Schreinemakers' projection of potential phase diagrams

Another method of reducing the number of axes is based on projection. By projecting all the features onto one side of the phase diagram, one will retain all the features, but the features of the highest dimensionality will no longer be visible because the dimensionality of a geometrical element will decrease by one unit by projection and they may thus overlap each other and also overlap features of the next-higher dimensionality. As an example, Fig. 10.1(b) shows a P, T diagram obtained by projection of Fig. 8.11 (shown again as Fig. 10.1(a)) in the μB direction. Such a P, T diagram is called Schreinemakers' projection [16]. In a system with c components it is obtained by projecting in the directions of c − 1 μi axes. It will show invariant equilibria with c + 2 phases as points, univariant equilibria with c + 1 phases as lines and in the angles between them there will be surfaces representing divariant equilibria with c phases. Using a short-hand notation developed by Schreinemakers, the coexistence lines for c − 1 phases are here identified also by giving in parentheses the phases from the invariant equilibrium which do not take part. For example, the (α) curve represents the α-absent equilibrium, i.e. β + γ + δ. By comparison with Fig. 10.1(a) it can be seen that the angle between (α) and (β) is covered by the γ + δ surface but also by the α + δ surface which extends to the (γ) line and by the β + γ surface which extends to the (δ) line.