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How old is it? How do we know? A review of dating methods--part two: radiometric dating: mineral, Isochron and Concordia methods.

This second in a three-part series on dating methods reviews important radiometric dating methods that are applicable only to geological situations. The basic theory behind radiometric dating is outlined. Details of the [sup.40]K-[sup.40]Ar, [sup.87]Rb-[sup.87]Sr, [sup.238]U-[sup.206]Pb, [sup.87]Rb-[sup.87]Sr whole-rock isochron, [sup.147]Sm-[sup.143]Nd whole-rock isochron, and U-Pb concordia methods are outlined and examples of results are provided.

Within a decade of the discovery of radioactivity in 1896, the chemical elements uranium, radium, polonium, and thorium were recognized as radioactive; the basic theory, mechanisms, and mathematics of radioactive decay were established; decay rates of some radioactive chemical elements were determined; and the ages of two mineral specimens were calculated for the first time from their uranium and helium contents. (1) Soon thereafter, isotopes were discovered, the decay constants of a few specific isotopes were determined, and mass spectroscopy was developed. (2) Ever since these early heady days in the study of radioactivity, numerous radiometric dating methods have been proposed. Some methods were eventually discarded, (3) but many others have not only withstood critical scrutiny but have been refined and improved. (4)

Radiometric Dating

To understand how radiometric dating works, consider a crystal of pure rubidium chloride (RbCl). Rubidium (Rb) consists of 72.2% [sup.85]Rb, an isotope that is not radioactive, and 27.8% 8[sup.7R]b, an isotope that is radioactive. A cubic crystal of RbCl with a volume of 1 [cm.sup.3] contains approximately 1018 atoms of [sup.87]Rb. Although we cannot predict exactly when any specific atom of [sup.87]Rb will undergo spontaneous radioactive decay, we know that extremely large quantities of any radioactive isotope will disintegrate in accord with a radioactive decay law that is expressed by the equations:

dN/dt = -[lambda]N (1)

N = [N.sub.0][e.sup.-[lambda]t] (2)

where dN/dt is the rate of decay of a radioactive isotope, N is the present amount of a radioactive isotope (called the parent) in a sample, [N.sub.0] is the amount of parent isotope in the sample at the time of its formation, t is the age of the sample, and [lambda] is the decay constant of the parent isotope. Equation (1) indicates that the decay rate, dN/dt, is proportional to the amount of radioactive isotope in a sample, an amount that decreases with time as indicated by the minus sign. Equation (2) is derived from equation (1) by rearrangement and integration. Decay constants (Table 1) have now been measured by a variety of methods. To date, experimental evidence indicates that decay constants vary by no more than a very small fraction of one percent, if they vary at all. (5) Decay constants are virtually independent of the chemical composition of compounds containing the radioactive isotope and of the effects of temperature, pressure, gamma radiation, x-rays, high energy particles, and electrical and magnetic fields. No geologically or archeologically important radioactive isotope shows any significant variation in its decay constant.

Closely related to the decay constant is the half-life ([[tau].sub.1/2]) of a radioactive isotope, defined as the amount of time during which the amount of parent radioactive isotope in a sample decreases by one half. The amount of isotope into which the parent is transformed (the so-called daughter) increases as the parent decreases. If the amount of parent isotope in a sample equals half the original amount (N = 1/2 [N.sub.0]), then the age equals one half-life (t = [[tau].sub.1/2]). By substitution of these expressions into equation (2), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

an expression from which we can calculate that [[tau].sub.1/2] = 0.693/[lambda]. The half-life, therefore, is inversely proportional to the decay constant of a radioactive isotope. Table 1 also lists half-lives of several important radioactive isotopes.

Figure 1 graphically shows the amount of decay of parent and increase in amount of daughter isotopes in a sample containing no initial daughter as a function of the number of half-lives of the parent isotope that have elapsed. For example, after one half-life has elapsed, the amount of parent has decreased to one half of the original amount, and the amount of daughter has increased from zero to an amount equal to that of the parent. After two half-lives have elapsed, the amount of parent has decreased to one half of one half, that is, one quarter of the original amount. The amount of daughter will now be three times as much as that of the parent. After a third half-life has elapsed, the amount has decreased to one half of one quarter, that is, one eighth of the original amount, and so on.

[FIGURE 1 OMITTED]

Radioactive isotopes with extremely large half-lives and correspondingly low decay constants are useful in dating extremely old materials, because it takes a very long time for the parent isotope in a sample to diminish to extremely small amounts that would be difficult to analyze. Moreover, in a brief span of time, an insufficient amount of daughter isotope would be formed to permit its precise measurement. For example, [sup.238]U with a half-life of nearly 4.5 billion years is eminently suitable for dating minerals and rocks that are hundreds of millions to billions of years old. Isotopes with short half-lives and correspondingly larger decay constants are useful in dating very recent materials. Such isotopes cannot be used to date old material because the parent isotope decreases sufficiently rapidly that it may be virtually immeasurable in a matter of years. By way of example, [sup.14]C with a half-life of 5,730 years is suitable for dating materials that are only hundreds to a few tens of thousands of years old, but not millions of years.

