1. Chapter 24. Stable Mineral Assemblages in Metamorphic Rocks
Equilibrium Mineral Assemblages
At equilibrium, the mineralogy (and the composition of each mineral) is determined by T, P, and X
“Mineral paragenesis” refers to such an equilibrium mineral assemblage
Relict minerals or later alteration products are excluded unless specifically stated
“Mineral assemblage” is used by some as a synonym for paragenesis, conventionally assuming equilibrium for the term
Impossible to prove that a mineral assemblage now at the Earth’s surface represents thermodynamic (chemical) equilibrium at prior elevated metamorphic conditions
Indirect textural and chemical support for such a conclusion is discussed in the text
In short, it is typically easy to recognize non-equilibrium minerals (retrograde rims, reaction textures, etc.)
We shall assume equilibrium mineral assemblages in the following discussion (will ignore retrograde…)

2. The Phase Rule in Metamorphic Systems
Phase rule, as applied to systems at equilibrium: F = C - f + 2 the phase rule (Eq 6.1)
f = the number of phases in the system
C = the number of components: the minimum number of chemical constituents required to specify every phase in the system
F = the number of degrees of freedom: the number of independently variable intensive parameters of state (such as temperature, pressure, the composition of each phase, etc.)
A typical sample from a metamorphic terrane
Likely select a sample from within a zone, and not from right on an isograd
Alternatively, pick a random point anywhere on a phase diagram
Likely point will be within a divariant field and not on a univariant curve or invariant point
The most common situation is divariant (F = 2), meaning that P and T are independently variable without affecting the mineral assemblage
In complex natural systems there may be one or more compositional variables as well, so that F may be greater than two
The common occurrence of certain metamorphic mineral assemblages worldwide supports this contention that F  2, because such assemblages are much more likely to represent variable P-T-X conditions than more restricted situations

3. The Phase Rule in Metamorphic Systems
If F  2 is the most common situation, then the phase rule may be adjusted accordingly: F = C - f + 2  2
f  C (Eq 24.1)
Goldschmidt’s mineralogical phase rule, or simply the mineralogical phase rule
The above simplified phase rule states that in the most common situation for a rock at equilibrium the number of phases is equal to or greater than the number of components
It has been called Goldschmidt’s mineralogical phase rule, or simply the mineralogical phase rule
It is useful in evaluating whether or not a rock is at equilibrium

4. The Phase Rule in Metamorphic Systems
Suppose we have determined C for a rock Consider the following three scenarios:
a) f = C
The standard divariant situation
The rock probably represents an equilibrium mineral assemblage from within a metamorphic zone
To some authors the mineralogical phase rule is phi = C

5. The Phase Rule in Metamorphic Systems
b) f < C
Common with mineral systems that exhibit solid solution
Plagioclase
Liquid
plus
For example, in the plagioclase or the olivine system we can see that under metamorphic conditions (below the solidus) these 2-C systems consist of a single mineral phase
phi = 1 and C = 2 so phi < C

6. The Phase Rule in Metamorphic Systems
c) f > C
A more interesting situation, and at least one of three situations must be responsible:
1)  F < 2
The sample is collected from a location right on a univariant reaction curve (isograd) or invariant point
The full phase rule (F = C - f + 2 ) indicates that, if C is constant, for every decrease in F, phi increases by one

7. The Phase Rule in Metamorphic Systems
Consider the following three scenarios:
C = 1
f = 1 common
f = 2 rare
f = 3 only at the specific P-T conditions of the invariant point
(~ 0.37 GPa and 500oC)
Figure The P-T phase diagram for the system Al2SiO5 calculated using the program TWQ (Berman, 1988, 1990, 1991). Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

8. The Phase Rule in Metamorphic Systems
2)  Equilibrium has not been attained
The phase rule applies only to systems at equilibrium, and there could be any number of minerals coexisting if equilibrium is not attained
Graywackes typically contain fragments of a number of different rocks in a matrix of grains derived from a rapidly eroding source
Dozens of minerals may be present, but they are not in equilibrium with one another
A number of igneous and metamorphic rocks also have retrograde reactions that begin, but do not run to completion

9. The Phase Rule in Metamorphic Systems
3)  We didn’t choose the # of components correctly
Some guidelines for an appropriate choice of C
Begin with a 1-component system, such as CaAl2Si2O8 (anorthite), there are 3 common types of major/minor components that we can add
a)  Components that generate a new phase
Adding a component such as CaMgSi2O6 (diopside), results in an additional phase: in the binary Di-An system diopside coexists with anorthite below the solidus
May sound silly, but a very real problem, and is difficult to evaluate
It is not sufficient to chemically analyze a rock and assume that each constituent is a component. This would lead to dozens of components, including trace elements
In order to manageably treat equilibrium mineral assemblages, it is helpful to limit the number of components and phases with which we must deal to those that are essential – this is the key: what is essential?
At end:
If a component goes only into a single phase (for example, P2O5  apatite, or TiO2  ilmenite) we can effectively ignore both the component and the phase because the phase rule is unaffected by reducing f and C by the same amount
As a result, many minor elements, such as P2O5 and TiO2, are often ignored along with the corresponding accessory phase so as to simplify the application of the phase rule
Otherwise components that generate additional phases must be considered

10. The Phase Rule in Metamorphic Systems
3)  We didn’t choose the # of components correctly
b) Components that substitute for other components
Adding a component such as NaAlSi3O8 (albite) to the 1-C anorthite system would dissolve in the anorthite structure, resulting in a single solid-solution mineral (plagioclase) below the solidus
Fe and Mn commonly substitute for Mg
Al may substitute for Si
Na may substitute for K
These components may just dissolve in phases (particularly if they are not present in great quantities)
We may decide to combine components that substitute for one another into a single, mixed component such as (Fe + Mg + Mn)
As a rule we don’t include trace elements in the phase rule, as they behave as substitutes only and are not present in sufficient quantity that they affect the number and nature of the mineral phases present
Although there is no need to ignore or combine major components like this (providing that we understand the nature of the mineral solutions in our rocks) we shall see that it is a handy device to limit the number of components when we wish to construct diagrams to display chemical-mineralogical data

