Uranium–lead dating - Wikipedia
The uranium-thorium (U-Th) method can constrain the age of cave art by providing dates of formation of calcite deposits from on top of paintings or calcite layers. For Uranium-Thorium dating, the initial ratio of Th/U at the time of The method assumes that the sample does not exchange Th or U with the. 86–91 defined, 86 lateral relationships, 86, 88 marine transgressions, 88–91 4–6 Uranium, 73, 74, 76, Uranium-lead dating, 77 Uranium-thorium dating.
The ions then travel through a magnetic field, which diverts them into different sampling sensors, known as " Faraday cups ", depending on their mass and level of ionization.
On impact in the cups, the ions set up a very weak current that can be measured to determine the rate of impacts and the relative concentrations of different atoms in the beams. Uranium—lead dating method[ edit ] Main article: Uranium—lead dating A concordia diagram as used in uranium—lead datingwith data from the Pfunze BeltZimbabwe.Uranium-lead dating
This scheme has been refined to the point that the error margin in dates of rocks can be as low as less than two million years in two-and-a-half billion years. Zircon has a very high closure temperature, is resistant to mechanical weathering and is very chemically inert. Zircon also forms multiple crystal layers during metamorphic events, which each may record an isotopic age of the event.
This can be seen in the concordia diagram, where the samples plot along an errorchron straight line which intersects the concordia curve at the age of the sample.
Samarium—neodymium dating method[ edit ] Main article: Samarium—neodymium dating This involves the alpha decay of Sm to Nd with a half-life of 1. Accuracy levels of within twenty million years in ages of two-and-a-half billion years are achievable. Potassium—argon dating This involves electron capture or positron decay of potassium to argon Potassium has a half-life of 1.
Rubidium—strontium dating method[ edit ] Main article: Rubidium—strontium dating This is based on the beta decay of rubidium to strontiumwith a half-life of 50 billion years.
This scheme is used to date old igneous and metamorphic rocksand has also been used to date lunar samples. Closure temperatures are so high that they are not a concern. Rubidium-strontium dating is not as precise as the uranium-lead method, with errors of 30 to 50 million years for a 3-billion-year-old sample.
Uranium—thorium dating method[ edit ] Main article: Uranium—thorium dating A relatively short-range dating technique is based on the decay of uranium into thorium, a substance with a half-life of about 80, years. It is accompanied by a sister process, in which uranium decays into protactinium, which has a half-life of 32, years.
While uranium is water-soluble, thorium and protactinium are not, and so they are selectively precipitated into ocean-floor sedimentsfrom which their ratios are measured. The scheme has a range of several hundred thousand years.
A related method is ionium—thorium datingwhich measures the ratio of ionium thorium to thorium in ocean sediment.
Radiocarbon dating method[ edit ] Main article: Carbon is a radioactive isotope of carbon, with a half-life of 5, years,   which is very short compared with the above isotopes and decays into nitrogen. Carbon, though, is continuously created through collisions of neutrons generated by cosmic rays with nitrogen in the upper atmosphere and thus remains at a near-constant level on Earth.
The carbon ends up as a trace component in atmospheric carbon dioxide CO2. A carbon-based life form acquires carbon during its lifetime. Plants acquire it through photosynthesisand animals acquire it from consumption of plants and other animals. When an organism dies, it ceases to take in new carbon, and the existing isotope decays with a characteristic half-life years.
The proportion of carbon left when the remains of the organism are examined provides an indication of the time elapsed since its death. This makes carbon an ideal dating method to date the age of bones or the remains of an organism.
The carbon dating limit lies around 58, to 62, years.
Everything Worth Knowing About Scientific Dating Methods | posavski-obzor.info
However, local eruptions of volcanoes or other events that give off large amounts of carbon dioxide can reduce local concentrations of carbon and give inaccurate dates. The releases of carbon dioxide into the biosphere as a consequence of industrialization have also depressed the proportion of carbon by a few percent; conversely, the amount of carbon was increased by above-ground nuclear bomb tests that were conducted into the early s.
Also, an increase in the solar wind or the Earth's magnetic field above the current value would depress the amount of carbon created in the atmosphere. Fission track dating method[ edit ] Main article: This involves inspection of a polished slice of a material to determine the density of "track" markings left in it by the spontaneous fission of uranium impurities.
The uranium content of the sample has to be known, but that can be determined by placing a plastic film over the polished slice of the material, and bombarding it with slow neutrons.
