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D ATING B ASED ON U-T H -P B
U and Th are, strictly speaking, rare earth elements, although they belong to the actinide series in- stead of the lanthanide series. The other rare earths we have met so far, Nd and Sm, are lanthanides. As in the lanthanide rare earths, an inner electron shell is being filled as atomic number increases in the actinides. Both U and Th generally have a valence of +4, but under oxidizing conditions, such as at the surface of the Earth, U has a valence of +6. In six-fold coordination, U 4+^ has an ionic radius of 89 pm 1 (100 pico meters = 1 Å); U 6+^ has an ionic radius of 73 pm in 6-fold and 86 pm in 8-fold coordination. Th 4+^ has an ionic radius of 94 pm. These radii are not particularly large, but the combination of some- what large radius and high-charge is not readily accommodated in crystal lattices of most common rock-forming minerals, so both U and Th are highly incompatible elements. Th is relatively immobile under most circumstances. In its reduced form, U 4+^ is insoluble and therefore fairly immobile, but in the U 6+^ form, which is stable under a wide range of conditions at the surface of the Earth, U forms the soluble oxyanion complex, UO 4 2–. As a result, U can be quite mobile. U and Th can form their own phases in sedimentary rocks, uranite and thorite, but they are quite rare. In igneous and metamorphic rocks, U and Th are either dispersed as trace elements in major phases, or concentrated in accessory minerals (when they are present) such as zircon (ZrSiO 4 ), which concentrates U more than Th, and monazite ([La,Ce,Th]PO 4 ) which concentrations Th more than U. These elements may be also concen- trated in other accessory phases such as apatite (Ca 5 (PO 4 ) 3 (OH)) and sphene (CaTi(SiO 4 )OH). However, zircon is by far and away the most important from a geochronological perspective. U and Th are refractory elements, and we can therefore expect the Th/U ratio of the Earth to be the same as chondrites or nearly so. There is, however, some debate about the exact terrestrial Th/U ratio, and we can be no more precise than to say it is 4±0.2 2. This ratio is 3.8 in the CI chondrite Orgueil, but may be low due to mobility of U in hydrous fluid in the CI parent body. The geochemical behavior of Pb is more complex than that of the elements we have discussed so far and consequently, less well understood. It is a relatively volatile element, so its concentration in the Earth is certainly much lower than in chondrites. It is also a chalcophile element. If the core contains, as some believe, S as the light element, it is possible that some of the Earth's Pb is in the core (it is, how- ever, difficult to distinguish loss of Pb from the Earth due to its volatility from loss of Pb from the sili- cate portion of the Earth due to extraction into the core). Pb can exist in two valence states, Pb 2+^ and Pb 4+. Pb 2+^ is by far the most common state; the Pb 4+^ state is rare and restricted to highly alkaline or oxi- dizing solutions. The ionic radius of Pb 2+^ is 119 pm in 6-fold coordination and 129 pm in 8-fold coordi- nation. As a result of its large ionic size, Pb is an incompatible element, though not as incompatible as U and Th (incompatibility seems to be comparable to the light rare earths). The most common Pb min- eral is galena (PbS). In silicates, Pb substitutes readily for K (ionic radius 133 pm) particularly in potas- sium feldspar, but less so in other K minerals such as biotite. Most naturally occurring compounds of Pb are highly insoluble under most conditions. As a result, Pb is usually reasonably immobile. How- ever, under conditions of low pH and high temperature, Pb forms stable and somewhat soluble chlo- ride and sulfide complexes, so that Pb can sometimes be readily transported in hydrothermal solutions.
(^1) In eight-fold coordination, the effective ionic radius of U 4+^ is 1.00Å. In zircon, a mineral which highly concentrations
U, U is in 8-fold coordination. This is probably a pretty good indication that 8-fold coordination is the preferred con- figuration. The figure for 6-fold coordination is given for comparison to other radii, which have been for 6-fold coor- dination. Th has a radius of 1.05Å in 8-fold coordination. (^2) The uncertainty results from the mobility of U. The CI carbonaceous chondrites experienced mild alteration in hy-
drous conditions on the parent body. U was mobilized under these conditions and thus the U/Th ratio varies in these meteorites. For this reason, they cannot be used to precisely determine the U/Th ratio of the Solar System and the Earth.
