Author: Chris Stassen Subject: FAQ: Isochron Dating Updated: 04/22/94 (added Section 6) CO

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======================================================================== Author: Chris Stassen Subject: FAQ: Isochron Dating Updated: 04/22/94 (added Section 6) ======================================================================== CONTENTS: 1. Generic Radiometric Dating 2. What's wrong with non-isochron dating methods? 3. Generic Isochron Dating 4. What's NOT wrong with isochron dating methods? 5. Some questions on isochron methods 6. A little math 7. For further information (some things to read) -------------------------------------------------------------------------- (1) Generic Radiometric Dating Generally, radiometric dating is done by performing a simple calculation on a sample, involving three measurements: a) The first "measurement" is actually a "known quantity" -- the half-life of the radioactive element used by the method. This value can be experimentally measured in a lab -- but since many experiments have failed to effect a noticeable change in the rates relevant to radiometric dating, it is usually taken from a table. b) The second measurement is the amount of "parent" element (the radioactive element used by the method). c) The third measurement is the amount of "daughter" element (the element that the radioactive one decays into). Since each atom of the parent element decays into one atom of the daughter element, we calculate that the original quantity of the parent element is the sum of the current amounts of parent and daughter elements. We then apply the following (frosh physics) equation (the infamous "radioactive decay" equation): P = P0 / (2 ^ (T / H) ) or P = (P + D) / (2 ^ (T / H) ) Where: P is the current amount of parent element P0 is the original amount of parent element (= P + D) T is time that has passed ("age" of the sample) H is the half-life of the element Solving for T, we calculate the sample's age as: T = H * log2 ( (P + D) / P) -------------------------------------------------------------------------- (2) What's wrong with non-isochron dating methods? Obviously, there are a few assumptions above which have been made for the sake of a simple expanation, but which will not always work in the real world. These include: a) The amount of daughter element at the time of formation of the sample is zero. Possible ways to avoid this problem include: work on a mineral that can't incorporate any of the daughter compound when it forms; somehow calculate the amount of initial daughter product and subtract it out. b) No parent element or daughter element has entered or left the sample since its time of formation. Possible ways to avoid this problem include: only date samples whose geological history does not appear to include events which might cause this problem; date several different parts of the same sample and only accept the result if they all agree because contamination is not likely to affect all parts of a large sample in the same way. The invention of isochron methods solves both of these problems at once. -------------------------------------------------------------------------- (3) Generic Isochron Dating Isochron dating requires a fourth measurement to be taken, which is the amount of a _different_ isotope of the daughter element. 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 perfect for this.) When any rock forms, minerals "choose" atoms for inclusion by their _chemical_ properties. Since the two isotopes of the daughter element have identical chemical properties, they will be mixed evenly when the sample forms. However, the parent element, with different properties, will not be mixed evenly relative to the daughter elements. So, at formation time, a sample would contain measurements like the following: --------- -------- -------- -------- Sample No. Parent Daughter Isotope --------- -------- -------- -------- 1 4 ppm 1 ppm 2 ppm 2 2 ppm 4 ppm 8 ppm 3 6 ppm 2 ppm 4 ppm --------- -------- -------- -------- Note that (for this example) there is always twice as much of the "isotope" as there is of the "daughter" in every mineral. Also note that the ratio of "parent" element to either one of the others varies (as the parent element has different chemical properties). After one half-life's worth of time has passed, the values will have changed (as half of the parent atoms in each mineral will have decayed into daughter compounds): --------- -------- -------- -------- Sample No. Parent Daughter Isotope --------- -------- -------- -------- 1 2 ppm 3 ppm 2 ppm 2 1 ppm 5 ppm 8 ppm 3 3 ppm 5 ppm 4 ppm --------- -------- -------- -------- Note that half of the amount in the Parent column has been taken away and added to the Daughter column for each mineral. Also note that the Isotope column, since it doesn't decay and isn't a decay product, doesn't change at all. I can do some math here, but it's easier to see it on a graph. The isochron graph is drawn by graphing D/Di vs. P/Di ("Di" is the other isotope of the same element as the daughter product). The first set of measurements results in: D/Di 1 - | | | - (2)................................