The Amazing Story of Quantum Mechanics

by Kakalios, James

  Depending on the nucleus and the nature of the unstable excited state it is in, the probability of decay may be very high or very low. In the lottery analogy, you may need to guess only one number from 0 to 9 in order to win the jackpot, or you may need to match seven random two-digit numbers in precise order. In the first case one would not need to play the game very long before winning, while in the second case it could take much longer than several lifetimes (if the lottery selected fresh numbers every day) before a winning match is obtained. Similarly, some elements’ unstable nuclei undergo radioactive decay within, on average, a few days or months, while others may take several billion years. However, in the first case there is no reason any given nucleus could not remain undecayed for a long time, while in the second situation there is no physical reason why any given nucleus could not decay almost immediately. It is possible to hit even a seven-digit lottery jackpot with your very first ticket, though I should be so lucky.

  If I start with a large number of radioactive atoms, then a plot of the number that avoid decaying into some other isotope as a function of time follows what’s termed an “exponential time dependence.” To understand this concept, imagine a car driving at sixty miles per hour that suddenly slams on the brakes. How long does it take the car to come to a complete stop? If we assume that the brakes provide a constant deceleration of ten miles per hour per second, then in six seconds the car will come to a rest. What if the brakes provided a deceleration that depends on how fast the car is moving at any instant? That is, when the car is moving very fast the brakes provide a large force, slowing you down. But if you were driving much more slowly, in a parking lot, say, then the brakes would provide a lower force. If the deceleration is proportional to the velocity, then it turns out that the car never comes to a full stop! (Well, for long times it may be moving so slowly that we could for all intents and purposes say that it had stopped, but if we were to measure the speed, we might find that it is very, very small, less than one millionth of a mile per hour, for example, but never truly zero.) In the first case, that of a constant deceleration, the auto’s speed decreases linearly with time. In the second situation, where the deceleration varies with the speed, initially the car slows down dramatically, as it is moving fast and that means the deceleration is large. But as it goes slower and slower, the braking force decreases, so that for long times it is moving very slowly, but the brakes are exerting only a very weak force. A plot of the car’s speed against time would be a concave curve called an “exponential decay function.”

  While the slowing automobile with velocity-sensitive brakes is artificial, the reverse phenomenon—exponential growth that leads to faster and faster increases—is more familiar, at least for those who have watched their savings grow through compound interest. A small amount deposited in the bank that earns a steady fixed interest rate, compounded continuously, will show a small increase initially. But as time progresses, both the original investment and the total interest earned will be subject to the same interest rate, and the returns will soon become much larger as your bank balance benefits from an exponential growth.

  Just such an exponential dependence is found for the decay of tritium, an unstable isotope of hydrogen. Normally hydrogen has one proton in its nucleus. The neutrons, participating in the strong force, are needed in larger nuclei to overcome the electrical repulsion between protons. As hydrogen has only one proton in its nucleus, it is the only element that does not need neutrons, though it is possible for neutrons to be present in the hydrogen nucleus. In hydrogen, one electron is electrostatically bound in a quantum mechanical “orbit” to the single proton in the nucleus. As the chemical properties of an atom are determined by the number of electrons it possesses, which in turn are set by the number of protons in its nucleus, one could form an alternative form of hydrogen containing one proton and one electron, with an extra neutron in the nucleus, and it would behave, for the most part, like ordinary hydrogen. We would call this isotope deuterium. If there were two neutrons and one proton in the nucleus, about which one electron “orbits,” this isotope is termed “tritium.”40

  As illustrated in Figure 24, tritium is unstable and, through a mechanism I describe in the next chapter, decays to form an isotope of helium, along with a high-speed electron (a beta ray) like those in Chapter 8 responsible for Dr. Manhattan’s blue glow. Figure 24 shows another page from Learn How Dagwood Splits the Atom, whereby the addition of two neutrons to a hydrogen nucleus (that is, a single proton) yields an unstable result. One of the neutrons converts to a proton and another electron, through a mechanism governed by the weak nuclear force, discussed in detail in the next chapter. The decay rate of tritium is very fast, such that for a given nucleus, after only about twelve and a half years, there is a fifty-fifty chance of the isotope decaying.

