Secular Equilibrium Calculator

Inputs

Parent half-life

Enter a positive number.

Parent unit

Daughter half-life

Should be much shorter for secular equilibrium.

Daughter unit

Initial parent atoms (Np0)

Any consistent atom count works.

Initial daughter atoms (Nd0)

Use 0 for a fresh daughter.

Elapsed time

Elapsed unit

Equilibrium tolerance (fraction)

0.01 means within 1% of Ad/Ap.

Max search time

Max unit

Example data table

Numbers below illustrate typical behavior where the daughter decays faster.

Parent half-life	Daughter half-life	Np0	Elapsed time	Expected Ad/Ap
30 days	6 hours	1.0e18	20 days	Near 1 after several daughter half-lives
5 years	2 days	5.0e16	30 days	Approaches 1 if λd ≫ λp
10 days	8 days	2.0e15	40 days	Often below 1 because rates are similar

Formula used

Decay constants: λ = ln(2) / T½
Parent atoms: Np(t) = Np0 · e^{−λp t}
Daughter atoms: Nd(t) = Nd0 · e^{−λd t} + (λp Np0 /(λd−λp)) · (e^{−λp t} − e^{−λd t})
Activities: Ap(t) = λp Np(t), Ad(t) = λd Nd(t)
Secular equilibrium idea: if λd ≫ λp and t is large, then Ad ≈ Ap

The time-to-equilibrium estimate searches for the earliest time meeting |Ad/Ap − 1| ≤ tolerance.

How to use this calculator

Enter parent and daughter half-lives using matching units.
Provide initial parent atoms and optional initial daughter atoms.
Set elapsed time to evaluate the chain at that moment.
Choose a tolerance to define “close enough” equilibrium.
Press Calculate to see results above the form.
Use the download buttons to export the latest result.

FAQs

1) What does secular equilibrium mean?

It is a state where the daughter activity becomes nearly equal to the parent activity. This typically happens when the daughter decays much faster than the parent and enough time has passed.

2) Why is λd/λp important?

The ratio compares decay speeds. When λd is much larger, the daughter quickly adjusts to the parent’s production rate. A ratio around ten or more often signals strong secular behavior.

3) Are the activities always in becquerels?

They are in becquerels if you input atom counts and time is handled in seconds internally. If you scale inputs differently, the ratio Ad/Ap remains meaningful even when absolute units change.

4) Can I start with nonzero daughter atoms?

Yes. Nd0 lets you model an existing daughter inventory. The solution includes both the decay of that inventory and the additional daughter atoms created from the parent’s decay over time.

5) What if the half-lives are similar?

If the half-lives are close, secular equilibrium is weak or absent. The daughter may never match the parent activity closely, and the Ad/Ap ratio can stabilize at a value noticeably different from one.

6) How is “time to equilibrium” computed?

The calculator searches forward in time until the activity ratio falls within your tolerance. It then refines the earliest time using bisection for a stable estimate without needing closed-form inversion.

7) What tolerance should I choose?

A common choice is 0.01 for within one percent. For quicker checks you can use 0.05. For strict analyses use 0.001, but the required time can increase substantially.

8) Does this handle branching decays?

This version models a simple two-member chain without branching. For branching, multiple daughters, or ingrowth from external sources, you would extend the equations or use a matrix decay system approach.