| The Age of the Solar System:
Introductory Remarks: Guessing Ages.
We have now considered the manifest ways in which the Solar System shows
and we will soon be developing an appreciation of
Before we do that, however, let us address one final consideration: that of the determination of the age of the Earth (and the other parts of the Solar System). You may -- and, I hope, already do -- know that the Solar System formed about 4.6 billion years ago. Where does this number come from? How do we know?
Let us start with a simple but not very useful analogy. How can we tell the age of another person? When you see me lecturing at the front of the theatre, I hope that you do not estimate that I am one hundred years old ; nor would you suspect that I am only fifteen . But if you make any guess at all, on what basis do you do so?
Surely the answer is that you already have an idea of how people of various ages look -- in other words, you compare me to other people whose ages you know. (Am I like your middle-aged uncle?) Moreover, we can rely on our memories of ourselves and others (or look at old photographs) to understand how people's appearances slowly change as the years roll on.
Unfortunately, no such technique is available to us for the Solar System. Although we have recently detected planets associated with stars other than the sun, we have had time to learn only a little about them. But even if we could study them in detail, we would not know their ages in any independent way, or how they change in appearance as aeons pass, so knowing about them does not solve the problem. (It just gives us more examples of solar systems of unknown ages!)
What we really need is some new indicator of age within our own Solar System, something which changes in a regular and well-understood fashion as time passes. Of course, it is not enough if that indicator merely allows us to count the passage of years into the future, since that would still leave the remote past a complete mystery. Whatever our new tool is, we will want to be able to apply it 'backwards,' to find out how long ago everything got started.
Let us turn back briefly to our analogy. For human beings, we could imagine discovering that there is some kind of precise in-built clock, some indicator of age, in every person. For instance, suppose we were born with a completely full-sized brain, but that the mere process of living gradually used it up, just as a fire consumes fuel, so that bits and pieces of it die away over one's lifetime. In that case, if a surgeon opened a person's skull (or used X-ray imaging) to determine that half of the person's brain had been consumed, the conclusion would be that his or her life is about half over. This would imply an age of about fifty years, for instance, if the average expectancy was one hundred years.
Of course, the brain doesn't behave in this fashion. I think you will agree, however, that there are such `clocks' for human beings, features which indicate the passage of years. Hair turns grey; the skin loses some of its elasticity and lines form; muscles and joints stiffen; and older people become infinitely wise. (You can dispute that last one if you like!) One problem is that these are not very precise indicators -- some people turn grey when still quite young, for instance -- but the second more serious problem is that the indicators provide no absolute numbers. You have to know some ages already on independent grounds in order to establish the typical timescale of physical change or decline.
We could say things like "Well, the Solar System must have been around for a long time for life on Earth to have evolved to the level of enormous complexity we see," but that is clearly pretty qualitative. We need something better.
Thinking About Cows.We do have the necessary tools: we can use the phenomenon of natural radioactive decay to determine the ages of our own planet and some of the other constituents of the solar system. To understand this, consider yet another analogy, one which is not original with me but which is very useful. Let us imagine a farmer who owns a herd of cows which he likes to move from field to field every couple of weeks. (It may be important to him, for instance, that they have a mixed diet -- some alfalfa, some grass, some clover.) One morning, he awakens with a mild case of amnesia and cannot remember when he last moved the cows. How can he make some reasonable estimate? The answer is actually quite simple. All he has to do is go and examine the field where the cows are now. If there is a lot of manure accumulated in the field, the cows have been there for some time, and might be in need of an immediate change. If there is no manure to be seen, the cows have presumably only just arrived. Of course, there are potential complications. Perhaps, for instance, the cows have been there for weeks, but a very heavy rain recently washed all the manure out of the field, leaving the impression that they have only just arrived. Conversely, perhaps the cows were only recently led into a field already full of manure from the last time around, so that we mistakenly believe that they have been there for weeks already. The lesson, of course, is that we need to know the processes and timescales over which manure gets cleaned out of the fields (by rain, through consumption by dung beetles, and so on) so that we have a good understanding of what we can safely interpret when we see cows in a field which is in a particular state. In geology, the role of the cows is played by the elements which are naturally radioactive - that is, they spontaneously spit out small bits of themselves (in the form of the so-called alpha rays, beta rays, and gamma rays) and thereby change, or transmute, into different elements. What is left, the so-called daughter product, is the analog of the accumulating manure. Uranium, for example, gradually changes into lead, in a well-understood way. If you start with a chunk of pure uranium, it will gradually turn into a chunk of pure lead. The important point is that the relative proportions of uranium and lead change with time in a way which provides us with a precise clock, or chronometer. The application of such techniques to the Earth itself is the science of geochronology.
