Decay Constant Calculator

The decay constant λ is the probability per unit time that a radioactive nucleus will decay. It is inversely proportional to the half‑life: λ = ln2 / T₁/₂. This parameter is fundamental to all decay calculations, including activity, remaining fraction, and dating. It is specific to each isotope and is independent of environmental conditions.

Written by the Vorzep Team

Decay ConstantHalf‑LifeRadioactivity

Decay Constant Nuclear Engineering · Nuclear Decay

λ = ln(2) / T_½

λ = ln(2) / T_½ · λ = decay constant · T_½ = half‑life

Decay constant λ is the probability per unit time that a radioactive nucleus will decay. It is related to the half‑life T_½ by λ = ln(2) / T_½. The decay constant is fundamental to all decay calculations, including activity, remaining fraction, and mean lifetime. It is used in nuclear physics, radiation protection, and radiometric dating.

Presets:

λ s⁻¹

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T_½ s

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Calculation Steps

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Decay Constant Reference λ = ln(2) / T_½

Isotope	Half‑life	λ (s⁻¹)

λ = ln(2) / T_½ · ln(2) ≈ 0.693147 · Units: λ in s⁻¹, T_½ in seconds

Variables

Symbol	Quantity	Unit
λ	Decay Constant	s⁻¹
T_{1/2}	Half‑Life	s

What it means

The decay constant is the rate parameter of the exponential decay law. A larger λ means faster decay.

Worked example

Co‑60 Decay Constant (λ = ln(2) / T_½)

Radiation Physics

Scenario: A radiation physicist needs the decay constant (λ) of ⁶⁰Co for a radiotherapy source calibration. The half‑life of ⁶⁰Co is 5.27 years. Using λ = ln(2)/T_½, the decay constant is calculated to determine the activity at any given time and to correct the source output for patient treatment planning.

Parameter	Value
Half‑life (T_½)	5.27 years
Natural Log of 2 (ln 2)	0.6931
Decay Constant (λ = ln(2)/T_½)	0.1315 yr⁻¹

1Obtain the half‑life of the radionuclide from reference data (e.g., 5.27 years for Co‑60).

2Divide ln(2) (≈ 0.6931) by the half‑life to get the decay constant in units of inverse time.

3Use λ in the activity formula A = A₀ e^−λt for source decay correction.

Decay Constant λ = 0.1315 yr⁻¹ This value is used to update the source activity weekly, ensuring accurate dose delivery in cancer treatments.

Common mistakes

Using T_½ in minutes when λ needs seconds: Units must be consistent; if T_½ is in years, λ is in year⁻¹; convert to s⁻¹ if needed for activity.
Confusing decay constant with decay rate: λ is a probability per unit time; the decay rate is A = λN, which depends on the number of nuclei.
Using half‑life instead of mean life: λ = ln(2)/T_½; using 1/T_½ (without ln 2) gives the mean life constant, which is different.

Applications

Radioactive source calibration: Determines how often sources need to be recalibrated due to decay.
Waste management: Calculates the decay rate to estimate when waste can be cleared from regulatory control.
Nuclear medicine: Schedules patient imaging based on the decay constant of the tracer.