Nuclei

Nuclei

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

The atomic nucleus is the dense, positively charged core made of protons (Z) and neutrons (N) called nucleons. Its mass number A = Z + N, and the radius follows R = R₀ · A^(1/3) with R₀ ≈ 1.2 fm, giving a near-constant nuclear density ≈ 2.3 × 10¹⁷ kg/m³. The mass defect (Δm) converts into binding energy via B.E. = Δm · c², and the binding energy per nucleon peaks near ⁵⁶Fe (~8.8 MeV), explaining why both fission of heavy nuclei and fusion of light nuclei release energy. Radioactivity is first-order decay: N(t) = N₀ e^(–λt), with half-life T₁/₂ = 0.693 / λ. NEET essentials: α-decay (ΔA = 4, ΔZ = 2), β-decay (ΔZ = ±1, A unchanged), γ-emission (only energy change), and conservation of A and Z in nuclear reactions.

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Composition and Size

Every nucleus is described by (Z, A). Protons repel each other electrostatically, so the nucleus would fly apart without the strong nuclear force — a short-ranged (≈ 1–3 fm), charge-independent force much stronger than the Coulomb force at femtometre distances but negligible beyond a few nucleon diameters. Because R ∝ A^(1/3), the volume V ∝ A, which means nuclear density ρ ≈ 2.3 × 10¹⁷ kg/m³ is roughly the same for every nuclide from hydrogen to uranium.

Mass Defect and Binding Energy

The mass defect Δm = [Z·m(¹H) + N·m(n) − m(atom)] is the mass lost when isolated nucleons assemble into a nucleus. By Einstein’s relation, this lost mass appears as binding energy B.E. = Δm · c². Dividing by A gives the binding energy per nucleon, the key indicator of stability.

The B.E./A Curve

B.E./A is small for very light nuclei (≈ 1.1 MeV in deuteron), rises sharply to a maximum of about 8.8 MeV around A ≈ 56 (Fe region), then slowly falls to ≈ 7.6 MeV for ²³⁵U. Consequence: moving a heavy nucleus toward the iron peak by fission, or moving two light nuclei toward it by fusion, both release energy.

Radioactive Decay

Radioactivity is spontaneous, statistical, and unaffected by temperature, pressure or chemical state. It obeys first-order kinetics:

N(t) = N₀ e^(–λt), Activity R = λN, T₁/₂ = ln 2 / λ = 0.693 / λ.

Decay	Emitted	ΔZ	ΔA	Mechanism
α	⁴₂He	–2	–4	Tunnelling out of heavy nucleus
β⁻	e⁻ + ν̄ₑ	+1	0	n → p + e⁻ + ν̄ₑ
β⁺ / EC	e⁺ (or K-capture)	–1	0	p → n + e⁺ + νₑ
γ	photon	0	0	De-excitation of nucleus

Nuclear Reactions and Q-value

For a reaction a + X → Y + b, A and Z are balanced on both sides. The Q-value = (m_initial − m_final)c²; Q > 0 ⇒ exoergic.

Worked Tip for NEET

Q: Mass of ⁴₂He = 4.0026 u, 2m(¹H) + 2m(n) = 4.0329 u. B.E. per nucleon = ? Δm = 4.0329 − 4.0026 = 0.0303 u → B.E. = 0.0303 × 931.5 ≈ 28.2 MeV → per nucleon ≈ 7.05 MeV.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Why the Nuclear Force is Special

The Yukawa picture treats the force as mediated by pions (mass ≈ 140 MeV/c²), giving a range R ≈ ℏ/(m_π c) ≈ 1.4 fm — consistent with experiment. The force is saturated (each nucleon binds only to a few neighbours, not all), spin-dependent, and slightly stronger between n–p pairs than n–n or p–p pairs, which is why stable light nuclei prefer nearly equal neutron and proton counts. Deviations produce β-instability: nuclei with too many neutrons undergo β⁻ decay; those with too few undergo β⁺ decay or electron capture.

Edge Cases in Binding Energy

The pairing term favours even–even nuclei (most stable, e.g. ¹²C, ⁵⁶Fe), while odd–odd nuclei (e.g. ²H, ¹⁴N, ⁴₀K) are rarest. Magic numbers (2, 8, 20, 28, 50, 82, 126) mark closed shells analogous to noble gases — nuclei such as ⁴He, ¹⁶O, ⁴⁰Ca and ²⁰⁸Pb sit at local B.E./A peaks.

Decay Chains and Secular Equilibrium

A heavy parent like ²³⁸U (T₁/₂ ≈ 4.5 × 10⁹ yr) cascades through α and β decays to stable ²⁰⁶Pb. When the parent half-life is much longer than any daughter’s, the secular equilibrium condition λ₁N₁ = λ₂N₂ holds, making the daughter’s activity equal to the parent’s.

Fission vs. Fusion Mechanics

In fission of ²³⁵U by thermal neutrons, the B.E./A difference between the parent and mid-mass fragments releases ≈ 200 MeV, plus 2–3 neutrons that sustain a chain reaction if the multiplication factor k ≥ 1 (controlled in reactors, uncontrolled in bombs). In fusion (e.g. D + T → ⁴He + n + 17.6 MeV), the reaction needs ≈ 10⁸ K to overcome Coulomb repulsion — this is the Sun’s pp-chain and the goal of tokamak reactors like ITER.

Common NEET Traps

1. Forgetting that γ-emission does not change Z or A — students often wrongly shift mass number.
2. Confusing activity (decays/sec) with decay constant λ (probability/sec).
3. Treating half-life as constant only when the parent has no replenishment; mixed sources do not simply add half-lives.
4. Missing that β⁺ decay needs 1.022 MeV minimum (2m_ec²) because the emitted positron pair-produces; electron capture can occur below this threshold.

Practice Prompts

• Show that nuclear density is independent of A using R = R₀A^(1/3), m ≈ A·u, and evaluate the numerical value.
• A sample has activity 6000 Bq at t = 0 and 375 Bq after 2 hours; find T₁/₂ and the decay constant, then the activity at t = 4 h.

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