Transcript 4550-15Lecture27x - Cornell Geological Sciences

Radiogenic
Isotope
Geochemistry II
Lecture 27
Beta Decay
Alpha & Gamma Decay
Basics of Radiogenic Isotope
Geochemistry
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•
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What makes radioactive decay useful to geochemists is that it occurs at
a rate that is constant and completely independent of external
influences*.
The probability that a nucleus will decay is expressed by the decay
constant, λ, which has units of inverse time and is unique to each
radioactive nuclide.
The rate of decay is given by the basic equation of radioactive decay:
dN
= -l N
dt
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•
where N is the number of radioactive nuclides. This is a first order rate
equation, like the ones we saw in kinetics. But unlike the rate constant, k,
of kinetics, λ is a true constant and independent of everything.
This is the only equation we need in radiogenic isotope geochemistry! .
Because we can derived a whole bunch of other equations from it. 
Basics of Radiogenic Isotope
Geochemistry
dN
= -l N
dt
• Rearrange and integrate:
• We start with:
• and get:
o Half-life:
ln
1
ln = - lt
2
N
= -lt
N0
t=
or
N
t
dN
òN N = ò0 - lt
0
N
N = N 0 e- l t
t
ln 2
l
• Our radionuclide will decay to a radiogenic
D
daughter so that D = N0 - N and
D = Nelt - N = N(elt -1)
• and usually there will have been some daughter
around to begin with, D0, so our equation is:
D = D0 + N(elt -1)
t
The Isochron Equation
D = D0 + N(elt -1)
o
(Aside: an approximation: for t<<1/λ, eλt = 1 + λt hence
D @ D0 + N lt
o
•
Essentially this says that for long-lived radionuclides, the growth of the daughter is a linear function
of t.)
It is more convenient to measure and work with ratios than with
absolute abundances. Consequently, we divide the abundance
of the radiogenic daughter by the abundance of a nonradiogenic daughter of that element, to form the ratio RD. Our
equation becomes:
RD = R0 + RP/D (elt -1)
•
where RP/D is the parent/daughter ratio. Hence for the Rb-Sr
system, we have:
87
87
87
Sr æ Sr ö
Rb l87t
=
+
(e -1)
86
Sr çè 86 Sr ÷ø 0 86 Sr
•
For reasons that will become apparent, we call this the Isochron
Equation.
Geochronology
RD = R0 + RP/D (elt -1)
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It is the constant rate of decay of the parent that gives rise to the
time dependence, but since it easier to measure what is there
rather than what is not, we tell time by measuring the
accumulation of the daughter.
The above equation tell us that a radiogenic isotope ratio is a
function of:
o
o
o
o
The decay constant, which we can determine experimentally (though not without uncertainty).
The parent daughter ratio (e.g., 87Rb/86Sr)
The initial ratio, R0, the radiogenic isotope ratio at t = 0
Time.
o
We can measure the radiogenic isotope ratio, the parent/daughter ratio and the decay constant,
but we still have two unknowns. How do we proceed?
In geochronology, we want to know t, time.
We measure these parameters in two or more samples for which t
and the initial ratio are the same. With two equations, we can
solve for both.
o
Measurement is by mass spectrometry. In the mass spectrometer, different isotopes of the same
element behave somewhat differently. This produces isotopic fractionation that degrade our
measurements. Fortunately, in most cases we can correct for this by measuring the ratio of a pair
of non-radiogenic isotopes, e.g., 84Sr and 88Sr, compare the measured ratio to the ‘known’ value
and apply a correction to the ratio of interest., e.g., 87Sr/86Sr. Since fractionation depends on mass,
the fractionation of the 87Sr/86Sr ratio will be half that of the 86Sr/88Sr ratio.
Isochrons
RD = R0 + RP/D (elt -1)
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This equation has the form y = a + bx.
Plotting the radiogenic isotope ratio
against the parent daughter ratio, the
intercept with be R0 and the slope will
be eλt -1.
Since the slope is a function of time, it
is called an isochron.
We determine the slope statistically
using linear regression (a more
sophisticated form than usually used).
From the slope, we easily solve for t:
t=
•
æ ∆R
ö
ln ç
+ 1÷
è ∆ RP/D ø
l
Most geochronology is based on this
approach.