11. The Phase Rule in Metamorphic Systems
3)  We didn’t choose the # of components correctly
c) “Perfectly mobile” components
Mobile components are either a freely mobile fluid component or a component that dissolves readily in a fluid phase and can be transported easily
The chemical activity of such components is commonly controlled by factors external to the local rock system
They are commonly ignored in deriving C for metamorphic systems

12. The Phase Rule in Metamorphic Systems
Consider the very simple metamorphic system, MgO-H2O
Possible natural phases in this system are periclase (MgO), aqueous fluid (H2O), and brucite (Mg(OH)2)
How we deal with H2O depends upon whether water is perfectly mobile or not
A reaction can occur between the potential phases in this system:
MgO + H2O  Mg(OH)2 Per + Fluid = Bru
retrograde reaction as written (occurs as the rock cools and hydrates)

13. Cool to the temperature of the reaction curve, periclase reacts with water to form brucite: MgO + H2O  Mg(OH)2
1) Suppose H2O is mobile and we ignore it as a component
If we don’t treat it as a component, we cannot treat it as a phase either (fluid is usually gone by the time we look at the rock anyway)
Begin with periclase above the reaction temperature
phi = 1 (water doesn’t count) and C = 1 (MgO), so the mineralogical phase rule holds
As we cool to the reaction curve, periclase reacts with water to form brucite: MgO + H2O  Mg(OH)2
Reaction: periclase coexists with brucite: phi = C + 1 = 2 F = 1 (#2 reason to violate the mineralogical phase rule)
To leave the curve, all the periclase must be consumed by the reaction, and brucite is the solitary remaining phase
phi = 1 and C = 1 again so phi = C
“Perfectly mobile” means that H2O can be added as it is needed, and any excess will leave as it is produced
Water = external reservoir, available enough to enable magnesium to exist as either periclase or brucite, depending on which is stable under the given hydrous P-T conditions
Imagine the diagram to be a field area, and the reaction an isograd
For any common sample phi = 1 (ignore H2O and don’t see fluid in the rock), and phi = C (mineralogical phase rule)
Only an uncommon sample, collected exactly on the isograd, will have phi > C
Figure P-T diagram for the reaction brucite = periclase + water. From Winter (2010). An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

14. 2) If water is not perfectly mobile, we must treat the system differently
Water is not a freely permeating phase: it may become a limiting factor in the reaction
Imagine cooling periclase until it intersects the reaction curve
At this point we stress an obvious, but commonly overlooked, aspect of chemical reactions:
A reaction involving more than one reactant will proceed until any one of the reactants is consumed
As the system cools, the reaction will proceed until either periclase or water is consumed
When one runs out, the other has nothing more to react with
Periclase can react only with the quantity of water that can diffuse into the system
If water is not perfectly mobile, and is limited in quantity, it will be consumed first
Once the water is gone, the excess periclase remains stable into the brucite stability field
Reactions such as this do represent absolute stability of brucite- the only reactant in the prograde reaction (brucite  periclase + water)
Reaction is not the absolute stability of periclase, or of water: either can be stable across the boundary if the other reactant is absent
Of course this is true for any reaction involving multiple phases
Figure P-T diagram for the reaction brucite = periclase + water. From Winter (2010). An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

15. We can now conclude that periclase can be stable anywhere on the whole diagram, if water is present in insufficient quantities to permit the reaction to brucite to go to completion
In water-deficient systems we are likely to find brucite and periclase coexisting under the conditions to the left of the univariant curve
We find that phi = 2 over a range of metamorphic grades, and thus over a significant portion of the field outcrops as well
C = 2 as well now since water is not perfectly mobile and it becomes a limiting factor and must therefore be counted
phi = C still holds
At any point (other than on the univariant curve itself) we would expect to find two phases, not one
phi = brucite + periclase below the reaction curve (if water is limited), or periclase + water above the curve
To the right of the curve we appear to have only periclase, but if we count water as a component, we must also include it as a phase: the fluid phase
Figure P-T diagram for the reaction brucite = periclase + water. From Winter (2010). An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

16. The Phase Rule in Metamorphic Systems
How do you know which way is correct?
The rocks should tell you
Phase rule = interpretive tool, not predictive
If only see low-f assemblages (e.g. Per or Bru in the MgO-H2O system) ® some components may be mobile
If many phases in an area it is unlikely that all is right on univariant curve, and may require the number of components to include otherwise mobile phases, such as H2O or CO2, in order to apply the phase rule correctly

17. Chemographic Diagrams
Chemographics refers to the graphical representation of the chemistry of mineral assemblages
A simple example: the plagioclase system as a linear C = 2 plot:
= 100 An/(An+Ab)
Binary Ab-An solid solution along a line
Metamorphic components typically expressed as molar quantities, and not as weight %
Any intermediate composition is simply plotted an appropriate distance along the line
An50 would plot in the center, An25 would plot ¼ of the way from Ab to An, etc.

18. Chemographic Diagrams
3-C mineral compositions are plotted on a triangular chemographic diagram as shown in Fig x, y, z, xz, xyz, and yz2
Suppose you had a small area of a metamorphic terrane in which the rocks correspond to a hypothetical 3-component system with variable proportions of the components x-y-z
The rocks in the area are found to contain 6 minerals with the fixed compositions x, y, z, xz, xyz, and yz2

19. Suppose that the rocks in our area have the following 5 assemblages:
x - xy - x2z
xy - xyz - x2z
xy - xyz - y
xyz - z - x2z
y - z - xyz
When applying such a chemographic diagram to the study of mineral assemblages in a particular area, phases that coexist at equilibrium in a rock are connected by tie-lines
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals. Minerals that coexist compatibly under the range of P-T conditions specific to the diagram are connected by tie-lines. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.

