Nucpaedia
Nucpaedia
Physics · Decay

Atomic Structure & Radioactive Decay

Snapshot

Unstable nuclei reach stability by emitting particles or photons. Nuclear medicine images the photons — γ-rays from isomeric transition/electron capture, and the 511 keV annihilation photons from β⁺ (positron) emitters. Decay is exponential, characterised by the half-life, and activity is measured in becquerels (SI) or curies.

A nuclide is defined by its protons (Z) and neutrons (N); the neutron-to-proton ratio determines stability and the decay pathway. Diagnostic tracers use γ or β⁺ emitters (imaged), while β⁻ and α emitters are used for therapy (short range, high local dose).

⁹⁹ᵐTc140 keV, T½ 6 h
1 mCi= 37 MBq
1/T_eff1/Tp + 1/Tb
Reference values
  • λ = ln2/T½ = 0.693/T½; mean life τ = 1/λ = 1.443·T½ (activity falls to 37% in one mean life).
  • Ten half-lives leave 2⁻¹⁰ ≈ 0.1% of the activity.
  • 1 Bq = 1 decay/s; 1 Ci = 3.7×10¹⁰ Bq = 37 GBq; 1 mCi = 37 MBq.
  • Half-lives (NNDC, rounded): ⁹⁹ᵐTc 6.01 h, ¹⁸F 110 min, ⁶⁸Ga 68 min, ¹²³I 13.2 h, ¹³¹I 8.02 d, ¹¹¹In 2.80 d, ²⁰¹Tl 3.04 d, ¹⁷⁷Lu 6.65 d, ⁹⁰Y 64.1 h, ²²³Ra 11.4 d.
  • β⁺ decay needs a parent–daughter atomic mass difference of at least 1.022 MeV (2 × 511 keV); below that only electron capture occurs.
  • ⁹⁹ᵐTc: 140.5 keV γ in 89% of decays; carrier-free specific activity 1.95×10¹⁴ Bq/mg (5.27×10⁶ mCi/mg).
  • Mean β energy is about ⅓ E_max for β⁻ (¹³¹I 182 keV; main branch E_max 606 keV) and about 0.4 E_max for β⁺ (¹⁸F 250 of 634 keV).
  • K-shell binding energy rises with Z: Tc 21.0 keV, I 33.2 keV, Pb 88.0 keV.
Worked example

Decaying ⁹⁹ᵐTc activity forward and back

Given. 5.55 GBq (150 mCi) ⁹⁹ᵐTc at 10:00; T½ = 6.01 h. Find the activity 4 h earlier and 5 h later.

  1. λ = 0.693/6.01 = 0.1153 h⁻¹.
  2. Earlier (× e^(+λt)): 5.55 × e^(0.461) = 5.55 × 1.586 = 8.80 GBq.
  3. Later (× e^(−λt)): 5.55 × e^(−0.577) = 5.55 × 0.562 = 3.12 GBq.

Answer. 8.8 GBq (238 mCi) at 06:00; 3.1 GBq (84 mCi) at 15:00.

Worked example

Effective half-life and residence

Given. 400 MBq of a ⁹⁹ᵐTc agent (T_phys 6.0 h) taken up instantly and cleared with T_biol 12 h.

  1. T_eff = T_phys·T_biol/(T_phys + T_biol) = 6 × 12/18 = 4.0 h.
  2. At 24 h: 2^(−24/4) = 1/64 = 1.6% remains (physical decay alone would leave 2^(−4) = 6.3%).
  3. Time-integrated activity à = 1.443 × A₀ × T_eff = 1.443 × 400 × 4 = 2 309 MBq·h.

Answer. T_eff = 4 h; 1.6% left at 24 h; Ã ≈ 2 310 MBq·h, the quantity multiplied by S-values in MIRD dosimetry.

