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Introduction

This page introduces the concept of the exponential decay law, that describes the evolution of a sample of unstable nuclei. The law is introduced first with a theoretical description and then with a model that has the same behavior. To model the decay of a nucleus we employ a die: we say that each die decays when the result is a 1. At the bottom of the page there is a simulation of such model. The simulation can throw a set of dice, check the decays and plot the evolution of the sample size.

The exponential decay law

Every unstable nucleus has a probability of decaying that is constant over time. Let us call \(\lambda\) the probability of decaying for a nucleus during a small time interval \(\text{d}t\). This means that the probability of seeing the nucleus decaying during the time interval \(\text{d}t\) is \(\lambda\). If there is a sample of \(N\) nuclei the number of decays that can be seen in the time span \(\text{d}t\) is then \begin{equation} n = \lambda N\, \text{d}t. \end{equation} Because the number of decays is proportional also to the length of the considered interval of time. What does the number of decays mean? It is the variation of the number of nuclei, because when they decay they transmute to another nucleus and therefore they "disappear" from the sample. Let us calculate the variation of the nuclei number \begin{equation} \text{d}N = N(t+\text{d}t) - N(t), \end{equation} since the number of nuclei is continuously decreasing this variation is negative The number of decays is a positive number and therefore we need to make \(\text{d}N\) positive by multiplying it by \(-1\). Therefore equaling the number of decays with the variation of the nuclei number we get \begin{equation} -\text{d}N = n = \lambda N\, \text{d}t. \end{equation} rearranging the equation we get that the derivative of the nuclei number is proportional to the nuclei number itself \begin{equation} \frac{\text{d}N}{\text{d}t} = - \lambda N \end{equation} This is a simple differential equation with solution \begin{equation} N(t) = N_0 \text{e}^{- \lambda t}. \end{equation} Oftentimes the mean lifetime is defined as the inverse of the probability \begin{equation} \tau = \frac{1}{\lambda} \end{equation} obtaining the function \begin{equation} N(t) = N_0 \text{e}^{- \frac{t}{\tau}}. \end{equation} The mean lifetime represent the characteristic time of the reduction of the nuclei number.

Decaying dice

A die has a constant probability of giving off a particular face each throw. We can therefore model the nuclear decay of a nucleus as a die giving off a particular face of choice. Instead of using time as the independent variable, we will use the number of throws \(T\) as a discrete independent variable. The time interval \(\text{d}t\) that we considered for a radioactive nucleus can be modeled as a single throw. In other words \begin{equation} \Delta T = 1\ \text{throw} \end{equation} For this model we can imagine to have a sample of dice, that we throw for a number of times \(\Delta T\). Whenever a die gives off a 1 we say that it is decayed and we remove it from the sample. In this model we have that the probability of having a decay for a number of throws \(\Delta T\) for a single die is \begin{equation} \lambda = \frac{1}{S}\cdot \frac{1}{\Delta T} \end{equation} where \(S\) is the number of sides of the die. In fact \(\frac{1}{S}\) is the probability of getting a 1 in a single throw. The mean lifetime of the dice can be expressed in terms of the number of sides \begin{equation} \tau = \frac{1}{\lambda} = S\cdot \Delta T \end{equation} Therefore the number of decays for a number of throws \(\Delta T\) of \(N\) dice is \begin{equation} n = \lambda N\cdot \Delta T \end{equation} and the decay law becomes \begin{equation} N(T) = N_0 \text{e}^{- \frac{T}{\tau}}. \end{equation} where \(T\) now is the throw number.

Simulation description

In this page we simulate the throw of \(N\) dice. Each die is white on all faces but one. When a dice lands with the red face up it is discarded as it has decayed. Subsequent throws are automatically carried out until the sample is less than two dice, or the number of throws equals the initial number of dice. The plot is automatically updated with the number of dice in each throw. The number of dice can be changed and the dice shape, by selecting the proper settings. It is possible to save the raw data as a CSV file, so it can be further processed. To start the simulation just press the Start button. To change settings, the simulation must be stopped and then restarted.
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