N = N 0 e -λt D* = N 0 -N D* = N 0 (1-e -λt ) or N(e λt -1) where N is number of parent atoms at time t, N 0 is initial number of parents, D* is number of radiogenic daughter atoms, and λ is the decay constant. D Total = D 0 + D* D* = N(e λt -1) D Total = D 0 + N(e λt -1) ISOCHRON EQUATION where D TOTAL is number of daughter atoms at time t, D 0 is initial number of daughters. For ease of measurement and reporting, Isochrons equations are normalized to a non-radiogenic isotope of the same element as the daughter. 87 Rb to 87 Sr, normalized to 86 Sr 147 Sm to 143 Nd normalized to 144 Nd 235 U to 207 Pb normalized to 204 Pb 238 U to 206 Pb normalized to 204 Pb
Key is to have variation in [Rb] while all samples have exactly the same 87 Sr/ 86 Sr 0. In igneous and metamorphic processes, isotopes of the same element move together, but different chemical elements do not. Get differential ingrowth of radiogenic parent isotope dependent on initial [Rb]. Follows paths of -1. D Total = D 0 + N(e λt -1) Isochron Equation Y = b + xm b = D 0 m = e λt -1 or t = ln(m+1)/λ
In some situations, can see through a phase of metamorphism. R minerals have high blocking temperature, don t reset. M minerals have lower blocking temperature, reset. Assumptions 1. Number of parent and daughter isotopes changes only by decay. 2. No fractionation of isotopes occurred at time of rock formation (isotopes move together). 3. You know λ. 4. The line being called an Isochron is not, in fact, a mixing line between two pools that formed at different times. I.e., you must use geochemical and petrologic logic to infer that the samples all formed at the same time. Volcanic rocks are best for this. 5. Measurement error is small relative to slopes.
U-Pb system is unique, in having two different parent isotopes of the same element that decay to isotopes of the same daughter. Assuming U isotopes travel together (i.e., are well mixed), this allows redundancy in age calculations and several unique approaches to Geochronology. Divide one equation by the other. Get an equation for Common Lead Dating. 1. Left side of equation is the slope of a radiogenic 207 Pb/ 206 Pb isochron in 207 Pb/ 204 Pb (y axis) vs. 206 Pb/ 204 Pb (x-axis) space. 2. Initial Pb must be known/modeled or you must chose minerals that have essentially no initial Pb (zircon, apatite, monazite). 3. 235 U/ 238 U known today (1/137.88) and invariant across most of planet. 4. Still a problem, the equation: e slope = 1/137.88( λ2t -1 ) e λ1t -1 cannot be solved analytically. Must be solved by chosing a t based on the slope, and then iterating several times until the answer approaches slope.
Even Zircon is not completely retentive of Pb and U. How can we date systems that have been altered? Concordia vs. Discordia George Wetherill (1925-2006) Concordia vs. Discorida All U-Pb bearing minerals on Earth that have remained closed to U or Pb loss or gain fall on the curved line. Loss of Pb forms a chord heading back to origin. Upper intersection with Concordia is age of crystallization
With addition of radiogenic Pb, Concordia grows away from the origin. Discordia rotates with Concordia. Can get age of metamorphism from lower intercept. This assume instantaneous (not continuous) Pb loss. Radioactive decay of U to Pb damages crystals. This damage leads to pathways for fluid flow and Pb migration. Can use separation methods to obtain differentially damaged crystals to: a) Find the best, most concordant crystals. b) Get the most spread in Discordia so that the upper intercept calculation is robust
Hadean = Eoarchean Oldest Minerals on Earth
Oldest Rocks on Earth? Extinct isotope (146Sm-142Nd) Standard isochron Cratonic Rocks
Archean Tectonics 1. Continents grow by accretion at their edges. 2. 80% of land mass formed in last 2.5x109 yrs. Crustal Growth Models
Mantle evolution and Crustal Extraction 0.697 Calculations of extraction age for lots of cratonic rock suggest rapid growth from 3.0 to 2.0x10 9 years ago. Seawater Strontium Evolution More continental weathering Less continental weathering Controlled by the balance between weathering of the craton (radiogenic) and weathering of seafloor basalt (which is similar to the mantle).
Formation of Continental and Oceanic Crust Crust: chemical definition. More Si rich. Less dense. Oceanic crust: ~10 km Continental crust: up to 70 km Lithosphere: rheologic definition. Outer, ridid part of Earth, consisting of crust and upper mantle; lies above asthenosphere. Base of crust is the Mohorovičić discontinuity Peridotite Basalt Granite Melting T (@ 1 bar) 1200 C 1250 C 700 C density 3.3 gm/cm3 3.0 gm/cm3 2.7 gm/cm3 SiO2 45 wt% 50 wt% 65 wt% Al2O3 3 wt% 11 wt% 15 wt% MgO 40 wt% 5 to 7 wt% 2.5 wt% FeO 8 wt% 10 wt% 5 wt%
Mantle melting temperature depends on volatiles Si-rich melts Si-poor melts Solidus: at equilibrium, line separating entirely solid phase from liquid/solid mixures. Liquidus: at equilibrium, line separating entirely liquid phase from liquid/solid mixtures. Mantle melting temperature depends on volatiles Dry mantle intersects suboceanic geothermal gradient. Produces a tiny bit of partial melt at high T (1250 C) and P/depth that rises and crystallizes to form gabbro and basalt.
Mantle melting temperature depends on volatiles Only wet mantle intersects the subcontinental geothermal gradient. Produces andesitic melts at lower T (700 c) and P/depth. Where does the water come from? Hydrated basalt = Serpentine Mg 3 Si 2 O 5 (OH) 4 Converts to eclogite: Mg 2 SiO 4 (olivine) + MgSiO 3 (enstatite)+ 2H 2 O
Oceans Serpentine Subduction Andesites Granites No water, no andesite/granite No granite, no continents No continents, no plate tectonics Continental growth curve indicates: Oceans way back into the Hadean By 3.0 to 2.0x10 9 years, certainly had subduction. Higher heat flow in the past 1. Higher production. Extinct isotopes 2. Heat of accretion/moon forming impact
Change in style of mafic/ultramafic volcanism Greenstone Belts: Small fragments More subduction Faster crust generation More hydrous alteration Why is all this hot stuff sinking?
Greenstone Ultramafic/Amphibolite Komatiite: 47 wt% SiO 2, 20 wt% MgO 3.2 gm/cm 3, 1600 C formation temperature 40 to 50% partial melting of peridotite/mantle Basalt: 50 wt% SiO 2, 5 to 7 wt% MgO 3.0 gm/cm 3, 1250 C formation temperature 20 partial melting of peridotite/mantle Dry Archean mantle (diff % partial melting) Modern mantle dry (right) vs. wet (left) Archean cooling/rising path Modern suboceanic geothermal gradient Modern path for rising magmas