An Introduction to Atoms.To explain geochronology, I begin with a simple discussion of the nature of the atom. I encourage you to imagine an atom as something like a tiny solar system, even though the analogy is imperfect in ways I will discuss later. Thus: in the Solar System, there is a massive lump (the sun) which you can visualise as being nearly at rest is the middle. Around it orbit the planets and asteroids, much smaller lumps which are held in the grip of the sun's gravitational influence. in the atom, there is a massive central nucleus, made up of sub-lumps called protons and neutrons. (You might like to visualise the nucleus as something like a handful of red and blue marbles.) Each [red] proton carries a positive electric charge, while each [blue] neutron is electrically neutral. around the nucleus orbit the electrons, each of which is much less massive than a proton or a neutron. (You may remember that the sun is about one thousand times as massive as Jupiter. Each proton or neutron is about two thousand times as massive as an electron.) Each electron carries a negative charge, and orbits the nucleus because of the electric force between itself and the positively-charged nucleus. Usually, there are as many electrons orbiting the nucleus as there are protons within it, so the atom as a whole is electrically neutral. the number of protons in the nucleus, the so-called atomic number, determines the kind of element you have. Hydrogen, for instance, is the simplest element: it has exactly one proton. Helium has two; lithium has three; and so on. Among the familiar elements, carbon has six protons; nitrogen has seven; oxygen has eight; and so on. Uranium, one of the most complex, has ninety-two protons in its nucleus. there is a second number we assign to every element: the atomic mass, which is simply the total number of protons and neutrons in the nucleus. Helium, for instance, contains two protons (which is what makes it helium) but also two neutrons, so its atomic mass is four while its atomic number is two .
Isotopes.A given element may exist in several different forms, known as isotopes. (The prefix `iso' means `the same.' On a weather map, isobars are lines which link up regions where the atmospheric [barometric] pressure is the same. Isometric exercises are those in which nothing changes position - the kind, for instance, where you press your hands hard against an immovable wall to tense the muscles.) But what exactly is the difference between two isotopes of a given element? Consider an atom of helium. It must contain two protons, or else it would not be helium at all, but the number of neutrons may not always be precisely two, as it is in the most common form of this element. There is also a rare `light' helium isotope in which there are two protons but only one neutron, so that its atomic mass is three rather than four. Likewise hydrogen, the simplest element. In its most common form, the hydrogen nucleus consists of a single proton and no neutrons at all, so that its atomic number and atomic mass are both one. But there is an isotope in which a single neutron accompanies the proton, doubling the mass. This isotope, which is quite rare in nature, is usually called heavy hydrogen, but it also goes by the name of deuterium. There is also an isotope with two neutrons; this is called tritium. At the other end of the scale, heavy elements can also exist in varied isotopic forms. Uranium, for instance, contains ninety-two protons (which is what makes it uranium), but there are several isotopes including one with one hundred and forty-three neutrons (callled U-235, since the atomic mass is 235) and another with one hundred and forty-six neutrons (U-238). In nature, these two isotopes are the most common, but they are all mixed up in the earth's rocks and ores, where they are not readily separated. This remark reminds me of an important piece of 20th century history. One of the uranium isotopes is a much more efficient source of energy in nuclear fission than is the other. Much technological effort during the Second World War was focussed on the best way to refine and separate these isotopes so that a working atomic bomb could be constructed. (This is a subject we will discuss in Phys 016 when we discuss the sources of energy within the stars. Perhaps I should say right now, however, that nuclear fission - the sort of thing that happens in an atomic bomb - is not the source of stellar energy.)
