12.480 Handout #6: Thermometry-Barometry Using Pyroxenes

12.480 Handout #6: Thermometry-Barometry Using Pyrox­ enes Reading Lindsley et al. (1981) Adv. in Physical Geochem., 149-175. Davidson and Lindsley (1985) Contrib. Min. Pet., 91:383-404. Supplementary Reading Lindsley and Andersen (1983) J. Geophys. Res. 88 (1983): A877-A906. Prewitt, Charles T. (ed.) Pyroxenes (Reviews in Mineralogy, Vol. 7). Washington, D. C.: Miner­ alogical Society of America, 1980. ISBN: 0939950146 This endeavor got its start thanks to an elegantly presented paper by Boyd (1973) GCA37 ­ 2533-2546. Boyd’s suggestion was that two equilibria be used to estimate the T&P of equilibration of garnet lherzolites. Boyd proposed that coexisting cpx-opx be used to infer temperature, and that exchange equilibria involving garnet-opx be used to infer pressure. We’ve come a long way from Boyd’s initial proposal to the thermometry calculations of today. Today we’ll look at progress and pitfalls encountered in this endeavor. opx-cpx — 2 phases with 2 associated G curves. Correct formulation treats each phase as a distinct solution. µ oDiopx o cpx µ En G o cpx µDi o opx µ En En ss En Di ss En + Di Di X At equilibrium: opx rxn µcpx En − µEn = ΔGEn = 0 opx rxn µcpx Di − µDi = ΔGDi = 0 1 We could also write an exchange reaction: Encpx + Diopx → Dicpx + Enopx ΔGexch = ΔGDi − ΔGEn opx opx cpx = µcpx Di + µEn − µDi − µEn Now we need to develop a realistic and workable solution model. We have at equilibrium: o,cpx µo,opx = RT ln En − µEn ΔGoEn = RT ln cpx cpx XEn γEn opx + RT ln opx XEn γEn cpx cpx XEn γEn opx + RT ln opx XEn γEn Now we may plug in what we know and solve for T . cpx XEn opx XEn ΔGoEn ΔGoDi cpx γDi cpx γEn ← asymmetric opx γDi opx γEn ← symmetric This is what Lindsley et al. do, using experimental data in Di − En. Note — This is the right way of doing things — other ways Assumption of Wood & Banno (1973) CMP 42 109-124 was that: activity coefficient ratio = 1 cpx γM g2 Si2 O6 opx γM g2 Si2 O6 =1 didn’t consider constraints from CaM gSi2 O6 . Warner & Luth (1973) Am. Min. 58, 998- assume opx-cpx obey single equation of state and used WGs as fitting parameters. Binary system - two phases equilibrium conditions: M g2 Si2 O6opx � M g2 Si2 O6cpx CaM gSi2 O6opx � CaM gSi2 O6cpx cpx µopx En − µEn = 0; opx cpx µDi − µDi =0 XDi dµDi + XEn dµEn = 0 Gibbs-Duhem relates Di & En components at equilibrium. Δµo cpx cpx � �� � XEn γEn cpx o,cpx o,opx µEn − µopx = µ − µ +RT ln + RT ln =0 opx opx En En En XEn γEn 2 cpx o,cpx o,opx µDi − µopx − µDi + RT ln Di = µDi cpx cpx XDi γDi opx + RT ln opx XDi γDi Use symmetric solution model for opx and asymmetric for cpx. o,cpx µo,opx = RT ln En − µEn cpx XEn opx opx 2 cpx cpx 2 cpx cpx 2 opx − WG (XDi ) + 2WG1 XEn (XDi ) + WG2 (1 − 2XEn ) (XDi ) XEn o,opx o,cpx µDi − µDi = RT ln cpx XDi opx opx 2 cpx cpx 2 cpx cpx 2 opx − WG (XEn ) + WG1 (1 − 2XDi ) (XEn ) + 2WG2 XDi (XEn ) XDi Lindsley, Grover, Davidson model: opx opx opx opx GXS, opx = WGopx XEn XDi = 25XEn XDi 2 2 cpx cpx cpx cpx GXS, cpx = WG1 XEn (XDi ) + WG2 XDi (XEn ) ↑ ↑ (31.216 − .0061P ) (25.484 + .0812P ) Calculation of temperature from opx-cpx pairs (binary system CaM gSi2 O6 − M g2 Si2 O6 ). Note: these two expressions should give you the same T . � cpx cpx 2 T ◦ K (M g2 Si2 O6opx � M g2 Si2 O6cpx ) = 3.561 + .0355P + 2WG1 XEn (XDi ) 2 2 cpx cpx opx +WG2 (XDi ) (1 − 2XEn ) − WGopx (XDi ) � � � X cpx / .0091 − R ln En opx XEn � cpx 2 cpx T ◦ K (CaM gSi2 O6cpx � CaM gSi2 O6cpx ) = −21.178 − .0908P + WG1 (XEn ) (1 − 2XDi ) 2 2 cpx cpx opx +2WG2 XDi (XEn ) − WGopx (XEn ) � � � X cpx / −.00816 − R ln Di opx XDi So, we’ve done the pure system Di − En, but the pyroxenes we want to study are almost always F e − Ca − M g solid solutions (at least!) — also Al, T i, Cr, N a, F e3+ etc. How do we generalize to these complex phases? 1. We must generalize activity approx. to multi-stite phase where the species can be ordered on sites. 2. We must generalize such an expression for two component systems to complex systems. This has been done by Davidson and Lindsley (1985). There are also several graphical/empirical thermometers for pyx solns. that can be used. 3