(3)...........(1) | | | +-------------|-------------|-------------|-------------| 0 0.5 1 1.5 2 P/Di Note that all of the samples lie on a straight, flat line. This is what we expect: they all have the same D/Di ratio, and hence the same Y-value. (Note: if the sample were homogeneously distributed with respect to parent and daughter, then all of the data points fall on the same point and no line can be derived.) The graph for the second set of measurements is: 2 - | | | - .(1) | .. | ..(3) | ... D/Di 1 - ... | .. | .. | (2) - | | | +-------------|-------------|-------------|-------------| 0 0.5 1 1.5 2 P/Di Once again, all the points lie on a straight line. And the slope of the line is 1. (I know it doesn't look like it on the screen, but that's because I used different units for X and Y -- you can calculate it for yourself from the table above.) We can make a simple table of slope of line versus age: Slope Age -------- ------------------- 0 0 1 1 half-life 3 2 half-lives 7 3 half-lives ... ... N log2( N + 1 ) half-lives -------------------------------------------------------------------------- (4) What's NOT wrong with isochron dating methods? Now that the mechanics of plotting an isochron have been described, we will return to address the problems that were mentioned before and describe why isochron methods don't fall prey to them. a) Initial daughter compound. Any amount of initial daughter compound is compensated by the isochron method. If one of the minerals happened to have none of the parent element (the Y-intercept of the line), then its amount of daughter compound wouldn't change over time -- because it has no parent atoms to produce daughter atoms. Regardless of whether there's a data point there or not, the Y-intercept of the line doesn't change as the slope of the line does. (You can verify this for yourself; the Y-intercept of both lines above is 0.5.) The Y-intercept of the isochron line actually gives the ratio of daughter to the other isotope at the time of formation. For each mineral, we can then measure the amount of the other isotope and calculate the amount of daughter product that was present when the sample formed. If we then subtract it out, we could derive a "traditional" age for each mineral by the equations described in the first section. Each such age would match the result given by the isochron. b) Random contamination (parent or daughter entering or leaving the system) For the sake of brevity, our example only included three data points. While isochrons are performed with that few data points, their value is not treated as seriously as those which have tens of points. Any non-systematic contamination is _extremely_ unlikely to leave all of the data points on the line. Even in our little example, any contamination of one of the minerals would require a specific contamination of one of the other two in order to keep all three points on the same line. When we get to an isochron with tens of data points, the suggestion that contamination "just happened to place the points on a (fake) isochron line" can be discarded out of hand. It's too unlikely. [Now, there is a form of isochron contamination, known as "mixing," which basically amounts to a _partial_ resetting of the isochron clock. However, there are tests to detect it.] c) General dating assumptions All radiometric dating methods must assume certain initial conditions and lack of contamination over time. The wonderful property of isochron methods is that *if one of these assumptions is violated*, it is nearly *certain* that the data will show that by the points not plotting on a line. -------------------------------------------------------------------------- (5) Some questions on isochron methods The following are interesting questions that were asked in about isochron dating. The names of the "questioners" have not been included because permission to use their names has not been obtained. Q1: # How do you tell the difference between radiogenic and non-radiogenic # Sr-87? For the Rb/Sr isochron method, the ratio of Sr-87 to Sr-86 (at time of formation) is not needed as an input to the equation; is given by the Y-intercept of the isochron line. It falls out of the computation of the method -- provided that all of the points fall on an isochron line. Q2: # What is isochron dating? a method, an equation, a graph ...? An "isochron" is a set of data points in a plot which all fall on a line representing a single age ("isochron" comes from: "isos" equal + "chronos" time). The line itself is also sometimes called an "isochron." The plot on which these data points appear is sometimes called an "isochron diagram" or "isochron plot." A dating method which uses such a plot to determine age is called an "isochron dating method." When "isochron dating" is mentioned in this FAQ, the intent is to cover the methodology which is common to all "isochron dating methods." Q3: # How is the half life of an element determined? For something that # takes 60 billion years to partially decay, how is an exact measure # of