Heavy Water.My mention of heavy hydrogen (deuterium) reminds me to tell you of the existence of heavy water. An ordinary molecule of water contains one atom of oxygen and two of hydrogen. In total, such a molecule contains ten protons (eight in the oxygen nucleus and one in each hydrogen), plus eight more neutrons (all in the oxygen) -- a total of eighteen nucleons, as these particles are collectively called. But if you now replace each hydrogen atom with a deuterium atom, which can be done since the chemical behaviour is exactly the same (water will still form), you add two neutrons to each molecule, and there are now twenty nucleons per molecule. The interesting point is that the individual molecules take up no more room than they did before, because the nuclear parts are simply tiny lumps at the very middle of the atoms, occupying a negligible volume. What holds the atoms and molecules apart are the interactions between the electrons which orbit the atoms and molecules, and these are essentially unaffected by the addition of extra neutrons. In short, the new deuterium-containing water will be just like ordinary water in its basic behaviour and properties, except that it will be about ten percent denser. For this reason, it has come to be called `heavy water.' I should perhaps emphasise that heavy water is not identical to ordinary water in absolutely all respects. There are small differences, for instance, in its physical behaviour at certain temperatures, such as the way it forms ice when it freezes. Moreover, if you drank a great deal of it, you might get ill since its behaviour in biological cells -- like the way it moves through cell walls by osmosis -- will be different. But the most important differences between light and heavy water are those which involve nuclear interactions. Sub-atomic particles which encounter heavy water may interact with the deuterium nuclei in ways which are quite different from what they would do if passing through a container filled with ordinary tap water. These differences are important in the design and control of `CANDU' nuclear reactors of the kind which we use in Canada to generate electric energy: the heavy water acts as a moderator , as it is called, to control the reaction rates. (I will not explain the details here.) For that reason, the Canadian government established, decades ago, a purification plant in Glace Bay, a facility in which heavy water is separated from ordinary water. Over the years, a large quantity of ultra-pure heavy water has been refined. This has an interesting consequence. Later in the year, in Phys 016, I will describe to you the Sudbury Neutrino Observatory, an exciting and important international scientific project based right here at Queen's (but with a working `observatory' deep in a mine in Sudbury). For reasons I will describe later, heavy water is used as the principal detector of the elusive neutrinos, and the SNO project has leased about a thousand tonnes of it (valued at hundreds of milions of dollars) from the Canadian government.
Isotopes of Many Kinds?At this stage, you may be wondering what limits the kinds of isotopes that might exist. Could there, for instance, be some kind of ultra-heavy hydrogen with, let us say, one proton but five hundred neutrons? Or is there some limitation on the allowed configurations? This is actually a fairly deep question in nuclear physics, but the short answer is that there are indeed limitations. Roughly speaking, any particular element may have some small number of isotopic forms (perhaps as many as six, say, but usually fewer). Moreover, the light elements often have about as many protons as neutrons (Carbon-12 has six protons and six neutrons; Carbon-14 has an extra couple of neutrons) whereas the heavy elements typically have a lot more neutrons than protons (look back at my comments on the U-235 and U-238 isotopes of uranium). Isotopes which do not obey these simple rules are not stable. Any newly-formed hypothetical `mega-hydrogen' with 500 neutrons, for example, would fall to bits instantly, for reasons I will not go into here. Only certain isotopes of certain elements can last an appreciable time. Experimental nuclear physics, coupled with our everyday experience, reveals which elements and isotopes these are, and their relative abundances.
Instabilities and Transmutations: The Alchemist's Dream.Isotopes which are unstable fall apart in a phenomenon known as radioactivity. In the simplest imaginable terms, what happens is that the nucleus of such an atom spits out a tiny particle (of a sort to be described in a moment) and turns into another element, a process called transmutation. It is interesting to note that, in this way, nature provides us with a path to the goal of the medieval alchemists. Their objective, using a mixture of chemistry, mysticism and magic, was to learn how to transmute base metals, such as lead, into gold. (Isaac Newton himself was a practioner.) We now know that no chemical treatment can succeed, but that spontaneous transmutations from one element to another can occur - although not in fact lead into gold. Moreover, it is possible to control the process within `atom smashers' (more correctly called `particle accelerators'). In such devices, a target which consists of a small lump of one kind of material is bombarded with rapidly-moving sub-atomic particles; the interactions can lead to transmutations. Of course, this is fantastically too expensive to serve as a plentiful source of a precious element like gold!
