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The simplest decay scheme is a two-level system. Let N_{1}
and N_{2} 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

dN_{2}(t) / dt = - R N_{2}

with the decay rate R as the proportionality constant. The solution to this first order, linear differential equation is

N_{2}(t) = N_{2}(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, t_{1/2} , the time for
half of the level 2 population to decay.

**Section 1 Exercises:**

- In the physlet above , what are the initial (t = 0) values of N
_{1}and N_{2}? - What is the sum of N
_{1}and N_{2}at any instant in time? Is the number of nuclei constant? - What is the rate equation for level 1? What is the solution to the level 1rate equation for zero initial population
- What is the lifetime of level 2? What is the half-life of level 2?
- Does t
_{1/2}= ln(2) t? - Suppose that instead of starting the clock when we did, we started it when
N
_{1}(t=0) = 100. Would the rate equations change? Write the solutions to the rate equations for level 1 and level 2.