20. Note that this subdivides the chemographic diagram into 5 sub-triangles, labeled (A)-(E)
x - xy - x2z
xy - xyz - x2z
xy - xyz - y
xyz - z - x2z
y - z - xyz
A diagram like this is a compatibility diagram, a type of phase diagram commonly employed by metamorphic petrologists
Any point within the diagram represents a specific bulk rock composition
The diagram determines the corresponding mineral assemblage that develops at equilibrium
For example, a point within the sub-triangle (E), the corresponding mineral assemblage corresponds to the corners = y - z - xyz
Any rock with a bulk composition plotting within triangle (E) will develop that same mineral assemblage

21. Common point corresponds to 3 phases, thus f = C
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals. Minerals that coexist compatibly under the range of P-T conditions specific to the diagram are connected by tie-lines. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.
The lever principle works too for proportions of the minerals
Compatibility diagrams explain clearly why rocks, even though equilibrated at the same metamorphic grade, often develop different mineral assemblages
A shift in bulk composition, even if only slightly, can vary the mineral assemblage
If move from (E) to (D) a rock will contain the mineral x2z (along with z and xyz) and no longer the mineral y that occurred in (E)
Only rocks with bulk compositions corresponding to sub-triangle (A), for example, develop the mineral x, because they are rich in the x component
Suppose some mineral, such as mineral xyz, is an important index mineral (e.g. garnet). The diagram can readily explain why all pelitic rocks need not contain garnet, even if they are within the garnet zone
Any rock with a bulk composition in sub-triangle (A) will be garnet-free
The diagram also demonstrates why some mineral pairs do not coexist. Mineral x cannot coexist at equilibrium with y, xyz, or z under the conditions represented, because tie-lines separate these pairs
Likewise mineral z cannot coexist with x, xz, or xyz

22. What happens if you pick a composition that falls directly on a tie-line, such as point (f)?
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals. Minerals that coexist compatibly under the range of P-T conditions specific to the diagram are connected by tie-lines. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.
In this case the mineral assemblage consists of xyz and z only (the ends of the tie-line), since by adding these two phases together in the proper proportion you can produce the bulk composition (f)
In such a situation phi = 2
…but so does C

23. In the unlikely event that the bulk composition equals that of a single mineral, such as xyz, then f = 1, but C = 1 as well
“compositionally degenerate”

24. Chemographic Diagrams
Valid compatibility diagram must be referenced to a specific range of P-T conditions, such as a zone in some metamorphic terrane, because the stability of the minerals and their groupings vary as P and T vary
Previous diagram refers to a P-T range in which the fictitious minerals x, y, z, xy, xyz, and x2z are all stable and occur in the groups shown
At different grades the diagrams change
Other minerals become stable
Different arrangements of the same minerals (different tie-lines connect different coexisting phases)

25. A diagram in which some minerals exhibit solid solution
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals, many of which exhibit solid solution. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.
Minerals x(y,z) and x2(y,z) show limited solid solution of components y and z on one type of lattice site.
Mineral x(y,z) allows more y in the lattice than does mineral x2(y,z)
Minerals (xyz)ss and zss (the subscript denotes solid solution) show limited solid solution of all three components
Click
Suppose a bulk rock composition is in the shaded field of the mineral (xyz)ss
phi = 1, but the system is not degenerate
Due to the variable nature of the composition of the phase, C must still equal 3 and the phase rule tells us that F = C - phi + 2 = 4
Thus P, T, and any 2 of the 3 components in the phase are independently variable
The shaded area of mineral (xyz)ss is thus an area (compositionally divariant), and our single-phase rock can have any composition within the shaded solid solution limits
phi < C in this case because of the solid solution and compositional variance

26. If Xbulk on a tie-line
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals, many of which exhibit solid solution. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.
As in the previous example, there are two coexisting phases (xyz)ss and zss
The composition of the two minerals that correspond to bulk rock composition (f) are indicated by the two shaded dots at the ends of the tie-line through (f)
phi = 2 and C is still 3, so F = = 3
Since P and T are independently variable, that means the composition of each phase is univariant, and must vary along the lines where the bundles of tie- lines end
Although the composition of (xyz)ss can vary anywhere in the shaded area, the composition of (xyz)ss that coexists with Zss is constrained to the edge of the area facing z
A degree of freedom is thus lost as a phase is gained
Likewise the composition of zss that coexists with (xyz)ss is constrained to a portion of the edge of the shaded zss area

27. Xbulk in 3-phase triangles F = 2 (P & T) so Xmin fixed
Figure Hypothetical three-component chemographic compatibility diagram illustrating the positions of various stable minerals, many of which exhibit solid solution. After Best (1982) Igneous and Metamorphic Petrology. W. H. Freeman.
In such situations phi = 3, and C = 3, as predicted by the mineralogical phase rule
Since F = 2 and corresponds to P and T the phase rule tells us that all of the compositional variables for each phase are fixed

28. Chemographic Diagrams for Metamorphic Rocks
Most common natural rocks contain the major elements: SiO2, Al2O3, K2O, CaO, Na2O, FeO, MgO, MnO and H2O such that C = 9
Three components is the maximum number that we can easily deal with in two dimensions
What is the “right” choice of components?
Some simplifying methods:
All 9 is clearly too complex. Must simplify if we want to display the system in a convenient graphical way
Four components requires 3-D tetrahedra, and we lose even a semi- quantitative sense of depth in the diagram
More than four components is much too complex to be useful