Alpha, Beta, Gamma: Three Kinds of Decay.Natural radioactivity was discovered accidentally, and the emission of particles from elements like uranium and radium was not fully understood for several decades thereafter. What quickly became apparent, however, was that there were three different kinds of particles emitted. These were named alpha, beta, and gamma particles (or rays) after the first three letters of the Greek alphabet. (You can recognize that the English word alpha-bet is itself derived from the first two of these.) Let us consider -- but not bother commiting to memory! -- a couple of exemplary decay events to see what we can learn: The unstable isotope of uranium-238 (one which contains 92 protons and 146 neutrons) decays by spitting out an alpha particle and turning into thorium-234 (a nucleus which has 90 protons and 144 neutrons). Remember the conservation of charge, one of our great Since there were 92 positively-charged protons in the original nucleus, but only 90 remain, the alpha particle must carry off two positive charges. Moreover, it must have a total mass of 4 atomic units. The conclusion is that the alpha particle must consist of two protons and two neutrons -- something which experiments confirm. Interestingly, this means that the alpha particles created in such decays are in fact the nuclei of helium! Most of the helium found on the Earth is created in just this fashion: there are large pockets of helium gas trapped in the natural gas fields of Texas, for instance, gas which was created underground by the radioactive decay of heavy elements in the Earth's rocks. (This is, however, definitely not how most of the helium in the stars and galaxies came into existence. We will learn more about that in Phys 016.) Consider another decay. The thorium-234 produced by the reaction above can itself decay by emitting a beta particle, leaving behind a nucleus of protactinium-234, one which contains 91 protons and 143 neutrons. This sounds impossible: thorium-234 has 90 protons, whereas the protactinium has 91. How can the departure of a beta particle lead to the addition of another proton? The answer lies in the fact that the protactinium has one neutron fewer than the thorium. In effect, one of the neutrons in the thorium nucleus turns into a proton (gaining a positive charge) by spitting out an electron (which carries off a negative charge. Note that the total charge remains unchanged, as is required by the conservation law). Indeed, experiments confirm that beta rays are simply electrons. This spontaneous breakdown of a neutron into a proton plus an electron is known as `beta decay,' and indeed isolated neutrons in empty space break apart in just this way surprisingly quickly. (In the confines of a heavy nucleus, they are less likely to do so.) What I have not told you yet is that such alpha and beta decays are often accompanied by the emission of very energetic photons, 'particles' of light. That is what gamma rays are, just pure radiant energy. It is in fact the gamma rays which are the most dangerous emissions from radioactive materials. They are typically energetic enough to cause radiation damage to cells in your body, and it was not uncommon for physicists early in this century to develop cancers and leukemias from handling elements like radium with insufficient protection. (The Nobel-prize winning Marie Curie died from leukemia, for instance.) Finally, I should tell you that the decays are sometimes accompanied by the emission of neutrinos, really lightweight particles which carry no charge. As I said earlier, there is a strong interest in neutrinos here at Queen's: we are involved in a special `observatory' deep in a mine in Sudbury, with a detector which detects and studies neutrinos from the sun. This will be a topic in the winter term.
Decays in Series.If you look back at the previous section, you will see that decays can run in series. In the examples given, first U-238 alpha-decayed to form Th-234, but the Th-234 then beta-decayed to form Pa-234. This sort of behaviour is not uncommon, and indeed a given unstable element may undergo a whole series of transmutations until it finally results in a stable nucleus (after which nothing changes). In the lecture, I showed one such chain of decays in which Uranium-238 passes through no fewer than thirteen other elemental and isotopic forms before winding up as the stable isotope of lead, Pb-206 (with 82 protons and 124 neutrons). There it is fated to stay forever.
