1. Introduction to the Nucleus

• ★The nucleus is the **densely packed center** of the atom containing almost all the mass (over 99.9%) and all the positive charge.
• ★The radius of the nucleus is about \(10^{-4}\) times smaller than the radius of the atom. This means the atom is **mostly empty space**.
• ★The volume of the nucleus is roughly \(10^{-12}\) times the volume of the atom.
• ★The nucleus contains **protons** and **neutrons**, collectively called **nucleons**.

2. Atomic Mass and Nuclear Composition

• ★Atomic Mass Unit (u): Atomic masses are extremely small, so the atomic mass unit is used for convenience.
• ★Definition: \(1~\text{u} = \frac{1}{12}\)th of the mass of a carbon-12 (\({}^{12}\text{C}\)) atom.
• ★Value: \(1~\text{u} = 1.660539 \times 10^{-27}~\text{kg}\)
• ★Constituents of the Nucleus:
• Proton (\(p\)): Mass = \(1.00727~\text{u}\), Charge = \(+e\), carries the positive charge of the nucleus.
• Neutron (\(n\)): Mass = \(1.00866~\text{u}\), Charge = 0 (neutral), discovered by James Chadwick in 1932. Free neutron is unstable.
• ★Nuclear Terms:
• Nuclide: represented as \({}_Z^A X\).
• Atomic Number (\(Z\)) = number of protons → determines element type.
• Neutron Number (\(N\)) = number of neutrons.
• Mass Number (\(A\)) = total nucleons → \(A = Z + N\).
• ★Isotopes, Isobars, and Isotones:
• Isotopes: Same \(Z\), different \(A\) → e.g., \({}_1^1\text{H}, {}_1^2\text{H}, {}_1^3\text{H}\).
• Isobars: Same \(A\), different \(Z\) → e.g., \({}_1^3\text{H}, {}_2^3\text{He}\).
• Isotones: Same \(N\), different \(Z\) → e.g., \({}_{80}^{198}\text{Hg}, {}_{79}^{197}\text{Au}\) (both \(N=118\)).

3. Size of the Nucleus

• ★Estimated via Rutherford α-particle scattering experiments: \(R < 4.0 \times 10^{-14}~\text{m}\).
• ★Nuclear radius formula: $$ R = R_0 A^{1/3}, \quad R_0 = 1.2 \times 10^{-15}~\text{m} = 1.2~\text{fm} $$
• ★Nuclear density is almost constant for all nuclei: $$ \rho \approx 2.3 \times 10^{17}~\text{kg/m}^3 $$

4. Mass-Energy Equivalence, Mass Defect, and Nuclear Binding Energy

• ★Mass-Energy Equivalence: Einstein showed that mass itself is a form of energy. Energy and mass are interchangeable. This is the foundation of nuclear energy.
• ★Mathematically: $$ E = mc^2, \quad c \approx 3 \times 10^8~\text{m/s} $$
• ★Mass Defect (\(\Delta M\)): When a nucleus forms, its mass is less than the sum of the masses of individual protons and neutrons. This difference is called the mass defect. $$ \Delta M = [Z m_p + (A-Z) m_n] – M $$
• ★Binding Energy (\(E_b\)): The mass defect is converted to energy as the nucleus forms. This energy holds the nucleus together. $$ E_b = \Delta M c^2 $$
• ★Energy equivalent of 1 u: \(1~\text{u} = 931.5~\text{MeV}\).
• ★Binding energy per nucleon: $$ E_{bn} = \frac{E_b}{A} $$
• ★Binding Energy Curve Insights:
  • Middle-mass nuclei (\(30 < A < 170\)) → \(E_{bn}\) almost constant → nuclear force is saturated.
  • Maximum stability → Iron-56 (\({}^{56}\text{Fe}\)), \(E_{bn} \approx 8.75~\text{MeV}\).
  • Light nuclei (\(A < 30\)) → lower \(E_{bn}\), fusion releases energy.
  • Heavy nuclei (\(A > 170\)) → lower \(E_{bn}\), fission releases energy.

5. Nuclear Force

• ★Strong attractive force between nucleons that binds the nucleus together.
• ★Much stronger than Coulomb or gravitational forces but acts over a very short range (a few femtometers).
• ★Potential energy minimum at separation \(r_0 \approx 0.8~\text{fm}\).
• ★Repulsive if nucleons get closer than \(r_0\) and attractive for separation greater than \(r_0\).
• ★Charge-independent: same force between p-p, n-n, and p-n.

6. Nuclear Energy and Reactions

• ★Energy released in nuclear reactions is in MeV, a million times higher than chemical reactions (~eV).
• ★Fission: Heavy nucleus splits into two or more smaller fragments, releasing energy.
  • Example: $$ {}_{0}^{1}n + {}_{92}^{235}\text{U} \rightarrow {}_{56}^{144}\text{Ba} + {}_{36}^{89}\text{Kr} + 3 {}_{0}^{1}n $$
  • Energy released ≈ 200 MeV per fission.
  • Applications: nuclear reactors (controlled fission), atomic bombs (uncontrolled fission).
• ★Fusion: Two light nuclei combine to form a heavier, more tightly bound nucleus.
  • Must overcome Coulomb repulsion → requires very high temperature (~\(3 \times 10^9\) K for protons).
  • Sun’s energy comes from proton-proton cycle: 4 H → 1 He + 26.7 MeV.

7. Radioactivity

• ★Spontaneous decay of unstable nuclei into more stable ones.
• ★Discovered by A. H. Becquerel in 1896.
• ★Types of decay:
  • α-decay: Emission of a helium nucleus \(({}_{2}^{4}\text{He})\).
  • β-decay: Emission of electrons (β−) or positrons (β+).
  • γ-decay: Emission of high-energy photons (γ-rays), no change in A or Z.