Nick Connor

What is Radiation

Most general definition is that radiation is energy that comes from a source and travels through some material or through space. Light, heat and sound are types of radiation. This is very general definition, the kind of radiation discussed in this article is called ionizing radiation. Most people connect the term radiation only with ionizing radiation, but it is not correct. Radiation is all around us. In, around, and above the world we live in. It is a natural energy force that surrounds us. It is a part of our natural world that has been here since the birth of our planet. We should distinguish between:

• Non-ionizing radiation. The kinetic energy of particles (photons, electrons, etc.) of non-ionizing radiation is too small to produce charged ions when passing through matter. The particles (photons) have only sufficient energy to change the rotational, vibrational or electronic valence configurations of target molecules and atoms. Sunlight, radio waves, and cell phone signals are examples of non-ionizing (photon) radiation. However, it can still cause harm, like when you get a sunburn.

• Ionizing radiation. The kinetic energy of particles (photons, electrons, etc.) of ionizing radiation is sufficient and the particle can ionize (to form ion by losing electrons) target atoms to form ions. Simply ionizing radiation can knock electrons from an atom.

The boundary is not sharply defined, since different molecules and atoms ionize at different energies. This is typical for electromagnetic waves. Among electromagnetic waves belong, in order of increasing frequency (energy) and decreasing wavelength: radio waves, microwaves, infrared radiation, visible light, ultraviolet radiation, X-rays and gamma rays. Gamma rays, X-rays, and the higher ultraviolet part of the spectrum are ionizing, whereas the lower ultraviolet, visible light (including laser light), infrared, microwaves, and radio waves are considered non-ionizing radiation.

Forms of ionizing radiation

Ionizing radiation is categorized by the nature of the particles or electromagnetic waves that create the ionizing effect. These particles/waves have different ionization mechanisms, and may be grouped as:

• Directly ionizing. Charged particles (atomic nuclei, electrons, positrons, protons, muons, etc.) can ionize atoms directly by fundamental interaction through the Coulomb force if it carries sufficient kinetic energy. These particles must be moving at relativistic speeds to reach the required kinetic energy. Even photons (gamma rays and X-rays) can ionize atoms directly (despite they are electrically neutral) through the Photoelectric effect and the Compton effect, but secondary (indirect) ionization is much more significant.
  • Alpha radiation. Alpha radiation consist of alpha particles at high energy/speed. The production of alpha particles is termed alpha decay. 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.
  • Beta radiation. Beta radiation consist of free electrons or positrons at relativistic speeds. Beta particles (electrons) are much smaller than alpha particles. They carry a single negative charge. They are more penetrating than alpha particles, but thin aluminum metal can stop them. They can travel several meters but deposit less energy at any one point along their paths than alpha particles.
• Indirectly ionizing. Indirect ionizing radiation is electrically neutral particles and therefore does not interact strongly with matter. The bulk of the ionization effects are due to secondary ionizations.
  • Photon radiation (Gamma rays or X-rays). Photon radiation consist of high energy photons. These photons are particles/waves (Wave-Particle Duality) without rest mass or electrical charge. They can travel 10 meters or more in air. This is a long distance compared to alpha or beta particles. However, gamma rays deposit less energy along their paths. Lead, water, and concrete stop gamma radiation. Photons (gamma rays and X-rays) can ionize atoms directly through the Photoelectric effect and the Compton effect, where the relatively energetic electron is produced. The secondary electron will go on to produce multiple ionization events, therefore the secondary (indirect) ionization is much more significant.
  • Neutron radiation. Neutron radiation consist of free neutrons at any energies/speeds. Neutrons can be emitted by nuclear fission or by the decay of some radioactive atoms. Neutrons have zero electrical charge and cannot directly cause ionization. Neutrons ionize matter only indirectly. For example, when neutrons strike the hydrogen nuclei, proton radiation (fast protons) results. Neutrons can range from high speed, high energy particles to low speed, low energy particles (called thermal neutrons). Neutrons can travel hundreds of feet in air without any interaction.

Shielding of Ionizing Radiation

Radiation shielding simply means having some material between the source of radiation and you (or some device) that will absorb the radiation. The amount of shielding required, the type or material of shielding strongly depends on several factors. We are not talking about any optimisation.

In fact in some cases an inappropriate shielding may even worsen the radiation situation instead of protecting people from the ionizing radiation.  Basic factors, which have to be considered during proposal of radiation shielding, are:

• Type of the ionizing radiation to be shielded
• Energy spectrum of the ionizing radiation
• Length of exposure
• Distance from the source of the ionizing radiation
• Requirements on the attenuation of the ionizing radiation – ALARA or ALARP principles
• Design degree of freedom
• Other physical requirements (e.g. transparence in case of leaded glass screens)

See also: Shielding of Ionizing Radiation

• Shielding of Alpha Radiation
• Shielding of Beta Radiation
• Shielding of Positrons
• Shielding of Gamma Radiation
• Shielding of Neutrons

Shielding in Nuclear Power Plants

Generally in nuclear industry the radiation shielding has many purposes. In nuclear power plants the main purpose is to reduce the radiation exposure to persons and staff in the vicinity of radiation sources. In NPPs the main source of radiation is conclusively the nuclear reactor and its reactor core. Nuclear reactors are in generall powerful sources of entire spectrum of types of ionizing radiation. Shielding used for this purpose is called biological shielding.

But this is not the only purpose of radiation shielding. Shields are also used in some reactors to reduce the intensity of gamma rays or neutrons incident on the reactor vessel. This radiation shielding protects the reactor vessel and its internals (e.g. the core support barrel) from the excessive heating due to gamma ray absorption fast neutron moderation. Such shields are usually referred to as thermal shields.

See also: Neutron Reflector

A little strange radiation shielding is usually used to protect material of reactor pressure vessel (especially in PWR power plants). Structural materials of pressure vessel and reactor internals are damaged especially by fast neutrons. Fast neutrons create structural defects, which in result lead to embrittlement of material of pressure vessel. In order to minimize the neutron flux at the vessel wall, also core loading strategy can be modified. In “out-in” fuel loading strategy fresh fuel assemblies are placed at the periphery of the core.  This configuration causes high neutron fluence at the vessel wall. Therefore the “in-out” fuel loading strategy (with low leakage loading patterns – L3P) has been adopted at many nuclear power plants. In contrast to “out-in” strategy, low leakage cores have fresh fuel assemblies in the second row, not at the periphery of the core. The periphery contains fuel with higher fuel burnup and lower relative power and serves as the very sophisticated radiation shield.

In nuclear power plants the central problem is to shield against gamma rays and neutrons, because the ranges of charged particles (such as beta particles and alpha particles) in matter are very short. On the other hand we must deal with shielding of all types of radiation, because each nuclear reactor is a significant source of all types of ionizing radiation.

See also:

Mass and Energy

See also:

Atomic and Nuclear Physics

See also:

Nuclear Stability

The atom consist of a small but massive nucleus surrounded by a cloud of rapidly moving electrons. The nucleus is composed of protons and neutrons. Total number of protons in the nucleus is called the atomic number of the atom and is given the symbol Z. The total electrical charge of the nucleus is therefore +Ze, where e (elementary charge) equals to 1,602 x 10^-19 coulombs. In a neutral atom there are as many electrons as protons moving about nucleus. It is the electrons that are responsible for the chemical bavavior of atoms, and which identify the various chemical elements.

