Nuclear Physics and Radioactivity

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Nuclear physics studies the atomic nucleus, while radioactivity describes the spontaneous decay of unstable nuclei emitting alpha (⁴He), beta (e⁻/e⁺) or gamma (photon) radiation. The nucleus contains Z protons and N = A − Z neutrons, where A is the mass number.

The must-know relations for ECAT:

Decay law: N(t) = N₀·e^(−λt); Activity: A(t) = λN(t) = A₀·e^(−λt)
Half-life: T½ = ln 2 / λ ≈ 0.693 / λ
Mass defect → Binding energy: E_b = Δm·c², with 1 u ≡ 931.5 MeV/c²

After n half-lives, the fraction of nuclei remaining is (1/2)ⁿ, and activity drops by the same factor. Mid-mass nuclei peak the binding-energy-per-nucleon curve near 8.5 MeV, which is why both fission of heavy nuclei and fusion of light nuclei release energy.

🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Atomic nucleus and binding energy

The nucleus holds Z protons and (A − Z) neutrons together against Coulomb repulsion via the strong nuclear force, which is short-ranged (≈1–3 fm). The mass defect Δm is the difference between the summed rest masses of the separated nucleons and the actual nuclear mass:

Δm = Z·m_p + (A − Z)·m_n − M_nucleus

Binding energy E_b = Δm·c². Dividing by A gives the binding energy per nucleon, which peaks near 8.5 MeV for A ≈ 56 (Fe region) and falls off for both very light and very heavy nuclei. This curve explains why fission of heavy nuclei (A > 200) and fusion of light nuclei (A < 20) release energy.

Radioactive decay modes

Alpha decay: parent → daughter + ⁴He²⁺; A drops by 4, Z by 2 (e.g., ²³⁸U → ²³⁴Th + α).
Beta-minus decay: n → p + e⁻ + anti-νₑ; A unchanged, Z increases by 1.
Gamma decay: excited nucleus drops to a lower energy state emitting a high-energy photon; A and Z unchanged.

Each nuclide has a characteristic decay constant λ (s⁻¹), the probability per unit time that any given nucleus decays. Activity A = λN therefore decreases with the same exponential law as N.

Worked decay calculation

A sample initially contains N₀ = 8.0 × 10¹⁰ nuclei of a nuclide with T½ = 6.0 h. After 18 h (three half-lives):

Remaining nuclei: N = N₀·(1/2)³ = 8.0×10¹⁰ × 0.125 = 1.0 × 10¹⁰
λ = 0.693 / (6.0 × 3600 s) ≈ 3.21 × 10⁻⁵ s⁻¹
Initial activity: A₀ = λN₀ ≈ 2.57 × 10⁶ Bq; after 18 h it is 1/8 of this, ≈ 3.2 × 10⁵ Bq.

Common ECAT traps

Trap	Correct treatment
λ and T½ are the same	T½ = 0.693/λ, so smaller λ ⇒ longer half-life
A and N confused	A = λN; both fall by (1/2)ⁿ after n half-lives
Mass in u, answer wanted in MeV	Multiply Δm (u) by 931.5 MeV/u
Alpha = electron or neutron	Alpha is specifically a ⁴He nucleus (2p + 2n)
N/Z = 1 always stable	Only true for light stable nuclei; heavy ones need N/Z ≈ 1.5

🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Q-value and reaction energetics

For any nuclear reaction, the Q-value is Q = (m_reactants − m_products)c². Q > 0 means the reaction is exoergic (releases kinetic energy), Q < 0 is endoergic and requires a threshold energy. Fission of ²³⁵U by thermal neutrons has Q ≈ 200 MeV distributed mostly among fission fragments; fusion of D + T → ⁴He + n releases 17.6 MeV. The asymmetry arises because mid-mass products sit higher on the binding-energy-per-nucleon curve than either heavy parents or light fusion reactants.

Half-life reasoning and statistics

Because N(t) is exponential, plotting ln N vs t yields a straight line of slope −λ. ECAT MCQs often disguise this as a graph-read or as “after how many half-lives does activity drop to 1%?” Since 1% ≈ (1/2)ⁿ gives n = log(0.01)/log(0.5) ≈ 6.64 half-lives, students who answer “7 half-lives ≈ 1.56%” or “6 half-lives ≈ 1.56%” must distinguish between the nearest integer and the exact n.

Chain reaction and critical mass

A self-sustaining chain reaction requires that, on average, at least one neutron from each fission event induces another fission. Below critical mass, too many neutrons escape the surface; the multiplication factor k < 1. Reactors control k ≈ 1 using moderators (graphite, water, heavy water) to slow neutrons so ²³⁵U captures them efficiently.

Edge cases examiners exploit

Isobars (same A, different Z), isotopes (same Z, different A), and isotones (same N) are routinely mixed in MCQ stems.
Background radiation (~2–3 mSv/yr natural) must be subtracted from raw counts to get true activity.
1 Ci = 3.7 × 10¹⁰ Bq (legacy curie unit), not 1 Bq — a known trap.

Practice prompts

A radioactive source has A₀ = 4.0 × 10⁸ Bq and T½ = 5 days. Find λ in s⁻¹ and A after 15 days.
For ⁷Li (M = 7.01600 u, Z = 3, A = 7), using m_p = 1.00728 u and m_n = 1.00867 u, compute the binding energy per nucleon in MeV.

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Nuclear Physics and Radioactivity