Radioactive Decay

What are the types of radioactive decay?

There are several types of decay depending on the emitted particles. Below we describe two of them: alpha decay and beta decay.

Alpha Decay

Here is the general form of an alpha decay equation:

\[^A_Z X \rightarrow ^{A - 4}_{Z - 2} Y + ^4_2 \alpha\]

Beta Decay

Beta minus decay

If the emitted particle is an electron, the proton number increases by one, and the disintegration process is called beta minus decay (β⁻). The simplified equation is:

\[^A_Z X \rightarrow ^{\space \space\space\space A}_{Z+1}Y + e^-\]

Beta plus decay

If the emitted particle is a positron, the proton number decreases by one, and the disintegration process is called beta plus decay (\(\beta^+\)). The simplified equation is:

\[^A_Z X \rightarrow ^{\space \space\space\space A}_{Z-1}Y + e^+\]

In these equations, X is a certain unstable element, Y is another element that may be stable, e⁺ is a positron, and e^- is an electron. The upper index denotes the nucleon number (protons + neutrons), while the lower index denotes the proton number.

Radioactive decay: from decay to stability

Radioactive decay happens until the element reaches a point where

• The excess of energy and particles has been released due to decay processes.
• The atoms have achieved a stable number of subatomic particles.

For most isotopes of elements, especially elements with a low number of protons, we find that stability is mainly achieved by alpha and beta decay.

A billet of highly enriched uranium

How can we calculate radioactive decay?

Radioactive decay is a random process. Since it is a quantum effect, we can only predict how likely an atom will decay in a certain period. The usual masses we deal with in laboratories have around 10²³ atoms, which means that our predictions will be, on average, almost perfectly fulfilled.

We can calculate the decay rate as the ratio of the decayed atoms in a sample of radioactive material divided by the time it took them to disintegrate.

Radioactive decay formula

The equation obeyed by different decaying elements usually takes a similar form. This equation relates the number of unstable nuclei in a sample at a certain time (t = 0) to the number of unstable nuclei at later times. This is the equation:

\[N(t) = N_0 \cdot e^{-\lambda \cdot t}\]

Here, \(\lambda\) is the decay constant, which is related to the probability of decay per unit time and is characteristic for each element and isotope. The letter t denotes time, and N₀ is the number of unstable atoms in our sample at t = 0.

This equation is important, and its properties are well known. Mass decays exponentially at the same rate due to the disintegration processes. The most relevant feature comes from comparing the sample content at two different times.

Suppose we look at a sample at a certain time t₁ > 0 and then at a later time t₂ > t₁. If we want to find the ratio of the number of unstable atoms in the sample, we only need to divide their expressions:

\[\frac{N(t_2)}{N(t_1)} = \frac{N_0 \cdot e^{-\lambda \cdot t_2}}{N_0 \cdot e^{-\lambda \cdot t_1}} = e^{-\lambda(t_2 - t_1)}\]

This relation gives two crucial facts:

1. The ratio between the numbers of unstable nuclei at two different times is independent of the initial number of unstable nuclei. Since the decay constant is given for a particular element, we know that for a specific time interval t₁ - t_2, the number of unstable nuclei will decrease in the same percentage (ratio).
2. Given that the percentage decrease of unstable nuclei is the same for a fixed interval, the decrease is much faster at earlier times because the total number of unstable nuclei is bigger.

The half-life of radioactive decay

The half life is the time it takes for a particular unstable element to have its number of unstable atoms halved. It depends only on the decay constant. Using the general decay equation, we can derive its expression:

\[T_{1/2} = \frac{\ln(2)}{\lambda}\]

Example of radioactive decay and carbon dating

Carbon plays a vital role in the functioning of organic beings. Although carbon-12 and carbon-13 are stable isotopes, the most abundant is carbon-12, which we usually find in every organic structure. On Earth, we also find an unstable isotope (carbon-14), which is continuously formed in the atmosphere due to the radiation from outer space.

It turns out that organic beings absorb this isotope, and both the production and absorption processes are very well studied. Here are two facts regarding this isotope:

• The ratio of carbon-12 and carbon-14 nuclei in alive organic structures is a well-known quantity.
• The absorption of carbon-14 stops when an organic structure is dead.

These facts give us the number of carbon-14 nuclei when an organic structure died, and by knowing the current amount, we can estimate the time passed since the organic structure’s death. As a result, we can accurately estimate the deaths of humans and animals or give very good estimations for making things with wood and paper. This technique works well in periods under 50,000 years.