Some chemographic relationships in n-component systems

A petrologic problem of fundamental importance is to determine whether 2 or more mineral assemblages can be related to one another by continuous or discontinuous facies changes, or whether their bulk compositions occupy non-overlapping regions of composition space. A general method is developed by which 2 regions of n-dimensional space whose vertices are defined by the phases present are tested for compositional overlap. This is accomplished by generating mass balance equations of the type:

$\text{∑}\text{i = 1}\text{m}\text{a}{}_{\mathrm{i}}\text{A}{}_{\mathrm{i}}\text{=}\text{∑}\text{j = 1}\text{k}\text{b}{}_{\mathrm{j}}\text{B}{}_{\mathrm{j}}$

where A_i is the ith phase in one region and B_j is the th phase in the other. If any such equation satisfies the requirement that the sign of each a_i is the same, and that the sign of each b_j is the opposite for all i, j such that: k + m = n + 1 then the 2 regions overlap in phase space.By eliminating all overlapping assemblages in a given set, the bivariant fields bounded by univariant equilibria in n-dimensional systems are completely specified. All bulk compositions are considered within the space defined by the phases that participate in the bounding reactions. An extension of the method generates in sequence all bivariant fields and associated reactions about any invariant point. A further extension is applied to multi-system analysis.