Bateman Equations Solver

Compute time-dependent activities in A to B and A to B to C radioactive decay chains using the analytical Bateman solution.

⚛️ Bateman Equations Solver
Initial Activity of A (A0)
Half-life of A (parent)
Half-life of B (daughter)
Half-life of C (granddaughter)
Evaluation Time (t)
Activity of B (daughter)
—
Activity of A (parent)
—
Total Activity (A + B)
—
Activity Ratio AB / AA
—
Fraction of A Remaining
—
B Atom Inventory
—
Activity of C (granddaughter)
—
Activity of A (parent)
—
Activity of B (intermediate)
—
Total Activity (A + B + C)
—
Fraction of A Remaining
—

⚛️ What Are the Bateman Equations?

The Bateman equations are the exact analytical solutions to the coupled first-order differential equations that describe how each nuclide in a radioactive decay chain grows and decays over time. Named after Harry Bateman who published the general solution in 1910, they form the mathematical backbone of every quantitative calculation in radioactive decay chain analysis, nuclear medicine generator physics, and reactor fuel burnup.

When nuclide A decays into daughter B, which itself decays into C, the atom counts of each species change simultaneously. The parent A simply follows the single-nuclide exponential law N_A(t) = N_A0 * exp(-lambda_A * t). The daughter B is more complex: it is both produced by the decay of A and consumed by its own decay. The Bateman equation resolves this competition exactly, giving a closed-form expression involving two exponential terms. The three-nuclide case for A to B to C has three exponential terms, and the pattern extends to any length chain.

This solver handles four special cases that would otherwise cause numerical errors. First, when the daughter is stable (lambda_B = 0), the solver switches to an atom-conservation formula. Second, when two decay constants are nearly equal (lambda_A ≈ lambda_B), the solver evaluates the mathematical limit analytically rather than computing a 0/0 fraction. Third, for very short evaluation times the exponential terms are computed with full floating-point precision. Fourth, the stable granddaughter case in 3-nuclide mode is handled by conservation across all three species.

The most famous application is the Mo-99/Tc-99m radionuclide generator used in nuclear medicine. Mo-99 (65.94 hr half-life) is loaded onto an alumina column; Tc-99m (6.006 hr half-life) grows in as Mo-99 decays and is eluted by the hospital physicist at regular intervals. The Bateman equations predict exactly how much Tc-99m is available at any point in the generator's life. Ra-226 in equilibrium with its short-lived daughters, Th-234 growing into Pa-234 after U-238 decay, and Bi-214 activity in environmental radon monitoring are other practical applications covered by this calculator.

📐 Bateman Equations

NA(t) = NA0 e−λAt
NB(t) = NA0 ⋅ λA / (λB − λA) ⋅ (e−λAt − e−λBt)
NC(t) = NA0 ⋅ λAλB ⋅ [e−λAt/((λB−λA)(λC−λA)) + e−λBt/((λA−λB)(λC−λB)) + e−λCt/((λA−λC)(λB−λC))]
NA0 = initial atom count of parent A = A0 / λA
λX = decay constant of nuclide X = ln(2) / t1/2,X (s−1)
AX(t) = activity of nuclide X = λX ⋅ NX(t) (Bq)
Special case (stable B): NB(t) = NA0 − NA(t)
Special case (λA = λB): NB(t) = NA0 ⋅ λA ⋅ t ⋅ e−λAt
Example: Mo-99 at t=24 hr: NA0 = 1000 MBq / λ(Mo-99) = 7.88×1013 atoms; AB(24) = 785.9 MBq

📖 How to Use This Calculator

Steps

1
Choose decay chain mode - click "A to B" for a two-nuclide chain or "A to B to C" for three nuclides. The calculator activates the relevant input fields automatically.
2
Enter the initial parent activity - type the activity of nuclide A at t=0 and select the unit (Bq, kBq, MBq, GBq, or TBq). Use the calibration-time activity for generator or source problems.
3
Enter all half-lives - provide the half-life of each nuclide and select the appropriate time unit (seconds through years). Select "Stable" from the unit dropdown for any nuclide that does not decay.
4
Set the evaluation time - enter the elapsed time since t=0 and choose the time unit. For a Mo-99/Tc-99m generator this is the time since last elution or since the generator was calibrated.
5
Read the results - the calculator shows the activity of each nuclide, the A_B/A_A activity ratio, the total chain activity, the fraction of the parent remaining, and the B atom inventory.

