Nuclear and Particle Physics: Binding, Moments, the Mass Formula, Nuclear Models, the Two-Nucleon Problem, Decays, Reactions, Detectors and the Quark Model

Nuclear radii follow R = r₀A^1/3 with r₀ ≈ 1.2 fm (3.6 fm for ²⁷Al), so the volume grows as A and the density, about 2.3 × 10¹⁷ kg/m³, is the same for all nuclei — the first sign that nuclear matter is like a liquid. The binding energy is B = [Zm_H + Nm_n − M(A, Z)]c²; for the deuteron, (1.007276 + 1.008665 − 2.013553) u × 931.494 MeV/u = 2.224 MeV (with the proton mass, since the deuteron mass is a nuclear mass). B/A rises steeply for light nuclei, peaks near ⁵⁶Fe–⁶²Ni at about 8.8 MeV, and falls slowly — so fusion of light nuclei and fission of heavy ones both release energy.

Nuclear magnetic moments are measured in nuclear magnetons μ_N = eħ/(2m_p), 1836 times smaller than μ_B. The proton has μ_p = +2.79 μ_N and the neutron μ_n = −1.91 μ_N; neither is the Dirac value (1 and 0), which shows the nucleons are composite. The electric quadrupole moment Q measures departure from spherical charge: Q > 0 for a prolate (cigar) nucleus, Q < 0 for oblate, and Q = 0 for spin 0 or ½. Nuclei far from closed shells are strongly deformed.

The liquid-drop model treats the nucleus as an incompressible charged drop, which gives the semi-empirical (Weizsäcker) mass formula B = a_vA − a_sA^2/3 − a_cZ(Z − 1)/A^1/3 − a_a(A − 2Z)²/A + δ. Each term has a reason: volume (saturation of short-range forces), surface (fewer neighbours at the surface), Coulomb repulsion, asymmetry (the Pauli principle favours N ≈ Z) and pairing (δ > 0 for even–even, 0 for odd A, < 0 for odd–odd). Setting ∂M/∂Z = 0 at fixed A gives the line of stability, which bends to N > Z for heavy nuclei as Coulomb repulsion grows. The same drop, deformed, explains fission: it happens when Coulomb energy wins over surface energy, roughly Z²/A ≳ 50 for spontaneous fission.

The shell model explains what the drop cannot: extra stability at the magic numbers 2, 8, 20, 28, 50, 82, 126, visible in binding energies, abundances and first excited states. Nucleons move independently in a mean potential; a harmonic-oscillator or Woods–Saxon well alone gives 2, 8, 20, 40, 70, 112, and the observed numbers need a strong spin–orbit term that pushes the j = l + ½ level of each high-l shell down into the shell below. Its prediction for an odd-A nucleus: the spin and parity are those of the last unpaired nucleon. ¹⁷O (one neutron beyond 16) has the neutron in 1d₅/₂, so 5/2⁺; ¹⁵N (a proton hole in 1p₁/₂) is 1/2⁻; parity is (−1)ˡ.

• Properties: short range (about 1–2 fm), strongly attractive at intermediate distance with a repulsive core below about 0.5 fm, charge-independent (pp, nn and np forces are nearly equal), spin-dependent, with a non-central tensor component, and saturating (B/A roughly constant).
• Yukawa’s meson theory: a force carried by a particle of mass m has range ħ/(mc); the pion (about 140 MeV/c²) gives about 1.4 fm, the range of the long-range attraction.
• The deuteron, the only bound two-nucleon state: B = 2.224 MeV, J^π = 1⁺, isospin 0; no excited bound state, and no bound pp or nn (the spin-singlet force is too weak). Its magnetic moment (0.857 μ_N) is close to but not equal to μ_p + μ_n = 0.880 μ_N, and its quadrupole moment is non-zero: the ground state is mostly ³S₁ with about 4% ³D₁ admixture, which only a tensor force can produce.

Decay is random with a constant probability per unit time λ: N = N₀e^−λt, activity A = λN, half-life t½ = ln 2/λ and mean life τ = 1/λ = t½/ln 2 (14.43 min for t½ = 10 min). An activity falling from 8000 Bq to 1000 Bq, a factor of 2³, has passed three half-lives. In a chain parent → daughter with λ_P ≪ λ_D, the daughter reaches secular equilibrium, A_D = A_P.