To determine when a mineral or rock sample originally crystallized, the sample must have remained closed to both parent and daughter isotopes subsequent to its crystallization. In other words, neither parent nor daughter isotope should have entered into the sample from its environment or escaped from the sample into its environment during the course of the sample's post-crystallization history. If a quantity of parent isotope escaped from, or a quantity of daughter isotope entered into, a sample, then the calculated age would be higher than the actual age. Conversely, if a quantity of parent entered, or a quantity of daughter escaped from, the sample, the calculated age would be lower than the actual age. To improve the chances of dating a sample that acted as a closed system, geochronologists collect specimens that lack evidence of chemical weathering, are not extensively fractured, and are not associated with faults or shear zones. If the specimen acted as a closed system experiencing no gains or losses of the relevant isotopes, then the sum of parent and daughter in a sample at present must equal that sum at the time of formation. Thus, we may express the closed system condition by the equation

N + D = [N.sub.0] + [D.sub.0] (4)

where D is the present amount of daughter isotope in a sample at present and D0 is the amount of daughter isotope in the sample at the time of formation. By rearranging equation (4) to the form N = [N.sub.0] + [D.sub.0] - D, substituting that form of equation (4) into equation (2), and further rearranging, we obtain

D = [D.sub.0] + N([e.sup.[lambda]t] - 1) (5)

Equation (5) is the fundamental equation for radiometric dating methods that assume a closed system and for which concentrations of the relevant isotopes can either be measured directly or calculated from appropriate measurements. Because absolute numbers of atoms of some isotopes are difficult to measure, and because ratios of isotopes with very similar atomic masses can be measured accurately and precisely with a mass spectrometer, equation (5) may be modified while preserving the equality of both sides of the equation by dividing each side by the identical amount of a stable non-radiogenic isotope (d) of the same element as the daughter isotope. A stable non-radiogenic isotope is not radioactive, and it is either not produced by decay of a radioactive isotope at all or else it is produced at an exceedingly slow rate by a radioactive isotope with a half-life that far exceeds the age of the universe. For example, in rubidium-strontium (Rb-Sr) dating, [sup.86]Sr in a sample will serve as the stable non-radiogenic (see equation 8). For purposes of isotope analysis, [sup.87]Sr/[sup.86]Sr ratios are measured. The amount of the stable non-radiogenic isotope in the sample does not change through time in a closed system so that the amount of the stable non-radiogenic isotope in the sample now is the same as it was when the sample was formed (d = d0). Thus, when we divide both sides of equation (5) by d, we obtain

D/d = [D.sub.0]/[d.sub.0] + (N/d) ([e.sup.[lambda]t] - 1) (6)

This is a perfectly valid general equation for the radioactive decay of a specific isotope within a closed system.

How do we use this equation to obtain the age of a sample? We can measure both the present-day ratios of D/d and N/d in a sample, and we know the value of the decay constant. But that leaves us with two unknowns, t, the age of the sample, and [D.sub.0]/[d.sub.0]. To calculate the age of the sample, therefore, we must also know [D.sub.0]/[d.sub.0], the ratio of daughter isotope and the stable non-radiogenic isotope in the sample at the time of its origin. But if no analyst was present when a mineral was formed millions of years ago, how can we possibly know the value of the initial ratio? We shall soon see.

Mineral Dating

Let's now look at some examples of various dating methods. In the process, we will see how the determination of original isotope ratios is addressed. Consider a crystal of a potassium-rich mineral like potassium feldspar or biotite in a volcanic ash deposit or a rhyolitic lava flow. We can ordinarily date the mineral by the [sup.40]K-[sup.40]Ar method. In this case, the problem of knowing the initial isotope ratios or amounts is avoided because essentially no daughter isotope is initially present ([sup.40]A[r.sub.0] = 0)6. Very little radiogenic daughter [sup.40]Ar is incorporated into most minerals when they crystallize because argon is an inert gas that is typically excluded from their crystal structures. Known exceptions are avoided in routine K-Ar dating. Any [sup.40]Ar already present in a cooling magma is also likely to be degassed during eruption or trapped in glass. Most 40K atoms decay to [sup.40]Ca by [beta] emission, but a small percentage of [sup.40]K atoms decays to [sup.40]Ar by electron capture. (7) The operative equation based on equation (5) is

[sup.40]Ar = [sup.40]A[r.sub.0] + ([[lambda].sub.ec]/[lambda])[sup.40]K ([e.sup.[lambda]t] - 1) (7)

where [[lambda].sub.ec]/[lambda] is the fraction of decays to [sup.40]Ar by electron capture. As noted, we assume that [sup.40]A[r.sub.0]=0. [sup.40]K is determined by measuring the total K concentration in the mineral by flame photometry or x-ray fluorescence and multiplying by 0.01167, the [sup.40]K/total K ratio in all K-bearing materials. The concentration of radiogenic [sup.40]Ar is calculated from argon isotope ratios that are measured by mass spectrometry. (8)

