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Paul Giem
Abstract
Paul Giem
Unfortunately, there is no way to deal with the subject without at least mentioning mathematics. This means that math phobics cannot be completely accommodated; they will at least have to see equations. It also means that those who are ignorant of mathematics will need to educate themselves regarding the equations, or else take them on faith.
We will begin with the concept of using radiometric dating to measure time. The underlying theory is that a given substance transforms into another, in a process called radioactive decay, at a rate which is proportional to the amount of the initial substance (sometimes called the parent substance). Writing this mathematically, we have
where λ is known as the decay constant and dP/dt is the instantaneous change in P with respect to time. One can integrate this formula^{1}
^{1}By rearranging, -dP/P = λdt. One then integrates, -∫ _{P0}dP/P = λ∫ _{t0}dt. |
where ln is the natural logarithm, P_{0} is the amount of parent substance at a starting time t_{0}, and P is the amount of parent substance at any subsequent time t. By convention, we usually define time since t_{0} as 0. This makes the equation
This equation can be transformed into
^{2}There is some evidence being published that the decay constant may have changed in the past, at least for uranium and lead. However, it is not clear exactly when that happened. Whether it happened during a Flood can be reasonably questioned. In order for a change in the decay constant to be helpful to a creationist arguing for a short age for fossiliferous strata, the decay constant would have to change during the Flood. However, this would mean that the radioactive elements inside the bodies of Noah, his family, the animals in the ark, and whatever animals survived outside the ark were spared from the otherwise general increase in radioactive decay, or that their bodies proved resistant to the effects of the increased radioactivity. This is not impossible, but does require extra intervention (or more change in the usual laws of physics). |
Measuring P is usually relatively easy. One simply measures the parent element and finds the percentage of the parent element that is of the desired isotope. In practice it is even easier, as almost all the time the percentage of parent that is of the desired isotope is fixed in nature. Thus, for example, in the case of rubidium-strontium dating, rubidium (Rb) in nature consists of 72.15% ^{85}Rb (rubidium-85) and 27.85% ^{87}Rb (the radioactive isotope). This means that if one wants to determine the amount of ^{87}Rb in a given sample, one simply measures the total amount of Rb and multiplies by 0.2785.
Measuring the original amount of parent in the sample, P_{0}, is much more complicated. In fact, it is technically impossible, as it would have to be done at the beginning of the time period in question, and obviously for the time periods we are considering that was never done. We can only infer it from other measurements.
The first approximation to P_{0} is made by noting that the parent isotope decays to one (or occasionally more) daughter isotopes. If we define D* as the daughter produced by radioactivity, then
The problem with this formula is the difficulty of measuring D*. The daughter product we measure now, D, may not be equal to D*. There may have been some D at time zero (D_{0}), there may have been some D added later (D_{A}), and there may have been some D lost later (D_{L}). So the correct formula is
In some cases the last formula is assumed to be enough. For example, in the case of potassium-argon dating, it is assumed that at a given time all the argon in a sample is driven off. One simply measures the daughter (^{40}Ar) isotope, subtracts out air argon, multiplies by a branching factor, and assumes that one then can calculate D* and therefore P_{0}. (A critique of potassium-argon dating is beyond the scope of this paper.) However, in many other methods of radiometric dating the assumption that the daughter isotope is driven off is clearly invalid. For example, ^{87}Sr (strontium-87), the daughter product of ^{87}Rb, is not volatile, and is chemically incorporated into minerals when a melt [4] cools. So the fact that we measure a given amount of ^{87}Sr does not mean that it is a product of decay that accumulated since the rock hardened. It could just as easily have been left in the melt, possibly from previous decay. So we have to have another way to find D*.