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Although Pb is clearly less incompatible that U and Th, these 3 elements have been extracted from the mantle and concentrated in the crust to approximately the same degree. The reason for this is not yet completely understood, and we will discuss the problem later in the course. The U-Th-Pb system is certainly the most powerful tool in the geochronologist's tool chest. Table 6. summarizes this decay system. The reason for the power is simply that there are three parents decay- ing to 3 isotopes of Pb, and in particular, there are two isotopes of U which decay to Pb with very dif- ferent half lives. This is important because chemical processes will not change the ratio of the two U isotopes to each other and will not change the ratio of the two Pb daughter isotopes to each other. The point is best illustrated as follows. First we write the decay equation for each of the two U decay sys- tems:
207 Pb * = 235 U ( e^ λ 5 t^ − 1) 3.
206 Pb * = 238 U ( e^ λ 8 t^ − 1) 6.
where the asterisk designates radiogenic^206 Pb and 207 Pb, and λ 5 and λ 8 are the decay constants for 235 U and 238 U respectively. If we divide 3.01 by 3.02, we have:
207 Pb *
206 Pb * =
235 U ( e^ λ 5 t^ − 1)
238 U ( e^ λ 8 t^ − 1)
Now if the ratio of the U isotopes is everywhere the same (as it is at the present day), 3.03 can be writ- ten as
207 Pb *
206 Pb * =^
( e
λ 5 t
137.88( e
λ 8 t
The nice thing about equation 3.04 is that the only variable on the right hand side is time; in other words the 207 Pb/ 206 Pb is a function only of time. TABLE 3.1. Parameters of the U-Th-Pb System Parent Decay Mode λ Half-life Daughter Ratio (^232) Th α,β 4.948 x 10 -11y -1 (^) 1.4 x 10 10 y 208 Pb, 8 4 He 208 Pb/ 204 Pb, 3 He/ 4 He (^235) U α,β 9.849 x 10 -10y -1 (^) 7.07 x 10 8 y 207 Pb, 7 4 He 207 Pb/ 204 Pb, 3 He/ 4 He (^238) U α,β 1.551 x 10 -10y -1 (^) 4.47 x 10 9 y 206 Pb, 6 4 He 206 Pb/ 204 Pb, 3 He/ 4 He
In practice what this means is that the age is independent of the parent/daughter ratio; i.e., we not need to measure the parent/daughter ratio. We shall see that this property actually allows us to somewhat relax our requirement that the system remain closed. We can also see that although we could write an equation similar to 80.3 using 232 Th and 208 Pb instead of 235 U and 207 Pb, there would be lit- tle advantage to doing so because Th and U are different elements and could well be lost or gained in different proportions. If Madison Avenue were given the task of selling the U-Th-Pb system, they would probably say that you get 4 dating methods for the price of one: 238 U- 206 Pb, 235 U- 207 Pb, 232 Th- 208 Pb, and 207 Pb- 206 Pb. In a cer- tain sense, this is true. However, if you bought the package, you would probably quickly discover that the first three above, applied independently, were not particularly powerful, at least in comparison to either the Pb-Pb technique or simultaneous use of a combination of several techniques (an exception might be the 232 Th- 208 Pb system, which might prove useful on separated Th-bearing minerals). The Pb- Pb method, as it is called, can be quite useful when applied independently, particularly where there is reason to believe that there has been some recent change in the parent/daughter ratio. We have men- tioned in an earlier lecture that the slope on a plot of 207 Pb/ 204 Pb vs. 206 Pb/ 204 Pb is proportional to age since:
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prudent to assume the ratio is actually constant for geochronological purposes. Furthermore, there is little reason to do so, since we can already compute the age using 207 Pb and 206 Pb. But it may be useful in some circumstances to turn the problem around and compute the Th/U ratio from the age and the slope of the data on a plot of 208 Pb/^204 Pb vs. 206 Pb/^204 Pb. The basis of this is as follows. We write the usual growth equations for 206 Pb and 208 Pb:
206 Pb / 204 Pb = ( 206 Pb / 204 Pb )
0 +^
238 U / 204 Pb ( e^ λ 8 t^ − 1 ) 3.
208 Pb / 204 Pb = ( 208 Pb / 204 Pb )
0 +^
232 Th / 204 Pb ( e^ λ 2 t^ − 1 ) 3.
Subtracting the initial ratio from each side of each equation and dividing 3.7 by 3.6 we have
∆( 208 Pb / 204 Pb )
∆( 206 Pb / 204 Pb )
232 Th / 204 Pb ( e^ λ 2 t^ − 1 )
238 U / 204 Pb ( e^ λ 8 t^ − 1 )
or
∆( 208 Pb / 204 Pb )
∆( 206 Pb / 204 Pb )
κ ( e
λ 2 t
( e
λ 8 t
where κ is used to designate the 232 Th/^238 U ratio. Using μ to designate the 238 U/^204 Pb ratio, the parent- daughter ratio of the Th-Pb system is the product μκ. Equation 3.09 tells us that the slope of a line on a plot of 208 Pb/^204 Pb vs. 206 Pb/^204 Pb is proportional to time and κ, provided that κ does not vary. If we can calculate t from the corresponding 207 Pb/^204 Pb– (^206) Pb/ (^204) Pb slope, we can solve 3.09 for κ. If, however, κ varies linearly with μ, a straight line will still result on the 208 Pb/^204 Pb vs. 206 Pb/^204 Pb plot and our estimate of κ will be incorrect.