Decays and Death Rates.Now that we know how certain unstable atoms can decay, we need to know what determines the chance that any particular atom will do so. In a lump of uranium, for example, not all of the atoms immediately decay into lead. In any given span of time, only a fraction do so; other atoms survive until later. What determines this sort of behaviour? To develop an understanding of this, think first about an analogy I presented in class. Suppose I present a public lecture about a planned Martian space probe to an 'Elderhostel' group of one hundred very senior citizens -- to be precise, let's suppose that they are all exactly ninety years old. Then, ten years later, I decide to present a follow-up lecture to describe the great scientific success of the Mars mission. I send out special invitations to those who came to the first lecture, only to find that essentially all of them have died in the meantime. If, instead, I had presented the first lecture to a high-school group of one hundred teen-agers, my second lecture might have drawn a good crowd (unless they had all lost interest, of course). What is the difference? The answer is obvious: older people have an increased probability of dying in the near future. Human beings do not remain unchanged through the passage of years; they age. Our bodies start to break down, the heart starts to weaken after a lifetime of toil, slowly-developing illnesses like cancer have time to get a firm hold, and so on. By contrast, atoms do not age. I cannot emphasise this strongly enough, as it is the essential point for a correct understanding of the use of radioactive chronometers! If you have a uranium atom in front of you, there is a given probability that it will decay into something else in the next five minutes -- perhaps a chance in a million, say. Now suppose that atom somehow lasts for a thousand years. What is the chance that it will decay in the next five minutes? It is still that one-in-a-million! The fact that the atom has already been around for a long time does not increase its chance of decaying. In this respect, radioactive decay is completely unlike the way in which living creatures die, a process which has a rapidly growing probability of happening as the complex internal metabolic and physiological systems break down. One way of understanding this is to imagine every uranium atom carrying a coin with it. Every so often, we metaphorically ask the atoms all to flip a coin. Those which throw `heads' decay; those which throw `tails' do not. We lose half the atoms each time we do this, and after a long time, the vast majority of the atoms will be gone, but for those which have survived there is still only a fifty-fifty chance of throwing heads (and decaying) at the next time of asking. The atoms have no 'memory' of how they have behaved in the past, and don't say to themselves: "Gee, I've flipped heads twenty times in a row, so I'd better flip a tail to 'even up the odds.'" It's still just a fifty-fifty proposition. Digression: misunderstanding the so-called 'Law of Averages' in this way is known as the Gambler's Fallacy. It's one very good reason that casinos make a lot of money! (The main reason, of course, is that the odds are always in their favour.) People watch a roulette wheel come up red ten times in a row and reason that it must come up black now, to `even the odds.' But the wheel has no memory, and the chances are still exactly fifty-fifty (ignoring the green `zero' and `double-zero' outcomes).
Constructing a Clock.Here is how to construct a clock based on an unstable radioactive element. There are two important but independent steps: calibration, and application. Calibration: First, we take an ultra-pure sample of the radioactive element - a lump containing a trillion trillion atoms, say. (We can tell how many atoms are in it just by weighing it.) We then use something like a Geiger counter to measure, in the laboratory, how many of these atoms decay in a particular time interval. (The Geiger counter tells us how many alpha particles and so on are being emitted by the lump, each one coming from the breakdown of a nucleus.) Please note, of course, that a bigger lump will produce more emissions than a smaller lump simply because it has more atoms in it. The interesting question is to calculate what fraction of the nuclei present will decay in a particular time interval. Measurements of this sort allow us to work out the half-life of a given element or isotope. The half-life is the time it would take for half of any surviving atoms still present to decay away. Please note one very important point. You do not have to monitor the material for a full half-life in order to work out what the half-life is! (Here's a simple analogy: if you discover that your five-hundred litre Jacuzzi spa is leaking a litre of water per minute, you don't have to wait around for four hours [two hundred and fifty minutes] to confirm that it will take about that long until it is only half full. You can figure it out mathematically.) Hypothetical Example: Let us suppose that your investigation reveals that the half life of the isotope you are studying turns out to be exactly ten minutes. (Let's call this exotic material mysterium, which is a name I have invented for the sake of this discussion, and assume that it decays into a daughter product which we will call tedium. ) The half-life of ten minutes means that if someone gives you a tiny speck of pure mysterium precisely at noon, one which contains (say) exactly 800 trillion atoms, then ten minutes later there will be about 400 trillion mysterium atoms remaining, with the other 400 trillion atoms having turned into tedium. (I say `about' because this is only a statistical prediction. The answer may be a little more or less, but it will be in that ballpark. If you flip a coin 100 times, you are not likely to get exactly 50 heads and 50 tails.) By the way, a lump containing 800 trillion atoms of even the heaviest known element would have a mass of less