Chapter 02Basic sciencesLast reviewed: · What's new
Radioactive Decay
This chapter's English text is a draft translation (prepared with Claude and checked against a terminology glossary and automatic checks); specialist review is in progress. Please report errors with the feedback button.
Learning objectives and modules3 modules · about 2 h 15 min
Learning objectives
After this chapter you will be able to:
State how Z and A change in alpha, β⁻ and β⁺ decay, electron capture and isomeric transition, and the accompanying emissions.
Explain the energy threshold for β⁺ decay and its competition with electron capture.
Calculate activity using the relations between decay constant, half-life, mean life and effective half-life.
Distinguish transient equilibrium, secular equilibrium and no equilibrium from the parent and daughter half-lives, and apply them to generators.
Read a decay scheme to extract transition types, energies and branching ratios.
Modules
The chapter is divided into study modules without changing the order of the text: read, watch the animations, then answer the questions linked to those headings.
Neutron excess → β⁻ decay → Z increases by 1, A unchanged
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Proton excess → β⁺ or EC → Z decreases by 1, A unchanged
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Heavy nucleus (Z > 83) → α decay → Z decreases by 2, A decreases by 4
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Excited nucleus → isomeric transition (IT) → Z and A unchanged
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β⁺ threshold: 1.022 MeV → below the threshold only EC occurs
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β⁻: n → p + e⁻ + ν̄ₑ → Z increases → an antineutrino is emitted
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β⁺: p → n + e⁺ + νₑ → Z decreases → a neutrino is emitted → threshold: 1.022 MeV
2.1 Alpha, Beta and Positron Decay
Radioactive decay is the process by which an unstable nucleus emits particles and/or energy to reach a more stable state. Decay is entirely random and spontaneous; it cannot be altered by external factors such as temperature, pressure or chemical bonding.
What it is: Momentum is conserved in alpha decay; the daughter nucleus recoils opposite to the alpha particle (shown in the Figure 2.1 animation).
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In numbers: Recoil energy ≈ Q × m(α) / M(daughter): ~100-160 keV (Ra-223 → Rn-219 ~108 keV; 105-160 keV along the Ac-225 chain). Chemical bond energies are a few eV, so the recoil energy is more than 1000 times larger; the daughter travels ~100 nm in water-like media.
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Why it matters clinically: With every decay the daughter breaks free of the chelator and leaves the targeting molecule. Parent nuclides with several alpha emitters in their chain (Ac-225, Ra-223, Pb-212) act as "in vivo generators": released daughters enter the circulation and accumulate in other organs; with Ac-225, free Bi-213 accumulating in the kidney can cause renal toxicity.
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In the exam: "Why does the daughter radionuclide not stay at the target in alpha therapy?" → The recoil energy far exceeds the bond energy.
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Sources: de Kruijff and Wolterbeek, Pharmaceuticals 2015; Kozempel et al., Molecules 2018; de Kruijff et al., Sci Rep 2019.
BOARD TIP
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Alpha particles are monoenergetic (discrete energy spectrum). This is the most important feature distinguishing them from the continuous energy spectrum of beta particles. If the board exam asks "which decay emits monoenergetic particles?" → think of alpha decay and internal conversion electrons.
Sr-89: T½ = 50.5 days → palliative treatment of bone pain
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Sm-153: T½ = 46.3 h → palliative treatment of bone pain
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P-32: T½ = 14.3 days, pure beta emitter → treatment of polycythaemia vera
BOARD TIP
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The continuous energy spectrum of beta particles results from the energy being shared between the beta particle and the antineutrino. The formula mean beta energy ≈ Emax/3 is frequently asked in board exams. Pure beta emitters (Y-90, P-32, Sr-89) emit no gamma rays → they cannot be imaged directly and are used only for therapy (for Y-90, bremsstrahlung imaging or PET is possible).
Decays are added one by one to a histogram: alpha lines (Ra-223, Ac-225, Th-227, At-211, Po-213, Po-212) are discrete, beta spectra (I-131, Lu-177, Y-90, P-32, F-18) continuous; the mean energy and the β⁻/β⁺ difference are shown.