Hydrogen (H), for example , consist of one electron and one proton. The number of neutrons in a nucleus is known as the neutron number and is given the symbol N. The total number of nucleons, that is, protons and neutrons in a nucleus, is equal to Z + N = A, where A is called the atomic mass number. The various species of atoms whose nuclei contain particular numbers of protons and neutrons are called nuclides. Each nuclide is denoted by chemical symbol of the element (this specifies Z) with tha atomic mass number as supescript.

Thus the symbol ¹H refers to the nuclide of hydrogen with a single proton as nucleus. ²H is the hydrogen nuclide with a neutron as well as a proton in the nucleus (2H is also called deuterium or heavy hydrogen). Atoms such as ¹H, ²H whose nuclei contain the same number of protons but different number of neutrons (different A) are known as isotopes. Uranium, for instance, has three isotopes occuring in nature – ²³⁸U, ²³⁵U and ²³⁴U. The stable isotopes (plus a few of the unstable isotopes) are the atoms that are found in the naturally occuring elements in nature. However, they are not found in equal amounts. Some isotopes of a given element are more abundant than others. For example 99,27% of naturally occuring uranium atoms are the isotope ²³⁸U, 0,72% are the isotope ²³⁵U and 0,0055% are the isotope ²³⁴U. Exact structure of atoms is described by Atomic Theory and Theory of Nuclear Structure.

Volume of an Atom and Nucleus

The atom consist of a small but massive nucleus surrounded by a cloud of rapidly moving electrons. The nucleus is composed of protons and neutrons. Typical nuclear radii are of the order 10⁻¹⁴ m. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r₀ . A^1/3

where r₀ = 1.2 x 10^-15 m = 1.2 fm

If we use this approximation, we therefore expect the volume of the nucleus to be of the order of 4/3πr³ or 7,23 ×10⁻⁴⁵ m³ for hydrogen nuclei or 1721×10⁻⁴⁵ m³ for ²³⁸U nuclei. These are volumes of nuclei and atomic nuclei (protons and neutrons) contains of about 99.95% of mass of atom.

Is an atom an empty space?

The volume of an atom is about 15 orders of magnitude larger than the volume of a nucleus. For uranium atom, the Van der Waals radius is about 186 pm = 1.86 ×10⁻¹⁰ m. The Van der Waals radius, r_w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom.  Assuming spherical shape, the uranium atom have volume of about  26.9 ×10⁻³⁰ m³. But this “huge” space is occupied primarily by electrons, because the nucleus occupies only about 1721×10⁻⁴⁵ m³ of space. These electrons together weigh only a fraction (let say 0.05%) of entire atom.

It may seem, that the space and in fact the matter is empty, but it is not. Due to the quantum nature of electrons, the electrons are not point particles, they are smeared out over the whole atom. The classical description cannot be used to describe things on the atomic scale. On the atomic scale, physicists have found that quantum mechanics describes things very well on that scale. Particle locations in quantum mechanics are not at an exact position, they are described by a probability density function. Therefore the space in an atom (between electrons and an atomic nucleus) is not empty, but it is filled by a probability density function of electrons (usually known as  “electron cloud“).

• Atomic Theory. Atomic theory is a scientific theory of the nature of matter, which states that matter is composed of discrete units called atoms. The word atom comes from the Ancient Greek adjective atomos, meaning “uncuttable”. Today it is known that also atoms are divisible. Atomic Theory consist of many models and discoveries, which gradually formed this theory.
• Theory of Nuclear Structure. Understanding the structure of the atomic nucleus is one of the central challenges in modern nuclear physics.

See also:

Fundamental Particles

See also:

Atomic and Nuclear Physics

See also:

Mass and Energy

Quarks and electrons are some of the elementary particles. A number of fundamental particles have been discovered in various experiments. So many, that researchers had to organize them, just like Mendeleev did with his periodic table. This is summarized in a theoretical model (concerning the electromagnetic, weak, and strong nuclear interactions) called the Standard Model. In particle physics, an elementary particle or fundamental particle is a particle whose substructure is unknown, thus it is unknown whether it is composed of other particles. Known elementary particles include the fundamental fermions and the fundamental bosons.

See also: Baryons

See also: Leptons

Fermions

The fermions are generally “matter particles” and “antimatter particles”:

Bosons

The bosons are generally “force particles” that mediate interactions among fermions: 

Subatomic particles

However, only a few of these fundamental particles (in fact, some of these are not fundamental particles – e.i. neutron consist of three quarks) are very important in nuclear engineering. Nuclear engineering or theory of nuclear reactors operates with much better known subatomic particles such as: 

Fundametal Particles by The Science Channel

See also:

Atomic and Nuclear Physics

See also:

Atomic and Nuclear Structure

Atomic and nuclear physics are not the same. The term atomic physics is often associated with nuclear power, due to the synonymous use of atomic and nuclear in standard English. However, physicists distinguish between atomic and nuclear physics. The atomic physics deals with the atom as a system consisting of a nucleus and electrons. The nuclear physics deals with the nucleus as a system consisting of a nucleons (protons and neutrons). Main difference is in the scale. While the term atomic deals with 1Å = 10^-10m, where Å is an ångström (according to Anders Jonas Ångström), the  term nuclear deals with  1femtometre = 1fermi = 10^-15m.

Atomic Physics

Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. It is primarily concerned with the arrangement of electrons around the nucleus and the processes by which these arrangements change. This includes ions as well as neutral atoms and, unless otherwise stated, for the purposes of this discussion it should be assumed that the term atom includes ions. Atomic physics also help to understand the physics of molecules, but there is also molecular physics, which describes physical properties of molecules.

Volume of an Atom and Nucleus

The atom consist of a small but massive nucleus surrounded by a cloud of rapidly moving electrons. The nucleus is composed of protons and neutrons. Typical nuclear radii are of the order 10⁻¹⁴ m. Assuming spherical shape, nuclear radii can be calculated according to following formula:

r = r₀ . A^1/3

where r₀ = 1.2 x 10^-15 m = 1.2 fm

If we use this approximation, we therefore expect the volume of the nucleus to be of the order of 4/3πr³ or 7,23 ×10⁻⁴⁵ m³ for hydrogen nuclei or 1721×10⁻⁴⁵ m³ for ²³⁸U nuclei. These are volumes of nuclei and atomic nuclei (protons and neutrons) contains of about 99.95% of mass of atom.

Is an atom an empty space?

The volume of an atom is about 15 orders of magnitude larger than the volume of a nucleus. For uranium atom, the Van der Waals radius is about 186 pm = 1.86 ×10⁻¹⁰ m. The Van der Waals radius, r_w, of an atom is the radius of an imaginary hard sphere representing the distance of closest approach for another atom.  Assuming spherical shape, the uranium atom have volume of about  26.9 ×10⁻³⁰ m³. But this “huge” space is occupied primarily by electrons, because the nucleus occupies only about 1721×10⁻⁴⁵ m³ of space. These electrons together weigh only a fraction (let say 0.05%) of entire atom.

It may seem, that the space and in fact the matter is empty, but it is not. Due to the quantum nature of electrons, the electrons are not point particles, they are smeared out over the whole atom. The classical description cannot be used to describe things on the atomic scale. On the atomic scale, physicists have found that quantum mechanics describes things very well on that scale. Particle locations in quantum mechanics are not at an exact position, they are described by a probability density function. Therefore the space in an atom (between electrons and an atomic nucleus) is not empty, but it is filled by a probability density function of electrons (usually known as  “electron cloud“).

Nuclear Physics

Nuclear physics is the field of physics that studies the constituents (protons and neutrons) and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation, but the modern nuclear physics contains also particle physics, which is taught in close association with nuclear physics. The nuclear physics  has provided application in many fields, including those in nuclear medicine (Positron Emission Tomography, isotopes production, etc.) and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.

These physical fundamentals consist of following topics:

In nuclear physics, nuclear fusion is a nuclear reaction in which two or more atomic nuclei collide at a very high energy and fuse together into a new nucleus, e.g. helium. If light nuclei are forced together, they will fuse with a yield of energy because the mass of the combination will be less than the sum of the masses of the individual nuclei. If the combined nuclear mass is less than that of iron at the peak of the binding energy curve, then the nuclear particles will be more tightly bound than they were in the lighter nuclei, and that decrease in mass comes off in the form of energy according to the Albert Einstein relationship. For elements like the uranium and thorium, fission will yield energy. Fusion reactions have an energy density many times greater than nuclear fission and fusion reactions are themselves millions of times more energetic than chemical reactions.

The fusion power offers the opportunity of an almost inexhaustible source of energy for future, but it the fusion technology presents a real scientific and

engineering challenges. For potential nuclear energy sources for humankind, the deuterium-tritium fusion reaction controlled by a magnetic confinement seems the most likely way. But nowadays also this way contains several insurmountable engineering challenges.

Fusion powers the Sun

The Sun is a hot star. Really hot star. But all of the heat and light coming from the Sun comes from the fusion reactions happening inside the core of the Sun. Inside the Sun, the pressure is million of times more than the surface of the Earth, and the temperature reaches more than 15 million Kelvin. Massive gravitational forces create the these conditions for nuclear fusion. On Earth, it is impossible to achieve such conditions.

The Sun burns hydrogen atoms, which fuse together to form helium nuclei, and a small amount of matter is converted into energy. In its core, the Sun consumes approximately 620 million metric tons of hydrogen each second. Hydrogen, heated to very high temperatures changes its state from a gaseous state to a plasma state. Normally, fusion is not possible because the strongly repulsive electrostatic forces between the positively charged nuclei prevent them from getting close enough together to collide and for fusion to occur. The mechanism, how to overcome the coulomb barrier is by the temperature and by the pressure. At close distances the attractive nuclear force allows the nuclei to fuse together.

Deuterium-Tritium Fusion

The fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future.
3T (d, n) 4He
The reaction yields ~17 MeV of energy per reaction but requires a enormous temperature of approximately 40 million Kelvins to overcome the coulomb barrier by the attractive nuclear force, which is stronger at close distances. The deuterium fuel is abundant, but tritium must be either bred from lithium or gotten in the operation of the deuterium cycle.

A nuclear reaction is considered to be the process in which two nuclear particles (two nuclei or a nucleus and a nucleon) interact to produce two or more nuclear particles or ˠ-rays (gamma rays). Thus, a nuclear reaction must cause a transformation of at least one nuclide to another. Sometimes if a nucleus interacts with another nucleus or particle without changing the nature of any nuclide, the process is referred to a nuclear scattering, rather than a nuclear reaction. Perhaps the most notable nuclear reactions are the nuclear fusion reactions of light elements that power the energy production of stars and the Sun. Natural nuclear reactions occur also in the interaction between cosmic rays and matter.

The most notable man-controlled nuclear reaction is the fission reaction which occurs in nuclear reactors. Nuclear reactors are devices to initiate and control a nuclear chain reaction, but there are not only manmade devices. The world’s first nuclear reactor operated about two billion years ago. The natural nuclear reactor formed at Oklo in Gabon, Africa, when a uranium-rich mineral deposit became flooded with groundwater that acted as a neutron moderator, and a nuclear chain reaction started.  These fission reactions were sustained for hundreds of thousands of years, until a chain reaction could no longer be supported. This was confirmed by existence of isotopes of the fission-product gas xenon and by different ratio of U-235/U-238 (enrichment of natural uranium).

See also: TALYS – a software for the simulation of nuclear reactions.

See also: JANIS – Java-based Nuclear Data Information System

Notation of Nuclear Reactions

Standard nuclear notation shows (see picture) the chemical symbol, the mass number and the atomic number of the isotope.

If the initial nuclei are denoted by a and b, and the product nuclei are denoted by c and d, the reaction can be represented by the equation:

 a + b → c + d

Instead of using the full equations in the style above, in many situations a compact notation is used to describe nuclear reactions. This style of the form a(b,c)d is equivalent to a + b producing c + d. Light particles are often abbreviated in this shorthand, typically p means proton, n means neutron, d means deuteron, α means an alpha particle or helium-4, β means beta particle or electron, γ means gamma photon, etc. The reaction above would be written as 10B(n,α)7Li.

Basic Classification of Nuclear Reactions

In order to understand the nature of neutron nuclear reactions, the classification according to the time scale of of these reactions has to be introduced. Interaction time is critical for defining the reaction mechanism.

There are two extreme scenarios for nuclear reactions (not only neutron reactions):

• A projectile and a target nucleus are within the range of nuclear forces for the very short time allowing for an interaction of a single nucleon only. These type of reactions are called the direct reactions.
• A projectile and a target nucleus are within the range of nuclear forces for the time allowing for a large number of interactions between nucleons. These type of reactions are called the compound nucleus reactions.

In fact, there is always some non-direct (multiple internuclear interaction) component in all reactions, but the direct reactions have this component limited.

Types of Nuclear Reactions

Although the number of possible nuclear reactions is enormous, nuclear reactions can be sorted by types. Most of nuclear reactions are accompanied by gamma emission. Some examples are:

• Elastic scattering. Occurs, when no energy is transferred between the target nucleus and the incident particle.

 208Pb (n, n) 208Pb

•  Inelastic scattering. Occurs, when energy is transferred. The difference of kinetic energies is saved in excited nuclide.

 40Ca (α, α’) 40mCa

• Capture reactions. Both charged and neutral particles can be captured by nuclei. This is accompanied by the emission of ˠ-rays. Neutron capture reaction produces radioactive nuclides (induced radioactivity).

 238U (n, ˠ) 239U

• Transfer Reactions. The absorption of a particle accompanied by the emission of one or more particles is called the transfer reaction.

4He (α, p) 7Li

• Fission reactions. Nuclear fission is a nuclear reaction in which the nucleus of an atom splits into smaller parts (lighter nuclei). The fission process often produces free neutrons and photons (in the form of gamma rays), and releases a large amount of energy.

235U (n, 3 n) fission products

• Fusion reactions.  Occur when, two or more atomic nuclei collide at a very high speed and join to form a new type of atomic nucleus.The fusion reaction of deuterium and tritium is particularly interesting because of its potential of providing energy for the future.

