6.4 Garnet-Bioite Mg-Fe Exchange Thermometer

One of the first experimentally calibrated geothermometers is based on Mg-Fe exchange between garnet (Grt) and biotite (Bt). Figure 6.07 shows Grt and Bt on a Thompson AFM projection. Red tie lines connect compositions of Grt and Bt in equilibrium. Click on the diagram to see a larger version and use your mouse to move the temperature slider. Modes and mineral compositions are shown below the diagram for the rock bulk composition, which is marked by a red dot. As you change the temperature with the slider, you will see the tie lines rotate and the Mg/(Mg+Fe) values for both Grt and Bt change.

John Ferry and Frank Spear (1978) equlibrated samples of Mg-Fe garnet and Mg-Fe biotite in the laboratory at 0.2 GPa to calibrate a geothermometer that uses Mg-Fe exchange between Grt and Bt, formally described by the following reaction:

[Fe₃Al₂(Si₃O₁₂)]^Grt + [KMg₃(AlSi₃O₁₀)(OH)2]^Bt =
[Mg₃Al₂(Si₃O₁₂)]^Grt + [KFe₃(AlSi₃O₁₀)(OH)2]^Bt (1)

Using the results of their experiments and thermodynamic data for the Fe and Mg end members of garnet and biotite they obtained the following equation relating Grt and Bt compositions to temperature and pressure:

52112 -19.51*T + 0.238*P + 3*R*ln(K_D) = 0 (2)

where T is temperature (K), P is pressure (bars), R is the gas constant (8.3145 J/K), and K_D is an exchange coefficient defined in terms of the Grt and Bt compositions:

Figure 6.08. Garnet-Biotite Fe-Mg Equilibrium Relations. Click on the image to see how the tie lines change with temperature and pressure on a larger animated version.

K_D = (Mg/Fe)^Grt/(Mg/Fe)^Bt (3)

in mole units. Figure 6.08 shows graphically the geothermometer, equation (2), with sliders to change temperature, pressure, and the exchange coefficient. Click on Figure 08 and move the sliders. Notice that the tie lines change more with T than with P. This is because there is little change of volume caused by the Mg-Fe exchange between Grt and Bt.

Figure 6.09. Grt-Bi Thermometer The red line shows the temperatures for K_D of 0.1. Click on the image to see a larger animated version.

Figure 6.09 shows the Ferry-Spear (1978) Grt-Bt geothermometer, equation (2), on a T-P diagram. A red line gives the possible temperatures and pressures for a K_D of 0.1. Click on the diagram to see a larger version with a slider to change K_D, either directly or by changing the Grt and Bt compositions. Notice that the red line is nearly verical, showing that the temperatures for a specific K_D do not change much with pressure (around 40°C per GPa). Notice also that small changes in the mineral compositions may lead to large changes in the calculated temperature. Other factors that increase the uncertainty of geothermeter temperatures will be discussed in the coming pages.

What about pressure? Are there ways to determine equilibrium pressures from mineral composition?

GASP Geobarometer