The standard way for rubidium-strontium dating, samarium-neodymium dating, lutetium-hafnium dating, potassium-calcium dating, and uranium-lead dating, to name a few dating methods, is to assume isotopic homogenization, or complete mixing. It works something like this in the case of rubidium-strontium dating: At the time of the melt, all the isotopes are assumed to be homogenized. That is, one assumes that initially the isotopic strontium composition was the same throughout a (presumably melted) rock. For example, the ^{87}Sr/^{86}Sr isotopic ratio^{3}
^{3}Strontium has three stable isotopes, ^{84}Sr, ^{86}Sr, and ^{88}Sr, which are present in constant ratios relative to each other, so that ^{84}Sr/^{86}Sr = 0.056584 and ^{86}Sr/^{88}Sr = 0.1194, which gives percentages in usual rock of 82.52% ^{88}Sr, 7.00% 87Sr, 9.86% ^{86}Sr, and 0.56% ^{84}Sr. The percentage of ^{87}Sr varies between 6.9% and 7.4%+, depending apparently on the past and/or present rubidium content of the rock. One could use the ^{87}Sr/^{88}Sr ratio or the ^{87}Sr/^{84}Sr ratio for our purposes, but the ^{87}Sr/^{86}Sr ratio is closer to 1, easier to work with, and the traditional one. |
If there is strontium in some mineral without any rubidium, this makes the calculations easy, because this mineral should still have the original ^{87}Sr/^{86}Sr ratio. Supposing that this ratio was 0.710. That means that if a given rubidium-containing [5] mineral now has a ^{87}Sr/^{86}Sr ratio of 0.720, then for every 1000 atoms of ^{86}Sr, 10 atoms of ^{87}Sr has been produced by radioactivity. If in this mineral the ^{87}Rb/^{86}Sr ratio is 0.40, then for every 1000 atoms of 86Sr there would be 400 atoms of ^{87}Rb. Thus the original ^{87}Rb concentration would have been 400 + 10, or 410 / 1000 atoms of ^{86}Sr. The formula for the age of the mineral would be
If ^{87}Sr/^{86}Sr is the ratio in the rubidium-containing rock, and (^{87}Sr/^{86}Sr)_{0} is the ratio of the rock with no rubidium and therefore the ratio at the time of homogenization, and ^{87}Rb/^{86}Sr is the ratio in the rubidium-containing rock, then the general formula for the age is
The problem with using this formula is that we rarely have a mineral with essentially no rubidium but enough strontium to determine the initial ^{87}Sr/^{86}Sr ratio. So what is usually done is to obtain several minerals with different degrees of rubidium enrichment so that they have different ^{87}Rb/^{86}Sr ratios. Then the ^{87}Rb/^{86}Sr ratios are plotted against the ^{87}Sr/^{86}Sr ratios.^{4}
^{4}The calculations done above can be reversed. That is to say, if the production rate in the given time is one atom of strontium per 41 atoms of rubidium, then for 615 atoms of ^{87}Rb per 1000 86Sr atoms originally one should have 15 extra atoms of ^{87}Sr now, for a total of 725, and 600 atoms of ^{87}Rb now. This gives different ratios and a different data point, namely, ^{87}Rb/^{86}Sr = 0.60 and ^{87}Sr/^{86}Sr = 0.725 If one starts with more ^{86}Sr, and therefore more ^{87}Sr, one obtains a different ratio. Say that 4000 atoms of ^{86}Sr and therefore 2840 atoms of ^{87}Sr were in the original sample, with 615 ^{87}Rb atoms. Then during this time period, there would still be 15 ^{87}Rb atoms which decayed to ^{87}Sr, for a total of 2855, and 600 ^{87}Rb atoms left. This leaves a ^{87}Rb/^{86}Sr ratio of 0.15 and a ^{87}Sr/^{86}Sr ratio of 0.71375. Both of these points, along with the point in the text, are shown on the graph. |
^{5}The derivation of the formula is as follows: | |
^{87}Rb_{0}= ^{87}Rb e^{λt} | (Assuming constant decay and no rubidium gain or loss) |
^{87}Rb_{0} = ^{87}Sr* + ^{87}Rb | (Decay products) |
^{87}Sr* = ^{87}Rb_{0} - ^{87}Rb = ^{87}Rb e^{λt} - ^{87}Rb = ^{87}Rb (e^{λt} - 1) | (Algebra) |
^{87}Sr = ^{87}Sr_{0} + ^{87}Sr* = ^{87}Sr_{0} + ^{87}Rb (e^{λt} - 1) | (Assuming no strontium gain or loss) |
^{87}Sr/^{86}Sr = (^{87}Sr/^{86}Sr)_{0} + ^{87}Rb/^{86}Sr (e^{λt} - 1) | (Assuming isotopic mixing initially) |
The formula is valid as long as the assumptions are fulfilled. |
^{6}Which is the change in the ^{87}Sr/^{86}Sr ratio divided by the change in the ^{87}Rb/^{86}Sr ratio. |
The general formula is
We only need 2 points to determine the straight line and thus the slope, but if there are more than 2 points, and all our assumptions are correct, the points should all lie on the same straight line. [7]
It is commonly felt that if all the points lie on a straight line, this is a good indication that the above assumptions are correct. For example, see The Age of the Earth, a book written by G. Brent Dalrymple,^{7}
^{7}Stanford, CA: Stanford University Press, 1991. |
However, this confident statement is an overstatement. There is a process which can routinely give a straight line on an isochron plot, while having essentially nothing to do with time. It is called a mixing line. If two rocks are mixed in varying proportions, and the ^{87}Rb/^{86}Sr ratio is plotted against the ^{87}Sr/^{86}Sr ratio, the result is always a straight line. The derivation is as follows.