  If the decay rate is so fast, why is there any tritium still around? Because it is constantly being created, when high-speed neutrons formed from cosmic rays collide with nitrogen atoms in the atmosphere. The now unstable nitrogen nuclei decay to form normal carbon and tritium. The tritium generated in the upper atmosphere can be captured by oxygen atoms and forms a version of “heavy water” (remember that aside from the heavier nucleus, tritium behaves chemically the same as normal hydrogen). This tritium-rich water reaches the ground in the form of raindrops. Because we know the decay curve of tritium, comparisons of water from the surface of the ocean to that obtained from greater depths enable determinations of the cycling time for oceanic circulation currents.

  Figure 24: Page from Learn How Dagwood Splits the Atom in which Dagwood, his son Junior, and his dog Daisy witness the radioactive transformation of a tritium nucleus into an isotope of helium.

  Ideally, in order to measure the time dependence of the tritium decay, one would like to have samples of rainwater from more than a hundred years ago, as well as more recent years all the way to the present. By measuring the fraction of tritium as a function of the age of the water, one could verify the exponential time dependence of its decay. The problem is that one does not have bottles of rainwater dating back more than a century. In a 1954 paper in the Physical Review, Sheldon Kauffman and Willard F. Libby did the next best thing and examined the tritium content of vintage wines. As shown in Figure 25, a plot of the tritium concentration per wine bottle as a function of time, determined from the vintage label, shows that, when measured in 1954, the tritium concentration was very high in a 1951 Hermitage Rhone, but the concentration was dramatically lower in a 1928 Chateaux Laujac Bordeaux. The full curve is very well described by an exponential time dependence. Based on this curve, if in 1954 we wanted a wine with a tritium concentration half as large as that in the 1951 Hermitage, we would decant a 1939 vintage, from which we conclude that the “half-life” of tritium is 12.5 years.

  Different radioactive nuclei have different decay rates. All unstable nuclei have exponential decay functions, but the time scale over which the decay occurs may be very different—from minutes to billions of years. Measurements of nuclei with short decay times, such as the tritium in wine bottles example, confirm that the number of nuclei that decay does indeed follow an exponential time dependence. The physics of the nucleus does not change depending on which element we are considering. For those nuclei that have very low decay rates, so that the time to decay is very long, we can nevertheless measure the initial portion of the exponential decay. Mathematical fitting of this curve indicates when the decay function is expected to reach the 50 percent point, and thus we can determine that the half-life of uranium, for example, is several billion years, even though we have not sat in the lab for this length of time to measure the full decay curve.

  Figure 25: Plot of the time dependence of tritium concentration in “heavy water” contained in wine bottles. The age of the water sample is determined by the vintage printed on the bottle’s label. The longer one waits, the less tritium is present, due to radioactive decay. The solid line is a fit to the data of an exponential time depende
nce, with a half-life of 12.5 years. Reprinted figure with permission from S. Kaufman and W.F. Libby, Physical Review 93, 1337 (1954).

  For a radioactive nucleus with a half-life of one year, if I start with a million atoms, after one year I will have approximately half a million remaining (there will typically be fluctuations about this average number of half a million, as the decays are probabilistic). As the decay rate is independent of the age of the atom, then in the next year, 50 percent of the remaining atoms will decay. That is from an initial number of one million, I will have approximately half a million after one year, a quarter of a million after two years, 125,000 after three years, and so on.