Continuous stream · no seeking
Key point: in alpha decay the energy is shared between two bodies, so the lines are discrete; in beta decay it is shared with a (anti)neutrino, the spectrum is continuous and the mean energy is about Emax/3.
Source: ENSDF (IAEA LiveChart, retrieved 10 November 2026): α lines, daughter levels and Q values, β branches; spectrum shape from a model with the relativistic Fermi function (model means within 2% of ENSDF values); β–ν angle W ∝ 1 + a·(v/c)·cos θ (Jackson, Treiman, Wyld 1957; allowed GT a = −1/3, forbidden transitions isotropic).
How to use: one click pauses and resumes; in the alpha and mean energy steps ↑ ↓ change the radionuclide; tap a histogram bar to see its energy range and number of decays; ← → change steps.
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Figure 2.1: Discrete (monoenergetic) energy spectrum of alpha particles and continuous spectrum of beta particles; mean beta energy ≈ Emax/3.
Beta-plus decay occurs in nuclei with a proton excess (below the band of stability). A proton turns into a neutron:
General equation:
p → n + e⁺ + νₑ (neutrino)
AZX → AZ−1Y + e⁺ + νₑ
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Mass number (A) is unchanged
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Atomic number (Z) decreases by 1
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A neutrino (νₑ) is emitted
Threshold condition:
For β⁺ decay to occur, the (atomic) mass difference between the parent and daughter atoms must be at least 1.022 MeV/c² (2 × 0.511 MeV). If this threshold is not met, the nucleus undergoes electron capture (EC) instead.
Fate of the positron – annihilation:
The emitted positron loses its kinetic energy through interactions with electrons in the medium (range of a few mm). As it comes to rest it combines with an electron (annihilation), producing two 511 keV photons. These photons are emitted in opposite directions at ~180°. PET imaging is based on detecting these coincidence photons.
Positronium:
Just before annihilating, the slowed positron may form with an electron a very short-lived bound system resembling a hydrogen atom: positronium (Ps). The mass of the two particles is then converted into two 511 keV annihilation photons emitted simultaneously at about 180° to each other. Three-photon annihilation is also possible from ortho-positronium; however, this pathway is very rare in tissue, and clinical PET imaging relies on two-photon annihilation.
Positron range and spatial resolution:
The distance travelled by the positron before annihilation (positron range) is one of the factors limiting the spatial resolution of PET:
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F-18: Mean positron range ~0.6 mm → best spatial resolution
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Ga-68: Mean positron range ~2.9 mm
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Rb-82: Mean positron range ~5.9 mm → poorest spatial resolution
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O-15: Mean positron range ~2.5 mm
WHY IT MATTERS · Positron range in a magnetic field (PET/MR)
What it is: In PET/MR the positron follows a helical path in the plane perpendicular to the main magnetic field because of the Lorentz force: the range perpendicular to the field (transaxial) shortens, the range along the field (axial) does not.
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In numbers: The effect grows with positron energy and field strength. At 3 T it is unimportant for F-18 (mean range ~0.6 mm); for high-energy positrons such as Ga-68 (~2.9 mm) and Rb-82 (~5.9 mm) transaxial resolution improves and axial resolution worsens. Positrons escaping into air travel along the field lines and annihilate on the first surface they meet: for a source in air, the ratio of spurious activity on a surface 3 cm away to the activity at the true location was 0.9 for F-18 and 5.7 for Ga-68.
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Why it matters clinically: Next to air cavities (trachea, larynx, lung, skin surface), symmetrical spurious foci ("shine-through") may appear on both sides of a lesion along the axis; for a lesion in the trachea the artefact-to-lesion ratio was 21-30%. With Ga-68 PET/MR this can lead to overstaging; foci lined up with the lesion along the axis and placed symmetrically are the clue.
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In the exam: "How does the magnetic field in PET/MR affect positron range?" → It shortens the range perpendicular to the field (transaxial resolution improves) but not along the field; the effect is marked for high-energy positrons.
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Sources: Kolb et al., J Nucl Med 2015; Conti and Eriksson, EJNMMI Phys 2016.