We will take two rocks, rock 1 and rock 2. Rock 1 contains p1 parent, d1 daughter, and r1 reference isotope (in the case of rubidium-strontium dating, p is ^{87}Rb, d is ^{87}Sr, and r is ^{86}Sr). Rock 2 contains p2 parent, d2 daughter, and r2 reference isotope (all of these are concentrations). Then we will mix them in proportion so that the proportion of our final rock that is rock 1 is a and the proportion of our final rock that is rock 2 is b. We have
that is, all the final rock is either a or b. Now we will assume that the rocks are not identical, and that both rocks have some r, so that p/r and d/r have some meaning at the endpoints. (If r = 0 at any point, p/r and d/r have no meaning at that point, and we cannot draw an isochron line either). We now have
d = ad_{1} + bd_{2}
r = ar_{1} + br_{2}
d/r = (ad_{1} + bd_{2}) / r
(p_{1}r_{2} - p_{2}r_{1}) d/r = (p_{1}r_{2} - p_{2}r_{1}) (ad_{1} + bd_{2}) / r
= (p_{1}r_{2}ad_{1} + p_{1}r_{2}bd_{2} - p_{2}r_{1}ad_{1} - p_{2}r_{1}bd_{2}) / r
= (ap_{1}d_{1}r_{2} + bp_{1}d_{2}r_{2} - ap_{2}d_{1}r_{1} - bp_{2}d_{2}r_{1}) / r
= (ap_{1}d_{1}r_{2} + bp_{1}d_{2}r_{2} - ap_{2}d_{1}r_{1} - bp_{2}d_{2}r_{1} + ap_{1}d_{2}r_{1} - ap_{1}d_{2}r_{1} + bp_{2}d_{1}r_{2} - bp_{2}d_{1}r_{2}) / r
= (ap_{1}d_{1}r_{2} - ap_{1}d_{2}r_{1} + bp_{2}d_{1}r_{2} - bp_{2}d_{2}r_{1} + ap_{1}d_{2}r_{1} - ap_{2}d_{1}r_{1} + bp_{1}d_{2}r_{2} - bp_{2}d_{1}r_{2}) / r
= (ap_{1}(d_{1}r_{2} - d_{2}r_{1}) + bp_{2}(d_{1}r_{2} - d_{2}r_{1}) + ar_{1} (p_{1}d_{2} - p_{2}d_{1}) + br_{2}(p_{1}d_{2} - p_{2}d_{1})) / r
= ((ap_{1} + bp_{2}) (d_{1}r_{2} - d_{2}r_{1}) + (ar_{1} + br_{2}) (p_{1}d_{2} - p_{2}d_{1})) / r
= (p(d_{1}r_{2} - d_{2}r_{1}) + r(p_{1}d_{2} - p_{2}d_{1}) / r
d/r (p_{1}r_{2} - p_{2}r_{1}) = (p/r) (d_{1}r_{2} - d_{2}r_{1}) + (p_{1}d_{2} - p_{2}d_{1})
which is in the form By = -Ax + C or Ax + By = C, which is a straight line, always. If p_{1}/r_{1} does not equal p_{2}/r_{2} (that is, the two rocks do not have the same proportion of parent to reference isotope), we may divide by p_{1}r_{2} - p_{2}r_{1}, and
d/r = (p/r) (d_{1}r_{2} d_{2}r_{1}) / (p_{1}r_{2} - p_{2}r_{1}) + ((p_{1}d_{2} - p_{2}d_{1})/ (p_{1}r_{2} - p_{2}r_{1}),
which is of the form d/r = S p/r + Y, where S is the slope and Y is the y-intercept. A mixing line not only can, but always does precisely mimic an isochron plot. This fact is well known in the field.
It is important to realize that a mixing line does not require thorough mixing of the two components, as long as we are doing whole rock dating. All that is required is that each component itself is homogeneous. One can even create a mixing line by, for example, taking a piece off of each component and simply putting those two pieces in a specimen container, measuring the relevant isotopes, and repeating the process. As long [9] as the pieces do not have precisely the same ratio of rock 1 to rock 2, we will get a mixing line. Any scatter in the line is evidence for gross post-mixing fractionation, contamination, and/or leaching, not incomplete mixing. One cannot tell a mixing line from a true isochron line on the basis of its straightness.
The question can be raised as to how sure we can be that a given line is an isochron plot rather than a mixing line. There are two ways. First, the date has to be believable on other grounds. Second, it has to be believable that the strontium isotopes in a given rock have indeed homogenized at the time.
