent nucleus has an extremely long half-life. Secular equilibrium is typical for natural radioactive series, such as the thorium series or the uranium series. For the uranium series with uranium-238 (with a half-life of 4.47 billion years), where all of the elements in the chain are in secular equilibrium, each of the descendants has built up to an equilibrium amount and all decay at the rate set by the original parent. The only exception is the final stable element (lead-206) on the end of the chain. Its number of atoms is constantly increasing. In any case, a radioactive equilibrium is not established immediately, but it only takes place after a transition period. This period is of the order of few half-lifes of the longest-lived nucleus in the decay chain (²³⁴U for the uranium series; ²³¹Pa for the actinium series). In case of radioactive decay chains, a radioactive equilibrium may be established between each member of the decay chain.

As can be seen,  the secular equilibrium is particularly important in nature. Over the 4.5 billion years of the Earth’s history, especially uranium 238, uranium 235 and thorium 232 and members of their decay chains have reached radioactive equilibria between the parent nucleus and the various descendants. The half-lives of all their descendants are all extremely variable, and it is difficult to represent a range of timescales going from individual seconds to millions of years. Since daughter radioisotopes have different half-lives then secular equilibrium is reached after some time. In the long decay chain for a naturally radioactive element, such as uranium-238, where all of the elements in the chain are in secular equilibrium, each of the descendants has built up to an equilibrium amount and all decay at the rate set by the original parent. If and when equilibrium is achieved, each successive daughter isotope is present in direct proportion to its half-life (or to its decay constant).

Since its activity is inversely proportional to its half-life, each nuclide in the decay chain finally contributes as many individual transformations as the head of the chain. Radioactive cascade in the secular equilibrium significantly influences radioactivity (disintegrations per second) of natural samples and natural materials. All the descendants are present, at least transiently, in any natural sample, whether metal, compound, or mineral. For example, pure uranium-238 is weakly radioactive (proportional to its long half-life), but a uranium ore is about 13 times more radioactive (approximately 13 descendants in the decay chain) than the pure uranium-238 metal because of its daughter isotopes (e.g. radon, radium etc.) it contains. Not only are unstable radium isotopes significant radioactivity emitters, but as the next stage in the decay chain they also generate radon, a heavy, inert, naturally occurring radioactive gas. Moreover the decay heat of uranium and its decay products (e.g. radon, radium etc.) contributes to heating of Earth’s core. Together with thorium and potassium-40 in the Earth’s mantle is thought that these elements are the main source of heat that keeps the Earth’s core liquid.

The secular equilibrium can occasionally be disrupted when one of the intermediary nuclei leaves the sample (e.g. radon released from soil) where its ancestors are confined. These local disruptions are important to consider in the use of dating techniques. Disruption of the secular equilibrium also influences radioactivity of fresh nuclear fuel. Most of PWRs use the uranium fuel, which is in the form of uranium dioxide. But this uranium dioxide must be chemically purified and most of its decay chain is not present in the fresh nuclear fuel.

Radioactive Equilibrium of Uranium-234

Isotope of uranium-234 is a member of the uranium series. This isotope has the half-life of only 2.46×10⁵ years and therefore it do not belong to primordial nuclides (unlike ²³⁵U and ²³⁸U). On the other hand, this isotope is still present in the Earth’s crust, but this is due to the fact ²³⁴U is an indirect decay product of ²³⁸U. ²³⁸U decays via alpha decay into ²³⁴U. ²³⁴U decays via alpha decay into 230Th, except very small fraction (on the order of ppm) of nuclei which decays by spontaneous fission.

In a natural sample of uranium, these nuclei are present in the unalterable proportions of the radioactive equilibrium of the ²³⁸U filiation at a ratio of one atom of ²³⁴U for about 18 500 nuclei of ²³⁸U. As a result of this equilibrium these two isotopes (²³⁸U and ²³⁴U) contribute equally to the radioactivity of natural uranium.

See also:

Radioactive Equilibrium

Positive Beta Decay – Positron Decay

In positron decay, a proton-rich nucleus emits a positron (positrons are antiparticles of electrons, and have the same mass as electrons but positive electric charge), and thereby reduces the nuclear charge by one unit. In this case, the process can be represented by: An annihilation occurs, when a low-energy positron collides with a low-energy electron.

