Magmatic residence times of zoned phenocrysts: introduction and application of the binary element diffusion modelling (BEDM) technique

Abstract

This paper describes a general technique, binary element diffusion modelling (BEDM), for determining single-crystal residence times in magmas that relies on modelling the diffusion of two or more elements in the crystal. BEDM has the advantage over other diffusion-based models in that it does not need a precisely defined initial compositional profile for the crystal at “zero time”, and instead requires that the concentrations of the two elements are correlated during crystallisation. Any differences subsequently observed between the two elements are caused by intracrystalline diffusion during residence in hot magma. These differences are removed by artificially ageing the slower-diffusing of the two elements, and the amount of time taken to “undo” the difference between the elements is simply related to the crystal residence time (=decoupling time) at high temperatures. The BEDM principle is demonstrated using artificial data and is then applied to literature data for Sr and Ba in a zoned sanidine crystal from the Bishop Tuff (Anderson et al., in J. Petrol 41(3):449–473, 2000). For this crystal, the method gives a residence time estimate of 114 ka at 800°C, which is then compared with estimates from other methods. In theory, the method can be further expanded for use as a geothermometer as well as geochronometer. However, this is not easily possible with the diffusivity data currently available.

Appendix

Theoretical extensions to BEDM: multi-element modelling

Should diffusivity data for a wide range of analysable elements become available, BEDM can be expanded to run with multiple pairs of elements for a better set of constraints. Notably, if the diffusivities show different responses to temperature increase, i.e. have different activation energies of diffusion, then it should be possible to use this method to obtain both the residence time and the magmatic temperature of equilibration simultaneously. In order to get strongly different activation energy for diffusion, a difference in ionic charge is often, but not always, necessary. Exceptions exist for minerals such as zircon, where activation energy is a strong function of ionic radius (Cherniak, et al. 1997a, b). It has been noted that site binding energy is essentially a function of ionic charge (Wood and Blundy 1997), and that site binding energy is a primary determinant of the activation energy of diffusion. This leads to the so-called “compensation law” where the lines of log D vs 1/T are parallel for ions of the same charge in the same site in the same mineral (Jaoul and Sautter 1999). Therefore, two ions of differing charge will have markedly different responses to temperature, and the ratio of the diffusivities will change as T is varied. In a set of three diffusing ions, if one has a valency different from the other two, running the three possible pairs within the binary element model will only produce the same answer if the temperature is correct and, therefore, the relative diffusivity ratios are also correct. In a graph showing ratios of residence time between element pairs versus temperature of equilibration, this will be the point at which the three lines from each set of modelled ions intersect, shown in Fig. 10.

Fig. 10

Simultaneous estimation of temperatures and residence time using BEDM. The magmatic temperature simulated was 1,073K (800°C). The model pins this down precisely at the intersection point of the three lines

As it currently stands, this approach will have limited application. Olivine may be a possible starting point (Costa and Dungan 2005), where Ca, Fe–Mg, Ni and Mn diffusion coefficients are known. However, as these authors show, although different elements do give consistent answers and imply the same zonation process (obeying BEDM assumptions), the uncertainties in the diffusion coefficients mean that the timescales are quite different. Therefore, using BEDM in the manner described here with these olivine diffusion coefficients would give quite a wide range in T–time estimates, within uncertainty. In addition, use of BEDM for temperature determinations requires some variation in diffusional activation energy and the listed elements are all quite similar. Therefore, alkali feldspar (for Rb, Sr and Ba diffusion) may be more productive, as these elements sit in the same site in the lattice despite different valency, and have different activation energies.