Nuclear Decay

Introduction

Nuclear decay refers to the spontaneous transformation of an unstable atomic nucleus into a more stable one, accompanied by the emission of particles or electromagnetic radiation. This process is fundamental to understanding nuclear physics and radioactivity.

Describing the Nucleus

Constituents of the Nucleus

• Protons: Positively charged particles with charge +e and mass kg.

• Neutrons: Electrically neutral particles with mass kg.

• Protons and neutrons are collectively called nucleons.

Nuclear Notation

• Nuclei are represented as , where:

  • = chemical symbol (e.g., H, He, N)

  • = atomic number (number of protons)

  • = mass number (number of protons + neutrons)

Periodic Table and Atomic Structure

The periodic table organizes elements by increasing atomic number. Each element's identity is determined by its number of protons ().

Mass-Energy Equivalence

• Einstein's equation:

• is the rest energy of a particle, representing the energy due to its mass alone.

• Mass and energy are interchangeable, provided total energy is conserved.

• Mass is not strictly conserved; it can be converted to energy and vice versa.

Common Units of Mass

• Atomic mass unit (u):

• By convention, C has a mass of exactly 12 u.

• MeV/:

• Rest mass energy:

Nuclear Structure

• Protons repel each other due to electrostatic (Coulomb) force.

• Strong nuclear force: Attractive force between nucleons, much stronger than electrostatic repulsion but only effective at very short ranges (≈ m).

Nuclear Stability

• Stable nuclei cluster near the line of stability (N ≈ Z for light elements).

• No stable nuclei exist for (bismuth).

• As increases, more neutrons are needed for stability.

Radioactivity

• Unstable nuclei spontaneously emit particles or photons to become more stable.

• Radioactive decay: Spontaneous emission of particles or high-energy photons from unstable nuclei.

The Shell Model

• Proposed by Maria Goeppert-Mayer (1949).

• Each nucleon moves independently in an average potential due to the strong force from all other nucleons.

• Energy levels (shells) exist for protons and neutrons, similar to electron shells in atoms.

Potential Energy Wells

• Neutron potential well depth: ≈ 50 MeV for all nuclei.

• Proton potential well is "lifted" by electrostatic repulsion, especially for high-Z nuclei.

Low-Z Nuclei

• Energy levels for neutrons and protons are nearly identical for .

• Example: C has closed n=2 shells for both protons and neutrons (6 each).

• For N and B, the extra nucleon occupies the next available shell, leading to beta decay for increased stability.

Beta Decay

• Two types: Beta-minus () and Beta-plus () decay.

• Beta-minus: Neutron transforms into a proton, emitting an electron and an antineutrino.

• Beta-plus: Proton transforms into a neutron, emitting a positron and a neutrino.

• Neutrinos are emitted to conserve momentum and energy.

General equations:

• Beta-minus:

• Beta-plus:

High-Z Nuclei and Alpha Decay

• High-Z nuclei have higher proton potential energy due to electrostatic repulsion.

• When too many nucleons are present, the nucleus may eject an alpha particle (He nucleus) to become more stable.

• Alpha decay is a quantum mechanical tunneling process.

Alpha decay equation:

Energy released:

Decay Series

• Some decay products are themselves radioactive, leading to a decay series until a stable isotope is reached.

• Multiple decay paths may exist.

Excited Nuclei

• Radioactive decay often leaves the nucleus in an excited state.

• Stability is achieved by emitting a high-energy photon (gamma decay).

• Typical half-life of excited nuclear state: s.

Decay Rate and Activity

Decay Rate (r)

• Probability per second that a nucleus will decay.

• Example: If Hz, there is a 10% chance of decay per second.

Activity (R)

• Number of decays per second in a sample.

Exponential Decay Law

• The number of undecayed nuclei at time :

• = initial number of nuclei

• = decay rate

Half-Life ()

• The time required for half the nuclei in a sample to decay.

• Relationship to decay rate:

Alternative form for number of nuclei remaining:

Summary Table: Types of Radioactive Decay

Applications and Examples

• Medical Imaging: Use of isotopes like I in nuclear medicine.

• Radiocarbon Dating: C decay used to date archaeological samples.

• Nuclear Power: Controlled fission reactions in reactors.

Additional info: The notes also include worked example questions and data tables for practice, as well as references to textbook appendices for nuclear data.