The K-Ar method is also useful for dating biotite, muscovite, hornblende, and feldspar in coarse-grained igneous rock bodies such as granite or gabbro that solidify far beneath Earth's surface. The coarse grains of such rocks result from the very slow cooling of magma in the subsurface. The rocks continue to cool very slowly after crystallization. As these rocks cool, [sup.40]Ar produced by radioactive decay of 40K easily diffuses out of the mineral structure in which it originally forms thanks to intense thermal vibration of atoms at high temperature. Consequently, an insignificant amount of Ar accumulates in minerals at very high temperatures. Not until the rock cools to a so-called closure (or blocking) temperature does a mineral begin to retain [sup.40]Ar atoms that were generated by 40K decay, because the atoms are no longer vibrating with sufficient energy to diffuse out of the mineral grains in significant quantities. Therefore, K-Ar ages of minerals in plutonic igneous rocks generally represent cooling ages that are somewhat younger than the age of original crystallization. Normally the cooling ages of different minerals in the same igneous rock body differ from one another because the closure temperatures for Ar retention are different in different minerals. (9)

As examples of K-Ar ages, potassium feldspar crystals in pumice fragments from the Bishop Tuff in California, a volcanic ash deposit, consistently yield ages of approximately 725,000 years. (10) K-Ar ages on biotite crystals from the Mount Givens Granodiorite and El Capitan Granite in the Sierra Nevada of California are 84, 86, and 89 million years for the former and 92 and 94 million years for the latter. (11)

Geochronologists formerly dated feldspar and the K-bearing mica minerals biotite, muscovite, and lepidolite by the [sup.87]Rb/[sup.87]Sr method.12 By substitution of [sup.87]Rb for N,

[sup.87]Sr for D, [sup.87]S[r.sub.0] for [D.sub.0], and [sup.86]Sr for d and d0 into equation (6), we obtain

[sup.87]Sr/[sup.86]Sr = ([sup.87]Sr/[sup.86]Sr)0 + ([sup.87]Rb/[sup.86]Sr) ([e.sup.[lambda]t] - 1) (8)

[sup.87]Sr/[sup.86]Sr is determined by mass spectrometry, and [sup.87]Rb/[sup.86]Sr is calculated from the measured Rb/Sr ratio and relative abundances of [sup.87]Rb/total Rb and [sup.86]Sr/total Sr, the latter determined by mass spectrometric measurements of several Sr isotope ratios. Again, how can we determine the original Sr isotope ratio? To address the problem, geochronologists commonly assumed an initial ([sup.87]Sr/[sup.86]Sr)0 ratio such as 0.704, a ratio that is commonly found in recently erupted oceanic lava flows. Because the initial ratio was assumed, the resulting ages were not as reliable as desired. In addition, Rb and Sr are sufficiently mobile elements that the closed system condition is not always met. As a result, this method is rarely if ever used today. Zircon grains, which are especially resistant to both chemical and physical weathering, are commonly dated by the U-Pb method. (13) Via a long chain of decays, 238U ultimately decays to [sup.206]Pb, and [sup.204]Pb is used as the stable non-radiogenic isotope. By substitution of 238U for N, [sup.206]Pb for D, [sup.206]P[b.sub.0] for [D.sub.0], and [sup.204]Pb for d and d0 into equation (6), we obtain

[sup.206]Pb/[sup.204]Pb = ([sup.206]Pb/[sup.204]Pb)0 + ([sup.238]U/[sup.204]Pb) ([e.sup.[lambda]t] - 1) (9)

The [sup.206]Pb/[sup.204]Pb ratio in zircon grains is typically measured by mass spectrometry. (14) 238U/[sup.204]Pb is calculated from measured [sup.238]U/total U and [sup.204]Pb/total Pb ratios. But how do we find the original lead ratio, [([sup.206]Pb/[sup.204]Pb).sub.0]? Consider that zircon (ZrSi[O.sub.4]) invariably contains some U because U4+ ions are sufficiently similar in size to [Zr.sup.4+] ions that they can substitute for Zr during crystallization of zircon. In contrast, [Pb.sup.2+] has a significantly different charge and is too large to substitute for [Zr.sup.4+]. Consequently, the initial lead content of most zircons would have been inconsequential. The original Pb isotope ratio term [([sup.206]Pb/[sup.204]Pb).sub.0] can, therefore, normally be ignored. But what if we attempt to date a mineral other than zircon? Other U-bearing minerals, such as uraninite, may contain some initial Pb. How do we assess the initial Pb isotope ratio in such cases? Here we may reasonably assume that the initial lead isotope ratio was identical to the lead isotope ratio in minerals such as feldspar and galena that coexist in the same rock with the mineral we wish to date. We can do this because both feldspar and galena lack radioactive uranium and would not have produced any [sup.206]Pb during their history. Consequently, the initial [sup.206]Pb/[sup.204]Pb ratio would not increase through time.