β⁺-emitting (PET) radionuclides in nuclear medicine:
C-11: T½ = 20.4 min → choline, methionine, raclopride
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N-13: T½ = 10 min → N-13 ammonia (cardiac perfusion)
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O-15: T½ = 2.04 min → O-15 water (myocardial blood flow, gold standard)
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Cu-64: T½ = 12.7 h → DOTATATE, research
BOARD TIP
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As positron energy increases, range increases and PET spatial resolution falls. F-18 has the lowest positron energy → shortest range → best spatial resolution. Rb-82 has the highest positron energy → longest range → poorest spatial resolution. This ranking is frequently asked in board exams!
A positron from a β⁺ emitter slows down in tissue with its real energy and speed on a common scale; it collides with tissue electrons, thermalises and annihilates; positronium formation (para and ortho) and the emission of two 511 keV photons in nearly opposite directions are shown.
Continuous stream · no seeking
Key point: positron range and the slight deviation of annihilation photons from 180° are physical limits of PET resolution.
Source: β⁺ branches and mean energies ENSDF (IAEA LiveChart); energy loss NIST ESTAR (water); the track is a multiple-scattering simulation whose mean net distance was compared with published Monte Carlo results (PeneloPET, GATE); e⁺–ν angular correlation Jackson-Treiman-Wyld 1957; mean ranges from the book; 125 ps / 142 ns, 0.5° FWHM from the book; three-photon fraction Mercurio 2006; ortho-positronium lifetime in water Mercolli 2025. Deflection over the first 20% of the track agrees with the Highland formula (PDG) within ±12%; the drawn kinked track uses the energy-loss (~25 keV) points.
How to use: one click pauses and resumes; in the first step ↑ ↓ or a row on the right change the radionuclide, in the annihilation step ↑ ↓ switch the ring diameter (80 / 30 cm); ← → change steps.
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Figure 2.2: Positron emission, positronium formation and annihilation (schematic); mean positron range of PET radionuclides.
Electron capture is an alternative to β⁺ decay in nuclei with a proton excess. The nucleus captures one of its inner orbital electrons (usually from the K shell), and a proton turns into a neutron:
General equation:
p + e⁻ → n + νₑ
AZX + e⁻ → AZ−1Y + νₑ
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Mass number (A) is unchanged
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Atomic number (Z) decreases by 1
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A monoenergetic neutrino is emitted
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Unlike β⁺ decay, there is no threshold energy (1.022 MeV is not required)
Radiations emitted after EC:
1. Characteristic X-rays: Arise from the transition of an outer-shell electron filling the K-shell vacancy
2. Auger electrons: Produced when the transition energy is transferred directly to a neighbouring electron without photon emission
3. Gamma rays: Emitted if the daughter nucleus is left in an excited state
EC versus β⁺ decay:
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Both decays occur in nuclei with a proton excess
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β⁺ decay requires a mass difference ≥ 1.022 MeV; EC has no such threshold
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If the mass difference is <1.022 MeV, only EC occurs
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If the mass difference is ≥ 1.022 MeV, both β⁺ and EC can occur (they compete)
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EC dominates in high-Z nuclei (the K-shell electron is closer to the nucleus)
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β⁺ dominates in low-Z nuclei
Radionuclides decaying by EC in nuclear medicine:
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Tl-201: Pure EC → imaged with Hg X-rays (69-83 keV), T½ = 73 h
I-125: EC → 27-35 keV X-rays and Auger electrons, T½ = 59.4 days
BOARD TIP
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If asked "which radionuclide decays only by electron capture?" → think of Tl-201 and I-125. I-123 also decays predominantly by EC (close to 100%). Ga-67 and In-111 also decay by EC. F-18, by contrast, decays 97% by β⁺ and 3% by EC → it is predominantly a positron emitter.
A Tl-201 nucleus captures a K electron and becomes Hg-201, emitting Hg X-rays; for seven radionuclides Q(EC) is compared with the 1.022 MeV threshold to show the β⁺ and EC shares.
Continuous stream · no seeking
Key point: electron capture can occur at any energy difference; β⁺ decay requires the energy difference between the nuclei to exceed 1.022 MeV.
Source: ENSDF (IAEA LiveChart): Q(EC) values and branching ratios.
How to use: one click pauses and resumes; in the β⁺ threshold step ↑ ↓ or the buttons change the radionuclide; ← → change steps.