💡 Example Calculations

Example 1 - Mo-99/Tc-99m Generator at 24 Hours

1000 MBq Mo-99 at calibration, elution after 24 hr

1
Convert: λA = ln(2) / (65.94 × 3600 s) = 2.920 × 10−6 s−1; NA0 = 109 Bq / 2.920×10−6 = 3.425×1014 atoms.
2
At t = 24 hr: NA = NA0 ⋅ exp(−λA × 86400) = NA0 ⋅ exp(−0.2521) = 0.7772 NA0; AA = 777.2 MBq.
3
NB from Bateman equation with λB = ln(2)/(6.006 × 3600); AB = λB ⋅ NB = 785.9 MBq. The Tc-99m activity exceeds the Mo-99 activity, confirming transient equilibrium is nearly reached.
Tc-99m Activity = 785.9 MBq  |  AB/AA = 1.011
Try this example →

Example 2 - Ra-226/Rn-222 Secular Equilibrium at 30 Days

1 kBq Ra-226 source, check Rn-222 after 30 days

1
Ra-226 half-life = 1600 yr; Rn-222 half-life = 3.8235 day. The ratio of half-lives is 1600 × 365.25 / 3.8235 = 152,700, far above the secular equilibrium threshold of 100.
2
At 30 days (about 7.8 Rn-222 half-lives), the Rn-222 has reached secular equilibrium. AA is essentially unchanged at 1000 Bq (parent half-life is 1600 yr). AB approaches AA.
3
Bateman equation gives AB = 995.6 Bq at 30 days (7.85 Rn-222 half-lives, 99.57% of secular equilibrium). The ratio AB/AA = 0.9957 reflects the remaining transient, not yet the full 0.9999 level reached after many more half-lives.
Rn-222 Activity = 995.6 Bq  |  AB/AA = 0.9957
Try this example →

Example 3 - Ra-228/Ac-228/Th-228 Three-Nuclide Chain at 1 Year

100 kBq Ra-228 source, three-nuclide chain at t = 1 yr

1
Ra-228: t1/2 = 5.75 yr; Ac-228: t1/2 = 6.25 hr; Th-228: t1/2 = 1.9116 yr. At 1 year, AA = 100 × exp(−ln2/5.75) = 88.5 kBq.
2
Ac-228 (6.25 hr half-life) quickly reaches secular equilibrium with Ra-228, so AB ≈ AA = 88.5 kBq within the first two days.
3
Th-228 (1.91 yr half-life) is still growing in after 1 yr. The Bateman 3-nuclide formula gives AC ≈ 28 kBq, not yet at equilibrium with AA.
AA ≈ 88.5 kBq  |  AB ≈ 88.5 kBq  |  AC ≈ 28 kBq
Try this example →

❓ Frequently Asked Questions

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.
Tags: Bateman Equations Radioactive Decay Chain Daughter Activity Calculator Secular Equilibrium Transient Equilibrium Nuclear Physics Calculator Formula Free Online

🔗 Related Calculators

⚛️
Secular/Transient Equilibrium Calculator
Science - Nuclear
⚛️
Radioactive Decay Calculator
Science - Nuclear
⏳
Nuclear Half-Life Calculator
Science - Nuclear
⚡
Neutron Flux and Reaction Rate Calculator
Science - Nuclear
⚛️
Moderator-to-Fuel Ratio Calculator
Science - Nuclear

What are the Bateman equations for radioactive decay chains?

The Bateman equations are analytical solutions for the number of atoms in each member of a serial decay chain as a function of time. For A to B with N_B(0)=0, the result is N_B(t) = N_A0 * lambda_A / (lambda_B - lambda_A) * (exp(-lambda_A*t) - exp(-lambda_B*t)). The three-nuclide case adds an extra exponential term for C.

What is secular equilibrium and when does it occur?

