Nuclear chemistry: decay modes, nuclear reactions, Q‑value, cross sections and photonuclear reactions

Nuclear Decay Modes

Radioactive decay is the spontaneous disintegration of an unstable atomic nucleus, releasing energy and transforming into a different nucleus. This process is governed by the fundamental forces of nature, primarily the strong nuclear force and the weak nuclear force. The instability of a nucleus arises from an unfavorable ratio of protons to neutrons or from being in an excited energy state. There are several primary modes of radioactive decay, each characterized by the type of particle or energy emitted.

Alpha (α) Decay

Alpha decay is a type of radioactive decay in which an atomic nucleus emits an alpha particle and thereby transforms or 'decays' into a different atomic nucleus. An alpha particle consists of two protons and two neutrons, making it identical to a helium-4 nucleus (⁴₂He). This emission reduces the atomic number (Z) of the parent nucleus by 2 and the mass number (A) by 4. Alpha decay is common in heavy nuclei, where the electrostatic repulsion between the numerous protons can destabilize the nucleus.

The general equation for alpha decay is:
^A_ZX → ^A-4_Z-2Y + ⁴₂He (α)
Where X is the parent nucleus and Y is the daughter nucleus.

For example, Uranium-238 (²³⁸₉₂U) decays via alpha emission to Thorium-234 (²³⁴₉₀Th):
²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He

Beta (β) Decay

Beta decay involves the transformation of a neutron into a proton or a proton into a neutron within the nucleus, accompanied by the emission of a beta particle and a neutrino or antineutrino. This process is mediated by the weak nuclear force. Beta decay changes the atomic number of the nucleus but leaves the mass number unchanged.

Beta-Minus (β^-) Decay

In beta-minus decay, a neutron transforms into a proton, emitting an electron (β^- particle) and an electron antineutrino (ν̄_e). This occurs in nuclei that have an excess of neutrons relative to protons. The atomic number increases by 1, while the mass number remains the same.

The general equation is:
^A_ZX → ^A_Z+1Y + e^- + ν̄_e
A neutron can be thought of as decaying: n → p + e^- + ν̄_e

An example is the decay of Carbon-14 (¹⁴₆C):
¹⁴₆C → ¹⁴₇N + e^- + ν̄_e

Beta-Plus (β⁺) Decay (Positron Emission)

In beta-plus decay, a proton transforms into a neutron, emitting a positron (β⁺ particle, the antiparticle of the electron) and an electron neutrino (ν_e). This occurs in nuclei that have an excess of protons relative to neutrons. The atomic number decreases by 1, while the mass number remains the same.

The general equation is:
^A_ZX → ^A_Z-1Y + e⁺ + ν_e
A proton can be thought of as decaying: p → n + e⁺ + ν_e

An example is the decay of Fluorine-18 (¹⁸₉F):
¹⁸₉F → ¹⁸₈O + e⁺ + ν_e

Electron Capture (EC)

Electron capture is a process where the nucleus of an atom absorbs an inner orbital electron (usually from the K or L shell). This captured electron combines with a proton in the nucleus to form a neutron and an electron neutrino. Like beta-plus decay, electron capture reduces the atomic number by 1 while keeping the mass number the same. It is a competing process with positron emission, especially in proton-rich nuclei.

The general equation is:
^A_ZX + e^- → ^A_Z-1Y + ν_e

An example is the decay of Potassium-40 (⁴⁰₁₉K):
⁴⁰₁₉K + e^- → ⁴⁰₁₈Ar + ν_e

Gamma (γ) Decay

Gamma decay is the emission of high-energy photons (gamma rays) from an atomic nucleus. This process typically occurs after alpha or beta decay, when the daughter nucleus is left in an excited, higher energy state. The nucleus transitions to a lower energy state by releasing this excess energy as a gamma photon. Gamma decay does not change the atomic number or the mass number of the nucleus; it only reduces the energy of the nucleus.

The general equation is:
^A_ZX* → ^A_ZX + γ
Where X* denotes the nucleus in an excited state.

