Radioactive decay is a ubiquitous process in nature. It is what causes nuclear substances (like uranium) to be dangerous, but it is also useful when investigating the age of certain entities. Ultimately, radioactive decay is how unstable elements regain stability.
The definition of radioactive decay
Radioactive decay is a random process by which unstable atoms (with an excess of particles and/or energy) emit radiation to achieve stability.
An excess of neutrons and protons can cause this instability, which leads to the emission of alpha particles, beta particles, or high-energy photons (gamma radiation). An atom undergoes decay processes until it reaches a stable form where no more radiation emission occurs.
See our article on Alpha, Beta and Gamma Radiation.
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
Alpha decay is the process by which an alpha particle is emitted from an unstable nucleus. Since an alpha particle consists of two protons and two neutrons, the proton number of the main nucleus is decreased by two, while the nucleon number (the sum of neutrons and protons) is decreased by four.
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 decay is the process by which a beta particle is emitted from the nucleus. A beta particle can either be an electron or a positron.
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.
These are simplified equations because we are just writing some of the particles involved in the process. A thorough analysis shows that there are also neutrinos and antineutrinos in these reactions. Although we will not dive into how these processes work, it’s important to note that there are conservation laws associated with them, like the conservation of electric charge.
The four kinds of radioactive decay are alpha, beta plus, beta minus, and gamma decay. We don’t discuss gamma decay in this explanation.
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.
An example of this decay process is the disintegration of a heavy element like uranium (with a high number of neutrons) into lead. The decay of heavy radioactive elements can take millions of years (as is the case with uranium-238), but it can also take just a few seconds for others.
A billet of highly enriched uranium
Isotopes = two or more types of atoms with the same atomic number (number of protons) and different nucleon numbers (number of protons and neutrons).
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.
Due to the accuracy of these predictions for a high number of atoms, we find that decay rates offer precise measurements of time. For instance, carbon dating techniques analyse fractions of disintegrated atoms in a sample of carbon of organic material and predict the time passed since the death of the being it belonged to.
Similar techniques with inorganic materials are used in astrophysics to study the age of certain astronomical objects like planets or comets.
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:
- 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 - t2, the number of unstable nuclei will decrease in the same percentage (ratio).
- 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.
See the graph below for a quick explanation of the above two facts for a certain value for the decay constant.
Radioactive decay as a function of time, Wikimedia Commons
Let’s start with 10 unstable nuclei for this example. The decay constant has been chosen so that the number of unstable nuclei has halved after one second.
Between 0 seconds and 1 second, we go from 10 unstable nuclei to 5. Between 1 and 2 seconds, we go from 5 unstable nuclei to 2.5 (it is impossible to have 2.5 unstable nuclei, this is just a statistical measure). Between 2 and 3 seconds, we go from 2.5 unstable nuclei to 1.25.
Furthermore, as we predicted, the rate of percentual decrease is constant (after each second, the amount of unstable nuclei decreases by 50%).
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.
Imagine we are given a mummy found in a prehistoric burial site, and we want to know when the body was buried. We are given a carbon-14 analyser. Due to theoretical models, we already know that the number of carbon-14 atoms present in the body before its death was \(6 \cdot 10^{26}\). With our equipment, we measure that the current number of carbon-14 atoms present in the mummy is \(9.77 \cdot 10^{25}\).
Theoretical models also tell us that the decay constant of carbon-14 is \(\lambda = 1.21 \cdot 10^{-4}\) years-1. We can solve the decay equation for t to find that:
\[t = - \frac{1}{\lambda} \cdot \ln(\frac{N}{N_0}) = -\frac{1}{1.21 \cdot 10^{-4} (years)^{-1}} \cdot \ln(\frac{9.77 \cdot 10^{25}}{6 \cdot 10^{26}}) \approx 15,000 \space years\]
As you can see, all we needed for this calculation was the initial number of carbon-14 (which can be estimated by biological models), the value of the decay constant (which is precisely known due to experimentation), and a device to measure the current amount of carbon-14 atoms.
Radioactive Decay - Key takeaways
Radioactive decay is when an unstable atom, which has an excess of particles, expels particles and/or radiation until it reaches a stable configuration.
The particle excess is emitted as radiation. This radiation can consist of particles (alpha and beta) or electromagnetic radiation (gamma).
The radioactive decay can be modelled as an exponential decay from the formalism of quantum mechanics. It is a statistical process and, when studied in samples with a very large number of unstable atoms, it is very accurate.
Due to the exponential law of decay and the contents of carbon of living bodies, the amount of carbon-14 (an unstable isotope of carbon) is used to measure the age of organic systems.
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