3T (d, n) 4He

• Spallation reactions. Occur, when a nucleus is hit by a particle with sufficient energy and momentum to knock out several small fragments or, smash it into many fragments.

• Nuclear decay (Radioactive decay). Occurs when an unstable atom loses energy by emitting ionizing radiation. Radioactive decay is a random process at the level of single atoms, in that, according to quantum theory, it is impossible to predict when a particular atom will decay. There are many types of radioactive decay:
  • Alpha radioactivity. Alha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its very large mass (more than 7000 times the mass of the beta particle) and its charge, it heavy ionizes material and has a very short range.

• • Beta radioactivity. 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. The production of beta particles is termed beta decay.

• • Gamma radioactivity. Gamma rays are electromagnetic radiation of an very high frequency and are therefore high energy photons. They are produced by the decay of nuclei as they transition from a high energy state to a lower state known as gamma decay. Most of nuclear reactions are accompanied by gamma emission.

• • Neutron emission. Neutron emission is a type of radioactive decay of nuclei containing excess neutrons (especially fission products), in which a neutron is simply ejected from the nucleus. This type of radiation plays key role in nuclear reactor control, because these neutrons are delayed  neutrons.

Conservation Laws in Nuclear Reactions

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

Law of Conservation of Lepton Number

In particle physics, the lepton number is used to denote which particles are leptons and which particles are not. Each lepton has a lepton number of 1 and each antilepton has a lepton number of -1. Other non-leptonic particles have a lepton number of 0. The lepton number is a conserved quantum number in all particle reactions. A slight asymmetry in the laws of physics allowed leptons to be created in the Big Bang.

The conservation of lepton number means that whenever a lepton of a certain generation is created or destroyed in a reaction, a corresponding antilepton from the same generation must be created or destroyed. It must be added, there is a separate requirement for each of the three generations of leptons, the electron, muon and tau and their associated neutrinos.

Consider the decay of the neutron. The reaction involves only first generation leptons: electrons and neutrinos:

Since the lepton number must be equal to zero on both sides and it was found that the reaction is a three-particle decay (the electrons emitted in beta decay have a continuous rather than a discrete spectrum),  the third particle must be an electron antineutrino.

Law of Conservation of Baryon Number

In particle physics, the baryon number is used to denote which particles are baryons and which particles are not. Each baryon has a baryon number of 1 and each antibaryonhas a baryon number of -1. Other non-baryonic particles have a baryon number of 0. Since there are exotic hadrons like pentaquarks and tetraquarks, there is a general definition of baryon number as:

where n_q is the number of quarks, and n_q is the number of antiquarks.

The baryon number is a conserved quantum number in all particle reactions.

The law of conservation of baryon number states that:

The sum of the baryon number of all incoming particles is the same as the sum of the baryon numbers of all particles resulting from the reaction.

For example, the following reaction has never been observed:

even if the incoming proton has sufficient energy and charge, energy, and so on, are conserved. This reaction does not conserve baryon number since the left side has B =+2, and the right has B =+1.

On the other hand, the following reaction (proton-antiproton pair production) does conserve B and does occur if the incoming proton has sufficient energy (the threshold energy = 5.6 GeV):

As indicated, B = +2 on both sides of this equation.

From these and other reactions, the conservation of baryon number 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. Thus, the decay would violate the conservation of baryon number. It must be added some theories have suggested that protons are in fact unstable with very long half-life (~10³⁰ years) and that they decay into leptons. There is currently no experimental evidence that proton decay occurs.

Law of Conservation of Electric Charge

The law of conservation of electric charge can be demonstrated also on positron-electron pair production. Since a gamma ray is electrically neutral and sum of the electric charges of electron and positron is also zero, the electric charge in this reaction is also conserved.

Ɣ → e^– + e⁺

It must be added, in order for electron-positron pair production to occur, the electromagnetic energy of the photon must be above a threshold energy, which is equivalent to the rest mass of two electrons. The threshold energy (the total rest mass of produced particles) for electron-positron pair production is equal to 1.02MeV (2 x 0.511MeV) because the rest mass of a single electron is equivalent to 0.511MeV of energy. If the original photon’s energy is greater than 1.02MeV, any energy above 1.02MeV is according to the conservation law split between the kinetic energy of motion of the two particles. The presence of an electric field of a heavy atom such as lead or uranium is essential in order to satisfy conservation of momentum and energy. In order to satisfy both conservation of momentum and energy, the atomic nucleus must receive some momentum. Therefore a photon pair production in free space cannot occur.

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.

Energetics of Nuclear Reactions – 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 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 = [m_a + m_A – (m_b + m_B)]c²

which is the same as the excess kinetic energy of the final products:

Q = T_final – T_initial

   = T_b + T_B – (T_a + T_A)

For reactions in which there is an increase in the kinetic energy of the products Q is positive. The positive Q reactions are said to be exothermic (or exergic). There is a net release of energy, since the kinetic energy of the final state is greater than the kinetic energy of the initial state.

For reactions in which there is a decrease in the kinetic energy of the products Q is negative. The negative Q reactions are said to be endothermic (or endoergic) and they require a net energy input.

The energy released in a nuclear reaction can appear mainly in one of three ways:

• Kinetic energy of the products
• Emission of gamma rays. Gamma rays are emitted by unstable nuclei in their transition from a high energy state to a lower state known as gamma decay.
• Metastable state. Some energy may remain in the nucleus, as a metastable energy level.

A small amount of energy may also emerge in the form of X-rays. Generally, products of nuclear reactions may have different atomic numbers, and thus the configuration of their electron shells is different in comparison with reactants. As the electrons rearrange themselves and drop to lower energy levels, internal transition X-rays (X-rays with precisely defined emission lines) may be emitted.

See also: Q-value Calculator

Exothermic Reactions

Endothermic Reactions

Example: Endothermic Reaction - Photoneutrons

In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcriticality control. Especially in nuclear reactors with D₂O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can be produced also in (γ, n) reactions and therefore they are usually referred to as photoneutrons.

A high energy photon (gamma ray) can under certain conditions eject a neutron from a nucleus. It occurs when its energy exceeds the binding energy of the neutron in the nucleus. Most nuclei have binding energies in excess of 6 MeV, which is above the energy of most gamma rays from fission. On the other hand there are few nuclei with sufficiently low binding energy to be of practical interest. These are: ²D, ⁹Be, ⁶Li, ⁷Li and ¹³C. As can be seen from the table the lowest threshold have ⁹Be with 1.666 MeV and ²D with 2.226 MeV.