  Because the time necessary for one half of the initial population of nuclei to decay is precisely known, we can use carbon dating to determine the age of archeological artifacts. Let’s say we start with a million unstable isotopes of carbon. Normally carbon has six protons (and a corresponding six electrons in quantum mechanical “orbits”) and six neutrons in its nucleus and is as stable as anything we know of. As there are twelve particles in its nucleus, this form of carbon is called carbon 12. Occasionally collisions with cosmic rays lead to the creation (through a process that we don’t have to worry about now) of a form of carbon with six protons but eight neutrons in its nucleus. As it has the same number of protons and electrons as carbon 12, this heavier isotope is chemically identical to normal carbon. However, this form of carbon with eight neutrons (called carbon 14) is unstable and beta decays into nitrogen 14.

  The rare heavier carbon 14 is constantly being created by random collisions with cosmic rays and is constantly decaying away into another element. A very small but constant percentage of the carbon in the world is heavy, unstable carbon 14. This holds for the food we eat, the clothes we wear, and pretty much everything that contains carbon atoms. Consequently, a small fraction of the carbon in our bodies is this unstable heavier form. The half-life for heavy carbon to decay is about 5,700 years. So normally we ingest heavy carbon by its random presence in the food we eat, and we lose heavy carbon through normal biological processes when we eliminate old cell material. This process comes to a rather abrupt stop when we die (the flux of cosmic rays on the Earth’s surface is low enough that we don’t have to worry about carbon 14 creation in our corpse). At death the amount of heavy carbon in our bodies, our skin, our tissues, and our bones is fixed. A future archaeologist, finding our skeletons, measures the quantity of carbon 14 and finds that it is only half of the normal amount of carbon 14. She can then confidently state that we died approximately 5,700 years ago. If the amount of heavy carbon is one quarter of the current level of carbon 14, then two half-lives must have passed, and our death is placed at roughly 11,400 years in the past. In this way any material containing organic matter, whether it be ancient bones or the shroud of Turin, can be dated from its last point of carbon intake. Willard Libby, who used old wine to obtain new measurements of tritium decays (Figure 25) shared the 1960 Nobel Prize in chemistry for developing carbon 14 dating.

  Longer-lived isotopes, such as uranium 235 and uranium 238, have half-lives of roughly billions of years. These two forms of uranium were generated in a supernova explosion that created all the atoms that went on to form the planets and moons in the solar system (more on this later). Assuming that initially they are created in equal concentrations, ascertaining their half-lives through independent measurements, and seeing the fraction of uranium 235 and uranium 238 present on the Earth today, we can calculate how long the Earth has been around to give the uranium isotopes a chance to decay to their present proportions.41 The answer turns out to be about 4.5 billion years.

  We thus know the age of the Earth through our understanding of quantum mechanics, the same quantum physics that underlies the field of solid-state physics. Without quantum mechanics, there would be no semiconductor revolution, and the nearly countless electronic devices we employ would not be possible. It is of course your right to believe that the Earth is actually much younger than its age determined by radioactive isotope dating, but to be consistent, you should stop believing in your cell phone, too!

  Elements that emit gamma rays, alpha particles, or beta particles are radioactive—while materials that are exposed to these nuclear ejections are described as irradiated. As mentioned at the start of this chapter, science fiction films in the 1950s ascribed to irradiation the mutation of animals and people into giants, though occasionally a miniaturization effect was possible. What exactly are the real, non-Hollywood movie, effects of exposure to radiation? Not all radioactivity is created equal, and some is more harmful than others.

  The emission of radioactivity results when a nucleus makes a quantum transition from a high energy state to a lower energy configuration. Recall that the energy spacing between quantum states in the nucleus is on the order of a million electron Volts, while electronic states in an atom are on the order of a few electron Volts. Electronic transitions involve energies in the ballpark of visible light, while the energy scale of nuclear quantum jumps is much larger. When the electrons in a neon atom make quantum transitions, they emit red light, which we associate with neon signs. When the neon atom’s nucleus makes a transition, the energy is about a million times greater and has the potential to do extensive damage. We evolved in a sea of visible and ultraviolet light, and aside from a sunburn (and the concomitant long-term skin damage), this radiation does not harm us. Light a million times more energetic is rare, and we are not equipped to shrug off such radiation.

