Radioactive Decay

Radioactive decay is the process by which an unstable nucleus loses energy by emitting radiation. This process allows a nucleus to reach a more stable state. The radiation emitted by an unstable nucleus may be a particle or energy in the form of electromagnetic radiation.

Certain isotopes are more stable than others. Their stability is determined by the ratio of the number of neutrons to the number of protons in the nucleus. If the ratios of neutrons to protons for all elements are plotted as a graph, a nuclear zone of stability can be identified. The zone of stability in the graph shows that lighter isotopes (atomic mass number under 20) are most stable when the neutron-to-proton ratio is 1:1. For these elements, the repulsive force within the nucleus is not strong enough to overcome the attractive force that holds the nucleus together. The nucleus is unstable if the neutron-to-proton ratio is greater than 1.5. This instability is typical for very heavy elements. The nuclear zone of stability can be used to predict how unstable isotopes will decay. A nuclide is a nucleus with specific proton and neutron numbers. The parent nuclide is the nucleus in radioactive decay that exists before the emission of radiation. The daughter nuclide is the nucleus in radioactive decay that remains after the emission of radiation. A radioisotope, or radioactive isotope, is an isotope with an unstable nucleus that experiences radioactive decay. An isotope is one of two or more atoms of an element that have the same number of protons but different numbers of neutrons. A radioisotope has a different number of neutrons from other, more stable, isotopes of the same element.

There are three main types of radioactive decay:

• Alpha decay ($\alpha$ decay) is radioactive decay that gives off an $\rm\alpha$ particle. An $\rm\alpha$ particle is composed of two protons and two neutrons. It is essentially a helium nucleus.
• Beta decay ($\beta$ decay) is radioactive decay that gives off a $\rm\beta$ particle. A $\rm\beta$ particle is a high-energy electron emitted from the nucleus rather than the electron cloud. In the nucleus, neutrons can decay into protons by emitting a $\rm\beta$ particle.
• Gamma decay ($\gamma$ decay) is radioactive decay that gives off a gamma ($\rm\gamma$) ray. A $\rm\gamma$ ray is electromagnetic radiation with the shortest wavelength ($\mathbf\lambda$), meaning it has the highest energy of all light.

There are other, less common types of radioactive decay. These include the emission of a positron (a nuclear particle with the mass of an electron but a charge of +1) and electron capture, in which the nucleus of an atom captures one of the atomic electrons and then emits a $\rm\gamma$ ray as well as an almost massless subatomic particle called a neutrino.

In $\rm\alpha$ and $\rm\beta$ decay, the nature of the daughter nuclide is fundamentally changed compared to the parent nuclide. A chemical equation can be written to show the transmutation (conversion of one type of nucleus into another) that occurs with radioactive decay. By convention, the notation for a radioactive element is to write the mass number in superscript to the left of the element's atomic symbol. For example, a carbon-14 isotope can be shown as ${}^{14}\rm{C}$. Alternatively the atomic symbol followed by a dash and the mass number can be used (e.g., C-14). Note that carbon-14 is more unstable than carbon-12 and is therefore more likely to undergo radioactive decay. A carbon-12 isotope is more stable and is much less likely to undergo radioactive decay.

For $\rm\alpha$ decay, the general equation is ${}_ Z^ A{\rm {X}}\rightarrow{}_{Z-2}^{A-4}{\rm {Y}}+{}_2^4{\rm{He}}$. The daughter nuclide is the element composed of two fewer protons than the parent nuclide—for example, thorium (Th) has two fewer protons than uranium (U). Thus, when ${}^{238}\rm{U}$ undergoes $\rm\alpha$ decay, the equation shows that the mass number is decreased by 4 and the atomic number is decreased by 2. For $\rm\beta$ decay, the general equation is ${}_Z^A{\rm {X}}\rightarrow{}_{Z+1}^{\;\;\;\;A}{\rm {Y}}+{}_{-1}^{\;\,\,0}\rm\beta$. Alternatively, the $\rm\beta$ particle, which is a high-energy electron, may be written as ${}_{-1}^{\,\,\;0}\rm {e}$, the term for an electron. The daughter nuclide is the element composed of one more proton than the parent nuclide—for example, sulfur (S) has one more proton than phosphorus (P). Thus, when ${}^{32}\rm{P}$ undergoes $\rm\beta$ decay, the equation shows that the mass number is unchanged and the atomic number is increased by 1. A subcategory of $\rm\beta$ decay is positron emission, radioactive decay that emits a positron. During positron emission, one proton in the nucleus transforms into a neutron. A positron is a particle with the same mass as an electron. The positron carries a positive charge with a magnitude equal to the negative charge on an electron. A positron is represented as ${}_{+1}^{\,\,\;0}\rm{e}$. The general formula for positron emission is ${}_Z^A{\rm {X}}{\rightarrow{}_{+1}^{\;\,\,0}{\rm{e}}+{}_{Z-1}^{\;\;\,\,\,A}\rm {Y}}$. Positron emission changes the atomic number, making the product a different element, but the mass number stays the same.