Electron capture, which is also typical for proton-rich nuclei, competes with positive beta decay, which is more common for lighter nuclei. Electron capture is the primary decay mode for isotopes with insufficient energy (Q < 2 x 511 keV) difference between the isotope and its prospective daughter for the nuclide to decay by emitting a positron. On the other hand, electron capture is always an alternative decay mode for radioactive isotopes that do have sufficient energy to decay by positron emission.

Positron Interactions

The coulomb forces that constitute the major mechanism of energy loss for electrons are present for either positive or negative charge on the particle and constitute the major mechanism of energy loss also for positrons. Whatever the interaction involves a repulsive or attractive force between the incident particle and orbital electron (or atomic nucleus), the impulse and energy transfer for particles of equal mass are about the same. Therefore positrons interact similarly with matter when they are energetic. The track of positrons in material is similar to the track of electrons. Even their specific energy loss and range are about the samefor equal initial energies.

At the end of their path, positrons differ significantly from electrons. When a positron (antimatter particle) comes to rest, it interacts with an electron (matter particle), resulting in the annihilation of the both particles and the complete conversion of their rest mass to pure energy (according to the E=mc² formula) in the form of two oppositely directed 0.511 MeV gamma rays (photons).

See also: Positron Interaction

See also: Shielding of Positrons

See also:

Beta Decay

Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β^– decay) can be characterized as:

Beta decay is governed by the weak interaction. During beta decay one of two down quarks changes into an up quark by emitting a W^– boson (carries away a negative charge). The W^– boson then decays into a beta particle and an antineutrino. This process is equivalent to the process, in which a neutrino interacts with a neutron.

As can be seen from the figure, the weak interaction changes one flavor of quark into another. Note that, the Standard Model counts six flavours of quarks and six flavours of leptons. The weak interaction is the only process in which a quark can change to another quark, or a lepton to another lepton (flavor change). Neither the strong interaction nor electromagnetic permit flavour changing. This fact is crucial in many decays of nuclear particles. In the fusion process, which, for example, powers the Sun, two protons interact via the weak force to form a deuterium nucleus, which reacts further to generate helium. Without the weak interaction, the diproton would decay back into two hydrogen-1 unbound protons through proton emission. As a result, the sun would not burn without it since the weak interaction causes the transmutation p -> n.

In contrast to alpha decay, neither the beta particle nor its associated 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. For either electron or positron emission to be energetically possible, the energy release (see below) or Q value must be positive.

See also:

Beta Decay

Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β^– decay) can be characterized as:

If a nucleus emits a beta particle, it loses an electron (or positron). In this case, the mass number of daughter nucleus remains the same, but daughter nucleus will form different element.

Example of Beta Decay

Free Neutron

A free neutron is a neutron that is not bounded in a nucleus. The free neutron is, unlike a bounded neutron, subject to radioactive beta decay.

It decays into a proton, an electron, and an antineutrino (the antimatter counterpart of the neutrino, a particle with no charge and little or no mass). A free neutron will decay with a half-life of about 611 seconds (10.3 minutes). This decay involves the weak interaction and is associated with a quark transformation (a down quark is converted to an up quark). The decay of the neutron is a good example of the observations which led to the discovery of the neutrino. Because it decays in this manner, the neutron does not exist in nature in its free state, except among other highly energetic particles in cosmic rays. Since free neutrons are electrically neutral, they pass through the electrical fields within atoms without any interaction and they are interacting with matter almost exclusively through relatively rare collisions with atomic nuclei.

It must be noted, a proton cannot  decay in free space. That is, a proton can undergo decay, but only inside a nucleus. It is due to the conservation of baryon number that has been established as a basic principle of physics. This principle provides basis for the stability of the proton. Since the proton is the lightest particle among all baryons, the hypothetical products of its decay would have to be non-baryons.

See also: Stability of Proton

Discovery of Neutrino

The study of beta decay provided the first physical evidence for the existence of the neutrino. The discovery of the neutrino is based on the law of conservation of energy during the process of beta decay.

See also: Discovery of Neutrino

Beta Decay of Tritium

Tritium is a radioactive isotope, bur it emits a very weak form of radiation, a low-energy beta particle that is similar to an electron. It is a pure beta emitter (i.e. beta emitter without an accompanying gamma radiation). The electron’s kinetic energy varies, with an average of 5.7 keV, while the remaining energy is carried off by the nearly undetectable electron antineutrino. Such a very low energy of electron causes, that the electron cannot penetrate the skin or even does not travel very far in air. Beta particles from tritium can penetrate only about 6.0 mm of air.

