What are the Bateman equations for radioactive decay chains?+
The Bateman equations are the exact analytical solutions for atom counts (and activities) in a serial radioactive decay chain as a function of time. For a two-nuclide chain A to B starting with only parent atoms, N_B(t) = N_A0 * lambda_A / (lambda_B - lambda_A) * (exp(-lambda_A*t) - exp(-lambda_B*t)). The three-nuclide case adds a third exponential term involving lambda_C. These formulas replace numerical integration and give exact results for any input values.
What is secular equilibrium and how long does it take to reach?+
Secular equilibrium occurs when the parent half-life is at least 100 times longer than the daughter half-life. Once the daughter has undergone about 7 half-lives (reaching 99.2% of equilibrium), A_B equals A_A and both activities decrease together at the slow rate of the parent. For Ra-226 (1600 yr) and Rn-222 (3.82 day), equilibrium is reached in about 27 days.
What is transient equilibrium and how does it differ from secular equilibrium?+
Transient equilibrium occurs when the parent half-life is longer than the daughter half-life but not by more than a factor of a few hundred. At equilibrium the daughter activity is slightly above the parent: A_B/A_A = t_{1/2,A} / (t_{1/2,A} - t_{1/2,B}). For Mo-99 (65.94 hr) and Tc-99m (6.006 hr) this ratio is 1.10. Unlike secular equilibrium, both activities eventually decrease together along the parent decay curve.
How do I calculate Tc-99m activity from a Mo-99/Tc-99m generator?+
Set A0 to the Mo-99 activity at calibration time (usually printed on the generator vial label in MBq or mCi), enter 65.94 hr for Mo-99 half-life, 6.006 hr for Tc-99m half-life, and enter the time elapsed since last elution or calibration. The calculator outputs the current Tc-99m activity, remaining Mo-99 activity, and total combined activity, which is what the dose calibrator will read.
What happens when the daughter nuclide is stable?+
When the daughter is stable, lambda_B = 0 and the standard Bateman formula divides by zero. The correct formula follows from conservation of nucleons: N_B(t) = N_A0 - N_A(t) = N_A0 * (1 - exp(-lambda_A * t)). As the parent decays completely, the stable daughter accumulates to N_A0 atoms. The activity of the stable daughter is zero (stable means no decay), so only its atom count can be reported.
At what time does the daughter activity reach its maximum?+
The daughter activity A_B(t) peaks when its rate of production equals its rate of decay. Setting dA_B/dt = 0 gives t_max = ln(lambda_B / lambda_A) / (lambda_B - lambda_A). For Mo-99/Tc-99m this is about 22.8 hours after a fresh generator is eluted. Beyond t_max the daughter activity decreases, following the parent exponential from above (secular) or below (transient, eventually converging).
Can the solver handle equal parent and daughter decay constants?+
Yes. When lambda_A = lambda_B, the denominator in the standard formula is zero, creating a 0/0 indeterminate form. Using L'Hopital's rule (or Taylor expanding the exponentials) gives the limiting formula N_B(t) = N_A0 * lambda_A * t * exp(-lambda_A * t). This degenerate case is uncommon in nature but arises in theoretical problems with constructed isotope pairs sharing the same half-life.
How do I use the three-nuclide mode for a real decay chain?+
Select the "A to B to C" tab. Enter the half-lives of all three nuclides. If the final product (C) is stable, select "Stable" from the C half-life unit dropdown. The calculator will then use atom conservation for C instead of the analytical formula. A good test case is Th-234 (A, 24.1 day) to Pa-234m (B, 1.17 min) to U-234 (C, 245,500 yr) where U-234 is effectively stable on laboratory timescales.
What does the activity ratio A_B/A_A represent physically?+
The ratio A_B/A_A tells you where the chain is in its approach to equilibrium. Before equilibrium is reached, the ratio is below the equilibrium value and rising. At secular equilibrium it equals 1. At transient equilibrium it equals t_{1/2,A} / (t_{1/2,A} - t_{1/2,B}), which is always greater than 1. After the daughter activity peaks and the chain decays together, the ratio stays at this constant equilibrium value.
How accurate are the Bateman equation results from this calculator?+
The Bateman equations are exact closed-form solutions. Numerical accuracy is limited only by floating-point arithmetic, which provides about 15 significant digits. For typical nuclear medicine inputs (half-lives from seconds to years, times from minutes to weeks), results are accurate to better than 0.01%. Precision degrades slightly only when two decay constants differ by less than one part in 10^12, which has no physical relevance.
Why does the calculator assume no daughter activity at time zero?+
The standard Bateman formulas assume a freshly separated pure parent at t=0 with N_B(0) = N_C(0) = 0. This describes generator systems after elution, freshly purified samples, and newly produced activation products. If your sample already contains some daughter activity at t=0, the total N_B(t) is the sum of (1) the Bateman contribution from parent ingrowth and (2) the simple exponential decay of the initial N_B(0) atoms.
Can I use this for more than three nuclides in the chain?+
This calculator covers up to three nuclides (A to B to C). For longer chains such as the uranium-238 decay series (14 members), the Bateman formula extends to N nuclides with N exponential terms, but the algebra grows rapidly. Dedicated nuclear inventory codes such as ORIGEN-2.2, FISPACT-II, or CINDER90 are the standard tools for full decay chains in reactor and waste management applications.