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Life-Time and Half-Life

Because particle decays are governed by quantum processes they are, like any other radioactive decay process, random events governed by overall probability laws.

Given a large number of identical particles, the time in which half of the particles will decay is a definite interval, which is characteristic of that particular particle. This interval is called the particle half-life.

After a second half-life interval again, one half of the remaining particles will decay, leaving 1/2*1/2 = 1/4 of the original sample. After three half-life intervals, 1/8 of the original particle remain, after four such intervals 1/16 remain, and so on. As long as the sample remains large enough that statistical methods apply, this pattern repeats.

If we plot this pattern on a graph we find the following distribution.

The curve that is drawn is the function:

where T is called the particle life-time. N(0) is the number of particles present at time, call it t=0, and N(t) is the number at some later time t>0.

Let us define the measured half-life, t_1/2, as the time in which 1/2 of the particles decay, thus:

which, with a little manipulation can be rewritten as

here "ln" indicates a natural logarithm, that is a logarithm with base e.

Relativistic Effects

One of the effects predicted by the special theory of relativity is a "time dilation" for fast-moving objects. This relativistic effect has been verified by comparing the measured half-life for the same type of particles moving at different velocities.

By convention, the half-life for any particle type is the half-life for that type of particle at rest relative to the observer. The measured half-life of high-energy particles must be corrected to find the actual half-life.

The half-life observed for a moving particle is longer by a factor gamma,

where t_measured is the half-life calculated from the measurement and t_half is the half-life of the same type of particle at rest in the laboratory.

Here . In the expression for (gamma), is the particle velocity and c is the speed of light. 

When gets close to c, this time dilation can be a large effect. It makes it possible to measure the half-lives of very short-lived particles by observing how far they travel before decaying.