This causes induced fission of U, as opposed to the spontaneous fission of U. The fission tracks produced by this process are recorded in the plastic film. The uranium content of the material can then be calculated from the number of tracks and the neutron flux.
This scheme has application over a wide range of geologic dates. For dates up to a few million years micastektites glass fragments from volcanic eruptionsand meteorites are best used.
Older materials can be dated using zirconapatitetitaniteepidote and garnet which have a variable amount of uranium content. The technique has potential applications for detailing the thermal history of a deposit. The residence time of 36Cl in the atmosphere is about 1 week. Thus, as an event marker of s water in soil and ground water, 36Cl is also useful for dating waters less than 50 years before the present.
Luminescence dating methods[ edit ] Main article: Luminescence dating Luminescence dating methods are not radiometric dating methods in that they do not rely on abundances of isotopes to calculate age.
Instead, they are a consequence of background radiation on certain minerals. Over time, ionizing radiation is absorbed by mineral grains in sediments and archaeological materials such as quartz and potassium feldspar.
The radiation causes charge to remain within the grains in structurally unstable "electron traps". There are quite a number of mechanisms in operation in a magma chamber.
I count at least three so far -- sorting by density, sorting by melting point, and sorting by how easily something is incorporated into minerals that form at the top of a magma chamber. Then you have to remember that sometimes one has repeated melting and solidification, introducing more complications.
There is also a fourth mechanism -- differences in solubilities. How anyone can keep track of this all is a mystery to me, especially with the difficulties encountered in exploring magma chambers.
These will be definite factors that will change relative concentrations of parent and daughter isotopes in some way, and call into question the reliability of radiometric dating. In fact, I think this is a very telling argument against radiometric dating. Another possibility to keep in mind is that lead becomes gaseous at low temperatures, and would be gaseous in magma if it were not for the extreme pressures deep in the earth. It also becomes very mobile when hot. These processes could influence the distribution of lead in magma chambers.
Let me suggest how these processes could influence uranium-lead and thorium-lead dates: The following is a quote from The Earth: The magnesium and iron rich minerals come from the mantle subducted oceanic plateswhile granite comes from continental sediments crustal rock. The mantle part solidifies first, and is rich in magnesium, iron, and calcium. So it is reasonable to expect that initially, the magma is rich in iron, magnesium, and calcium and poor in uranium, thorium, sodium, and potassium.
Later on the magma is poor in iron, magnesium, and calcium and rich in uranium, thorium, sodium, and potassium. It doesn't say which class lead is in. But lead is a metal, and to me it looks more likely that lead would concentrate along with the iron. If this is so, the magma would initially be poor in thorium and uranium and rich in lead, and as it cooled it would become rich in thorium and uranium and poor in lead.
Thus its radiometric age would tend to decrease rapidly with time, and lava emitted later would tend to look younger. Another point is that of time. Suppose that the uranium does come to the top by whatever reason.
Perhaps magma that is uranium rich tends to be lighter than other magma. Or maybe the uranium poor rocks crystallize out first and the remaining magma is enriched in uranium. Would this cause trouble for our explanation?
It depends how fast it happened. Some information from the book Uranium Geochemistry, Mineralogy, Geology provided by Jon Covey gives us evidence that fractionation processes are making radiometric dates much, much too old.
The half life of U is 4. Thus radium is decaying 3 million times as fast as U At equilibrium, which should be attained inyears for this decay series, we should expect to have 3 million times as much U as radium to equalize the amount of daughter produced. Cortini says geologists discovered that ten times more Ra than the equilibrium value was present in rocks from Vesuvius. They found similar excess radium at Mount St. Helens, Vulcanello, and Lipari and other volcanic sites.
The only place where radioactive equilibrium of the U series exists in zero age lavas is in Hawiian rocks. We need to consider the implications of this for radiometric dating.
How is this excess of radium being produced? This radium cannot be the result of decay of uranium, since there is far too much of it. Either it is the result of an unknown decay process, or it is the result of fractionation which is greatly increasing the concentration of radium or greatly decreasing the concentration of uranium.
Thus only a small fraction of the radium present in the lava at most 10 percent is the result of decay of the uranium in the lava. This is interesting because both radium and lead are daughter products of uranium. If similar fractionation processes are operating for lead, this would mean that only a small fraction of the lead is the result of decay from the parent uranium, implying that the U-Pb radiometric dates are much, much too old. Cortini, in an article appearing in the Journal of Volcanology and Geothermal Research also suggests this possibility.
