Contr. Mineral. and Petrol. 40, 259--274 (1973) 9 by Springer-Verlag 1973 Closure Temperature in Cooling Geochronological and Petrological Systems Martin I{. Dodson Department of Earth Sciences, The University, Leeds Received March 5, 1973 Abstract. 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 (D = Doe--E/RT ) it is given by T c = R/[E In (A ~Do/a2)] (i) 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 = R/(Ed T-1/dt) = -- R T~/(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 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. 1. Definition of Closure Temperature When the "age" of a rock or mineral is calculated from its accumulated products of radioactive decay, whether those products be crystal structural changes caused by fission fragments, or radiogenic isotopes such as sTSr or a~ the result ideally represents a point in time at which a completely mobile daughter product became completely immobile. "Mobility", in this context, means either rapid diffusion from the lattice site at which a radiogenic isotope is formed, or very fast an- nealing of a disturbed crystal lattice. At one time it was believed that the change in mobility of a radiogenic isotope could always be identified with either the crystallisation of an igneous rock from a melt, or recrystallisation during meta- morphism. In recent years, however, it has become increasingly clear that for some methods of age determination, notably the dating of separated minerals by the Rb--Sr and K--Ar methods, such a simple interpretation is inadequate; radiogenic argon and strontium evidently are mobile in some minerals at tem- peratures well below that of crystallisation. The best evidence for this view comes from the Rb--Sr and K--Ar age pattern on micas from the central Alps (J/tger, 1965; J/iger et al., 1967; Armstrong et al., 1966), for which the simplest interpretation is that closure of the Rb--Sr and K--Ar systems occurred during260 M.H. Dodson: post-metamorphic cooling. A comparable interpretation of age patterns in the British Caledonides has been presented by Harper (1967), while Armstrong (1966) developed similar concepts in a review of K--Ar dating of orogenic belts. Workers on fission track-dating have been led to similar conclusions (Fleischer et al., 1968; Wagner and Rcimer, 1972), because track loss occurs by annealing of the crystal structure at rather low temperatures in some minerals. Fig. 1 shows how calculated ages are related to the real situation in a cooling radiogenic system. At high temperatures the daughter product escapes as fast as it is formed, and so cannot accumulate. At low temperatures its rate of escape is negligible, so that it can accumulate undisturbed. There is a continuous transi- tion from one extreme to the other. Calculation of the apparent age corresponds to extrapolating the low-temperature portion of the accumulation curve back to the time axis. The effective closure temperature, Tc, can therefore be defined as the temperature of the system at the time represented by its apparent age. The value of T c will depend on the exact cooling history of a particular system, but it should be independent of the starting temperature if the latter is sufficient- ly high. Thermodynamically we can consider a crystal in which radiogenic isotopes are accumulating to be out of equilibrium with its environment. The equilibrium concentration of the radiogenic daughter product may for many purposes be considered to be zero. Thus, at high temperatures, the loss of isotopes by rapid diffusion can be regarded as the maintenance of an equilibrium state, and there is a close resemblance to other petrological situations. For example, oxygen isotopes in a cooling mineral assemblage commonly record a temperature well below that at which erystallisation occurred, because at higher temperatures isotopic equilibrium was continuously maintained. A formal mathematical similarity between this kind of situation and geochronological closure temperature is established in Section 5. Little work has been published on the mathematics of diffusion in a cooling solid. Gentner et al. (1954) presented a theoretical analysis of argon and helium diffusion in a slowly cooling cubic crystal, and used the results to interpret observed relationships between apparent age and grain size in sylvite. Their primary differential equation was criticised by Amirkhanoff et al. (1961), but their principal solution appears to be correct. Wood (1964) and Goldstein and Short (1967) determined cooling rates of iron meteorites from the nickel concen- tration distributions in neighbouring lamellae of kamaeite and taenite, using finite difference methods. Damon (1970) discussed the relative importance of radiogenic production of argon-40 and its loss by volume diffusion in various geological environments, but did not attempt to analyse the consequences of steady cooling. In the present paper some simple equations which relate closure temperature to cooling rates and diffusion parameters are derived, and their application to palaeo-thermometry is discussed. 2.