For example, Cobalt-60 (⁶⁰₂₇Co) undergoes beta decay to an excited state of Nickel-60 (⁶⁰₂₈Ni*), which then emits gamma rays:
⁶⁰₂₇Co → ⁶⁰₂₈Ni* + e^- + ν̄_e ⁶⁰₂₈Ni* → ⁶⁰₂₈Ni + γ

Spontaneous Fission

Spontaneous fission is a form of radioactive decay in which a heavy nucleus splits into two or more smaller nuclei, along with the emission of neutrons and a large amount of energy. This mode of decay is observed only in very heavy elements, typically those with atomic numbers greater than 90. The nucleus splits because the repulsive electrostatic forces between the protons overcome the attractive nuclear forces.

For example, Californium-252 (²⁵²₉₈Cf) undergoes spontaneous fission:
²⁵²₉₈Cf → Fission fragments (e.g., ¹⁴⁰₅₄Xe + ¹⁰⁸₄₄Ru) + neutrons + energy

Nuclear Reactions

Nuclear reactions involve the transformation of atomic nuclei due to interactions with other particles, such as protons, neutrons, alpha particles, or other nuclei. Unlike radioactive decay, nuclear reactions are typically induced. These reactions can result in the formation of new isotopes, elements, or the release of significant amounts of energy. The fundamental principles of conservation of mass-energy, charge, and nucleon number (protons + neutrons) apply to all nuclear reactions.

Types of Nuclear Reactions

Nuclear reactions can be broadly classified based on the incident particle and the resulting products.

Neutron-Induced Reactions

These are very important in nuclear technology, especially for nuclear reactors and weapons. Neutrons are effective projectiles because they are electrically neutral and are not repelled by the positive charge of the nucleus.

• Neutron Capture: A nucleus absorbs a neutron, often followed by gamma emission. This results in an isotope of the same element with a higher mass number.
  ^A_ZX + n → ^A+1_ZX + γ
  Example: ⁵⁶₂₆Fe + n → ⁵⁷₂₆Fe + γ
• Fission: The absorption of a neutron by a heavy nucleus (like Uranium-235) can induce fission, splitting the nucleus into smaller fragments and releasing more neutrons, which can sustain a chain reaction.
  ²³⁵₉₂U + n → Fission fragments + 2-3n + Energy
• Transmutation: Neutron bombardment can lead to the formation of a different element. For example, in a fast breeder reactor, Plutonium-238 can capture a neutron and undergo beta decay to become Plutonium-239.
  ²³⁸₉₄Pu + n → ²³⁹₉₄Pu → ²³⁹₉₅Am + e^- + ν̄_e (This is a two-step process: capture then decay)

Charged Particle-Induced Reactions

Reactions induced by protons, alpha particles, or heavier ions. These require high energies to overcome the Coulomb repulsion between the incident particle and the target nucleus.

• Artificial Transmutation: Ernest Rutherford performed the first artificial nuclear reaction in 1919 by bombarding Nitrogen with alpha particles, producing Oxygen and a proton.
  ¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁p
• Fusion: The combining of two light nuclei to form a heavier nucleus, releasing immense energy. This is the process that powers stars.
  ²₁H + ³₁H → ⁴₂He + n + Energy

Photonuclear Reactions

Reactions induced by high-energy photons (gamma rays). These are discussed in more detail later in this section.

Conservation Laws in Nuclear Reactions

In any nuclear reaction, several quantities are conserved:

• Conservation of Mass Number (A): The total number of nucleons (protons + neutrons) remains constant.
  Sum of A on the left side = Sum of A on the right side.
• Conservation of Atomic Number (Z) / Charge: The total charge remains constant.
  Sum of Z on the left side = Sum of Z on the right side.
• Conservation of Energy and Momentum: These are always conserved, though energy can be converted between kinetic energy and mass-energy.

Q-value of Nuclear Reactions

The Q-value of a nuclear reaction represents the energy released or absorbed during the reaction. It is determined by the mass difference between the reactants and the products. The Q-value is calculated using Einstein's mass-energy equivalence principle, E = mc².

If the total mass of the products is less than the total mass of the reactants, mass is converted into energy, and the reaction is exothermic (Q > 0). If the total mass of the products is greater than the total mass of the reactants, energy must be supplied for the reaction to occur, and the reaction is endothermic (Q < 0).