In case of deuterium, neutrons can be produced by the interaction of gamma rays (with a minimum energy of 2.22 MeV) with deuterium:

The reaction Q-value is calculated below:

The atom masses of the reactant and products are:

m(²D) = 2.01363 amu

m(¹n) = 1.00866 amu

m(¹H) = 1.00728 amu

Using the mass-energy equivalence, we get the Q-value of this reaction as:

Q = {2.01363 [amu] – (1.00866+1.00728) [amu]} x 931.481 [MeV/amu]

= -0.00231 x 931.481 = -2.15 MeV

Example: Endothermic Reaction - (α,n) reaction

Calculate the reaction Q-value of the following reaction:

7Li (α, n) 10B

The atom masses of the reactants and products are:

m(⁴He) = 4.0026 amu

m(⁷Li) = 7.0160 amu

m(¹n) = 1.0087 amu

m(¹⁰B) = 10.01294 amu

Using the mass-energy equivalence, we get the Q-value of this reaction as:

Q = {(7.0160+4.0026) [amu] – (1.0087+10.01294) [amu]} x 931.481 [MeV/amu]

= 0.00304 x 931.481 = -2.83 MeV

Test your Knowledge – Nuclear Reactions

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Radioactive Decay

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Atomic and Nuclear Physics

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Binding Energy

Basics of Nuclear Fission

Basics of Nuclear Fission

There are nuclei that can undergo fission on their own spontaneously, but only certain nuclei, like uranium-235, uranium-233 and plutonium-239, can sustain a fission chain reaction. This is because these nuclei release neutrons when they break apart, and these neutrons can induce fission of other nuclei. Free neutrons released by each fission play very important role as a trigger of the reaction.

Chain Reaction

Chain reaction

A nuclear chain reaction occurs when one single nuclear reaction causes an average of one or more subsequent nuclear reactions, thus leading to the possibility of a self-propagating series of these reactions. The “one or more” is the key parameter of reactor physics. To raise or lower the power, the amount of reactions must be changed (using the control rods) so that the number of neutrons present (and hence the rate of power generation) is either reduced or increased.

Youtube animation

Principles of Nuclear Fission

In general, the neutron-induced fission reaction is the reaction, in which the incident neutron enters the heavy target nucleus (fissionable nucleus), forming a compound nucleus that is excited to such a high energy level (E_excitation > E_critical) that the nucleus splits into two large fission fragments. A large amount of energy is released in the form of radiation and fragment kinetic energy. Moreover and what is crucial, the fission process may produce 2, 3 or more free neutrons and these neutrons can trigger further fission and a chain reaction can take place. In order to understand the process of fission, we must understand processes, that occur inside the nucleus to be fissioned. At first, the nuclear binding energy must be defined.

Nuclear Binding Energy

During the nuclear splitting or nuclear fusion, some of the mass of the nucleus gets converted into huge amounts of energy and thus this mass is removed from the total mass of the original particles, and the mass is missing in the resulting nucleus. The nuclear binding energies are enormous, they are on the order of a million times greater than the electron binding energies of atoms.

For a nucleus with A (mass number) nucleons, the binding energy per nucleon E_b/A can be calculated. This calculated fraction is shown in the chart as a function of them mass number A. As can be seen, for low mass numbers E_b/A increases rapidly and reaches a maximum of 8.8 MeV at approximately A=60. The nuclei with the highest binding energies, that are most tightly bound belong to the “iron group” of isotopes (⁵⁶Fe, ⁵⁸Fe, ⁶²Ni). After that, the binding energy per nucleon decreases. In the heavy nuclei (A>60) region, a more stable configuration is obtained, when a heavy nucleus splits into two lighter nuclei. This is the origin of the fission process. It may seem that all the heavy nuclei may undergo fission or even spontaneous fission. In fact, for all nuclei with atomic number greater than about 60, fission occurs very rarely. In order to fission process to take place, a sufficient amount of energy must be added to the nucleus and no matter how. The energetics and binding energies of certain nucleus are well described by the Liquid Drop Model, which examines the global properties of nuclei.

Liquid Drop Model

One of the first models which could describe very well the behavior of the nuclear binding energies and therefore of nuclear masses was the mass formula of von Weizsaecker (also called the semi-empirical mass formula – SEMF), that was published in 1935 by German physicist Carl Friedrich von Weizsäcker. This theory is based on the liquid drop model proposed by George Gamow.

According to this model, the atomic nucleus behaves like the molecules in a drop of liquid. But in this nuclear scale, the fluid is made of nucleons (protons and neutrons), which are held together by the strong nuclear force. The liquid drop model of the nucleus takes into account the fact that the nuclear forces on the nucleons on the surface are different from those on nucleons in the interior of the nucleus. The interior nucleons are completely surrounded by other attracting nucleons. Here is the analogy with the forces that form a drop of liquid.

In the ground state the nucleus is spherical. If the sufficient kinetic or binding energy is added, this spherical nucleus may be distorted into a dumbbell shape and then may be splitted into two fragments. Since these fragments are a more stable configuration, the splitting of such heavy nuclei must be accompanied by energy release. This model does not explain all the properties of the atomic nucleus, but does explain the predicted nuclear binding energies.

The nuclear binding energy as a function of the mass number A and the number of
protons Z based on the liquid drop model can be written as:

This formula is called the Weizsaecker Formula (or the semi-empirical mass formula). The physical meaning of this equation can be discussed term by term.

Volume term

Volume term – a_V.A. The first two terms describe a spherical liquid drop of an incompressible fluid with a contribution from the volume scaling with A and from the surface, scaling with A^2/3. The first positive term a_V.A is known as the volume term and it is caused by the attracting strong forces between the nucleons. The strong force has a very limited range and a given nucleon may only interact with its direct neighbours. Therefore this term is proportional to A, instead of A². The coefficient a_V is usually about ~ 16 MeV.

Surface term

Surface term – a_sf.A^2/3. The surface term is also based on the strong force, it is, in fact, a correction to the volume term. The point is that particles at the surface of the nucleus are not completely surrounded by other particles. In the volume term, it is suggested that each nucleon interacts with a constant number of nucleons, independent of A. This assumption is very nearly true for nucleons deep within the nucleus, but causes an overestimation of the binding energy on the surface. By analogy with a liquid drop this effect is indicated as the surface tension effect. If the volume of the nucleus is proportional to A, then the geometrical radius should be proportional to A^1/3 and therefore the surface term must be proportional to the surface area i.e. proportional to A^2/3.

Coulomb term

Coulomb term – a_C.Z².A^-⅓. This term describes the Coulomb repulsion between the uniformly distributed protons and is proportional to the number of proton pairs Z²/R, whereby R is proportional to A^1/3. This effect lowers the binding energy because of the repulsion between charges of equal sign.

Asymmetry term

Asymmetry term – a_A.(A-2Z)²/A. This term cannot be described as ‘classically’ as the first three. This effect is not based on any of the fundamental forces, this effect is based only on the Pauli exclusion principle (no two fermions can occupy exactly the same quantum state in an atom). The heavier nuclei contain more neutrons than protons. These extra neutrons are necessary for stability of the heavier nuclei. They provide (via the attractive forces between the neutrons and protons) some compensation for the repulsion between the protons. On the other hand, if there are significantly more neutrons than protons in a nucleus, some of the neutrons will be higher in energy level in the nucleus. This is the basis for a correction factor, the so-called symmetry term.

Pairing term

Pairing term – δ(A,Z). The last term is the pairing term δ(A,Z). This term captures the effect of spin-coupling. Nuclei with an even number of protons and an even number of neutrons are (due to Pauli exclusion principle) very stable thanks to the occurrence of ‘paired spin’. On the other hand, nuclei with an odd number of protons and neutrons are mostly unstable.

