Fission Product Inventory Calculator
Estimates the radioactive inventory of a specific fission product in a reactor core after a given operating period. Used by nuclear engineers to assess source terms, plan refueling, or evaluate accident scenarios.
About this calculator
When a fissile nucleus splits, each fission produces radioactive daughter products at a characteristic yield. The inventory of a particular isotope at time t depends on its production rate and its simultaneous radioactive decay. The production rate equals fissionRate × (fissionYield / 100) × (isotope / 100), where isotope is the fractional branching to the target nuclide. As the reactor operates, the inventory grows toward a saturation value governed by the Bateman equation for a single-step decay chain: N(t) = [fissionRate × fissionYield × (isotope / 100) × (1 − e^(−λt))] / λ, where λ is the decay constant in day⁻¹ and t is operating time in days. At long times (t ≫ 1/λ) the exponential term vanishes and inventory saturates; for short-lived nuclides this equilibrium is reached quickly, while long-lived products accumulate nearly linearly.
How to use
Suppose a reactor runs at a fission rate of 3 × 10¹⁸ fissions/s, fission yield = 6.3%, target isotope branching = 100%, decay constant λ = 0.693 day⁻¹ (half-life ≈ 1 day), and operating time = 3 days. First convert: production rate = 3×10¹⁸ × (6.3/100) × (100/100) = 1.89×10¹⁷ atoms/s. Then: N = 1.89×10¹⁷ × (1 − e^(−0.693 × 3)) / 0.693 = 1.89×10¹⁷ × (1 − 0.125) / 0.693 ≈ 2.39×10¹⁷ atoms. At 3 half-lives the inventory is 87.5% of its saturation value.
Frequently asked questions
What does fission yield mean in the fission product inventory calculator?
Fission yield is the percentage of all fission events that produce a specific mass-chain fragment. For uranium-235 thermal fission, values range from near zero for some masses to about 6–7% for the most probable products near mass 90 and 140. Entering the correct chain yield ensures the production rate is scaled accurately. Some calculators also accept the independent yield of a specific nuclide rather than the cumulative chain yield, so confirm which convention your data source uses.
Why does fission product inventory reach a saturation limit over time?
Saturation occurs because the isotope is simultaneously produced by fission and destroyed by radioactive decay. As the inventory grows, the decay rate (λN) also grows until it exactly equals the production rate, at which point dN/dt = 0. This equilibrium is reached after roughly five half-lives of continuous irradiation. Short-lived fission products like iodine-131 (t½ ≈ 8 days) saturate within weeks, while long-lived products like cesium-137 (t½ ≈ 30 years) accumulate for years before approaching equilibrium.
How does decay constant affect the calculated fission product inventory?
The decay constant λ = ln(2) / t½ appears in both the denominator and the exponent of the inventory formula, so it has a dual effect. A larger λ (shorter half-life) reduces the saturation inventory (production rate / λ is smaller) but also causes the inventory to reach that lower saturation value more quickly. Conversely, a small λ means a very large potential inventory but one that builds up slowly. This interplay is why short-lived, high-yield isotopes like iodine-131 dominate early accident source terms while long-lived species like cesium-137 dominate long-term contamination.