Chart-of-nuclides diagram showing how beta-minus, beta-plus or electron capture, alpha decay and isomeric transition change proton and neutron numbers, with nuclear medicine examples.
Figure. Decay modes on the chart of nuclides. β⁻ decay moves the nucleus up one proton and down one neutron, β⁺ decay and electron capture do the reverse, α decay removes two of each, and isomeric transition changes only the energy state; the photon emitters are used for imaging and the particle emitters for therapy.
Decay curve halving every half-life, a log-scale plot showing that the effective half-life is shorter than both physical and biological half-lives, and a bar chart of physical half-lives from gallium-68 to radium-223.
Figure. Radioactive decay is exponential: activity halves every physical half-life and plots as a straight line on a log scale. Biological clearance adds to physical decay, so the effective half-life is shorter than either (in the example, a biological half-life twice the physical one gives T_eff = ⅔ T_phys). Physical half-lives are from the NNDC evaluated nuclear data, rounded.

Decay modes

ModeEmissionUse
Isomeric transitionγ-ray (e.g. ⁹⁹ᵐTc 140 keV)Imaging
Electron captureCharacteristic X-rays and γ-rays (¹²³I, ²⁰¹Tl, ¹¹¹In)Imaging
β⁺ (positron)Two 511 keV annihilation photons (¹⁸F, ⁶⁸Ga)PET imaging
β⁻Electron + antineutrino (¹³¹I, ¹⁷⁷Lu, ⁹⁰Y)Therapy
αHelium nucleus (²²⁵Ac, ²²³Ra)Targeted α therapy

Half-life & activity

  • Decay is exponential: A = A₀·e^(−λt), with λ = ln2 / T½.
  • Physical half-life is fixed; biological half-life reflects excretion; effective half-life combines them: 1/T_eff = 1/T_phys + 1/T_biol.
  • Activity units: 1 Bq = 1 decay/s; 1 Ci = 3.7×10¹⁰ Bq; 1 mCi = 37 MBq.
  • An ideal diagnostic radionuclide has a half-life comparable to the study, γ emission of about 100–200 keV and no particulate emission.

Why ⁹⁹ᵐTc dominates

  • 140 keV γ-ray (89% of decays) with no β⁻ or β⁺ emission: well matched to NaI detection and gives a low patient dose.
  • 6-hour half-life balances image quality against radiation burden.
  • Generator-produced (⁹⁹Mo/⁹⁹ᵐTc) with versatile cold-kit chemistry; used in about 80% of diagnostic procedures.