29. 1) Simply “ignore” components Trace elements
Elements that enter only a single phase (we can drop both the component and the phase without violating the phase rule)
Perfectly mobile components
Some minerals are stable over a large range of metamorphic conditions, and thus are not useful as indicators of metamorphic grade
Even if present in significant quantities, they may be eliminated from consideration without impairing the analysis, particularly if they are the only phase containing a particular component
This is often the case for albite, K-feldspar, magnetite, ilmenite, sphene, and apatite

30. 2) Combine components
Components that substitute for one another in a solid solution: (Fe + Mg)
3) Limit the types of rocks to be shown
Only deal with a sub-set of rock types for which a simplified system works
4) Use projections
I’ll explain this shortly

31. The phase rule and compatibility diagrams are rigorously correct when applied to complete systems
A triangular diagram thus applies rigorously only to true (but rare) 3-component systems
If drop components and phases, combine components, or project from phases, we face the same dilemma we faced using simplified systems in Chapters 6 and 7
Gain by being able to graphically display the simplified system, and many aspects of the system’s behavior become apparent
Lose a rigorous correlation between the behavior of the simplified system and reality
The advantages of being able to see what is going on by adopting an appropriate and easily visualized system, generally outweighs the loss due to simplification
When we simplify, however, the diagrams may at times exhibit some “inconsistencies”
If we choose a good simplified system, inconsistencies will be minimal
When inconsistencies occur, careful analysis using a lesser degree of simplification usually reveals the reason for it

32. The ACF Diagram
Illustrate metamorphic mineral assemblages in mafic rocks on a simplified 3-C triangular diagram
Concentrate only on the minerals that appeared or disappeared during metamorphism, thus acting as indicators of metamorphic grade
Eskola (1915) proposed the ACF diagram as a way to illustrate metamorphic mineral assemblages in mafic rocks on a simplified 3-C triangular diagram
He wanted to concentrate only on the minerals that appeared or disappeared during metamorphism, thus acting as indicators of metamorphic grade

33. Figure 24. 4. After Ehlers and Blatt (1982). Petrology. Freeman
Figure After Ehlers and Blatt (1982). Petrology. Freeman. And Miyashiro (1994) Metamorphic Petrology. Oxford.
Fig illustrates the positions of several common metamorphic minerals on the ACF diagram. Note: this diagram is presented only to show you where a number of important phases plot.
It is not specific to a P-T range and therefore is not a true compatibility diagram, and has no petrological significance

34. The ACF Diagram
The three pseudo-components are all calculated on an atomic basis:
A = Al2O3 + Fe2O3 - Na2O - K2O
C = CaO P2O5
F = FeO + MgO + MnO
To calculate A for a mineral, we combine the atomic proportions Al and Fe3+ in the mineral formula, and then subtract Na and K
If you begin with an ideal mineral formula Fe3+ is rare, except in a few minerals
In real mafic minerals, Fe is probably in both valence states, but generally is dominated by Fe2+
Microprobe analyses cannot distinguish Fe oxidation states, but you can often estimate the relative proportions of ferric and ferrous iron by calculating the charge balance

35. The ACF Diagram A = Al2O3 + Fe2O3 - Na2O - K2O Why the subtraction?
Na and K in the average mafic rock are typically combined with Al to produce Kfs and Albite
In the ACF diagram, we are interested only in the other K- bearing metamorphic minerals, and thus only in the amount of Al2O3 that occurs in excess of that combined with Na2O and K2O (in albite and K-feldspar)
Because the ratio of Al2O3 to Na2O or K2O in feldspars is 1:1, we subtract from Al2O3 an amount equivalent to Na2O and K2O in the same 1:1 ratio
Although it appears as though we are subtracting sodium and potassium from aluminum, we are really subtracting from Al an amount of Al that is equivalent to the amounts of Na + K, and then dealing with the Al that is left over
As we’ll see in Section , this subtraction is more appropriately considered as a projection from K-feldspar and albite, thereby eliminating K2O and Na2O as components
Combing Al and Fe3+ is not very rigorous, and projecting from K-feldspar is only justified when this phase is present, which is rare in mafic rocks
The amounts of Fe3+ and K2O are usually minor, however, and this is seldom a major problem
Projecting from Ab is more reliable, because this component is commonly present in plagioclase

36. The ACF Diagram C = CaO - 3.3 P2O5 F = FeO + MgO + MnO
C is formulated in a similar way to A
All the P2O5 in most rocks is combined with CaO (in the ratio 1:3.3) to create apatite
Apatite is an ubiquitous accessory mineral, and in the ACF diagram we are interested only in the CaO that exists in excess of that captured by P2O5 to create apatite
Thus we subtract from CaO an amount equal to 3.3 times the amount of P2O5 to eliminate P2O5 as a component and apatite as a phase without altering CaO in the rest of the system
F is a combined pseudo-component based on the common exchangeability of Fe, Mg, and Mn in solid solution on the octahedral sites of most mafic minerals
If they behave in such a similar fashion, it makes some sense to treat them as a single component

37. The ACF Diagram
By creating these three pseudo-components, Eskola reduced the number of components in mafic rocks from 8 to 3
Water is omitted under the assumption that it is perfectly mobile
Note that SiO2 is simply ignored
We shall see that this is equivalent to projecting from quartz
In order for a projected phase diagram to be truly valid, the phase from which it is projected must be present in the mineral assemblages represented
Thus, to be valid, the ACF diagram must have both quartz, alkali feldspar, and plagioclase present
It may work anyway when these phases are lacking, but the result may also violate the mineralogical phase rule on occasion

38. The ACF Diagram An example: Anorthite CaAl2Si2O8
A = = 1, C = = 1, and F = 0
Provisional values sum to 2, so we can normalize to 1.0 by multiplying each value by ½, resulting in
A = 0.5
C = 0.5
F = 0
Anorthite thus plots half way between A and C on the side of the ACF triangle, as shown in Fig. 24.4
Where does Ab plot? Plagioclase?