than a billionth of a gram - a speck of material far too small to see. Any sensibly-sized lump of material will contain many, many more atoms, but I have picked this number because of the numerical simplicity it provides in the following discussion. Since the radioactive decays continue as time passes, by 12:20 PM, there will be only about 200 trillion atoms of mysterium left, and 600 trillion atoms of tedium will have accumulated; by 12:30, there will be about 100 trillion mysterium atoms and 700 trillion atoms of tedium; by 12:40, there will only be about 50 trillion atoms of mysterium left, with 750 trillion atoms of tedium; and so on. Notice one immediate implication: As time passes, there are fewer and fewer decays in each unit of time because there are fewer of the unstable radioactive atoms still present. In the first ten minutes, for instance, 400 trillion alpha particles were emitted -- one from each disintegrating atom. In the ten minutes between 12:30 and 12:40, only 50 trillion were emitted; and fewer and fewer will be emitted in the future. This is why dangerous radioactive elements become less of a threat as time passes. Application: Now for the clock. Suppose an interested investigator walks into the room at 12:40 PM, and wants to know how long that lump of mysterium-plus-tedium has been sitting around. She does not need to measure the radioactivity using a Geiger counter, and she does not need to monitor any further change in the material. All she has to do is carry out a simple chemical test to determine the composition of the lump of material. As we saw, it contains 750 trillion atoms of stable tedium and 50 trillion atoms of unstable mysterium at 12:40 PM, but she will not actually directly count the atoms, of course! Instead, she will express the results of her compositional analysis in terms of proportions. In this example, she will discover that the material is only 6.25% mysterium but 93.75% tedium (the daughter product). She already knows that the half-life of mysterium is ten minutes, because this was independently determined, using a different lump of mysterium , in the calibration step described earlier. So she can work backwards, reasoning like so: Ten minutes ago, there must have been twice as much mysterium as now -- that's what the half-life means! -- so the lump must have been 12.5% mysterium (and consequently 87.5% tedium) at 12:30 PM. Similarly, at 12:20, the mineral must have been 25% mysterium (and 75% tedium); and at 12:10 the compostion must have been 50% mysterium (and 50% tedium); and at 12:00 the lump must have been pure mysterium. The clever investigator has now worked out that the sample of mysterium was given to you, in ultra-pure form, exactly 40 minutes ago. Summary: There are two particularly important things to note. 1 The original calibration, a once-only exercise, required us to watch the rate at which the material decayed over time. As I noted above, we didn't actually have to watch it during an entire half-life (remember the Jacuzzi!) but we did have to watch it for a while, to determine the rate of decay. (Actually, maybe you and I didn't even have to bother -- all that matters is that somebody worked it out! The half-life of a give radioactive species is a constant of nature. If Scientist Joe Blow, say, has done me the favour of working out the half-life of uranium in his lab, we can simply use the value he has provided -- assuming we trust his work!) In any event, the actual application does not require any monitoring at all. 2 Secondly, you can see that you would get the age wrong if there had been some contaminating daughter product in the material to start with. (This is analogous to putting the cows into a field which already has manure in it.) In the example given, suppose that someone stealthily mixed a bit of tedium into the original sample before delivering it to you. Your age calculation will then be biassed -- you will overestimate the age. You have to have some independent way of determining the extent of contamination by any 'extra' daughter product present at the time the mineral was formed. This consideration has an immediate consequence in geological age dating (in which we use, among others, the decay of uranium to lead). It would be extremely convenient if lumps of pure uranium had been deposited naturally, like gold nuggets embedded in a surrounding sea of granite, when the Earth was formed. The age would then be easy to work out. Just hack out one of those lumps and melt it down to determine the relative proportions of unstable uranium and the stable daughter product (lead). If the lump is now exactly half lead, it was formed in a pure state one uranium half-life ago; if it is three-quarters lead, it was formed two half-lives ago; and so on. But the Earth did not form with these readily identifiable lumps in absolutely pure form. Out of the magma and elements present in the early days, complex minerals formed. Granite, for instance, contains minerals which are made up of a bit of silicon, a dollop of oxygen, an atom or two of potassium, and so on, all in very complex arrangments and mixtures. In any given sample of rock, therefore, we may find some lead atoms which were present as a chemical contaminant from the very beginning, rather than being produced by the slow decay of uranium as the eons passed. In other words, we can only use radioactive elements as reliable clocks if we know the original composition of the minerals in which they are found, so that we can correct for the irrelevant original inclusion of some daughter product which was not produced through radioactive decay in the subsequent centuries. Fortunately, there are ways of dealing with this. The technique depends upon an analysis of those isotopes of the element which are not radioactive. Because they all isotopes behave identically chemically, we can use the abundances of those species to get a good estimate of the proportions of the various elements in the original mix which formed the mineral.
