Contamination & Internal bateman_calc
Bateman Parent-Daughter Decay Kinetics & Equilibrium
Solves coupled Bateman differential equations for parent-daughter radioactive decay chains, modeling transient equilibrium, secular equilibrium, and generator milking cycles.
A₂(t) = A₁(0) × [λ₂ / (λ₂ − λ₁)] × (e^(−λ₁ t) − e^(−λ₂ t)) + A₂(0) e^(−λ₂ t)
RADCALC HEALTH PHYSICS & NUCLEAR ENGINEERING WORKSHEET
Bateman Parent-Daughter Decay Kinetics & Equilibrium (Bateman Decay)
Domain: Contamination, Air Sampling & Internal Dosimetry · Tool ID:
bateman_calc · Engine: 100% Local Browser WASM A₂(t) = A₁(0) × [λ₂ / (λ₂ − λ₁)] × (e^(−λ₁ t) − e^(−λ₂ t)) + A₂(0) e^(−λ₂ t)
Operational Notes:
Guided Stepper Workflow All Input Parameters & Solver
1
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Select your target output variable. The stepper will guide you through only the required inputs.
2
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Pre-fill realistic health physics values, then refine individual parameters or jump straight to the result.
All Parameters Entered
Review Inputs & Calculated Result
Solved Output Live WASM Readout
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Entered Input Parameters (Tap any row to edit)
Calculated Result & Visual Analysis
Nominal Solved Value Updated automatically
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CLIENT-SIDE 3D SIMULATION Bateman Parent-Daughter Decay Kinetics & Equilibrium 3D Transport Field
Interactive 3D Geometry & Particle Transport
WebGPU / WebGL2 Ready
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Checking Browser 3D Capabilities… Probing WebGPU adapter, WebGL 2.0, and WebGL 1.0 contexts locally.
Geometry & Field Diagram
Interactive Model
Sensitivity Curve
Normalized Base-Unit & WebMCP Tool Payload bateman_calc
1. Input & Solved Parameter Ledger
| Role | Symbol | Parameter Name | Entered / Display Value | Normalized Base Unit | Definition |
|---|
2. Radiopharmacy & Decay Kinetics Ledger
Governing Equation & Parameters
Daughter Activity = A1(0) × [λ2 / (λ2 − λ1)] × (exp(−λ1 t) − exp(−λ2 t)); Peak Time t_max = ln(λ2/λ1) / (λ2 − λ1)
| Symbol | Parameter | Default Unit | Definition |
|---|---|---|---|
| A₁(0) | Initial Parent Activity | mCi | Initial activity of the parent radionuclide at time zero in millicuries. |
| A₂(0) | Initial Daughter Activity | mCi | Initial activity of the daughter radionuclide present at time zero in millicuries. |
| T_½,1 | Parent Half-Life | hr | Radioactive half-life of parent isotope in hours. |
| T_½,2 | Daughter Half-Life | hr | Radioactive half-life of daughter isotope in hours. |
| t | Elapsed Ingrowth Time | hr | Elapsed duration since purification or start of ingrowth in hours. |
| BR | Decay Branching Ratio | fraction | Fractional branching probability yielding the daughter state (0 to 1.0). |
| A₂(t) | Daughter Current Activity | mCi | Resulting daughter radionuclide activity at time t in millicuries. |
| A₁(t) | Parent Remaining Activity | mCi | Remaining parent radionuclide activity at time t in millicuries. |
| t_max | Time of Peak Daughter Activity | hr | Time of maximum daughter activity (generator milking optimum) in hours. |
Health Physics Notes & Related Calculators
Operational Guidance & Physics Assumptions
- Secular Equilibrium (T_1/2,parent >> T_1/2,daughter): Daughter activity rises until it precisely matches parent activity (e.g., Cs-137/Ba-137m, Sr-90/Y-90, Ra-226/Rn-222).
- Transient Equilibrium (T_1/2,parent > T_1/2,daughter): Daughter activity exceeds parent activity by the factor λ2 / (λ2 - λ1) after reaching peak activity t_max (e.g., Mo-99/Tc-99m).
- Time of maximum daughter activity (when A_2(0) = 0): t_max = ln(λ2 / λ1) / (λ2 - λ1).
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