text: In nuclear science, the decay chain refers to a series of radioactive decays of different radioactive decay products as a sequential series of transformations. It is also known as a "radioactive cascade". The typical radioisotope does not decay directly to a stable state, but rather it decays to another radioisotope. Thus there is usually a series of decays until the atom has become a stable isotope, meaning that the nucleus of the atom has reached a stable state.
  Decay stages are referred to by their relationship to previous or subsequent stages. A parent isotope is one that undergoes decay to form a daughter isotope. One example of this is uranium (atomic number 92) decaying into thorium (atomic number 90). The daughter isotope may be stable or it may decay to form a daughter isotope of its own. The daughter of a daughter isotope is sometimes called a granddaughter isotope. Note that the parent isotope becomes the daughter isotope, unlike in the case of a biological parent and daughter.
  The time it takes for a single parent atom to decay to an atom of its daughter isotope can vary widely, not only between different parent-daughter pairs, but also randomly between identical pairings of parent and daughter isotopes. The decay of each single atom occurs spontaneously, and the decay of an initial population of identical atoms over time t, follows a decaying exponential distribution, e, where λ is called a decay constant. One of the properties of an isotope is its half-life, the time by which half of an initial number of identical parent radioisotopes can be expected statistically to have decayed to their daughters, which is inversely related to λ. Half-lives have been determined in laboratories for many radioisotopes (or radionuclides). These can range from nearly instantaneous (less than 10 seconds) to more than 10 years.
  The intermediate stages each emit the same amount of radioactivity as the original radioisotope (i.e., there is a one-to-one relationship between the numbers of decays in successive stages) but each stage releases a different quantity of energy. If and when equilibrium is achieved, each successive daughter isotope is present in direct proportion to its half-life; but since its activity is inversely proportional to its half-life, each nuclide in the decay chain contributes as many individual transformations as the head of the chain, though not the same energy. For example, uranium-238 is weakly radioactive, but pitchblende, a uranium ore, is 13 times more radioactive than the pure uranium metal because of the presence of shorter-lived decay products, such as radium and the noble gas radon. Rock containing thorium and/or uranium (such as some types of granite) emits radon gas, which tends to accumulate in enclosed places such as basements or underground mines due to its high density.
  The quantity of isotopes in the decay chains at a certain time are calculated with the Bateman equation.