The Q-value is calculated as:
Q = (Σm_reactants - Σm_products) c²
Where:

• Σm_reactants is the sum of the rest masses of the reactant nuclei and particles.
• Σm_products is the sum of the rest masses of the product nuclei and particles.
• c is the speed of light.

The Q-value is often expressed in mega-electron volts (MeV). The conversion factor is approximately 1 atomic mass unit (u) ≈ 931.5 MeV/c². Therefore, the Q-value can also be calculated as:
Q (in MeV) = (Σm_reactants - Σm_products) × 931.5 MeV/u
Where masses are in atomic mass units (u).

Exothermic Reactions (Q > 0)

These reactions release energy. The released energy appears as kinetic energy of the products and/or gamma rays. Examples include nuclear fission and fusion.
Example: Deuterium-Tritium fusion ²₁H + ³₁H → ⁴₂He + n
Mass of reactants: m(²₁H) + m(³₁H) = 2.014102 u + 3.016049 u = 5.030151 u
Mass of products: m(⁴₂He) + m(n) = 4.002603 u + 1.008665 u = 5.011268 u
Mass difference = 5.030151 u - 5.011268 u = 0.018883 u
Q = 0.018883 u × 931.5 MeV/u ≈ 17.59 MeV. This energy is released.

Endothermic Reactions (Q < 0)

These reactions require an input of energy to proceed. The minimum energy required is called the threshold energy. This energy must be supplied, usually as kinetic energy of the incident particle, to overcome the mass defect of the products.
Example: Rutherford's first artificial transmutation ¹⁴₇N + ⁴₂He → ¹⁷₈O + ¹₁p
Mass of reactants: m(¹⁴₇N) + m(⁴₂He) = 14.003074 u + 4.002603 u = 18.005677 u
Mass of products: m(¹⁷₈O) + m(¹₁p) = 16.999132 u + 1.007825 u = 18.006957 u
Mass difference = 18.005677 u - 18.006957 u = -0.001280 u
Q = -0.001280 u × 931.5 MeV/u ≈ -1.19 MeV. This reaction is endothermic and requires at least 1.19 MeV of kinetic energy from the alpha particle.

Cross Sections (σ)

In nuclear physics, the cross section (symbol σ) is a measure of the probability that a particular nuclear reaction will occur when a target nucleus is bombarded by a projectile particle. It can be visualized as the effective area that a target nucleus presents to the incident particle for a specific reaction to take place. A larger cross section means a higher probability of reaction.

The unit of cross section is the 'barn' (b), where 1 barn = 10^-28 m² = 10^-24 cm². The name 'barn' originated during the Manhattan Project as a code word, implying a large area ("as big as a barn door").

The number of reactions (N_R) occurring per unit time is given by:
N_R = Φ × N × σ
Where:

• Φ is the incident particle flux (number of incident particles per unit area per unit time).
• N is the number of target nuclei per unit volume.
• σ is the cross section for the specific reaction.

Cross sections are dependent on:

• The type of reaction (e.g., scattering, absorption, fission).
• The energy of the incident particle.
• The specific isotopes involved (both projectile and target).

Types of Cross Sections

Cross sections can be defined for specific types of interactions:

• Total Cross Section (σ_total): The probability of any interaction occurring.
• Scattering Cross Section (σ_s): The probability of the projectile being scattered by the target nucleus. This can be further divided into elastic scattering (kinetic energy conserved) and inelastic scattering (kinetic energy not conserved, often leading to excitation of the nucleus).
• Absorption Cross Section (σ_a): The probability of the projectile being absorbed by the target nucleus. This includes capture reactions, fission, and other processes where the projectile effectively disappears into the target.
• Fission Cross Section (σ_f): The probability of a fission-fission reaction occurring.
• Capture Cross Section (σ_c): The probability of the projectile being captured by the target nucleus, usually resulting in gamma emission.

These are related by: σ_total = σ_s + σ_a, and σ_a = σ_c + σ_f + ...

