Formula & Calculator
Mean Lifetime Formula
The mean lifetime (τ) is the average time a radioactive nucleus exists before decaying. It is the reciprocal of the decay constant. The mean lifetime is longer than the half‑life by a factor of 1/ln2 ≈ 1.443. It is used in statistical descriptions of decay and in calculations of integrated activity (e.g., total energy deposition).
Mean LifetimeDecay ConstantRadioactivity
Mean Lifetime
Nuclear Engineering · Nuclear Decay
τ = 1/λ
τ =
1 / λ
·
τ = mean lifetime ·
λ = decay constant
Mean lifetime τ (also called average lifetime) is the average time a radioactive nucleus
survives before decaying. It is the reciprocal of the decay constant λ.
The mean lifetime is related to the half‑life by τ = t½ / ln(2) ≈ 1.443 · t½.
This quantity is fundamental in nuclear physics and is used in decay calculations, reactor physics,
and radiation protection.
Presets:
—
—
Decay input:
Solve for:
Calculation Steps
Ready| Step | Operation | Value |
|---|---|---|
| Enter values and press Calculate | ||
Mean Lifetime Reference
τ = 1/λ = t½ / ln(2)
| Isotope | Half‑life | λ (s⁻¹) | τ (s) |
|---|
τ = 1/λ · τ = t½ / ln(2) ≈ 1.443 · t½ · Used in decay statistics and reactor kinetics
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| τ | Mean Lifetime | s |
| λ | Decay Constant | s⁻¹ |
What it means
Mean lifetime is the average time before decay. For a sample, the total number of decays over all time is N₀, and the total activity integrated is N₀ τ.
Worked example
Cs‑137 Mean Lifetime (τ = 1 / λ)
Nuclear Physics
Scenario: A radiation protection officer needs to know the mean lifetime (τ) of 137Cs, a fission product with a half‑life of 30.08 years. The decay constant λ = ln(2) / T½. The mean lifetime is used for predicting long‑term decay behaviour and waste management strategies.
| Parameter | Value |
|---|---|
| Half‑life (T½) | 30.08 years |
| Decay Constant (λ = ln(2)/T½) | 2.30×10⁻² yr⁻¹ |
| Mean Lifetime (τ = 1/λ) | 43.4 years |
1Determine the half‑life of the radionuclide from reference data.
2Calculate the decay constant λ = ln(2) / T½.
3Compute the mean lifetime τ = 1 / λ — this is the average time a nucleus exists before decaying.
Mean Lifetime
τ = 43.4 years
Cs‑137 has a mean lifetime about 1.44 times its half‑life, important for waste disposal planning.
F‑18 Mean Lifetime (τ = 1 / λ)
Medical Physics
Scenario: A cyclotron facility produces 18F‑FDG for PET scans. The half‑life of 18F is 109.8 minutes. The mean lifetime (τ) is used to estimate the activity at the time of injection, accounting for decay during transport and synthesis.
| Parameter | Value |
|---|---|
| Half‑life (T½) | 109.8 minutes |
| Decay Constant (λ = ln(2)/T½) | 6.31×10⁻³ min⁻¹ |
| Mean Lifetime (τ = 1/λ) | 158.4 minutes |
1Record the half‑life from the nuclide's decay data.
2Compute λ using the standard formula.
3Calculate τ — the mean lifetime helps to determine the optimal time window for imaging.
Mean Lifetime
τ = 158.4 minutes
After 2.5 hours, the activity drops to about e⁻¹ (37%) of the initial, so logistics must be tight.
Muon Mean Lifetime (τ = 1 / λ)
Particle Physics
Scenario: In a particle physics experiment, the decay of muons is observed. The muon half‑life is 2.197 µs. The mean lifetime (τ = 1/λ) is used to calculate the decay probability per unit time and to test the standard model.
| Parameter | Value |
|---|---|
| Half‑life (T½) | 2.197 µs |
| Decay Constant (λ = ln(2)/T½) | 0.315 µs⁻¹ |
| Mean Lifetime (τ = 1/λ) | 3.17 µs |
1Measure the half‑life from the exponential decay curve of muon events.
2Derive λ from the half‑life.
3Compute τ — the mean lifetime is a more direct measure of the particle's stability.
Mean Lifetime
τ = 3.17 µs
The muon mean lifetime is about 1.44 times its half‑life; this is a fundamental constant in physics.
Common mistakes
- Confusing mean lifetime with half‑life: Many mistakenly use τ = T½; the relationship is τ = T½ / ln(2) ≈ 1.44 × T½.
- Using τ in the exponential decay formula: The decay law uses λ (i.e., e‑t/τ works), but if you use τ instead of T½, you must use e‑t/τ.
- Forgetting the units: τ has the same units as 1/λ (e.g., seconds, years).
Applications
- Calculating total decays over time: The total number of decays in an interval is A₀ × τ × (1 – e‑t/τ).
- Decay heat calculations: Used in reactor physics to estimate residual heat from fission products.
- Statistical physics: The mean life is the expectation value of the lifetime distribution.