Closure temperature (Tc) of a geochronological system may be defined as its temperature at the time corresponding to its apparent age. For thermally activated diffusion
in which R is the gas constant, E the activation energy, z the time constant with which the diffusion coefficient D diminishes, a is a characteristic diffusion size, and A a numerical constant depending on geometry and decay constant of parent. The time constant z is related to cooling rate by
(ii)
Eq. (i) is exact only if T -1 increases linearly with time, but in practice a good approximation is obtained by relating T to the slope of the cooling curve at T c.
If the decay of parent is very slow, compared with the cooling time constant, A is 55, 27, or 8.7 for volume diffusion from a sphere, cylinder or plane sheet respectively. Where the decay of parent is relatively fast, A takes lower values. Closure temperatures of 280-300 ~ C are calculated for Rb--Sr dates on Alpine biotites from measured diffusion parameters, assuming a grain size of the order 0.5 mm.
The temperature recorded by a "frozen" chemical system, in which a solid phase in contact with a large reservoir has cooled slowly from high temperatures, is formally identical with geochronological closure temperature.