Q values, two-body kinematics, and conservation ledgers

Q = (Σmᵢₙᵢₜᵢₐₗ − Σm_final)c², with 1 u c² = 931.494 MeV. Mass tables list neutral atoms, so every entry carries Z electrons, and a Q value is right only once they balance. In α decay they do: Z on the left, Z − 2 in the daughter atom plus 2 in the helium atom. In β⁻ decay the daughter atom's extra electron is the one emitted, so Q = [M(Z) − M(Z + 1)]c² as it stands. Electron capture takes its electron from the parent's own cloud, so again the bare difference is Q, less the binding energy of that electron, which returns as X-rays or Auger electrons. β⁺ decay is the exception: the daughter atom has one electron too many and a positron is created as well, so 2mₑc² = 1.022 MeV comes off. For ¹⁸F → ¹⁸O the atomic difference is 0.0017777 u, or 1.656 MeV: that is Q for capture, while β⁺ gets only 0.634 MeV. Below 1.022 MeV, β⁺ is closed altogether: ⁷Be → ⁷Li has 0.862 MeV and decays by capture alone.

A parent at rest has zero momentum, so two fragments leave back to back with equal momenta. Nonrelativistically K = p²/2m, so K₁/K₂ = m₂/m₁, and with K₁ + K₂ = Q the light fragment takes K₁ = Q m₂/(m₁ + m₂). For ²²⁶Ra → ²²²Rn + α, Q = 4.8706 MeV; the α leaves with 4.8706 × 222.018/226.020 = 4.784 MeV and the radon recoils with 0.086 MeV. Two things follow. The α energy is fixed, not merely likely: every decay to the same final state emits the same energy, so α spectra are lines, and a line below the ground-state value labels an excited level of the daughter. And the recoil is small only beside Q: 86 keV is some 10⁴ times a chemical bond energy, so the daughter atom is torn out of whatever molecule held its parent.

Relativity keeps the logic and changes only the energy–momentum relation. With E₁ + E₂ = Mc², equal momenta, and E² − p²c² = m²c⁴ for each fragment, E₁ = (M² + m₁² − m₂²)c²/2M, so K₁ = Q(Q + 2m₂c²)/(2Mc²). When Q ≪ m₂c² this tends to Q m₂/(m₁ + m₂), the nonrelativistic split, as correspondence demands; for ²²⁶Ra the two differ by about one part in 10⁵. Where a fragment is massless the old formula fails outright. In π⁺ → μ⁺ + ν_μ, Q = (139.570 − 105.658) MeV = 33.912 MeV, and with m₂ = 0 the exact result is K_μ = Q²/(2Mc²) = 4.12 MeV, while Q m₂/(m₁ + m₂) would give zero. A third body breaks the argument differently: three momenta can sum to zero in endlessly many ways, so β decay gives a continuous electron spectrum, and kinematics fixes only its endpoint.

Fire projectile a at target X at rest. The invariant s = Eₜₒₜ² − pₜₒₜ²c² is the squared energy in the centre-of-mass frame, and here s = (mₐc² + m_Xc²)² + 2m_Xc² Kₐ. The reaction is just possible when √s equals the products' rest energy, all of them created at rest in that frame. Solving gives Kₜₕ = −Q(1 + mₐ/m_X) + Q²/(2m_Xc²), exactly. The first term is the familiar result: the extra fraction mₐ/m_X pays for the kinetic energy of the centre of mass, which momentum conservation will not let the reaction spend. For ¹⁴N(α, p)¹⁷O, Q = −1.192 MeV, so Kₜₕ = 1.192 × (1 + 4.003/14.003) = 1.533 MeV, and the relativistic term adds only 54 eV. In particle physics it matters: p + p → p + p + π⁰ has Q = −135.0 MeV and a threshold of 279.7 MeV, 3.6% above the nonrelativistic 270.0 MeV. An exothermic reaction has no kinematic threshold at all, though a Coulomb barrier can still bar the way.

Three counts must balance in every process here: charge, nucleon number A, and electron lepton number Lₑ, which is +1 for e⁻ and νₑ and −1 for e⁺ and ν̄ₑ. The ledger names the neutrino: n → p + e⁻ + ν̄ₑ balances Lₑ as 0 = 1 − 1, with Q = [M(n) − M(¹H)]c² = 0.782 MeV. It also separates forbidden from merely uphill. A free proton cannot decay as p → n + e⁺ + νₑ, yet every ledger balances; the obstacle is Q = −1.804 MeV, and inside ¹⁸F the nucleus pays it. By contrast p → e⁺ + π⁰ would release 803 MeV and balances charge, but breaks nucleon number and lepton number, and the measured lower limit on its lifetime is above 10³⁴ years. What no ledger gives is a rate: ²¹²Po (Q = 8.95 MeV) and ²³²Th (Q = 4.08 MeV) pass every ledger, yet their α half-lives, 0.3 μs and 1.4 × 10¹⁰ years, differ by a factor of 10²⁴.