39. Figure 24. 4. After Ehlers and Blatt (1982). Petrology. Freeman
Figure After Ehlers and Blatt (1982). Petrology. Freeman. And Miyashiro (1994) Metamorphic Petrology. Oxford.
Where does K-feldspar plot??
For KAlSi3O8 A = 0.5 + 0 - 0.5 = 0, C = 0, and F = 0
Kspar doesn’t plot on the ACF diagram
If you try this for albite you will find that it doesn’t plot either
The ACF formula eliminates Na and K from the diagram (and thus these phases) without altering the remaining Al after the removal
All Ca-bearing plagioclase feldspars, regardless of the K and Na content, plot at the anorthite point

40. A typical ACF compatibility diagram, referring to a specific range of P and T (the kyanite zone in the Scottish Highlands)
Plot all phases and connect coexisting ones with tie-lines
The composition of most mafic rocks fall in the hornblende-plagioclase field or the hornblende- plagioclase-garnet triangle, and thus most metabasaltic rocks occur as amphibolites or garnet amphibolites in this zone
More aluminous rocks develop kyanite and/or muscovite and not hornblende
More calcic rocks lose Ca-free garnet, and contain diopside, grossularite, or even calcite (if CO2)
We again see how the diagram allows us to interpret the relationship between the chemical composition of a rock and the equilibrium mineral assemblage
Figure After Turner (1981). Metamorphic Petrology. McGraw Hill.

41. The AKF Diagram In the AKF diagram, the pseudo-components are:
Because pelitic sediments are high in Al2O3 and K2O, and low in CaO, Eskola proposed a different diagram that included K2O to depict the mineral assemblages that develop in them
In the AKF diagram, the pseudo-components are:
A = Al2O3 + Fe2O3 - Na2O - K2O - CaO
K = K2O
F = FeO + MgO + MnO
Note that CaO is now subtracted from Al2O3 in calculating A
This eliminates CaO and plagioclase from the diagram by projecting
We are now interested only in the Al that occurs in excess of that combined with K, Na, and Ca to make any feldspar
As a result, no feldspar, including anorthite or any intermediate plagioclase, plots on the AKF diagram
Only one atom of Ca is subtracted from two of Al because the Al:Ca ratio in anorthite is 1:2.

42. Figure 24.6. After Ehlers and Blatt (1982). Petrology. Freeman.
Another figurative AKF to show where minerals plot

43. AKF compatibility diagram (Eskola, 1915) illustrating paragenesis of pelitic hornfelses, Orijärvi region Finland
Figure After Eskola (1915) and Turner (1981) Metamorphic Petrology. McGraw Hill.
Al-poor rocks contain biotite, and may contain an amphibole if sufficiently rich in Mg and Fe, or microcline if not
Rocks richer in Al contain andalusite and cordierite

44. Three of the most common minerals in metapelites: andalusite, muscovite, and microcline, all plot as distinct points in the AKF diagram
And & Ms plot as the same point in the ACF diagram, and Micr doesn’t plot at all, so the ACF diagram is much less useful for pelitic rocks (rich in K and Al)

45. Projections in Chemographic Diagrams
When we explore the methods of chemographic projection we will discover:
Why we ignored SiO2 in the ACF and AKF diagrams
What that subtraction was all about in calculating A and C
It will also help you to better understand the AFM diagram and some of the shortcomings of projected metamorphic phase diagrams

46. Projection from Apical Phases
Example- the ternary system: CaO-MgO-SiO2 (“CMS”)
Straightforward: C = CaO, M = MgO, and S = SiO2… none of that fancy subtracting business!
Let’s plot the following minerals:
Fo - Mg2SiO4 Per - MgO
En - MgSiO3 Qtz - SiO2
Di - CaMgSi2O6 Cc - CaCO3
A naturally simple system that occurs in metamorphosed siliceous dolomites (+ CO2) and some ultramafic rocks (with some Fe and other relatively minor components)

47. Projection from Apical Phases
Fo - Mg2SiO4 Per - MgO En - MgSiO3
Qtz - SiO2 Di - CaMgSi2O6 Cc - CaCO3
For the purpose of illustration, let’s now project the composition of Di from the Ca apex of the triangle to the Mg-Si side. To do this we draw a line from C through the Di point to the M-S side of the triangle (labeled Di’)

48. The line intersects the M-S the side at a point equivalent to 33% MgO
The line intersects the M-S the side at a point equivalent to 33% MgO 67% SiO2
Note that any point on the dashed line from C through Di to the M-S side has a constant ratio of Mg:Si = 1:2
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.

49. Projection from Apical Phases
Fo - Mg2SiO4 Per - MgO En - MgSiO3
Qtz - SiO2 Di - CaMgSi2O6 Cc - CaCO3
Pseudo-binary Mg-Si diagram in which Di is projected to a 33% Mg - 66% Si
+ Cal
By projecting we have reduced the ternary system to a pseudo-binary one, thus making it simpler
The geometric projection of diopside from CaO is mathematically equivalent to ignoring CaO in the formulas
The ratio of MgO to SiO2 in Di is 1:2 or 33:67
We could calculate and plot Di in the pseudo-binary system directly without going through the complicated procedure of plotting the 3-C system, drawing the (dashed) projection line, and then redrawing the result
MgO
SiO 2 Fo En
Di' Q
Per