As examples of [sup.238]U/[sup.206]Pb dating, zircon from the Mount Givens Granodiorite in the Sierra Nevada yielded ages of 87.9 and 87.6 million years, and zircons from the El Capital Granite in Yosemite National Park have ages of 102.8 and 96.9 million years. (15) Note that these U-Pb ages are higher than the K-Ar ages mentioned above, a finding that is consistent with the idea that the U-Pb data represent crystallization ages and the K-Ar data represent cooling ages.

Isochron Methods

To obtain crystallization ages, geochronologists also commonly employ the graphical isochron and concordia methods. We begin with the isochron method. The [sup.87]Rb/[sup.87]Sr and [sup.147]Sm/[sup.143]Nd decay systems are particularly amenable to treatment by the isochron method. (16) With [sup.87]Rb/[sup.87]Sr as an example, let's rearrange equation (8) slightly by switching the terms on the right side to obtain

[sup.87]Sr/[sup.86]Sr = ([sup.87]Rb/[sup.86]Sr) ([e.sup.[lambda]t] - 1) + [([sup.87]Sr/[sup.86]Sr).sub.0] (10)

This equation has the same form as the equation for a straight line: y = mx + b. Thus, if we plot a straight line on a diagram with [sup.87]Sr/[sup.86]Sr as the vertical y-axis (ordinate) versus [sup.87]Rb/[sup.86]Sr as the horizontal x-axis (abscissa), [([sup.87]Sr/[sup.86]Sr).sub.o] is the y-intercept, b, and ([e.sup.[lambda]t] -1) is the slope, m, of the straight line (Figure 2). Notice that the slope of the straight line for equation (10) is a function of t, the age of a sample containing Rb and Sr. Obviously then it would be nice to generate a straight line based on information about Rb and Sr so that we could obtain an age from the slope of the line. How do we do that? We produce this line is by determining the present [sup.87]Sr/[sup.86]Sr and [sip.87]Rb/[sup.86]Sr ratios of several rock samples from an igneous intrusion or a mass of metamorphic rocks such as gneiss and then plotting these ratios on a diagram of [sup.87]Sr/[sup.86]Sr versus [sup.87]Rb/[sup.86]Sr. The straight line that is generally obtained is termed a whole-rock isochron. Attainment of a straight line plot from the analytical data suggests that the rocks we analyzed were essentially closed to Rb and Sr diffusion and that all samples had the same initial ratio at the time of igneous or metamorphic crystallization.

[FIGURE 2 OMITTED]

To understand why the whole-rock isochron method works and why we might obtain a straight line that yields age information, let's try to visualize the major geologic process of magmatic crystallization. Envision a mass of magma, that is, intensely hot molten (normally silicate) rock, in Earth's crust that contains trace amounts of [sup.87]Rb, [sup.87]Sr, and [sup.86]Sr. Different isotopes of a chemical element are likely to fractionate during geologic processes such as evaporation, crystallization from magma, or diffusion in a temperature gradient in a magma if the masses of the isotopes differ from one another significantly. For example, hydrogen isotopes strongly fractionate during evaporation of water on Earth's surface. Light hydrogen ([sup.1]H) atoms are preferentially incorporated into water vapor whereas heavy hydrogen atoms known as deuterium (2H) are preferentially incorporated into liquid water. As a result, the [sup.1]H/[sup.2]H ratio in water vapor that coexists with liquid water is much higher than the [sup.1]H/[sup.2]H ratio in the liquid water. This strong isotopic fractionation occurs because of the great difference in mass between the two isotopes of hydrogen. A heavy hydrogen isotope is twice as massive as a light hydrogen isotope.

In contrast, the atomic masses of [sup.87]Sr and [sup.86]Sr are very similar. Because a [sup.87]Sr atom is only 1.2% heavier than a [sup.86]Sr atom, Sr isotopes should fractionate only very slightly during geologic processes. Therefore, the ratio of these two isotopes should normally have much the same value throughout a body of cooling magma. As the magma cools, minerals eventually begin to crystallize. Each mineral has a different chemical composition and crystal structure. [Rb.sup.1+] ions readily substitute for [K.sup.1+] ions in growing crystals because the two ions have similar sizes (ionic radii) and identical electrical charges, and [Sr.sup.2+] ions likewise substitute for [Ca.sup.2+] ions. There will be variation in [sup.87]Rb/[sup.86]Sr ratios throughout the newly crystallized igneous rock mass because of variation in mineral abundances. For example, K-rich and Ca-poor domains of a magma body will crystallize to rocks with high [sup.87]Rb/[sup.86]Sr initial ratios, and K-poor and Ca-rich domains will produce rocks with low [sup.87]Rb/[sup.86]Sr initial ratios. If we could analyze the isotope ratios of several chunks of rock as soon as the magma crystallized and plot the results on an isochron diagram, we would normally obtain a virtually horizontal line (Figure 2). The slope of a horizontal straight line is zero, indicating a zero age for a suite of rocks that have just crystallized.