Secular equilibrium occurs when the parent half-life is much longer than the daughter half-life, by a factor of at least 100. In this regime the daughter activity closely tracks the parent, so A_B equals A_A at equilibrium. Examples include Ra-226 (1600 yr) building up its short-lived decay products.

What is transient equilibrium in radioactive decay?

Transient equilibrium occurs when the parent half-life is longer than, but not vastly greater than, the daughter half-life. At equilibrium A_B/A_A equals t_{1/2,A} / (t_{1/2,A} - t_{1/2,B}). The Mo-99/Tc-99m generator is the classic example: equilibrium activity ratio is about 1.12.

How do I calculate Tc-99m activity from a Mo-99 generator?

Enter the Mo-99 activity at time zero, set its half-life to 65.94 hr, set the Tc-99m half-life to 6.006 hr, and enter the time since last elution. The calculator outputs the current Tc-99m activity, remaining Mo-99 activity, and total combined activity.

What happens when the daughter nuclide is stable?

When the daughter is stable, the standard Bateman formula has a singularity. This calculator handles it via conservation: N_B(t) = N_A0 minus N_A(t), so the stable daughter accumulates as the parent decays and reaches N_A0 atoms when the parent is fully consumed.

How do I find the time of maximum daughter activity?

Maximum daughter activity occurs at t_max = ln(lambda_B / lambda_A) / (lambda_B - lambda_A). You can find it by increasing the evaluation time in steps and watching when A_B peaks before turning over. This marks the point where the daughter's production rate equals its own decay rate.

Can the calculator handle equal parent and daughter decay constants?

Yes. When lambda_A equals lambda_B, the standard formula has a 0/0 singularity. The calculator uses the mathematical limit: N_B(t) = N_A0 * lambda_A * t * exp(-lambda_A*t). This degenerate case occurs when two nuclides happen to share the same half-life.

What is the difference between activity and atom count in decay chains?

Activity A equals lambda * N, measured in becquerels (decays per second). Atom count N is the number of radioactive atoms present. For a stable daughter, lambda equals zero so only N can be reported. This calculator shows stable daughter inventories in scientific notation (e.g. 3.61 x 10^14 atoms).

How accurate are the Bateman equation results?

The Bateman equations are exact analytical solutions to the coupled first-order differential equations governing serial decay chains. Numerical precision exceeds 0.01% for typical inputs. The only approximation is floating-point arithmetic, which loses a few digits of precision only when two decay constants are extremely close.

Can I model more than three nuclides in the chain?

This calculator supports up to three nuclides. The Bateman equations extend naturally to N nuclides with N exponential terms, but formula complexity grows rapidly. For chains of four or more nuclides, dedicated nuclear inventory codes such as ORIGEN or FISPACT are the standard tools.

What does the A_B divided by A_A ratio represent at equilibrium?

The ratio A_B/A_A shows how the daughter activity compares to the parent at the chosen time. At secular equilibrium this ratio approaches 1. At transient equilibrium it approaches t_{1/2,A} / (t_{1/2,A} - t_{1/2,B}). A ratio below 1 means equilibrium has not yet been reached or the chain has passed its peak.

Why does the calculation assume N_B equals zero at time zero?

The Bateman solution assumes a freshly separated pure parent sample with no daughter or granddaughter present at t=0. This is the standard assumption for generator systems and freshly prepared radiopharmaceuticals. If you have a mixed starting inventory, superpose two separate Bateman calculations.

📌 Quick Tips

💡For Mo-99/Tc-99m generators, set A0 to the Mo-99 activity at calibration, enter half-lives 65.94 hr and 6.006 hr, and read off the Tc-99m activity at your elution time.
💡Secular equilibrium is reached when the parent half-life is at least 100 times longer than the daughter half-life. At equilibrium, A_B equals A_A.
💡If the daughter half-life is shorter than the parent, the ratio A_B/A_A reaches a fixed value called transient equilibrium. This equals t_{1/2,A} divided by (t_{1/2,A} minus t_{1/2,B}).
💡Use the 3-nuclide mode for chains like Ra-228 to Ac-228 to Th-228, where the intermediate nuclide has a much shorter half-life than its parent or grandparent.
💡The Fraction of A Remaining result shows how much of the original parent activity survives, useful for confirming that the simulation time is physically reasonable.