Empirical constraints on closure temperatures from a single diffusion coefficient |
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| Authors: | J K W Lee |
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| Institution: | (1) Department of Geological Sciences, Queen's University, Kingston, Ontario K7L 3N6, Canada e-mail: lee@geol.queensu.ca, CA |
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| Abstract: | 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 solution of the closure temperature
(T
c
*) can only be obtained if the diffusion coefficient D of the diffusion process is measured at two or more different temperatures. If the diffusion coefficient is known at only
one temperature, however, the true closure temperature (T
c
*) cannot be calculated analytically because there exist an infinite number of possible (apparent) closure temperatures (Tˉ
c
) which can be generated by this single datum. By introducing further empirical constraints to limit the range of possible
closure temperatures, however, mathematical analysis of a modified form of the closure temperature equation shows that it
is possible to make both qualitative and quantitative estimates of T
c
* given knowledge of only one diffusion coefficient D
M
measured at one temperature T
M
. Qualitative constraints of the true closure temperature T
c
* are obtained from the shapes of curves on a graph of the apparent T
c
(Tˉ
c
) vs. activation energy E, in which each curve is based on a single diffusion coefficient measurement D
M
at temperature T
M
. Using a realistic range of E, the concavity 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 dimensionless parameters ln
êRT^
c
vs. T
c
*/T
M
] derived 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 dated K–Ar minerals –
biotite, phlogopite, muscovite, hornblende and orthoclase – in spite of differences in their diffusion parameters. A common
curve for Ar diffusion can be derived by least-squares fitting of all the Ar diffusion 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 °C and a corresponding activation energy of about 65 kcal/mol, given a grain size of 200 μm and a cooling rate
of 5 °C/Ma. Curves for He diffusion in silicates (augite, quartz and sanidine) also overlap to a significant degree, both
among themselves and with the Ar model curve, suggesting that a single model curve may be a good representation of noble gas
closure temperatures in silicates. An analogous model curve for a selection of 18O 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/charge and crystal
structure may influence the closure temperatures of diffusing cations. An indication of the degree of overlap among the various
curves for Ar, He, 18O and cations is also obtained by considering the dimensionless parameter E/RT
c
*; for the noble gases and 18O, E/RT
c
* values for the respective minerals are very similar, whereas for cations, there is significant dispersion. Given these constraints,
this may be a potential method of estimating closure temperatures for certain diffusing species when there are limited diffusion
data.
Received: 1 July 1999 / Accepted: 24 March 2000 |
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| Keywords: | |
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