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Figure 2.3: Electron capture and the radiations emitted afterwards; the 1.022 MeV threshold relationship between β⁺ and EC (schematic).
2.2.2 Isomeric Transition (IT)
Isomeric transition is the return of an excited (metastable) nucleus to its ground state by emitting its excess energy as a gamma ray. Z and A do not change in this process; only the energy state of the nucleus changes.
General equation:
AZX* (or AmZX) → AZX + γ
Features:
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No particle is emitted → Z and A do not change
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The emitted gamma ray is monoenergetic
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The metastable state is denoted by "m" (e.g. Tc-99m)
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The half-life of the metastable state is measurably long (from nanoseconds to hours)
Tc-99m → Tc-99 transition:
The most important example of isomeric transition in nuclear medicine:
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Tc-99m becomes Tc-99 by emitting a 140 keV gamma ray
Internal conversion is an alternative to isomeric transition. Instead of being emitted as a gamma photon, the excitation energy of the nucleus is transferred to one of the inner orbital electrons, which is ejected from the atom.
Energy of the internal conversion electron:
EIC electron = Eγ − Ebinding
Features:
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Internal conversion electrons are monoenergetic (gamma energy − binding energy)
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Competes with gamma emission
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Internal conversion coefficient (α): IC/gamma ratio
Large ΔL (change in angular momentum) → α increases
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The inner-shell vacancy left after IC leads to emission of a characteristic X-ray or an Auger electron
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IC and Auger electrons have a very short range in tissue and deposit their energy right where they are produced. They therefore do not contribute to image formation; they only increase the patient's dose.
BOARD TIP
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The internal conversion coefficient (α) is frequently asked in board exams. Three factors increase α: (1) high Z, (2) low transition energy, (3) large change in angular momentum (ΔL). In Tc-99m the IC fraction is 11.5% → in every 100 decays ~88 gamma photons and ~12 IC electrons are emitted.
The transition of Tc-99m from the 142.7 keV isomeric level to the ground state; whether it proceeds by γ emission or an internal conversion electron, and the factors that raise the conversion coefficient, are shown.
Continuous stream · no seeking
Key point: in an isomeric transition the energy leaves either as a γ photon or is transferred to a shell electron; internal conversion dominates in low-energy, high-multipolarity transitions.
How to use: one click pauses and resumes; in the last step ↑ ↓ or tapping a row highlights one example (tap again to show all three); ← → change steps.
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Figure 2.4: Competition between isomeric transition (γ) and internal conversion in Tc-99m, and factors that increase the internal conversion coefficient (schematic).
2.3 Decay Constant, Half-Life, Mean Life
2.3.1 Law of Radioactive Decay
Radioactive decay is a first-order kinetic process. The number of atoms decaying in a given time interval is proportional to the number of atoms present at that moment:
Differential form:
dN/dt = −λN
Solution (exponential decay equation):
N(t) = N₀ × e−λt
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N(t): Number of atoms at time t
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N₀: Initial number of atoms (t=0)
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λ: Decay constant (s⁻¹, min⁻¹ or h⁻¹)
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e: Euler's number (≈ 2.718)
2.3.2 Decay Constant (λ)
The decay constant expresses the probability that an atom decays per unit time:
The half-life is the time required for half of the radioactive atoms to decay:
N(T½) = N₀ / 2
Half-lives and remaining activity:
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1 half-life → 50% remains
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2 half-lives → 25% remains
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3 half-lives → 12.5% remains
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4 half-lives → 6.25% remains
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5 half-lives → 3.125% remains
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7 half-lives → ~1% remains (exactly 0.78%)
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10 half-lives → ~0.1% remains (exactly 0.098%)
Tc-99m atoms decay at random; the measured stepped curve lies around the theoretical exponential; half-life marks, the 10 half-life rule and the mean life are shown.
Continuous stream · no seeking
Key point: the remaining activity halves with each half-life and falls below one thousandth after 10 half-lives; the mean life is 1.443 × T½.
How to use: one click on the atom panel pauses and resumes; move over or drag along the curve to read the fraction remaining at that time; ← → change steps.
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Figure 2.5: Radioactive decay curve: remaining activity by number of half-lives, mean life (τ) and the 10 half-life rule.