50. Projection from Apical Phases
Could project Di from SiO2 and get C = 0.5, M = 0.5
Di’ plots at the midpoint of the Ca-Mg side
We get the same result mathematically by ignoring CaO in the diopside formula
Note that periclase, forsterite, and enstatite all stack up on the same point in this projection, impeding its usefulness
+ Qtz
MgO
CaO
Di'
Cal
Per, Fo, En

51. Projection from Apical Phases
MgO
SiO 2 Fo En
Di' Q
Per
In accordance with the mineralogical phase rule (f = C) get any of the following 2-phase mineral assemblages in our 2-component system:
Per + Fo Fo + En
En + Di Di + Q
Suppose next that we were to treat the pseudo-binary as a compatibility diagram for some rocks belonging to the CMS system
End:
Because this diagram is projected from CaO, we must include calcite in each of the above assemblages if we were to consider the full 3-C (and hence 3-f) system from which we projected
It appears that quartz cannot coexist with enstatite, nor can diopside coexist with forsterite
Imagine our surprise as we check back with our field notes and discover that enstatite and quartz commonly coexist, as do diopside and forsterite

52. Projection from Apical Phases
What’s wrong?
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
In order to find out, we need to go back to the more rigorously correct 3-C system (hypothetical)
In the complete diagram we see that our projection from CaO collapsed several important tie-lines, and that calcite does not coexist with each of the assemblages listed above
To project from some point assumes that a phase plotting at that point coexists with the projected phases
Thus the projected diagram assumes calcite is present
In 3-C we have Di + En + Fo (coexisting diopside and forsterite) and Di + En + Q (coexisting enstatite and quartz), each of which is a calcite-free 3-phase assemblage
By referring back to the “real” system, we understand why the projected system gave us some spurious results
The apparent incompatibility of En + Q and Di + Fo is an artifact of the projection, which assumes that calcite is the third phase in each projected assemblage!
Because calcite cannot coexist with forsterite or enstatite, these phases should not be projected from calcite at the CaO apex to the MgO-SiO2 system
Projected from Calcite
+ Cal
MgO
SiO 2 Fo En
Di' Q
Per

53. Projection from Apical Phases
What’s wrong?
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Better to have projected from Diopside
+ Di
MgO
SiO 2
Per Fo En Q

54. Projection from Apical Phases
ACF and AKF diagrams eliminate SiO2 by projecting from quartz
Math is easy: projecting from an apex component is like ignoring the component in formulas
The shortcoming is that these projections compress the true relationships as a dimension is lost
To be properly applicable, quartz must therefore be present in every mineral assemblage on the ACF and AKF diagrams (or SiO2 must at least be present to the point of saturation)
Why create projected pseudo- systems when they lead to such inconsistencies?
3-C -> 2-C does not provide much of a benefit, since both are easily illustrated in 2D diagrams
When > 3 components, the systems are no longer so easily visualized
Reducing the components from 8 or 9 to three provides a motivation to combine components
As seen in the ACF and AKF diagrams, the ability to visualize the geometric relationships is often an advantage that outweighs the problems

55. Projection from Apical Phases
Two compounds plot within the ABCQ compositional tetrahedron,
x (formula ABCQ)
y (formula A2B2CQ)
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
To better understand how this applies to projecting from SiO2 to triangular ACF or AKF diagrams, consider the fictitious 4-component system ABCQ illustrated in Fig

56. Projection from Apical Phases
x = ABCQ
y = A2B2CQ
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
If minerals coexisting within a metamorphic zone are connected by tie-lines, the tetrahedron becomes subdivided into triangular-sided sub-polyhedra: AYQX, BYQX, etc
A bulk rock composition that falls within any one polyhedral volume will consist of the 4- phase equilibrium assemblage represented by the 4 corners of the sub-polyhedron

57. Projection from Apical Phases
x = ABCQ
y = A2B2CQ
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Now visualize a projection from the Q apex to the ABC plane
The point X is in the center of the tetrahedron and the point Y is within the ABQX volume
The projection process works the same way as in the ternary example above
Projecting from Q projects points X and Y to the ABC base as points X' and Y’
Along any single projection line, such as Q-X-X' or Q-Y-Y', the ratio of A:B:C is constant
We can mathematically perform the projection by simply ignoring Q in the formula of the projected phase and normalize A+B+C to 100
For X the result is A:B:C = 1:1:1 = 33:33:33 so that X plots as X' in the center of the projected pseudo- ternary diagram
Y also plots as Y' with A:B:C = 2:2:1 = 40:40:20, resulting in the pseudo-ternary ABC system illustrated in Fig

58. Projection from Apical Phases
x plots as x' since A:B:C = 1:1:1 = 33:33:33
y plots as y' since A:B:C = 2:2:1 = 40:40:20
x = ABCQ
y = A2B2CQ
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
The resulting ABC projection is simplified with respect to the quaternary system, and we need no longer attempt to estimate the depth of points X and Y
But the projection suffers from the same pitfalls that we saw above

59. Projection from Apical Phases
If we remember our projection point (q), we conclude from this diagram that the following assemblages are possible:
(q)-b-x-c
(q)-a-x-y
(q)-b-x-y
(q)-a-b-y
(q)-a-x-c
The assemblage a-b-c appears to be impossible

60. Projection from Apical Phases
Figure Winter (2010) An Introduction to Igneous and Metamorphic Petrology. Prentice Hall.
Note the sub-polyhedron a-b-c-x in 4-C
Phases a, b, and c are truly on the base and x (and y) are above it
The diagram in Fig is not truly ternary, but a projection, which must be remembered!
The pseudo-ternary triangle is really part of a quaternary system
Because it is projected from q, it should be clearly referenced to coexisting q!