THE U-TH-PB SYSTEM: ZIRCON DATING
Zircon (ZrSiO 4 ) is a mineral with a number of properties that make it extremely useful for geochro- nologists (Figure 3.02). First of all, it is very hard (hardness 7^1 / 2 ), which means it extremely resistant to mechanical weathering. Second, it is extremely resistant to chemical weathering and metamorphism. For geochronological purposes, these properties mean it is likely to remain a closed system. Third, it concentrates U (and Th to a lesser extent) and excludes Pb, resulting in typically very high 238 U/^204 Pb ra- tios. It is quite possibly nature's best clock. Finally, it is reasonably common as an accessory phase in a variety of igneous and metamorphic rocks. The very high 238 U/^204 Pb ratios in zircon (and similar high μ minerals such as sphere and apatite) pro- vide some special geochronological opportunities and a special diagram, the concordia diagram , has been developed to take advantage of them. The discussion that follows can be applied to any other system with extremely high 238 U/^204 Pb ratios, but in practice, zircons constitute the principle target for Pb geo- chronologists. A concordia diagram is simply a plot of 206 Pb/^238 U vs. 207 Pb/^235 U. You should satisfy yourself that both of these ratios are proportional to time. In essence, the concordia diagram is a plot of the 238 U– (^206) Pb age against the 235 U– (^207) Pb age. The ‘concordia’ curve on such a diagram that is the locus of points where the 238 U–^206 Pb age equals the 235 U–^207 Pb age. Such ages are said to be concordant. Figure 3.3 is an example of a concordia diagram. The best way to think about evolution of Pb/U ratios is to imagine that the diagram itself evolves with time, along with its axes, while the actual data point stays fixed. Let’s take a 4.0 Ga old zircon as an example. When it first formed, or “closed”, it would have plotted at the origin, because had anyone been around to analyze it, they would have found the 207 Pb/^235 U and 206 Pb/^238 U ratios to be 0. Ini- tially, 207 Pb/^235 U would have increased rapidly, while the 206 Pb/^238 U would have been increasing only slowly. This is because 4.0 Ga ago there was a lot of 235 U around (recall that 235 U has a short half-life). As time passed, the increase in 207 Pb*/^235 U would have slowed as the 235 U was ‘used up’. So imagine
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that the diagram initially 'grows' or 'expands' to the left, expanding downward only slowly. Had someone been around 3.0 Ga ago to deter- mine 'zircon' ages, he would have drawn it as it appears in Figure 3.04 (of course, he would have labeled the 3.0 Ga point as 0, the 4.0 Ga point as 1.0, etc.). Any zircon that has re- mained as a completely closed system since its crystallization must plot on the concordia line. What happens when a zircon gains or looses U or Pb? Let’s take the case of Pb loss, since that is the most common type of open-system behavior in zircons. The zircon must lose 207 Pb and 206 Pb in exactly the proportions they exist in the zircon because the two are chemically identical. In other words, a zircon will not lose (^206) Pb in preference to 207 Pb or visa versa. Let’s take the specific case of a 4.0 Ga zircon that experi- enced some Pb loss during a metamorphic event at 3.0 Ga. If the loss was complete, the zircon would have been reset and would have plotted at the origin in Figure 3.04. We could not distinguish it from one that formed 3.0 Ga. Suppose now that that zircon had lost only half its Pb at 3.0 Ga. During the Pb loss, the 206 Pb/^238 U and (^207) Pb/ (^235) U would have both de- creased by half. Consequently, the point would have migrated half way to the origin. At 3.0 Ga, therefore, it would have plotted on a ‘cord’, i.e., a straight line, between its initial position on the concordia curve, the 4.0 Ga point, and the origin (Figure 3.5a) at 3.0 Ga. Had it lost some other amount of Pb, say 30% or 80%, it would have plot- ted on the same cord, but further or nearer the origin. The line is straight because the loss of 207 Pb is al- ways directly proportional to the loss of 206 Pb. The origin in Figure 9.3a corresponds to the 3.0 Ga point
Figure 3.02. Upper. Separated Zircon crystals. Notice the zoning.
Lower. Strongly zoned zircon showing differing ages of spots ana- lyzed by ion probe.
