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Simple, complete substitution exists when two or more ions of similar radii and the same charge may substitute for one another in a coordination site in any proportions. In such cases, it is convenient to define end members or components that have only one type of ion in the structural site in question. The olivine group illustrates complete substitution. In the olivine group, (Mg,Fe)₂SiO₄, Mg⁺² (radius = 0.66 Å) and Fe⁺² (radius = 0.74 Å) can substitute for one another in the octahedral site in any proportion. The two end members are the pure magnesium silicate component called forsterite [(Mg)₂SiO₄] and the pure iron silicate component called fayalite [(Fe)₂SiO₄]. Since these two end members can substitute for one another in any proportion in olivine, a complete solid solution series exists between them. As a result, the composition of any olivine can be expressed in terms of the proportions of forsterite (Fo) and/or fayalite (Fa). Simple two‐component, complete solid solution series are easily represented by a number line called a tie line between the two end members (Figure 3.2).

Compositions of any olivine can be represented in a number of different ways. For example, pure magnesium olivine can be represented by (1) a formula (Mg₂SiO₄), (2) a name (forsterite), (3) its position on the tie line (far right) or (4) the proportion of either end member (Fo₁₀₀ or Fa₀). Similarly, pure iron olivine can be represented by a formula (Fe₂SiO₄), a name (fayalite), its position on the tie line (far left) or the proportion of either end member (Fo₀ or Fa₁₀₀). Any composition in the olivine complete solid solution series can be similarly represented. For example, the composition of an olivine with equal amounts of the two end member components can be represented by the formula [(Mg_0.5,Fe_0.5)₂SiO₄], its position on the tie line (halfway between the ends), or the proportions of either end member (Fo₅₀ or Fa₅₀). Typically the forsterite component is used (e.g., Fo₅₀) and the fayalite (Fa) component (100 – Fo) is implied.

Figure 3.2 Olivine complete substitution solid solution series.

In cases where three ions substitute freely for one another in the same coordination site, it is convenient to define three end member components. Each of these end member components contains only one of the three ions in the structural site in which substitution occurs. For example, ferrous iron (Fe⁺²), magnesium (Mg⁺²), and manganese (Mn⁺²) can all substitute for one another in any proportions in the cation site of rhombohedral carbonates. The general formula for such carbonate minerals can be written as (Fe,Mg,Mn)CO₃.The three end member components are the “pure” minerals siderite (FeCO₃), magnesite (MgCO₃), and rhodochrosite (MnCO₃). On a three‐component diagram, the three pure end member components are plotted at the three apices of a triangle (Figure 3.3).

Figure 3.3 Compositions of carbonate minerals expressed in terms of the proportions of iron, magnesium, and manganese; that is of the three components: siderite (Sd), magnesite (Ms), and rhodochrosite (Rc) plotted on a ternary diagram.

Points on the apices of the triangle represent “pure” carbonate minerals with only one end member component. Percentages of any component decrease systematically from 100% at the apex toward the opposite side of the triangle where its percentage is zero. Each side of the triangle is a tie line connecting two end members. Points on the sides represent carbonate solid solutions between two end member components. Point A on Figure 3.3 lies on the side opposite the magnesite apex and so contains no magnesium. Because it lies halfway between rhodochrosite and siderite, its composition may be written as Rc₅₀Sd₅₀ or as (Mn_0.5,Fe_0.5)CO₃. Any point that lies within the triangle represents a solid solution that contains all three end member components. The precise composition of any three‐component solid solution can be determined by the distance from the point to the three apices of the triangle. Point B in Figure 3.3 lies closest to the rhodochrosite apex and farthest from the magnesite apex and so clearly contains more Mn than Fe and more Fe than Mg. Its precise composition can be expressed as Sd₁₀Ms₂Rc₈₈ or as (Fe_0.10,Mg_0.02,Mn_0.88)CO₃. Many other examples of three‐component systems with complete solid solution exist; all may be represented in a similar fashion by their position on a triangular diagram.