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The simplest decay scheme is a two-level system. Let N1 and N2 denote the number of nuclei in levels 1 and 2, respectively. When a large number of nuclei are in the upper level, a certain number of them will decay in a given time interval. The number that remain in their excited state, level 2, decreases at a rate that is proportional to the number present at a particular moment. The rate equation for level 2 is
dN2(t) / dt = - R N2
with the decay rate R as the proportionality constant. The solution to this first order, linear differential equation is
N2(t) = N2(t=0) * exp(-R t) .
In the time t = 1/R, the population of level 2 will reach 1/e of its initial value. This characteristic time is called the lifetime, or mean life, of the level. Another characteristic time that is often used to describe the system is the half-life, t1/2 , the time for half of the level 2 population to decay.
Section 1 Exercises: