Half-life of Radioactive Material

The half-life of a radioactive species, denoted $T_{1/2}$, is defined as the time required for the number of parent nuclei to fall to exactly HALF its initial value. Setting $N(T_{1/2})=N_0/2$ in the decay law $N(t)=N_0e^{-\lambda t}$ gives $\frac{1}{2}=e^{-\lambda T_{1/2}}$, which rearranges (taking the natural logarithm of both sides) to $T_{1/2}=\frac{\ln 2}{\lambda}=\frac{0.693}{\lambda}$ -- a species with a larger decay constant $\lambda$ (decaying 'faster' in a statistical sense) therefore has a SHORTER half-life, and vice versa.

A subtle but important point about half-life is what happens after MORE than one half-life has elapsed. After one half-life, the population has fallen from $N_0$ to $N_0/2$ -- but it does NOT then fall all the way to zero over the next half-life; instead, exactly the same FRACTIONAL reduction applies again, so after a second half-life the population has fallen only to half of $N_0/2$, i.e. to $N_0/4$, not to zero. This same halving rule applies to any equal interval of length $T_{1/2}$, no matter how far along the decay process has already progressed -- af …