The isochron method
Many radioactive dating methods are based on minute additions of daughter products to a rock or mineral in which a considerable amount of daughter-type isotopes already exists. These isotopes did not come from radioactive decay in the system but rather formed during the original creation of the elements. In this case, it is a big advantage to present the data in a form in which the abundance of both the parent and daughter isotopes are given with respect to the abundance of the initial background daughter. The incremental additions of the daughter type can then be viewed in proportion to the abundance of parent atoms. In mathematical terms this is achieved as follows. It has already been shown—equation 7—that the number of daughter atoms present from radioactive decay D* can be related to the number of parent atoms remaining P by the simple expression:
When some daughter atoms are initially present (designated D0), the total number D is the sum of radiogenic and initial atoms, so that
To establish the condition that both parent and daughter abundances should be relative to the initial background, a stable isotope S of the daughter element can be chosen and divided into all portions of this equation; thus,
This equation has the form y = b + xm, which is that of a straight line on x–y coordinates. The slope m is equal to (eλt − 1), and the intercept is equal to (D/S)0. This term is called the initial ratio. The slope is proportional to the geologic age of the system.
In practice, the isochron approach has many inherent advantages. When a single body of liquid rock crystallizes, parent and daughter elements may separate so that, once solid, the isotopic data would define a series of points in time that can be plotted along a horizontal line reflecting a common value for the initial daughter isotope ratio (D/S)0. With time, each would then develop additional daughter abundances in proportion to the amount of parent present. If a number of samples are analyzed and the results are shown to define a straight line within error, then a precise age is defined because this is only possible if each is a closed system and each has the same initial ratio and age. The uncertainty in determining the slope is reduced because it is defined by many points. A second advantage of the method relates to the fact that under high-temperature conditions the daughter isotopes may escape from the host minerals. In this case, a valid age can still be obtained, provided that they remain within the rock. Should a point plot below the line, it could indicate that a particular sample was open to migration of the dating elements or that the sample was contaminated and lay below the isochron when the rock solidified.
Rubidium-strontium (Rb-Sr) dating was the first technique in which the whole-rock isochron method was extensively employed. Certain rocks that cooled quickly at the surface were found to give precisely defined linear isochrons, but many others did not. Some studies have shown that rubidium is very mobile both in fluids that migrate through the rock as it cools and in fluids that are present as the rock undergoes chemical weathering. Similar studies have shown that the samarium-neodymium (Sm-Nd) parent-daughter pair is more resistant to secondary migration but that, in this instance, sufficient initial spread in the abundance of the parent isotope is difficult to achieve.
Analysis of separated minerals
When an igneous rock crystallizes, a wide variety of major and trace minerals may form, each concentrating certain elements and radioactive trace elements within the rock. By careful selection, certain minerals that contain little or no daughter element but abundant parent element can be analyzed. In this case, a graph can be set up in which the slope of the line may be computed from an assumed value for the initial ratio, and it is usually possible to show that uncertainties related to this assumption are negligible. This is possible in potassium-argon (K-Ar) dating, for example, because most minerals do not take argon into their structures initially. In rubidium-strontium dating, micas exclude strontium when they form but accept much rubidium. In uranium-lead (U-Pb) dating of zircon, the zircon is found to exclude initial lead almost completely. Minerals too are predictable chemical compounds that can be shown to form at specific temperatures and remain closed up to certain temperatures if a rock has been reheated or altered. A rock, on the other hand, may contain minerals formed at more than one time under a variety of conditions. Under such circumstances the isolation and analysis of certain minerals can indicate at what time these conditions prevailed. If a simple mineral is widespread in the geologic record, it is more valuable for dating as more units can be measured for age and compared by the same method. However, if a single parent-daughter pair that is amenable to precise analysis can be measured in a variety of minerals, the ages of a wide variety of rock types can be determined by a single method without the need for intercalibration. In some cases the discovery of a rare trace mineral results in a major breakthrough as it allows precise ages to be determined in formerly undatable units. For example, the minerals baddeleyite, an oxide of zirconium (ZrO2), and zirconolite (CaZrTi2O7), have been shown to be widespread in small amounts in mafic igneous rocks (i.e., those composed primarily of one or more ferromagnesian, dark-coloured minerals). Here, a single uranium-lead isotopic analysis can provide an age more precise than can be obtained by the whole-rock isochron method involving many analyses. When single minerals are analyzed, each grain can be studied under a microscope under intense side light so that alterations or imperfections can be revealed and excluded. If minerals are used for dating, the necessary checks on the ages are achieved by analyzing samples from more than one location and by analyzing different grain sizes or mineral types that respond differently to disturbing events. It can be said that minerals provide a high degree of sample integrity that can be predicted on the basis of experience gained through numerous investigations under a variety of geologic conditions. An ideal mineral is one that has sufficient parent and daughter isotopes to measure precisely, is chemically inert, contains little or no significant initial daughter isotopes, and retains daughter products at the highest possible temperatures. A specific datable mineral like rutile, which can be linked to a specific event such as the formation of a mineral deposit, is especially important.