Rb-Sr isochron for a
achondritic meteorite.
Individual points are
individual minerals. t =
4.57±0.02 Ga.
Isochrons
RD = R0 + RP/D (elt -1)
•
Assumptions inherent in
geochronology:
o
o
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All analyzed samples forming part of the
isochron had the same radiogenic
isotope ratio at t = 0. (System was in
isotopic equilibrium). This is most likely to
occur as a result of a thermal event
(allowing for high diffusion rates) - so
dating is restricted it igneous and
metamorphic events.
The system (and analyzed ‘subsystems’
such as minerals) remained closed to loss
or gain of both parent and daughter
since time 0.
Accuracy and precision of
the age depends on
o
o
o
Fit of the isochron (analytical, geological
disturbance).
Range of isotope (and parent-daughter)
ratios - the greater the spread, the more
accurate the age.
Uncertainty in decay constant.
Rb-Sr isochron for a
achondritic meteorite.
Individual points are
individual minerals. t =
4.57±0.02 Ga.
Isotope Geochemistry
87
Sr æ 87 Sr ö
Rb
= ç 86 ÷ + 86 (el87t -1)
86
Sr è Sr ø 0
Sr
87
o
The best summary statement of isotope geochemistry was given by Paul Gast in a
1960 paper:
• In a given chemical system the isotopic abundance of 87Sr is
determined by four parameters: the isotopic abundance at a
given initial time, the Rb/Sr ratio of the system, the decay
constant of 87Rb, and the time elapsed since the initial time.
The isotopic composition of a particular sample of strontium,
whose history may or may not be known, may be the result of
time spent in a number of such systems or environments. In
any case the isotopic composition is the time-integrated result
of the Rb/Sr ratios in all the past environments. Local
differences in the Rb/Sr will, in time, result in local differences in
the abundance of 87Sr. Mixing of material during processes will
tend to homogenize these local variations. Once
homogenization occurs, the isotopic composition is not further
affected by these processes.
o
This statement applies to other decay systems, many of which were ‘developed’ well
after 1960.
Decay Systems of Geologic
Interest
Parent
Decay mode
 , yr-1
Half-life, yr
Daughter
Ratio
Parent
40K
+, e.c., -
5.549 × 10-10
1.25 × 109
40Ar, 40Ca
40Ar/36Ar
40K
87Rb
-
1.42 × 10-11
4.88 × 1010
87Sr
87Sr/86Sr
87Rb
138La
-
2.67 × 10-12
2.60 × 1011
138Ce
138Ce/142Ce
138La
147Sm

6.54 × 10-12
1.06 × 1011
143Nd
176Lu
-
1.867 × 10-11
3.71 × 1010
187Re
-
1.64 × 10-11
232Th

235U

238U
 s.f.
,
138Ce/136Ce
143Nd/144Nd
147Sm
176Hf
176Hf/177Hf
176Lu
4.23 × 1010
187Os
187Os/188Os
187Re
4.948 × 10-11
1.40 × 1010
208Pb, 4He
208Pb/204Pb,
232Th
9.849 × 10-10
7.04 × 108
207Pb, 4He
207Pb/204Pb,
235U
1.5513 × 10-10
4.47 × 109
206Pb, 4He
206Pb/204Pb,
238U
3He/4He
3He/4He
3He/4He
Decay Systems
The Rb-Sr System
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Both elements incompatible
(Rb more so than Sr).
Both soluble and therefore
mobile (Rb more so than Sr).
Range of Rb/Sr is large,
particularly in crustal rocks
(good for geochronology).
Subject to disturbance by
metamorphism and
weathering.
Both elements concentrated
in crust relative to mantle - Rb
more so than Sr.
87Sr/86Sr evolves to high values
in the crust, low ones in the
mantle.
Sr Isotope
Chronostratigraphy
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We can’t generally radiometricly
date sedimentary rocks, but there is
an exception of sorts.
87Sr/86Sr has evolved very non-linearly
in seawater. This is because the
residence time of Sr in seawater is
short compared to 87Rb half-life, so
87Sr/86Sr is controlled by the relative
fluxes of Sr to the oceans:
o
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Rivers and dust from the continents
The mantle, via oceanic crust and hydrothermal
systems.
Changes in these fluxes result in
changes in 87Sr/86Sr over time.