Of course, the first criterion immediately raises questions. Believable by whom? Believable for what reasons? Some may be tempted to question whether such criteria are ever used. However, it appears that they are. Gunter Faure, in his classic book Principles of Isotope Geology,^{8}
^{8}2nd ed. New York: John Wiley and Sons, 1986. |
^{9}Pp. 145-7. His examples follow: Pleistocene to Recent (<1.6 million years old) lava with a Rb/Sr age of 773 million years (Bell K, Powell JL: "Strontium isotopic studies of alkalic rocks: The potassium-rich lavas of the Birunga and Toro-Ankole Regions, east and central Africa." J Petrol 1969;10:536-72); upper Miocene to Pliocene (5-9 million years old by K/Ar dating) lava with a Rb/Sr age of 31-39 million years (Dickinson DR, Dodson Mn, Gass IG, Rex DC: "Correlation of initial ^{87}Sr/^{86}Sr with Rb/Sr in some late Tertiary volcanic rocks of south Arabia." Earth Planet Sci Lett 1969;6:84-90); Pliocene to Holocene (<5.3 million years old) lava giving Rb/Sr ages of 570 and 870 million years (the 570 million year "isochron" is apparently from <3000 year old lava). Leeman WP, Manton WI: "Strontium isotopic composition of basaltic lavas from the Snake River Plain, southern Idaho." Earth Planet Sci Lett 1971;11:420-34); and Miocene to Holocene (<24 million years old) volcanic rock with a Rb/Sr age of 1.2 billion years (Duncan RA, Compston W: "Sr-isotopic evidence for an old mantle source region for French Polynesian vulcanism." Geology 1976;4:728-32). An additional report has been made of Pliocene to Holocene (<5.3 million years old) lava with a Rb/Sr age of 1.5 billion years (Leeman WP: "Late Cenozoic alkali-rich basalt from the western Grand Canyon area, Utah and Arizona: Isotopic composition of strontium." Bull Geol Soc Am 1974;85: 1691-6). |
^{10}For an example, see Dasch EJ, Green DH: "Strontium isotope geochemistry of lherzolite inclusions and host basaltic rocks, Victoria, Australia." Am J Sci 1975;275:461-9. |
Of course, this raises at least the possibility that if a creationist did not believe the standard geologic time scale, he/she is not obligated to believe any of the rubidium-strontium dates in the literature, or any of the other dates that are "confirmed" by the isochron method. This leaves us in an uncomfortable position. Science does not like to leave matters where either of two widely disparate theories are compatible with exactly [10] the same phenomena. Hopefully somewhere they will make differing predictions, and can thus be distinguished by testing.
One might suppose that having most of the dates match the standard geologic time scale would be a prediction of evolutionary theory, whereas short-age creationists would expect the ages not to match. However, it is known that some but not all of the dates match. What percentage is enough to validate the standard geologic time scale? 95%? 60%? 30%? 5%? And how do you determine the percentage? By the statistical limits of error? By being within 20% of the predicted value? And do all published data count, or the raw data from a sample of laboratories? I personally think the project would be a fascinating one. But at the end of the day (or rather, project) I doubt that either side would concede based on this evidence.
One might also suppose that if several different dating methods all got the same age, this adds credibility to the date obtained. There are three problems with this approach. First, one must then subtract credibility if the methods disagree. This is not always done. Second, it is predictable that if two magmas that have been either naturally or artificially aged to the same degree are mixed, all their "isochron lines" will match without the match having anything to do with time since deposition, even if the magmas in question have been melted.^{11}
^{11}To illustrate, supposing that one originally had two magmas, both with the same ^{87}Sr/^{86}Sr ratio and the same ^{143}Nd/^{144}Nd ratios but differing contents of ^{87}Rb and ^{147}Sm. If one "aged" the two magmas, part of the ^{87}Rb would change to ^{87}Sr, and part of the ^{147}Sm would change to ^{143}Nd. If one mixed the two magmas, one would then get straight lines which would appear to be isochron lines precisely matching in "age", while the mixing could have been done fifteen minutes before the measurement. Basically, the two magmas would behave as two points of a true isochron line with respect to each isotope system. Since they are aged in the same way, they will give the same dates. The mathematics of mixing will guarantee that the various mixtures of the two magmas will fall along the same line as the magmas themselves, thus giving the same dates as if one simply measured the two magma sources and used the line determined by their two points for the date. For concordant flattening of "isochrons" which are really mixing lines, see below. |
^{12}For example, see Odin GS (ed): Numerical Dating in Stratigraphy. Chinchester, UK: John Wiley and Sons, 1982, especially chapters 12 and 24. |
One can approach the problem another way. Mixing lines are supposed to give mixing hyperbolae. We will try to address the question whether mixing hyperbolae can reliably distinguish mixing lines from isochrons. For completeness, the technical details are given, but since they are, or at least should be, non-controversial, they will be given in smaller print (some readers may wish to skip over them to the conclusion).