the decay rate determined in a few hours? These experiments don't necessarily take only "a few hours." Davis et al. (See Faure p. 119) measured the decay rate of Rb-87 (48.9 +/- 0.4 billion years) by counting the accumulation of Sr-87 over a period of nineteen years. While it may take "60 billion years" for half of the atoms to decay, in a large enough sample there will be many decays over a shorter span of time. If the sample's size can be measured accurately and is large enough (in terms of number of atoms) to be statistically significant, and the number of decays can be counted accurately, then the half-life can be computed accurately. That's the basis for the "direct counting experiments" from which half-lives are calculated. Q4-a: # The line is telling us that no matter what size sample we take we # always have the same ratio of parent to daughter. [...] # So let's say that when the rocks were formed, certain amounts of # both the parent and daughter were present. But in the process of # forming, everything got evenly distributed. You would get your # nice straight isochron line, but still not know the age of your # sample. This statement would be correct if the isochron plot were quantity of parent (P) versus quantity of daughter (D). But, it's not. Three measurements are taken, where the third is the quantity of a different isotope of the daughter product (Di). The graph isn't P vs D, it's P/Di vs D/Di. That's not the same as P vs D; Di will vary over different minerals. A plot of P/Di vs D/Di might form a line when P vs D does not. It's easy to understand how different minerals in a rock could get different P/Di ratios. P and Di have different chemical properties, and P will fit better into some minerals than Di (and vice versa). This explains why data plots don't all hit the same X-value. However, it's less easy to understand how different minerals in a rock could different D/Di ratios. The chemical properties of D and Di are the same. If the pool of matter were homogeneously distributed, then all minerals should get the same D/Di ratio. How do the data points end up with different Y-values? What the isochron plot shows is that there is a perfect correlation between minerals enriched in D/Di -- exactly proportional to how they are enriched in P/Di. This pattern is easily explained if the enriched D is a result of the enriched P decaying over time in a closed system. (If the matter didn't start out homogeneously distributed, then there is no reason for those minerals which got extra D/Di to get such a precise extra amount of P/Di in proportion.) Q4-b: # Let's say we have a bunch of red marble and blue marbles. If we put # the marbles in a jar with blue first and then red then whenever # you grab a handful, depending on how deep you go in the jar, you # will probably get different ratios of blue and red. However, if # you stir the marbles so that they are evenly distributed, then you # will always grab basically the same ratio now matter how large or # small your sample. Thus graphing your samples always yields a # line. Let's modify this analogy to make it more like isochron dating (per the response to Q4-a, above). It gets a lot more complicated, unfortunately. Suppose that you have a barrel of three kinds of marbles. Some are made of metal (P). The other two kinds are lighter (but of density equal to each other), made of wood. Some are ivory colored (Di) and others are black (D). Now, you mix up the barrel thoroughly. Since the metal marbles are slightly heavier, they will be present in higher proportion near the bottom and in lower proportion near the top. The wooden marbles will vary relative to the metal marbles by depth. But since the properties of the two kinds of wooden marbles are identical, they are always present in the same ratio to each other. If you divide the barrel into quarters and count the marbles in each quarter, you will find something like this: -------- -- -- -- ---- ---- ---- location P D Di P/Di D/Di P/D -------- -- -- -- ---- ---- ---- top 30 40 80 0.375 0.5 0.75 2nd qtr 60 30 60 1 0.5 2 3rd qtr 90 20 40 2.25 0.5 4.5 bottom 120 10 20 6 0.5 12 -------- -- -- -- ---- ---- ---- Note that these do *not* have the same P/D ratio, but they *do* have the same D/Di ratio. For this reason, when you make an isochron plot (P/Di vs D/Di), they all have the same Y-value and the result is a *horizontal* line. The equation for the line is: D/Di = 0.5 Now suppose that you keep the quarters of the barrel separate, as if the "atoms" (marbles) were now pretty much stuck with their individual "minerals" (quarters of the barrel). And in each quarter you take half of the metal marbles (P) and substutite in their place an equal number of black wood marbles (D). (This simulates radioactive decay.) You get: -------- -- -- -- ---- ---- ---- location P D Di P/Di D/Di P/D -------- -- -- -- ---- ---- ---- top 15 55 80 0.1875 0.6875 0.2727 2nd qtr 30 60 60 0.5 1.0 0.5 3rd qtr 45 65 40 1.125 1.625 0.6923 bottom 60 70 20 3 3.5 0.8571 -------- -- -- -- ---- ---- ---- When you make an isochron plot now, you will still get a line, but this time it has