Nuclear Decay

Why nuclei decay

The chart of nuclides — protons (Z) on one axis, neutrons (N) on the other — shows a narrow valley of stability. About 254 nuclides are stable; thousands of others fall above, below, or beyond the valley. Each unstable nucleus moves toward the valley by a specific decay mode determined by where it sits.

• Above the valley (too many neutrons): β⁻ decay converts a neutron to a proton.
• Below the valley (too many protons): β⁺ or electron capture converts a proton to a neutron.
• Beyond Z = 82 (lead): the Coulomb repulsion of so many protons exceeds the strong force's reach, and α decay sheds tightly-bound He-4 nuclei.
• Excited states (right energy, wrong configuration): γ emission drops the nucleus to its ground state.

The Q-value of any decay — the energy released — is the parent-daughter mass difference, converted to energy via E = Δm·c². Decay is allowed if Q > 0; the rate depends on quantum-mechanical transition probabilities and barrier penetration.

Alpha decay

An alpha particle is a ⁴He nucleus: 2 protons + 2 neutrons, very tightly bound (28.3 MeV total binding). Alpha decay shifts the parent by Z → Z−2, A → A−4:

      ²³⁸_{92}U  →  ²³⁴_{90}Th  +  ⁴_{2}He   (Q = 4.27 MeV)
      ²²⁶_{88}Ra →  ²²²_{86}Rn  +  ⁴_{2}He   (Q = 4.87 MeV)

The kinetic energy is shared between the alpha (≈ 98 %, since it's much lighter than the daughter) and the recoiling daughter (≈ 2 %). Alpha energies are discrete — typically 4–9 MeV — because the two-body decay leaves no continuous spectrum.

Quantum tunneling sets the rate. The alpha must escape the Coulomb barrier holding it in. Gamow's 1928 calculation showed that

      log₁₀(t_{1/2}) ≈ a·Z/√Q + b      (Geiger–Nuttall law)

so a small change in Q-value swings half-life over many orders of magnitude. ²¹²Po decays in 0.3 µs at Q = 8.95 MeV; ²³²Th lives 1.4 × 10¹⁰ years at Q = 4.08 MeV — five orders of magnitude in Q gives 23 orders of magnitude in half-life.

Penetration: alpha particles are stopped by a sheet of paper or 5 cm of air. Internal exposure is the danger — Po-210 ingested by Alexander Litvinenko in 2006 (estimated 4.4 µg, 1 GBq) caused fatal alpha damage to bone marrow.

Beta-minus decay

A neutron in a neutron-rich nucleus converts to a proton, emitting an electron (β⁻) and an antineutrino:

      n  →  p  +  e⁻  +  ν̄_e

      ¹⁴_{6}C  →  ¹⁴_{7}N  +  e⁻  +  ν̄_e   (Q = 0.156 MeV, t_{1/2} = 5,730 y)
      ¹³⁷_{55}Cs →  ¹³⁷ᵐ_{56}Ba + e⁻ + ν̄_e (Q = 0.514 MeV; daughter γ-decays)
      ⁹⁰_{38}Sr →  ⁹⁰_{39}Y  +  e⁻  +  ν̄_e  (Q = 0.546 MeV)

Mass number A stays the same; Z increases by 1. The decay energy is shared between the electron and antineutrino, giving a continuous energy spectrum from 0 up to Q for the electron. (The antineutrino was first proposed by Pauli in 1930 specifically to fix the apparent energy non-conservation.)

Penetration: a few mm of aluminum or 1 m of air for typical 1 MeV β⁻. Sr-90 and its daughter Y-90 are bone-seekers in fallout because Sr substitutes for Ca in bone; the high-energy beta from Y-90 (Q = 2.28 MeV) reaches up to 11 mm in tissue.

Beta-plus (positron) decay

In proton-rich nuclei, a proton converts to a neutron, emitting a positron (β⁺) and a neutrino:

      p  →  n  +  e⁺  +  ν_e

      ¹⁸_{9}F  →  ¹⁸_{8}O   +  e⁺  +  ν_e   (Q = 0.634 MeV, t_{1/2} = 109.7 min)
      ²²_{11}Na →  ²²_{10}Ne  +  e⁺  +  ν_e   (t_{1/2} = 2.602 y)
      ¹¹_{6}C  →  ¹¹_{5}B   +  e⁺  +  ν_e   (t_{1/2} = 20.4 min)

Z decreases by 1; A unchanged. There is a 1.022 MeV threshold — the parent mass must exceed the daughter mass by at least 2·m_e (the rest mass of the created e⁺ pair). Below this threshold only electron capture is energetically allowed.

