Bateman Decay Chain Calculator
Nuclear radiochemistry: Solve the Bateman equations for parent-daughter-granddaughter decay chains, secular/transient equilibrium, and peak ingrowth time.
Decay Chain Radionuclides
Decay Kinetics & Activity Summary
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Frequently Asked Questions
What is the Bateman equation and how does it model decay chains?
Formulated by Harry Bateman in 1910, the Bateman equations are an analytical solution to the coupled system of linear first-order differential equations that govern serial radioactive decay (dN_i/dt = lambda_{i-1} N_{i-1} - lambda_i N_i). They calculate the exact inventory of each daughter nuclide over time based on decay constants.
What is the difference between Secular Equilibrium and Transient Equilibrium?
In Secular Equilibrium (e.g. Ra-226 decaying to Rn-222), parent half-life is thousands of times longer than daughter (T1 ≫ T2). The parent activity remains virtually constant while daughter activity rises until daughter decay rate equals daughter production rate (A2 = A1). In Transient Equilibrium (e.g. Mo-99 to Tc-99m), T1 is only moderately longer than T2. Daughter activity peaks, then declines in lockstep with the parent, maintaining a constant ratio A2/A1 = lambda2 / (lambda2 - lambda1).
Why is the Mo-99 / Tc-99m generator eluted every 24 hours?
In a medical "technetium cow" generator, Mo-99 (T1/2 = 66 h) produces Tc-99m (T1/2 = 6 h). Evaluating Bateman kinetics shows daughter Tc-99m activity reaches its theoretical maximum at t_max = 22.9 hours. Eluting (milking) the generator once every 24 hours extracts the maximum yield of diagnostic gamma-emitting Tc-99m.