Decay chain 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 finally 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 radium and other daughter isotopes it contains. Not only are unstable radium isotopes significant radioactivity emitters, but as the next stage in the decay chain they also generate
Rare-earth element produced by supernova nucleosynthesis or the s-process in asymptotic giant branch stars. In nature, spontaneous fission of uranium-238 produces trace amounts of radioactive promethium, but most promethium is synthetically produced in nuclear reactors. Due to their chemical similarity, the concentrations of rare earths in rocks are only slowly changed by geochemical processes, making their proportions useful for geochronology and dating fossils. Rare-earth element cerium is actually the 25th most abundant element in Earth's crust, having 68 parts per million (about as common as copper). Only the highly unstable and radioactive promethium "rare earth" is quite scarce. The rare-earth elements are