Model ages
Since Earth was formed, the abundance of daughter product isotopes has increased through time. For example, the ratio of lead of mass 206 relative to that of mass 204 has changed from an initial value of about 10 present when Earth was formed to an average value of about 19 in rocks at the terrestrial surface today. This is true because uranium is continuously creating more lead. A lead-rich mineral formed and isolated early in Earth’s history would have a low ratio of lead-206 to lead-204 because it did not receive subsequent additions by the radioactive decay of uranium. If Earth’s interior were a simple and homogeneous reservoir with respect to the ratio of uranium to lead, a single sample extracted by a volcano would provide the time of extraction. This would be called a model age. No parent-daughter value for a closed system is involved—rather, just a single isotopic measurement of lead viewed with respect to the expected evolution of lead on and in Earth. Unfortunately, the simplifying assumption in this case is not true, and lead model ages are approximate at best. Other model ages can be calculated using neodymium isotopes by extrapolating present values back to a proposed mantle-evolution line. In both cases, approximate ages that have a degree of validity with respect to one another result, but they are progressively less reliable as the assumptions on which the model is calculated are violated.
The progressive increase in the abundance of daughter isotopes over time gains a special significance where the parent element is preferentially enriched in either the mantle or the crust. For example, rubidium is concentrated in the crust, and as a result the present-day continents, subjected to weathering, have an elevated radiogenic to stable isotope ratio (87Sr/86Sr) of 0.720. In contrast, modern volcanic rocks in the oceans imply that much of the mantle has a value between about 0.703 and 0.705. Should crustal material be recycled, the strontium isotopic signature of the melt would be diagnostic.
Multiple ages for a single rock: the thermal effect
Fossils record the initial, or primary, age of a rock unit. Isotopic systems, on the other hand, can yield either the primary age or the time of a later event, because crystalline materials are very specific in the types of atoms they incorporate, in terms of both the atomic size and charge. An element formed by radioactive decay is quite different from its parent atom and thus is out of place with respect to the host mineral. All it takes for such an element to be purged from the mineral is sufficient heat to allow solid diffusion to occur. Each mineral has a temperature at which rapid diffusion sets in, so that, as a region is slowly heated, first one mineral and then another loses its daughter isotopes. When this happens, the isotopic “clock” is reset to zero, where it remains until the mineral cools below the blocking temperature. (This is the temperature below which a mineral becomes a closed chemical system for a specific radioactive decay series. Accordingly, the parent-daughter isotope ratio indicates the time elapsed since that critical threshold was reached.) In this case, the host mineral could have an absolute age very much older than is recorded in the isotopic record. The isotopic age then is called a cooling age. It is even possible by using a series of minerals with different blocking temperatures to establish a cooling history of a rock body—i.e., the times since the rock body cooled below successively lower temperatures. Such attempts can be complicated by the fact that a mineral may “grow” below the blocking temperature rather than simply become closed to isotopic migration. When this happens, the age has little to do with the cooling time. Another problem arises if a region undergoes a second reheating event. Certain minerals may record the first event, whereas others may record the second, and any suggestion of progressive cooling between the two is invalid. This complication does not arise when rapid cooling has occurred. Identical ages for a variety of minerals with widely different blocking temperatures is unequivocal proof of rapid cooling.
Fortunately for geologists, the rock itself records in its texture and mineral content the conditions of its formation. A rock formed at the surface with no indication of deep burial or new mineral growth can be expected to give a valid primary age by virtue of minerals with low blocking temperatures. On the other hand, low-blocking-point minerals from a rock containing minerals indicative of high temperatures and pressures cannot give a valid primary age. Such minerals would be expected to remain open until deep-level rocks of this sort were uplifted and cooled.
Given these complicating factors, one can readily understand why geochronologists spend a great deal of their time and effort trying to see through thermal events that occurred after a rock formed. The importance of identifying and analyzing minerals with high blocking temperatures also cannot be overstated. Minerals with high blocking temperatures that form only at high temperatures are especially valuable. Once formed, these minerals can resist daughter loss and record the primary age even though they remained hot (say, 700 °C [1,292 °F]) for a long time. The mineral zircon datable by the uranium-lead method is one such mineral. The mica mineral biotite dated by either the potassium-argon or the rubidium-strontium method occupies the opposite end of the spectrum and does not retain daughter products until cooled below about 300 °C (572 °F). Successively higher blocking temperatures are recorded for another mica type known as muscovite and for amphibole, but the ages of both of these minerals can be completely reset at temperatures that have little or no effect on zircon.
Taken in perspective, it is evident that many parts of Earth’s crust have experienced reheating temperatures above 300 °C—i.e., reset mica ages are very common in rocks formed at deep crustal levels. Vast areas within the Canadian Shield, which have identical ages reflecting a common cooling history, have been identified. These are called geologic provinces. By contrast, rocks that have approached their melting point—say, 750 °C (1,382 °F), which can cause new zircon growth during a second thermal event—are rare, and those that have done this more than once are almost nonexistent.
