Empirical constraints on closure temperatures from a single diffusion coefficient

The elucidation of thermal histories by geochronological and isotopic means is based fundamentally on solid-state diffusion and the concept of closure temperatures. Because diffusion is thermally activated, an analytical solu tion of the closure temperature (T-c*) can only be obtained if the diffusio n coefficient D of the diffusion process is measured at two or more differe nt temperatures. If the diffusion coefficient is known at only one temperat ure, however, the true closure temperature (T-c*) cannot be calculated anal ytically because there exist an infinite number of possible (apparent) clos ure temperatures ((T) over bar(c)) which can be generated by this single da tum. By introducing further empirical constraints to limit the range of pos sible closure temperatures, however, mathematical analysis of a modified fo rm of the closure temperature equation shows that it is possible to make bo th qualitative and quantitative estimates of T-c* given knowledge of only o ne diffusion coefficient D-M measured at one temperature T-M. Qualitative c onstraints of the true closure temperature T-c* are obtained From the shape s of curves on a graph of the apparent T-c((T) over bar(c)) vs. activation energy E, in which each curve is based on a single diffusion coefficient me asurement D-M at temperature T-M. Using a realistic range of E. the concavi ty of the curve shows whether T-M is less than, approximately equal to. or greater than T-c*. Quantitative estimates are obtained by considering two d imensionless parameters [ln (E) over cap R (T) over cap(c) vs. T-c*/T-M] de rived from these curves. When these parameters are plotted for known argon diffusion data and for a given diffusion size and cooling rate, it is found that the resultant curves are almost identical for all of the commonly dat ed K-Ar minerals - biotite, phlogopite, muscovite, hornblende and orthoclas e - in spite of differences in their diffusion parameters. A common curve f or Ar diffusion can be derived by least-squares fitting of all the Ar diffu sion data and provides a way of predicting a "model" closure temperature T- cm from a single diffusion coefficient D-M at temperature T-M. Preliminary diffusion data for a labradorite lead to a T-cm of 507 +/- 17 degrees C and a corresponding activation energy of about 65 kcal/mol, given a grain size of 200 mu m and a cooling rate of 5 degrees C/Ma. Curves for He diffusion in silicates (augite, quartz and sanidine) also overlap to a significant de gree, both among themselves and with the Ar model curve, suggesting that a single model curve may be a good representation of noble gas closure temper atures in silicates. An analogous model curve for a selection of O-18 data can also be constructed, but this curve differs from the Ar model curve. A single model curve for cationic species does not appear to exist, however, suggesting that chemical bonding relationships between the ionic size/charg e and crystal structure may influence the closure temperatures of diffusing cations. An indication of the degree of overlap among the various curves f or Ar, He, O-18 and cations is also obtained by considering the dimensionle ss parameter E/RTc*; for the noble gases and O-18, E/RTc* values for the re spective minerals are very similar, whereas for cations, there is significa nt dispersion. Given these constraints, this may be a potential method of e stimating closure temperatures for certain diffusing species when there are limited diffusion data.