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References Generic Radiometric Dating The simplest form of isotopic age computation involves substituting three measurements into an equation of four variables, and solving for the fourth. The equation is the one which describes radioactive decay: The variables in the equation are: Pnow - The quantity of the parent isotope that remains now. This is measured directly. Porig - The quantity of the parent isotope that was originally present. This is computed from the current quantity of parent isotope plus the accumulated quantity of daughter isotope. Standard values are used, based on direct measurements. Solving the equation for "age," and incorporating the computation of the original quantity of parent isotope, we get: Potential problems for generic dating Some assumptions have been made in the discussion of generic dating, for the sake of keeping the computation simple. Such assumptions will not always be accurate in the real world. The amount of daughter isotope at the time of formation of the sample is zero or known independently and can be compensated for. No parent isotope or daughter isotope has entered or left the sample since its time of formation. If one of these assumptions has been violated, the simple computation above yields an incorrect age. Note that the mere existence of these assumptions do not render the simpler dating methods entirely useless. In many cases, there are independent cues such as geologic setting or the chemistry of the specimen which can suggest that such assumptions are entirely reasonable. However, the methods must be used with care -- and one should be cautious about investing much confidence in the resulting age Isochron methods avoid the problems which can potentially result from both of the above assumptions. Isochron methodology Isochron dating requires a fourth measurement to be taken, which is the amount of a different isotope of the same element as the daughter product of radioactive decay. For brevity's sake, hereafter I will refer to the parent isotope as P, the daughter isotope as D, and the non-radiogenic isotope of the same element as the daughter, as Di. In addition, it requires that these measurements be taken from several different objects which all formed at the same time from a common pool of materials. Rocks which include several different minerals are excellent for this. Each group of measurements is plotted as a data point on a graph. The X-axis of the graph is the ratio of P to Di. The Y-axis of the graph is the ratio of D to Di. What does it mean? The intent of the plot is to assess a correlation between: Meaning of the plot axes. If the data points on the plot are colinear, and the line has a positive slope, it shows an extremely strong correlation between: The amount of P in each sample, and The extent to which it is enriched in D, relative to Di. This is a necessary and expected consequence, if the additional D is a product of the decay of P in a closed system over time. It is not easily explained, in the general case, in any other way. Why isochron data are colinear The data points would be expected to start out on a line if certain initial conditions were met. Consider some molten rock in which isotopes and elements are distributed in a reasonably homogeneous manner. Its composition would be represented as a single point on the isochron plot: Global composition of the melt. As the rock cools, minerals form. They "choose" atoms for inclusion by their chemical properties. This results in an identical Y-value for the data points representing each mineral matching the Y-value of the source material. There are minor differences between isotopes of the same element, and in relatively rare circumstances it is possible to obtain some amount of differentiation between them. This is known as isotope fractionation. The effect is almost always a very small departure from homogeneous distribution of the isotopes -- perhaps enough to introduce an error of 0. In contrast, P is a different element with different chemical properties. This results in a range of X-values for the data points representing individual minerals. Since the data points have the same Y-value and a range of X-values, they initially fall on a horizontal line: Differential migration of elements as minerals form. A horizontal line represents "zero age. In most cases, any age less than about P half-lives will include zero within its range of uncertainty. The range of uncertainty varies, and may be as much as an order of magnitude different from the approximate value above. It depends on the accuracy of the measurements and the fit of the data to the line in each individual case. That encompasses the entire young-Earth timescale thousands of times over. As more time passes and a significant amount of radioactive decay occurs, the quantity of P decreases by a noticeable amount in each sample, while the quantity of D increases by the same amount. This results in a movement of the data points to the left decreasing P and upwards increasing D. Since each atom of P decays to one atom of D, the data point for each sample will move along a path with a slope of As a result, the data points with the most P the right-most ones on the plot move the greatest distance per unit time. The data points remain colinear as time passes, but the slope of the line increases: Movement of data points as decay occurs. The slope of the line is the ratio of enriched D to remaining P. Miscellaneous notes Age "uncertainty" When a "simple" dating method is performed, the result is a single number. There is no good way to tell how close the computed result is likely to be to the actual age. An additional nice feature of isochron ages is that an "uncertainty" in the age is automatically computed from the fit of the data to a line. A routine statistical operation on the set of data yields both a slope of the best-fit line an age and a variance in the slope an uncertainty in the age. The better the fit of the data to the line, the lower the uncertainty. For further information on fitting of lines to data also known as regression analysis , see: York , a short technical overview of a technique specially designed for assessing isochron fits. Note that the methods used by isotope geologists as described by York are much more complicated than those described by Gonick. This will be discussed in more detail in the section on Gill's paper below. The "generic" method described by Gonick is easier to understand, but it does not handle such necessities as: Unfortunately, one must wade through some hefty math in order to understand the procedures used to fit isochron lines to data. General comments on "dating assumptions" All radiometric dating methods require, in order to produce accurate ages, certain initial conditions and lack of contamination over time. The wonderful property of isochron methods is: This topic will be discussed in much more detail below. Where the simple methods will produce an incorrect age, isochron methods will generally indicate the unsuitability of the object for dating. Avoidance of generic dating's problems Now that the mechanics of plotting an isochron have been described, we will discuss the potential problems of the "simple" dating method with respect to isochron methods. Initial daughter product The amount of initial D is not required or assumed to be zero. The greater the initial D-to-Di ratio, the further the initial horizontal line sits above the X-axis. But the computed age is not affected. If one of the samples happened to contain no P it would plot where the isochron line intercepts the Y-axis , then its quantity of D wouldn't change over time -- because it would have no parent atoms to produce daughter atoms. Whether there's a data point on the Y-axis or not, the Y-intercept of the line doesn't change as the slope of the isochron line does as shown in Figure 5. Therefore, the Y-intercept of the isochron line gives the initial global ratio of D to Di. For each sample, it would be possible to measure the amount of the Di, and using the ratio identified by the Y-intercept of the isochron plot calculate the amount of D that was present when the sample formed. That quantity of D could be subtracted out of each sample, and it would then be possible to derive a simple age by the equation introduced in the first section of this document for each sample. Each such age would match the result given by the isochron. Contamination - parent isotope Gain or loss of P changes the X-values of the data points: Gain or loss of P. In order to make the figures easy to read and quick to draw , the examples in this paper include few data points. While isochrons are performed with that few data points, the best ones include a larger quantity of data. If the isochron line has a distinctly non-zero slope, and a fairly large number of data points, the nearly inevitable result of contamination failure of the system to remain closed will be that the fit of the data to a line will be destroyed. For example, consider an event which removes P. The data points will tend to move varying distances, for the different minerals will have varying resistance to loss of P, as well as varying levels of Di: Loss of P in all samples The end result is that the data are nearly certain not to remain colinear: Loss of P destroys the fit to a line. Even in our simple four-data-point example isochron, a change to two of the samples Migration of parent in two data points. Specific loss of P required to yield a different colinear plot. The two samples must each change by the indicated amount -- no more and no less -- if the data are to remain colinear. In the special case where the isochron line has a zero slope indicating zero age , then gain or loss of P may move the data points, but they will all still fall on the same horizontal line. In other words, random gain or loss of P does not affect a zero-age isochron. This is an important point.
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