Exponential decay: radioactivity

Suppose we have a population of radioactive nuclei which we study at regular intervals. Each interval lasts a time δt, and at the end of any interval a further fraction f of the nuclei has decayed. We might say that f is the probability of decay in time δt.

Mathematically we might formulate this as follows. Let the number of undecayed nuclei after n intervals be an; then the number decaying in the next interval is, and the number remaining after the next interval is. This sets up a simple mathematical model of the decay of the nuclei that has three input parameters:

The initial number of nuclei a0;

The time interval δt;

The fraction f of decays per time interval.

Create a spreadsheet in which these input parameters appear near the top. (Hint: you might want to name the cells containing the input parameters, to make it easier to refer to them). Initially you should take the following values of the parameters:

a0 =106; f=0.05; δt=0.1.

Now prepare a spreadsheet to generate the sequence of nuclear populations at different times. The recommended way to do this is to prepare several spreadsheet columns. As a minimum, you will need:

  1. A column containing the elapsed time at the beginning of the timestep;
  2. A column containing the number of nuclei remaining undecayed at the start of the timestep;
  3. A third column containing the number of nuclei expected to decay during the forthcoming timestep.

Choose 50 to 100 time intervals. Make sure you label your columns so you can remember what the data in them are supposed to signify. (The beginning of the spreadsheet may look similar to that below).

Parameters / Values
Undecayed nuclei a0 / 1000000
Time interval / 0.1
Fraction decayed f / 0.05
Timestep (n) / Time / Population remaining at start of interval / Population decaying in interval
0 / 0.0000 / 1000000 / 50000
1 / 0.1000 / 950000 / 47500
2 / 0.2000 / 902500 / 45125

What is the value of the half-life,, of the nuclei, (the time taken for the initial population to decay to half its initial value) if you take the input parameters above?

Now you have set up a model of the decay process, you can study the effect of changing the parameters. If you have designed your spreadsheet properly, the whole sequence should automatically update whenever you change one of the parameters. In particular, notice the effect of changing the fraction fdecaying per timestep.

Plot graphs of the results. First select an X-Y "scatter plot" (as it is called in Excel) of the number of undecayed nuclei (on the y-axis) versus the elapsed time (on the x-axis). Notice that the plot should change automatically if you change the parameters.

Next plot a second graph in which the number of undecayed nuclei is represented on a (natural) logarithmic scale. You should find this graph is a straight line; calculate its gradient.Determine the mean lifetime from the data. (See the mathematical background below.)

Save this spreadsheet, includingthe graphs as “username-exponential-decay.xls”.

Mathematical background

The mathematical discussion of the decay process is described by a first-order differential equation

,

with time t treated as a continuous variable. This equation has solution

,

where is the decay rate, τ is the mean life.

The decay has been approximated by a discrete process in the spreadsheet with a discrete time interval δt, so that

which leads to a difference equation with solution

.

The discrete process has used a particular finite difference approximation for the derivative,

,

implying that . However this is not strictly correct.

Consider the half-life of the decay. In the discrete approximation,

In the continuous form

The two expressions for are not equal if we take. In fact from the differential equation the number remaining after a time δt is giving

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