Parent-Daughter Decay Calculator

The formula assumes the daughter isotope was not present at the time of sample formation (t=0). If initial daughter is present, the equation becomes N_daughter(t) = N_daughter(0) + N_parent(0)(1 - e^{-λt}). Without accounting for initial daughter, the calculated age will be too old. In geochronology, this is avoided by using isochron methods that determine both the initial daughter ratio and the age from multiple samples. If initial daughter is known (e.g., from common lead corrections in U-Pb dating), it can be subtracted before using the simple formula.

Since each parent decay produces one daughter (for a simple decay chain), the number of daughter atoms at time t is simply the number of parents that have decayed: N_daughter(t) = N_parent(0) - N_parent(t) = N_parent(0)(1 - e^{-λt}). This is a direct consequence of mass conservation in the parent-daughter system, assuming the daughter is stable and no other loss or gain mechanisms. The formula is essentially the decay law rearranged for the daughter product.

For a chain where the daughter is also radioactive, the simple equation fails because the daughter decays further. The correct description uses the Bateman equations, which account for the sequential decays. The formula given is only valid for a single-step decay (parent → stable daughter). For example, in the U-238 decay series, you cannot use this formula to calculate the amount of Pb-206 from U-238 decay without including the intermediate decays, though if the chain is in secular equilibrium, the activity of each daughter equals the parent's activity, and the total daughter production is still governed by the parent's decay, but the time to reach equilibrium matters.

In practice, you measure the current numbers of parent (P) and daughter (D) atoms. The formula gives D = P₀(1 - e^{-λt}) and P = P₀e^{-λt}. Dividing gives D/P = e^{λt} - 1, so t = (1/λ) ln(1 + D/P). This is the standard dating equation used in Rb-Sr, Sm-Nd, and other systems. If initial daughter is present, the isochron method is used to account for it. The simple formula assumes D₀ = 0; otherwise, you use D - D₀ = P₀(1 - e^{-λt}) and solve for t using the initial daughter correction.

The formula N_daughter = N_parent(0)(1 - e^{-λt}) describes the growth of a stable daughter from a single parent. In a chain with a radioactive daughter, secular equilibrium occurs when the activity of the daughter equals the activity of the parent (λ_d N_d = λ_p N_p). The simple formula does not apply to the radioactive daughter; instead, it applies to the final stable end product. For example, in the U-Pb system, the formula applies to the accumulation of Pb (stable) from U decay, with the intermediate daughters ignored because they are short-lived relative to U's half-life.

Yes, it can be used to estimate the amount of a stable daughter produced from a parent fission product (e.g., ⁹⁰Sr → ⁹⁰Zr). However, in a reactor, the parent is also being produced (burnup) and may be removed by neutron capture, so the simple formula is only valid for a closed system with no production of the parent. For spent fuel, you can use the formula to calculate the ingrowth of stable daughters from the inventory of long-lived parents, but you must know the initial parent concentration at the time of discharge. For example, the ingrowth of ⁹⁰Zr from ⁹⁰Sr (t₁/₂=28.8 yr) can be calculated to assess changes in radiotoxicity over time.

Measuring absolute numbers of atoms is difficult and depends on sample size; ratios are more precise. The formula can be rewritten in terms of a stable reference isotope (e.g., ⁸⁶Sr) that does not change: (⁸⁷Sr/⁸⁶Sr) = (⁸⁷Sr/⁸⁶Sr)₀ + (⁸⁷Rb/⁸⁶Sr)(e^{λt} - 1). This is the isochron equation, which is linear in (⁸⁷Rb/⁸⁶Sr). It still contains the same exponential factor, but the slope gives (e^{λt} - 1). The simple formula N_daughter = N_parent(0)(1 - e^{-λt}) becomes a ratio equation by dividing by the reference isotope. This is the standard way to apply the decay law in geochronology without needing absolute atom counts.

The closure temperature is the temperature below which the daughter isotope becomes trapped in the mineral lattice and cannot escape by diffusion. Above this temperature, the mineral is 'open' to daughter loss, and the formula's assumption of a closed system is violated. When a rock cools below the closure temperature, the radioactive clock starts. The formula gives the time since cooling below the closure temperature, not the time of crystallization. Different minerals have different closure temperatures (e.g., biotite ~300°C for K-Ar, zircon ~900°C for U-Pb), so dating different minerals in the same rock can yield different ages reflecting different cooling stages.

If the parent has branching, the total decay constant λ = λ₁ + λ₂, where λ₁ and λ₂ are the partial decay constants for each branch. For the production of daughter 1 (e.g., ⁴⁰Ar), the formula becomes N_daughter1 = (λ₁/λ) N_parent(0) (1 - e^{-λt}). This is because only a fraction of decays lead to that daughter. In the K-Ar system, the age is calculated using t = (1/λ) ln(1 + (λ/λ_Ar)(⁴⁰Ar*/⁴⁰K)). Thus, the simple formula must be multiplied by the branching ratio (λ₁/λ) when only one branch is monitored.

