Decay chain

In nuclear science a decay chain refers to the predictable series of radioactive disintegrations undergone by the nuclei of certain unstable chemical elements.
Radioactive isotopes do not usually decay directly to stable isotopes, but rather into another radioisotope. The isotope produced by this radioactive emission then decays into another, often radioactive isotope. This chain of decays always terminates in a stable isotope, whose nucleus no longer has the surplus of energy necessary to produce another emission of radiation. Such stable isotopes are then said to have reached their ground states.
The stages or steps in a decay chain are referred to by their relationship to previous or subsequent stages. Hence, a parent isotope is one that undergoes decay to form a daughter isotope. For example element 92, uranium, has an isotope with 144 neutrons and it decays into an isotope of element 90, thorium, with 142 neutrons. The daughter isotope may be stable or it may itself decay to form another daughter isotope. ²³²Th does this when it decays into radium-228. The daughter of a daughter isotope, such as ²²⁸Ra, is sometimes called a granddaughter isotope. ²²⁸Ra in turn undergoes a further eight decays and transmutations until a stable isotope, ²⁰⁸Pb, is produced, terminating the decay chain of ²³⁶U.
The time required for an atom of a parent isotope to decay into its daughter is fundamentally unpredictable and varies widely. For individual nuclei the process is not known to have determinable causes and the time at which it occurs is therefore completely random. The only prediction that can be made is statistical and expresses an average rate of decay. This rate can be represented by adjusting the curve of a decaying exponential distribution with a decay constant particular to the isotope. On this understanding the radioactive decay of an initial population of unstable atoms over time t follows the curve given by e^−λt.
One of the most important properties of any radioactive material follows from this analysis, its half-life. This refers to the time required for half of a given number of radioactive atoms to decay and is inversely related to the isotope's decay constant, λ. Half-lives have been determined in laboratories for many radionuclides, and can range from nearly instantaneous—hydrogen-5 decays in less time than it takes for a photon to go from one end of its nucleus to the other—to fourteen orders of magnitude longer than the age of the universe: tellurium-128 has a half-life of.
The Bateman equation predicts the relative quantities of all the isotopes that compose a given decay chain once that decay chain has proceeded long enough for some of its daughter products to have reached the stable end of the chain. A decay chain that has reached this state, which may require billions of years, is said to be in equilibrium. A sample of radioactive material in equilibrium produces a steady and steadily decreasing quantity of radioactivity as the isotopes that compose it traverse the decay chain. On the other hand, if a sample of radioactive material has been isotopically enriched, meaning that a radioisotope is present in larger quantities than would exist if a decay chain were the only cause of its presence, that sample is said to be out of equilibrium. An unintuitive consequence of this disequilibrium is that a sample of enriched material may occasionally increase in radioactivity as daughter products that are more highly radioactive than their parents accumulate. Both enriched and depleted uranium provide examples of this phenomenon.

