Structure and transformations of atomic nuclei: binding energy, radioactive decay law, fission and fusion, nuclear cross-sections, criticality, and reactor kinetics. Provides the physical substrate for the reactor-safety demo.
nuclear-physics
Radioactive decay law
A population of unstable nuclei decays exponentially: N(t) = N₀·e^(−λt), with decay constant λ and half-life t_½ = ln 2 / λ.
Nuclear binding energy (SEMF)
Semi-empirical mass formula: B(A,Z) = a_V·A − a_S·A^{2/3} − a_C·Z(Z−1)/A^{1/3} − a_A·(A−2Z)²/A + δ(A,Z). Predicts binding-energy curve with…
Nuclear cross-section
σ (units: barn = 10⁻²⁸ m²) encodes the effective target area for a nuclear reaction channel, a function of incident energy. Reaction rate =…
Nuclear fission
A heavy nucleus (e.g. ²³⁵U, ²³⁹Pu) captures a neutron and splits into lighter fragments plus ~2–3 prompt neutrons plus ~200 MeV of energy.…
Nuclear fusion
Light nuclei combine (e.g. D+T → ⁴He + n + 17.6 MeV) releasing energy because binding energy per nucleon rises toward iron. Requires…
Neutron flux
Φ = n·v (neutrons·cm⁻²·s⁻¹) is the neutron-density-times-speed scalar field that drives reaction rates. Primary directly measurable…
Criticality condition (k_eff = 1)
The effective multiplication factor k_eff = (neutrons produced in gen n+1) / (neutrons produced in gen n). k<1: subcritical (decaying);…
Delayed-neutron fraction β
Fraction of fission neutrons emitted by beta-decay of fission products rather than promptly. β ≈ 0.0065 for ²³⁵U. The slow timescale of…
Point-kinetics equations
Coupled ODEs for reactor power P and precursor concentration C_i: dP/dt = (ρ−β)/Λ·P + Σ λ_i C_i, dC_i/dt = β_i/Λ·P − λ_i C_i. ρ is…
Effective multiplication factor k_eff
Ratio of neutrons produced in one fission generation to neutrons lost to absorption + leakage in the previous generation. k_eff < 1…
Nuclear shell model
Independent-particle model of nucleons in a mean-field potential with strong spin–orbit coupling; explains magic numbers 2, 8, 20, 28, 50,…
Semi-empirical mass formula (liquid-drop)
Bethe–Weizsäcker mass formula: B(A,Z) = a_V A − a_S A^{2/3} − a_C Z²/A^{1/3} − a_A (A−2Z)²/A ± δ(A,Z). Reproduces bulk trends in binding…
Gamow factor (tunneling)
Exponential suppression of charged-particle fusion rates by Coulomb-barrier tunnelling: P ∝ exp(−2πη), η = Z_1 Z_2 e² / (ℏ v). Controls…
Semi-empirical mass (Weizsäcker) formula
B(A,Z) = a_V A - a_S A^(2/3) - a_C Z(Z-1)/A^(1/3) - a_A(N-Z)²/A ± δ/A^(1/2); ~1% accuracy; predicts valley of stability.
Nuclear shell model (magic numbers)
Spin-orbit splitting gives magic N,Z = 2,8,20,28,50,82,126; Mayer & Jensen (1963 Nobel); predicts spins, parities, moments.
Bohr–Mottelson collective model
Rotations and surface vibrations of deformed nuclei; E(I) = ℏ²I(I+1)/(2J); β, γ shape parameters; 1975 Nobel.
Gamow theory of α-decay
Gamow's 1928 tunneling theory of alpha decay: the half-life of an alpha emitter is controlled by the transmission coefficient through the…
Fermi theory of β-decay
4-fermion contact interaction with G_F; Fermi function F(Z,E) corrects spectrum; Kurie plot linearity confirms mass(ν)~0.
Fermi vs Gamow–Teller β transitions
ΔJ=0 (F, vector) and ΔJ=0,±1 (GT, axial); sum rule N-Z for F; Ikeda sum rule for GT; super-allowed 0⁺→0⁺ pin V_ud.
Double β-decay (2νββ and 0νββ)
2νββ observed τ ~ 10¹⁹-10²¹ yr (e.g. ⁷⁶Ge, ¹³⁶Xe); 0νββ would probe Majorana neutrino mass; KamLAND-Zen, GERDA, CUORE searches.
Liquid-drop fission
Surface-Coulomb competition: barrier disappears for Z²/A > 50; saddle, scission; mass distribution bimodal from shell effects.
Nuclear fusion (tokamak, ICF)
D+T → ⁴He+n+17.6 MeV; Lawson criterion nτT > 3×10²¹ keV·s/m³; ITER (2026) target Q=10; NIF achieved ignition 2022.
Giant resonances
Collective dipole, quadrupole, monopole oscillations at E ~ 79A^(-1/3) MeV; exhaust Thomas-Reiche-Kuhn sum rule.
Halo nuclei (¹¹Li, ⁶He)
Weakly bound valence nucleons form extended low-density halo; large reaction cross-sections; Efimov states in three-body subsystems.
Superheavy elements (Z ≥ 104)
Hot and cold fusion paths; island of stability near Z=114-126; Og (Z=118) confirmed; aiming for deformed doubly-magic ²⁹⁸Fl.
Neutron-star matter equations of state
χEFT constraints at nuclear densities; transition to hyperons, quark matter at 2-8 n_0; tidal deformability from GW170817.
Mössbauer effect
Recoilless γ emission in solids; μeV linewidth; measured Pound–Rebka gravitational redshift, hyperfine shifts in chemistry.
Nuclear magnetic resonance (physics)
Zeeman splitting of nuclear spins in B₀; ω_L = γB₀; chemical shift, J-coupling; Bloch equations for macroscopic magnetization.
Hartree–Fock–Bogoliubov for nuclei
Self-consistent mean field with pairing; Skyrme / Gogny effective interactions; basis of modern energy-density functional approach.
Nuclear isomers
High-K or spin-trap states with μs-yr lifetimes; ¹⁷⁸m²Hf (31 yr), ²²⁹mTh (8 eV) thorium isomer for nuclear clock.
Pygmy dipole resonance
Low-energy E1 strength in neutron-rich nuclei (~E_x ≈ 8 MeV); interpreted as neutron-skin oscillation; affects r-process rates.