Tritium decays via negative beta decay into helium-3 with half-life of 12.3 years.

See also:

Beta Decay

Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β^– decay) can be characterized as:

Conservation Laws in Beta Decay

In analyzing nuclear reactions, we apply the many conservation laws. Nuclear reactions are subject to classical conservation laws for charge, momentum, angular momentum, and energy (including rest energies).  Additional conservation laws, not anticipated by classical physics, are:

• Law of Conservation of Lepton Number
• Law of Conservation of Baryon Number
• Law of Conservation of Electric Charge

Certain of these laws are obeyed under all circumstances, others are not. We have accepted conservation of energy and momentum. In all the examples given we assume that the number of protons and the number of neutrons is separately conserved. We shall find circumstances and conditions in which  this rule is not true. Where we are considering non-relativistic nuclear reactions, it is essentially true. However, where we are considering relativistic nuclear energies or those involving the weak interactions, we shall find that these principles must be extended.

Some conservation principles have arisen from theoretical considerations, others are just empirical relationships. Notwithstanding, any reaction not expressly forbidden by the conservation laws will generally occur, if perhaps at a slow rate. This expectation is based on quantum mechanics. Unless the barrier between the initial and final states is infinitely high, there is always a non-zero probability that a system will make the transition between them.

For purposes of analyzing non-relativistic reactions, it is sufficient to note four of the fundamental laws governing these reactions.

1. Conservation of nucleons. The total number of nucleons before and after a reaction are the same.
2. Conservation of charge. The sum of the charges on all the particles before and after a reaction are the same
3. Conservation of momentum. The total momentum of the interacting particles before and after a reaction are the same.
4. Conservation of energy. Energy, including rest mass energy, is conserved in nuclear reactions.

Reference: Lamarsh, John R. Introduction to Nuclear engineering 2nd Edition

Energy Spectrum of Beta Decay

In both alpha and gamma decay, the resulting particle (alpha particle or photon) has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. For example, in case of alpha decay, when a parent nucleus breaks down spontaneously to yield a daughter nucleus and an alpha particle, the sum of the mass of the two products does not quite equal the mass of the original nucleus (see Mass Defect). As a result of the law of conservation of energy, this difference appears in the form of the kinetic energy of the alpha particle. Since the same particles appear as products at every breakdown of a particular parent nucleus, the mass-difference should always be the same, and the thermodynamics/what-is-energy-physics/what-is-kinetic-energy/”>kinetic energy of the alpha particles should also always be the same. In other words, the beam of alpha particles should be monoenergetic.

It was expected that the same considerations would hold for a parent nucleus breaking down to a daughter nucleus and a beta particle. Because only the electron and the recoiling daughter nucleus were observed beta decay, the process was initially assumed to be a two body process, very much like alpha decay. It would seem reasonable to suppose that the beta particles would form also a monoenergetic beam.

To demonstrate energetics of two-body beta decay, consider the beta decay in which an electron is emitted and the parent nucleus is at rest, conservation of energy requires:

Since the electron is much lighter particle it was expected that it will carry away most of the released energy, which would have a unique value T_e-.

But the reality was different. The 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, however. Moreover virtually all of the emitted beta particles have energies below that predicted by energy conservation in two-body decays. The electrons emitted in beta decay have a continuous rather than a discrete spectrumappeared to contradict conservation of energy, under the then-current assumption that beta decay is the simple emission of an electron from a nucleus. When this was first observed, it appeared to threaten the survival of one of the most important conservation laws in physics!

To account for this energy release, Pauli proposed (in 1931) that there was emitted in the decay process another particle, later named by Fermi the neutrino. It was clear, this particle must be highly penetrating and that the conservation of electric charge requires the neutrino to be electrically neutral. This would explain why it was so hard to detect this particle. The term neutrino comes from Italian meaning “little neutral one” and neutrinos are denoted by the Greek letter ν (nu). In the process of beta decay the neutrino carries the missing energy and also in this process the law of conservation of energy remains valid.

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See also:

Beta Decay

Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β^– decay) can be characterized as:

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Beta Decay – Q-value

In nuclear and particle physics the energetics of nuclear reactions is determined by the Q-value of that reaction. The Q-value of the reaction is defined as the difference between the sum of the rest masses of the initial reactants and the sum of the masses of the final products, in energy units (usually in MeV).