"About 0.1% of the activity remains after 10 half-lives" is critical for radiation safety questions → the "10 half-life rule" is applied in waste management.
Figure 2.6: Physical half-lives of important radionuclides in nuclear medicine (logarithmic scale, by field of use).
The mean life is the average lifetime of all the radioactive atoms:
τ = 1/λ = T½ / 0.693 = 1.443 × T½
The mean life is always longer than the half-life (1.443 times).
Importance of the mean life:
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Used in cumulated activity (Ã) calculations → dosimetry (MIRD formalism)
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à = A₀ × τ = A₀ × T½ / 0.693 = 1.443 × A₀ × T½
2.3.5 Physical, Biological and Effective Half-Life3 questions
Radiopharmaceuticals decrease in the body by both physical decay and biological elimination (kidney, liver, etc.):
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Physical half-life (Tp): Half-life due to radioactive decay
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Biological half-life (Tb): Half-life due to biological elimination
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Effective half-life (Te): Combined effect of both processes
Formula:
1/Te = 1/Tp + 1/Tb or Te = (Tp × Tb) / (Tp + Tb)
Important rules:
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The effective half-life is always shorter than both the physical and the biological half-life
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If one is much shorter than the other, the effective half-life approaches the shorter one
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If Tp >> Tb → Te ≈ Tb (biological elimination dominates)
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If Tb >> Tp → Te ≈ Tp (physical decay dominates)
BOARD TIP
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The effective half-life formula is a staple of board exams. Key point: Te is always shorter than both Tp and Tb. Example: if Tp = 6 h and Tb = 3 h → Te = (6×3)/(6+3) = 18/9 = 2 h. And for "which affects the effective half-life most?" → the shorter half-life dominates.
Atoms in an organ are lost by physical decay or biological excretion; physical, biological and effective retention curves and the area under the activity–time curve (cumulated activity) are shown.
Continuous stream · no seeking
Key point: the effective half-life is shorter than both the physical and biological half-lives (1/Teff = 1/Tphys + 1/Tbiol); cumulated activity is the area under the curve, Ã = 1.443 × A₀ × Teff.
Source: book (example Tphys 6 h, Tbiol 24 h → Teff 4.8 h); A₀ = 100 MBq and f = 1 schematic; Tc-99m 6.01 h, I-123 13.2 h, I-131 8.02 d.
How to use: one click on the organ scene pauses and resumes; move over or drag along the graph; in the free step ↑ ↓ change the biological half-life smoothly; ← → change steps.
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Figure 2.7: Relationship between physical, biological and effective half-life: the effective half-life always approaches the shorter one.
Activity (A) is the number of atoms decaying per unit time:
A = λN = (0.693 / T½) × N
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A: Activity
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λ: Decay constant
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N: Number of radioactive atoms
Activity and emission rate are different concepts:
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Activity: the number of decays (disintegrations) per second in the source
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Emission rate: the number of a given radiation (e.g. the 140 keV gamma photon) emitted per second
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The two quantities are linked through the gamma yield or branching ratio of that radiation
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Example: 1 MBq of Tc-99m undergoes 10⁶ decays per second; because of internal conversion only ~0.89 × 10⁶ of these are emitted as 140 keV photons
BOARD TIP
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It is a mistake to equate activity with the number of photons emitted. Activity gives the decay rate (Bq); the number of photons contributing to the image is obtained by multiplying the activity by the gamma yield. For low-yield emissions (e.g. the 208 keV photon of Lu-177, ~10%) this distinction is especially important for counting and dosimetry calculations.
1 Ci = 3.7 × 10¹⁰ disintegrations per second (dps)
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Historically equal to the activity of 1 gram of Ra-226
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1 mCi = 3.7 × 10⁷ dps
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1 µCi = 3.7 × 10⁴ dps
Becquerel (Bq) – SI unit:
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1 Bq = 1 disintegration per second (dps)
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1 kBq = 10³ dps
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1 MBq = 10⁶ dps
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1 GBq = 10⁹ dps
Conversion:
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1 mCi = 37 MBq
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1 µCi = 37 kBq
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1 Ci = 37 GBq
BOARD TIP
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The conversion 1 mCi = 37 MBq must be known for board exams. Quick calculation: multiply mCi by 37 to obtain MBq. Example: 20 mCi = 740 MBq, 30 mCi = 1110 MBq. You may also be asked that 1 Ci was defined by reference to Ra-226.