History

The chemical elements came into being in two phases. The first commenced shortly after the Big Bang. From ten seconds to 20 minutes after the beginning of the universe the earliest condensation of light atoms was responsible for the manufacture of the four lightest elements. The vast majority of this primordial production consisted of the three lightest isotopes of hydrogen—protium, deuterium and tritium—and two of the nine known isotopes of helium—helium-3 and helium-4. Trace amounts of lithium-7 and beryllium-7 were likely also produced.
So far as is known, all heavier elements came into being starting around 100 million years later, in a second phase of nucleosynthesis that commenced with the birth of the first stars. The nuclear furnaces that power stellar evolution were necessary to create large quantities of all elements heavier than helium, and the r- and s-processes of neutron capture that occur in stellar cores are thought to have created all such elements up to iron and nickel. The extreme conditions that attend supernovae explosions are capable of creating the elements between oxygen and rubidium. The creation of heavier elements, including those without stable isotopes—all elements with atomic numbers greater than lead's, 82—appears to rely on r-process nucleosynthesis operating amid the immense concentrations of free neutrons released during neutron star mergers.
Most of the isotopes of each chemical element present in the Earth today were formed by such processes no later than the time of our planet's condensation from the solar protoplanetary disc, around 4.5 billion years ago. The exceptions to these so-called primordial elements are those that have resulted from the radioactive disintegration of unstable parent nuclei as they progress down one of several decay chains, each of which terminates with the production of one of the 251 stable isotopes known to exist. Aside from cosmic or stellar nucleosynthesis, and decay chains the only other ways of producing a chemical element rely on atomic weapons, nuclear reactors or the laborious atom-by-atom assembly of nuclei with particle accelerators.
Unstable isotopes decay to their daughter products at a given rate; eventually, often after a series of decays, a stable isotope is reached: there are 251 stable isotopes in the universe. In stable isotopes, light elements typically have a lower ratio of neutrons to protons in their nucleus than heavier elements. Light elements such as helium-4 have close to a 1:1 neutron:proton ratio. The heaviest elements such as uranium have close to 1.5 neutrons per proton. No nuclide heavier than lead-208 is stable; these heavier elements have to shed mass to achieve stability, mostly by alpha decay. The other common way for isotopes with a high neutron to proton ratio to decay is beta decay, in which the nuclide changes elemental identity while keeping the same mass number and lowering its n/p ratio. For some isotopes with a relatively low n/p ratio, there is an inverse beta decay, by which a proton is transformed into a neutron, thus moving towards a stable isotope; however, since fission almost always produces products which are neutron heavy, positron emission or electron capture are rare compared to electron emission. There are many relatively short beta decay chains, at least two for every discrete weight up to around 207 and some beyond, but for the higher mass elements there are only four pathways which encompass all decay chains. This is because there are just two main decay methods: alpha radiation, which reduces the mass number by 4, and beta, which leaves it unchanged. The four paths are termed 4n, 4n + 1, 4n + 2, and 4n + 3; the remainder from dividing the atomic mass by four gives the chain the isotope will follow in its decay. There are other decay modes, but they invariably occur at a lower probability than alpha or beta decay. For example, the third atom of nihonium-278 synthesised underwent six alpha decays down to mendelevium-254, followed by an electron capture to fermium-254, and then a seventh alpha to californium-250, upon which it would have followed the 4n + 2 chain as given in this article. However, the heaviest superheavy nuclides synthesised do not reach the four decay chains, because they reach a spontaneously fissioning nuclide after a few alpha decays that terminates the chain: this is what happened to the first two atoms of nihonium-278 synthesised, as well as to all heavier nuclides produced.
Three of those chains have a long-lived isotope near the top; this long-lived nuclide is a bottleneck in the process through which the chain flows very slowly, and keeps the chain below them "alive" with flow. The three long-lived nuclides are uranium-238, uranium-235 and thorium-232. The fourth chain has no such long-lasting bottleneck nuclide near the top, so that chain has long since decayed down to the last before the end: bismuth-209. This nuclide was long thought to be stable, but in 2003 it was found to be unstable, with a very long half-life of 20.1 billion billion years; it is the last step in the chain before stable thallium-205. Because this bottleneck is so long-lived, very small quantities of the final decay product have been produced, and for most practical purposes bismuth-209 is the final decay product.
In the past, during the first few million years of the history of the Solar System, there were more unstable high-mass nuclides in existence, and the four chains were longer, as they included nuclides that have since decayed away. Notably, ²⁴⁴Pu, ²³⁷Np, and ²⁴⁷Cm have half-lives over a million years and would have then been bottlenecks higher in the 4n, 4n+1, and 4n+3 chains respectively - ²⁴⁴Pu and ²⁴⁷Cm have been identified as having been present. Today some of these formerly extinct isotopes are again in existence as they have been manufactured. Thus they again take their places in the chain: plutonium-239, used in nuclear weapons, is the major example, decaying to uranium-235 via alpha emission with a half-life 24,500 years. There has also been large-scale production of neptunium-237, resurrecting the extinct fourth chain. The tables below hence start the four decay chains at isotopes of californium with mass numbers from 249 to 252.

Name of series	Thorium	Neptunium	Uranium	Actinium
Mass numbers	4n	4n+1	4n+2	4n+3
Long-lived nuclide	²³²Th	²⁰⁹Bi	²³⁸U	²³⁵U
Half-life	14.1		4.463	0.704
End of chain	²⁰⁸Pb	²⁰⁵Tl	²⁰⁶Pb	²⁰⁷Pb

These four chains are summarised in the chart in the following section.