This is especially important if we wish to graphically display chemical rock and mineral data in a way that is easily visualized. In fact, the best way to visualize such data is to attempt to reduce the number of components to 3, so that we can plot the compositions of rocks and minerals on a triangular composition diagram. Before discussing how this is done, we first review some of the principles of three component compositional diagrams.

General Three Component Compositional Diagrams

For the moment we will assume that we are in a true 3 component system and the minerals that we display on a triangular composition diagram is a set of mineral that are in equilibrium over a narrow range of temperature and pressure.

We first look at how one of these diagrams might appear in the hypothetical system A, B, C.

F= C + 2 - P

and with F=2 & C=3, P = 3, which tells us that for this divariant assemblage, we will have three coexisting phases.

Mineral phases that coexist with each other at this temperature and pressure are connected by lines, called tie lines.

Within each compositional triangle, we can use the lever rule to estimate the proportions of the phases that will occur in the rock. For example, a rock with composition x will have the same mineral assemblage as a rock with composition z, but rock z will have a much higher proportion of the mineral A.

In a rare case, note that it is possible for a rock composition to lie exactly on a tie line, for example the rock with composition w. Such a rock would consist of approximately equal proportions of the minerals ABC and A₂C. Note that if we apply the phase rule to this situation, C=3 and P = 2, we find F = 3. But - in reality, because the number of components is defined as the minimum number required to form all minerals, we can see that rock w really lies in a 2 component system, rather than a 3 component system. So, F is still equal to 2.

Similarly if a rock had exactly the same composition as mineral ABC, then it would only have 1 phase, ABC, present at this temperature and pressure and would be considered a 1 component system.

We next illustrate what might happen to the mineral assemblages if we change the pressure and temperature condition to such that some of the mineral assemblages change. The same compositional triangle is shown in the diagram below, but note the shift in the tie lines. Instead of minerals AB and A₂C coexisting, at the new conditions minerals ABC and A coexist. This forms new compositional triangles.

The chemical reaction breaks the tie line to form a new one, and the reaction can be written as:

AB + A₂C => 2A + ABC

Note that after the reaction has occurred, new mineral assemblages develop in the rock. Rock x is now composed of AB, ABC and A, rock y is now composed of A, A₂C, and ABC, and rock z (which had the same assemblage as x in the previous diagram) now has the same assemblage as rock y, although, again, the proportions of the three minerals will be different in rocks y and z..

A₂C => 2A + C

Note that now rock w, along with rocks y and z are in the three phase triangle A, ABC, C and will contain the minerals A, ABC, and C, although the minerals will occur in different proportions.

Furthermore, rock x still has the same mineral assemblage - AB, ABC, A, as it had in the previous diagram.

2ABC => AB + AC + BC

Note that now, rocks x, y, and z all lie within the same composition triangle and have the same mineral assemblage - AB, AC, and A, although again, each rock has different proportions of these minerals.

Rock w, is now in the 2 component system AB - AC.

In this diagram the mineral X(Y,Z) shows limited solid solution with variable amounts of Z substituting for Y. This is shown by a solid line extending from pure XY into the ternary diagram.

Similarly, mineral X₂(Z,Y) shows limited solid solution of Y substituting for Z.

The minerals XYZ_ss and Z_ss show a range of possible compositions that are represented by a shaded field on the diagram.

For rocks with compositions that plot in the 3 phase triangles, like b and c in the diagram, the mineral assemblage would consist of the three phases at the corner of the triangles. If solid solution minerals occur at one or more of the apices of the compositional triangle, then the composition of the solid solution mineral will be that found at the apex of the triangle. These solid solution compositions are shown as white filled circles.

For example, a rock with composition c at the pressure and temperature of the diagram will be made up of Y, Z_ss, and XYZ_ss. A rock with composition b will be made up of X(Y,Z), X₂(Z,Y), and X.

Common Triangular Plots Used in Metamorphic Rocks

As stated above, common metamorphic rocks contain many more than the 3 components we would like to have for easy graphical description of composition and mineral assemblage. The 13 major elements (expressed as oxides) in most (not all) metamorphic rocks are SiO₂, Al₂O₃, TiO₂, FeO, Fe₂O₃, MnO, CaO, MgO, K₂O, Na₂O, P₂O₅, H₂O, and CO₂. Obviously if a rock consists only of one or two of these constituents, the problem is easy. For example, a pure Quartz sandstone would contain only SiO₂, or a pure calcite limestone would contain only CaO and CO₂. But, the more common rocks like shales, basalts, siliceous dolomites, or granites, are not so simple.

There are various methods available to reduce the number of components to a workable number. Among these are:

1. Ignore some components. This is probably OK if the component occurs in small amounts or is always present in a the rocks. Also, a constituent may be ignored if it occurs in a very mobile phase, that could always be present, such as H₂O and CO₂. Or, as mentioned above, we could ignore a component if its occurrence was always within a particular phase, for example Na₂O is usually only found in albite, or K₂O is usually only found in K-spar.
   
   
2. Combine some components that are known to freely substitute for one another. For example Fe and Mg or Fe and Mn could be combined as (Mg,Fe, Mn)O because they readily substitute for one another in the ferromagnesian silicates.
   
   
3. Limit the range of composition that a diagram would apply to. In other words construct diagrams that only deal with a subset of rocks, limited by composition, and specifically state that the diagrams only apply to this subset.
   
   
4. Use projection. That is assume that a constituent will always be present and project compositions from that constituent in a four or five component system to the 3 component system. We have seen this done to a limited extent in our study of phase diagrams in igneous rocks and we will see more detail on this in the discussions that follow.