If there are initially 20 mCi of Tc-99m (T½ = 6 h), what is the activity after 18 h?
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n = 18/6 = 3 half-lives
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A(18) = 20 × (1/2)³ = 20 × 1/8 = 2.5 mCi
2.4.4 Specific Activity
Specific activity is the activity per unit mass:
SA = A/m (Bq/g or Ci/g)
Carrier-free specific activity:
The theoretical maximum specific activity of a radionuclide in the carrier-free state (containing no stable isotopes of the same element):
SAmax = λ × Nₐ / M = (0.693 × 6.022 × 10²³) / (T½ × M)
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Nₐ: Avogadro's number (6.022 × 10²³)
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M: Atomic mass (g/mol)
Important rules:
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Short-lived radionuclides have a high specific activity
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Long-lived radionuclides have a low specific activity
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The carrier-free state gives the highest specific activity
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Cyclotron products are usually carrier-free → high specific activity
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Products of reactor neutron activation usually contain carrier → low specific activity
BOARD TIP
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If asked "what is the most important factor affecting specific activity?" → the half-life and the presence of carrier. Short T½ → high SA. Carrier-free → maximum SA. Cyclotron products (target of one element → product of a different element) are usually carrier-free. Reactor neutron activation products (target and product of the same element) contain carrier → low SA. This distinction is covered in detail in Chapter 11 (Radionuclide Production).
2.5 Radioactive Equilibrium (Transient and Secular)
Radioactive equilibrium describes special situations arising in systems where a parent radionuclide decays to produce a daughter radionuclide. The type of equilibrium depends on the relationship between the half-lives of the parent and daughter.
2.5.1 General Parent–Daughter Relationship
When a parent radionuclide (P) decays to a daughter radionuclide (D), the daughter's activity depends on the balance of two processes:
1. Production: The daughter is produced by decay of the parent 2. Decay: The daughter decays with its own half-life
Most important clinical example: the Mo-99 / Tc-99m generator
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Mo-99: T½ = 66 h (parent)
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Tc-99m: T½ = 6 h (daughter)
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tmax ≈ 23 h (time at which Tc-99m activity reaches its maximum)
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At equilibrium: theoretical ratio = 66/(66−6) ≈ 1.1; because only 87.6% of Mo-99 decays to Tc-99m, the actual ratio ≈ 0.96
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After elution of the generator, Tc-99m builds up again and reaches its maximum again at ~23 h
BOARD TIP
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The Mo-99/Tc-99m generator is the classic example of transient equilibrium. Key points for the board exam: (1) Theoretical ratio 1.1; because of 87.6% branching, actually ≈0.96. (2) After elution Tc-99m reaches its maximum at ~23 h. (3) At equilibrium Tc-99m decreases with the half-life of Mo-99 (66 h). (4) At tmax, ideally daughter activity = parent activity (for Mo/Tc ≈0.88 × because of branching).
Condition: the parent half-life is very much longer than the daughter half-life.
T½(parent) >> T½(daughter) (a difference of at least 100-1000 times)
Features:
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The parent activity remains practically constant (no significant decay during the observation period)
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The daughter activity increases and eventually equals the parent activity
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At equilibrium: AD = AP (daughter and parent activities are equal)
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Daughter/parent activity ratio = 1
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The daughter reaches equilibrium with its own half-life (equilibrium is established in ~5-6 daughter half-lives)
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After equilibrium the daughter activity remains constant (as long as the parent activity is constant)
Clinical examples:
Sr-82 / Rb-82 generator:
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Sr-82: T½ = 25.4 days (parent)
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Rb-82: T½ = 75 seconds (daughter)
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T½ ratio ≈ 29,000 → secular equilibrium
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Rb-82 reaches equilibrium within a few minutes
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Used for cardiac PET perfusion imaging
Ge-68 / Ga-68 generator:
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Ge-68: T½ = 271 days (parent)
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Ga-68: T½ = 68 minutes (daughter)
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T½ ratio ≈ 5,700 → secular equilibrium
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Ga-68 reaches equilibrium within a few hours
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Used for DOTATATE and PSMA PET imaging
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The relationship between Ra-226 (T½ ~1600 years) and Rn-222 (T½ 3.8 days) is the well-known example of secular equilibrium in nature. Secular equilibrium is also established in the Rb-81 (4.6 h) / Kr-81m (13 s) generator.
