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?
Basic Equations Basic Equation Rearrange and integrate: Half-life:
Case where some daughter already present D, daughter; D* =N 0 – N, radiogenically produced daughter Since some daughter is initially present (D 0 ): 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:
Isochron Dating In the general case: we have two unknowns, R 0 and t (we measure R and R 0 ). We need two equations. If we have two samples with (presumably) identical R 0 and t, subtracting one from the other, we have: Rearranging: or
Isochrons From equ. 2.20, we see that t is proportional to the slope on a plot of R vs. R P/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.
Assumptions We are assuming each sample analyzed has the same value of R 0 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 R P/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).
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.
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). http://bgc.org/isoplot_etc/isoplot.html. http://bgc.org/isoplot_etc/isoplot.html
The K-Ca-Ar System 40 K is can decay to either 40 Ca (by β – decay) or 40 Ar 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 10 -10 yr -1 corresponding to a half-life of ~1.28 Ga. Most (~90%) decays to 40 Ca, but 40 Ca is doubly magic and a very abundant nuclide. Thus the radiogenic fraction is small. 40 Ar, on the other hand is a rare gas and fairly rare (on the Earth anyway). Thus mostly we are interested in the decay to 40 Ar. 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.
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 adsorped atmospheric Ar, whose isotopic composition is well known. Thus our equation becomes:
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.
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: E A 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. Figure 2.2 Ar in biotite
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?
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
Ar loss from biotite Figure 2.3. 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.
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 T c 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 T c occurs both in and out of the log, but it can be solved by indirect methods (MatLab, Solver in Excel).
40 Ar– 39 Ar Dating You might wonder what this is all about. 39 Ar has a 269 yr half-life and does not occur naturally. 40 Ar– 39 Ar dating is simply a specific analytical technique for 40 K– 40 Ar dating. The sample is irradiated with neutrons in a reactor and 39 Ar is created from 39 K by: 39 K(n,p) 39 Ar. Since the amount of 39 Ar is proportional to the amount of 39 K and that is in turn proportional to the amount of 40 K, the 39 Ar/ 40 Ar ratio is a proxy for the 40 K/ 40 Ar ratio.
40 Ar– 39 Ar Technique The amount of 39 Ar produced is a function of the amount of 39 K present, the reaction cross-section (analogous to the neutron capture cross-section), the neutron flux, neutron energies, and the irradiation time: The 40 Ar*/ 39 Ar 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:
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 ratios. In fact, this is what is typically done. Even newer techniques involve spot heating with a laser, allowing for high spatial resolution.
A Textbook Plateau Figure 2.4. Here, there is been some diffusional loss from the edges of the biotite, giving a younger age, but subsequent temperature steps all give the same age.
Partial reseting in contact metamorphic aereole Figure 2.5
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.
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.
40 Ar- 39 Ar 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 39 Ar from 39 K is the same and 40 K/ 39 K ratios are constant, we may substitute 39 Ar × C for 40 K: Figure 2.8
Inverse Isochrons The problem with that approach is 36 Ar 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 39 Ar/ 40 Ar. 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
Two Trapped Components Figure 2.10 after correction for inherited Ar
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.
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
Sm-Nd characteristics Long half-life (~100 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.
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 143 Nd/ 144 Nd in the Earth should follow that of chondrites. Furthermore, variations in 143 Nd/ 144 Nd are small. Thus the ε notation was introduced by DePaolo and Wasserburg (1976):
Figure 2.16 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 ε Nd. Continental crust is Nd enriched relative to Sm, has high Sm/Nd, evolves to lowε Nd.
Figure 2.17. Garnet bearing granulite from Dabie UHP metamorphic belt in China.
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, e x ≈ x+1 and linearize this equation: On a plot of 143 Nd/ 144 Nd vs. t, the slope = 147 Sm/ 144 Ndλ We want to know t assuming 143 Nd/ 144 Nd 0 is the mantle value at the time the material was added to crust. We project back along the slope defined by 147 Sm/ 144 Nd to the point of intersection on the mantle evolution curve.
Figure 2.18 Sm-Nd Model Ages (aka Crustal Residence Times)
Lu-Hf System 176 Lu is another odd-odd nuclide. Decays to 176 Hf with half-life of 36 Ga (possible it might also decay to 176 Yb, 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 (ZrSiO 4 ) – a highly resistance accessory mineral in many crustal rocks.
epsilon Hf notation 176 Hf/ 177 Hf ratio commonly represented as ε Hf – exactly analogous to ε Nd :
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 ε Hf and ε Nd. 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 ε Hf. This together with U-Pb ages of zircons provides analogous information on provinance of sediments.
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.
Re-Os 187 Re decays to 187 Os with a half-live of 42 Ga. Also a decay of 190 Pt to 186 Os 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 186 Os/ 188 Os.
Gamma notation In a manner somewhat analogous to the epsilon notation, Os isotope ratios are often reported as γ Os : percent deviations from Primitive Upper Mantle: