A first numerical problem: Radioactive Decays

In this first lecture we will discuss solutions for ordinary differential equations. Such equations are frequently encountered when discussing physics problems; examples include particle motion, harmonic oscillators, etc... While some of these issues will be discussed in the following lectures, this lecture uses the example of radioactive decay to introduce some elementary techniques for solving differential equations.

This lecture is treated in Giordano & Nakanishi, chapter 1.

The physical problem and its analytical solution

Many nuclei are unstable and decay into lighter nuclei under the emission of various other particles. Typical radiation includes helium-nuclei, α-radiation, electrons, β-radiation, or photons, γ-radiation. Due to their quantum nature, such processes are not deterministic but random. While it is impossible to predict exactly when an individual nucleus will decay, it is possible to make a probabilistic statement about its decay.

This can be described by giving the average time for such a decay to happen. This time, typically called the half-life, can range from very short -- fractions of seconds -- to very long -- billions of years. Due to its probabilistic behaviour it is perfectly reasonable to assume that we are talking about statistically large samples of such atoms, with their number N decreasing over time. Using a time constant τ, the behaviour of the decay is governed by the differential equation:

\[ \frac{\mathrm{d}N}{\mathrm{d}t}=-\frac{N}{\tau}\,\, . \]
It should be noted here that the time constant τ is different from the half-life time, $T_{1/2}$.
This mathematical problem can be classified as an ordinary differential equation of first order with initial value(s). There are in fact many solution strategies for this kind of problem, both analytical and numerical.
The solution of the equation above, of course, is given by:
\[ N(t) = N(0)\, e^{-t/\tau}\,,\]
if N(0) is the number of atoms at the start time $t_0=0$.
Despite the easy analytical solution presented here, it is worthwhile to use this example to introduce some concepts employed in numerical problem solving in physics (and other sciences). In the following some computational methods are discussed, which will be further refined while this lecture series proceeds.

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Frank Krauss and Daniel Maitre
Last modified: Tue Oct 3 14:43:58 BST 2017