Nuclear Reactions

Article Summary & FAQs

What is a nuclear reaction?

nuclear reaction is considered to be the process in which two atomic nuclei or subatomic particles interact to produce one or more new particles or gamma rays.

Key Facts

  • Perhaps the most notable nuclear reactions are the nuclear fusion reactions of light elements that power the energy production of stars and the Sun.
  • The most notable man-controlled nuclear reaction is the fission reaction which occurs in nuclear reactors.
  • Classification of nuclear reactions is according to the time scale of these reactions.
    • In direct nuclear reactions, a projectile and a target nucleus are within the range of nuclear forces for a very short time allowing for an interaction of a single nucleon only.
    • In compound nuclear 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.
  • 10B(n,alpha)7Li is typical notation of nuclear reactions.
  • Energetics of nuclear reactions is determined by the Q-value of that reaction.
What are main types of nuclear reactions?
What are main types of nuclear reactions?

Although the number of possible nuclear reactions is enormous, nuclear reactions can be sorted by types:

  • Elastic scattering
  • Inelastic scattering
  • Capture reactions
  • Transfer reactions
  • Fission reactions
  • Fusion reactions
  • Spallation reactions
  • Nuclear decay
What is typical example of a nuclear reaction?
What is a typical example of a nuclear reaction?

10B(n, alpha)7Li is a typical nuclear reaction in nuclear reactors. This reaction is mainly used to control nuclear reactors because it absorbs neutrons without fission.

(n,alpha) reactions of 10B

What is the main characteristics of nuclear reaction?
What are the main characteristics of nuclear reactions?

In nuclear physics, the nuclear cross-section of a nucleus is commonly used to characterize the probability that a nuclear reaction will occur. The cross-section is typically denoted σ and measured in units of the area [m2]. The standard unit for measuring a nuclear cross-section is the barn, equal to 10−28 m² or 10−24 cm².

What is an exothermic nuclear reaction?
What is an exothermic nuclear reaction?

Q is positive. The positive Q reactions are said to be exothermic (or exergic). An exothermic nuclear reaction is a reaction in which there is an increase in the kinetic energy of the products. 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.

A nuclear reaction is a process when two atomic nuclei or subatomic particles interact to produce one or more new particles or 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 as 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 also occur in the interaction between cosmic rays and matter.

The most notable man-controlled nuclear reaction is the fission reaction which occurs in nuclear reactorsNuclear reactors are devices to initiate and control a nuclear chain reaction, but there are not only artificial 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 be supported no longer. This was confirmed by the existence of isotopes of the fission-product gas xenon and by different ratios 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

See also: Neutron Reactions

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

boron-neutron reaction
This equation describes neutron capture in the boron, which is diluted in the coolant. Boric acid is used in nuclear power plants as a long-term compensator of nuclear fuel reactivity.
Notation of nuclei
Notation of nuclei
Source: chemwiki.ucdavis.edu

Instead of using the full equations in the style above, a compact notation describes nuclear reactions in many situations. 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

To understand the nature of neutron nuclear reactions, the classification according to the time scale 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 a very short time allowing for an interaction of a single nucleon only. These types of reactions are called direct reactions.
  • A projectile and a target nucleus are within the range of nuclear forces, allowing for many interactions between nucleons. These types 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.

Basic Characteristics of Direct Reactions
  • The direct reactions are fast and involve a single-nucleon interaction.
  • The interaction time must be very short (~10-22 s).
  • The direct reactions require incident particle energy larger than ∼ 5 MeV/Ap. (Ap is the atomic mass number of a projectile)
  • Incident particles interact on the surface of a target nucleus rather than in the volume of a target nucleus.
  • Products of the direct reactions are not distributed isotropically in angle, but they are forward-focused.
  • Direct reactions are of importance in measurements of nuclear structure.
Basic Characteristics of Compound Nucleus Reactions
  • The compound nucleus is a relatively long-lived intermediate state of the particle-target composite system.
  • The compound nucleus reactions involve many nucleon-nucleon interactions.
  • A large number of collisions between the nucleons leads to a thermal equilibrium inside the compound nucleus.
  • The time scale of compound nucleus reactions is 10-18 s – 10-16 s.
  • The compound nucleus reactions are usually created if the projectile has low energy.
  • Incident particles interact in the volume of a target nucleus.
  • Products of the compound nucleus reactions are distributed near isotropically in angle (the nucleus loses memory of how it was created – Bohr’s hypothesis of independence).
  • The decay mode of the compound nucleus does not depend on how the compound nucleus is formed.
  • Resonances in the cross-section are typical for the compound nucleus reaction.