As the granite becomes older, [sup.87]Rb in each volume of rock spontaneously decays to [sup.87]Sr. The [sup.87]Rb/[sup.86]Sr ratio of each volume of granite decreases and the [sup.87]Sr/[sup.86]Sr ratio increases in each sample through time because [sup.87]Sr is being added to the rock by decay of [sup.87]Rb. Data points on the isochron diagram would shift their positions as shown by the lines with arrows in Figure 2. All data points would shift through time along paths with the same slope. Points representing rock samples with the highest initial [sup.87]Rb/[sup.86]Sr ratios would shift the most because they have higher [sup.87]Rb contents, and a greater amount of [sup.87]Rb decays into a greater amount of [sup.87]Sr. As the granite body ages, the farther the data points would shift. Through time, the isochron would pivot around the y-intercept, [([sup.87]Sr/[sup.86]Sr).sub.0], and the slope of the isochron would become steeper, indicating that the rocks are getting older. Although we cannot observe this shift in position of the isochron in real time (we have not lived long enough!), we can obtain information about the age of the granite from knowledge of the isotope ratios that we analyze today. From the diagram, we can calculate the time of crystallization of the granite from the slope of the whole-rock isochron, find the initial [sup.87]Sr/[sup.86]Sr ratio from the y-intercept, and infer lack of significant diffusion of Rb or Sr from strong co-linearity of points. Significant scatter of analytical data points suggests that there may have been substantial diffusion and that the age calculated from the slope of a poorly defined isochron is somewhat suspect. Some variation in initial Sr isotope ratio may also produce slight scatter of data points that nonetheless define a straight line with reasonably good fit.

As an example, a Rb-Sr whole-rock isochron for the Stolzburg pluton in the Barberton Mountains of South Africa, indicates a probable crystallization age of 3.48 billion years (Figure 3). (17) The slight scatter of data points in this isochron plot suggests possible minor diffusion or slight variation in initial Sr isotope ratios. As another example, a remarkably tightly constrained whole-rock Rb-Sr isochron for 38 samples of several different stony meteorites indicates a meteorite crystallization age of 4.498 billion years (Figure 4). (18)

[FIGURES 3-4 OMITTED]

Internal or mineral isochrons are constructed from isotopic analyses of individual minerals that have been extracted from a single large rock sample. Such isochrons sometimes include an analysis of the whole-rock sample. Internal isochrons are especially useful for determining crystallization ages of individual meteorite fragments and fragments of returned lunar samples. Figure 5 shows an internal isochron from a lunar basalt fragment that indicates an age of 3.09 billion years. (19)

[FIGURE 5 OMITTED]

Conceivably, straight lines on an isochron diagram could be generated simply by mixing two magmas in various proportions and allowing them to crystallize. If so, the resulting straight line would tell us nothing about the age of the rocks. The vast majority of straight lines of an isochron plot, however, cannot be explained by simple mixing for several reasons. We mention only two. For example, evidence of extensive magma mixing can generally be detected from variations in chemical composition of an igneous rock body as well as from field evidence. For most igneous rock bodies, however, field and chemical evidence of significant mixing of two or more magmas is lacking. In addition, the mixing hypothesis fails to account for the very wide range of values of Sr isotope ratios that have been observed on all isochron diagrams of igneous rocks.

The Sm-Nd decay is also amenable to treatment by isochron methods. Sm-Nd isochrons have an advantage over Rb-Sr isochrons in that rare-earth elements (REE) like Sm and Nd are relatively immobile during geologic heating events, and, therefore, are unlikely to diffuse into or out of rocks. As a result, the isotope ratios of these elements are not significantly changed during geologic disturbances such as metamorphism. Evidence of the original crystallization age of an igneous intrusion, therefore, is normally preserved even though that body may have been metamorphosed subsequently. This characteristic feature means that Sm-Nd whole-rock isochron dating is especially important for dating exceptionally old rocks in the one to four billion year range. As an example, Sm-Nd whole-rock isochron data obtained from metamorphosed volcanic rocks in the Barberton Mountains indicate an age of 3.53 billion years (Figure 6). (20)

[FIGURE 6 OMITTED]

Several additional isochron methods have been developed in recent decades based on the [beta] decay of [sup.138]La to [sup.138]Ce, of [sup.187]Re to [sup.187]Os, and of [sup.176]Lu to [sup.176]Hf. (21) Although none of these methods is as widely useful as Sm-Nd or Rb-Sr, they offer promise in the investigation of igneous rocks rich in iron-magnesium minerals, iron meteorites, or certain ore samples--geological materials that are not so readily datable by the Rb-Sr and Sm-Nd isochron method.

