0.0001209681
1/selected time unit
8,266.642584
selected time unit
180,825,048,000
s
3.8332475961e-12
s^-1
260,875,400,018.10876
s
0.0001209681
1/selected time unit
8,266.642584
selected time unit
180,825,048,000
s
3.8332475961e-12
s^-1
260,875,400,018.10876
s
The Decay Constant Calculator converts a radioactive half-life into the decay constant (λ) and mean lifetime (τ), providing the fundamental parameters needed for all radioactive decay calculations. The decay constant is the probability per unit time that a given atom will decay, and it is the key parameter in the exponential decay law \(N(t) = N_0 e^{-\lambda t}\).
The relationship between half-life and decay constant is straightforward: \(\lambda = \ln 2 / t_{1/2}\). Despite its simplicity, this conversion is performed constantly in nuclear physics, radiation safety, medical physics, and geochemistry, making a dedicated calculator valuable for quick reference.
The mean lifetime (\(\tau = 1/\lambda\)) represents the average time an atom survives before decaying. It is related to the half-life by \(\tau = t_{1/2} / \ln 2 \approx 1.443 \times t_{1/2}\). While half-life is more commonly used in nuclear physics and chemistry, mean lifetime is preferred in particle physics and quantum mechanics because it appears naturally in decay rate expressions and Breit-Wigner resonance formulas.
This calculator also provides the decay constant in per-second units (s⁻¹), regardless of the input time unit, which is essential for computing activity in becquerels (decays per second). The flexibility to work in different time units—seconds, minutes, hours, days, or years—accommodates the enormous range of half-lives encountered in nuclear science.
From technetium-99m with its 6-hour half-life used in medical imaging, to cesium-137's 30-year half-life relevant for nuclear accident cleanup, to uranium-238's 4.47-billion-year half-life used in geological dating, the decay constant calculator bridges the gap between the half-life you look up in a table and the mathematical parameter you need for calculations.
Understanding the decay constant is also essential for calculating specific activity, branching ratios in complex decay schemes, and reaction rates in nuclear reactors where neutron-induced transmutations compete with natural decay.
The fundamental relationships are:
Decay constant from half-life:
$$\lambda = \frac{\ln 2}{t_{1/2}} = \frac{0.693147}{t_{1/2}}$$
Mean lifetime:
$$\tau = \frac{1}{\lambda} = \frac{t_{1/2}}{\ln 2} \approx 1.4427 \times t_{1/2}$$
Unit conversion to seconds:
$$\lambda_{s^{-1}} = \frac{\ln 2}{t_{1/2} \times f}$$
where \(f\) is the conversion factor to seconds (1 for seconds, 60 for minutes, 3600 for hours, 86400 for days, 31557600 for years).
A larger decay constant means faster decay (shorter half-life). For example, λ = 0.693 per second means the half-life is 1 second. The mean lifetime is always about 44.3% longer than the half-life. The decay constant in per-second units is needed to compute activity in becquerels: A(Bq) = λ(s⁻¹) × N(atoms).
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Results
Carbon-14 has λ ≈ 1.21×10⁻⁴ per year (or 3.83×10⁻¹² per second) and a mean lifetime of 8,267 years.
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Results
Iodine-131 with t½ = 8.02 days has λ ≈ 0.0864 per day, or about 10⁻⁶ per second, and a mean lifetime of 11.6 days.
The decay constant λ represents the probability per unit time that any individual atom will decay. For example, if λ = 0.01 per second, each atom has a 1% chance of decaying in any given second. It is a fundamental quantum mechanical property of the nucleus, determined by the nuclear force, energy levels, and decay mode.
Half-life is the median survival time (when 50% have decayed), while mean lifetime is the average survival time. Because exponential decay has a long tail—some atoms survive much longer than average—the mean is pulled higher than the median. Mathematically, τ = t½/ln(2) ≈ 1.443 × t½.
Mean lifetime is preferred in particle physics (where it appears in Breit-Wigner formulas and Feynman diagram calculations), quantum optics (photon emission rates), and any context where the natural exponential form e^(-t/τ) is more convenient than the half-life form (1/2)^(t/t½). Half-life is more common in nuclear physics, chemistry, and applied radiation science.
This calculator uses the Julian year (365.25 days = 31,557,600 seconds), which is the standard in astronomy and nuclear physics. The actual calendar year varies (365 or 366 days), but the Julian year provides a consistent, well-defined conversion factor used in all scientific reference tables.
While theoretically possible, in practice every isotope has a unique decay constant determined by its specific nuclear structure. Even isotopes with very similar half-lives (like ¹⁴C at 5,730 years and ¹⁶⁰Tb at 5,810 years) have distinct decay constants. The decay constant is as characteristic of an isotope as its mass or charge.
Activity A = λN, where N is the number of radioactive atoms. In SI units, activity is measured in becquerels (Bq = decays/second), so you need λ in s⁻¹. This calculator provides λ in per second regardless of your input time unit, making activity calculations straightforward.
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