61. Projection from Apical Phases
Although A, B, and C are separated by several tie-lines in Fig , and appear to be incompatible together, they can coexist, but with x and without q!
We’re not just dropping Q when we create the projection in Fig
The sub-triangles in the figure are only rigorously valid groupings when phase Q is included as a part of each mineral assemblage
Conscientious workers include the message “+Qtz” in the corner of most diagrams, such as ACF, AKF, etc. that are projected from quartz
Multiple projections are possible, each successive one reducing the number of remaining components
The ACF diagram is projected from SiO2 and then from alkali feldspar
In such cases all projection phases should be noted in the diagrams

62. J.B. Thompson’s A(K)FM Diagram
An alternative to the AKF diagram for metamorphosed pelitic rocks
Although the AKF is useful in this capacity, J.B. Thompson (1957) noted that Fe and Mg do not partition themselves equally between the various mafic minerals in most rocks
Pelites are sensitive to T & P, and undergo many mineralogical changes as grade increases
Good to have a comprehensive diagram to display these changes

63. J.B. Thompson’s A(K)FM Diagram
Figure Partitioning of Mg/Fe in minerals in ultramafic rocks, Bergell aureole, Italy After Trommsdorff and Evans (1972). A J Sci 272,
Talc is always richer in Mg than coexisting diopside, diopside more so than tremolite, etc
All but anthophyllite are more Mg-rich than olivine
The same type of partitioning occurs in virtually all mafic minerals, including those in pelitic rocks
Garnet, staurolite, and chloritiod have low Mg/Fe ratios, whereas cordierite has a higher ratio
Thompson concluded that it would be advantageous in many situations to account for this unequal distribution, since it plays a role in mineral stability
-> AFM diagram, in which Fe and Mg are no longer combined as a single component
He neglected minor components in pelitic rocks and considered water to be perfectly mobile
He eliminated SiO2 by projecting from quartz
Since quartz is nearly always present in metapelites, this requirement of the projection is generally met

64. J.B. Thompson’s A(K)FM Diagram
A = Al2O3
K = K2O
F = FeO
M = MgO
Four principal components thus remain (A = Al2O3, K = K2O, F = FeO, M = MgO)
In order to avoid dealing with a three-dimensional tetrahedron, Thompson projected the phases in the system to the AFM face, thereby eliminating K2O
He recognized that projecting from K2O would not work, since no phase corresponds to this apical point, and projections would cross important tie-lines

65. J.B. Thompson’s A(K)FM Diagram
Project from a phase that is present in the mineral assemblages to be studied
Since muscovite is the most widespread K-rich phase in metapelites, he decided to project from muscovite (Mu) to the AFM base as shown
Projecting from muscovite can lead to a strange looking AFM projections
Note that Mu is still rather K-poor, and only mineral phases in the volume A-F-M-Mu in Fig will be projected to points within the AFM face of the AKFM tetrahedron
Biotite is outside this volume, and projecting it from Mu causes it to plot as a band (of variable Fe/Mg) outside the AFM triangle
Figure AKFM Projection from Mu. After Thompson (1957). Am. Min. 22,

66. J.B. Thompson’s A(K)FM Diagram
At high grades muscovite dehydrates to K-feldspar as the common high-K phase
Then the AFM diagram should be projected from K-feldspar
When projected from Kfs, biotite projects within the F-M base of the AFM triangle
Muscovite may be absent in some pelitic rocks, particularly at higher grades when it dehydrates, giving way to K-feldspar as the common high-K phase
When this is the case, the AFM diagram should be projected from K-feldspar if the mineral assemblages of the diagram are to have significance
When projected from Kfs, biotite projects within the F-M base of the AFM triangle
Figure AKFM Projection from Kfs. After Thompson (1957). Am. Min. 22,

67. J.B. Thompson’s A(K)FM Diagram
A = Al2O3 - 3K2O (if projected from Ms)
= Al2O3 - K2O (if projected from Kfs)
F = FeO
M = MgO
We avoid having to create diagrams like this, and attempting 3-D projections, by calculating the projected components of the AFM diagram mathematically, using the following formulas:
We can then plot any composition as a point directly on the AFM diagram
When A, F, and M are all positive, plotting a point is straightforward
When A is negative, however, as with biotite, the process requires an extra step:

68. J.B. Thompson’s A(K)FM Diagram
Biotite (from Ms):
KMg2FeSi3AlO10(OH)2
A = 0.5 - 3 (0.5) =  - 1
F = 1
M = 2
To normalize we multiply each by 1.0/(2 + 1 - 1) = 1.0/2 = 0.5
Thus A = -0.5
F = 0.5
M = 1
To plot the point, we extend a line from A at a constant M/F ratio. Since M/(F+M) = 0.66
We next extend a vertical line from A a distance equal to half the distance from A to the F-M base, but we extend this distance beyond the base, because A is negative
Thus it extends to A/(A+F+M) = and not 0.5
Where a horizontal line at this value of A intersects the first line of constant F:M is the location of our biotite on the AFM diagram
The broad biotite field in Fig is due to Al-Al substitution for (Fe-Mg)-Si in biotite
K-feldspar, when projected from muscovite, projects away from the AFM diagram
In order to include Kfs in the AFM diagram it is assigned a position at negative infinity

69. J.B. Thompson’s A(K)FM Diagram
Figure AFM Projection from Ms for mineral assemblages developed in metapelitic rocks in the lower sillimanite zone, New Hampshire After Thompson (1957). Am. Min. 22,
Mg-enrichment typically in the order: cordierite > chlorite > biotite > staurolite > garnet
Due to extensive Mg-Fe solid solution in biotite and garnet, much of the area is dominated by 2-phase fields with tie- lines (really 4-phase when we include the Qtz and Mu projection phases)
Although we can easily plot ideal mineral formulas on the ACF and AKF diagrams, in order for a real mafic phase to be plotted on an AFM diagram we must know Mg/(Fe+Mg), which can only be determined by chemical analysis of the minerals, generally performed using the electron microprobe
If analyses are unavailable, we can approximate the correct positions on the basis of typical relative Mg/(Fe+Mg), based on our knowledge of numerous analyses of these minerals available in the literature. From these we know that Mg-enrichment occurs typically in the order: cordierite > chlorite > biotite > staurolite > garnet

