Chapter 42: Nuclear Physics – Structure, Stability, and Radioactivity

Nuclear Structure and Properties

The Atomic Nucleus

The nucleus is the dense central core of an atom, containing protons and neutrons (collectively called nucleons). It is much smaller than the atom itself, with a typical nuclear radius on the order of $10^{-14}$ m, compared to the atomic radius of about $10^{-10}$ m.

• Protons have a positive charge (+e), while neutrons are neutral.

• The atomic number (Z) is the number of protons, defining the element.

• The mass number (A) is the total number of nucleons: $A = Z + N$ (N = number of neutrons).

Atomic Mass and Isotopes

Atomic masses are measured in atomic mass units (u), where $1\ \text{u} = 1.6605 \times 10^{-27}\ \text{kg} = 931.49\ \text{MeV}/c^2$. Isotopes are atoms of the same element (same Z) with different mass numbers (A). Isobars are nuclei with the same A but different Z and N.

• Most isotopes are radioactive; only 266 are stable in nature.

• The chemical atomic mass is a weighted average of all naturally occurring isotopes.

Nuclear Size and Density

The radius of a nucleus is given by $R = r_0 A^{1/3}$, where $r_0 \approx 1.2 \times 10^{-15}$ m. The volume is proportional to the number of nucleons, implying a nearly constant nuclear density for all nuclei.

• Nuclear matter density: $\rho_{nuc} \approx 2.3 \times 10^{17}\ \text{kg/m}^3$.

• Nucleons are tightly packed and incompressible.

Nuclear Stability and Binding Energy

Stability of Nuclei

Stable nuclei cluster along the line of stability in a plot of neutron number (N) vs. proton number (Z). For light elements (Z < 16), stability occurs when N ≈ Z. For heavier elements, more neutrons are needed for stability. No nuclei with Z > 83 are stable.

Nuclear Binding Energy

The binding energy is the energy required to disassemble a nucleus into its constituent protons and neutrons. It is calculated from the mass defect (the difference between the mass of the nucleus and the sum of the masses of its nucleons):

• $B = [Zm_p + Nm_n - m_{nuc}] \times 931.49\ \text{MeV/u}$

• The binding energy per nucleon peaks at about 8.8 MeV for nuclei near iron ($^{56}$Fe), indicating maximum stability.

The Strong Nuclear Force

Nature of the Strong Force

The strong nuclear force is the fundamental force that binds nucleons together. It is:

• Attractive between any two nucleons (proton-proton, neutron-neutron, or proton-neutron).

• Short-ranged (acts over distances ~1-3 femtometers).

• Much stronger than the electrostatic repulsion between protons at short distances.

Nuclear Models

The Shell Model

The shell model treats nucleons as moving independently in an average potential well created by the strong force. Energy levels are quantized, similar to electrons in atoms. Closed shells correspond to particularly stable nuclei.

• Proposed by Maria Goeppert-Mayer (Nobel Prize, 1963).

• Neutrons and protons fill separate shells.

Radioactivity and Nuclear Decay

Types of Radiation

Radioactive decay emits three main types of radiation:

Radiation	Identification	Charge	Stopped by
Alpha (α)	$^4$He nucleus	+2e	Sheet of paper
Beta (β)	Electron	−e	Few mm of aluminum
Gamma (γ)	High-energy photon	0	Many cm of lead

Ionizing Radiation and Detection

Ionizing radiation can break molecular bonds and ionize atoms, which is the principle behind radiation detectors like the Geiger counter.

Nuclear Decay and Half-Life

The probability that a nucleus will decay in a given time is characterized by its decay rate (r) and lifetime ($\tau = 1/r$). The half-life ($t_{1/2}$) is the time for half the nuclei in a sample to decay. The number of nuclei remaining at time t is:

• $N = N_0 e^{-t/\tau}$

• Or, in terms of half-life: $N = N_0 (1/2)^{t/t_{1/2}}$

Radioactive Dating

Carbon-14 dating uses the known half-life of $^{14}$C (5730 years) to date organic materials by comparing the $^{14}$C/$^{12}$C ratio in a sample to that in living organisms.

Types of Nuclear Decay

Alpha Decay

In alpha decay, a nucleus emits an alpha particle ($^4$He nucleus), reducing its atomic number by 2 and mass number by 4:

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

• The energy released is primarily carried by the alpha particle.

Beta Decay

There are two types of beta decay:

• Beta-minus decay: A neutron converts to a proton, emitting an electron ($e^-$) and an antineutrino ($\bar{\nu}$): $^A_ZX \rightarrow ^A_{Z+1}Y + e^- + \bar{\nu}$

• Beta-plus decay: A proton converts to a neutron, emitting a positron ($e^+$) and a neutrino ($\nu$): $^A_ZX \rightarrow ^A_{Z-1}Y + e^+ + \nu$

Gamma Decay

In gamma decay, an excited nucleus releases energy by emitting a high-energy photon (gamma ray), without changing its atomic or mass number.

Decay Series

Some radioactive nuclei decay through a series of steps, producing a sequence of daughter nuclei until a stable isotope is reached. This is called a decay series.

Radiation Dose and Applications

Radiation Dose and Biological Effects

The absorbed dose is the energy deposited per kilogram of tissue (measured in grays, Gy). The dose equivalent (in sieverts, Sv) accounts for the type of radiation using the relative biological effectiveness (RBE):

• $\text{Dose equivalent (Sv)} = \text{Absorbed dose (Gy)} \times \text{RBE}$

• Alpha particles have a much higher RBE than x-rays or gamma rays.

Medical and Industrial Applications

• Radiation therapy uses focused gamma rays to kill cancer cells while minimizing damage to healthy tissue.

• Magnetic Resonance Imaging (MRI) uses the magnetic properties of protons in a magnetic field to produce detailed images of internal organs.

Summary Table: Properties of Protons and Neutrons

Property	Proton	Neutron
Number	Z	N
Charge (q)	+e	0
Spin (s)	1/2	1/2
Mass (u)	1.00728	1.00866

Additional info: This guide covers the essential concepts of nuclear structure, stability, radioactivity, and applications, as outlined in a typical college-level physics curriculum.