Table of Calculated Binding Energies

In order to calculate the binding energy, the coefficients a_V, a_S, a_C, a_A and a_P must be known. The coefficients have units of megaelectronvolts (MeV) and are calculated by fitting to experimentally measured masses of nuclei. They usually vary depending on the fitting methodology. According to ROHLF, J. W., Modern Physics from α to Z0 , Wiley, 1994., the coefficients in the equation are following:

Using the Weizsaecker formula, also the mass of an atomic nucleus can be derived and is given by:

m = Z.m_p +N.m_n -E_b/c²

where m_p and m_n are the rest mass of a proton and a neutron, respectively, and E_b is the nuclear binding energy of the nucleus.

From the nuclear binding energy curve and from the table it can be seen that, in the case of splitting a ²³⁵U nucleus into two parts, the binding energy of the fragments (A ≈ 120) together is larger than that of the original ²³⁵U nucleus.
According to the Weizsaecker formula, the total energy released for such reaction will be approximately 235 x (8.5 – 7.6) ≈ 200 MeV.

See also: Liquid Drop Model

Critical Energy – Threshold Energy for Fission

In principle, any nucleus, if brought into sufficiently high excited state, can be splitted. For fission to occur, the excitation energy must be above a particular value for certain nuclide. The minimum excitation energy required for fission to occur is known as the critical energy (E_crit) or threshold energy.

The critical energy depends on the nuclear structure and is quite large for light nuclei with Z < 90. For heavier nuclei with Z > 90, the critical energy is about 4 to 6 MeV for A-even nuclei, and generally is much lower for A-odd nuclei. It must be noted, some heavy nuclei (eg. ²⁴⁰Pu or ²⁵²Cf) exhibit fission even in the ground state (without externally added excitation energy). This phenomena is known as the spontaneous fission. This process occur without the addition of the critical energy by the quantum-mechanical process of quantum tunneling through the Coulomb barrier (similarly like alpha particles in the alpha decay). The spontaneous fission contributes to ensure sufficient neutron flux on source range detectors when reactor is subcritical in long term shutdown.

See also: Critical Energy – Threshold Energy for Fission

Energy Release from Fission

In general, the nuclear fission results in the release of enormous quantities of energy. The amount of energy depends strongly on the nucleus to be fissioned and also depends strongly on the kinetic energy of an incident neutron. In order to calculate the power of a reactor, it is necessary to be able precisely identify the individual components of this energy. At first, it is important to distinguish between the total energy released and the energy that can be recovered in a reactor.

The total energy released in fission can be calculated from binding energies of initial target nucleus to be fissioned and binding energies of fission products. But not all the total energy can be recovered in a reactor. For example, about 10 MeV is released in the form of neutrinos (in fact antineutrinos). Since the neutrinos are weakly interacting (with extremely low cross-section of any interaction), they do not contribute to the energy that can be recovered in a reactor.

In order to understand this issue, we have to first investigate a typical fission reaction such as the one listed below.

Using this picture, we can identify and also describe almost all the individual components of the total energy energy released during the fission reaction.

As can be seen from the description of the individual components of the total energy energy released during the fission reaction, there is significant amount of energy generated outside the nuclear fuel (outside fuel rods). Especially the kinetic energy of prompt neutrons is largely generated in the coolant (moderator). This phenomena needs to be included in the nuclear calculations.

For LWR, it is generally accepted that about 2.5% of total energy is recovered in the moderator. This fraction of energy depends on the materials, their arrangement within the reactor, and thus on the reactor type.

Fission Fragments – Products of Nuclear Fission

Nuclear fission fragments are the fragments left after a nucleus fissions. Typically, when uranium 235 nucleus undergoes fission, the nucleus splits into two smaller nuclei, along with a few neutrons and release of energy in the form of heat (kinetic energy of the these fission fragments) and gamma rays. The average of the fragment mass is about 118, but very few fragments near that average are found. It is much more probable to break up into unequal fragments, and the most probable fragment masses are around mass 95 (Krypton) and 137 (Barium).

Most of these fission fragments are highly unstable (radioactive) and undergo further radioactive decays to stabilize itself. Fission fragments interact strongly with the surrounding atoms or molecules traveling at high speed, causing them to ionize.

See also: Interaction of Heavy Charged Particles with Matter

Prompt and Delayed Neutrons

It is known the fission neutrons are of importance in any chain-reacting system. Neutrons trigger the nuclear fission of some nuclei (²³⁵U, ²³⁸U or even ²³²Th). What is crucial the fission of such nuclei produces 2, 3 or more free neutrons.

But not all neutrons are released at the same time following fission. Even the nature of creation of these neutrons is different. From this point of view we usually divide the fission neutrons into two following groups:

• Prompt Neutrons. Prompt neutrons are emitted directly from fission and they are emitted within very short time of about 10^-14 second.
• Delayed Neutrons. Delayed neutrons are emitted by neutron rich fission fragments that are called the delayed neutron precursors. These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo neutron emission. The fact the neutron is produced via this type of decay and this happens orders of magnitude later compared to the emission of the prompt neutrons, plays an extremely important role in the control of the reactor.

See also: Prompt Neutrons

See also: Delayed Neutrons

See also: Reactor control with and without delayed neutrons – Interactive chart

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Key Characteristics of Prompt Neutrons

• Prompt neutrons are emitted directly from fission and they are emitted within very short time of about 10^-14 second.

• Most of the neutrons produced in fission are prompt neutrons – about 99.9%.

• For example a fission of ²³⁵U by thermal neutron yields 2.43 neutrons, of which 2.42 neutrons are prompt neutrons and 0.01585 neutrons are the delayed neutrons.

• The production of prompt neutrons slightly increase with incident neutron energy.

• Almost all prompt fission neutrons have energies between 0.1 MeV and 10 MeV.

• The mean neutron energy is about 2 MeV. The most probable neutron energy is about 0.7 MeV.

• In reactor design the prompt neutron lifetime (PNL) belongs to key neutron-physical characteristics of reactor core.

• Its value depends especially on the type of the moderator and on the energy of the neutrons causing fission.

• In an infinite reactor (without escape) prompt neutron lifetime is the sum of the slowing down time and the diffusion time.

• In LWRs the PNL increases with the fuel burnup.

• The typical prompt neutron lifetime in thermal reactors is on the order of 10^-4 second.

• The typical prompt neutron lifetime in fast reactors is on the order of 10^-7 second.

Key Characteristics of Delayed Neutrons

• The presence of delayed neutrons is perhaps most important aspect of the fission process from the viewpoint of reactor control.

• Delayed neutrons are emitted by neutron rich fission fragments that are called the delayed neutron precursors.

• These precursors usually undergo beta decay but a small fraction of them are excited enough to undergo neutron emission.

• The emission of neutron happens orders of magnitude later compared to the emission of the prompt neutrons.

• About 240 n-emitters are known between ⁸He and ²¹⁰Tl, about 75 of them are in the non-fission region.

• In order to simplify reactor kinetic calculations it is suggested to group together the precursors based on their half-lives.

• Therefore delayed neutrons are traditionally represented by six delayed neutron groups.

• Neutrons can be produced also in (γ, n) reactions (especially in reactors with heavy water moderator) and therefore they are usually referred to as photoneutrons. Photoneutrons are usually treated no differently than regular delayed neutrons in the kinetic calculations.

• The total yield of delayed neutrons per fission, v_d, depends on:
  • Isotope, that is fissioned.
  • Energy of a neutron that induces fission.

