its parent. Eight methods are listed in the table below. The U/U method is based on the fact that U is dissolved preferentially over U because when a U atom decays by emitting an alpha ray the daughter atom is displaced from its normal position in the crystal by atomic recoil. This produces a Th atom which quickly becomes a U atom. Once the uranium is deposited, the ratio of U to U goes back down to its secular equilibrium (at which the radioactivities of the two are equal), with the distance from equilibrium decreasing by a factor of 2 every 245,000 years. A material balance gives, for some unknown constant , these expressions for activity rations (assuming that the Th starts at zero): We can solve the first equation for in terms of the unknown age, : Putting this into the second equation gives us an equation to be solved for : Unfortunately there is no closed-form expression for the age, , but it is easily found using equation solving algorithms. Dating limits Uranium–thorium dating has an upper age limit of somewhat over 500,000 years, defined by the half-life of thorium-230, the precision with which one can measure the thorium-230/uranium-234 ratio in a sample, and the accuracy to which one knows the half-lives of thorium-230 and uranium-234. Using this technique to calculate an age, the ratio of uranium-234 to its parent isotope uranium-238 must also be measured. Precision U-Th dating yields the most accurate results if applied to precipitated calcium carbonate, that is in stalagmites, travertines, and lacustrine limestones. Bone and shell are less reliable. Mass spectrometry can achieve a precision of ±1%. Conventional alpha counting's precision is ±5%. Mass spectrometry also uses smaller samples. See also Radiocarbon dating References External links Radiometric dating Thorium Uranium