A nuclear reaction that is related to positron emission is electron capture, in which the nucleus captures a high-energy electron. This transforms one proton into a neutron. The general formula for electron capture is ${}_Z^A{\rm {X}}{+{}_{-1}^{\;\,\,0}{\rm{e}}\rightarrow{}_{Z-1}^{\;\;\;\,A}\rm {Y}}$. In electron capture, the atomic number and the identity of the element changes, but the mass number stays the same. Electron capture and positron emission are two ways through which a proton transforms to a neutron in the nucleus. Electron capture is technically not radioactive decay, as there is no particle emission.

For $\rm\gamma$ decay, no general equation exists. This is because the makeup of the nucleus is not fundamentally changed—the emission is a high-energy photon, not a subatomic particle composed of protons and neutrons. However, $\rm\gamma$ emission often accompanies other types of radiation. When a $\rm\gamma$ ray is emitted during radioactive decay, it is added to the products as $\rm\gamma$ or ${}_0^0\gamma$. For example, when ${}^{238}\rm{U}$ decays to ${}^{234}{\rm{Th}}$, two $\rm\gamma$ rays of different energies are emitted. They can be added to the $\rm\alpha$ decay equation. Notice that the sum of the mass numbers of thorium and helium is equal to the mass number of uranium, and the sum of the atomic numbers of thorium and helium is equal to the atomic number of uranium. The $\rm\gamma$ decay does not contribute to mass number or atomic number changes. When writing nuclear decay equations, keep in mind that the total mass numbers (the top numbers) of the products and reactants must be equal, as must the total atomic numbers (the bottom numbers). The charge of atoms is not considered because the electrons do not take part in the nuclear reaction.

Some unstable elements decay into other unstable elements, which in turn decay into other elements. This forms a decay series, a cascading series of radioactively decaying elements and their daughter nuclides. Uranium-238 is the first element in a well-known series, the uranium decay series. This series begins with ${}^{238}\rm{U}$ and goes through decay at least 14 times before finally reaching lead-206 (${}^{206}{\rm{Pb}}$), a stable element. An important consideration when studying radioactive isotopes is half-life (t_1/2), which is the time it takes for half the nuclei of a sample of a radioactive element to decay. Half-lives can range from a tiny fraction of a second to many trillions of years. Half-life does not imply that half of the original sample disappears; rather, it indicates how long it takes for half of the original, unstable sample to decay into a daughter nuclide. For example, the half-life of iodine-131 is about eight days. It undergoes beta decay to produce xenon-131. Thus, a 100.0 g sample of ${}^{131}\rm{I}$ will become 50.0 g ${}^{131}\rm{I}$ and 50.0 g ${}^{131}{\rm{Xe}}$ after eight days. Importantly, if the daughter nuclide is itself unstable, it will have its own half-life, and thus the decay product will be a mixture of the unstable daughter nuclide and the products of its decay. A useful equation exists for calculations involving half-lives. N(t) is the amount of sample at time t, N₀ is the original amount of sample, t is the amount of time elapsed, and t_1/2 is the half-life. For example, this equation can be used to calculate how much ${}^{131}\rm{I}$ will remain from an initial mass of 100.0 g after 32.0 days, given that its half-life is 8.06 days.
Half-lives can be used to determine the age of a sample containing a radioisotope. The process of using a radioisotope to determine the age of a sample of material is called radiometric dating. The most common form of radiometric dating is radiocarbon dating, also called carbon dating, which is the process in which the isotope used to determine the age of a sample is carbon-14 (${}^{14}\rm{C}$). This isotope forms naturally in the atmosphere as cosmic rays combine with atmospheric nitrogen. The newly created ${}^{14}\rm{C}$ combines with atmospheric oxygen to form radioactive carbon dioxide, which is taken up by plants and algae during photosynthesis. Animals then eat these plants and algae, incorporating the ${}^{14}\rm{C}$ into their own tissues. Thus, living organisms maintain a fairly constant amount of ${}^{14}\rm{C}$ in their bodies until they die. Upon death, their uptake of ${}^{14}\rm{C}$ ceases, leaving a fixed amount in their remains. The half-life of ${}^{14}\rm{C}$ is known to be 5,730 years, which means it can be used to date organisms that have been dead for thousands of years. Radiocarbon dating is accurate to about 50,000 years ago. Some methods, such as using large sample sizes or long measurement times, may allow the accuracy to be extended to up to 75,000 years ago.

Importantly, radiocarbon dating can only be used to determine the age of once-living things. Nonliving things do not take in atmospheric carbon and thus cannot be dated in this manner, although rock strata may be dated based on the ages of dead organisms embedded in them. For dating nonliving matter, other radiometric dating methods are used. One such method is uranium-lead (U-Pb) dating. The mineral zircon incorporates uranium and thorium, but not lead, into its crystal structure. Analysis of the amount of lead in these crystals gives information that can be used to date the rocks in which they are found. U-Pb dating is precise to about 2 million years and can be used for rocks as old as 4.5 billion years. The method was initially created in order to accurately determine the age of Earth. Other radiometric dating methods, such as samarium-neodymium, potassium-argon, and rubidium-strontium, are also used for similar purposes.