Consider a typical reaction, in which the projectile a and the target A gives place to two products, B and b. This can also be expressed in the notation that we used so far, a + A → B + b, or even in a more compact notation, A(a,b)B.

See also: E=mc2

The Q-value of this reaction is given by:

Q = [ma + mA – (mb + mB)]c²

When describing beta decay (reaction without projectile), the disintegrating nucleus is usually referred to as the parent nucleus and the nucleus remaining after the event as the daughter nucleus. The emission of a beta particle, either an electron, β^–, or a positron, β⁺, changes the atomic number of the nucleus without affecting its mass number. The total rest mass of the daughter nucleus and of the nuclear radiation released in a beta disintegration, m_Daughter + m_Radiation, is always less than that of the parent nucleus, m_parent.

The mass-energy difference,

Q = [m_parent – (m_Daughter + m_Radiation)]c²

appears as the disintegration energy, liberated in the process. For example, the Q-value of typical beta decay is:

In the process of beta decay, either an electron or a positron is emitted. This emission is accompanied by the emission of antineutrino (β- decay) or neutrino (β+ decay), which shares energy and momentum of the decay. The beta emission has a characteristic spectrum. This characteristic spectrum is caused by the fact that either a neutrino or an antineutrino is emitted with emission of beta particle. The shape of this energy curve depends on what fraction of the reaction energy (Q value-the amount of energy released by the reaction) is carried by the massive particle. Beta particles can therefore be emitted with any kinetic energy ranging from 0 to Q. After an alpha or beta decay, the daughter nucleus is often left in an excited energy state. In order to stabilize itself, it subsequently emits high-energy photons, γ-rays.

See also:

Beta Decay

Graf 1

Graf 2

Alpha decay (or α-decay and also alpha radioactivity) represents the disintegration of a parent nucleus to a daughter through the emission of the nucleus of a helium atom. This transition can be characterized as:

As can be seen from the figure, alpha particle is emitted in alpha decay. Alpha particles are energetic nuclei of helium. Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Alpha particles are relatively large and carry a double positive charge. They are not very penetrating and a piece of paper can stop them. They travel only a few centimeters but deposit all their energies along their short paths.

In practice, this mode of decay has only been observed in nuclides considerably heavier than nickel, with the lightest known alpha emitters being the lightest isotopes (mass numbers 106–110) of tellurium (element 52). In nuclear reactors alpha decay occurs for example in the fuel (alpha decay of heavy nuclei). Alpha particles are commonly emitted by all of the heavy radioactive nuclei occuring in the nature (uranium, thorium or radium), as well as the transuranic elements (neptunium, plutonium or americium).

Theory of Alpha Decay – Quantum Tunneling

Among the variety of channels in which a nucleus decays, alpha decay has been one of the most studied. The alpha decay channel in heavy and super heavy nuclei has provided information on the fundamental properties of nuclei far from stability, such as their ground state energies and the structure of their nuclear levels.

Alpha decay is a quantum tunneling process. In order to be emitted, the alpha particle must penetrate a potential barrier. This is similar to cluster decay, in which an atomic nucleus emits a small “cluster” of neutrons and protons (e.g. ¹²C).

The height of the Coulomb barrier for nuclei of A « 200 is about 20-25 MeV. The alpha particles emitted in nuclear decay have typical energies of about 5 MeV. On the one hand an incoming 5 MeV alpha particle is scattered from a heavy nucleus and it cannot penetrate the Coulomb barrier and get sufficiently close to the nucleus to interact via the strong force. On the other hand, a 5 MeV alpha particle bound in a nuclear potential well is able to tunnel that same Coulomb barrier.

By 1928, George Gamow (and independently by Ronald Gurney and Edward Condon) had solved the theory of alpha decay via quantum tunneling. They assumed that the alpha particle and the daughter nucleus exist within the parent nucleus prior to its dissociation, namely the decay of quasistationary states (QS).  A quasistationary state is defined as a long-lived state that eventually decays. Initially, the alpha cluster oscillates in the potential of the daughter nucleus, with the Coulomb potential preventing their separation. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of “tunneling” through the barrier and appearing on the other side to escape the nucleus. Using the tunneling mechanism, Gamow, Condon and Gurney calculated the penetrability of the tunneling α particle through the Coulomb barrier, finding the lifetimes of some α emitting nuclei. The main success of this model was the reproduction of the semi-empirical Geiger-Nuttall law that expresses the lifetimes of the α emitters in terms of the energies of the released α particles. It must be noted, that other common forms of decay (e.g. beta decay) are governed by the interplay between both the nuclear force and the electromagnetic force.