a slope of one. The equation for the line is: D/Di = 1.0 * P/Di + 0.5 In fact, regardless of the fraction you select for replacing P with D (as long as you replace the same fraction in each quarter), the data points will remain on a line. The slope of the line will vary, however, depending on the size of fraction which you substitute. The equation for slope (m) as a function of that fraction (f) is: m = -1 + ( 1 / (1-f) ) The computation works in the other direction, too. From the slope of the line, you can compute what fraction of P marbles have been replaced with D marbles. The equation is: f = 1 - ( 1 / (m+1) ) Or, to turn this analogy back to the world of dating methods: from the slope of the line you can figure out what fraction of parent element has been replaced with daughter element, from which you can calculate the amount of time required for that to happen via radioactive decay. Q-5: # If an area is homogeneously mixed, then you will always get the same # ratio of everything you grab. And they will all be equally related to # each other. [...] # In a few thousand years the decay is insignificant, so the isochron # line would just represent uniform mixing during formation. The situation you describe here won't result in an isochron line. If there is no chemical separation of P vs (D and Di), then all data points will have the same P/Di and D/Di ratios. To go back to your marble analogy, you'll get something like this (allowing for samples to be different sizes, which I didn't include the last time for sake of simplifying the data): -------- -- -- -- ---- ---- location P D Di P/Di D/Di -------- -- -- -- ---- ---- top 30 15 30 1 0.5 2nd qtr 60 30 60 1 0.5 3rd qtr 50 25 50 1 0.5 bottom 60 30 60 1 0.5 -------- -- -- -- ---- ---- ... every data point plots at ( 1 , 0.5 ), and no isochron line results. One depends on some chemical differentiation between P and (D and Di) in order to get a spread of data along the X-axis. Q-6: # But when scientists get data for something that appears contaminated, # what do they do with it? If data does not conform to the isochron # method and fall along a line it is interpretted as contamination, I # presume, as your FAQ also says. Why keep around bad samples? It sounds as if you are suggesting that geologists might keep trying isochron plots on a single item until they get one where the data points line up, which probably isn't representative of its "real" age, and only that one gets published. (This is about one pace away from some pretty heavy-duty "conspiracy-theorizing.") Four reasons why I doubt that this is done: First, that is recognized as being dishonest. If a geologist were to plot 30 data points, and then throw away the ten which fell furthest from the least-squares-fit isochron line, s/he would be in deep doo-doo when anyone else found out. (And the next person to attempt to replicate the work *would* find out.) The same would be true of someone who buried evidence of many bad plots in favor of one good one. Second, that would be outlandishly expensive -- in terms of both time and money. Performing just one isochron age requires quite a few measurements that are with rather expensive equipment. The ICR has spent over $30,000 in getting *one* Rb/Sr isochron "age" for their "Grand Canyon Dating Project." (While that is a little extreme, it's not cheap in any case.) If attempt number one showed the object to be unsuitable, chances are that attempts number two through N would merely give the same result. Who has that sort of time or money to waste? Third, negative results often get published. Even when the plot does not yield the age directly, it is often possible to ascertain useful information about the history of the object from the data. (For example, see Faure p. 126.) Fourth -- and most importantly -- if it were the case that isochron ages were essentially random, fictitious numbers, then we would not expect any sort of agreement between different methods, results published by different researchers, etc. For example, assuming you have Dalrymple's "Open-File Report 86-100", see page 44. Several different investigators using several different dating methods all get about the same age for the amitsoq gneiss. This is easily explained (indeed, required) if these methods yield the actual age of the formation. How is it explained if the "ages" are essentially random numbers? Suppose that the first researcher publishes an age of X years for the amitsoq gneiss. Do you think that the next guy is going to keep repeating the isochron method until he gets a result that both plots as a line and agrees with the first guy's data? Heck, it's no skin off *his* nose if the first researcher's "age" is a half billion years off. -------------------------------------------------------------------------- (6) A little math The author of this FAQ prefers to deal mainly in "intuitive" arguments, but there have been some requests for something more rigorous. Here is a mathematical derivation of isochron methodology, written by Dave Thomas (P.O. Box 1017 / Peralta, NM 87042): Assume a variety of minerals (k=1,2,3,...N) with