Pitfalls

  • Longer-lived or higher-energy radionuclidic impurities (e.g. ⁹⁹Mo in ⁹⁹ᵐTc eluate, ¹²⁴I in cyclotron ¹²³I) add patient dose and degrade images through septal penetration.
  • Particulate (β/α) emissions deliver dose without contributing to the image.
  • Confusing physical with effective half-life over- or under-estimates residence time and absorbed dose.
In the clinic — why the physics matters
  • Internal conversion steals imageable photons — for ⁹⁹ᵐTc (α_T ≈ 0.11) only ~90% of transitions yield a 140-keV γ.
  • Two 511-keV photons at 180° are the physical basis of PET coincidence detection.
  • Effective half-life (not physical) governs how long a tracer actually delivers dose: 1/T_eff = 1/T_phys + 1/T_biol.
  • ⁹⁹Mo (66 h) → ⁹⁹ᵐTc (6 h) transient equilibrium is what makes the generator work; daughter peaks ~23 h after elution.
Modes of radioactive decay
ModeNuclear changeΔZΔAExample / note
Alphaemits He nucleus−2−4²²³Ra; range in tissue <0.1 mm (a few cell diameters)
β⁻n → p + β⁻ + ν̄+10¹³¹I → ¹³¹Xe; continuous spectrum, mean ≈ ⅓ E_max
β⁺p → n + β⁺ + ν−10needs ≥1.022 MeV; ¹⁸F, ⁶⁸Ga
Electron capturep + e⁻ → n + ν−10¹¹¹In, ¹²³I, ²⁰¹Tl; characteristic X-rays and Auger electrons
Isomeric transitionexcited → lower nuclear state00⁹⁹ᵐTc → ⁹⁹Tc, 140 keV
Common pitfalls & misconceptions
  • Physical ≠ effective ≠ biological half-life — effective is always shorter than the shorter of the other two.
  • Mean life ≠ half-life: τ = 1.44·T½, so it's longer, not equal.
  • A β⁻ particle comes from the nucleus (n→p), not from the orbital shells.
  • β⁻ particles don't all carry E_max — the spectrum is continuous, averaging ~⅓ E_max.
In depth
  • Transient equilibrium (T_parent > T_daughter): after several daughter half-lives A_d/A_p = BR·λ_d/(λ_d − λ_p). For ⁹⁹Mo/⁹⁹ᵐTc, 1.10 × 0.87 ≈ 0.96, and daughter activity peaks at t_max = ln(λ_d/λ_p)/(λ_d − λ_p) ≈ 23 h after elution.
  • Secular equilibrium (T_parent ≫ T_daughter): A_d ≈ A_p, as in the ⁶⁸Ge/⁶⁸Ga (271 d/68 min) and ⁸²Sr/⁸²Rb (25.3 d/75 s) generators.
  • Carrier-free specific activity = λNA/A, so it rises as half-life falls: ¹⁸F reaches about 63 TBq/µmol. Tracer masses are therefore nanomolar or less and have no pharmacological effect.
  • Internal conversion transfers the nuclear energy to an orbital electron instead of a γ-ray (conversion coefficient α = e⁻/γ). For the 140 keV ⁹⁹ᵐTc transition α ≈ 0.1, so about 89% of decays give an imageable photon; conversion and Auger electrons add local dose.
  • β⁺ and electron capture compete: ¹⁸F is 96.9% β⁺ and ⁶⁸Ga 88.9% β⁺. The positron fraction must be applied whenever PET counts are converted to activity.
  • ⁶⁸Ga positrons (E_max 1.90 MeV) have a mean range in water of 3.5 mm against 0.6 mm for ¹⁸F (E_max 0.63 MeV), and 13% of ⁸²Rb positrons are accompanied by a 777 keV prompt γ.
  • ²⁰¹Tl is imaged mainly through mercury K X-rays of 69–83 keV emitted after electron capture; its 135 keV and 167 keV γ-rays are much less abundant (about 3% and 10%).
  • ⁹⁰Y β⁻ particles (E_max 2.28 MeV) penetrate up to about 11 mm of tissue (mean about 2.5 mm), whereas ²²³Ra α particles travel less than 0.1 mm: range sets the scale of cross-fire and dose heterogeneity.

Sources: Cherry, Sorenson & Phelps, 4th ed. (2012) · NNDC NuDat 3 decay data · ICRP 107 (2008) · Conti & Eriksson, EJNMMI Phys 2016 (PMID 27271304) · ⁹⁰Y microsphere and ²²³Ra product information

Sources

  1. Cherry SR, Sorenson JA, Phelps ME. Physics in Nuclear Medicine. 4th ed. Philadelphia: Elsevier Saunders; 2012.
  2. Saha GB. Physics and Radiobiology of Nuclear Medicine. 4th ed. New York: Springer; 2013.
  3. Bushberg JT, Seibert JA, Leidholdt EM Jr, Boone JM. The Essential Physics of Medical Imaging. 4th ed. Philadelphia: Wolters Kluwer; 2021.
  4. National Nuclear Data Center, Brookhaven National Laboratory. NuDat 3: evaluated nuclear structure and decay data.
  5. ICRP. Nuclear decay data for dosimetric calculations. ICRP Publication 107. Ann ICRP. 2008;38(3).
  6. Conti M, Eriksson L. Physics of pure and non-pure positron emitters for PET: a review and a discussion. EJNMMI Phys. 2016;3:8.
← BackPhysics & Instrumentation