Fractional crystallization forward problem

Let us consider the behavior of the ith among m trace elements upon partitioning between a homogeneous liquid (labeled liq) and the n phases (labeled j) of the cumulate in a system of finite size. These phases are usually considered as mineral phases but liquid trapped in the cumulate can be handled as an additional phase with partition coefficients equal to unity (Greenland, 1970 Albarede, 1976). 

The solution has been worked out in Section 1.5 and given in its differential form as equation (1.5.3) 

The Rayleigh equation (9.3.3) rests on the assumption of constant D/s. This problem was discussed by Allegre et al. (1977). The mineralogy of the cumulate during fractional crystallization varies nearly step-wise provided phase boundaries remain nearly linear 

A related problem is the evolution of a trace-element ratio with crystallization 

Incompatible-element ratios (e.g., Th/La, Nb/Zr, Ce/Yb in basalts) are therefore expected to be very insensitive to mineral separation from the melt and, for differentiated lavas, can be used as a parameter characteristic of their parent liquid see below). 

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