ABSTRACT

Analytical solutions are proposed predicting the change in concentration and in isotopic composition of trace elements in magmatic liquids evolving by concurrent processes of fractional crystallization and wallrock assimilation (AFC), in the case the concentration of the trace element in the assimilate is assumed to change according to Shaw’s (1970) model of fractional melting of the wallrock. This particular AFC model, was first suggested and numerically approached by Spera and Bohrson (2001) in their energy-constrained AFC model (EC-AFC), without however giving analytical solutions.The solutions here proposed are ‘natural extensions’ of the simpler solutions given by DePaolo (1981) and by Taylor and Sheppard (1986), turning to them if the distribution coefficient of the trace element between bulk residual wallrock and instantaneous melt is 1. However, as also shown by Bohrson and Spera’s calculation examples, these solutions generally predict concentration and isotopic evolution paths which are very different from those predicted by DePaolo’s and Taylor and Sheppard’s model. This is because, in addition to the trace element distribution coefficient between the residual solid and the instantaneous melt produced by melting of the wallrock, the analytical solutions contain a new parameter, which is the mass ratio between the initial magmatic liquid and the wallrock involved in the melting process.