Unit 2: Beta & Gamma Decays and Nuclear Reactions

1. Beta Decay: Kinematics and Spectrum

Beta decay is a radioactive process in which a nucleus transforms by emitting an electron (beta-minus) or a positron (beta-plus). This process occurs to adjust the neutron-to-proton ratio for better stability.

Energy Kinematics

In beta-minus decay, a neutron transforms into a proton, an electron, and an antineutrino. The energy released (Q-value) is shared among these three particles.

The Beta Spectrum

Unlike alpha particles, which have discrete energies, beta particles are emitted with a continuous range of energies. The energy varies from zero up to a maximum value (the endpoint energy), which corresponds to the total Q-value of the reaction.

2. Positron Emission and Electron Capture

These are alternative modes of beta decay primarily observed in proton-rich nuclei.

• Positron Emission (Beta-plus): A proton inside the nucleus converts into a neutron, a positron, and a neutrino.
• Electron Capture: An inner-shell electron (usually from the K-shell) is captured by the nucleus, combining with a proton to form a neutron and emitting a neutrino.

3. Neutrino Hypothesis and Detection

The continuous nature of the beta spectrum and the apparent violation of energy and angular momentum conservation led Wolfgang Pauli to propose the Neutrino Hypothesis in 1930.

Definition: A neutrino is an elementary particle with extremely small (or zero) mass, no electric charge, and spin 1/2. It carries away the "missing" energy and momentum in beta decay.

Detection of Neutrino

Due to their weak interaction with matter, neutrinos are incredibly difficult to detect. The Rein & Cowans experiment (1956) provided the first experimental evidence of neutrinos using a large tank of water and detecting the results of inverse beta decay.

4. Gamma Decay and Internal Conversion

After alpha or beta decay, the daughter nucleus is often left in an excited state. It reaches the ground state by emitting high-energy electromagnetic radiation called Gamma Rays.

Gamma Ray Emissions

Gamma rays are emitted with discrete energies corresponding to the difference between nuclear energy levels.

Internal Conversion

Instead of emitting a gamma ray, the excited nucleus may transfer its energy directly to an inner orbital electron (K or L shell), which is then ejected from the atom. These are called Conversion Electrons.

5. Nuclear Reactions: Types and Conservation Laws

A nuclear reaction occurs when a target nucleus is bombarded by a projectile particle, resulting in a change in the nucleus.

Types of Reactions

• Elastic Scattering: Projectile and target remain unchanged; only kinetic energy is redistributed.
• Inelastic Scattering: The target is left in an excited state.
• Transmutation: The identity of the nucleus changes.
• Fission & Fusion: Splitting or joining of nuclei.

Conservation Laws

In every nuclear reaction, the following must be conserved:

1. Total Mass-Energy
2. Linear and Angular Momentum
3. Electric Charge (Z)
4. Nucleon Number (A)
5. Parity and Isospin (in certain cases)

6. Kinematics and Q-Value Analysis

The Q-value is the net energy released or absorbed in a nuclear reaction.

• Exoergic (Exothermic): Q > 0, energy is released.
• Endoergic (Endothermic): Q < 0, energy is absorbed. For these reactions, the projectile must have a minimum Threshold Energy to proceed.

7. Expression of Scattering Cross-Section

The Scattering Cross-Section (σ) is a measure of the probability that a specific nuclear reaction will occur when a target is hit by a beam of particles.

Concept: It is imagined as an "effective area" presented by the target nucleus to the incoming projectile. Its unit is the barn (1 barn = 10^-28 m^2).