Sr is concentrated in carbonates
precipitated from seawater. By
comparing the 87Sr/86Sr of carbonates
with the evolution curve, an age can
be assigned.
This quite accurate in the Tertiary
(and widely used by oil companies),
less so in earlier times.
The Sm-Nd System
147Sm
alpha decays to 143Nd with a half-life of 106 billion years.
Both are rare earths and behave similarly.
In addition, 146Sm decays to 142Nd with a half-life of 68 103 million
years. As a consequence of its short half-life, 146Sm no longer exists
in the solar system or the Earth. But it once did, and this provides
some interesting insights.
Sm-Nd and εNd
• Because Sm and Nd, like all rare earths, are refractory
lithophile elements, and because their relative abundances
vary little in chondritic meteorites, it is reasonable to suppose
that the Sm/Nd ratio of the Earth is the same as chondrites.
• This leads to a notation of 143Nd/144Nd ratios relative to the
chondritic value, εNd:
e Nd
æ 143 Nd / 144 Ndsample - 143 Nd / 144 Ndchondrites ö
=ç
143
144
÷ ´10000
Nd
/
Nd
è
ø
chondrites
• While we usually use present-day values in this equation, we
can calculate εNd (t) for any time, using the appropriate values
for that time.
• There are several advantages:
o
o
o
ε values are generally numbers between ~+10 and -20.
If the Earth has chondritic Sm/Nd, then the 143Nd/144Nd of the Earth is chondritic and
εNd of the bulk Earth is 0 both today and at any time in the past.
The 142Nd/144Nd of the modern observable Earth differs from chondrites slightly (by
about 20 ppm), which raises the question of whether the Earth’s Sm/Nd ratio is in fact
exactly chondritic. The notation survives, however.
Sm-Nd Evolution of the
Earth
•
Sm and Nd are incompatible
elements (Nd more so that Sm).
o
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By converting to εNd, our
evolution diagram rotates such
that a chondritic uniform
reservoir always evolves
horizontally (εNd always 0).
o
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Consequently, the crust evolves to low
143Nd/144Nd while the mantle evolves to high
143Nd/144Nd.
The mantle evolves to positive εNd, the crust
to negative εNd.
Both Sm and Nd are insoluble
and not very mobile, so it is in
many ways a more robust
chronometer than Rb-Sr.
Unfortunately, the range in
Sm/Nd ratios in crustal rocks is
usually small, limiting the use of
the system for geochronology.
Sm-Nd model ages or
‘Crustal Residence Times”
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A relatively large fractionation of Sm/Nd is
involved in crust formation. But after a
crustal rock is formed, its Sm/Nd ratio tends
not to change.
This leads to another useful concept, the
model age or crustal residence time. From
143Nd/144Nd and 147Sm/144Nd, we can
estimate the “age” or crustal residence
time, i.e., the time the rock has spent in the
crust. We assume:
o
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o
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We
along a line whose slope corresponds to
the measured present 147Sm/144Nd ratio until
it intersects the chondritic growth line. The
model age is this age at which these lines
intersect:
o
o
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The crustal rock or its precursor was derived from the
mantle
The 147Sm/144Nd of the crustal rock did not change.
We know how the mantle evolved.
project the 143Nd/144Nd ratio back
CHUR model age (τCHUR).
Depleted mantle model age is used (τDM).
In either case, the model age is calculated
by extrapolating the 143Nd/144Nd ratio back
to the intersection with the mantle growth
curve.
Our isochron equation was:
147
147
æ 143 Nd ö
Nd æ 143 Nd ö
Sm lt
Sm
= ç 144 ÷ + 144 (e -1) @ ç 144 ÷ + 144 l t
144
Nd è Nd ø 0
Nd
Nd
è Nd ø 0
143
If we plot the radiogenic isotope ratio
against t, then the slope is RP/Dλ.
(note that x-axis label should be ‘age’,
not t, in the sense of the equation).
Model Age Calculations
• To calculate the model
age, we note that the
point where the lines
intersect is the point
where (143Nd/144Nd)0 of
both the crustal rock and
the mantle (CHUR or DM)
are equal.
• We write both growth
equations and set the
(143Nd/144Nd)0 values
equal, then solve for t.
• See Example 8.3.
Sr-Nd Systematics of the
Earth