8.7.5.2. Lifetime

Table of Contents

What Lifetime Means

In particle physics, lifetime tells us how long an unstable particle typically exists before it decays into other particles. Many particles are not permanent objects. They are created in reactions, travel for some time, and then transform.

A lifetime is not usually the exact time for one single particle. Instead, it is a statistical quantity. If you create many identical unstable particles under the same conditions, some decay very quickly, some survive a bit longer, and the average behavior is described by the lifetime.

The lifetime is usually written as $\tau$, the Greek letter tau.

The lifetime $\tau$ of an unstable particle is the characteristic time scale for its decay.
It is not the guaranteed survival time of one particle.

Decay as a Random Process

Radioactive and particle decays are random. Even if two particles are identical, one may decay almost immediately while the other lasts much longer. Physics does not usually predict the exact decay time of one particle. It predicts probabilities.

The key idea is that in a very small time interval $dt$, the probability that a particle decays is proportional to that interval. This leads to exponential decay.

If $N(t)$ is the number of particles still undecayed at time $t$, then

$$
N(t) = N_0 e^{-t/\tau}
$$

where $N_0$ is the initial number of particles.

This equation means the number of surviving particles decreases exponentially with time.

Meaning of the Average Lifetime

The lifetime $\tau$ is the mean decay time. If you could measure the decay time of a very large number of identical particles and average all those times, the result would be $\tau$.

So lifetime is an average over many events, not a clock built into a single particle.

A useful interpretation comes from the survival law. At time $t = \tau$,

$$
N(\tau) = N_0 e^{-1} \approx 0.368 N_0
$$

This means that after one lifetime, about $36.8\%$ of the original particles are still undecayed.

Lifetime and Decay Constant

Lifetime is closely related to the decay constant, usually written as $\lambda$. The decay constant gives the probability per unit time that a particle decays.

Their relationship is

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

or equivalently,

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

Important relation:
$$
\tau = \frac{1}{\lambda}
$$
A large decay constant means a short lifetime.
A small decay constant means a long lifetime.

With this notation, the decay law can also be written as

$$
N(t) = N_0 e^{-\lambda t}
$$

Probability View

The probability that a particle survives until time $t$ is

$$
P_{\text{survive}}(t) = e^{-t/\tau}
$$

The probability that it decays between time $t$ and $t + dt$ is

$$
dP = \frac{1}{\tau} e^{-t/\tau} dt
$$

This is the decay time distribution. It shows that short decay times are more common than long ones, but long survival times are still possible.

Comparison with Half-Life

Lifetime and half-life are related, but they are not the same thing. The half-life is the time required for half of the particles in a sample to decay. It is often written as $t_{1/2}$.

Using the exponential law,

$$
\frac{N(t_{1/2})}{N_0} = \frac{1}{2}
$$

$$
e^{-t_{1/2}/\tau} = \frac{1}{2}
$$

Taking the natural logarithm gives

$$
t_{1/2} = \tau \ln 2
$$

Since $\ln 2 \approx 0.693$,

$$
t_{1/2} \approx 0.693 \tau
$$

$$
\tau \approx 1.44 t_{1/2}
$$

Lifetime and half-life are different quantities:
$$
t_{1/2} = \tau \ln 2
$$
Do not confuse the mean lifetime $\tau$ with the half-life $t_{1/2}$.

Short-Lived and Long-Lived Particles

Different particles have very different lifetimes. Some decay so quickly that they exist only for tiny fractions of a second. Others survive long enough to travel measurable distances in detectors.

The following table shows the general idea.

Particle type	Typical behavior	Lifetime character
Very unstable resonance	Decays almost immediately after being created	Extremely short
Muon	Can travel through matter before decaying	Moderate
Neutron, free	Survives much longer than many subatomic particles	Relatively long

A short lifetime usually means the particle is highly unstable. A long lifetime means it is comparatively more stable.

Lifetime and Distance Traveled

If a particle moves with speed $v$, then a simple estimate of the distance it travels before decaying is

$$
d \approx v \tau
$$

This is only a basic estimate. For very fast particles, relativistic effects become important, and the observed lifetime in the laboratory can be longer than the lifetime measured in the particle's own rest frame. The detailed treatment belongs to relativity, but the main idea is that lifetime affects how far unstable particles can travel before decaying.

Visualizing Exponential Decay

A graph of the number of surviving particles versus time slopes downward quickly at first and then more gradually.

Rendering scientific sketch…

Exponential decay and lifetime

At $t = \tau$, the curve has dropped from $N_0$ to $N_0/e$.

Experimental Measurement of Lifetime

Lifetimes can be measured in different ways depending on how long they are.

For particles with relatively long lifetimes, physicists can measure the time between creation and decay, or the distance traveled before decay.

For extremely short-lived particles, direct timing is often impossible. In such cases, the lifetime is inferred indirectly from other measurable properties. One important connection is with decay width, which is treated separately.

Summary

Lifetime is a statistical measure of how long an unstable particle exists before decaying. It is denoted by $\tau$ and describes the average decay time for a large collection of identical particles. Decay follows an exponential law,

$$
N(t) = N_0 e^{-t/\tau}
$$

and lifetime is related to decay constant by

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

It is also related to half-life by

$$
t_{1/2} = \tau \ln 2
$$

Core formulas for lifetime:
$$
N(t) = N_0 e^{-t/\tau}
$$
$$
\tau = \frac{1}{\lambda}
$$
$$
t_{1/2} = \tau \ln 2
$$

Understanding lifetime is essential because it tells us how unstable particles behave in experiments and how long they can exist before transforming into other particles.

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