Two State Decay Theory

If the number of radionuclides N is large, the rate of decay from a radioactive state is proportional to N and the following differential equation is obtained.

This equation can be solved to give the well-known exponential decay function

where N0 is the initial number of radionuclides. The time required for the radioactivity to decrease to 1/e of its original value is the lifetime τ of the radionuclide.

An easy to understand and commonly used parameter is the half-life which is the time taken for half the initial number of radionuclides to decay.

Discrete decay model

The differential equation model produces a smooth continuous exponential decay curve but hides the random and discrete nature of the underlying processes. A stochastic model that "rolls the dice" to determine if a radionuclide decays shows that the continuous model is only an approximation. The probability that a radionuclide survives for a time Δt without decaying is

The Two State Nuclear Decay model uses this probability to compute the number of decay events within each time step Δt. At every time step, the simulation checks each nucleus to see if it has decayed. If it has not decayed, the model computes a random number r in the interval [0,1]. If r is less than P(Δt) the particle decays and its color is changed. The simulation counts the number of decay events within Δt and stops when there are no radionuclides remaining.