2.5.4 No Equilibrium
Condition: the daughter half-life is longer than the parent half-life.
T½(daughter) > T½(parent)
Features:
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The parent decays rapidly and is exhausted
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The daughter accumulates and then decays with its own half-life
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Equilibrium is never established
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The daughter activity first rises, reaches a maximum and then decreases with its own half-life
Example:
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I-131 (T½ = 8 days) → Xe-131 (stable) → the concept of equilibrium does not apply (stable daughter)
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Conceptually, however: with a short-lived parent and a long-lived daughter, no equilibrium forms
2.5.5 Comparison of Generator Systems
Generator
Parent (T½)
Daughter (T½)
Type of Equilibrium
Clinical Use
Mo-99/Tc-99m
66 h
6 h
Transient
SPECT imaging
Ge-68/Ga-68
271 days
68 min
Secular
PET (DOTATATE, PSMA)
Sr-82/Rb-82
25.4 days
75 s
Secular
Cardiac PET perfusion
Sn-113/In-113m
115 days
100 min
Secular
Quality control (historical)
Ti-44/Sc-44
59.1 years
4.04 h
Secular
Research
BOARD TIP
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In secular equilibrium A(daughter) = A(parent); in transient equilibrium A(daughter) > A(parent). This distinction is one of the most commonly confused topics in board exams. Mnemonic: secular → same, transient → ten percent more (theoretical). For Mo-99/Tc-99m the actual ratio is ≈0.96 because of 87.6% branching.
Parent and daughter activities are plotted over time: transient equilibrium (Mo-99/Tc-99m), secular equilibrium (Ge-68/Ga-68) and no equilibrium (Te-131/I-131).
Continuous stream · no seeking
Key point: if the parent's half-life is somewhat longer than the daughter's, transient equilibrium forms; if much longer, secular equilibrium; if shorter, no equilibrium.
How to use: one click on the atom columns pauses and resumes; move over or drag along the time graph; ← → change steps.
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Figure 2.8: Parent and daughter activity curves in transient equilibrium (Mo-99/Tc-99m), secular equilibrium (Ge-68/Ga-68) and no equilibrium.
2.6 Decay Schemes
Decay schemes are diagrams that show the decay process of a radionuclide graphically. Being able to read and interpret decay schemes is critical in board exams.
In-111 (T½ = 67.3 h) decays 100% by electron capture
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Daughter: Cd-111
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Two principal gammas: 171 keV (91%) and 245 keV (94%)
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A separate energy window is used for each photopeak
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Board exams may show a decay scheme and ask "which radionuclide is this?" or "which type of decay is this?". Watch the direction of the arrow: down to the right → β⁻ (Z increases), down to the left → β⁺/EC (Z decreases), vertically down → IT/gamma. You must also know that the principal gamma energy of I-131 is 364 keV (81%) and its principal beta energy is Emax = 606 keV.
Figure 2.10: Simplified decay schemes of Mo-99/Tc-99m, F-18 and I-131 (not to scale, schematic).
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2.6.3 Branching Ratios
Some radionuclides can decay by more than one route. The probability of each route is expressed as the "branching ratio":
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F-18: 96.7% β⁺ + 3.3% EC = 100%
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Cu-64: 17.6% β⁺ + 38.5% β⁻ + 43.9% EC = 100% → a rare radionuclide capable of both β⁺ and β⁻ decay
Alpha decay has two bodies: the α particle and the daughter nucleus. In each decay the parent nucleus appears first, then decays: as the α leaves in one direction the whole daughter nucleus recoils the other way. The momenta are equal and opposite and the energy is shared in inverse proportion to mass: for Ra-223 the α carries 5716 keV and the Rn-219 recoil ~104 keV (E_recoil ≈ E_α·4/219); the daughter moves ~55 times more slowly, its path is drawn 10 times longer. The rest of the energy is the daughter's excitation (marked *; released by γ emission or internal conversion); the bar below shows α + recoil + excitation = Q. Because the energy is always shared the same way for a given transition, the α has a single energy and the histogram shows separate lines (a discrete spectrum); several lines mean transitions to different daughter levels. Change the radionuclide (↑ ↓): Ac-225 and Th-227 also have many lines; 58% of At-211 decays are EC (no α; the daughter Po-211 emits a 7450 keV α 0.5 s later); Po-213 (Bi-213 chain) and Po-212 (Pb-212 chain) have a single, more energetic line.