What Ages Anyway?There is yet another complication. Imagine a mineral which formed early in the life of the Solar System, one with a lot of uranium in it. As the aeons pass, some of the uranium turns into lead. But then there is a local disturbance - an earthquake shatters the rocks, and a volcanic eruption leads to local melting. The uranium and lead (and all the other bits and pieces) are put back into a molten slurry out of which new minerals congeal as the temperature slowly falls again. The problem is that the new minerals may have quite a different distribution of elements than the original material did. The propensity for various elements to form compounds and minerals is a complex function of the composition of the magma, the rate at which the temperature falls, and so on. There is absolutely no guarantee that the proportions of uranium and lead will be anything like they were before the disruption, and in general the use of radioactive chronometers can only allow you to work out the time which has passed since the last major episode of remelting and recrystallization, a process which `resets the clocks.' You can see that this is a real problem here on the Earth, where there is a very active geology. Volcanoes erupt; continental drift carries old crustal material down into subduction zones, where it is remelted; there are active and vigorous erosional processes at the surface; and so on. For all of these reasons, you would not expect the very oldest rocks on the Earth to date from the very moments of formation of the planet. (Indeed, the oldest rocks we know are only about 3.5 billion years old.) We can do better, in fact, by studying the composition of meteorites and rocks from the moon (which has no very active geology). Indeed, one of the objectives of the Apollo lunar program was to find examples of so-called `Genesis' rocks - those which date back as closely as possible to the formation of the Solar System. It is through the study of such samples that we can say, with confidence, that the Solar System was formed about 4.6 billion years ago.
What Chronometers Do We Use?There is one more issue we need to consider. I talked about a hypothetical element called mysterium, with a half-life of ten minutes. Suppose I had even a kilogram of such a material in absolutely pure form, with something like ten trillion trillion nuclei in it. (The precise number of atoms depends on how massive each nucleus is, but it will be something like this.) After ten minutes, only half are left; another ten minutes cuts that in half again; and so on. You may be interested to learn, and can confirm if you like, that after just a little more than eighty half-lives, or only about twelve hours, the `mysterium' will be all gone, and we will have only a lump of absolutely pure tedium, which will undergo no further changes. Whether we look at it a day from now, a week from now, a year from now, or a century from now, it will have undergone no further change. Clearly this is no good for studying ages of lumps which have been around for a long time. The half-life is too short to be useful. Now consider instead a radioactive isotope with an enormous half-life -- five billion years, say. (Uranium has one such isotope.) If I give you a chunk of this uranium in absolutely pure form at 10 A.M., some of the atoms within it will convert to lead as time passes. But by 4:00 P.M. that same day, only a tiny fraction of the atoms will have done so, since less than one trillionth of a half-life has passed by. Only a microscopically tiny number of nuclei will have decayed away. The composition may have changed from 100% uranium to 99.999999999999% uranium and 0.000000000001% lead. Measuring such tiny differences would be difficult: the slightest error would lead to unacceptably large uncertainties in the deduced age. So elements which have very long half-lives are utterly inappropriate for studying lifetimes which are short. The conclusion is this: in astronomy and geophysics, where we want to determine the ages of very old rocks, we need to use a radioactive chronometer which has a very long half-life. Several are known, including: radioactive potassium (K-40), which changes to argon (Ar-40) with a half life of 1.3 billion years; rubidium (Rb-87), which decays to strontium (Sr-87) with a half-life of 47 billion years; U-235, which decays to lead (Pb-207) with a half-life of 700 million years; and U-238, which decays to Pb-206 with a half-life of 4.5 billion years.
