1. Radiogenic Isotope Geochemistry II
Lecture 27

2. Beta Decay

3. Alpha & Gamma Decay

4. Basics of Radiogenic Isotope Geochemistry
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:
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. 

5. Basics of Radiogenic Isotope Geochemistry
We start with:
Rearrange and integrate:
and get: or
Half-life:
Our radionuclide will decay to a radiogenic daughter so that D = N0 - N and
and usually there will have been some daughter around to begin with, D0, so our equation is: D t

6. The Isochron Equation
(Aside: an approximation: for t<<1/λ, eλt = 1 + λt hence
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 non-radiogenic daughter of that element, to form the ratio RD. Our equation becomes:
where RP/D is the parent/daughter ratio. Hence for the Rb-Sr system, we have:
For reasons that will become apparent, we call this the Isochron Equation.

7. Geochronology
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:
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.
In geochronology, we want to know t, time.
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?
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.
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.

8. Isochrons
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:
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.

9. Isochrons Assumptions inherent in geochronology:
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
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.

10. Isotope Geochemistry
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.
This statement applies to other decay systems, many of which were ‘developed’ well after 1960.

11. Decay Systems of Geologic Interest
Parent
Decay mode
λ, yr-1
Half-life, yr
Daughter
Ratio
40K
β+, e.c., β-
5.549 × 10-10 
1.25 × 109 
40Ar, 40Ca
40Ar/36Ar
87Rb β-
1.42 × 10-11
4.88 × 1010 
87Sr
87Sr/86Sr
138La
2.67 × 10-12
2.60 × 1011 
138Ce
138Ce/142Ce , 138Ce/136Ce
147Sm α
6.54 × 10-12
1.06 × 1011 
143Nd
143Nd/144Nd
176Lu
1.867 × 10-11
3.71 × 1010 
176Hf
176Hf/177Hf
187Re
1.64 × 10-11
4.23 × 1010 
187Os
187Os/188Os
232Th
4.948 × 10-11
1.40 × 1010 
208Pb, 4He
208Pb/204Pb, 3He/4He
235U
9.849 × 10-10
7.04 × 108 
207Pb, 4He
207Pb/204Pb, 3He/4He
238U
α, s.f.
× 10-10
4.47 × 109 
206Pb, 4He
206Pb/204Pb, 3He/4He

12. Decay Systems

13. The Rb-Sr System 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.

14. Sr Isotope Chronostratigraphy
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:
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.

15. 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 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.

16. 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:
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:
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.

17. Sm-Nd Evolution of the Earth
Sm and Nd are incompatible elements (Nd more so that Sm).
Consequently, the crust evolves to low 143Nd/144Nd while the mantle evolves to high 143Nd/144Nd.
By converting to εNd, our evolution diagram rotates such that a chondritic uniform reservoir always evolves horizontally (εNd always 0).
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.

18. Sm-Nd model ages or ‘Crustal Residence Times”
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:
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.
We project the 143Nd/144Nd ratio back 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:
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:
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).

19. 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.

20. Sr-Nd Systematics of the Earth