The emitted positron travels a few mm in tissue, then annihilates with an electron to produce two 511 keV photons emitted back-to-back. PET (positron emission tomography) detects these coincident gamma pairs to localize tracer concentration. ¹⁸F-FDG (fluorodeoxyglucose) is the most common PET tracer; cancer cells accumulate it because of high glucose uptake.

Electron capture

An inner-shell (usually K) electron is absorbed by the nucleus; a proton converts to a neutron and a neutrino is emitted:

      p  +  e⁻  →  n  +  ν_e

      ⁷_{4}Be  +  e⁻  →  ⁷_{3}Li  +  ν_e        (t_{1/2} = 53.22 d)
      ⁵⁵_{26}Fe + e⁻  →  ⁵⁵_{25}Mn + ν_e        (t_{1/2} = 2.74 y)
      ⁴⁰_{19}K + e⁻   →  ⁴⁰_{18}Ar + ν_e        (10.7 % branch; rest is β⁻)

Like β⁺, Z drops by 1 and A is unchanged. Unlike β⁺, there is no 1.022 MeV threshold — EC is the only option for low-Q proton-rich decays. The K-shell hole left behind fills with an outer electron, emitting a characteristic X-ray (or, by the Auger effect, ejecting a second electron). EC daughters are detected by these X-rays.

EC is the only decay whose rate has a (tiny) chemical sensitivity: changing the electron density at the nucleus by altering the daughter's chemical bonding shifts the rate by ~1 %. ⁷Be in metallic Be vs in BeF₂ has measurably different t_1/2 — about 0.83 % difference, the largest known chemical effect on a decay rate.

Gamma decay

An excited nucleus drops to a lower-energy state, emitting a gamma photon. Z and A unchanged:

      ⁹⁹ᵐ_{43}Tc  →  ⁹⁹_{43}Tc  +  γ (140.5 keV)   (t_{1/2} = 6.01 h)
      ⁶⁰_{28}Ni*  →  ⁶⁰_{28}Ni  +  γ (1.17, 1.33 MeV)  (after ⁶⁰Co β⁻)

Most gamma emission follows alpha or beta decay within picoseconds — the daughter lands excited, then de-excites. A few "metastable" excited states (isomers) live longer because the transition is angular-momentum forbidden. Tc-99m's 6-hour metastable lifetime makes it the most-used isotope in nuclear medicine; injected, imaged, and gone within a day.

Gamma photons from MeV-scale transitions are highly penetrating: a few cm of lead halve them. They are the source of external dose from ¹³⁷Cs and ⁶⁰Co in radiation oncology and food irradiation.

The five decay modes side by side

Mode	What's emitted	ΔZ	ΔA	Driving force	Penetration	Canonical example
Alpha (α)	⁴He nucleus (2p+2n)	−2	−4	Z > 82, Coulomb repulsion vs. tight α binding (28.3 MeV)	Sheet of paper / 5 cm air	²³⁸U → ²³⁴Th + ⁴He, 4.27 MeV
Beta-minus (β⁻)	e⁻ + antineutrino	+1	0	Excess neutrons (above stability valley)	~ 1 cm Al / few m air	¹⁴C → ¹⁴N + e⁻ + ν̄, 0.156 MeV
Beta-plus (β⁺)	e⁺ + neutrino	−1	0	Proton excess; Q > 1.022 MeV required	Annihilation: 2 × 511 keV γ	¹⁸F → ¹⁸O + e⁺ + ν, 0.634 MeV
Electron capture (EC)	Neutrino + characteristic X-ray	−1	0	Proton excess; Q < 1.022 MeV (β⁺ forbidden)	Detect via emitted X-ray	⁷Be + e⁻ → ⁷Li + ν, t1/2 = 53.2 d
Gamma (γ)	High-energy photon	0	0	Excited nuclear state from prior α or β decay	Few cm lead halves; very penetrating	⁹⁹ᵐTc → ⁹⁹Tc + γ at 140 keV
Spontaneous fission (rare)	Two daughter nuclei + 2–3 neutrons	varies	varies	Z² /A > 47 (very heavy nuclei)	Two large fragments, neutrons	²⁵²Cf, t1/2 = 2.65 y, 3 % SF branch

Decay chain: ²³⁸U series

  ²³⁸U  ──α──►  ²³⁴Th ──β⁻──►  ²³⁴Pa ──β⁻──►  ²³⁴U  ──α──►  ²³⁰Th
   4.5e9 y       24.1 d         6.7 h           245,500 y      75,400 y

         ──α──►  ²²⁶Ra ──α──►  ²²²Rn ──α──►  ²¹⁸Po ──α──►  ²¹⁴Pb
                  1,600 y      3.82 d         3.10 min       26.8 min