By analogy with the behaviour of Ra, Th and U it can be suggested that Pb, owing to its large mobility, was also fed to the magma by fluids. This can and must be tested. The open-system behaviour of Pb, if true, would have dramatic consequences On the other hand, even if such a process is not operating for lead, the extra radium will decay rapidly to lead, and so in either case we have much too much lead in the lava and radiometric dates that are much, much too ancient!
It is also a convincing proof that some kind of drastic fractionation is taking place, or else an unknown process is responsible. He says this is inexplicable in a closed-system framework and certainly invalidates the Th dating method. And it is also possible that something similar is happening in the U decay chain, invalidating U based radiometric dates as well.
In fact, U and Th both have isotopes of radium in their decay chains with half lives of a week or two, and 6. Any process that is concentrating one isotope of radium will probably concentrate the others as well and invalidate these dating methods, too. Radium has a low melting point degrees K which may account for its concentration at the top of magma chambers. What radiometric dating needs to do to show its reliability is to demonstrate that no such fractionation could take place.
Can this be done? With so many unknowns I don't think so. How Uranium and Thorium are preferentially incorporated in various minerals I now give evidences that uranium and thorium are incorporated into some minerals more than others. This is not necessarily a problem for radiometric dating, because it can be taken into account. But as we saw above, processes that take place within magma chambers involving crystallization could result in a different concentration of uranium and thorium at the top of a magma chamber than at the bottom.
This can happen because different minerals incorporate different amounts of uranium and thorium, and these different minerals also have different melting points and different densities. If minerals that crystallize at the top of a magma chamber and fall, tend to incorporate a lot of uranium, this will tend to deplete uranium at the top of the magma chamber, and make the magma there look older.
Concerning the distribution of parent and daughter isotopes in various substances, there are appreciable differences. Faure shows that in granite U is 4. Some process is causing the differences in the ratios of these magmatic rocks. Depending on their oxidation state, according to Faure, uranium minerals can be very soluble in water while thorium compounds are, generally, very insoluble. These elements also show preferences for the minerals in which they are incorporated, so that they will tend to be "dissolved" in certain mineral "solutions" preferentially to one another.
More U is found in carbonate rocks, while Th has a very strong preference for granites in comparison. I saw a reference that uranium reacts strongly, and is never found pure in nature. So the question is what the melting points of its oxides or salts would be, I suppose. I also saw a statement that uranium is abundant in the crust, but never found in high concentrations. To me this indicates a high melting point for its minerals, as those with a low melting point might be expected to concentrate in the magma remaining after others crystallized out.
Such a high melting point would imply fractionation in the magma. Thorium is close to uranium in the periodic table, so it may have similar properties, and similar remarks may apply to it. It turns out that uranium in magma is typically found in the form of uranium dioxide, with a melting point of degrees centrigrade. This high melting point suggests that uranium would crystallize and fall to the bottom of magma chambers.
Geologists are aware of the problem of initial concentration of daughter elements, and attempt to take it into account. U-Pb dating attempts to get around the lack of information about initial daughter concentrations by the choice of minerals that are dated. For example, zircons are thought to accept little lead but much uranium. Thus geologists assume that the lead in zircons resulted from radioactive decay. But I don't know how they can be sure how much lead zircons accept, and even they admit that zircons accept some lead.
Lead could easily reside in impurities and imperfections in the crystal structure. Also, John Woodmorappe's paper has some examples of anomalies involving zircons. It is known that the crystal structure of zircons does not accept much lead.
However, it is unrealistic to expect a pure crystal to form in nature. Perfect crystals are very rare. In reality, I would expect that crystal growth would be blocked locally by various things, possibly particles in the way. Then the surrounding crystal surface would continue to grow and close up the gap, incorporating a tiny amount of magma.
I even read something about geologists trying to choose crystals without impurities by visual examination when doing radiometric dating. Thus we can assume that zircons would incorporate some lead in their impurities, potentially invalidating uranium-lead dates obtained from zircons. Chemical fractionation, as we have seen, calls radiometric dates into question.
But this cannot explain the distribution of lead isotopes. There are actually several isotopes of lead that are produced by different parent substances uraniumuraniumand thorium. One would not expect there to be much difference in the concentration of lead isotopes due to fractionation, since isotopes have properties that are very similar.
