1. Chapter 2 K-Ar, Rb-Sr, Sm-Nd, Lu-Hf and Re-Os
Geochronology I
Chapter 2
K-Ar, Rb-Sr, Sm-Nd, Lu-Hf and Re-Os

2. Figure 2.1

3. Radioactive Decay
Unstable nuclei decay to stable ones at rates independent of all environmental influences.
Modes
Beta
electron or positron emission plus neutrino
Electron capture
functionally equivalent to positron emission - neutrino emitted; x-ray also emitted
Alpha
emission of 4He nucleus
Fission
rare - effectively only 238U
Gamma ray emission (energetic photon) generally accompanies all of the above
Radioactive decay obeys all conservation laws: mass-energy, spin, momentum.
Mass of daughter and particles plus energy released must sum to mass of parent.
Radioactive decay is a major source of the Earth’s internal energy driving plate tectonics and other geologic processes.

4. Decay Systems
Geochronologists have a variety of decay systems to work with. Each is in some ways unique. Success in dating depends on choosing the right tool. Factors to consider:
Half-life: shorter half-live better for younger samples
Chemical behavior of parent and daughter: specifically
abundance
fractionation of parent and daughter in different minerals
mobility of parent and daughter
‘closure temperature’ of system: at what temperature is the clock reset?

6. Basic Equations
Basic Equation
Rearrange and integrate:
Half-life:

7. Case where some daughter already present
D, daughter; D* =N0 – N, radiogenically produced daughter
Since some daughter is initially present (D0):
Its more convenient to work with ratios than absolute numbers, so we divide by the amount of another, non- radioactive, non-radiogenic isotope. For Sr:

8. Isochron Dating In the general case:
we have two unknowns, R0 and t (we measure R and R0).
We need two equations. If we have two samples with (presumably) identical R0 and t, subtracting one from the other, we have:
Rearranging: or

9. Isochrons
From equ. 2.20, we see that t is proportional to the slope on a plot of R vs. RP/D.
Looking at equation 2.19 with this in mind:
we see that it has the form y=a +xb, where a is the intercept and b the slope.
So we can determine the age by determining the slope through a set of cogenetic samples.
This line is called an isochron.

10. Assumptions
We are assuming each sample analyzed has the same value of R0 and t.
In other words, they are cogenetic: they formed at the same time with the same isotope ratio at the time.
The latter requires isotopic homogeneity (isotope equilibrium). Typically achieved by diffusion (± convective mixing), requiring elevated temperature. Hence we are usually dating thermal events.
We also assume the only change in R and RP/D is due to radioactive decay. In other words, we assume the system has remained closed, no migration of parent or daughter in or out (we’ll see some ‘work-arounds’ for this).

11. Additional Considerations
In addition to a closed system that was initially isotopically homogenized, an accurate date requires:
Large amount of radiogenic isotope to have been produced (essentially requiring a large parent/daughter ratio)
For isochron dating, we want a large range of parent- daughter ratios, which minimizes the uncertainly on the isochron.

12. Determining the slope
Relations between observations are commonly determined using regression (commonly included in calculators, Excel, etc.).
Classical regression assumes x values are known absolutely – not the case with analytical data.
We should take errors in both x and y into account in computing our slope – this is done by weighting each point inversely by its associated analytical error.
Known as two-error regression, mathematically a bit more complicated and requires iteration.
In practice, many geochronologists use the Isoplot Excel add-in from the Berkeley Geochronology Laboratory (unfortunately, latest version runs only on Excel for Windows).

13. The K-Ca-Ar System
40K is can decay to either 40Ca (by β– decay) or 40Ar by electron capture (or more rarely β+ decay).
The ratio of electron captures to beta decays is called the branching ratio and is defined as:
Total decay constant, sum of these two, is 5.5 x yr-1 corresponding to a half-life of ~1.28 Ga.
Most (~90%) decays to 40Ca, but 40Ca is doubly magic and a very abundant nuclide. Thus the radiogenic fraction is small. 40Ar, on the other hand is a rare gas and fairly rare (on the Earth anyway). Thus mostly we are interested in the decay to 40Ar.
The usually large K/Ar ratio in rocks and the relatively short half-life makes this a good choice for many dating applications, particularly young events.
Because Ar is a gas and mobile (K also readily mobilized), the system is readily reset. This can be a good thing for dating low temperature events.