Special Reference: W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467.

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Beta decay or β decay represents the disintegration of a parent nucleus to a daughter through the emission of the beta particle. This transition (β^– decay) can be characterized as:

If a nucleus emits a beta particle, it loses an electron (or positron). In this case, the mass number of daughter nucleus remains the same, but daughter nucleus will form different element.

Beta particles are high-energy, high-speed electrons or positrons emitted by certain types of radioactive nuclei such as potassium-40. The beta particles have greater range of penetration than alpha particles, but still much less than gamma rays. The beta particles emitted are a form of ionizing radiation also known as beta rays. There are the following forms of beta decay:

• Negative Beta Decay – Electron Decay. In electron decay, a neutron-rich nucleus emits a high-energy electron (β^– particle). The electrons are negatively charged almost massless particles Due to the law of conservation of electric charge, the nuclear charge must increase by one unit. In this case, the process can be represented by: 
• Positive Beta Decay – Positron Decay. In positron decay, a proton-rich nucleus emits a positron (positrons are antiparticles of electrons, and have the same mass as electrons but positive electric charge), and thereby reduces the nuclear charge by one unit. In this case, the process can be represented by: An annihilation occurs, when a low-energy positron collides with a low-energy electron.
• Inverse Beta Decay – Electron Capture. Electron capture, known also as inverse beta decay is sometimes included as a type of beta decay, because the basic nuclear process, mediated by the weak interaction, is the same. In this process, a proton-rich nucleus can also reduce its nuclear charge by one unit by absorbing an atomic electron. 

Theory of Beta Decay – Weak Interaction

Beta decay is governed by the weak interaction. During beta decay one of two down quarks changes into an up quark by emitting a W^– boson (carries away a negative charge). The W^– boson then decays into a beta particle and an antineutrino. This process is equivalent to the process, in which a neutrino interacts with a neutron.

As can be seen from the figure, the weak interaction changes one flavor of quark into another. Note that, the Standard Model counts six flavours of quarks and six flavours of leptons. The weak interaction is the only process in which a quark can change to another quark, or a lepton to another lepton (flavor change). Neither the strong interaction nor electromagnetic permit flavour changing. This fact is crucial in many decays of nuclear particles. In the fusion process, which, for example, powers the Sun, two protons interact via the weak force to form a deuterium nucleus, which reacts further to generate helium. Without the weak interaction, the diproton would decay back into two hydrogen-1 unbound protons through proton emission. As a result, the sun would not burn without it since the weak interaction causes the transmutation p -> n.

In contrast to alpha decay, neither the beta particle nor its associated 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. For either electron or positron emission to be energetically possible, the energy release (see below) or Q value must be positive.

Energy Spectrum of Beta Decay

In both alpha and gamma decay, the resulting particle (alpha particle or photon) has a narrow energy distribution, since the particle carries the energy from the difference between the initial and final nuclear states. For example, in case of alpha decay, when a parent nucleus breaks down spontaneously to yield a daughter nucleus and an alpha particle, the sum of the mass of the two products does not quite equal the mass of the original nucleus (see Mass Defect). As a result of the law of conservation of energy, this difference appears in the form of the kinetic energy of the alpha particle. Since the same particles appear as products at every breakdown of a particular parent nucleus, the mass-difference should always be the same, and the thermodynamics/what-is-energy-physics/what-is-kinetic-energy/”>kinetic energy of the alpha particles should also always be the same. In other words, the beam of alpha particles should be monoenergetic.

It was expected that the same considerations would hold for a parent nucleus breaking down to a daughter nucleus and a beta particle. Because only the electron and the recoiling daughter nucleus were observed beta decay, the process was initially assumed to be a two body process, very much like alpha decay. It would seem reasonable to suppose that the beta particles would form also a monoenergetic beam.

To demonstrate energetics of two-body beta decay, consider the beta decay in which an electron is emitted and the parent nucleus is at rest, conservation of energy requires:

Since the electron is much lighter particle it was expected that it will carry away most of the released energy, which would have a unique value T_e-.