The term (1 - e^{-λt}) represents the fraction of the original parent atoms that have decayed by time t. At t=0, it is 0 (no decays). As t → ∞, e^{-λt} → 0, so the fraction approaches 1, meaning all parent atoms have decayed. In that limit, N_daughter = N_parent(0), i.e., all parents have become daughters. This reflects the complete transformation of the parent isotope into the stable daughter, which is the end state of the decay process. For practical dating, we rarely reach this limit except for very old samples or short-lived isotopes.

Using the formula, N_daughter = N_parent(0) - N_parent(t). You know N_daughter = 500 and N_parent(0) = 1000, so N_parent(t) = 500. Then the decay law gives N_parent(t) = N_parent(0) e^{-λt} → 500 = 1000 e^{-λt} → e^{-λt} = 0.5 → λt = ln(2) → t = t₁/₂ = 10 years. Alternatively, using the daughter formula: 500 = 1000(1 - e^{-λt}) → 0.5 = 1 - e^{-λt} → e^{-λt} = 0.5 → same result. So the age is exactly one half-life. In general, you would solve for t using t = (1/λ) ln(1 + N_daughter / N_parent(t)).

Common pitfalls include: (1) assuming no initial daughter when in fact the daughter was present at formation (e.g., common lead in U-Pb), leading to overestimated ages; (2) assuming the system remained closed to gain or loss of parent or daughter (e.g., argon loss in K-Ar due to heating, uranium leaching in groundwater); (3) assuming the decay constant is accurately known (some are known to 0.1-1%, but for some systems, like ⁴⁰K, there is a slight uncertainty in the branching ratio); (4) assuming the sample has not been partially reset by later thermal events; (5) using the formula for a chain with radioactive daughter without branching corrections; (6) mixing of different materials with different initial ratios (can be detected by isochron scatter). Always validate with isochron plots or multiple decay systems if possible.

In ¹⁴C dating, the parent is ¹⁴C, which is continuously produced in the atmosphere and incorporated into organic matter. At the time of death, the organism stops exchanging carbon, and the ¹⁴C decays (parent) to ¹⁴N (daughter). The formula N_daughter = N_parent(0)(1 - e^{-λt}) applies, but the daughter is stable nitrogen, which is abundant in the sample and cannot be measured as a separate component. Instead, ¹⁴C dating measures the remaining parent activity relative to a modern standard (N_parent(t)/N_parent(0)). The age is calculated from the parent decay: N_parent(t) = N_parent(0)e^{-λt}. The daughter formula is not directly used because the daughter (¹⁴N) is not radiogenic in the sense of a distinct isotope; it's the common isotope. So ¹⁴C dating uses the parent decay formula, not the daughter growth formula.

In waste disposal, the ingestion or inhalation radiotoxicity of a radionuclide is often estimated by considering the decay chain. For a parent that decays to a stable daughter, the simple formula gives the amount of stable daughter produced, but the radiotoxicity of the parent decreases with time. However, for a chain with radioactive daughters, the Bateman equations are needed. The formula N_daughter = N_parent(0)(1 - e^{-λt}) can be used to calculate the ingrowth of a stable daughter that might have different chemical behavior (e.g., plutonium to uranium), affecting the mobility in the environment. For example, the ingrowth of ²³⁹Pu's stable daughter (²³⁵U, though ²³⁹Pu decays to ²³⁵U via ²³⁹U with short half-lives) is negligible for radiotoxicity, but for long-lived actinides like ²⁴⁴Cm, the ingrowth of ²⁴⁰Pu (which is also radioactive) requires the full chain. The simple formula is a starting point for very long timescales where the parent's decay is the only relevant process.

The decay constant λ is the probability per unit time of decay. It has a well-determined value for most isotopes (e.g., λ for ⁸⁷Rb is 1.397×10⁻¹¹ yr⁻¹ with ~0.5% uncertainty). The age t is derived from t = (1/λ) ln(1 + D/P). The relative uncertainty in t due to λ is approximately (σ_λ/λ) × (ln(1 + D/P) / (1 + D/P)? Actually, error propagation shows σ_t ≈ (σ_λ/λ) × t. So the uncertainty in age is directly proportional to the fractional uncertainty in λ. For young samples (small D/P), the uncertainty from λ is less significant than the measurement uncertainty in D/P. For very old samples, the λ uncertainty becomes a limiting factor. This is why constants like ⁸⁷Rb are carefully measured; any improvement in λ precision directly improves age accuracy.