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concordia diagram in the same manner as Pb loss; the two processes are essentially indistinguishable on the concordia diagram. U loss, on the other hand, moves the points away from the origin at the time of the loss (Figure 3.6). In this case, the zircons lie on an extension of a cord above the concordia. As is the case for Pb loss, the upper intercept of the cord gives the initial age and the lower intercept gives the age of U loss. U loss in less common than Pb loss. This is true for two reasons. First, U is happy in the zircon, Pb is not. Second, Pb will occupy a site damaged by the alpha decay process, particularly the recoil of the atom as it undergoes alpha decay, making diffusion out of this site easier. Radiation damage to the crystal lattice is a significant problem in zircon geochronology, and one of the main rea- sons ages can be imprecise. U-rich zircons are particularly subject to radiation damage. Heavily dam- aged crystals are easily recog- nized under the microscope and are termed metamict. Pb gain in zircons is not pre- dictable because the isotopic composition of the Pb gained need not be the same as the composition of the Pb in the zircon. Thus Pb gain would destroy any age relationships. However, Pb gain is much less likely than other open system behaviors. Zircons that have suffered multiple episodes of open sys- tem behavior will have U-Pb systematics that are difficult to interpret and could be incor- rectly interpreted. For exam- ple, zircons lying on a cord be-
Figure 3.05. (a) Concordia diagram as it would have appeared at 3.0 Ga. Three zircons that experience variable amounts of Pb loss move from the 4.0 Ga point on the concordia curve (their crystallization age) toward the origin. (b) The same three zircons as they would plot at present. The three define a cord be- tween 3.0 Ga and 4.0 Ga. A possible interpretation of this result would be that 4.0 Ga is the crystallization age and 3.0 Ga is the metamorphic age.
Figure 3.06. A concordia plot showing hypothetical zircons that crys- tallized at 4.0 Ga and lost U during metamorphism at 3.0 Ga.
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tween 4.0 and 3.0 Ga that subsequently lose Pb and move on a second cord toward the 2.0 Ga could be interpreted as having a metamorphic age of 2.0 Ga and a crystallization age of between 4.0 and 3.0 Ga. Continuous Pb loss from zircons can also complicate the task of interpretation. The reason is that in continuous Pb loss, zircons do not define a straight-line cord, but rather a slightly curved one. Again imagining that the concordia diagram grows with time, a zircon loosing Pb will always move toward the origin. However, the position of the origin relative to the position of the zircon moves with time in a non-linear fashion. The result is a non-linear evolution of the isotopic composition of the zircon. Thus although it is possible to “see through” partial resetting of the zircon “clock”, this process nev- ertheless limits the accuracy with which initial crystallization ages can be determined. Several tech- niques have been developed to get around this problem. Quite often, zircons are zoned with the out- most-layers being the richest in U. This can be a result of increasing U concentration during fractional crystallization of a magma or overgrowth of U-rich layers during metamorphism. Regardless, the out- most zones are most likely to be subject to radiation damage and consequent U-loss. Krogh (1982) showed that these out zones could be removed by abrasion using small air abrasion chambers designed and built expressly for this purpose. He demonstrated that abraded zircons were typically much more concordant than unabraded ones. Consequently, age uncertainties were considerably reduced. In ad- dition to physical abrasion, numerous attempts have been made to chemically abrade, or leach, zircons, the idea being to remove the radiation damaged regions of the crystal. The most success- ful of these methods has been that of Mattinson (2005), which involves first annealing the crystals at 800 - 1000˚C for 48 hours before step-wise partial dissolution in acid at progres- sively higher temperatures. This step-wise dissolution al- lows for an approach similar to step-wise heating in 40 Ar-^39 Ar dating, as illustrated in Figure 3.07. The uncertainty in this age is less than 0.1%, a level of accuracy otherwise unattain- able. Figure 3.08 shows an- other example of how this technique improves accurate determination of crystalliza- tion ages. Three different frac- tionations of untreated zircons from a Finnish tonalite are dis- cordant, but define a cord with an upper concordia intercept of about 1870 Ma. The chemi- cally abraded zircon fraction is nearly concordant at this age, allowing a much more precise determination. In this dia- gram, the analyses are plotted
Figure 3.07. a. Ages calculated from step-wise dissolution of zircons from the Sierra Nevada batholith that had been abraded, then an- nealed at 850˚C for 48 hours. Corresponding U concentrations in the step-wise dissolution. The outer zones of the zircons are the most U- rich and consequently the most radiation-damaged and discordant.