But the reality was different. The 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, however. Moreover virtually all of the emitted beta particles have energies below that predicted by energy conservation in two-body decays. The electrons emitted in beta decay have a continuous rather than a discrete spectrumappeared to contradict conservation of energy, under the then-current assumption that beta decay is the simple emission of an electron from a nucleus. When this was first observed, it appeared to threaten the survival of one of the most important conservation laws in physics!

To account for this energy release, Pauli proposed (in 1931) that there was emitted in the decay process another particle, later named by Fermi the neutrino. It was clear, this particle must be highly penetrating and that the conservation of electric charge requires the neutrino to be electrically neutral. This would explain why it was so hard to detect this particle. The term neutrino comes from Italian meaning “little neutral one” and neutrinos are denoted by the Greek letter ν (nu). In the process of beta decay the neutrino carries the missing energy and also in this process the law of conservation of energy remains valid.

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See also:

Radioactive Decay

Gamma decay or γ decay represents the disintegration of a parent nucleus to a daughter through the emission of gamma rays (high energy photons). This transition (γ decay) can be characterized as:

As can be seen, if a nucleus emits a gamma ray, atomic and mass numbers of daughter nucleus remain the same, but daughter nucleus will form different energy state of the same element. Note that, nuclides with equal proton number and equal mass number (thus making them by definition the same isotope), but in a different energy state are known as nuclear isomers. We usually indicate isomers with a superscript m, thus: ^241mAm or ^110mAg.

In most practical laboratory sources, the excited nuclear states are created in the decay of a parent radionuclide, therefore a gamma decay typically accompanies other forms of decay, such as alpha or beta decay. Typically after a beta decay (isobaric transition), nuclei usually contain too much energy to be in its final stable or daughter state.

Gamma rays are high-energy photons with very short wavelengths and thus very high frequency. Gamma rays from radioactive decay are in the energy range from a few keV to ~8 MeV, corresponding to the typical energy levels in nuclei with reasonably long lifetimes. As was written, they are produced by the decay of nuclei as they transition from a high energy state to a lower state. Since the gamma rays are in substance only a very high-energy photons, they are very penetrating matter and are thus biologically hazardous. Gamma rays can travel thousands of feet in air and can easily pass through the human body.

In contrast to alpha and beta radioactivity, gamma radioactivity is governed by an electromagnetic interaction rather than a weak or strong interaction. As in atomic transitions, the photon carries away at least one unit of angular momentum (the photon, being described by the vector electromagnetic field, has spin angular momentum of ħ), and the process conserves parity.

Prompt Gamma Decay

As was written, gamma decay may follow nuclear reactions such as neutron capture, nuclear fusion, or nuclear fission. Most of nuclear reactions produce extremely unstable nuclei that decay as soon as they are formed in nuclear reactions (half-life less than 10^-11s) and are not generally classified as nuclear isomers. Moreover, these nuclei usually produce a cascade of gamma rays and the gamma-ray cascade concludes when all excess energy of the excited nucleus is released.

For example, following a nuclear fission prompt gamma rays are emitted from fission fragments. Most of prompt gamma rays are emitted after prompt neutrons. The fission reaction releases approximately ~7 MeV in prompt gamma rays and additional ~7 MeV (for 235U) in delayed gamma rays. This is a significant portion of energy (~7 % of fission energy released) and it must be considered in many fields of reactor design

Isomeric Transition

In certain cases, the excited nuclear state that follows the emission of a beta particle or other type of excitation, are able to stay in metastable state for a long time (hours, days and sometimes much longer) before undergoing gamma decay, in which they emit a gamma ray. These long-lived excited nuclei are known as isomeric states (or isomers) and their decays are termed isomeric transitions. The process of isomeric transition is therefore similar to any gamma emission, but differs in that it involves the intermediate metastable excited state(s) of the nuclei.

Metastable nuclei are often characterized by high nuclear spin, requiring a change in spin of several units or more with gamma decay, instead of a single unit transition that occurs in only 10⁻¹² seconds. The rate of gamma decay is also slowed when the energy of excitation of the nucleus is small. An example is the decay of the isomer or metastable state of protactinium:

Extremely unstable nuclei that decay as soon as they are formed in nuclear reactions (half-life less than 10^-11s) are not generally classified as nuclear isomers. Isomeric transitions must occur by higher order multipole transitions (in contrast to gamma emission that occurs by dipole radiation) that occur on a longer time-scale.

See also:

Radioactive Decay