So one could argue that any variations in Pb ratios would have to result from radioactive decay. However, the composition of lead isotopes between magma chambers could still differ, and lead could be incorporated into lava as it traveled to the surface from surrounding materials. I also recall reading that geologists assume the initial Pb isotope ratios vary from place to place anyway.
Later we will see that mixing of two kinds of magma, with different proportions of lead isotopes, could also lead to differences in concentrations. Mechanism of uranium crystallization and falling through the magma We now consider in more detail the process of fractionation that can cause uranium to be depleted at the top of magma chambers.
Uranium and thorium have high melting points and as magma cools, these elements crystallize out of solution and fall to the magma chamber's depths and remelt. This process is known as fractional crystallization. What this does is deplete the upper parts of the chamber of uranium and thorium, leaving the radiogenic lead. As this material leaves, that which is first out will be high in lead and low in parent isotopes. This will date oldest.
Magma escaping later will date younger because it is enriched in U and Th. There will be a concordance or agreement in dates obtained by these seemingly very different dating methods. This mechanism was suggested by Jon Covey.
Tarbuck and Lutgens carefully explain the process of fractional crystallization in The Earth: An Introduction to Physical Geology. They show clear drawings of crystallized minerals falling through the magma and explain that the crystallized minerals do indeed fall through the magma chamber.
Further, most minerals of uranium and thorium are denser than other minerals, especially when those minerals are in the liquid phase. Crystalline solids tend to be denser than liquids from which they came. But the degree to which they are incorporated in other minerals with high melting points might have a greater influence, since the concentrations of uranium and thorium are so low. Now another issue is simply the atomic weight of uranium and thorium, which is high.
Any compound containing them is also likely to be heavy and sink to the bottom relative to others, even in a liquid form. If there is significant convection in the magma, this would be minimized, however.
At any rate, there will be some effects of this nature that will produce some kinds of changes in concentration of uranium and thorium relative to lead from the top to the bottom of a magma chamber.
Some of the patterns that are produced may appear to give valid radiometric dates. The latter may be explained away due to various mechanisms. Let us consider processes that could cause uranium and thorium to be incorporated into minerals with a high melting point. I read that zircons absorb uranium, but not much lead. Thus they are used for U-Pb dating. But many minerals take in a lot of uranium. It is also known that uranium is highly reactive.
To me this suggests that it is eager to give up its 2 outer electrons. This would tend to produce compounds with a high dipole moment, with a positive charge on uranium and a negative charge on the other elements.
This would in turn tend to produce a high melting point, since the atoms would attract one another electrostatically.
I'm guessing a little bit here. There are a number of uranium compounds with different melting points, and in general it seems that the ones with the highest melting points are more stable. I would suppose that in magma, due to reactions, most of the uranium would end up in the most stable compounds with the highest melting points. These would also tend to have high dipole moments.
Now, this would also help the uranium to be incorporated into other minerals. The electric charge distribution would create an attraction between the uranium compound and a crystallizing mineral, enabling uranium to be incorporated. But this would be less so for lead, which reacts less strongly, and probably is not incorporated so easily into minerals.
So in the minerals crystallizing at the top of the magma, uranium would be taken in more than lead. These minerals would then fall to the bottom of the magma chamber and thus uranium at the top would be depleted. It doesn't matter if these minerals are relatively lighter than others. The point is that they are heavier than the magma. Two kinds of magma and implications for radiometric dating It turns out that magma has two sources, ocean plates and material from the continents crustal rock.
This fact has profound implications for radiometric dating. Mantle material is very low in uranium and thorium, having only 0. The source of magma for volcanic activity is subducted oceanic plates.
Subduction means that these plates are pushed under the continents by motions of the earth's crust. While oceanic plates are basaltic mafic originating from the mid-oceanic ridges due to partial melting of mantle rock, the material that is magma is a combination of oceanic plate material and continental sediments. Subducted oceanic plates begin to melt when they reach depths of about kilometers See Tarbuck, The Earth, p. In other words, mantle is not the direct source of magma.
Further, Faure explains that uraninite UO sub2 is a component of igneous rocks Faure, p. Uraninite is also known as pitchblende. According to plate tectonic theory, continental crust overrides oceanic crust when these plates collide because the continental crust is less dense than the ocean floor. As the ocean floor sinks, it encounters increasing pressures and temperatures within the crust. Ultimately, the pressures and temperatures are so high that the rocks in the subducted oceanic crust melt.