The Concordia Method

The concordia method, another powerful graphical method, is very useful in evaluating radiometric data for the U-Th-Pb system. U- and Th-bearing minerals can be dated by analyzing 238U and its ultimate daughter product [sup.206]Pb, [sup.235]U and its ultimate daughter product [sup.207]Pb, and [sup.232]Th and its ultimate daughter product [sup.208]Pb. In addition, the lead-lead method provides another calculation that is based on the ratio of radiogenic [sup.207]Pb to radiogenic [sup.206]Pb. (22) There are, therefore, four independent methods for calculating the time of crystallization of a U-Th-bearing mineral such as zircon in an igneous rock body. The difficulty is that most U-Th-Pb analyses of the same sample yield discordant, i.e., different, ages by the four different methods because lead can be lost from U-bearing minerals either by slow continuous leakage or in discontinuous episodes, most likely caused by heating. The Pb/Pb method typically yields higher ages than those obtained by the other three methods. The reason is that the [sup.207]Pb/[sup.206]Pb ratio of any lead that is lost from a U-bearing sample should be virtually the same as the isotope ratio of lead that is retained in the sample. The masses of the two Pb isotopes are so similar that there would be negligible isotope fractionation during lead loss. Because the age in this method is calculated from the [sup.207]Pb/[sup.206]Pb ratio, a reliable age should be obtained even if some lead is lost. In the other three methods, however, if lead is lost at all, a mineral will yield a spuriously young age because a daughter isotope, either [sup.208]Pb, [sup.207]Pb, or [sup.206]Pb, is lost, and the ages are based on the ratios of 238U/[sup.206]Pb, 235U/[sup.207]Pb, or [sup.232]Th/[sup.208]Pb.

The concordia method was developed to deal with these problems of lead loss and discordance among ages obtained for the same sample by different U-Th-Pb methods. 23 The most important concordia diagram plots [sup.206]Pb/238U (y-axis) versus [sup.207]Pb/235U (x-axis). The diagram (Figure 7) includes a curve, termed concordia, that shows how the values of these two ratios change continuously over time for any hypothetical sample that has been completely closed to U and Pb and for which all of the Pb is radiogenically derived from U decay. Because both 238U and 235U decrease and [sup.206]Pb and [sup.207]Pb increase through time due to radioactive decay, the [sup.206]Pb/[sup.238]U and [sup.207]Pb/[sup.235]U ratios must both increase in a closed system, resulting in the positive slope of concordia.

[FIGURE 7 OMITTED]

Now suppose that a U-bearing zircon is analyzed for U and Pb, that any necessary correction is made for original Pb isotopes in the sample, that the appropriate Pb/U ratios are plotted on the diagram, and that the data point for the sample analysis plots exactly on concordia. A reasonable conclusion would be that the position of the point on concordia indicates the crystallization age of the zircon. In many studies, however, data points obtained from several analyses of a suite of zircons from a rock body define a straight line, called discordia that intersects concordia at two points (Figure 7). Even if no data points lie on concordia, the upper intercept of discordia indicates the time of original crystallization of the zircons, and the lower intercept of the straight line with concordia presumably indicates a time of disturbance when Pb was lost in varying amounts from the analyzed zircons. If lead had been lost from individual zircons during some geologic event in the past, their Pb/U ratios would decrease, and the more lead that they lost, the lower their Pb/U ratios would become even though the ratio of lead isotopes ([sup.207]Pb/[sup.206]Pb) would not change. If the zircons were analyzed immediately after an episode of lead loss, the data points would define a line intersecting the origin where both Pb/U ratios are zero. Any zircon that lost all of its lead (Pb = 0) would plot at the origin because the Pb/U ratios would both be zero. After the episode of lead loss is over, the radioactive decay of U continues and accumulation of Pb in the zircons resumes, and both Pb/U ratios of all zircons begin to increase again. Through time, the straight line representing the time of lead loss rotates into the position defined by the zircon analyses at present (Figure 7). In this scenario, we obtain the original crystallization age and the age of lead loss episode.

There are other instances in which lead may have been lost continuously or in two or more discrete episodes of lead loss. These cases are much more complex and will not be treated here. Suffice it to say, however, that the concordia method demonstrates that, for U-Th-Pb systems, the closed system condition does not always have to be met to make use of it for dating purposes.

As examples of the concordia method, consider the following. Zircon separates from the Duffer Formation in Western Australia (Figure 8) define a discordia that intersects concordia at 3.45 billion years. (24) Five zircon separates from the Gamitagama Lake Complex, an igneous intrusion in southern Canada, define an excellent straight-line discordia that intersects concordia at 2.67 billion years, and five zircons from another intrusion, the nearby Kasubeck Creek Granodiorite, also define a very good straight line that intersects concordia at 2.67 billion years.25 These ages are close to those of volcanic rocks in the area. One additional example concerns the Qorqut Granite from western Greenland. (26) Seventeen zircon separates define a very good straight line discordia that intersects concordia at 2.53 billion years, whereas zircons from the Amitsoq gneisses that were intruded by the granite give an intersection age of 3.595 billion years.

[FIGURE 8 OMITTED]

Geologists now have at their disposal a wide array of radiometric methods that are capable of yielding highly reliable information about the ages of igneous crystallization, metamorphic recrystallization, cooling, lead-loss, and other events for many kinds of samples provided that the methods are used in conjunction with careful sample collection, precise analysis, understanding of the geologic conditions that affected the samples being dated, and judicious interpretation of the geologic significance of the age that is obtained. My final article, to appear in the September 2007 issue, will examine thermochronometric methods as well as methods based on the radioactive decay of cosmogenically produced isotopes. It will conclude with a brief reflection on the theological implications of belief in a very ancient Earth.