  If you were wandering around a nuclear weapon testing site, you would be exposed to the fallout—radioactive isotopes that are created as the by-products of the fission reaction. A variety of secondary, hazardous unstable nuclei can be generated depending on the nature of the atomic blast. They, in turn, emit radioactivity as they relax to lower energy states. You would wish that the radioactivity present would be alpha particles, consisting of two protons and two neutrons, rather than beta rays, or high-speed electrons. The energy, in the form of kinetic energy, of either the alpha particle or beta ray is about a million electron volts. The mathe-matical expression for kinetic energy is KE = (1/2) mv2, where m is the mass of the object and v is its velocity. Therefore, for a given kinetic energy, the larger the mass, the smaller the velocity. Protons and neutrons have a mass roughly two thousand times larger than that of an electron, so the alpha particle is nearly eight thousand times more massive than an electron. Thus, if both an alpha particle and a beta particle have comparable energies, as they both arise from nuclear quantum transitions, the alpha will be moving nearly ninety times slower than the beta. The slower a particle moves through matter, the more time it spends near each of the atoms in the object, increasing the likelihood of losing energy through collisions with the electrons surrounding each atom. Slower alpha particles can be stopped by a single sheet of paper, and they almost never penetrates a person’s clothing, while it takes a quarter-inch-thick sheet of aluminum to stop much faster beta rays, and they can indeed get under your skin.

  Figure 26: A sketch of a 1936 Buck Rogers toy ray gun and nuclear radioactivity consisting of alpha particles, beta rays, and gamma ray photons. (The toy shown is a Buck Rogers Disintegrator Pistol, model XZ-38, and did not actually emit high-energy sub-atomic particles). All three types of radiation have roughly the same energy of a few million electron Volts, but they are stopped by different levels of shielding. The alphas are blocked by a sheet of paper, which the betas can penetrate, but are themselves stopped by a thin aluminum sheet. Highenergy light in the form of gamma rays passes through both, and a thick block of lead is needed to stop them.

  Gamma-ray photons are high-energy light, several hundred times more energetic than X-rays. As gamma rays are photons of light and uncharged, they do not interact directly with the electronic charges in atoms, which makes them much harder to stop. It takes about one-half inch to an inch of dense material such as lead to stop gammas, and they can penetrate through an entire person. Some nuclei (such
as uranium 238) may also emit neutrons42 that in themselves are not harmful, but when they collide with hydrogen atoms in the body, the resulting high-speed ricocheting protons can be damaging.

  If you are unfortunate enough to ingest an unstable nucleus, so that it is inside you when it undergoes radioactive decay, then even alpha particles can be deadly. Rather than striking the dead skin cells on your epidermis, which you slough off naturally, alpha particles inside you have a direct path to your internal organs. In this case the alpha particles prove to be very efficient in stripping electrons from the atoms they strike, disrupting the chemical bonds within the cell and causing extensive chromosomal damage.

  In 2006 Russian journalist Alexander Litvinenko was murdered when he drank tea that was spiked with polonium 210. This unstable nucleus has a half-life of just over 138 days and emits high-energy alpha particles when it decays. A pound of polonium 210 releases energy at the rate of nearly 64,000 Watts. Because its probability of radioactive decay is so high, even 0.05 micrograms of polonium 210 is considered to be lethal (it is believed that Litvinenko had 10 micrograms in his body at the time of death). So, alphas on the outside, not too much of a problem—on the inside, a rather big problem. Which is probably why it was fortunate that Gilbert stopped marketing their U-238 Atomic Energy Lab in 1952. I mentioned in the last chapter that this kit contained a mini- cloud chamber. Part of the radioactive elements supplied with this chamber was a small piece of potent, though short-lived, radioactive polonium 210!