14. K-Ar Dating Our relevant decay equation is:
where λis the total decay constant.
Most rocks have little Ar; lavas, for example, almost completely degas. What little Ar is present is generally adsorbed atmospheric Ar, whose isotopic composition is well known. Thus our equation becomes:
296.16

15. Diffusion, Cooling and Closure Temperature
We mentioned radiogenic chronometers are generally reset by thermal events.
This occurs when diffusion is sufficiently rapid to isotopically homogenize our system.
Or, the case of K-Ar, Ar is able to diffuse out of the rock.
The temperature at which the chronometer is reset is known as the closure temperature. It differs for each decay system, mineral, and, as we’ll see, cooling rate.
Let’s first consider diffusion.

16. Temperature Dependence of Diffusion
The diffusio flux is given by Fick’s first law:
where D is the diffusion coefficient and ∂C/∂x the concentration gradient.
Diffusion coefficient in solids depends on temperature according to:
EA is the activation energy and D is the frequency factor.
We can determine these by making measurements at multiple temperatures, taking the log of the above equation, then plotting up the results.
Ar in biotite
Figure 2.2

17. Closure Temperature
Using the data in the previous figure, we would find there is no significant loss of Ar at 300˚C even on geologic time scales. At 600˚C, loss would be small but significant. At 700˚C, about 1/3 of Ar would be lost from a 100 µ biotite in 2 to 3 weeks!
If the rock cools rapidly from 700˚C, it will quickly close. If it cools slowly, closure will come much later.
Think about Ar in a cooling intrusive igneous or metamorphic rock. Unlike a lava, cooling will occur on geologic time scales. At first, most Ar is lost, but as the rock cools, loss slows.
What is the closure temperature?

18. Diffusion Calculations
To determine the distribution of a diffusion species with time c(x,t), we use Fick’s Second Law:
Solutions depend on circumstances. Easy way to solve it is to look in Crank (1975) who gives this equation for diffusion out of a cyclinder of radius a
where ƒ is the fraction lost

19. Ar loss from biotite
Figure Fraction of Ar lost from a 150 µ cylindrical crystal as a function of temperature for various heating times. All Ar is lost in 10 Ma at 340°C, or in 1 Ma at 380° C.

20. Dodson’s Closure Temperature
Dodson (1973) derived an equation for ‘closure temperature’ (also sometimes called blocking temperature) as a function of diffusion parameters, grain size and shape, and cooling rate:
where Tc is the closure temperature, τ is the cooling rate, dT/dt (for cooling, this term will be negative), a is the characteristic diffusion dimension (e.g., radius of a spherical grain), and A is a geometric factor (equal to 55 for a sphere, 27 for a cylinder, and 9 for a sheet) and temperatures are in Kelvins.
Unfortunately, this is not directly solvable since Tc occurs both in and out of the log, but it can be solved by indirect methods (MatLab, Solver in Excel).

21. 40Ar–39Ar Dating
You might wonder what this is all about. 39Ar has a 269 yr half-life and does not occur naturally.
40Ar–39Ar dating is simply a specific analytical technique for 40K–40Ar dating.
The sample is irradiated with neutrons in a reactor and 39Ar is created from 39K by: 39K(n,p)39Ar.
Since the amount of 39Ar is proportional to the amount of 39K and that is in turn proportional to the amount of 40K, the 39Ar/40Ar ratio is a proxy for the 40K/40Ar ratio.

22. 40Ar–39Ar Technique
The amount of 39Ar produced is a function of the amount of 39K present, the reaction cross-section (analogous to the neutron capture cross-section), the neutron flux, neutron energies, and the irradiation time:
The 40Ar*/39Ar is then:
This is way too much nuclear physics for simple geochemists. The trick is to combine several of these terms in a single term, C:
then determine C by irradiating a ‘standard’ of known age and solving this equation for C:

23. Advantages
The parent-daughter ratio can be determined simply by determining the isotopic composition of Ar in the irradiated sample (rather than having to separately measure K).
Ar can be extracted from a sample simply by heating it in vacuum. This can be done in temperature steps, allowing for multiple cases and multiple isotope ratio measurements.
In fact, this is what is typically done.
Even newer techniques involve spot heating with a laser, allowing for high spatial resolution.