Notes

(1) Henri Becquerel discovered radioactivity in 1896 while working with uranium compounds. Marie and Pierre Curie discovered the radioactive nature of the chemical elements thorium, radium, and polonium, and Pierre Curie coined the term "radioactivity." In 1903, Ernest Rutherford and Frederick Soddy worked out the general theory and mathematics of radioactive decay, recognized the mechanisms of [alpha] and [beta] particle emission, and postulated that helium is an important by-product of radioactive decay. Two years later, Rutherford calculated the ages of two mineral samples from measurements of their uranium and helium contents.

(2) In 1907, Bertram Boltwood separated "ionium," a substance chemically indistinguishable from thorium but with a different atomic mass. In 1912, Soddy applied the term "isotope" to such apparently different forms of the same chemical element, and J. J. Thomson provided experimental confirmation of the existence of isotopes. In 1919, Frederick Aston constructed the first mass spectrograph and by 1927 had determined the isotope composition of more than fifty chemical elements.

(3) For a discussion of some of the earlier, but now abandoned, methods see Henry Faul, ed., Nuclear Geology: A Symposium on Nuclear Phenomena in the Earth Sciences (New York: John Wiley and Sons, 1954).

(4) For details on geologically important radiometric dating systems, see Gunter Faure and Teresa M. Mensing, Isotopes: Principles and Applications, 3d ed. (Hoboken, NJ: Wiley, 2005) and Alan P. Dickin, Radiogenic Isotope Geology (Cambridge: Cambridge University Press, 1997). Also of great value are the summaries in G. Brent Dalrymple, The Age of the Earth (Stanford, CA: Stanford University Press, 1991) and G. Brent Dalrymple, Ancient Earth, Ancient Skies: The Age of Earth and Its Cosmic Surroundings (Stanford, CA: Stanford University Press, 2004).

(5) For reviews of experimental studies of changes in rates of radioactive decay, see G. T. Emery, "Perturbation of Nuclear Decay Rates," Annual Review of Nuclear Science 22 (1972): 165-202 and H.-P. Hahn, H. J. Born, and J. I. Kim, "Survey on the Rate Perturbation of Nuclear Decay," Radiochimica Acta 23 (1976): 23-37. As early as 1903, Rutherford and Soddy performed experiments under a wide range of chemical and physical conditions and detected no change in rates of radioactive decay.

(6) OnK-Ar dating, see G. Brent Dalyrmyple and Marvin A. Lanphere, Potassium-Argon Dating: Principles, Techniques, and Applications to Geochronology (San Francisco: W. H. Freeman, 1969). See also chapter 6 in Faure and Mensing, Isotopes: Principles and Applications.

(7) In [beta] decay, a neutron of a radioactive isotope such as [sup.40]K is transformed to a proton plus an electron ([beta] particle) which is emitted from the nucleus. As a result, the atomic number (Z) of a daughter isotope such Ca (Z = 20) is one more than that of the parent K (Z = 19) whereas the atomic mass remains the same (40) because the sum of protons and neutrons remains constant, a neutron being replaced by a proton. In electron capture, an electron in an inner shell is captured by the nucleus. The combination of electron with a proton creates a neutron. As a result, the atomic number of the daughter is reduced by one with the loss of the proton but the atomic mass remains the same as the proton is replaced by a neutron. Thus, [sup.40]Ar produced by decay of [sup.40]K due to electron capture has Z = 18.

(8) In mass spectrometry, isotopes of interest in either gaseous or solid samples are ionized. The beam of positively charged ions is deflected by a magnetic field as it is passed through a curved electromagnet. The radius of curvature of beam paths of each isotope is dependent on isotopic mass. The paths of heavier isotopes are deflected less than those of lighter isotopes. The magnetically separated ion beams are focused onto a collecting cup where beam intensities are measured. Mass spectrometric analysis of isotope ion beams is carried out in a near vacuum. Older mass spectrographs measured isotope abundances photographically.

(9) The K-Ar closure (blocking) temperature of biotite is approximately 325[degrees]C and that of hornblende is approximately 475[degrees]C. The closure temperatures are a function of the rate of cooling and grain size of the mineral. In actuality, as a K-bearing mineral cools, it passes through a temperature range during which Ar retentivity increases until below the closure temperature the Ar retention is 100%.

(10) G. Brent Dalrymple, "K-Ar Ages of the Friant Pumice Member of the Turlock Lake Formation, the Bishop Tuff and the Tuff of Reds Meadow, Central California," New Mexico Bureau of Mines and Mineral Resources, Isochron/West, no. 28 (1980): 3-5.

(11) T. W. Stern, P. C. Bateman, B. A. Morgan, M. F. Newell, and D. L. Peck, "Isotopic U-Pb Ages of Zircon from the Granitoids of the Central Sierra Nevada," California United States Geological Survey Professional Paper 1185 (1981).

(12) For fundamentals of Rb-Sr dating, see Louis H. Ahrens, "Measuring Geologic Time by the Strontium Method," Geological Society of America Bulletin 60 (1949): 217-66.