Once the rocks melt, a plume of molten material begins to rise in the crust. As the plume rises it melts and incorporates other crustal rocks. This rising body of magma is an open system with respect to the surrounding crustal rocks. It is possible that these physical processes have an impact on the determined radiometric age of the rock as it cools and crystallizes.
Time is not a direct measurement. The actual data are the ratios of parent and daughter isotopes present in the sample. Time is one of the values that can be determined from the slope of the line representing the distribution of the isotopes. Isotope distributions are determined by the chemical and physical factors governing a given magma chamber.
Uranium is believed to be able to incorporate itself as a trace material in many other minerals of low density, and so be relatively highly concentrated in the crust.
A lower mantle concentration of uranium is inferred because if the mantle contained the same uranium concentration as the crust, then the uranium's heat of radiactive decay would keep the crust molten. Rhyolites in Yellowstone N. Most genetic models for uranium deposits in sandstones in the U. Most of the uranium deposits in Wyoming are formed from uraniferous groundwaters derived from Precambrian granitic terranes.
Uranium in the major uranium deposits in the San Juan basin of New Mexico is believed to have been derived from silicic volcanic ash from Jurassic island arcs at the edge of the continent. From the above sources, we see that another factor influencing radiometric dates is the proportion of the magma that comes from subducted oceanic plates and the proportion that comes from crustal rock. Initially, we would expect most of it to come from subducted oceanic plates, which are uranium and thorium poor and maybe lead rich.
Later, more of the crustal rock would be incorporated by melting into the magma, and thus the magma would be richer in uranium and thorium and poorer in lead. So this factor would also make the age appear to become younger with time. There are two kinds of magma, and the crustal material which is enriched in uranium also tends to be lighter. For our topic on radiometric dating and fractional crystallization, there is nothing that would prevent uranium and thorium ores from crystallizing within the upper, lighter portion of the magma chamber and descending to the lower boundaries of the sialic portion.
The upper portion of the sialic magma would be cooler since its in contact with continental rock, and the high melting point of UO sub 2 uranium dioxide, the common form in granite: The same kind of fractional crystallization would be true of non-granitic melts.
I think we can build a strong case for fictitious ages in magmatic rocks as a result of fractional cystallization and geochemical processes. As we have seen, we cannot ignore geochemical effects while we consider geophysical effects. Sialic granitic and mafic basaltic magma are separated from each other, with uranium and thorium chemically predestined to reside mainly in sialic magma and less in mafic rock.
Here is yet another mechanism that can cause trouble for radiometric dating: As lava rises through the crust, it will heat up surrounding rock. Lead has a low melting point, so it will melt early and enter the magma. This will cause an apparent large age. Uranium has a much higher melting point.
It will enter later, probably due to melting of materials in which it is embedded. This will tend to lower the ages. Mechanisms that can create isochrons giving meaningless ages: Geologists attempt to estimate the initial concentration of daughter product by a clever device called an isochron. Let me make some general comments about isochrons. The idea of isochrons is that one has a parent element, P, a daughter element, D, and another isotope, N, of the daughter that is not generated by decay.
One would assume that initially, the concentration of N and D in different locations are proportional, since their chemical properties are very similar.
Note that this assumption implies a thorough mixing and melting of the magma, which would also mix in the parent substances as well. Then we require some process to preferentially concentrate the parent substances in certain places. Radioactive decay would generate a concentration of D proportional to P. By taking enough measurements of the concentrations of P, D, and N, we can solve for c1 and c2, and from c1 we can determine the radiometric age of the sample.
Otherwise, the system is degenerate. Thus we need to have an uneven distribution of D relative to N at the start. If these ratios are observed to obey such a linear relationship in a series of rocks, then an age can be computed from them.
The bigger c1 is, the older the rock is. That is, the more daughter product relative to parent product, the greater the age. Thus we have the same general situation as with simiple parent-to-daughter computations, more daughter product implies an older age.
This is a very clever idea. However, there are some problems with it. First, in order to have a meaningful isochron, it is necessary to have an unusual chain of events. Initially, one has to have a uniform ratio of lead isotopes in the magma. Usually the concentration of uranium and thorium varies in different places in rock. This will, over the assumed millions of years, produce uneven concentrations of lead isotopes.
To even this out, one has to have a thorough mixing of the magma. Even this is problematical, unless the magma is very hot, and no external material enters. Now, after the magma is thoroughly mixed, the uranium and thorium will also be thoroughly mixed.
