Earth Materials. John O'Brien

Figure 3.13 Progressive change in the proportions of radioactive parent (N) and daughter (D) isotopes over time, in terms of number of half‐lives.

      More generally, the age of any sample may be calculated from the following equation:

      where t = age, λ = the decay constant, d = number of stable daughter atoms and p = number of radioactive parent atoms. Where stable daughter atoms were present in the original sample, a correction must be made to account for them, as explained below.

Many radioactive to stable isotope decay series, each with a unique decay constant, can be used, often together, to determine robust formation ages for Earth materials, especially for older materials. Table 3.3 summarizes some common examples. Three of these are discussed in more detail in the sections that follow.

       Uranium–lead systematics

      Uranium (U) occurs in two radioactive isotopes, both of which decay in steps to different stable isotopes of lead (Pb). The closely related actinide element thorium (Th) decays in a similar fashion to yet another stable isotope of lead. The essential information on the radioactive and stable isotopes involved for these three decay series (²³⁸U →²⁰⁶Pb,²³⁵U →²⁰⁷Pb, and²³²Th → ²⁰⁸Pb) is summarized in Table 3.3. This includes the types of decay processes, their half‐lives, and the useful age range for dating by each of these methods.

      Of the three decay series, the most commonly used is²³⁸U → ²⁰⁶Pb. This is because²³⁸U is much more abundant than the other two radioactive isotopes and thus it and its daughter product are easier to measure accurately. Typically, heavy minerals, such as zircon, sphene, and/or monazite, are used for analyses because they contain substantial actinides, are relatively easy to separate and resist chemical alteration.

      A problem arises because some²⁰⁶Pb occurs naturally in Earth materials, without radioactive decay, so that measured²³⁸U/²⁰⁶Pb ratios include both daughter and nondaughter²⁰⁶Pb. This must be corrected if an accurate age is to be determined. The correction can be accomplished because yet another lead isotope,²⁰⁴Pb, is not produced by radioactive decay. The ratio of²⁰⁴Pb/²⁰⁶Pb in meteorites, formed at the time Earth formed, is known and is taken to be the original ratio in the rock whose age is being determined. The correction is then accomplished by measuring the amount of²⁰⁴Pb in the sample and subtracting the meteoritic proportion of²⁰⁶Pb from total²⁰⁶Pb to arrive at the amount of²⁰⁶Pb that is the daughter of²³⁸U.

In most analyses, the ratio of²³⁵U/²⁰⁷Pb is also determined. The²³⁵U/²⁰⁷Pb decay series has a much shorter half‐life than the²³⁸U/²⁰⁶Pb series, so that the U/Pb ratios are much smaller for a given sample age. Once the ratios of both parent uranium isotopes to the corrected daughter isotopes (²³⁸U/²⁰⁶Pb,²³⁵U/²⁰⁷Pb) have been determined, sample ages are generally determined on a concordia plot (Figure 3.14) that displays the expected relationship between the reciprocal ratios²⁰⁶Pb/²³⁸U and²⁰⁷Pb/²³⁵U as a function of sample age.

Figure 3.14 Uranium–lead concordia plot (blue) showing sample ages as a function²⁰⁶Pb/²³⁸U and²⁰⁷Pb/²³⁵U ratios and samples with lower²⁰⁶Pb/²³⁸U and²⁰⁷Pb/²³⁵U ratios (red), reset during a 2.5 Ga metamorphic event.

      Either decay series is capable of producing a sample age on its own. For example, a²⁰⁶Pb/²³⁸U ratio of 0.5 indicates an age of approximately 2.6 Ga (Figure 3.14), while a²⁰⁷Pb/²³⁵U ratio of 40 indicates an age of approximately 3.7 Ga. One advantage of these decay series is that the measurement of multiple, closely related isotopes permits ages to be checked against each other and provides robust sample ages when both methods yield a closely similar age. For example, if a sample possesses a²⁰⁶Pb/²³⁸U ratio of 0.7 and a²⁰⁷Pb/²³⁵U ratio of 32, a sample age near 3.5 Ga (Figure 3.14) is highly likely. When two age determinations are in agreement, their ages are said to be concordant. Another way to state this is that samples that plot on the concordia line when their²⁰⁶Pb/²³⁸U versus²⁰⁷Pb/²³⁵U ratios are plotted yield concordant ages. Such dating techniques are especially powerful in determining robust crystallization ages of igneous rocks that crystallized from magmas and lavas (Chapters 8 and 9).

A significant problem arises from the fact that subsequent events can alter the geochemistry of rocks, so that age determinations are no longer concordant. For example, when rocks undergo metamorphism, lead is commonly mobilized and lost from the minerals or rocks in question. This results in lower than expected amounts of daughter lead isotopes, lower Pb/U ratios and points that fall below the concordant age line on²⁰⁶Pb/²³⁸U versus²⁰⁷Pb/²³⁵U diagrams. When samples of similar real ages, affected differently during a single metamorphic event, are plotted on a concordia diagram, they fall along a straight line below the concordant age line. One interpretation of such data is that the right end of the dashed straight line intersects the concordant age line at the original age of the sample (4.0 Ga in Figure 3.14) and that the left end intersects the concordant age line at the time of metamorphism (2.5 Ga in Figure 3.14). Wherever possible, such interpretations should be tested against other data so that a robust conclusion can be drawn. One such independent radiometric decay series, with widespread application, is described in the section that follows.

       Rubidium–strontium systematics

      Several isotopes of the relatively rare element rubidium (Rb) exist; some 27% of these are radioactive⁸⁷Rb. Most Rb⁺¹ is concentrated in continental crust, especially in substitution for potassium ion (K⁺¹) in potassium‐bearing minerals such as potassium feldspar, muscovite, biotite, sodic plagioclase, and amphibole. Radioactive⁸⁷Rb is slowly (half‐life = 48.8 Ga) transformed by beta decay into strontium‐87 (⁸⁷Sr). Unfortunately, the much smaller strontium ion (Sr⁺²) does not easily substitute for potassium ion (K⁺¹) and therefore tends to migrate into minerals in the rock that contain calcium ion (Ca⁺²) for which strontium easily substitutes. This makes using⁸⁷Rb/⁸⁶Sr for age dating a much less accurate method than the U/Pb methods explained previously.

      One basic concept behind rubidium–strontium dating is that the original rock has some initial amount of⁸⁷Rb, some initial ratio of⁸⁷Sr/⁸⁶Sr and some initial ratio of⁸⁷Rb/⁸⁶Sr. These ratios evolve through time in a predictable manner. Over time, the amount of⁸⁷Rb decreases and the amount of⁸⁷Sr increases by radioactive decay so that the⁸⁷Sr/⁸⁶Sr