To create a phase diagram for these rocks, the number of independent chemical constituents must first be defined. In this case, 4 elements are considered, which account for more than 95% of the total analyses, and all other elements with a content of less than 1% are neglected.

On the other hand, it is known that iron and magnesium, elements of equivalent volume, replace each other in ferro-magnesium minerals, which are called solid solutions. Thus olivine of composition (Fe, Mg)2SiO4 is a combination (a solid solution) between a pure iron pole (Fe2SiO4, called fayalite) and a pure magnesium pole (Mg2SiO4, called forsterite). In a pure magnesium system, the olivine is forsterite. If iron is added to the system, the olivine becomes a Fe-Mg solid solution, but no new phase is formed as would be expected according to the phase rule.

Iron and magnesium can therefore be considered as a single element. Thus, the compositions of C1 and C2 can be represented in a simplified system of 3 elements: SiO2 - Al2O3 - (Fe, Mg)O. The variations of this system can be represented graphically in a triangle:

The rocks C1 and C2 are plotted as two blue dots (C1 on the right and C2 on the left) according to the respective proportions of the 3 elements. The minerals in these rocks of course contain the same elements and can also be represented on this triangle (red dots). The quartz (Q) of composition SiO2 is positioned at the SiO2 vertex of the triangle; sillimanite, of composition Al2SiO5, which can be rewritten as Al2O3,SiO2, is located in the middle of the SiO2-Al2O3 side; and so on for spinel (Sp), orthopyroxene (Opx) and Sapphirine (Saph).

The phase rule states that the paragenesis ("divariant assemblage") of a rock contains the same number of minerals M as chemical constituents. In this case, C=3, so M= 3. Partial triangles are drawn in the triangle SiO2 - Al2O3 - (Fe, Mg)O with a mineral at each apex: a rock in one of these triangles contains these 3 minerals.

The ties lines between minerals in equilibrium must never intersect. Considering the position of the different minerals, 2 solutions are possible. Each solution represents a divariant domain DP-DT. These domains are separated by an univariant assemblage. This univariant assemblage consists of the 4 minerals (M=C+1) of the 2 connecting lines which replace each other: Opx - Sill - Q - Saph.

Depending on the proportions of the 3 chemical constituents, different rocks have different parageneses: Q-Opx-Sill ; Opx-Sill-Saph ; Opx-Sp-Saph ; Sill-Sp-Saph in the left divariant domain. Q-Opx-Saph ; Q-Sill-Saph ; Opx-Sp-Saph ; Sill-Sp-Saph in the right divariant domain. Note that 2 parageneses are the same in the 2 divariant fields.

The Opx + Sill + Q paragenesis of the C1 rock shows that it was formed under the P-T conditions of the left divariant domain. The Opx + Saph + Q paragenesis of the C2 rock shows that it was formed under the P-T conditions of the right divariant domain.Thus, the simple reading of this figure indicates that these 2 rocks were not formed under the same P-T conditions.

This example can be illustrated by a small mapping exercise.

The sample shown below (under the microscope, in PPL) shows the transition between the 2 divariant domains (red arrow):

The large sapphirine and quartz crystals were initially in contact, in equilibrium in the right divariant domain. They are now separated by a corona of sillimanite and orthopyroxene, suggesting a transition into the left divariant domain.

Unfortunately, this example is oversimplified, as it is rare that only 3 chemical constituents account for the composition of a rock. Classical graphical representations are used to interpret the parageneses of the metabasites (triangular diagram ACF) and of the métapelites (triangular diagrams A'KF et AFM).

See also the construction of a petrogenitic grid and the rest of this exercise.

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