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N = N 0 e - λt
D* = N0 -N
D* = N0(1-e-λt) or N(eλt-1)
where N is number of parent atoms at time t, N0 is initial number of parents, D* is number
of radiogenic daughter atoms, and λ is the decay constant.
DTotal = D0 + D*
D* = N(eλt-1)
DTotal = D0 + N(eλt -1) ISOCHRON EQUATION
where DTOTAL is number of daughter atoms at time t, D0 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.
87Rb
to 87Sr, normalized to 86Sr
147Sm to 143Nd normalized to 144Nd
235U to 207Pb normalized to 204Pb
238U to 206Pb normalized to 204Pb
Key is to have variation in
[Rb] while all samples
have exactly the same
87Sr/86Sr .
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.
DTotal = D0 + N(eλt-1) Isochron Equation
Y
= b + xm
b = D0
m = eλt-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 207Pb/206Pb isochron
in 207Pb/204Pb (y axis) vs. 206Pb/204Pb (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. 235U/238U known today (1/137.88) and invariant across most of planet.
4. Still a problem, the equation:
slope = 1/137.88(
eλ2t-1
)
eλ1t-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 2.0x109 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 Mg3Si2O5(OH)4
Converts to eclogite: Mg2SiO4 (olivine) + MgSiO3 (enstatite)+ 2H2O
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 2.0x109 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% SiO2, 20 wt% MgO
3.2 gm/cm3, 1600°C formation temperature
40 to 50% partial melting of peridotite/mantle
Basalt: 50 wt% SiO2, 5 to 7 wt% MgO
3.0 gm/cm3, 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