         ──β⁻──► ²¹⁴Bi ──β⁻──► ²¹⁴Po ──α──►  ²¹⁰Pb ──β⁻──► ²¹⁰Bi
                  19.9 min     163 µs          22.3 y         5.01 d

         ──β⁻──► ²¹⁰Po ──α──►  ²⁰⁶Pb (stable)
                  138.4 d

14 steps from ²³⁸U to stable ²⁰⁶Pb, alternating α (8 steps) and β⁻ (6 steps), shedding 8·4 + 0 = 32 amu and 8·2 − 6·1 = 10 protons (matching A = 238 − 32 = 206 and Z = 92 − 10 = 82). Half-lives span 12 orders of magnitude. Radon-222 in basements is a member: it diffuses out of uranium-bearing rock and lodges in lungs, where its α-emitting daughters cause an estimated 21,000 lung cancer deaths per year in the US (EPA 2003 estimate).

Real isotopes and where they live

• ²¹⁰Po (Litvinenko, 2006). Pure α emitter, t_1/2 = 138.4 d, Q = 5.41 MeV. About 4.4 µg (1 GBq) was lethal because alpha damage in bone marrow is extreme when the source is internal.
• ¹³¹I (Chernobyl, 1986). β⁻ + γ, t_1/2 = 8.02 d. Decayed within months but caused thousands of pediatric thyroid cancers in Belarus and Ukraine. Iodine prophylaxis (KI tablets) saturates the thyroid with stable iodine.
• ¹³⁷Cs (Chernobyl/Fukushima long-term). β⁻ + γ via metastable ¹³⁷ᵐBa, t_1/2 = 30.07 y. Dominates persistent contamination; 40 years post-Chernobyl, ¹³⁷Cs activity is at 0.5⁴⁰⁄³⁰·⁰⁷ ≈ 40 % of 1986 levels.
• ⁶⁰Co (radiotherapy, sterilization). β⁻ + 1.17 + 1.33 MeV γ, t_1/2 = 5.27 y. Standard external-beam radiotherapy source from the 1950s through 1980s, still used in lower-resource countries; now mostly supplanted by linac X-ray.
• ⁹⁹ᵐTc (medical imaging). Pure isomeric γ at 140 keV, t_1/2 = 6.01 h. Made from ⁹⁹Mo (t_1/2 = 66 h) cow generators; used in ~ 30 million scans per year worldwide.
• ¹⁸F-FDG (PET). β⁺, t_1/2 = 109.7 min. Annihilation gammas at 511 keV imaged in coincidence; standard for cancer staging and brain metabolism.
• ⁴⁰K (banana radiation). 89.3 % β⁻ to ⁴⁰Ca, 10.7 % EC to ⁴⁰Ar, t_1/2 = 1.248 × 10⁹ y. Naturally present in all biological potassium; ~ 15 Bq per banana.

Conservation laws in nuclear decay

Every nuclear decay conserves: (i) electric charge — total Z + emitted lepton charge balanced; (ii) baryon number — A counts protons + neutrons, conserved across α, β, γ; (iii) lepton number — the antineutrino in β⁻ matches the electron, the neutrino in β⁺ matches the positron; (iv) energy — Q-value equals total kinetic energy plus rest masses of products; (v) momentum — daughter recoils opposite the emitted particle(s); (vi) angular momentum and parity — selection rules govern allowed vs. forbidden transitions and explain isomers.

Common misconceptions

• "Beta decay emits an electron from the nucleus." The electron is created at the moment of decay (along with the antineutrino), not pre-existing in the nucleus. Same for the positron in β⁺.
• Confusing β⁺ and EC. Both increase neutron count and decrease Z, but β⁺ requires Q > 1.022 MeV and emits a detectable positron; EC has no threshold and emits only a neutrino plus characteristic X-rays.
• Treating gamma decay as separate. Pure isomeric transitions exist (e.g., Tc-99m), but most gammas are de-excitations of daughters from prior α or β. The cesium-137 "γ source" is really β⁻ to metastable Ba-137 followed by γ.
• Equating activity with danger. A tiny ²¹⁰Po speck has lower activity than a smoke detector's ²⁴¹Am, but ingested ²¹⁰Po is far more dangerous because of the dose distribution. Penetration, route of exposure, and biological half-life all matter.
• Assuming all alpha emitters are heavy. Mostly true (Z > 82), but a few light nuclei α-decay too: ⁵Li → ⁴He + ¹H (unbound), and ⁸Be → 2 × ⁴He (Q = 92 keV, t_1/2 = 8 × 10⁻¹⁷ s, the basis of stellar carbon synthesis).
• Confusing decay rate with half-life sensitivity. A 1 % shift in alpha Q-value can change t_1/2 by several orders of magnitude (Geiger–Nuttall). Beta decay rates are much less sensitive to Q.