Ep. 2  Radioactive Decay

Mathematics articles that help in reading this article
・Numerical Computation：Ep. 1, Ep. 2, Ep. 5, Ep. 6, Ep. 7, Ep. 8

　Radioactive atoms are unstable, so they decay in an attempt to become more stable atoms. During this process, radiation is emitted.

　Atoms are the basic particles that make up matter, consisting of a nucleus surrounded by electrons. The nucleus itself is made up of protons and neutrons. Different combinations of protons and neutrons give rise to different types of atoms. This can be illustrated as follows:

　\( ^{4}_2 \rm He \) represents a helium atom, and \( ^{3}_{1} \rm H \) is tritium (also called hydrogen-3), which is one of the substances found in the treated water released from the Fukushima nuclear plant. \( \rm He \) is the chemical symbol for helium, and \( \rm H \) is the symbol for hydrogen. Elements are categorized by the number of protons in their nuclei. Even when atoms have the same number of protons, if they have different numbers of neutrons, they are called isotopes of the same element. Among isotopes, those that possess radioactivity are called radioactive isotopes, or radioactive atoms. For example, naturally occurring hydrogen has three isotopes, each with a different number of neutrons:

・Protium \( ^{1}_{1} \rm H \)
・Deuterium \( ^{2}_{1} \rm H \)
・Tritium \( ^{3}_{1} \rm H \)

Of these, tritium is a radioactive isotope.
　Here, the number in the upper left of the element symbol indicates the total number of protons and neutrons in the nucleus (called the mass number), while the number in the lower left indicates the number of protons (the atomic number).

　The letter \( \rm E \) in the table represents scientific (exponential) notation.

・\( \rm E + \it x \) means \( \times 10^x \)
・\( \rm E - \it x \) means \( \times 10^{-x} \)

The values in the table show the radioactivity when exactly 1 gram of each radioactive isotope is present. If the amount of the isotope is less than 1 gram due to decay over time, the actual radioactivity will also be lower than the value listed. The following equation expresses the relationship between radioactivity and the mass of the radioactive isotope:

　The decay constant is a value specific to each radioactive isotope. When the mass of a radioactive isotope is exactly 1 gram, the decay constant is numerically equal to the radioactivity, so the values in the previous table can also be interpreted as decay constants. Radioactivity represents the number of radioactive atoms that decay per unit time. This can also be understood as the decay rate of the radioactive isotope. Therefore, the previous relationship can be rewritten as the following differential equation:

This is essentially the same as the differential equation in the Malthusian model, except for the negative sign on the right-hand side. Therefore, its solution is given as follows:

A graph of the radioactive isotopes from the earlier table would look like the one shown below.

　The decay of iodine-131 is rapid, and almost all of it disappears within a year. However, this does not mean that any biological damage caused by the radiation emitted during its decay will be repaired after one year. What it means is that the substance iodine-131, which was released into the environment, will be nearly gone after a year due to radioactive decay.
　On the other hand, plutonium-239 decays extremely slowly — even after 100 years, most of it still remains, so it requires long-term management and monitoring.
　The decay constant determines how fast a radioactive isotope decays, but there is also a more intuitive time-based indicator called the half-life. The half-life is the time it takes for the mass of a radioactive isotope to decrease to half its original amount due to decay. For example:

・The half-life of iodine-131 is 8 days.
・The half-life of tritium (hydrogen-3) is 12.4 years.

The half-life can be calculated using the following formula:

　As this equation shows, the half-life \( T \) is determined solely by the decay constant \( \lambda \) and does not depend on the initial mass. In other words, the half-life is a fixed value that depends on the type of radioactive isotope.

　Among the radioactive isotopes released into the atmosphere during the Fukushima Daiichi nuclear accident, cesium-137, which was released in relatively large quantities, has a half-life of approximately 30 years. How much a radioactive substance must decay before people stop worrying about it may vary depending on the individual, but as long as we talk about a "half-life of 30 years," it is likely to be treated as a non-negligible issue for at least that long. In fact, even now—14 years after the accident—Japan’s Ministry of the Environment continues to monitor radioactive substances annually in public water bodies and groundwater throughout the country.

　The chemist Willard Libby, who developed this dating method, was awarded the Nobel Prize in Chemistry in 1960 for his pioneering work.