KAHIBARO
Table of Contents
Understanding Mean Lifetime
Mean lifetime is a way to describe how long radioactive nuclei survive before they decay. While half life tells us how long it takes for half of a sample to decay, mean lifetime tells us the average waiting time for the decay of a single nucleus.
This idea is very useful because radioactive decay is random for each individual nucleus. We cannot predict exactly when one nucleus will decay, but we can describe the average behavior of many nuclei very well.
Average lifetime of a nucleus
Imagine a large collection of identical unstable nuclei. Some decay very soon, some last longer, and a few survive for a very long time. If we average the lifetimes of all these nuclei, we get the mean lifetime, usually written as $\tau$.
The mean lifetime is directly related to the decay constant $\lambda$. The relation is
The mean lifetime $\tau$ and decay constant $\lambda$ are related by
$$\tau = \frac{1}{\lambda}$$
So, if the decay constant is large, the nucleus decays quickly and the mean lifetime is short. If the decay constant is small, the nucleus decays more slowly and the mean lifetime is long.
Connection with the decay law
The number of undecayed nuclei at time $t$ is
$$N(t) = N_0 e^{-\lambda t}$$
Since $\tau = 1/\lambda$, this can also be written as
$$N(t) = N_0 e^{-t/\tau}$$
This form is often very helpful because it shows clearly that $\tau$ sets the natural time scale for decay.
When $t = \tau$,
$$N(\tau) = N_0 e^{-1} \approx 0.368 N_0$$
So after one mean lifetime, about $36.8\%$ of the original nuclei remain undecayed.
After one mean lifetime,
$$N(\tau) = \frac{N_0}{e} \approx 0.368 N_0$$
This means about $63.2\%$ have already decayed.
Relation between mean lifetime and half life
Mean lifetime and half life describe the same decay process, but in different ways. Their relation comes from the exponential decay law:
$$T_{1/2} = \frac{\ln 2}{\lambda}$$
Using $\tau = 1/\lambda$, we get
$$T_{1/2} = \tau \ln 2$$
or equivalently,
$$\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 T_{1/2}$$
So the mean lifetime is always longer than the half life.
Important relations:
$$\tau = \frac{1}{\lambda}$$
$$T_{1/2} = \tau \ln 2$$
$$\tau = \frac{T_{1/2}}{\ln 2} \approx 1.44 T_{1/2}$$
Physical meaning
The mean lifetime does not mean that every nucleus survives exactly for time $\tau$. Radioactive decay is random. Some nuclei decay much earlier than $\tau$, and some much later. The mean lifetime is only the statistical average over many nuclei.
This is similar to the average result of rolling a die. A single roll can give many different outcomes, but the average over many rolls has a definite value. In the same way, individual nuclear decay times vary, but the average lifetime is well defined.
Simple comparison table
| Quantity | Symbol | Meaning | Formula |
|---|---|---|---|
| Decay constant | $\lambda$ | Probability rate of decay | $\lambda = 1/\tau$ |
| Mean lifetime | $\tau$ | Average lifetime of a nucleus | $\tau = 1/\lambda$ |
| Half life | $T_{1/2}$ | Time for half the sample to decay | $T_{1/2} = \tau \ln 2$ |
Example
Suppose a radioactive isotope has decay constant
$$\lambda = 0.20\ \text{s}^{-1}$$
Then its mean lifetime is
$$\tau = \frac{1}{0.20} = 5.0\ \text{s}$$
Its half life is
$$T_{1/2} = \tau \ln 2 = 5.0 \times 0.693 \approx 3.47\ \text{s}$$
So this isotope survives on average for $5.0\ \text{s}$, even though half of a large sample disappears in only $3.47\ \text{s}$.
Visual picture of decay
The number of undecayed nuclei drops exponentially with time. The half life marks the point where the number has fallen to one half of the original value. The mean lifetime marks the point where the number has fallen to $1/e$ of the original value.
Final idea
Mean lifetime is the average time an unstable nucleus exists before decaying. It is one of the most natural ways to describe exponential decay, and it is simply the reciprocal of the decay constant.
For radioactive decay, the key definition is
$$\tau = \frac{1}{\lambda}$$
and the decay law can be written as
$$N(t) = N_0 e^{-t/\tau}$$
This makes mean lifetime a central quantity in nuclear physics and in all processes that follow exponential decay.