Types of Nuclear Reactions

Although the number of possible nuclear reactions is enormous, nuclear reactions can be sorted by type. Most nuclear reactions are accompanied by gamma emissions. 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 an excited nuclide.

 40Ca (α, α’) 40mCa

  • Capture reactions. Nuclei can capture both charged and neutral particles. The emission of ˠ-rays accompanies this. 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, 3n) 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 exciting because of its potential of providing energy for the future.

3T (d, n) 4He

  • Spallation reactions. Occur occurs when a particle hits a nucleus with sufficient energy and momentum to knock out several small fragments or smash them 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. Alpha particles consist of two protons and two neutrons bound together into a particle identical to a helium nucleus. Because of its huge mass (more than 7000 times the mass of the beta particle) and its charge, it heavily ionizes material and has a very short range.
    • Beta radioactivity. Beta particles are high-energy, high-speed electrons or positrons emitted by specific radioactive nuclei such as potassium-40. The beta particles emitted are a form of ionizing radiation, also known as beta rays. The beta particles have a greater range of penetration than alpha particles but still much less than gamma rays. The production of beta particles is termed beta decay.
    • Gamma radioactivity. Gamma rays are electromagnetic radiation of a very high frequency and are therefore high-energy photons. Nuclei decay produces them as they transition from a high-energy state to a lower state known as gamma decay. Most nuclear reactions are accompanied by gamma emissions.
    • Neutron emissionNeutron 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 a key role in nuclear reactor control because these neutrons are delayed neutrons.
Notation of nuclear reactions - radioactive decays
Radioactive decays
Source: chemwiki.ucdavis.edu

Main Characteristics of Nuclear Reactions

In nuclear physics, the nuclear cross-section of a nucleus is commonly used to characterize the probability that a nuclear reaction will occur. The cross-section is typically denoted σ and measured in units of the area [m2]. The standard unit for measuring a nuclear cross-section is the barn, equal to 10−28 m² or 10−24 cm². It can be seen that the concept of a nuclear cross-section can be quantified physically using a “characteristic target area,” where a larger area means a larger probability of interaction.

nuclear cross-sections, microscopic cross-sectionsFor a given event, the cross-section σ is given by

σ = μ/n

where

  • σ is the cross-section of this event [m2],
  • μ is the attenuation coefficient due to the occurrence of this event [m-1],
  • n is the density of the target particles [m-3].

 

Conservation Laws in Nuclear Reactions

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

Lepton Number. Conservation of Lepton NumberIn 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, and 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:

lepton-number-neutron-decay

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.

Spoiler title
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 antibaryon has 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:

baryon-number-equation

where nq is the number of quarks, and nq 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:

baryon-number-example-violation

even if the incoming proton has sufficient energy and charge, energy, and so on, are conserved. This reaction does not conserve the 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):

baryon-number-pair-production

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

From these and other reactions, the conservation of the baryon number has been established as a basic principle of physics.

This principle provides the 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 the baryon number. It must be added some theories have suggested that protons are, in fact, unstable with a 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 the gamma-ray is electrically neutral and the 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, for electron-positron pair production to occur, the electromagnetic energy of the photon must be above 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 equals 1.02MeV (2 x 0.511MeV) because the rest mass of a single electron is equivalent to 0.511MeV of energy. If the origin 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 to satisfy the conservation of momentum and energy. To satisfy the conservation of momentum and energy, the atomic nucleus must accept some momentum. Therefore a photon pair production in free space cannot occur.

Certain of these laws are obeyed under all circumstances, and others are not. We have accepted the 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, we shall find that these principles must be extended when we consider relativistic nuclear energies or those involving weak interactions.

Some conservation principles have arisen from theoretical considerations, and 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.

