Nuclear 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

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.

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. In elastic scattering, the kinetic energy of a particle is conserved in the center-of-mass frame, but its direction of propagation is modified. There is no energy transferred into nuclear excitation in an elastic scattering reaction. It is a crucial reaction for neutron moderators in nuclear reactors. To be an effective moderator, the probability of an elastic reaction between the neutron and the nucleus must be high.

 1H (n, n) 1H

•  Inelastic scattering. In inelastic scattering, the particle is absorbed and then re-emitted. The difference of kinetic energies is saved in an excited nuclide. An inelastic scattering plays an important role in slowing down neutrons, especially at high energies and by heavy nuclei.

238U (n, n’) 238U*

• Capture reaction. The capture reaction is one of the two possible absorption reactions that may occur. Capture reactions result in the loss of a neutron coupled with the production of one or more gamma rays. The resulting nucleus may also undergo a subsequent decay, such as beta decay in this example, which is a very important reaction in nuclear fuel.

 238U (n, ˠ) 239U

• Transfer Reaction. Transfer reactions are nuclear reactions in which one or more nucleons are transferred to the other nucleus. Transfer reactions can occur from the projectile to the target; stripping reactions, or from the target to the projectile; pick-up reactions. These reactions are common in particle accelerators and astrophysics.

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. Spallation reaction 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 spallation is one of the processes by which a particle accelerator may be used to produce a beam of neutrons.

209Bi (α,xn)213-xAt

• 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 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 a key role in nuclear reactor control because these neutrons are delayed neutrons.

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.

For 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 laws. Nuclear 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:

• 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, 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:

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 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:

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 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):

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 (m_C = 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.

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.

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

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 = [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 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.

Exothermic Reactions

Example: Exothermic Reaction - DT fusion

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

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.

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:

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