1. Assessment of reaction textures. The purpose of this step is to identify which minerals are early, which are late, and which are part of a stable assemblage. Early minerals are likely to be inclusions or broken, late minerals may be in cracks or strain shadows, and minerals that are in textural equilibrium should not be separated by reaction zones. Back-scattered electron imaging is often a convenient and powerful way to study textures:
   
   

   This back-scattered electron image shows zoning in titanite from a Norwegian gneiss:
   
   

2. Identification of zoning. Compositional zoning in minerals can be done by tuning the electron microprobe spectrometers--or focusing the ICP mass spectrometer detectors--to specific elements and then making element maps:
   
   

   An electron microprobe works by measuring element-characteristic x-rays emitted by a sample being bombarded with electrons on a 2-micron spot:
   
   

   Zoning can be quantified over an entire grain with good coverage of spot analyses:
   
   

   Such zoning can quantified using line scanning:
   
   

3. Application of thermodynamics. Experiments have been used to determine the PT positions of many different kinds of endmember reactions.
   For example, experiments have shown that the distribution of Fe and Mg between garnet and biotite is a function of inverse temperature:
   
   

   thermometry: The assessment of metamorphic temperatures.
   Quantitative thermometry typically is based on the exchange of Fe and Mg between pairs of minerals. For example, the exchange of Fe and Mg between garnet and biotite can be written as a reaction among the Fe and Mg endmembers of these minerals:

   • Fe₃Al₂Si₃O₁₂ + KMg₃AlSi₃O₁₀(OH)₂ = Mg₃Al₂Si₃O₁₂ + KFe₃AlSi₃O₁₀(OH)₂
     
   
   

   barometry: The assessment of metamorphic pressures.
   Qualitative barometry is based on mineral assemblages-such as the coexistence of kyanite + staurolite.
   Quantitative barometry typically is based on net transfer reactions
   For example, garnet - aluminumsilicate - silica - plagioclase (GASP)

   • 3CaAl₂Si₂O₈ = Ca₃Al₂Si₃O₁₂ + 2Al₂SiO₅ + SiO₂
   • 3 anorthite = grossular + 2 kyanite + quartz
   
   

Examples of P-T Determinations

Let's look at some examples of quantitative determinations of metamorphic pressures and temperatures.
Pressures and temperatures of equilibration in lower crustal xenoliths from Tibet were constrained using compositions of garnet, pyroxene, and plagioclase.

• almandine + diopside = pyrope + ferrosilite.

• CaMgSi₂O₆ diopside + CaAl₂SiO₆ 'cats' = Ca₂MgAl₂Si₃O₁₂ garnet or
• CaFeSi₂O₆ hedenbergite + CaAl₂SiO₆ 'cats' = Ca₂FeAl₂Si₃O₁₂ garnet

• grossular + 2 pyrope + 3 quartz = 3 enstatite + 3 anorthite or
• grossular + 2 almandine + 3 quartz = 3 ferrosilite + 3 anorthite.

A Detailed Example

Here is an example, using a rock from the Pamir of Tajikistan, of the result of an integrated approach to thermobarometry that uses phase-stability fields, exchange thermometry, net-transfer reactions, and garnet zoning to determine a P-T history.

Quantitative Geothermometry and Geobarometry

Clapeyron Relation

• _rG = V_rdP - S_rdT

• _rVdP = _rSdT

• =

Popular thermometers include garnet-biotite (GARB), garnet-clinopyroxene, garnet-hornblende, and clinopyroxene-orthopyroxene; all of these are based on the exchange of Fe and Mg, and are excellent thermometers because _rV is small, such that

• =

Calculating the PT Position of a Reaction Because rG = 0 at equilibrium, we can write the following approximation (ignoring fluids and variable heat capacities): 0 = rH1,Tref - TrS1,Tref + rVsP and thus we can calculate the pressure of a reaction at different temperatures by P = rH1,Tref - TrS1,Tref / -rVs and we can calculate the temperature of a reaction at different pressures by T = Tref + rH1,Tref + rVsP / rS1,Tref Let's do this for the albite = jadeite + quartz reaction at T = 400 K and T = 1000 K: P = (15,860 - 5147 * 1000) / -1.7342 = 20.6 kbar Assuming that dP/dT is constant (incorrect in detail), the reaction looks like this This is fine and dandy for determining P and T based on the appearance and disappearance of minerals, but how do we determine P and T based on changes in mineral composition?

Equilibrium Constant

• aA + bB = cC + dD

• K = (a_C^c a_D^d/a_A^a a_B^b)

• _rG° = -RT ln K

• ln K = -_rG°/RT

• ln K = - (_rH_1,Tref - T_rS_Tref + _rV_sP)/ RT

Let's see what K looks like for jadeite + quartz = albite at 800 K and 20 kbar:

• = -(15,860 - 800 * 51.47 + 1.7342 * 20,000)/(8.314*800) = -1.4

Solid Solutions

Activity Models (Activity-Composition Relations)