The non-relativistic reactions are governing by four of the fundamental laws and to analyze them is necessary to follow them:

  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 is 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.

Elastic Nuclear Collision

A neutron (n) of mass 1.01 u traveling with a speed of 3.60 x 104m/s interacts with a carbon (C) nucleus (mC = 12.00 u) initially at rest in an elastic head-on collision.

What are the velocities of the neutron and carbon nucleus after the collision?

Solution:

This is an elastic head-on collision of two objects with unequal masses. We have to use the conservation laws of momentum and kinetic energy and apply them to our system of two particles.

conservation-laws-elastic-collisions

We can solve this equation system, or we can use the equation derived in the previous section. This equation stated that the relative speed of the two objects after the collision has the same magnitude (but opposite direction) as before the collision, no matter what the masses are.

solution-elastic-collision

The minus sign for v’ tells us that the neutron scatters the back of the carbon nucleus because the carbon nucleus is significantly heavier. On the other hand, its speed is less than its initial speed. This process is known as neutron moderation, and it significantly depends on the mass of moderator nuclei.

Energetics of Nuclear Reactions – Q-value

Q-value of DT fusion reaction
Q-value of DT fusion reaction

In nuclear and particle physics, the energetics of nuclear reactions are determined by the reaction’s Q-value. 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 target A place to two products, B and b. This can also be expressed in the notation we have 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

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

Q = Tfinal – Tinitial

   = Tb + TB – (Ta + TA)

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 product, Q is negative. The negative Q reactions are endothermic (or endoergic), and they require net energy input.

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

  • The 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

Example: Exothermic Reaction - DT fusion
Q-value of DT fusion reaction
Q-value of DT fusion reaction

The DT fusion reaction of deuterium and tritium is particularly interesting because of its potential to provide future energy. Calculate the reaction Q-value.

3T (d, n) 4He

The atom masses of the reactants and products are:

m(3T) = 3.0160 amu

m(2D) = 2.0141 amu

m(1n) = 1.0087 amu

m(4He) = 4.0026 amu

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

Q = {(3.0160+2.0141) [amu] – (1.0087+4.0026) [amu]} x 931.481 [MeV/amu]

= 0.0188 x 931.481 = 17.5 MeV

Example: Exothermic Reaction - Tritium in Reactors
Cross-section of 10B(n,2alpha)T reaction.
Cross-section of 10B(n,2alpha)T reaction.

Tritium is a byproduct of nuclear reactors. Most of the tritium produced in nuclear power plants stems from boric acid, commonly used as a chemical shim to compensate for an excess of initial reactivity. The main reaction in which the tritium is generated from boron is below:

10B(n,2*alpha)T

This neutron reaction with an isotope 10B is the main way radioactive tritium in the primary circuit of all PWRs is generated. Note that this reaction is a threshold reaction due to its cross-section.

Calculate the reaction Q-value.

The atom masses of the reactants and products are:

m(10B) = 10.01294 amu

m(1n) = 1.00866 amu

m(3T) = 3.01604 amu

m(4He) = 4.0026 amu

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

Q = {(10.0129+1.00866) [amu] – (3.01604+2 x 4.0026) [amu]} x 931.481 [MeV/amu]

= 0.00036 x 931.481 = 0.335 MeV

Endothermic Reactions

Example: Endothermic Reaction - Photoneutrons
In nuclear reactors the gamma radiation plays a significant role also in reactor kinetics and in a subcritical control. Especially in nuclear reactors with D2O moderator (CANDU reactors) or with Be reflectors (some experimental reactors). Neutrons can also be produced 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 over 6 MeV, above the energy of most gamma rays from fission. On the other hand, few nuclei with sufficiently low binding energy are of practical interest. These are 2D, 9Be, 6Li, 7Li, and 13C. As can be seen from the table, the lowest threshold has 9Be with 1.666 MeV and 2D with 2.226 MeV.

Photoneutron sources
Nuclides with low photodisintegration
threshold energies.

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

Photoneutron - 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

quiz - nuclear reactions

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References:
Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
  7. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  8. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  9. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See previous:

Radioactive Decay

See above:

Atomic and Nuclear Physics

See next:

Binding Energy