Other Applications: Medicine, and Carbon Dating.Isotopes with relatively short lifetimes can be important in varied contexts, not necessarily just in determining ages. Let us consider a couple of examples. Medicine: The absence of iodine in the diet can lead to a goitre, the enlargement of the thyroid gland in the throat. (Canadian regulations require that milk be iodized so that all children get enough iodine.) To test the proper functioning of the thyroid, doctors can have a patient drink a fluid which is laced with a `tracer' of a radioactive isotope of iodine (not enough to do any damage). A detector like a Geiger counter then allows one to determine where the iodine goes and in particular if it is being absorbed efficiently by the thyroid. The important thing is that such a tracer has to have a short half-life. You might think that this is to be sure that it all decays away quickly to some harmless form, but that is not really the issue. The principal reason is that you want to be sure that there is a good chance that lots of nuclei will be decaying away at the same time, emitting lots of alpha particles (or whatever it is that is given off in the process) so that the iodine can be tracked efficiently. Iodine with a half-life of just a few minutes can be used for this purpose. By the way, this discussion may leave you completely puzzled -- at least, I hope that it does! You should be asking, "If radioactive iodine has such a short half-life, why is there any of it around at all? If it formed along with the rest of the elements in the Solar System or the universe, surely by now it has all decayed away to some other daughter form? Where do we find any to put in the drink the doctor gives the patient?" If you spotted this, I congratulate you! But there is in fact no problem. The answer is that the medical people have to make new radioactive iodine when they need it, by putting some other element or isotope into a nuclear reactor and bombarding it to carry out the necessary transmutations. Quite a lot of the business carried out at the Atomic Energy Commission in Chalk River, or other reactors here and there, including one at McMaster University, is of exactly this sort, preparing isotopes for medical or laboratory purposes. Carbon Dating: Radioactive carbon (C-14) has a half-life of about 5700 years, and it is therefore useful as an indicator of age for things which are tens of thousands of years old. Obviously it is important for archaeologists in particular, and served a useful purpose in the study of for instance. (By the way, the more common form of non-radioactive carbon is C-12.) Clearly, a half-life of 5700 years is short enough that all the C-14 created at the time the stars and solar system came into existence has long since decayed away. How, then, can such a tool work? The ancient people did not obligingly use nuclear reactors to leave helpful amounts of C-14 scattered around in their ancient homes! The answer is as follows: Cosmic rays (energetic particles streaming through the cosmos) collide with atoms high in the Earth's atmosphere, and convert some fraction of them to C-14. In this sense, the universe as a whole acts as an `atom smasher' which maintains the supply of C-14 at some level. Some of the C-14 combines with oxygen to from carbon dioxide, and some of this in turn gets converted by plants into cellulose, sugars, and so on through the process of photosynthesis. In turn, some of these plant forms get ingested by animals, who use them to build body mass. The continuing cycle of photosynthesis and animal metabolism maintains a fairly constant level of C-14 in the plants or animals while they still live, but that activity stops when they die and the C-14 proportion falls as the ensuing centuries pass. Just as for the geological samples, then, the proportions of C-14 and the stable daughter product tell us the age of the archaeological sample. The samples may be vegetable material - the wooden shaft of a spear, perhaps - or animal, like a piece of bone. Previous chapter:Next chapter
0: Physics 015: The Course Notes, Fall 2004 1: Opening Remarks: Setting the Scene. 2: The Science of Astronomy: 3: The Importance of Scale: A First Conservation Law. 4: The Dominance of Gravity. 5: Looking Up: 6: The Seasons: 7: The Spin of the Earth: Another Conservation Law. 8: The Earth: Shape, Size, and State of Rotation. 9: The Moon: Shape, Size, Nature. 10: The Relative Distances and Sizes of the Sun and Moon: 11: Further Considerations: Planets and Stars. 12: The Moving Earth: 13: Stellar Parallax: The Astronomical Chicken 14: Greek Cosmology: 15: Stonehenge: 16: The Pyramids: 17: Copernicus Suggests a Heliocentric Cosmology: 18: Tycho Brahe, the Master Observer: 19: Kepler the Mystic. 20: Galileo Provides the Proof: 21: Light: Introductory Remarks. 22: Light as a Wave: 23: Light as Particles. 24: Full Spectrum of Light: 25: Interpreting the Emitted Light: 26: Kirchhoff's Laws and Stellar Spectra. 27: Understanding Kirchhoff's Laws. 28: The Doppler Effect: 29: Astronomical Telescopes: 30: The Great Observatories: 31: Making the Most of Optical Astronomy: 32: Adaptive Optics: Beating the Sky. 33: Radio Astronomy: 34: Observing at Other Wavelengths: 35: Isaac Newton's Physics: 36: Newtonian Gravity Explains It All: 37: Weight: 38: The Success of Newtonian Gravity: 39: The Ultimate Failure of Newtonian Gravity: 40: Tsunamis and Tides: 41: The Organization of the Solar System: 42: Solar System Formation: 43: The Age of the Solar System: 44: Planetary Structure: The Earth. 45: Solar System Leftovers: 46: The Vulnerability of the Earth: 47: Venus: 48: Mars: 49: The Search for Martian Life: 50: Physics 015 - Parallel Readings.
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(Wednesday, 20 January, 2021.)