One of the first uses of these types of diagrams was by Eskola (1915) who employed a diagram known as the ACF diagram in his study of metamorphic rocks. To plot a rock on the ACF diagram, the chemical analysis of the rock is first recalculated to molecular proportions by dividing the molecular weight of each oxide constituent by the molecular weight of that oxide.

Nominally, the ACF diagram plots the following components:

A = Al₂O₃

C = CaO, and

F = FeO + MgO

However, the A value we want is the value of excess Al₂O₃ left after allotting Na₂O and K₂O to form alkali feldspar. The CaO value we want is the excess CaO after allotting P₂O₅ to form apatite, assuming that any P₂O₅ in the rock will suck up CaO to form apatite. We will assume then that all mineral assemblages plotted may also contain alkali feldspar and quartz (and apatite).

So, to obtain the plotting parameters, we calculate the following, where the bracket symbols [ ] indicate the molecular proportions of the oxides.

a = [Al₂O₃ + Fe₂O₃] - [Na₂O + K₂O]

c = [CaO] - 3.33[P₂O₅]

f = [FeO + MgO + MnO]

Since we are only plotting these 3 components, they have to be normalized so that they add up to 1 (or 100 if we are plotting %).

if t = a + c + f, then the plotting parameters are:

A = 100 * a/t

C = 100 * c/t

F = 100 * f/t

Most shales will plot in the field of Pelitic Rocks. Quartzo-feldspathic rocks like feldspathic sandstones, granites, and rhyolites will plot in the Quartzo-Feldspathic field. Basic igneous rocks, like basalts and gabbros will plot in the field of Basic Rocks, and siliceous limestones and dolomites will plot in the field of Calcareous Rocks.

This diagram and the fields shown will become an important part of later discussions, so it is wise to know approximately where the fields of these different chemical types occur on the ACF diagram.

a = 0

c = 0

f = 1

t =1

so, the plotting parameters become

A = 100 * 0/1 = 0

C = 100 * 0/1 = 0

F = 100 * 1/1 = 100%

and we see that hypersthene would plot at the F corner of the ACF diagram.

As a second example, look at the formula for tremolite - Ca₂(Mg,Fe)₅Si₈O₂₂(OH)₂

We can rewrite this formula as 2CaO 5(Mg,Fe)O 8SiO₂ H₂O

then:

a = 0

c = 2

f = 5

t = 7

so the plotting parameters become:

A = 100 * 0/7 = 0

C = 100 * 2/7 = 28.57%

F = 100 * 5/7 = 71.43%

Other minerals are more complicated. For example, the formula for chlorite is (Mg,Al,Fe)₁₂(Si,Al)₈O₂₀(OH)₁₆. Thus we have several possibilities for writing the formula for chlorite, and depending on which formula we use, chlorite will plot at different locations on the diagrams. You will explore the possible solid solution ranges in a lab on this topic.

In AKF diagrams we assume that both alkali feldspar and plagioclase feldspar can be present, thus the amount of Al₂O₃ that we use is the excess Al₂O₃ left after allotting it to all of the feldspars. To obtain the plotting parameters for ACF diagrams, calculate the following:

a = [Al₂O₃ + Fe₂O₃] - [Na₂O + K₂O + CaO]

k = [K₂O]

f = [FeO + MgO + MnO]

Let t = a + k + f, then the plotting parameters in % are:

A = 100 * a/t

K = 100 * k/t

F = 100 * f/t

a = ½ - ½ = 0

k = ½

f = 0

t = ½

So,

A = 100 * 0/½ = 0%

K = 100 * ½ /½ = 100%

F = 100 * 0/½ = 0%

Another example is muscovite KAl₃Si₃O₁₀(OH)₂ or 1/2K₂O 3/2Al₂O₃ 3SiO₂ H₂O. For muscovite:

a = 3/2 - 1/2 = 1

k = 1/2

f = 0

t = 1½ = 1.5

So,

A = 100 * 1/1.5 = 66.7%

K = 100 * 0.5/1.5 = 33.33%

F = 100 * 0 = 0%

AKFM Projection onto AFM

The ACF and AKF diagrams discussed so far, are fairly simple, but useful. One of the problems associated with ACF and AKF diagrams is that Fe and Mg are assumed to substitute for one another and act as a single component. We know, however, that in natural minerals the composition of Fe - Mg solid solutions is very much dependent on temperature and pressure. Thus, in treating Fe and Mg as a single component, we lose some information. Realizing this, J.B. Thompson developed a projected diagram that takes into account possible variation in the Mg/(Mg+Fe) ratios in ferromagnesium minerals, and has proven very useful in understanding metamorphosed pelitic sediments.

Thompson starts with the 5 component system SiO₂ - Al₂O₃ - K₂O - FeO - MgO and ignores minor components in pelitic rocks like CaO and Na₂O. Because quartz is a ubiquitous phase in metamorphosed pelitic rocks, the five component system is projected into the four component system Al₂O₃ - K₂O - FeO - MgO as shown below.

Minerals that contain no K₂O like andalusite, kyanite and sillimanite plot at the A corner of the diagram, and minerals like staurolite, chloritoid (Ctd), chlorite, and garnet plot on the front face of the diagram.

Biotite, however, does contain K₂Oand has varying amounts of Al₂O₃ and thus is a solid solution that lies in the four component system. Because muscovite is relatively K - poor, this results in biotite being projected to negative values of Al₂O₃.

Any rock composition, like composition a, shown in the diagram, will also project to the front face, and may or may not plot at negative values of Al₂O₃.