(13) For details of the U-Pb method, see Chapter 10 in Faure and Mensing, Isotopes: Principles and Applications.

(14) Small mineral grains such as tiny zircon crystals can be analyzed with the aid of a Sensitive High-Resolution Ion Microprobe (SHRIMP), an instrument that bombards a mineral surface with high-energy ions that cause atoms to be sputtered from the surface of the sample. These atoms are then analyzed by mass spectrometry.

(15) Stern, et al., Isotopic U-Pb Ages of Zircon from the Granitoids of the Central Sierra Nevada, California.

(16) For details on the Rb-Sr and Sm-Ndisochronmethods, see chapters 5 and 9 in Faure and Mensing, Isotopes: Principles and Applications.

(17) J. M. Barton, Jr., L. J. Robb, C. R. Anhaeusser, and D. A.Van Nierop, "Geochronologic and Sr-Isotopic Studies of Certain Units in the Barberton Granite-Greenstone Terrane, South Africa," in C. R. Anhaeusser, ed., Contributions to the Geology of the Barberton Mountain Land (The Geological Society of South Africa Special Publication No. 9, 1983), 63-72.

(18) J.-F. Minster, J.-L. Birck, and C. J. Allegre, "Absolute Age of Formation of Chondrites Studied by the [sup.87]Rb-[sup.87]Sr Method," Nature 300 (1982): 414-9.

(19) L. E. Nyquist, C.-Y. Shih, J. L. Wooden, B. M. Bansal, and H. Wisemann, "The Sr andNdIsotopic Record of Apollo 12 Basalts: Implications for Lunar Geochemical Evolution," Proceedings of the 10th Lunar and Planetary Science Conference (1979): 77-114.

(20) P. J. Hamilton, N. M. Evensen, R. K. O'Nions, H. S. Smith, and A. J. Erlank, "Sm-Nd Dating of Onverwacht Group Volcanics, Southern Africa," Nature 279 (1979): 298-300.

(21) On the Lu-Hf, Re-Os, and La-Ca methods, see chapters 12-14 in Faure and Mensing, Isotopes: Principles and Applications.

(22) On various U-Th-Pb methods, see Chapters 10 and 11 in Faure and Mensing, Isotopes: Principles and Applications. Both the U and Th decay chains include [alpha] and [beta] decays. An [alpha] particle consists of two protons and two neutrons, in essence the nucleus of a helium atom. Consequently, helium is an important by-product of both U and Th decay. The initial step in the decay chain of [sup.238]U is the decay of 238U to [sup.234] Th in which the atomic number of the daughter is two less than that of the parent (90 vs. 92) because of the loss of two protons, and the atomic mass decreases by four (234 from 238) because of the loss of two protons and two neutrons.

(23) George Wetherill, "Discordant Uranium-Lead Ages," Transactions of the American Geophysical Union 37 (1956): 320-36.

(24) R. T. Pidgeon, "3450-m.y.-old Volcanics in the Archaean Layered Greenstone Succession of the Pilbara Block, Western Australia," Earth and Planetary Science Letters 37 (1978): 421-8.

(25) T. E. Krogh and A. Turek, "Precise U-Pb Zircon Ages from the Gamitagama Greenstone Belt, Southern Superior Province," Canadian Journal of Earth Sciences 19 (1982): 859-67.

(26) H. Baadsgaard, "Further U-Pb dates on Zircons from the Early Precambrian Rocks of the Godthaabsfjord Area, West Greenland," Earth and Planetary Science Letters 33 (1976): 261-7.

After graduating from Princeton University, Davis A. Young received a M.S. degree from Penn State and a Ph.D. degree in geological sciences from Brown University in 1969. Dave then taught geology for five years at New York University, five years at the University of North Carolina at Wilmington, and 26 years at Calvin College. Upon retirement from Calvin in 2004, Dave and his wife Dottie moved to Tucson, Arizona, where they enjoy the view of the Santa Catalina Mountains and Dave vigorously pursues his hobby of birding. A fellow of ASA, Dave continues service as a member of the Editorial Board of Perspectives on Science and Christian Faith. He and his wife are members of Catalina Foothills Presbyterian Church in America.
Table 1. Decay Constants and Half-Lives
Geologically Important Radioactive Isotopes

Radioactive Decay Constant Half-Life
Isotope ([years.sup.-1]) (years)

[sup.14]C 0.1209 x [10.sup.-3] 5730

[sup.40]K 0.581 x [10.sup.-10] 1.19 x [10.sup.10]

[sup.87]Rb 1.42 x [10.sup.-11] 48.8 x [10.sup.9]

[sup.147]Sm 0.654 x [10.sup.-12] 1.06 x [10.sup.11]

[sup.232]Th 4.9475 x [10.sup.-11] 14.01 x [10.sup.9]

[sup.235]U 9.8485 x [10.sup.-10] 0.7038 x [10.sup.9]

[sup.238]U 1.55125 x [10.sup.-10] 4.468 x [10.sup.9]
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