Radionuclide:
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β⁺ emission and slowing down
In a proton-rich nucleus a proton becomes a neutron; a positron (e⁺) and a neutrino (ν) are emitted and share the energy. Every positron therefore has a different energy (from the real β⁺ spectrum): mean 250 keV for F-18, 1.5 MeV for Rb-82. The positron loses energy in collisions with tissue electrons (ionisation) and changes direction mostly by scattering in the field of atomic nuclei. While fast its path is nearly straight; as it slows down scattering grows and the path becomes tortuous; once thermalised it annihilates with an electron and two 511 keV photons leave in opposite directions. All radionuclides are drawn on the same scale and in real time: a high-energy positron travels a longer path and range. The close-up is the same for every radionuclide (×6): short tracks appear whole, long ones show their start. The track is drawn as a line that bends at each visible collision; the small direction changes in between show up together at the corner. Measured: in the first 20% of the track the corners are ~19° for F-18 and ~6° for Rb-82; in the last 20%, ~54° and ~30°. Nucleus (sized ∝ A^1/3) and electron symbols are not to scale; positron and electron are drawn the same size.
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Electron capture
A proton-rich nucleus captures an inner-shell electron (usually K): p + e⁻ → n + ν. Z falls by one, A is unchanged: Tl-201 → Hg-201. Only a neutrino leaves, and with two bodies its energy is a single value. The K vacancy is filled from an outer shell and a characteristic X-ray (or an Auger electron) is emitted: the 69-83 keV photons used in Tl-201 imaging are mercury X-rays.
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Isomeric transition
Tc-99m is an excited (metastable) nucleus; it gives up its extra energy without emitting a particle, as a γ ray, and becomes Tc-99: Z and A do not change. The isomeric level is actually at 142.7 keV: it first drops 2.2 keV to the 140.5 keV level (this step proceeds almost entirely by internal conversion, releasing a 1.6-2.1 keV electron absorbed on the spot), then the 140.5 keV γ used for imaging follows. T½ 6.01 h.
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Random decay and half-life
Each Tc-99m atom decays at a random moment; which one decays when cannot be predicted. Yet the population falls regularly: each hour about 11% of the remaining atoms decay (λ = 0.693 / T½ = 0.115 / h). The number halves at 6.01 h (T½) and falls to a quarter at 12 h. The stepped curve of 100 atoms fluctuates around the theoretical exponential (dashed).
The radiopharmaceutical in the organ decreases in two ways: physical decay (the atom flashes and fades in place) and biological excretion (the atom leaves the organ). Because both act together, the organ activity falls faster than with either alone: 1/Teff = 1/Tphys + 1/Tbiol. The effective half-life is always shorter than both and closer to the shorter one (6 h and 24 h → 4.8 h).
PhysicalBiologicalEffective
Example from the book: Tphys = 6 h, Tbiol = 24 h → Teff = (6 × 24) / (6 + 24) = 4.8 h. In the cumulated activity step A₀ = 100 MBq and organ fraction f = 1 (schematic). Atoms are not to scale; each atom either decays physically or is excreted from the organ.
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Transient equilibrium: Mo-99 → Tc-99m
The parent half-life is about ten times the daughter's (Mo-99 66 h, Tc-99m 6.01 h). After elution Tc-99m builds up, peaks at about 23 h and then decays together with Mo-99 at the Mo-99 half-life. At equilibrium the daughter activity should slightly exceed the parent (theoretical ratio 66/60 ≈ 1.1); since only 87.6% of Mo-99 decays feed Tc-99m the real ratio is ≈ 0.96.
Hover over the graph or drag with your finger: parent and daughter activity at that moment appear.