What has to happen next to get an isochron is that the uranium or thorium has to concentrate relative to the lead isotopes, more in some places than others.
So this implies some kind of chemical fractionation. Then the system has to remain closed for a long time. This chemical fractionation will most likely arise by some minerals incorporating more or less uranium or thorium relative to lead. Anyway, to me it seems unlikely that this chain of events would occur. Another problem with isochrons is that they can occur by mixing and other processes that result in isochrons yielding meaningless ages.
Sometimes, according to Faure, what seems to be an isochron is actually a mixing line, a leftover from differentiation in the magma. Fractionation followed by mixing can create isochrons giving too old ages, without any fractionation of daughter isotopes taking place. To get an isochron with a false age, all you need is 1 too much daughter element, due to some kind of fractionation and 2 mixing of this with something else that fractionated differently.
Since fractionation and mixing are so common, we should expect to find isochrons often. How they correlate with the expected ages of their geologic period is an interesting question. There are at least some outstanding anomalies. Faure states that chemical fractionation produces "fictitious isochrons whose slopes have no time significance.
As an example, he uses Pliocene to Recent lava flows and from lava flows in historical times to illustrate the problem.
He says, these flows should have slopes approaching zero less than 1 million yearsbut they instead appear to be much older million years. Steve Austin has found lava rocks on the Uinkeret Plateau at Grand Canyon with fictitious isochrons dating at 1.
Then a mixing of A and B will have the same fixed concentration of N everywhere, but the amount of D will be proportional to the amount of P. This produces an isochron yielding the same age as sample A. This is a reasonable scenario, since N is a non-radiogenic isotope not produced by decay such as leadand it can be assumed to have similar concentrations in many magmas.
Magma from the ocean floor has little U and little U and probably little lead byproducts lead and lead Magma from melted continental material probably has more of both U and U and lead and lead Thus we can get an isochron by mixing, that has the age of the younger-looking continental crust. The age will not even depend on how much crust is incorporated, as long as it is non-zero. However, if the crust is enriched in lead or impoverished in uranium before the mixing, then the age of the isochron will be increased.
If the reverse happens before mixing, the age of the isochron will be decreased. Any process that enriches or impoverishes part of the magma in lead or uranium before such a mixing will have a similar effect. So all of the scenarios given before can also yield spurious isochrons. I hope that this discussion will dispel the idea that there is something magical about isochrons that prevents spurious dates from being obtained by enrichment or depletion of parent or daughter elements as one would expect by common sense reasoning.
So all the mechanisms mentioned earlier are capable of producing isochrons with ages that are too old, or that decrease rapidly with time. The conclusion is the same, radiometric dating is in trouble. I now describe this mixing in more detail. Suppose P p is the concentration of parent at a point p in a rock. The point p specifies x,y, and z co-ordinates. Let D p be the concentration of daughter at the point p. Let N p be the concentration of some non-radiogenic not generated by radioactive decay isotope of D at point p.
Suppose this rock is obtained by mixing of two other rocks, A and B. Suppose that A has a for the sake of argument, uniform concentration of P1 of parent, D1 of daughter, and N1 of non-radiogenic isotope of the daughter. Thus P1, D1, and N1 are numbers between 0 and 1 whose sum adds to less than 1.
Suppose B has concentrations P2, D2, and N2. Let r p be the fraction of A at any given point p in the mixture. So the usual methods for augmenting and depleting parent and daughter substances still work to influence the age of this isochron. More daughter product means an older age, and less daughter product relative to parent means a younger age. In fact, more is true. Any isochron whatever with a positive age and a constant concentration of N can be constructed by such a mixing.
It is only necessary to choose r p and P1, N1, and N2 so as to make P p and D p agree with the observed values, and there is enough freedom to do this.
Anyway, to sum up, there are many processes that can produce a rock or magma A having a spurious parent-to-daughter ratio. Then from mixing, one can produce an isochron having a spurious age.
This shows that computed radiometric ages, even isochrons, do not have any necessary relation to true geologic ages. Mixing can produce isochrons giving false ages.
But anyway, let's suppose we only consider isochrons for which mixing cannot be detected. How do their ages agree with the assumed ages of their geologic periods? As far as I know, it's anyone's guess, but I'd appreciate more information on this. I believe that the same considerations apply to concordia and discordia, but am not as familiar with them. It's interesting that isochrons depend on chemical fractionation for their validity.