Nuclear Decay and Radioactivity: Structure, Stability, and Decay Processes

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 $1.673 \times 10^{-27}$ kg.

• Neutrons: Electrically neutral particles with mass $1.675 \times 10^{-27}$ kg.

• Protons and neutrons are collectively called nucleons.

Nuclear Notation

• Nuclei are represented as $^{A}_{Z}X$, where:

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

  • $Z$ = atomic number (number of protons)

  • $A$ = 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 ($Z$).

Mass-Energy Equivalence

• Einstein's equation: $E_0 = mc^2$

• $E_0$ 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): $1\ \text{u} = 1.661 \times 10^{-27}\ \text{kg}$

• By convention, $^{12}$C has a mass of exactly 12 u.

• MeV/$c^2$: $1\ \text{u} = 931.5\ \text{MeV}/c^2$

• Rest mass energy: $E_0 = (1\ \text{u})c^2 = 931.5\ \text{MeV}$

Particle	kg	u	MeV/$c^2$
Proton	1.67262 × 10−27	1.00727	938.27
Neutron	1.67493 × 10−27	1.00866	939.57
Electron	9.10939 × 10−31	0.0005486	0.511

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 (≈ $10^{-15}$ m).

Nuclear Stability

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

• No stable nuclei exist for $Z > 83$ (bismuth).

• As $Z$ 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.

Radiation	Identification	Charge	Stopped by
Alpha, $\alpha$	$^4$He nucleus	+2e	Sheet of paper
Beta, $\beta$	Electron or positron	±e	Few mm of aluminum
Gamma, $\gamma$	High-energy photon	0	Many cm of lead

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 $Z < 8$.

• Example: $^{12}$C has closed n=2 shells for both protons and neutrons (6 each).

• For $^{12}$N and $^{12}$B, the extra nucleon occupies the next available shell, leading to beta decay for increased stability.

Beta Decay

• Two types: Beta-minus ($\beta^-$) and Beta-plus ($\beta^+$) 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: $^{A}_{Z}X \rightarrow ^{A}_{Z+1}Y + e^- + \bar{\nu}$

• Beta-plus: $^{A}_{Z}X \rightarrow ^{A}_{Z-1}Y + e^+ + \nu$

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 ($^4$He nucleus) to become more stable.

• Alpha decay is a quantum mechanical tunneling process.

Alpha decay equation:

• $^{A}_{Z}X \rightarrow ^{A-4}_{Z-2}Y + \alpha + \text{energy}$

Energy released:

• $\Delta E \approx K_{\alpha} = (m_X - m_Y - m_{He})c^2$

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: $\sim 10^{-10}$ s.

Decay Rate and Activity

Decay Rate (r)

• Probability per second that a nucleus will decay.

• Example: If $r = 0.1$ Hz, there is a 10% chance of decay per second.

Activity (R)

• Number of decays per second in a sample.

• $R(t) = r N(t)$

• $R(t) = -\frac{dN}{dt}$

Exponential Decay Law

• The number of undecayed nuclei at time $t$:

$N(t) = N_0 e^{-rt}$

• $N_0$ = initial number of nuclei

• $r$ = decay rate

Half-Life ($t_{1/2}$)

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

• Relationship to decay rate:

$t_{1/2} = \frac{\ln(2)}{r}$

Alternative form for number of nuclei remaining:

$N(t) = N_0 \left(\frac{1}{2}\right)^{t/t_{1/2}}$

Summary Table: Types of Radioactive Decay

Decay Type	Process	Particles Emitted	Example
Alpha ($\alpha$)	Loss of 2 protons, 2 neutrons	$^4$He nucleus	$^{238}_{92}$U $\rightarrow$ $^{234}_{90}$Th + $^4$He
Beta-minus ($\beta^-$)	Neutron $\rightarrow$ proton	Electron, antineutrino	$^{14}$C $\rightarrow$ $^{14}$N + $e^-$ + $\bar{\nu}$
Beta-plus ($\beta^+$)	Proton $\rightarrow$ neutron	Positron, neutrino	$^{11}$C $\rightarrow$ $^{11}$B + $e^+$ + $\nu$
Gamma ($\gamma$)	De-excitation of nucleus	Photon	$^{60}$Co* $\rightarrow$ $^{60}$Co + $\gamma$

Applications and Examples

• Medical Imaging: Use of isotopes like $^{131}$I in nuclear medicine.

• Radiocarbon Dating: $^{14}$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.