[11]
What is a mixing hyperbola? We need to note first that most plots on a two-component mixture (TCM) are straight lines. If one plots the amounts of any two substances x and y in a given TCM, one gets a straight line. That is because for any given component x we have the concentration of x expressed by
where a is the fraction of a given rock that is from rock 1, x_{1} is the concentration of x in rock 1, and x_{2} is the concentration of x in rock 2. This eqation can be further manipulated:
or, if one assumes that x_{1} does not equal x_{2} (which we will routinely assume--we are interested only in variable components),
For a second component y we have
We thus have
y-y_{2} = [(y_{1}-y_{2})/(x_{1}-x_{2})] (x-x_{2}), and
y = [(y_{1}-y_{2})/(x_{1}-x_{2})] x - x_{2} [(y_{1}-y_{2})/(x_{12})] + y_{2},
which is of the form y = mx + b, which is a straight line.
If one plots x/z versus y/z for a TCM, one also obtains a straight line. The derivation of this was given above, if one substitutes p for x, d for y, and r for z. And if one plots x/w versus y/z, where there is some constant k such that w = kz, one still gets a straight line. One can simply define a new variable v = kx, and use the same derivation, as v/w now becomes kx/kz = x/z, which was shown above to give a straight line. If v/w versus y/z gives a straight line, then kx/w versus y/z gives the same straight line, and x/w versus y/z gives a straight line also, only with a changed slope. (This means that a TCM will always give a straight line on a discordia plot.)
But if one plots x versus y/z, one now gets a hyperbolic plot, assuming that z_{1} does not equal z_{2}. What do we mean by a hyperbolic plot? It is a plot where a variable, say x, is inversely proportional to another variable, say y, so that in the simplest case
or in a more complicated case
y = c_{1} + c_{2}/(x-c_{3}).
In this case c_{1} determines the distance from the origin to the y- (conventionally horizontal) asymptote, c_{3} determines the distance from the origin to the x- (conventionally vertical) asymptote, and c_{2} determines the distance from the crossing of the asymptotes to the curve itself.
The reason we have a hyperbolic plot for x versus y/z is because on the x-axis we have
which means that z is linearly dependent on x. At the same time
This means that
which plots against z as a hyperbola.
But it also plots against x as a hyperbola, for
which is of the form y/z = c_{1} + c_{2}/(x-c_{3}).
For x/w versus y/z (assuming z is not proportional to w), the math is more complicated, but the result is the same. Solving the relationships between y/z and w, we have [12]
x = m_{3}w + b_{3}, so that
x/w = m_{3} + b_{3}/w.
This means that
Therefore
y/z = m_{2} + b_{2}/(m_{1}w + b_{1}) = m_{2} + (b_{2}/m_{1})/([b_{3}/(x/w - m_{3})] + b_{1}/m_{1})
= m_{2} + (x/w - m_{3})(b_{2}/m_{1})/(b_{3} + (x/w - m_{3})b_{1}/m_{1})
= m_{2} + (x/w - m_{3})(b_{2}/b_{1})/(b_{3}m_{1}/b_{1} + x/w - m_{3})
= m_{2} + [(x/w(b_{2}/b_{1}) - m_{3}(b_{2}/b_{1})]/(x/w + b_{3}m_{1}/b_{1} - m_{3})
= m_{2} + [(x/w(b_{2}/b_{1}) + (b_{3}m_{1}/b_{1} - m_{3})(b_{2}/b_{1}) - (b_{3}m_{1}/b_{1} - m_{3})(b_{2}/b_{1}) - m_{3}(b_{2}/b_{1})]/(x/w + b_{3}m_{1}/b_{1} - m_{3})
= m_{2} + b_{2}/b_{1} - (b_{3}m_{1}/b_{1})(b_{2}/b_{1})/(x/w + b_{3}m_{1}/b_{1} - m_{3}),
which is of the form y/z = c_{1} + c_{2}/(x-c_{3}).
The point of all this is that both the straight lines and the mixing hyperbolas are direct consequences of the linear relationships of various components to each other. Although a mixing line is the easiest way to produce this relationship, any process that will produce the same relationship will produce mixing plots.