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Subsequently even older zircons (would be more correct to say parts of zircons) were discovered in the Jack Hills of Australia (Wilde, et al., 2001). An ion probe date on one part of one of these zircons (Figure 3.09) is shown 4.404 Ga ±8 Ma. Thus the oldest known terrestrial materials are approaching the oldest ages from other planetary bodies, including the Moon, Mars, and asteroids (as represented by meteorites). They remain, however, significantly younger than the age of the Solar System, which is 4.556 Ga. Nevertheless, these very old ages seem to demonstrate that it is zircons, not diamonds, that “are forever”.
U-DECAY SERIES DATING
Up to now, we have been discussing decay schemes that are based on measuring the amount of a sta- ble daughter nuclide relative to the amount of the radioactive parent. Since the decay of the parent takes place at an invariant rate, this ratio of daughter to parent is proportional to time (in a closed sys- tem). In this lecture, we will discuss a method of dating in which both parent and daughter are radio- active. U-decay series dating differs in a very fundamental way from the conventional techniques we have been discussing. It does, however, share two features. First, the time we are measuring is the time since the system closed, and second, an accurate date requires the system to have remained closed.
Figure 3.09. Photomicrograph of the oldest zircon known. Light area are regions of ion probe analysis. Ages determined on these areas are shown. From Wilde et al. (2001).
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Basic Principles
The fundamental principal involved in U decay series dating is that, in a closed system, the ratio of parent to daughter will tend toward an equilibrium state in which the rate of decay of the parent is equal to the rate of decay of the daughter. A closed system will approach this equilibrium state at a predictable rate with the ratio of the parent to daughter being proportional to time , provided equilibrium has not been reached. Once equilib- rium has been achieved, the ratio of parent to daughter no longer depends on the time elapsed, and we can calculate only a minimum age*. U and Th do not decay directly to Pb; rather the transition from U to Pb passes through many inter- mediate radioactive daughters (Figure 3.10). Most of these daughters have very short half-lives, rang- ing from milliseconds to hours, and are of little use in study of the Earth. However, a number of these intermediate daughters have half-lives ranging from days to hundreds of thousands of years and do provide useful information about geological processes. Table 3.01 lists half-lives and decay constants of some of the most useful of these isotopes. As one might guess from the short half-lives, U-decay series isotopes are used to date relatively recent events.
- (^) In principle, a system will approach equilibrium asymptotically, and will only achieve equilibrium after an infinite
amount of time. In practice, an effective equilibrium is achieved when the difference between the measured ratio of parent to daughter is less than the analytical uncertainty of the measurement. This typically will occur after a maxi- mum of 7-10 half-lives of the nuclide with the shortest half-life.
Figure 3.10. Part of the chart of the nuclides showing the series of decays that occur as
238 U,
235 232 U, and Th are transformed to
206 Pb,
207 Pb, and
208 Pb respectively.
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0 = λ P NP − λ D ND 3.
We substitute the dN/dt for the λN terms in 7.04, rearrange and we obtain 3.12; QED. The second demonstration is a thought experiment. Imagine a hopper, a grain hopper for example, with an open top and a door in the bottom. The door is spring-loaded such that the more weight placed on the door, the wider it opens. Suppose we start dropping marbles into the hopper at a con- stant rate. The weight of marbles accumulating in the hopper will force the door open slightly and marbles will start falling out at a slow rate. Because the marbles are falling out more slowly than they are falling in, the number and weight of marbles in the hopper will continue to increase. As a result, the door will continue to open. At some point, the door will be open so wide that marbles are falling out as fast as they are falling in. This is the steady, or equilibrium, state. Marbles no longer accumulate in the hopper and hence the door is not forced to open any wider. The marbles falling into the door are like the decay of the parent isotope. The marbles in the hopper represent the population of daughter isotopes. Their decay is represented by their passing through the bottom door. Just as the number of marbles passing through the door depends on the number of marbles in the hopper, the activity (number of decays per unit time) of an isotope depends on the number of atoms present. If the rate of marbles dropping into the hopper decreases for some reason, marbles