• Variation among individual group yields is much greater than variation among group periods.

• In reactor kinetic calculations it is convenient to use relative units usually referred to as delayed neutron fraction (DNF).

• At the steady state condition of criticality, with k_eff = 1, the delayed neutron fraction is equal to the precursor yield fraction β.

• In LWRs the β decreases with fuel burnup. This is due to isotopic changes in the fuel.

• Delayed neutrons have initial energy between 0.3 and 0.9 MeV with an average energy of 0.4 MeV.

• Depending on the type of the reactor, and their spectrum, the delayed neutrons may be more (in thermal reactors) or less effective than prompt neutrons (in fast reactors). In order to include this effect into the reactor kinetic calculations the effective delayed neutron fraction – β_eff must be defined.

• The effective delayed neutron fraction is the product of the average delayed neutron fraction and the importance factor β_eff = β . I.

• The weighted delayed generation time is given by τ = ∑_iτ_i . β_i / β = 13.05 s, therefore the weighted decay constant λ = 1 / τ ≈ 0.08 s^-1.

• The mean generation time with delayed neutrons is about ~0.1 s, rather than ~10^-5 as in section Prompt Neutron Lifetime, where the delayed neutrons were omitted.

• Their presence completely changes the dynamic time response of a reactor to some reactivity change, making it controllable by control systems such as the control rods.

Capture-to-Fission Ratio

The probability that a neutron that is absorbed in a fissile nuclide causes a
fission is very important parameter of each fissile isotope. In terms of cross-sections, this probability is defined as:

σ_f / (σ_f + σ_γ) = 1 / (1 + σ_γ/σ_f) = 1 / (1 + α),

where α = σ_γ/σ_f is referred to as the capture-to-fission ratio. The capture-to-fission ratio may be used as an indicator of “quality” of fissile isotopes. The lower C/F ratio simply means that an absorption reaction will result in the fission rather than in the radiative capture. The ratio depends strongly on the incident neutron energy. In the fast neutron region, C/F ratio decreases. It is determined by the steeper decrease in radiative capture cross-section (see chart).

For ²³⁵U and ²³³U the thermal neutron capture-to-fission ratios are typically lower than those for fast neutrons (for mean energy of about 100 keV). It must be noted, the neutron flux of most fast reactors tends to peak around 200 keV, but the mean energy is between 100-200 keV depending on certain reactor design.

Further increase in neutron energy causes conversely a decrease in C/F ratio. This is not the case of ²³⁹Pu, for 100 keV neutrons, the C/F ratio is lower than for thermal neutrons. For the fissile isotopes (²³³U, ²³⁵U and ²³⁹Pu), a small capture-to-fission ratio is an advantage, because neutrons captured onto them are lost.

Source: JANIS (Java-based Nuclear Data Information Software); The JEFF-3.1.1 Nuclear Data Library

Nuclear Fission Chain Reaction

The chain reaction can take place only in the proper multiplication environment and only under proper conditions. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the non-fission neutron absorption in the system (e.g. to insert control rods). Moreover, this multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is exponential.

On the other hand, if one neutron causes less than one further fission, the number of neutrons in the multiplication system will decrease in time and the reactor power (reaction rate) will also decrease in time. In order to sustain the chain reaction, it is necessary to decrease the non-fission neutron absorption in the system (e.g. to withdraw control rods).

In fact, there is always a competition for the fission neutrons in the multiplication environment, some neutrons will cause further fission reaction, some will be captured by fuel materials or non-fuel materials and some will leak out of the system.

In order to describe the multiplication system, it is necessary to define the infinite and finite multiplication factor of a reactor. The method of calculations of multiplication factors has been developed in the early years of nuclear energy and is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutron-physical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.

Distinction between Fissionable, Fissile and Fertile

In nuclear engineering, fissionable material (nuclide) is material  that is capable of undergoing fission reaction after absorbing either thermal (slow or low energy) neutron or fast (high energy) neutron. Fissionable materials are a superset of fissile materials. Fissionable materials also include an isotope ²³⁸U that can be fissioned only with high energy (>1MeV) neutron. These materials are used to fuel thermal nuclear reactors, because they are capable of sustaining a nuclear fission chain reaction.

Fissile materials undergoes fission reaction after absorption of the binding energy of thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into additional excitation energy of the compound nucleus. On the other hand, the binding energy released by compound nucleus of (²³⁸U + n) after absorption of thermal neutron is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consist of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consist of fissionable isotopes that are capable of undergoing nuclear fission only after capturing a thermal neutron. ²³⁸U is not fissile isotope, because ²³⁸U cannot be fissioned by thermal neutron. ²³⁸U does not meet also alternative requirement to fissile materials. ²³⁸U is not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of ²³⁸U have lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials: ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu.
• Fertile material consist of isotopes that are not fissionable by thermal neutrons, but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

Fissile materials undergoes fission reaction after absorption of the binding energy of thermal neutron. They do not require additional kinetic energy for fission. If the neutron has higher kinetic energy, this energy will be transformed into additional excitation energy of the compound nucleus. On the other hand, the binding energy released by compound nucleus of (²³⁸U + n) after absorption of thermal neutron is less than the critical energy, so the fission reaction cannot occur. The distinction is described in the following points.

• Fissile materials are a subset of fissionable materials.
• Fissionable material consist of isotopes that are capable of undergoing nuclear fission after capturing either fast neutron (high energy neutron – let say >1 MeV) or thermal neutron (low energy neutron – let say 0.025 eV).   Typical fissionable materials: ²³⁸U, ²⁴⁰Pu, but also ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu
• Fissile material consist of fissionable isotopes that are capable of undergoing nuclear fission only after capturing a thermal neutron. ²³⁸U is not fissile isotope, because ²³⁸U cannot be fissioned by thermal neutron. ²³⁸U does not meet also alternative requirement to fissile materials. ²³⁸U is not capable of sustaining a nuclear fission chain reaction, because neutrons produced by fission of ²³⁸U have lower energies than original neutron (usually below the threshold energy of 1 MeV). Typical fissile materials: ²³⁵U, ²³³U, ²³⁹Pu, ²⁴¹Pu.
• Fertile material consist of isotopes that are not fissionable by thermal neutrons, but can be converted into fissile isotopes (after neutron absorption and subsequent nuclear decay). Typical fertile materials: ²³⁸U, ²³²Th.

See also: Neutron cross-section

Comparison of cross-sections

Source: JANIS (Java-based nuclear information software)  http://www.oecd-nea.org/janis/

Fissile / Fertile Material Cross-sections. Comparison of total fission cross-sections.

Uranium 238. Comparison of total fission cross-section and cross-section for radiative capture.

See also:

See also:

Neutron Nuclear Reactions

See also:

What is Nuclear Energy

Nuclear energy comes either from spontaneous nuclei conversions or induced nuclei conversions. Among these conversions (nuclear reactions) belong for example nuclear fission, nuclear decay and nuclear fusion. Conversions are associated with mass and energy changes. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible, one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula:

, where M is the small amount of mass and C is the speed of light.

What that means? If the nuclear energy is generated (splitting atoms, nuclear fussion), a small amount of mass (saved in the nuclear binding energy) transforms into the pure energy (such as kinetic energy, thermal energy, or radiant energy).

The energy equivalent of one gram (1/1000 of a kilogram) of mass is equivalent to:

• 89.9 terajoules
• 25.0 million kilowatt-hours (≈ 25 GW·h)
• 21.5 billion kilocalories (≈ 21 Tcal)
• 85.2 billion BTUs

or to the energy released by combustion of the following:

• 21.5 kilotons of TNT-equivalent energy (≈ 21 kt)
• 568,000 US gallons of automotive gasoline

Any time energy is generated, the process can be evaluated from an E = mc² perspective.

Nuclear Binding Energy – Mass Defect

Conservation of Mass-Energy

At the beginning of the 20th century, the notion of mass underwent a radical revision. Mass lost its absoluteness. One of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc². In words, energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number, the formula implies that any small amount of matter contains a very large amount of energy. The mass of an object was seen to be equivalent to energy, to be interconvertible with energy, and to increase significantly at exceedingly high speeds near that of light. The total energy of an object was understood to comprise its rest mass as well as its increase of mass caused by increase in kinetic energy.

In special theory of relativity certain types of matter may be created or destroyed, but in all of these processes, the mass and energy associated with such matter remains unchanged in quantity. It was found the rest mass an atomic nucleus is measurably smaller than the sum of the rest masses of its constituent protons, neutrons and electrons. Mass was no longer considered unchangeable in the closed system. The difference is a measure of the nuclear binding energy which holds the nucleus together. According to the Einstein relationship (E = mc²) this binding energy is proportional to this mass difference and it is known as the mass defect.

E=mc² represents the new conservation principle – the conservation of mass-energy.

Nuclear Binding Curve

If the splitting releases energy and the fusion releases the energy, so where is the breaking point? For understanding this issue it is better to relate the binding energy to one nucleon, to obtain nuclear binding curve. The binding energy per one nucleon is not linear. There is a peak in the binding energy curve in the region of stability near iron and this means that either the breakup of heavier nuclei than iron or the combining of lighter nuclei than iron will yield energy.

The reason the trend reverses after iron peak is the growing positive charge of the nuclei. The electric force has greater range than strong nuclear force. While the strong nuclear force binds only close neighbors the electric force of each proton repels the other protons.

Example: Mass defect of a 63Cu

Calculate the mass defect of a ⁶³Cu nucleus if the actual mass of ⁶³Cu in its nuclear ground state is 62.91367 u.

⁶³Cu nucleus has 29 protons and also has (63 – 29) 34 neutrons.

The mass of a proton is 1.00728 u and a neutron is 1.00867 u.

The combined mass is: 29 protons x (1.00728 u/proton) + 34 neutrons x (1.00867 u/neutron) = 63.50590 u

The mass defect is Δm = 63.50590 u – 62.91367 u =  0.59223 u

Convert the mass defect into energy (nuclear binding energy).

(0.59223 u/nucleus) x (1.6606 x 10^-27 kg/u) = 9.8346 x 10^-28 kg/nucleus

ΔE = Δmc²

ΔE = (9.8346 x 10^-28 kg/nucleus) x (2.9979 x 10⁸ m/s)² = 8.8387 x 10^-11 J/nucleus

The energy calculated in the previous example is the nuclear binding energy.  However, the nuclear binding energy may be expressed as kJ/mol (for better understanding).

Calculate the nuclear binding energy of 1 mole of ⁶³Cu:

(8.8387 x 10^-11 J/nucleus) x (1 kJ/1000 J) x (6.022 x 10²³ nuclei/mol) = 5.3227 x 10¹⁰ kJ/mol of nuclei.

One mole of ⁶³Cu (~63 grams) is bound by the nuclear binding energy (5.3227 x 10¹⁰ kJ/mol) which is equivalent to:

• 14.8 million kilowatt-hours (≈ 15 GW·h)
• 336,100 US gallons of automotive gasoline

Example: Mass defect of the reactor core

Calculate the mass defect of the 3000MW_th reactor core after one year of operation.

It is known the average recoverable energy per fission is about 200 MeV, being the total energy minus the energy of the energy of antineutrinos that are radiated away.

The reaction rate per entire 3000MW_th reactor core is about  9.33×10¹⁹ fissions / second.

The overall energy release in the units of joules is:

200×10⁶ (eV) x 1.602×10^-19 (J/eV) x 9.33×10¹⁹ (s^-1) x 31.5×10⁶ (seconds in year) = 9.4×10¹⁶ J/year

The mass defect is calculated as:

Δm = ΔE/c²

Δm = 9.4×10¹⁶ / (2.9979 x 10⁸)² = 1.046 kg

That means in a typical 3000MWth reactor core about 1 kilogram of matter is converted into pure energy.

Note that, a typical annual uranium load for a 3000MWth reactor core is about 20 tonnes of enriched uranium (i.e. about 22.7 tonnes of UO₂). Entire reactor core may contain about 80 tonnes of enriched uranium.

Mass defect directly from E=mc²

The mass defect can be calculated directly from the Einstein relationship (E = mc²) as:

Δm = ΔE/c²

Δm = 3000×10⁶ (W = J/s) x 31.5×10⁶ (seconds in year) / (2.9979 x 10⁸)² = 1.051 kg

Nuclear Energy and Electricity Production

Today we use the nuclear energy to generate useful heat and electricity. This electricity is generated in nuclear power plants. The heat source in the nuclear power plant is a nuclear reactor. As is typical in all conventional thermal power stations the heat is used to generate steam which drives a steam turbine connected to a generator which produces electricity. In 2011 nuclear power provided 10% of the world’s electricity. In 2007, the IAEA reported there were 439 nuclear power reactors in operation in the world, operating in 31 countries. They produce base-load electricity 24/7 without emitting any pollutants into the atmosphere (this includes CO2).

Nuclear Energy Consumption – Summary

Consumption of a 3000MWth (~1000MWe) reactor (12-months fuel cycle)

It is an illustrative example, following data do not correspond to any reactor design.

• Typical reactor may contain about 165 tonnes of fuel (including structural material)
• Typical reactor may contain about 100 tonnes of enriched uranium (i.e. about 113 tonnes of uranium dioxide).
• This fuel is loaded within, for example, 157 fuel assemblies composed of over 45,000 fuel rods.
• A common fuel assembly contain energy for approximately 4 years of operation at full power.
• Therefore about one quarter of the core is yearly removed to spent fuel pool (i.e. about 40 fuel assemblies), while the remainder is rearranged to a location in the core better suited to its remaining level of enrichment (see Power Distribution).
• The removed fuel (spent nuclear fuel) still contains about 96% of reusable material (it must be removed due to decreasing k_inf of an assembly).

• Annual natural uranium consumption of this reactor is about 250 tonnes of natural uranium (to produce of about 25 tonnes of enriched uranium).

• Annual enriched uranium consumption of this reactor is about 25 tonnes of enriched uranium.

• Annual fissile material consumption of this reactor is about 1 005 kg.

• Annual matter consumption of this reactor is about 1.051 kg.

• But it corresponds to about 3 200 000 tons of coal burned in coal-fired power plant per year.

See also: Fuel Consumption

See also:

Atomic and Nuclear Structure

See also:

Atomic and Nuclear Physics

See also:

Radiation