Stability analysis of invariant points using Euler spheres, with an application to FMAS granulites

Alternative assignment of invariant point stabilities in a possible P – T  phase diagram is given by a family of grids that derives from a form of the Euler equation. Invariant points are represented by great circles that divide the surface of a sphere (the Euler sphere) into polygonal regions that correspond to the number of potential solutions or grids in n -component systems with n +3 non-degenerate phases. A particular invariant point is stable in all grids on one side of the great circle and metastable on the other. The advantage of this representation is the ease and efficiency by which all grids consistent with experimental and theoretical constraints can be identified. The method is well suited for systems of n +3 phases in which the thermochemical data necessary for direct calculation of the phase diagram is either uncertain or non-existent for one or more of the phases. The mass balance equations among the n +3 phases of interest define the Euler sphere for any particular system. There is a unique Euler sphere for unary systems, and another for binary systems. Ternary and quaternary systems have four and 11 different types of Euler spheres, respectively. In the ternary case with six phases, the 16 non-degenerate chemographies belong to four groups that are associated with the four Euler spheres. An analysis of those groups shows a close relationship between the topologies of the chemographies and the topologies of the grids represented on the Euler sphere. Euler spheres for degenerate chemographies are characterized by a smaller number of spherical polygons. A useful application of the Euler sphere concept is the systematic derivation of possible FMAS petrogenetic grids from subsystem constraints. Assumption of just one stable invariant point in each of MAS and FAS systems is consistent with seven FMAS grids involving cordierite, garnet, hypersthene, quartz, sapphirine, sillimanite and spinel.