will fall out of the hopper faster than the fall in. The number of marbles in the hopper will decrease, as a result the weight on the door decreases and it starts to close. It continues to close (as the number of marbles decreases) until the rate at which marbles fall out equals the rate at which marbles fall in. At that point, there is no longer a change in the number of marbles in the hopper and the position of the door stabilizes. Again equilibrium has been achieved, this time with fewer marbles in the hopper, but nevertheless at the point where the rate of marbles going in equals the rate of marbles going out. The analogy to radioac- tive decay is exact. Thus when a system is disturbed by addition or loss of parent or daughter, it will ultimately return to equilibrium. The rate at which it returns to equilibrium is determined by the decay constants of the parent and daughter. If we know how far out of equilibrium the system was when it was disturbed, we can determine the amount of time which has passed since it was disturbed by measuring the present rate of decay of the parent and daughter. 234
U-
238
U dating
An example of this is 234 U-^238 U dating of sediments. As it turns out, (^234 U) and (^238 U) in seawater are not in equilibrium, i.e., the (^234 U/^238 U) ratio is not 1. It is uniform†, however, with a ratio of about 1.14±0.03. The reason the ratio is higher than 1 is that 234 U is preferentially leached from rocks because (^234) U is located in crystals of rocks in damaged lattice sites. It occupies the site of a 238 U atom that has undergone α-decay. The α-particle and the recoil of the nucleus damage this site. Since it occupies a damaged site, it is more easily removed from the crystal by weathering than 238 U. The oceans collect this 'leachate', hence they are enriched in 234 U. When U precipitates from seawater into, for example, the calcium carbonate in a coral skeleton, it will initially have the same (^234 U/^238 U) as seawater, but 234 U will decay faster than it is created by decay of 238 U, so (^234 U/^238 U) will slowly return to the equilibrium condition where (^234 U/^238 U) = 1. Let’s see how we can take advantage of this to determine geologic time. We can divide the activity of (^234) U in a sample into that which is supported by 238 U, i.e. that amount in radioactive equilibrium with (^238) U and that amount which is excess, i.e. unsupported by 238 U:
(^234 U) = (^234 U)s + (^234 U)u 3.
† (^) The ratio is uniform in space, but there is evidence to suggest it has varied slightly with time, particularly between
glacial and interglacial periods.
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where we are using parentheses to denote activity and subscripts s and u denote supported and unsup- ported abundances respectively. The activity of the excess 234 U de- creases with time according to equation 2.04, which we can re- write as:
( 234 U ) u = ( 234 U ) u^ o^ e −^ λ^234 t^ 3.
where the superscript naught de- notes the initial activity (at time t = 0). We can also write:
(^234 U)u^ o^ = (^234 U)o^ – (^234 U)s 3.
which just says that the initial un- supported activity of 234 U is equal to the total initial activity of 234 U less the (initial) supported activity of 234 U. Since to a very good ap- proximation the activity of the parent, (^238) U, does not change over times on the order of the half-life of 234 U or even ten half-lives of 234 U, the present 238 U ac- tivity is equal to the activity at t = 0 (we make the usual assumption that the system is closed). And by definition the supported activity of 234 U is equal to the activity of 238 U, both now and at t = 0. Hence, 3.14 can be expressed as:
(^234 U) = (^238 U) + (^234 U)u 3.
and 3.16 becomes
(^234 U ) u^ o^ = (^234 U ) o^ − (^238 U ) 3.
Substituting 3.18 into 3.15 yields:
( 234 U ) U = ( 234 U ) o^ − ( 238 U ) e −^ λ^234 t^ 3.
Substituting 3.19 into 3.17, we have:
( 234 U ) = ( 238 U ) + ( 234 U ) o^ − ( 238 U ) e −^ λ^234 t^ 3.
Just as for other isotope systems, it is generally most convenient to deal with ratios rather than absolute activities (among other things, this allows us to ignore detector efficiency provided the detector is equally efficient at all energies of interest‡), hence we divide by the activity of 238 U:
234 U
238 U )^ =^1 +^
( 234 U ) o^ − ( 238 U )
( 238 U )
e
− λ 234 t
‡ (^) In the case of 238 U and 234 U the α energies are quite similar (4.2 and 4.7 MeV). Interestingly, the energies of α parti-
cles are approximately inversely proportional to half-life. From our discussion of α decay, you should be able to surmise why.
Figure 3.11. Hypothetical (^230 Th/^232 Th) as a function of depth in a core. (a) is normal, constant sedimentation rate, (b) reflects a change in sedimentation rate, (c) reflects bioturbation mixing the sediment at the top of the core, (d) is a case of dominance by U- supported 230 Th. From Faure (1986).
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230 Th
232
Th
^ u
230 Th
232
Th
^ u
o
e −^ λ^230 t^ 3.
