Calculate the time evolution of nuclear count and activity for an entire decay chain at once using the Bateman equation – not just for a single parent-daughter pair. Choose a known decay series (e.g., U-238 → Th-234 → Pa-234m → U-234 → …) or build a custom chain of up to 8 isotopes and view activity curves directly in the chart.
Many radioactive isotopes do not decay directly into a stable end product, but rather through an entire chain of daughter isotopes that are themselves radioactive – e.g., Uranium-238 decays through 14 intermediate steps until stable Lead-206 is formed.
A simple half-life calculator only describes the decay of a single isotope. To calculate how much of each chain member is present at a given time, the Bateman equation is needed, accounting for both production (from the predecessor) and decay (into the successor) of each member simultaneously.
Bateman Equation (General Form)
Nk(t) = N₀ · (λ₁···λk−1) · Σᵢ [ e−λᵢt / Πⱼ≠ᵢ(λⱼ − λᵢ) ]
Number of Nuclei (N)
Number of atomic nuclei of the respective isotope at time t. Initially (t=0), N > 0 only for the parent isotope; all daughter isotopes start at 0.
Activity (A)
Number of decays per second (in Becquerel), calculated as A = λ·N. A stable end isotope has no activity by definition.
Decay Constant (λ)
λ = ln(2) / T½ describes how fast an isotope decays. The shorter the half-life T½, the larger λ and the faster the decay.
In the Uranium-238 series, U-238 (half-life 4.468 billion years) decays first into short-lived Th-234 (24.1 days) and then into short-lived Pa-234m (1.17 minutes) before forming U-234.
Observation
Because Th-234 and Pa-234m decay extremely quickly compared to U-238, secular equilibrium is established after just a few months.
Result
In secular equilibrium, the activities of all short-lived chain members are practically identical to the (nearly constant) activity of the long-lived parent isotope U-238 – even though their absolute nuclear numbers N are tiny.
Most commonly, half-lives in different time units (e.g., minutes and years) are entered directly without realizing it – which is not an issue here since each row has its own unit. Another typical error is confusing the initial value with the final value of a daughter isotope: all daughter isotopes always start at N = 0.
Recommended Approach
Start with one of the preconfigured decay series to understand the behavior, then step-by-step replace the values with your own isotopes. Select a time period on the order of the shortest interesting half-life in the chain – otherwise the interesting dynamics at the start of the curve will be missed.
All isotopes used in this calculator can also be found with additional properties in our Periodic Table.
Where do the half-lives of the predefined decay series come from?
The half-lives are taken from the nuclear data of the National Nuclear Data Center (NNDC, Brookhaven National Laboratory, ENSDF database) or the Wikipedia overviews derived from them regarding the decay series of uranium-238, thorium-232, and uranium-235. For the uranium-238 series, values were additionally cross-checked using the publication "Radiological and Chemical Properties of Uranium" by the U.S. Nuclear Regulatory Commission (NRC). As with all nuclear data, individual values may vary slightly in the last decimal place depending on the source and measurement campaign – for high-precision applications, a direct look at the ENSDF database is recommended.