Suppose one had a true isochron plot. If it is a 2-component plot, standard theory holds that either it is a two-component mixture (TCM) where the strontium isotopes have equilibrated, or it is a completely homogenized (presumably melted) rock which has separated into two phases, presumably by crystallization. If it is a TCM, then the distributions of all components including rubidium and strontium (or other parent isotopes and daughter product) are linear, the distributions of parent and daughter isotopes should stay linear while decay is taking place, and all plots should be indistinguishable from mxing lines and/or hyperbolae. If it crystallized from a melt into two phases, we have the mathematical equivalent of a TCM, and the plots should again be indistinguishable from mixing lines and/or hyperbolae. So isochrons in a two-component system are not distinguishable from a two-component mixing line.
One is tempted to suggest that perhaps the differentiation between a mixing line and an isochron line is easier if there are 3 (or 4 or more) components to the rock suite. A mixing line with more than two components should not give a straight line on an isochron plot, and a rock suite isochron with more than 2 mineral types should not give mixing hyperbolae (because most other components should not have straight lines colinear with the isochron line components). So if we find a 3 (or more) mineral suite of rocks with an "isochron" line, it should be unusual for it to be a mixing line, and we should be able to tell the difference by seeing whether mixing hyperbolae are present. Mixing lines should have them, and isochrons should not.
However, there are several problems with this way of differentiating mixing lines from isochron lines. First, minor components may add too little mass and/or amounts of the minerals being measured to change either the "isochron" line or the mixing [13] hyperbola. Alternatively, the other components might be too evenly distributed in the rock to change the essential mathematics. So a mixing line might still be a good straight imitation of an isochron plot, and a true isochron might still show a good "mixing hyperbola".
Second, if the third (and fourth or more if present) component of a mixing line has, for example. a ^{87}Rb/^{86}Sr ratio that plots against its ^{87}Sr/^{86}Sr ratio on a straight line with the respective ratios of the first two components, one will still get a straight "isochron" line which is really a mixing line. In fact, the other components do not even have to be exactly on the line. They only have to be close enough so that the confidence limits of their measurements overlap the line. In this case we can have a mixing line without necessarily having good mixing hyperbolae, mimicking an isochron plot.
In fact, according to Faure (p. 151) "However, suites of samples formed by mixing of two components may not fit a single mixing equation because the end members may have had variable isotopic and chemical compositions. Moreover, both chemical and isotopic compositions of rocks may be changed by processes occurring subsequent to mixing, such as fractional crystallization, contamination by third components, and alteration by hydrothermal fluids or chemical weathering. Therefore a certain amount of scatter of data points above and below the mixing equation is commonly observed for suites of geologic samples that are in fact binary mixtures." (italics his) Thus Faure is not confident that a two-component mixture will always give good mixing hyperbolae.
Finally, the third (and fourth or more if present) component of a true cogenetic suite of rocks (giving a true isochron) might happen to have, for example, their Rb/Sr ratios roughly colinear with their absolute Sr concentrations. In that case they will also give "mixing hyperbolae" even though they are not mixing lines.
It will be theoretically interesting to test mixing lines for mixing hyperbolae, and also lines that are considered true isochrons by long-age geochronologists, and see what proportion of each has mixing hyperbolae. To my knowledge this has not been done. I am working on getting that done on at least a few samples. However, the results are not likely to be regarded as conclusive by either side.
The final possible difference between the short-age and long-age interpretations of isochron dating focuses on a crucial assumption. Is it believable that, for example, the strontium isotopes have homogenized?
Intuitively, it would seem that strontium isotopes should be easier to homogenize within the minerals of a given rock specimen than across rock suites. This is especially true if the rock suites extend across kilometers. So when rubidium-strontium dating is done, one might expect that minerals within a given rock would be used preferentially. [14]
However, this is apparently not usually the case. One of the statements that stunned me when I first read about Rb-Sr dating was found in Faure as his first paragraph on experimental results (pp. 120-121):
Igneous rocks of granitic composition may contain both mica minerals and K-feldspar, all of which can be dated by the Rb-Sr method. Ideally, all minerals of an igneous rock should indicate the same date which can then be regarded as the age of the rock. When mineral dates obtained from one rock specimen or from a suite of cogenetic igneous rocks are in agreement, they are said to be "concordant." Unfortunately, "discordance" of mineral dates is more common than "concordance." The reason is that the constituent minerals of a rock may gain or lose radiogenic ^{87}Sr as a result of reheating during regional or contact metamorphism after crystallization from a magma. In such cases, the mineral dates generally are not reliable indicators of the age of the rock. We must therefore turn to the rocks themselves if we want to determine their ages.It is much easier for me to visualize equilibration of strontium isotopes in the minerals in a rock than in whole rock samples in a batholith, for example. Unfortunately, we have almost no direct experimental evidence here. The indirect evidence I can find (admittedly not exhaustive, and for that reason very possibly not representative, but reasonably representative of the evidence I have seen), is in four forms:
