Closure temperature in cooling geochronological and petrological systems

Abstract

Closure temperature (T _c ) of a geochronological system may be defined as its temperature at the time corresponding to its apparent age. For thermally activated diffusion (D=D _o e ^−E/RT it is given by

$$T_c = R/[E ln (A \tau D_0 /a^2 )]$$

(i) in which R is the gas constant, E the activation energy, τ 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 τ is related to cooling rate by

$$\tau = R/(Ed T^{ - 1} /dt) = - RT^2 /(Ed T/dt).$$

(ii) Eq. (i) is exact only if T ⁻¹ increases linearly with time, but in practice a good approximation is obtained by relating τ 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.