Two-level System

Note to student:  You may right-click with the graph to make a copy.

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, t_1/2 , the time for half of the level 2 population to decay.

Section 1 Exercises:

1. In the physlet above , what are the initial (t = 0) values of N₁ and N₂?
2. What is the sum of N₁ and N₂ at any instant in time?  Is the number of nuclei constant?
3. What is the rate equation for level 1?  What is the solution to the level 1rate equation for zero initial population
4. What is the lifetime of level 2?  What is the half-life of level 2? 
5. Does t_1/2 = ln(2) t?
6. Suppose that instead of starting the clock when we did, we started it when N₁(t=0) = 100.  Would the rate equations change?  Write the solutions to the rate equations for level 1 and level 2.