JEE Physics Nuclei Notes Set 03

Radioactivity

Radioactivity is the spontaneous emission of radiation from the nucleus of an unstable atom. It was discovered by Becquerel and studied further by Marie and Pierre Curie. It is a nuclear property: it does not depend on temperature, pressure or chemical combination, and a given nuclide emits only one kind of radiation at a time in a particular decay.

Alpha, Beta and Gamma Rays

When the radiation from a source is passed through an electric or magnetic field it splits into three parts. Alpha rays bend toward the negative plate, beta rays toward the positive plate, and gamma rays go straight.

Radioactive Decays and Displacement Rules

1. Alpha decay: \( {}^{A}_{Z}\mathrm{X} \rightarrow {}^{A-4}_{Z-2}\mathrm{Y} + {}^{4}_{2}\mathrm{He} \). A falls by 4 and Z by 2. The energy released is Q = (m_X − m_Y − m_He)c², shared by the α-particle and the recoiling daughter in inverse ratio to their masses.
2. Beta-minus decay: \( {}^{A}_{Z}\mathrm{X} \rightarrow {}^{A}_{Z+1}\mathrm{Y} + e^{-} + \bar{\nu} \). A neutron changes into a proton (n → p + e⁻ + ν̄), so the n/p ratio decreases.
3. Beta-plus decay: \( {}^{A}_{Z}\mathrm{X} \rightarrow {}^{A}_{Z-1}\mathrm{Y} + e^{+} + \nu \). A proton changes into a neutron, so n/p increases. Electron capture (K-capture) has the same effect and also gives X-rays.
4. Gamma decay: an excited nucleus drops to a lower level by emitting a photon, with no change in A or Z. It usually follows α or β decay.

To find the numbers of particles emitted when a nucleus changes from \( {}^{A}_{Z}\mathrm{X} \) to \( {}^{A'}_{Z'}\mathrm{Y} \), first find the number of α-particles n_α = (A − A′)/4, then the number of β⁻ particles n_β = Z′ − (Z − 2n_α). For example, from U-238 to Pb-206: n_α = 8 and n_β = 82 − (92 − 16) = 6.

The neutrino and antineutrino are neutral, have almost zero rest mass and spin 1/2. Pauli proposed them to save the conservation of energy, momentum and angular momentum in β-decay, and they explain the continuous β spectrum.

Law of Radioactive Decay

The rate of decay at any instant is proportional to the number of undecayed nuclei: −dN/dt = λN, where λ is the decay constant, a constant of the nuclide that is independent of time and of any physical or chemical condition. Integrating gives N = N₀e^−λt. The number that has decayed in time t is N₀(1 − e^−λt). The decay constant is the probability of decay per nucleus per unit time.

Half-life and Mean Life

1. Half-life T_1/2 is the time in which half of the nuclei decay: T_1/2 = ln2/λ = 0.693/λ. After n half-lives, N = N₀/2ⁿ = N₀(1/2)^t/T.
2. Mean (average) life τ is the average lifetime of a nucleus: τ = 1/λ = T_1/2/0.693 = 1.44 T_1/2.
3. After one half-life 50% remains; after one mean life 37% remains and 63% has decayed.
4. If a nuclide can decay in two independent ways, the decay constants add: λ = λ₁ + λ₂, and so 1/T = 1/T₁ + 1/T₂.

Example. A sample has 640 nuclei at t = 0 and T_1/2 = 10 min. After 30 min (3 half-lives) N = 640/8 = 80. Nuclei decayed = 560, i.e. 87.5%. For a fraction 1/10 left the time is t = T log₂10 = 3.32 T.

Activity

Activity R is the number of decays per second: R = −dN/dt = λN = R₀e^−λt. Its SI unit is the becquerel (1 Bq = 1 decay per second); 1 curie = 3.7 × 10¹⁰ Bq. Activity halves in every half-life. For m grams of a substance of molar mass M, N = (m/M)N_A, so R = λ(m/M)N_A. Two samples have equal activity when N₁/T₁ = N₂/T₂.

Example. For 1 g of radium-226 (T_1/2 = 1600 years): N = 6.02 × 10²³/226 = 2.66 × 10²¹, λ = 0.693/(1600 × 3.15 × 10⁷ s) = 1.38 × 10⁻¹¹ s⁻¹, so R = λN = 3.7 × 10¹⁰ Bq = 1 curie.

Radioactive Equilibrium

In a chain A → B → C, the daughter B is formed at the rate at which A decays and decays at the rate λ_BN_B. When these two rates are equal, N_B stays constant and the activities of the members are equal: λ_AN_A = λ_BN_B. A stable end product has λ = 0 (infinite half-life).

Uses of Radioactive Isotopes

1. Radiocarbon dating: the fraction of carbon-14 left in a fossil, compared with living matter, gives its age (for example 1/8 left means 3 half-lives, about 17190 years).
2. Uranium and potassium-40 are used to date rocks and the Earth.
3. Cobalt-60 (γ-rays) is used for cancer therapy, iodine-131 for the thyroid, sodium-24 as a tracer of blood circulation.
4. Phosphorus-32 is used as a tracer in agriculture, and Geiger–Müller counters detect α and β particles.

Key Facts to Remember

1. N = N₀e^−λt, R = λN, T_1/2 = 0.693/λ, τ = 1/λ.
2. Number of α-particles from the change in A, then β from the change in Z.
3. α is positive and deflects toward the negative plate; β deflects toward the positive plate; γ is not deflected.
4. Decay constant and half-life cannot be altered by temperature, pressure or chemical state.
5. Parent and daughter in β-decay are isobars; in an (α + 2β) combination they are isotopes.