Although it is impossible to predict when a particular atom will change, given a sufficient number of atoms, the rate of their decay is found to be constant. The situation is analogous to the death rate among human populations insured by an insurance company. Even though it is impossible to predict when a given policyholder will die, the company can count on paying off a certain number of beneficiaries every month.
The recognition that the rate of decay of any radioactive parent atom is proportional to the number of atoms N of the parent remaining at any time gives rise to the following expression:. Converting this proportion to an equation incorporates the additional observation that different radioisotopes have different disintegration rates even when the same number of atoms are observed undergoing decay.
Solution of this equation by techniques of the calculus yields one form of the fundamental equation for radiometric age determination,. Two alterations are generally made to equation 4 in order to obtain the form most useful for radiometric dating. In the first place, since the unknown term in radiometric dating is obviously t , it is desirable to rearrange equation 4 so that it is explicitly solved for t. Half-life is defined as the time period that must elapse in order to halve the initial number of radioactive atoms. The half-life and the decay constant are inversely proportional because rapidly decaying radioisotopes have a high decay constant but a short half-life.
With t made explicit and half-life introduced, equation 4 is converted to the following form, in which the symbols have the same meaning:. Alternatively, because the number of daughter atoms is directly observed rather than N , which is the initial number of parent atoms present, another formulation may be more convenient. Since the initial number of parent atoms present at time zero N 0 must be the sum of the parent atoms remaining N and the daughter atoms present D , one can write:. Substituting this in equation 6 gives. If one chooses to use P to designate the parent atom, the expression assumes its familiar form:.
This pair of equations states rigorously what might be assumed from intuition , that minerals formed at successively longer times in the past would have progressively higher daughter-to-parent ratios. This follows because, as each parent atom loses its identity with time, it reappears as a daughter atom. Equation 8 documents the simplicity of direct isotopic dating. The time of decay is proportional to the natural logarithm represented by ln of the ratio of D to P. In short, one need only measure the ratio of the number of radioactive parent and daughter atoms present, and the time elapsed since the mineral or rock formed can be calculated, provided of course that the decay rate is known.
Likewise, the conditions that must be met to make the calculated age precise and meaningful are in themselves simple:.
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The rock or mineral must have remained closed to the addition or escape of parent and daughter atoms since the time that the rock or mineral system formed. It must be possible to correct for other atoms identical to daughter atoms already present when the rock or mineral formed. The measurement of the daughter-to-parent ratio must be accurate because uncertainty in this ratio contributes directly to uncertainty in the age. Different schemes have been developed to deal with the critical assumptions stated above.
In uranium-lead dating , minerals virtually free of initial lead can be isolated and corrections made for the trivial amounts present. In whole-rock isochron methods that make use of the rubidium- strontium or samarium - neodymium decay schemes, a series of rocks or minerals are chosen that can be assumed to have the same age and identical abundances of their initial isotopic ratios.
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The results are then tested for the internal consistency that can validate the assumptions. In all cases, it is the obligation of the investigator making the determinations to include enough tests to indicate that the absolute age quoted is valid within the limits stated. In other words, it is the obligation of geochronologists to try to prove themselves wrong by including a series of cross-checks in their measurements before they publish a result. Such checks include dating a series of ancient units with closely spaced but known relative ages and replicate analysis of different parts of the same rock body with samples collected at widely spaced localities.
The importance of internal checks as well as interlaboratory comparisons becomes all the more apparent when one realizes that geochronology laboratories are limited in number. Because of the expensive equipment necessary and the combination of geologic, chemical, and laboratory skills required, geochronology is usually carried out by teams of experts.
Most geologists must rely on geochronologists for their results. In turn, the geochronologist relies on the geologist for relative ages. In order for a radioactive parent-daughter pair to be useful for dating, many criteria must be met. This section examines these criteria and explores the ways in which the reliability of the ages measured can be assessed.
Because geologic materials are diverse in their origin and chemical content and datable elements are unequally distributed, each method has its strengths and weaknesses. Of these, only the radioisotopes with extremely long half-lives remain. It should be mentioned in passing that some of the radioisotopes present early in the history of the solar system and now completely extinct have been recorded in meteorites in the form of the elevated abundances of their daughter isotopes.
Analysis of such meteorites makes it possible to estimate the time that elapsed between element creation and meteorite formation. Natural elements that are still radioactive today produce daughter products at a very slow rate; hence, it is easy to date very old minerals but difficult to obtain the age of those formed in the recent geologic past.
This follows from the fact that the amount of daughter isotopes present is so small that it is difficult to measure. The difficulty can be overcome to some degree by achieving lower background contamination, by improving instrument sensitivity, and by finding minerals with abundant parent isotopes. Geologic events of the not-too-distant past are more easily dated by using recently formed radioisotopes with short half-lives that produce more daughter products per unit time.
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Two sources of such isotopes exist. In one case, intermediate isotopes in the uranium or thorium decay chain can become isolated in certain minerals because of differences in chemical properties and, once fixed, can decay to new isotopes, providing a measure of the time elapsed since they were isolated. To understand this, one needs to know that though uranium U does indeed decay to lead Pb , it is not a one-step process. In fact, this is a multistep process involving the expulsion of eight alpha particles and six beta particles , along with a considerable amount of energy.
There exists a series of different elements, each of them in a steady state where they form at the same rate as they disintegrate.
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The number present is proportional to their decay rate, with long-lived members being more abundant. Because all these isotopes have relatively short half-lives, none remains since the formation of the elements, but instead they are continuously provided by the decay of the long-lived parent.
This type of dating, known as disequilibrium dating, will be explored below in the section Uranium-series disequilibrium dating. The amounts produced, although small, provide insight into many near-surface processes in the geologic past. The most widely used radioactive cosmogenic isotope is carbon of mass 14 14 C , which provides a method of dating events that have occurred over roughly the past 60, years. This time spans the historic record and a significant part of the prehistoric record of humans.
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