In a radioactive decay chain, what is the name of the special balance reached when the rate at which a short-lived 'child' isotope is created equals the rate at which it disappears?

The special balance reached in a radioactive decay chain, when the rate at which a short-lived 'child' isotope is created equals the rate at which it disappears, is known as radioactive equilibrium. This phenomenon occurs within a radioactive decay chain, which is a sequence of nuclear transformations where an unstable atomic nucleus, called a 'parent' isotope, decays into another, often also unstable, 'child' isotope. An isotope refers to atoms of the same chemical element that have the same number of protons but different numbers of neutrons. The 'rate' of creation or disappearance refers to the number of atoms decaying or being formed per unit of time, also known as activity. The time it takes for half of the radioactive atoms in a sample to decay is called its half-life.

There are two primary types of radioactive equilibrium, distinguished by the relative half-lives of the parent and child isotopes:

1. Secular Equilibrium: This occurs when the half-life of the parent isotope is significantly longer than the half-life of the child isotope, typically by a factor of 1,000 or more. In this state, the activity (decay rate) of the child isotope builds up until it becomes equal to the activity of its parent isotope. Since the parent's half-life is so long, its activity appears essentially constant over many half-lives of the child. The child isotope is created at a steady rate by the decaying parent, and it accumulates until its own rate of disappearance (decay) exactly matches this creation rate. Consequently, the activity of the child isotope then appears to decay with the parent's much longer half-life, even though the child's intrinsic half-life is short. For example, in the decay of Uranium-238, which has a half-life of billions of years, several short-lived daughters can reach secular equilibrium with it.

2. Transient Equilibrium: This occurs when the half-life of the parent isotope is longer than the half-life of the child isotope, but not by a very large factor (e.g., typically 10 to 100 times longer). In this situation, the parent's activity is noticeably decreasing, but still slower than the child's intrinsic decay rate. As the child isotope is formed, its activity initially increases, reaches a maximum, and then decreases. Once transient equilibrium is established, the ratio of the child's activity to the parent's activity becomes constant, and both isotopes appear to decay with the half-life of the parent isotope. The number of child atoms adjusts so that its instantaneous decay rate keeps pace with its instantaneous creation rate from the decaying parent. The child's activity will be greater than the parent's activity in this state, but their ratio remains fixed. A common example is the decay of Molybdenum-99 (parent, half-life of 66 hours) to Technetium-99m (child, half-life of 6 hours), where Technetium-99m activity reaches a peak and then decays with the 66-hour half-life of Molybdenum-99.