24. A Textbook Plateau
Figure 2.4. Here, there has been some diffusional loss from the edges of the biotite, giving a younger age, but subsequent temperature steps all give the same age.

25. Partial reseting in contact metamorphic aureole
Figure 2.5. Ar release spectra for hornblendes taken from varying distance from a 114 million year old intrusion. The crystallization age of the samples is 367 Ma.
Curves show calculated release spectra expected for samples that lost 31, 57, and 78% of their argon.

26. Figure 2.6. Ar release spectrum of a hornblende in a Paleozoic gabbro reheated in the Cretaceous by the intrusion of a granite. Anomalously old apparent ages in the lowest temperature release fraction results from diffusion of radiogenic Ar into the hornblende during the Cretaceous reheating.

27. Figure 2.7. Ar release spectrum from a calcic plagioclase from Broken Hill, Australia. Low temperature and high temperature fractions both show erroneously old ages. This peculiar saddle shaped pattern, which is common in samples containing excess Ar, results from the excess Ar being held in two different lattice sites.

28. 40Ar-39Ar Isochrons
The data from various temperature re­lease steps are essentially indepen­dent observations of Ar isotopic composition. Because of this, they can be treated much the same as in conventional isochron treatment.
Since for all re­lease fractions of a sample the efficiency of production of 39Ar from 39K is the same and 40K/39K ratios are constant, we may substitute 39Ar × C for 40K:
Figure 2.8

29. Inverse Isochrons
The problem with that approach is 36Ar will not be very abundance and there will be a relatively large error in measurement - not something we want when it occurs as both denominators.
We can invert that ratio and plot it vs 39Ar/40Ar.
The x intercept is then the age (case of no trapped Ar) and the y-intercept gives the isotopic composition of trapped Ar.
Figure 2.9

30. Inverse Isochrons
The problem with that approach is that 36Ar is not very abundance and there will be a relatively large error in measurement - not something we want when it occurs as both denominators.
We can invert that ratio and plot it vs 39Ar/40Ar.
The x intercept then gives the age (case of no unradiogenic Ar) and the y-intercept gives the isotopic composition of trapped Ar.
Figure 2.9

31. Two Trapped Components
after correction for inherited Ar
Figure 2.10

32. Rb-Sr System
Rb: alkali; soluble, mobile, highly incompatible, substitutes for K
Sr: alkaline Earth; soluble, somewhat mobile, incompatible, substitutes for Ca
Both concentrated in the Earth’s crust; particularly Rb. High Rb/Sr in granitic rocks and their derivatives.

33. Sr chronostratigraphy
Sr present in relatively high concentration is seawater.
Also concentrated in carbonates (abundant marine bio-sediment).
Long residence time; therefore
Sr isotopic composition of open ocean water is uniform (in space).
Sr isotope ratio varies in time, mainly due to changes in the relative fluxes from the continents (erosion) and mantle (ridge-crest hydrothermal activity).
Particularly in the Tertiary, Sr isotope ratio of marine sediment can be used to date the horizon..
Figure 2.12

34. Sm-Nd System
Figure 2.13

35. Sm-Nd characteristics
Long half-life (~106 Ga).
Sm and Nd both rare earths elements (REE), similar behavior, typically small variation in Sm/Nd
generally more variation in mafic igneous rocks that granitic ones and their derivatives
garnet has quite high Sm/Nd.
REE behavior well understood.
Both form 3+ ions and are quite insoluble and immobile.
High closure temperature.

36. Cosmic Rare Earth Abundances
Figure 2.14

37. Normalized abundances
Figure 2.15

38. Ionic Radii and Partition Coefficients

39. The Epsilon Notation
Because Sm and Nd are refractory lithophile elements (condensed at high T in solar nebula and partition into silicate part of the planet) and because the Sm/Nd differs little in nebular materials (i.e., chondritic meteorites), it was assumed the Earth had a chondritic Sm/Nd ratio and therefore that the evolution of 143Nd/144Nd in the Earth should follow that of chondrites. Furthermore, variations in 143Nd/144Nd are small. Thus the ε notation was introduced by DePaolo and Wasserburg (1976):