1. Sedimentary rocks, deposited under water, do not homogenize their strontium if the grain size, at least of illite clay, is 2 microns or larger (and maybe not then--the limit is based on the assumption that long-age dates are correct).^{13}
^{13} Faure, p. 130, cites some examples. |
2. Whole rock dating is sometimes assumed to be reasonably accurate when the whole rocks are separated by miles. In fact, in one case, because the Rb-Sr dates are younger than the other dates, it is assumed that without further physical mixing, the strontium isotopes somehow equilibrated across the same distances after the rock was emplaced. And this without affecting the argon content of the rocks, so that the K-Ar age was still correct!^{14}
^{14}Lanphere MA, Wasserburg GJF, Albee AL, Tilton GR: "Redistribution of strontium and rubidium isotopes during metamorphism, World Beater Complex, Panamint Range, California." In: Craig H, Miller SL, Wasserburg GJ (eds): Isotopic and Cosmic Chemistry. Amsterdam: North-Holland Publishing Company, 1964, pp. 269-320. |
3. There are several examples of lava which flowed in recent to Miocene times whose Rb-Sr dates are in the 500-1,500 Ma range.^{15}
^{15}See note 8 for some examples. |
4. The experimental data are against easy migration of strontium atoms, at least according to Hanson and Gast.^{16}
^{16}Hanson GN, Gast PW: "Kinetic studies in contact metamorphic zones." Geochim et Cosmochim Acta 1967;31:1119-53. |
For what it is worth, where I have seen the mineral dates reported, they have generally been about an order of magnitude younger than the corresponding whole rock dates.
It would appear from the foregoing that the theory of strontium isotopic equilibration does not have the support of the available experimental evidence. This should be tested, as it may make it possible to distinguish experimentally between the predictions of short-age and long-age interpretations of life on earth, at least if short-age theories assume no significant change in radioactive time constants during the Flood. Further evaluation may also clarify the validity of other methods of isochron dating.
If all the magma on the earth is "aged" (either naturally or artificially) to the same extent, would one not get the same age for all isochrons, unless some are true isochrons? In other words, if all magmas were initially to lie on the same isochron line, would not all mixtures of these magmas give either points or "isochron" lines with identical "ages" when they are mixed?
If the geology of earth were simple, this would be the case. However, it is not simple, and several kinds of processes can flatten the original isochron without requiring millions of years. For example, let us take two magmas, one with a ^{87}Rb/^{86}Sr ratio of 0.64 and a ^{87}Sr/^{86}Sr ratio of 0.70, and one with a ^{87}Rb/^{86}Sr ratio of 0.16 and a ^{87}Sr/^{86}Sr ratio of 0.70. The two magmas have the same ^{87}Sr/^{86}Sr ratio, as they are primordial. We then age the two magmas 4.54 billion years (naturally or artificially--it does not matter which). We now have a ^{87}Rb/^{86}Sr ratio of 0.60 and a ^{87}Sr/^{86}Sr ratio of 0.74 for the first magma and a ^{87}Sr/^{86}Sr ratio of 0.15 and a ^{87}Sr/^{86}Sr ratio of 0.71 for the second magma. We now send both magmas up through country rock with a composition identical to the second magma. The composition of the second magma does not change. However, if the first magma dissolves enough country rock so that the ^{86}Sr content of the country rock is equal to the ^{86}Sr content of the first magma, the new magma will have a ^{87}Rb/^{86}Sr ratio of 0.375 and a ^{87}Sr/^{86}Sr of 0.725. This would still be on the original isochron line. But if we allow the magma to fractionally crystallize so that the ^{87}Rb/^{86}Sr ratio of the more fluid part rises [16] back to 0.60,^{17}
^{17}17We have assumed (a reasonable first approximation) that the ratio of rubidium in the rubidium-rich fraction to rubidium in the rubidium-poor fraction has remained constant. We have also assumed that the ratio of strontium in the rubidium-rich fraction to strontium in the rubidium-poor fraction has not changed. Finally, we have assumed that the volume of rubidium-rich rock and that of rubidium-poor rock have not changed. Slightly different numbers will be obtained if we assume that the numbers can change, but since rubidium is a very tiny fraction of the rock, we should not expect the changes to be great. In fact, the situation is more complicated. An adjustment should be made for the fact that there is now more calcium and less potassium than previously. The potassium will change by roughly one part in 1,000, and the calcium (on the average) by somewhat less. Other elements will also change slightly. At some point making the corrections becomes not worthwhile. |
So the answer to the last question is, one can fairly easily flatten "isochrons" that are really mixing lines, and give straight lines, as long as one allows for mixing and fractional crystallization. It does not require large amounts of time.
