### Question

According to QM, a radioactive atom will decay randomly. But, how many atoms does it take to enforce a predictable half life? And why?

Asked by:
Paul Donohue

### Answer

This is actually more of a statistics question than a physics one. A group of atoms of a given isotope each have a certain chance of decaying in a given period of time, and as you indicate in your question, the larger the population of atoms, the closer to that proability you can expect to be (in percentage terms, not absolute number of decay events).

The closer you want to be to the expected half life, and the higher the degree of certainty you desire, the more atoms you will need.

Answered by:
Rob Landolfi, Science Teacher, Washington, DC

Because of the nature of statistics, there is no hard limit on the number of atoms that will guarantee a predictable behavior. For practical purposes, however, we can find the probability of various outcomes.

The decay constant (lambda) for an atom is related to the half life by:

lambda = (ln 2)/half-life

The probability that an atom decays in a time t is given by:

Pd(t) = 1 - exp(-lambda*t).

If we have a population of N atoms, we can find the probability that any given percentage of them decay in a given time. Since each decay is statistically independent, we can multiply the individual probabilites. For examples, the probability that all N atoms decay in time t is:

P100(t) = (1 - exp(-lambda*t))^{N}.

Note that this probability is very small for large N and small t, and it continues to decrease as you increase N, but it never goes away completely. In practice, we can neglect it once it's small enough for our purposes.

Answered by:
Edward Faulkner, Electrical Engineering Grad Student