The simplest type of useful activity model is the ionic model, wherein we assume that mixing occurs on crystallographic sites. For a Mg-Fe-Ca-Mn garnet with mixing on one site, which we can idealize as (A,B,C,D)Al2Si3O12, the activities are aprp = Mg3XMg3 aalm = Fe3XFe3 agrs = Ca3XCa3 asps = Mn3XMn3 In general, for ideal mixing in a mineral with a single crystallographic site that can contain ions, ai = Xj where a, the activity of component i, is the mole fraction of element j raised to the power. Exchange Reactions Many thermometers are based on exchange reactions, which are reactions that exchange elements but preserve reactant and product phases. For example: Fe3Al2Si3O12 + KMg3AlSi3O10(OH)2 = Mg3Al2Si3O12 + KFe3AlSi3O10(OH)2 almandine + phlogopite = pyrope + annite We can reduce this reaction to a simple exchange vector: (FeMg)gar+1 = (FeMg)bio-1 Let's write the equilibrium constant for the GARB exchange reaction K = (aprpaann)/(aalmaphl) thus rG = -RT ln (aprpaann)/(aalmaphl) This equation implies that the activities of the Fe and Mg components of biotite and garnet are a function of Gibbs free energy change and thus are functions of pressure and temperature. If we assume ideal behavior ( = 1) in garnet and biotite and assume that there is mixing on only 1 site aalm = Xalm3 = [Fe/(Fe + Mg + Ca + Mn)]3 aprp = Xprp3 = [Mg/(Fe + Mg)]3 aann = Xann3 = [Fe/(Fe + Mg)]3 aphl = Xphl3 = [Mg/(Fe + Mg)]3 Thus the equilibrium constant is K = (XMggar XFebio)/(XFegar XMgbio) or K = (Xprp3 Xann3)/(Xalm3 Xphl3) Long before most of you were playground bullies (1978) a couple of deities named John Ferry and Frank Spear measured experimentally the distribution of Fe and Mg between biotite and garnet at 2 kbar and found the following relationship: If you compare their empirical equation ln K = -2109 / T + 0.782 this immediately reminds you of ln K = - (rG° / RT) = -(rH / RT) - (PrV / RT) + (rS / R) Molar volume measurements show that for this exchange reaction rV = 0.238 J/bar, thus rH = 52.11 kJ/mol The full equation is then 52,110 - 19.51*T(K) + 0.238*P(bar) + RT ln K = 0 To plot the K lines in PT space Net-Transfer Reactions Net-transfer reactions are those that cause phases to appear or disappear. Geobarometers are often based on net-transfer reactions because rV is large and relatively insensitive to temperature. A popular one is GASP: 3CaAl2Si2O8 = Ca3Al2Si3O12 + 2Al2SiO5 + SiO2 anorthite = grossular + kyanite + quartz which describes the high-pressure breakdown of anorthite. For this reaction rG = -RT ln [(aqtzaky2agrs) / aan3] = -RT ln agrs / aan3 (the activities of quartz and kyanite = 1 because they are pure phases). A best fit through the experimental data for this reaction by Andrea Koziol and Bob Newton yields P(bar) = 22.80 T(K) - 7317 for rV = -6.608 J/bar. Again, if we use ln K = -(rH / RT) - (PrV / RT) + (rS / R) and set ln K = 0 to calculate values at equilibrium, we can rewrite the above as (PrV / R) = -(rH / R) + (TrS / R) or P = TrS / rV - rH / rV if TrS / rV = 22.8 then rS = -150.66 J/mol K if rH / rV = 7317 then rH = -48.357 kJ/mol So, we can write the whole shmear as 0 = -48,357 + 150.66 T(K) -6.608 P (bar) + RT ln K Contours of ln K on a PT diagram for GASP look like this: A Complete Example Let's suppose you analyze a group of minerals that are in equilibrium and find the following compositions: garnet: Ca0.42Mg0.51Fe2.04Mn0.03Al2Si3O12 biotite: KMg1.62Fe1.38AlSi3O10(OH)2 plagioclase:Na0.64Ca0.36Al1.36Si2.64O8 kyanite quartz Let's determine the equilibrium P and T. Determine mole fractions Xgrs = 0.14 Xprp = 0.17 Xalm = 0.68 Xann = 0.46 Xphl = 0.54 Xan = 0.36 Determine activities, assuming ideal behavior agrs = 0.143 = 0.0027 aprp = 0.173 = 0.0049 aalm = 0.683 = 0.31 aann = 0.463 = 0.097 aphl = 0.543 = 0.16 aan = 0.36 Calculate equilibrium constants: KGASP = agrs / aan3 = 0.0025 / 0.363 = 0.054 KGARB = (aprpaann) / (aalmaphl) = (0.0049 * 0.097) / (0.31 * 0.16) = 0.0096 Calculate P and T: GASP: P (bar) = ( -48,357 + 150.66 * T(K) + RT ln K ) / 6.608 GARB: P (bar) = ( 52,110 - 19.51 * T(K) + RT ln K ) / -0.238 Choose T = 873 K: GASP P = 1.7 kbar GARB P = -7.7 kbar Choose T = 1073 K: GASP P = 13.2 kbar GARB P = 40.6 kbar These two lines intersect at ~955K and 10.9 kbar. Part 1. Overview of Metamorphism and Tectonics Part 2. Introduction to Metamorphism Part 3. Physical Processes of Metamorphism Part 4. Introductory Phase Equilibria and Thermodynamics Part 5. Ultramafic Rocks Part 6. Mafic Rocks Part 7. Pelitic Rocks Part 8. Diffusion Part 9. Thermobarometry Part 10. Kinetics Part 11. Interaction Between Metamorphism and Deformation Part 12. Metamorphism and Geochronology Part 13. Metamorphism and Tectonics I Part 14. Metamorphism and Tectonics II Thermodynamics Notes