(^232 Th, although non-radiogenic, is radioactive, of course. However, with a half-life of 14Ga, its abun- dance will not change on time scales comparable to the half-life of 230 Th.) In seawater, and in its oxidized state generally, U is quite soluble. Th, however, is quite insoluble: its seawater residence time is 300 years or less compared to about 500,000 year for U. (It should be noted here that solubility in seawater does not control concentrations or residence times. Nevertheless solu- bility is a good guide to both of these.) Once a 234 U atom decays to 230 Th, it is quickly absorbed onto particles that in turn are quickly incorporated into sediment. As a result, relatively high concentrations of unsupported 230 Th can be removed (by leaching) from some sediments (mainly relatively slowly ac- cumulating ones). In cases where the amount of leached 234 U is negligible and where the (^230 Th/^232 Th) is known (from, for example, zero-age sediment at the seawater-sediment interface), equation 7.14 can be used to determine the age of the sediment. If we can't neglect 234 U then we can still determine the age by taking this into account, though the mathematics becomes more complex (see Dickin, 1995). The point is that by determining 230 Th/^232 Th ratios in a series of sediments down a core, sedimentation rate can be determined. Various curves of (^230 Th/^232 Th) vs. depth suggest various processes, as is illustrated in Fig. 3.11. In systems where we can assume (^234 U) = (^238 U), high-temperatures systems for example, the mathe- matics simplifies somewhat. With this assumption, we can treat the production of 230 Th as if it were the direct decay product of 238 U. We write an equation analogous to 7.05 and from it derive:
230 Th
232
Th
^
230 Th
232
Th
o
e −^ λ^230 t^ +
238 U
232
Th
^
(^1 −^ e −^ λ^230 t ) 3.
(the tricks to this derivation are to make the approximations λ 230 – λ 238 = λ 230 and e-λ^238 t^ = 1; i.e., assume λ 238 ≈ 0; this is the mathematical equivalent of assuming the activity of 238 U does not change with time). The first term on the right describes the decay of unsupported 230 Th while the second term describes the growth of supported 230 Th. Note that this equation has the form of a straight line in (^230 Th/^232 Th) — (^238 U/^232 Th) space, where the first term is the intercept and (1 – e-λ^230 t) is the slope. This is illustrated in Figure 3.12. Let’s consider an example of a low-temperature application of U-Th dating. The tops of manganese nodules grow by precipitation of Mn-Fe oxides and hydroxides from seawater. They are known to grow very slowly, but how slowly? If we assume the rate of growth is constant then depth in the nod- ule should be proportional to time. If z is the depth in the nodule, and s is the growth (sedimentation) rate, then:
t = z/s 3.
and 3.25 becomes:
230 Th
232
Th
^
230 Th
232
Th
o
e −^ λ^230 z^ /^ s^ +
238 U
232
Th
^
(^1 −^ e −^ λ^230 z^ /^ s ) 3.
At the surface of the nodule, z = 0, so the exponential terms both go to 1 and the measured activity ratio is initial activity ratio. Having a value for (^230 Th/^232 Th)°, equ. 3.27 can then be solved for s , the growth rate if measurements are made at some other depth. In practice, however, it is difficult to obtain a sample exactly at the surface: a finite amount of ma- terial is required for analysis, and this often translates into a layer of several mm thickness. Equation 3.27 is solved in that instance by less direct means. For example, consider the data shown in Figure 3. on a Pacific manganese nodule reported by Huh and Ku (1984). In this plot of log (^230 Th/^232 Th) vs. depth, the initial ratio is the intercept of the best-fit line through the data. A growth rate was obtained
Chapter 3: Geochronology II Spring 2011
81 February 28 , 2011
by obtaining an initial guess of the initial (^230 Th/^232 Th), then iteratively refining the solution to 3.27 by minimizing the differ- ence between computed and observed ac- tivity ratios. A growth rate of 49.5 mm/Ma and a (^230 Th/^232 Th) of 77.7 was found to best fit the observations. Some of the most successful applications of 230 Th dating have been in determining the age of corals. As we noted, carbonates, in- cluding corals, concentrate U and exclude Th. This leads to (^230 Th/^238 U) ratios much smaller than 1 (the equilibrium value); in- deed, 230 Th/^238 U in modern corals ap- proaches 0. However, the problem of de- termining ages, specifically the mathemat- ics, is complicated by the disequilibrium that will generally exist between 234 U and (^238) U. The relevant equation is:
230 Th
234 U
^
1 − e −^ λ^230 t
( 234 U / 238 U )
( 234 U / 238 U )
1 −^ e
−( λ 230 − λ 234 ) t ( ) 3.
