Half Life

What is half-life?

The half-life symbol and half-life equation

Suppose we look at a sample at a specific 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:

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

This relation gives us two important (related) 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 for a specific element is given, we know that for a specific time interval t1 - t2, 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.

An example showing radioactive decay as a function of time where the y-axis gives the number of particles as a percentage of the initial value

If we now focus on a ratio of one-half, we can find the expression for the half-life. The symbol for half-life is usually \(\tau_{1/2}\)_.

\[e^{-\lambda \tau_{1/2}} = \frac{1}{2} \rightarrow \tau_{1/2} = \frac{\ln(2)}{\lambda}\]

This expression confirms that the time it takes for a radioactive sample to lose half of its unstable nuclei depends only on the isotope (decay constant) and not on the number of unstable nuclei. Thus, it is constant.

Below is a table with some values for the half-lives of certain isotopes.

Here you can see that some isotopes have a very short half-life. This means they decay very fast and almost don’t exist in nature. However, like uranium-236, others have a very long half-life, making them dangerous (like the radioactive waste from nuclear power plants).

What are some applications of half-life?

Half-life is a valuable indicator of the age of a sample or the needed containment time of a particular material. Let’s look at this in more detail.

Carbon-14 dating techniques

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

If you refer to our explanation on Radioactive Decay, you can find more information and examples about carbon-14 dating. Just know that we can accurately estimate the deaths of humans and animals using carbon-14 dating.

Storage of dangerous materials

The decay equation helps calculate how long radioactive materials need to be stored so that they no longer emit large amounts of radiation. There are three kinds of waste:

• Low-level waste from hospitals and industry. These emit low levels of ionising radiation, which are still enough to pose some environmental threat. This waste may require some combination of shielding, incineration, or compacting for shallow burial. The half life of materials of this kind can reach approximately five years.
• Intermediate-level waste, such as sludge, fuels, and chemical waste. These materials require shielding; solidification in concrete, bitumen, or silica; and burial in relatively shallow nuclear storage sites (repositories). The half life of materials of this kind range from five to 30 years.
• High-level waste, like heavy atomic elements (uranium, for example) and materials involved in nuclear fission. These products must be cooled first and then subjected to deep geological burial in concrete and metal containers for a very long time. The half lives of these kinds of materials are typically over 30 years.

Nuclear dry cask storage

Tracers

Gamma emitters are used as tracers because their radiation is not very dangerous and can be accurately detected by specific devices. Some tracers are used to trace a substance’s distribution in a medium, like fertilisers in the soil. Others are used for exploring the human body, which means that they don’t have a very long half-life (they don’t emit radiation for a long time inside the body and damage it).

Decay calculations can also determine whether a radioisotopic tracer is fit for use. Tracers can neither be highly radioactive nor not radioactive enough because, in the latter case, radiation would not reach the measuring devices, and we would not be able to detect or “trace” them. In addition, the half-life allows us to classify them by the rate of decay.