Nuclei, radioactivity, and particles

A nucleus packs Z protons and N neutrons inside a radius of roughly 1.2 A¹⁄³ fm — 7.4 fm for ²³⁸U, some 10⁴ times smaller than the atom around it. Bring two protons within 2 fm and their electrostatic energy is ke²/r = 1.44 MeV⋅fm ÷ 2 fm = 0.72 MeV, so whatever binds a nucleus has to beat that. Weigh the result and it comes out light. Two hydrogen atoms and two neutrons total 4.032980 u; the ⁴He atom measures 4.002603 u; the deficit of 0.030377 u at 931.494 MeV per u is 28.30 MeV of binding energy, or 7.07 MeV per nucleon. Set that beside the 13.6 eV needed to ionise hydrogen and the ratio is 5.2 × 10⁵. Same accounting, new scale — and that is why a nucleus dropping between its own levels emits a MeV gamma ray where an atom gave you visible light.

Total binding energy just tracks size, so divide it by A. B/A rises steeply through the light nuclei, peaks near A ≈ 56–62 at about 8.8 MeV per nucleon — ⁵⁶Fe is bound by 492.3 MeV, so 8.79 per nucleon — then sags to 7.57 MeV per nucleon at ²³⁸U, whose total is 1801.7 MeV. Read the curve as a landscape in which downhill means toward the peak. Fuse light nuclei and the product sits higher, so mass is released: assembling one ⁴He from four hydrogen atoms sheds 0.028697 u, or 26.7 MeV, which is what the Sun runs on. Split a heavy nucleus and the fragments land near 8.5 MeV per nucleon, releasing about 0.85 MeV × 236 ≈ 200 MeV per fission. Both directions release energy because the peak sits between them. Nothing in this survey explains the shape of that curve; measuring it is enough to say which way energy flows.

Take the parent's mass, subtract the products', multiply by c²: that is Q, and a decay is energetically allowed exactly when Q > 0. For ²³⁸U → ²³⁴Th + ⁴He, 238.050788 − (234.043601 + 4.002603) = 0.004584 u, so Q = 4.27 MeV. Momentum conservation then fixes the split: the parent is at rest, the fragments leave with equal and opposite momenta, so the alpha takes Q × 234/238 = 4.20 MeV and the thorium keeps 0.07 MeV — and 4.20 MeV is the line detectors actually record. Beta decay refused that pattern. For n → p + e⁻, Q = 939.565 − 938.272 − 0.511 = 0.782 MeV, yet the electrons emerge with every energy from zero up to that ceiling, which no two-body decay can produce. A third product carries the balance: the antineutrino. Gamma emission changes neither A nor Z, being a nucleus falling between its own MeV-spaced levels.

²³⁸U has Q = 4.27 MeV and a half-life of 4.5 × 10⁹ years, so λ = 4.9 × 10⁻¹⁸ s⁻¹. A free neutron has the smaller Q of 0.782 MeV and a half-life near 610 s. Released energy does not set the rate. What experiment supplies instead is a decay constant: every surviving nucleus has probability λ dt of decaying in any interval dt, independent of its age and of its neighbours. That one assumption gives dN/dt = −λN, hence N = N₀e^(−λt), t½ = ln2/λ, and activity A = λN. For ¹⁴C, t½ = 5700 y means λ = 0.693/(1.80 × 10¹¹ s) = 3.85 × 10⁻¹² s⁻¹; a gram of modern carbon holds about 6.5 × 10¹⁰ ¹⁴C nuclei and so ticks at 0.25 Bq. A sample down to 0.40 of that ratio has been decaying for (5700/ln2) × ln(1/0.40) = 7.5 × 10³ years.

A nuclear reaction or decay must conserve electric charge, nucleon number A, energy–momentum, angular momentum, and lepton number. Check n → p + e⁻ + ν̄ₑ: charge 0 = +1 − 1, A gives 1 = 1, and lepton number 0 = +1 − 1 once the antineutrino counts −1. That ledger is why the neutrino had to exist decades before anyone detected one. Momentum does more work than a Q value alone suggests. Rutherford's ¹⁴N + ⁴He → ¹⁷O + ¹H has Q = −1.19 MeV, so it is endothermic — but firing an alpha at a stationary ¹⁴N nucleus with exactly 1.19 MeV still fails, because the products must carry the incoming momentum and therefore keep kinetic energy. The lab threshold is |Q|(1 + m_α/m_N) = 1.19 × 1.286 = 1.53 MeV. Balance the ledgers first: they rule out most of what you might write down and cost nothing to check.

Nucleons are not fundamental. The current inventory is six quarks and six leptons in three generations, plus the carriers — gluon, photon, W±, Z⁰ — and the Higgs. Quarks carry charge +2/3 or −1/3 and appear only inside composites: baryons of three quarks, mesons of a quark and an antiquark. A proton is uud, a neutron udd, and β-minus decay is one d becoming a u by emitting a W⁻ that turns into e⁻ + ν̄ₑ, which is why A holds while Z rises by one. The quark rest masses sum to about 9 MeV/c², near 1% of the proton's 938 MeV/c²; the rest is field energy. Now the limits. Nothing here predicts λ, which needs barrier tunnelling and transition-rate theory; nothing here says which nuclides are bound, which needs the nuclear force and shell structure; cross sections and branching ratios sit outside it too. This is conservation-law accounting on measured inputs, and it should be quoted as such.