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8.2.1.3. Decay Constant

Table of Contents

Meaning of the Decay Constant

The decay constant is a number that tells us how quickly a radioactive substance tends to decay. It is usually written with the Greek letter $\lambda$.

A radioactive sample contains many unstable nuclei. Each nucleus has some chance to decay, but we cannot predict exactly when one specific nucleus will decay. What we can describe is the overall tendency of the whole group. The decay constant measures that tendency.

If $\lambda$ is large, nuclei decay more rapidly. If $\lambda$ is small, they decay more slowly.

The decay constant $\lambda$ is the probability per unit time that an individual unstable nucleus will decay.
It does not mean that a fixed fraction of nuclei decays at one exact moment. It means that during each small time interval, each undecayed nucleus has the same chance to decay.

Probability Interpretation

Suppose a sample contains $N$ undecayed nuclei at some time. In a very short time interval $\Delta t$, the number that decays is proportional to both $N$ and $\Delta t$. This is written as

$$
\Delta N \propto -N \Delta t
$$

The minus sign shows that the number of undecayed nuclei decreases. Introducing the decay constant gives

$$
\Delta N = -\lambda N \Delta t
$$

In differential form, this becomes

$$
\frac{dN}{dt} = -\lambda N
$$

This equation expresses the core meaning of the decay constant. The rate of decay is proportional to how many undecayed nuclei are still present.

Units of the Decay Constant

Because $\lambda$ represents probability per unit time, its unit is inverse time.

QuantitySymbolTypical unit
Decay constant$\lambda$$\text{s}^{-1}$
Time$t$s
Number of undecayed nuclei$N$dimensionless count

Sometimes other time units are used, such as $\text{day}^{-1}$ or $\text{year}^{-1}$, depending on the process being studied.

The SI unit of the decay constant is
$$
[\lambda] = \text{s}^{-1}
$$
A larger value of $\lambda$ means faster decay.

Physical Picture

The decay constant does not count how many nuclei have already decayed. Instead, it describes how likely decay is for each nucleus that remains undecayed.

Imagine two radioactive samples with the same number of nuclei. If sample A has a larger decay constant than sample B, then in the same short time interval, a larger fraction of nuclei in sample A will decay.

This makes $\lambda$ a property of the radioactive isotope. Different isotopes have different decay constants because they have different nuclear stability.

Relation to Fractional Decay

From

$$
\frac{dN}{dt} = -\lambda N
$$

we can divide by $N$ and write

$$
\frac{1}{N}\frac{dN}{dt} = -\lambda
$$

This shows that the decay constant is the fractional rate of decrease of the undecayed nuclei.

In words, $\lambda$ tells us what fraction of the remaining nuclei tends to disappear per unit time.

For a very short interval $\Delta t$, the fraction that decays is approximately

$$
\frac{|\Delta N|}{N} \approx \lambda \Delta t
$$

when $\Delta t$ is small.

For a small time interval $\Delta t$,
$$
\text{fraction decayed} \approx \lambda \Delta t
$$
This approximation is valid only when $\Delta t$ is small.

Visual Interpretation

A large decay constant gives a steep drop in the number of undecayed nuclei. A small decay constant gives a gentler drop.

Large and small decay constants

The red curve falls faster, so it corresponds to a larger decay constant.

Connection with Other Decay Quantities

The decay constant is closely related to other common measures of radioactive decay, especially half-life and mean lifetime. Those topics are treated separately, but the main relations are

$$
T_{1/2} = \frac{\ln 2}{\lambda}
$$

and

$$
\tau = \frac{1}{\lambda}
$$

where $T_{1/2}$ is the half-life and $\tau$ is the mean lifetime.

These formulas show again that a large decay constant means a short-lived isotope.

Important relations:
$$
T_{1/2} = \frac{\ln 2}{\lambda}, \qquad \tau = \frac{1}{\lambda}
$$
So, increasing $\lambda$ decreases both half-life and mean lifetime.

Example of Interpretation

If a nucleus has decay constant

$$
\lambda = 0.002\ \text{s}^{-1}
$$

then in a short time interval of $1\ \text{s}$, the fraction of undecayed nuclei expected to decay is approximately

$$
\lambda \Delta t = 0.002 \times 1 = 0.002
$$

which is $0.2\%$.

This does not mean exactly $0.2\%$ must decay every second forever in a strict mechanical way. It means each remaining nucleus has that decay tendency per second, and the actual number is governed statistically.

Summary

The decay constant $\lambda$ is the central parameter that describes the speed of radioactive decay. It gives the probability per unit time that a nucleus will decay, has units of inverse time, and appears in the basic decay law

$$
\frac{dN}{dt} = -\lambda N
$$

A larger decay constant means a less stable nucleus and a faster decrease in the number of undecayed nuclei.

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