70. Choosing the Appropriate Chemographic Diagram
Example, suppose we have a series of pelitic rocks in an area. The pelitic system consists of the 9 principal components: SiO2, Al2O3, FeO, MgO, MnO, CaO, Na2O, K2O, and H2O
How do we lump those 9 components to get a meaningful and useful diagram?
The ACF, AKF, and AFM diagrams are the most commonly used chemographic diagrams, but several others have been employed for rocks of different compositions
The choice of components and projection point depends on the system and mineral assemblages under study

71. Choosing the Appropriate Chemographic Diagram
Each simplifying step makes the resulting system easier to visualize, but may overlook some aspect of the rocks in question
MnO is commonly lumped with FeO + MgO, or ignored, as it usually occurs in low concentrations and enters solid solutions along with FeO and MgO
In metapelites Na2O is usually significant only in plagioclase, so we may often ignore it, or project from albite
As a rule, H2O is sufficiently mobile to be ignored as well
We are left with SiO2, Al2O3, K2O, CaO, FeO, and MgO
Which combination and diagram is best?

72. Choosing the Appropriate Chemographic Diagram
Common high-grade mineral assemblage:
Sil-St-Mu-Bt-Qtz-Plag
Figure AFM Projection from Ms for mineral assemblages developed in metapelitic rocks in the lower sillimanite zone, New Hampshire After Thompson (1957). Am. Min. 22,
A bulk rock composition corresponding to this assemblage is marked with an x in the AFM diagram shown previously

73. Choosing the Appropriate Chemographic Diagram
Sil-St-Mu-Bt-Qtz-Plag
Figure After Ehlers and Blatt (1982). Petrology. Freeman.
In the AFM diagram Sil, St, and Bt plot (with Mu + Qtz as associated phases, required by the projection)
In the ACF diagram Ky, St, and Plag are all that plot
In both the ACF and AFM diagrams C = P = 3, so we have “valid” divariant assemblages as far as the mineralogical phase rule is concerned
The ACF diagram is less useful for pelites, because plagioclase is the only Ca-bearing phase, and is stable over a wide range of conditions
Plag is thus not much of an indicator of metamorphic grade
Biotite, which is an important pelitic mineral that is often indicative of both grade and composition, does not appear
Although our assemblage plots as a 3-phase sub-triangle, other pelitic assemblages plot on the A-F edge, and most of the area of the diagram is wasted
The ACF diagram is more useful for mafic rocks

74. Choosing the Appropriate Chemographic Diagram
Sil-St-Mu-Bt-Qtz-Plag
We don’t have equilibrium
There is a reaction taking place (F = 1)
We haven’t chosen our components correctly and we do not really have 3 components in terms of AKF
The AKF diagram contains 4 phases and the Sil-Bt and Mu-St tie-lines cross
Crossing tie-lines can be a troublesome problem, because we cannot choose a single sub-triangle for our rock composition x
Remember, there are three possible explanations for this seeming violation of the mineralogical phase rule now that f > C
We don’t have equilibrium
There is a reaction taking place (F = 1)
We haven’t chosen our components correctly and we do not really have 3 components in terms of AKF
Figure After Ehlers and Blatt (1982). Petrology. Freeman.

75. Choosing the Appropriate Chemographic Diagram
Sil-St-Mu-Bt-Qtz-Plag
That Mu+Ky+Bt+St is a common assemblage argues against first two possibilities
If we compare the AKF and AFM diagrams we see Fe/Mg partitioning between Bt & St is moderately strong, and should not be ignored
Adding more Mg to the biotite than the staurolite effectively lifts Bt higher than St from the pure Fe end of the AKF base
As a result, the 4 phases appearing in the AKF diagram are not really coplanar in a more comprehensive AKFM 4-component diagram
Thus the Sil-Bt and Mu-St tie-lines do not intersect in space, so that Sil+Bt is not compositionally equivalent to Mu+St
We conclude that the AFM diagram is the best choice for the assemblage in question
It may not be the best for other assemblages, however, and the AFM and AKF diagrams are often used in combination to depict the relationship between rock composition, mineral assemblage, and metamorphic grade in metamorphosed pelitic rock sequences
Figure After Ehlers and Blatt (1982). Petrology. Freeman.

76. Choosing the Appropriate Chemographic Diagram
Myriad chemographic diagrams have been proposed to analyze paragenetic relationships in various metamorphic rock types
Most are triangular: the maximum number that can be represented easily and accurately in two dimensions
Some natural systems may conform to a simple 3- component system, and the resulting metamorphic phase diagram is rigorous in terms of the mineral assemblages that develop
Other diagrams are simplified by combining components or projecting
These diagrams are not as rigorous, but the ability to compare chemical and mineralogical variations usually outweighs the disadvantages, particularly when the method of simplification is clearly understood
Anomalies, such as phi appearing to be greater than C and crossed tie-lines may thus be resolved by referring to the more comprehensive system

77. Choosing the Appropriate Chemographic Diagram
Variations in metamorphic mineral assemblages result from:
1) Differences in bulk chemistry
2) differences in intensive variables, such as T, P, PH2O, etc (metamorphic grade)
A good chemographic diagram permits easy visualization of the first situation
The second can be determined by a balanced reaction in which one rock’s mineral assemblage contains the reactants and another the products
These differences can often be visualized by comparing separate chemographic diagrams, one for each grade