Sections#

Review & Summary#

Discovering the Nucleus#

Rutherford scattering of alpha particles from thin metal foils showed that the atom has a small, dense, positively charged nucleus. Most alpha particles pass through with small deflection; the few large-angle scatterings indicate a compact core containing most of the mass.

Some Nuclear Properties#

A nucleus has \(Z\) protons and \(N\) neutrons. The mass number is \(A = Z + N\). The atomic mass unit is \(1\,\mathrm{u} = 931.5\,\mathrm{MeV}/c^2\). The binding energy \(E_b\) is the energy required to separate the nucleus into its nucleons:

(432)#\[ E_b = (\Delta m)c^2 \]

where \(\Delta m\) is the mass defect (difference between the sum of nucleon masses and the nuclear mass). The semi-empirical mass formula describes how \(E_b\) varies with \(A\) and \(Z\).

Radioactive Decay#

The number \(N\) of undecayed nuclei decreases exponentially with time:

(433)#\[ N = N_0 e^{-\lambda t} \]

where \(\lambda\) is the decay constant. The half-life is \(T_{1/2} = (\ln 2)/\lambda = 0.693/\lambda\). The activity (decay rate) is \(R = \lambda N = -dN/dt\), measured in becquerels (Bq; 1 Bq = 1 decay/s).

Alpha Decay#

A nucleus emits an alpha particle (\(^4\mathrm{He}\) nucleus). The decay energy \(Q\) is the total kinetic energy released. Quantum tunneling through the Coulomb barrier explains how alpha particles escape despite having energy less than the barrier height.

Beta Decay#

Beta-minus decay: \(n \to p + e^- + \bar{\nu}_e\). Beta-plus decay: \(p \to n + e^+ + \nu_e\). Electron capture: \(p + e^- \to n + \nu_e\). The neutrino (or antineutrino) is required to conserve energy, momentum, and lepton number.

Radioactive Dating#

Carbon-14 dating: Living organisms maintain a ratio \(^{14}\mathrm{C}/^{12}\mathrm{C}\); after death, \(^{14}\mathrm{C}\) decays (half-life 5730 years) and the ratio decreases. Other isotopes (e.g., uranium series) date older materials.

Measuring Radiation Dosage#

Absorbed dose: gray (Gy) = 1 J/kg of energy deposited. Equivalent dose: sievert (Sv) = Gy × RBE (relative biological effectiveness). Activity: becquerel (Bq) = 1 decay/s.

Nuclear Models#

Liquid-drop model: treats the nucleus as an incompressible fluid; explains fission and the semi-empirical mass formula. Shell model: nucleons occupy orbitals; explains magic numbers (2, 8, 20, 28, 50, 82, 126) and nuclear spin.