40. Nd isotopic evolution of the Earth
CHUR: “Chondritic Uniform Reservoir” – (silicate Earth if it is chondritic)
Mantle is Nd depleted relative to Sm, has high Sm/Nd, evolves to high εNd.
Continental crust is Nd enriched relative to Sm, has high Sm/Nd, evolves to low εNd.
Figure 2.16

41. Figure 2.17. Garnet-bearing granulite from Dabie UHP metamorphic belt in China.

42. Crustal Residence Times
Basic idea is that there is a relatively large fractionation between Sm and Nd during melting to form new additions to crust. Subsequent crustal processing produces little change in Sm/Nd.
Consider our isochron equation:
Since λt << 1, we can use the approximation that for x<<1, ex ≈ x+1 and linearize this equation:
On a plot of 143Nd/144Nd vs. t, the slope = 147Sm/144Ndλ
We want to know t assuming 143Nd/144Nd0 is the mantle value at the time the material was added to crust.
We project back along the slope defined by 147Sm/144Nd to the point of intersection on the mantle evolution curve.

43. Sm-Nd Model Ages (aka Crustal Residence Times)
Figure 2.18

44. Lu-Hf System

45. Lu-Hf in Chondrites
Figure 2.19

46. Lu-Hf System 176Lu is another odd-odd nuclide.
Decays to 176Hf with half-life of 36 Ga (possible it might also decay to 176Yb, but extremely infrequently – not demonstrated.
Lu and Hf both refractory lithophile elements -implies silicate Earth has chondritic Lu/Hf (?).
Lu slightly incompatible, Hf moderately incompatible.
Lu has 3+ valance state, Hf 4+ valance state, both quite insoluble and immobile.
Strong chemical similarity of Hf to Zr; therefore Hf strongly partitioned into zircon (ZrSiO4) – a highly resistance accessory mineral in many crustal rocks.

47. epsilon Hf notation
176Hf/177Hf ratio commonly represented as εHf – exactly analogous to εNd:

48. Hf in the crust
Unlike Sm/Nd, the Lu/Hf ratio does change significantly during weathering and other crustal processes – mainly related to zircon.
Zircon is extremely resistant chemically and physically and when weathering occurs will go into the sand fraction (taking Hf with it).
Lu will mainly go in the clay fraction.
Thus sedimentary processes act to fractionate Lu/Hf.
Thus sediments sometimes deviate from the otherwise strong correlation between εHf and εNd.
So there is no analogous Lu-Hf crustal residence time.
On the other hand, zircons have very low Lu/Hf, so preserve, or nearly so, their initial εHf. This together with U-Pb ages of zircons provides analogous information on provenance of sediments.

49. Figure 2.20

50. Lu-Hf in dating Lu and Hf are immobile - good for dating older rocks.
Lu half-life relatively short.
Lu/Hf ratio more variable that Sm/Nd.
Consequently, larger variation in Hf isotope ratios than Nd isotope ratios.
Lu very strongly concentrated in garnet, so this system again very useful for dating garnet-bearing rocks.

51. Figure 2.21

52. Re-Os 187Re decays to 187Os with a half-live of 42 Ga.
Also a decay of 190Pt to 186Os with very long half-life (450 Ga), so that resulting variation is usually not detectable
Unlike most elements we’ve considered, Re and Os are siderophile (and also chalcophile) meaning they are concentrated in the Earth’s core.
Consequently, Re and Os concentrations in the crust and mantle are very low (ppb and usually lower).
Within the silicate part of the Earth, Os behaves as a very compatible element (hence remains in the mantle with very low concentrations in the crust), while Re is moderately incompatible and concentrates in the crust.
Large variation in Re/Os leads to quite large variations in 186Os/188Os.

53. Gamma notation
In a manner somewhat analogous to the epsilon notation, Os isotope ratios are often reported as γOs : percent deviations from Primitive Upper Mantle:

54. Figure 2.22

55. Figure 2.23: Re/Os isochron for a komattite from Munro Township, Ontario.

56. Re-Os dating of diamond inclusions
Figure 2.24: sulfide inclusion in diamond

57. Tracing Os isotope evolution of seawater from hydrogenous sediments and black shales.
Figure 2.26

58. Dating Hydrocarbon Generation (?)
Figure 2.27