different initial rubidium-87/strontium 86 ratios P0(k) (parent, horizontal axis), but all with the same initial strontium-87/strontium-86 ratios D0(k) (daughter, vertical axis). It is important to normalize the ratios to an isotope of strontium (namely Sr-86) that is *not* a decay product of rubidium-87. For one of these minerals (say, k=5), the amount of the parent isotope would decrease from the initial ratio due to radioactive decay: P(T) = P0*exp(-lambda*T) (Equation 1) where exp is the exponential function, lambda is the Rb decay rate (for rubidium-87, decay rate lambda = 1.39E-11 / year ), and T is the elapsed time since rock crystallization. For the same mineral (k=5), the amount of the daughter would increase from the initial ratio: D(T) = D0 + P0*(1-exp(-lambda*T)) (Equation 2) Note that P(T) + D(T) = D0 + P0 = constant for all ages T. At age T=0, P(T) = P0, and D(T) = D0+P0*(1-exp(0)) = D0+P0*(1-1) = D0. At age T > 0, P(T) < P0, and D(T) > D0. At age T approaching infinity, P(T) goes to zero, and D(T) to P0+D0. For a different mineral (say, k=13), let's use the prime (') symbol: P'(T) = P0'*exp(-lambda*T) (Equation 1') D'(T) = D0' + P0'*(1-exp(-lambda*T)) (Equation 2') Because of mixing prior to crystallization, the initial daughter ratios for both minerals are identical: D0 = D0' (*important*). Now, the slope of the line connecting point (P,D) to point (P',D') is: D'(T) - D(T) slope m = ------------ (Equation 3) P'(T) - P(T) Putting equations 1,2,1', and 2' into equation 3, we obtain: (D0'+P0'*(1-exp(-lambda*T))) - (D0+P0*(1-exp(-lambda*T))) slope m = --------------------------------------------------- P0'*exp(-lambda*T) - P0*exp(-lambda*T) which can be simplified by noting that initial strontium ratios D0' = D0 (*important*), and multiplying top and bottom by exp(+lambda*T) : P0'*(exp(+lambda*T)-1) - P0*(exp(+lambda*T)-1) slope m = -------------------------------------------------- P0' - P0 which further simplifies to (P0' - P0) * (exp(+lambda*T)-1) slope m = -------------------------------- ( P0' - P0 ) or just slope m = (exp(+lambda*T)-1) (Equation 4) Equation 4 is the basic connection between age T and isochron slope m. It can be inverted to give the age T as a function of the measured slope on an isochron plot: T = log_e( m+1 ) / lambda (Equation 5) where log_e is the natural logarithm. We can also express decay rate lambda in terms of half-life, H. From Equation 1, P(H) = P0/2 = P0*exp(-lambda*H) (for age T = Halflife H) which reduces to log_e(2) = lambda*H, or lambda = log_e(2) / H (Equation 6) Putting Equation 6 into Equation 5, we get: T = H * log_e( m+1 ) / log_e(2) (Equation 7) Multiplying Eqtn. 7 by log_e(2), and using log_e(x^n) = n*log_e(x), we get T*log_e(2) = log_e(2^T) = H*log_e( m+1 ) = log_e( (m+1)^H ), or 2^T = (m+1)^H , or just 2^(T/H) = (m+1) (Equation 8). For m = 1, m+1=2 = 2^1 so that T = H = 1 half-life. For m = 3, m+1=4 = 2^2 so that T = 2H = 2 half-lives. For m = 7, m+1=8 = 2^3 so that T = 3H = 3 half-lives. The method requires minerals with a spread of initial rubidium-87 ratios (P0 not equal to P0'), else the denominator of Equation 3 (and subsequent equations leading to Equation 4) would be zero, indicating an undefined slope. -------------------------------------------------------------------------- (7) For further information, see: G. Faure. 1986. _Principles of Isotope Geology_ Second Edition. John Wiley and Sons. ISBN 0-471-86412-9. 589 pp. Faure is a textbook/handbook on isotope dating. It is relatively technical and has many references to the scientific literature. Isochron methods are first detailed in Chapter 8: "The Rb-Sr Method of Dating." G. B. Dalrymple. 1991. _The Age of the Earth_. Stanford University Press. ISBN 0-8047-1569-6. 474 pp. Dalrymple is an excellent work on the age of the earth, which does not discuss creationist arguments but details the evidence and mainstream science's position. It is written for the layman, but is well-referenced. Isochron methods are introduced in a section titled "Age-Diagnostic Diagrams" (pp. 102-124). G. B. Dalrymple. 1982. _Radiometric Dating, Geologic Time, and the Age of the Earth: A Reply to "Scientific" Creationism_ (U.S. Geological Survey Open-File Report 86-110). 76 pp. Available from Wesley Elsberry ( for the cost of postage. Brent Dalrymple has given permission for this paper to be copied and distributed. The paper details radiometric dating methods, creationist criticisms of radiometric dating methods, and critiques creationists' own means for deriving an age for the earth. Isochron methods are introduced with the Rb/Sr isochron method (pp. 31-33). A. N. Strahler. 1987. _Science and Earth History_. Prometheus Books. ISBN 0-87975-414-1. 552 pp. Strahler is aimed at discussing creationism and creationist arguments, and only devotes a few pages to radiometric dating. It is on this list because many of the readers of this FAQ are likely to have the book (for other reasons). Strahler discusses isochron dating on pp. 132-134. ========================================================================


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