The derivation is given in Dickin (1995). Note that because Th is essentially absent in corals, it is no longer useful to divide by the activity of 232 Th. 230 Th dating of fossil coral reefs has been extremely useful in determining the rise and fall of sea level over the last glacial cycle (140 ka). Now let’s consider an example of a high temperature application. Imagine a crystallizing magma with homogeneous (^230 Th/^232 Th) and (^238 U/^232 Th) ratios. Th and U will partition into different minerals to different degrees. The minerals will have homogeneous (^230 Th/^232 Th), (assuming crystallization oc- curs quickly compared to the half-life of Th) since these two isotopes are chemically identical, but vari- able (^238 U/^232 Th) ratios. Thus the minerals will plot on a horizontal line in Figure 3.12. After the system closes, 238 U and 230 Th will begin to come to radioactive equilibrium (either 230 Th will decay faster than it is produced or visa versa, depending on whether (^230 Th/^238 U) is greater than or less than 1, the equilib- rium value). Thus the original horizontal line will rotate, as in a conventional isochron diagram. The rotation occurs about the point where (^230 Th/^232 Th) = (^238 U/^232 Th) known as the equipoint. As t ap- proaches infinity, the exponential term approaches 1 and:
lim
t →∞
230 Th
232
Th
^
238 U
232
Th
^
Thus the equilibrium situation, the situation at t =∞, is (^230 Th/^232 Th) = (^238 U/^232 Th). In this case, all the minerals will fall on a line, having a slope of 1. This line is known as the equiline. In the past, activities were measured by alpha-counting and fairly large quantities of material were necessary. But improvements in mass spectrometry have made it possible to measure the 230 Th/^232 Th ratio (which is of the order of 10-^6 in igneous rocks) on relatively small quantities of material (and with better precision than α counting). The geochronological work done so far is mainly on cogenetic lavas, where the fractionation of U and Th presumably results from partial melting. Some interesting work on Mount Etna has been done this way. There is some hope that when these more sensitive analytical techniques are fully worked out, Th disequilibrium dating may provide a geochronological tool which will provide insights into volcanic and tectonic processes on relatively short (<100,000 yrs) time scales, for example, processes at mid-ocean ridges.
Figure 3.13. 230 Th/^232 Th as a function in depth in a man- ganese nodule from MANOP Site H. After Huh and Ku (1984).
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83 February 28 , 2011
We can write an equation analogous to the one equa- tion we wrote for unsupported 230 Th (3.23):
( 210 Pb ) u = ( 210 Pb ) u^ o^ e −^ λ^210 t^ 3.
Again letting t = z/s, we have:
( 210 Pb ) u = ( 210 Pb ) u^ o^ e −^ λ^210 z^ /^ s^ 3.
Thus we would expect the unsupported 210 Pb to decay exponentially with depth. It is generally not useful to ratio 210 Pb to one of the stable isotopes of Pb for several reasons. First, a decay- ing 238 U will pass through a stage as 222 Rn, a gas, before decaying to 210 Pb. Some of this gas will escape into the atmosphere. However, its half-life is only 4 days and the nuclides between 222 Rn and 210 Pb are all very short- lived, so this Rn will quickly decay to Pb. The Pb will be adsorbed on particles and washed out of the atmos- phere by rain. Thus the pathway of some 210 Pb to sediment will be different that that of normal Pb. Sec- ond, there has been a very substantial input of anthro- pogenic (stable) Pb to lakes and streams from leaded gasoline over the 20th^ century. For both these reasons, the variation of stable Pb with depth in a core will likely be quite different than that of 210 Pb. The first step in determining sediment ages using (^210) Pb is to estimate the supported (^210) Pb so it can be sub- tracted that from the total 210 Pb activity to determine unsupported activity. One easy solution results from the fact that 210 Pb should reach equilibrium in 5 to 10 half-lives. Thus sediment deposited more than 100- 200 years ago should be in equilibrium, so that the total ac- tivity is the supported activity. This requires the as- sumption that the flux of 210 Pb has been constant over the last 100-200 years. Alternatively, the activity of supported Pb can be determined by determining the ac- tivity of 226 Ra. Once the unsupported activity is calcu- lated, we can solve equation 3.31 for s , the sedimenta- tion rate. This can be done as follows. By taking the log of equation 3.31, we have
ln( 210 Pb ) u = ln( 210 Pb ) u^ o^ +
s
z 3.
Equation 3.32 is the equation of a straight line on a plot of ln(^210 Pb) vs depth, where ln(^210 Pb)^0 is the in- tercept and - λ 210 /s is the slope. Applying linear regression to the data in this form, we can determine both the slope and the intercept. From the slope, we can easily solve for s. Figure 3.156 illustrates an example calculated in this manner.
Figure 3.15. Unsupported 210 Pb activity measured on sediment samples from a box core taken from Cayuga Lake. The upper 4- 5 cm are probably disturbed, either by bio- turbation or in the coring process. The line shows the calculated decay of unsupported (^210) Pb assuming ( (^210) Pb) (^0) = 5.19 pCi and s= 0.319 cm/yr. This was calculated by aver- aging the first 2 points and discarding the last 3. Unpublished data of D. Engstrom.
