Topological relations in multisystems of more than n+ 3 phases 1

Multisystems of n+k (k > 3) phases are very complicated and knowledge of them has suffered as a result. The successful solution of the topological relationships in n+ 3 phase multisystems by Zen (1966, 1967) and Zen & Roseboom (1972) has aroused much interest regarding what will happen in a multisystem of more than n+ 3 phases. Since 1979, some important research results on this topic have been published. These results have expounded the substantial rules governing the appearance of phase relations in phase diagrams of n - k (k > 3) phase multisystems. The most significant conclusions include: (1) It is impossible to incorporate all the possible phase relations in an n+k (k > 3) phase multisystem in a single closed net. Therefore, it is no longer enough to use only a single closed net to depict the topological relations involved in these types of multisystems. Instead, one or more groups of closed nets, namely the complete system(s) of closed nets are necessary for this purpose. (2) A principle called the Combination Principle has been proposed and proved. It states: Any closed net of one n+k (k > 3) phase multisystem must be a combination of two or more distinct n+ 3 order submultisystem closed nets belonging to the given n+k phase multisystem, if it is not one of the n+ 3 order submultisystem closed nets itself. The combination principle provides both a theoretical basis and a practical method for the construction of closed nets and, hence, for the derivation of the real phase diagrams for any n+k (k > 3) phase multisystem. (3) A theorem on divariant-assemblage-characteristic-stability-polygons is also important to our understanding of the n+k (k± 3) phase multisystem closed nets. This theorem can be stated as follows: A divariant assemblage of an n+k (k± 3) phase multisystem will be stable in an l-polygon lacking diagonals in an appropriate set of closed-net-diagrams, and this l-polygon may be at least a triangle, and at most a k-polygon. In addition, the closed-net-diagrams of unary and binary n+ 4 phase multisystems derived respectively by Guo (1980b, 1980c, 1981a) and by Roseboom & Zen (1982) have also been summarized. The combination principle is applied to a practical petrological problem in this paper, dealing with 7 phases in the system FeO-Fe₂O₃-SiO₂.