• α decay: (A, Z) → (A − 4, Z − 2) + α. Momentum sharing gives the α a kinetic energy T_α = Q(A − 4)/A — 4.78 MeV for ²²⁶Ra with Q = 4.87 MeV. The α tunnels through the Coulomb barrier (Gamow), so the half-life depends exponentially on Q^−1/2: the Geiger–Nuttall law, which spans more than twenty orders of magnitude in half-life.
• β decay: n → p + e⁻ + ν̄_e (β⁻), p → n + e⁺ + ν_e inside a nucleus (β⁺), and electron capture. The electron spectrum is continuous because three bodies share Q — the reason Pauli proposed the neutrino. Allowed transitions carry no orbital angular momentum: Fermi (ΔJ = 0, spins antiparallel) and Gamow–Teller (ΔJ = 0, ±1, not 0 → 0), both with no parity change. The weak interaction violates parity (Wu, ⁶⁰Co).
• γ transitions between nuclear levels are classified by multipolarity, EL or ML, with parity change (−1)ᴸ for EL and (−1)^L+1 for ML; rates fall steeply with L, so high-spin differences give long-lived isomers. Internal conversion ejects an atomic electron instead of a photon and is the only route for 0 → 0.

A point charge ze of kinetic energy E scattering from a nucleus Ze obeys dσ/dΩ = (zZe²/(4πε₀ · 4E))² / sin⁴(θ/2), and in a head-on collision it stops at the distance of closest approach d = zZe²/(4πε₀E): for a 5 MeV α on gold, d = 2 × 79 × 1.44/5 = 45.5 fm, far outside the nucleus, which is why Rutherford saw a point. Deviations at higher energy measure the nuclear radius.

• Conservation in reactions a + A → b + B: charge, baryon number (hence nucleon number), energy, momentum, angular momentum and parity (the strong and electromagnetic interactions conserve parity).
• Q-value Q = (m_a + m_A − m_b − m_B)c². An endoergic reaction (Q < 0) has a laboratory threshold E_th = −Q(1 + m_a/m_A) non-relativistically, because part of the projectile’s energy must go into the motion of the centre of mass. For ¹⁴N(α, p)¹⁷O, Q = −1.19 MeV and E_th = 1.19 × 18/14 = 1.53 MeV.
• Compound nucleus: at low energy the projectile is absorbed, the energy is shared, and the nucleus "forgets" how it was formed before decaying — which gives sharp resonances.
• Fission of ²³⁵U by a thermal neutron releases about 200 MeV, mostly as fragment kinetic energy, and 2–3 neutrons that sustain a chain reaction. Fusion of light nuclei needs temperatures of order 10⁸ K to overcome the Coulomb barrier by tunnelling; D + T → ⁴He + n releases 17.6 MeV, and the Sun burns hydrogen by the pp chain.

• Cyclotron: a particle in B circles at f = qB/(2πm), independent of radius, so a fixed-frequency field across the dees accelerates it every half-turn — 15.2 MHz for protons in 1 T. The energy is limited by the relativistic mass increase, which the synchrocyclotron (falling frequency) and synchrotron (rising field, fixed orbit) overcome. Linacs accelerate in a straight line; colliders give the highest centre-of-mass energy.
• Gas detectors: an ionisation chamber collects the primary ions; a proportional counter multiplies them in proportion to the deposited energy; a Geiger–Müller counter discharges fully, so its pulse size is independent of energy — it counts but cannot measure energy.
• Scintillation detectors (NaI(Tl)) convert energy to light read by a photomultiplier; semiconductor detectors (HPGe, cooled) create electron–hole pairs at about 3 eV each, and so resolve γ-ray energies far better than NaI.
• Cherenkov detectors: a charged particle faster than light in the medium (v > c/n) emits a cone of light at cos θ = 1/(nβ) — 48.2° for β → 1 in n = 1.5 — which gives a velocity threshold. Cloud and bubble chambers show tracks; calorimeters measure total energy.

Quarks carry charges +2/3 (u, c, t) and −1/3 (d, s, b), baryon number 1/3 and colour; hadrons are colour singlets. The Gell-Mann–Nishijima relation Q = I₃ + Y/2, with hypercharge Y = B + S, organises them into isospin multiplets: (p, n) is an I = ½ doublet, (π⁺, π⁰, π⁻) an I = 1 triplet. Isospin symmetry — u and d nearly degenerate — is why nuclear forces are charge-independent; it is conserved by the strong interaction and broken by electromagnetism.

C swaps particles and antiparticles, P inverts space and T reverses time. The weak interaction violates P and C separately (neutrinos are only left-handed, antineutrinos right-handed), CP is violated slightly, and the combination CPT is conserved by every local Lorentz-invariant field theory, which makes particle and antiparticle masses and lifetimes equal. Strange particles are made in pairs by the strong interaction (π⁻ + p → K⁰ + Λ, "associated production") and decay slowly by the weak one (Λ → p + π⁻).