00 Foundations
Parent–daughter decay (secular / transient)
Bateman solution for daughter activity starting from a pure parent.
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Simulation
Parent–daughter decay (secular / transient) — Change the numbers; the scene follows.
Where it works
Hot lab
Generator
At the Mo-99/Tc-99m generator — elution yield, secular equilibrium, Mo breakthrough.
Open this machineFormula
Typical values
Variables
Results
A₁
Parent activity
77.7203MBq
A₂
Daughter activity
78.5799MBq
A₂/A₁
Ratio
1.0111
Curve
Explanation
What it means
A parent nuclide decays to a radioactive daughter. If λ₂ ≫ λ₁ (half-life of the daughter much shorter) the system reaches secular equilibrium: A₂ ≈ A₁ after a few daughter half-lives, and they then decay together with the parent’s T½ — the ⁹⁹Mo/⁹⁹ᵐTc generator and ²²⁶Ra/²²²Rn. If the half-lives are comparable (λ₂ > λ₁ but not ≫) the equilibrium is transient: A₂ / A₁ = λ₂ / (λ₂ − λ₁) > 1 (¹³²Te/¹³²I). This is a working relation in Radiation physics.
Where it is used
Clinically it sits on the Hot lab — Generator. At the Mo-99/Tc-99m generator — elution yield, secular equilibrium, Mo breakthrough. Radiation physics lives at the x-ray target, the linac head, and inside the patient: how a photon is born, how it scatters, and how it dies. Use these relations before you trust a spectrum, a wall, or a kV-versus-MV contrast argument.
Hot lab · Open this machineHow to use it
Enter A₁(0), the two half-lives and elapsed time, in the same time unit. Read A₁, A₂ and the ratio. Presets: Mo-99/Tc-99m (secular-ish, T½ 66 h / 6 h) and Te-132/I-132 (transient). Change one input and watch the curve and the simulation follow.
Symbols
- A₁(0)Parent activity100 MBq
- T½₁Parent half-life66 h
- T½₂Daughter half-life6.02 h
- tElapsed time24 h
Worked example
A typical case from the default values: A₁(0) = 100 MBq (Parent activity); T½₁ = 66 h (Parent half-life); T½₂ = 6.02 h (Daughter half-life); t = 24 h (Elapsed time). Substituting into the relation gives A₁ = 77.7203 MBq; A₂ = 78.5799 MBq; A₂/A₁ = 1.0111. These are teaching numbers — align them with your machine.
Typical values give
- A₁ = 77.7203MBq
- A₂ = 78.5799MBq
- A₂/A₁ = 1.0111
Where it comes from
The displayed formula is the working relation. Bateman solution for daughter activity starting from a pure parent. Usual reference: Bateman 1910 / Cherry, Sorenson & Phelps. Derive it in the specialty lesson, then return here to pin the numbers.
Reference: Bateman 1910 / Cherry, Sorenson & Phelps
Assumptions & limits
Single daughter, branching ratio 1, no grand-daughter, A₂(0)=0. Mo-99 → Tc-99m actually has BR ≈ 0.86 to the metastable state. Generator elution resets A₂.
Pitfalls
Do not mix free-electron Compton kinematics with photoelectric-dominated kV imaging. Check keV versus MeV, and never treat a spectrum as one photon. Single daughter, branching ratio 1, no grand-daughter, A₂(0)=0. Mo-99 → Tc-99m actually has BR ≈ 0.86 to the metastable state. Generator elution resets A₂.
Keep this
Photons do not deposit dose; the electrons they set in motion do. Single daughter, branching ratio 1, no grand-daughter, A₂(0)=0. Mo-99 → Tc-99m actually has BR ≈ 0.86 to the metastable state. Generator elution resets A₂.
In this specialty