00 Foundations

Mean life and decay constant

τ = 1/λ = T½ / ln 2 ≈ 1.443 T½. Mean life, not half-life.

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Simulation

Mean life and decay constant — Change the numbers; the scene follows.

Where it works

Hot lab

Dose calibrator

In the dose calibrator well — assayed activity, decay between two times, Marinelli.

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Formula

\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2} \approx 1.443\,T_{1/2}

Variables

T½ · Half-lifeh

Results

τ
Mean life
8.685h
λ
Decay constant
0.115141h⁻¹

Explanation

\tau = \frac{1}{\lambda} = \frac{T_{1/2}}{\ln 2} \approx 1.443\,T_{1/2}

What it means

Half-life is the time for activity to fall to 50%. Mean life τ is the average lifetime of an atom, equal to 1/λ, and is about 44% longer than T½. Cumulated activity for a pure exponential is A₀τ = 1.443 A₀ T½ — the factor that appears in MIRD. This is a working relation in Radiation physics.

Where it is used

Clinically it sits on the Hot lab — Dose calibrator. In the dose calibrator well — assayed activity, decay between two times, Marinelli. Radiation physics lives at the x-ray target, the linac head, and inside the patient: how a photon is born, how it scatters, and how it dies. Use these relations before you trust a spectrum, a wall, or a kV-versus-MV contrast argument.

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How to use it

Enter T½ in hours. Use τ when integrating activity to infinity (Ã = A₀ τ) or converting a decay constant from a tabulated half-life. Change one input and watch the curve and the simulation follow.

Symbols

T½Half-life6.02 h

Worked example

A typical case from the default values: T½ = 6.02 h (Half-life). Substituting into the relation gives τ = 8.685 h; λ = 0.115141 h⁻¹. These are teaching numbers — align them with your machine.

Typical values give

τ = 8.685h
λ = 0.115141h⁻¹

Where it comes from

The displayed formula is the working relation. τ = 1/λ = T½ / ln 2 ≈ 1.443 T½. Mean life, not half-life. Usual reference: Cherry. Derive it in the specialty lesson, then return here to pin the numbers.

Reference: Cherry

Assumptions & limits

Applies to a single exponential (one radionuclide, no biological clearance). For effective decay, replace T½ by T_e.

Pitfalls

Do not mix free-electron Compton kinematics with photoelectric-dominated kV imaging. Check keV versus MeV, and never treat a spectrum as one photon. Applies to a single exponential (one radionuclide, no biological clearance). For effective decay, replace T½ by T_e.

Keep this

Photons do not deposit dose; the electrons they set in motion do. Applies to a single exponential (one radionuclide, no biological clearance). For effective decay, replace T½ by T_e.

In this specialty

Radiation physics