If there is a characteristic fractionation coefficient for each of several isochron dating systems, and some process of magma mixing and re-differentiation occurs, the same percentage flattening can happen with each system. One therefore can obtain matching reduced "isochron" dates, which actually are produced by mixing lines. Whether this process is believable depends on the precision of the measurements and the precise parameters of mixing. As far as I know, one cannot find this derivation in textbooks, and so those with the proper competence are urged to follow the math closely looking for errors.
Suppose we take a rock and differentiate it into two fractions, fraction 1 containing p_{1} parent, d_{1} daughter, and r_{1} reference isotopes, and fraction 2 containing p_{2} parent, d_{2} daughter, and r_{2} reference isotopes. We note that
but assume that
(so that the rock can be dated). This means that
and that p_{1}r_{2} - p_{2}r_{1} ≠ 0. [17]
We now age it (either naturally or artificially) so that a fraction f of the parent in each rock turns into daughter. We now have, for rock 1, p_{1} - fp_{1} parent and d_{1} + fp_{1} daughter, and for rock 2 p_{2} - fp_{2} parent and d_{2} + fp_{2} daughter. The slope of the line is
This makes sense, for since t = ln (p_{0}/p) / k, and
then t = ln (1 - slope) / k, which is the traditional formula for the age of an isochron.
We now mix a unit amount of fraction 1 with x amount of fraction 2, and a unit amount of fraction 2 with y amount of fraction 1. The new concentrations will be, for parent,
for daughter,
and for reference isotope,
The slope for this line is
which is identical to the previous slope. We expect this because this process is only a special case of a mixing line.
However, if we now allow our two magmas to re-differentiate with the same ratios of elements as in the differentiation of the original magma, we now have a slope of
1+x(r_{1}/r_{2})_{1}+y/(r_{1}/r_{2})_{1}+xy = 1+x(r_{1}/r_{2})_{2}+y/(r_{1}/r_{2})_{2}+xy
x(r_{1}/r_{2})_{1}+y/(r_{1}/r_{2})_{1} = x(r_{1}/r_{2})_{2}+y/(r_{1}/r_{2})_{2}
x(r_{1}/r_{2})_{1}-x(r_{1}/r_{2})_{2} = y/(r_{1}/r_{2})_{2}-y/(r_{1}/r_{2})_{1}
x(r_{1}/r_{2})_{1}^{2}(r_{1}/r_{2})_{2}-x(r_{1}/r_{2})_{1}(r_{1}/r_{2})_{2}^{2} = y(r_{1}/r_{2})_{1}-y(r_{1}/r_{2})_{2}
x(r_{1}/r_{2})_{1}(r_{1}/r_{2})_{2}[(r_{1}/r_{2})_{1}-(r_{1}/r_{2})_{2}] = y[(r_{1}/r_{2})_{1}-(r_{1}/r_{2})_{2}].
Assuming (r_{1}/r_{2})_{1} ≠ (r_{1}/r_{2})_{2},
(If x = y, this means that (r_{1}/r_{2})_{1} = (r_{2}/r_{1})_{2}.)
However, three further points need to be made. First, this model may prove in the end to be no more reliable than the one which posited a mixing hyperbola, about which Faure had his doubts. Second, it is not necessary for either the fractionation ratios to be identical or the x and y values to precisely match their differences. All that has to happen is that they be close enough so that the age reductions match to within the limits of statistical error. Some idea of the behavior of these variables may be obtained from the following observations. If x and y are varied while holding r constant one gets a graph something like the one on the next page.
[20]
If one holds y constant while allowing x and r to vary, the results look like this:
Another complication is that slope reduction does not automatically translate linearly into "age" reduction. The formula for the reduction of an "age" is
Note that for samarium-neodymium and rubidium-strontium dating the transformation is nearly linear, whereas for uranium-238 dating the transformation is significantly non-linear, and for uranium-235 dating the transformation is grossly non-linear. It is of interest that usually uranium-235 dates are generally "older" than uranium-238 dates, which is predicted by this model. Also, neither generally match thorium-232 dates.
Finally, I will suggest that this model should be tested, first on whole-rock systems that everyone can agree are mixing lines, and then with systems where there is a disagreement between creationists and evolutionists about whether they are isochron lines or mixing lines. The results could conceivably be helpful in resolving the impasse between the two sides. Then again, the results may not, as the side that has more difficulty explaining the evidence may simply retrench. At least the exercise may help to elucidate more clearly the nature of the debate.
Having said all that, I do not believe that creationists should quit either believing in a short time for life on earth, or in the possibility that the presently available radiometric data can be explained using a short-age model, at least not at this time. Now is not the time for us to throw in the towel.