Energy Dependence of Cross Sections

The cross section is a function of the incident particle's energy, often denoted as σ(E). This relationship is crucial for designing nuclear reactors and other applications.

• Low Energy (Resonance Region): At specific low energies, the cross section can exhibit sharp peaks called resonances. These occur when the incident particle's energy matches an excited energy level of the compound nucleus formed by the interaction.
• High Energy (High Energy Region): At very high energies, the cross section generally decreases as the projectile interacts more like a particle passing through a target, rather than being 'captured' by the nuclear potential well.
• Fission Cross Section Behavior: For fissile materials like Uranium-235, the fission cross section is particularly high for slow (thermal) neutrons and decreases significantly at higher energies. This is why moderators are used in reactors to slow down neutrons.

Photonuclear Reactions

Photonuclear reactions are nuclear reactions induced by high-energy photons, typically gamma rays or X-rays. When a photon interacts with a nucleus, it can transfer its energy, potentially leading to nuclear transformations. These reactions are important in astrophysics, radiation physics, and medical imaging.

Types of Photonuclear Reactions

The most common types of photonuclear reactions involve the ejection of nucleons from the nucleus. The minimum energy required for these reactions is related to the binding energy of the emitted particle.

• (γ, n) Reaction (Photoneutron Emission): The absorption of a photon leads to the emission of a neutron. This is the most common photonuclear reaction because neutrons are not bound by the Coulomb force and are relatively easy to eject. The threshold energy for this reaction is approximately equal to the neutron binding energy in the nucleus.
  Example: ¹⁶₈O + γ → ¹⁵₈O + n The threshold energy is around 12.55 MeV for Oxygen-16.
• (γ, p) Reaction (Photoproton Emission): The absorption of a photon leads to the emission of a proton. This reaction requires a higher energy than (γ, n) because protons are repelled by the positively charged nucleus, meaning the threshold energy is higher (binding energy + Coulomb barrier).
  Example: ¹⁶₈O + γ → ¹⁵₇N + p The threshold energy is around 10.2 MeV for Oxygen-16, but often higher due to Coulomb effects.
• (γ, α) Reaction (Alpha Particle Emission): The absorption of a photon leads to the emission of an alpha particle. This requires even higher energies due to the larger binding energy and Coulomb repulsion of the alpha particle.
  Example: ²⁰₁₀Ne + γ → ¹⁶₈O + ⁴₂He
• (γ, 2n) Reaction: Emission of two neutrons. This occurs in heavier nuclei where the binding energy per nucleon is lower.
  Example: ²³⁸₉₂U + γ → ²³⁶₉₂U + 2n
• Photodisintegration: A general term for the breakup of a nucleus induced by photons, often referring to the emission of nucleons or clusters of nucleons.

Giant Dipole Resonance (GDR)

A significant feature of photonuclear reactions is the phenomenon of the Giant Dipole Resonance. This is an excited state of the nucleus that occurs when the protons and neutrons oscillate collectively out of phase. This oscillation creates an oscillating electric dipole moment, making the nucleus highly responsive to electromagnetic radiation. The GDR typically occurs in the energy range of 10-30 MeV for most nuclei.

The absorption of photons in the GDR region leads to a sharp increase in the cross section for photonuclear reactions, particularly (γ, n) and (γ, 2n) reactions. This absorption is highly efficient and is responsible for most of the photonuclear interactions at these energies.

The peak energy of the GDR is approximately proportional to Z^-1/3A^-1/6, where Z is the atomic number and A is the mass number. This means heavier nuclei tend to have GDR peaks at lower energies.

Applications of Photonuclear Reactions

• Radiochemical Analysis: Photonuclear reactions can be used for elemental analysis. For example, the (γ, n) reaction on an isotope can produce a radioisotope that can be detected and quantified.
• Medical Applications: Photonuclear reactions are used in some medical imaging techniques and in radiation therapy. For instance, the production of isotopes for PET scans can involve these reactions.
• Nuclear Astrophysics: Photonuclear reactions play a role in nucleosynthesis in stars, particularly in the formation of heavier elements.
• Industrial Applications: High-energy photon beams from linear accelerators can be used for non-destructive testing (radiography) of dense materials or for industrial radiography.