Great Calculations: A Surprising Look Behind 50 Scientific Inquiries

by Colin Pask

  Figure 11.6. Quark model results for hadron masses as published by Dürr and his collaborators. The horizontal bars indicate experimental errors; the vertical bars indicate uncertainties in the calculated results. The π, K, and Ξ masses were used to set quark data. Reprinted with permission of AAAS, from S. Dürr, et al., “Ab Initio Determination of Light Hadron Masses,” Science 322, no. 5905 (2008).

  11.5 INTRODUCING THE NUCLEUS

  Before describing the final two calculations for this chapter, it is useful to set out a few facts about nuclei. (To go further, the book by Dunlap is a good reference and the National Research Council (US) book gives an excellent introduction to modern topics in nuclear physics.)

  An atom has atomic number Z, meaning that it has Z electrons orbiting a nucleus containing Z protons. The nucleus also contains N neutrons, and A = Z + N is known as the mass number. Protons and neutrons are known collectively as nucleons; the nucleus contains A nucleons. When necessary, an element with the name X will be written as NZXA.

  The nucleons interact through the strong nuclear force (which can be related to the exchange of mesons or the interaction of quarks through gluons). The nuclear forces, which tell us how two protons, or two neutrons, or a proton and a neutron interact, are all similar and have a very short range, falling off exponentially. The electric force between the protons is much weaker than the nuclear force, but it is a long-range force and so dominates the p-p interaction at large distances. It is the electric force between protons and electrons that holds the atom together, but it is the nuclear force in competition with the p-p electric force that holds the nucleus together.

  The state of the nucleus is given by solving the Schrödinger equation with the appropriate particles and forces included. However, that is a formidable mathematical problem, not particularly simple even when A = 3, never mind when ten or even hundreds of nucleons are involved. Physicists have resorted to models like the shell model (which puts nucleons into orbits around an effective center like the electrons in an atom) or the liquid-drop model (which will be used in a later section).

  Nuclei vary from hydrogen (with A = 1 and Z = 1) up to uranium with A = 238 and beyond to the transuranic elements known up to A = 277. For small numbers Z of protons, the nuclei have roughly the same number of neutrons, but for larger Z, say beyond 25, the trend is for increasingly more neutrons than protons. The number of neutrons in the nucleus for a given Z may vary to give the isotopes, not all of which will be stable; for example, carbon with six protons has isotopes CA ranging from C9 to C17 although only C12 and C13 are stable. We have already referred to the α-decay and β-decay processes.

  11.5.1 Binding Energies

  The next two calculations both refer to the energy contained in nuclei, so some basic information is required. By Einstein's formula, the energy E of a nucleus containing Z protons and N neutrons is given by

  E(Z,N) = Zmpc2 + Nmnc2–B.

  B is the (positive) binding energy for the nucleus, and it occurs as–B because energy must be supplied to counter the binding and destroy the nucleus. B depends on the nucleons involved, so we must write B(Z,N).

  A study over the range of nuclei shows that the binding energy per nucleon, B/A, follows a definite trend: as A increases, B/A increases to reach a maximum around the element iron with A = 53, after which it steadily decreases. Using this data, we can see what happens when we have one nucleus with a given A value compared with two nuclei with mass numbers A1 and A2 and A1 + A2 = A. The nuclear reaction will favor the lower energy state, and energy will be given out possibly to electrons, positrons, or photons. This gives us the processes known as fusion and fission:

  Fusion: for A < 53, two nuclei with mass numbers A1 and A2 combine to form a nucleus with mass number A = A1 + A2 accompanied by energy production.

  Fission: for A > 53, a nucleus with mass number A breaks up to form two nuclei with mass numbers A1 and A2 with A1 + A2 = A accompanied by energy production.

  An example of fusion given in section 7.3.2 is F19 + H1 → Ne20 + hν where hν represents the energy output in a photon.

  Energy changes in nuclear reactions are governed by Einstein's E = mc2 equation. In 1932, John Cockcroft and Ernest Walton reported their observations of the reaction

  Li7 + p → He4 + He4 + (8MeV in energy for each He4)

  with the comment that “the evolution of energy on this view is about 16 million electron volts per disintegration, agreeing approximately with that to be expected from the decrease of atomic mass involved in such a disintegration.”13 This is probably the earliest test of this kind and could easily have made my list of important calculations.

  (An aside on energies: in atomic physics, a suitable unit of energy is the electron volt eV, which is the energy acquired by an electron in being accelerated through a potential difference of one volt. In terms of other units, 1eV = 1.602 × 10–19 Joules = 1.602 × 10–12ergs. In nuclear physics, the appropriate unit is one million electron volts or MeV.)

  These nuclear reactions seem to bring us close to the old transmutation-of-elements dream However, not everyone liked to see it like that; apparently when Lord Rutherford's assistant Frederick Soddy mentioned such things, Rutherford responded that Soddy must not call it transmutation or they would be accused of being alchemists.

  11.6 SUNSHINE

  Life on Earth depends on the energy, heat, and light radiated by our sun. The sun emits a staggering 3.8 × 1026 watts. The questions of how the sun generates such energy and how long it can last are naturally questions of great importance. The same questions apply to all other stars. The answers take us through a whole set of calculations.

  The sun has a mass of around 2 × 1030 kilograms. Gravitational forces have caused the sun to form a dense, hot core with a temperature about ten million degrees Kelvin, reducing to about six thousand degrees Kelvin at the surface. The physics developed late in the nineteenth century and early in the twentieth allowed the properties of such a body to be described in terms of densities and particle motions. There is an internal pressure that counteracts gravitational forces to form a stable structure and prevent further collapsing. A series of calculations allows the internal structure of stars to be understood. (See the book by King for a modern, short introduction to stars.) However, those calculations did not explain how a star produces its enormous energy output.

  The situation as it was in 1926 is beautifully set out by Sir Arthur Eddington (1882–1944), one of the great astrophysicists of the twentieth century. (The book by Kilmister gives a short biography and excerpts from Eddington's most important work.) In his Internal Constitution of Stars, Eddington explains the problem with the then-current theories and offers a revolutionary solution:

  The energy radiated by the Sun into space amounts to 1.19 × 1041 ergs per year. Its present store of heat energy is as follows

  Radiant energy…………………….. 2.83 × 1047 ergs

  Translatory energy of atoms and electron……26.9 × 1047 ergs

  Energy of ionization and excitation………< 26.9 × 1047 ergs

  This constitutes 47 million years’ supply at the most. We do not, however, think that this capital is being used for expenditure; it is being added to rather than exhausted.

  It is now generally agreed that the main source of a star's energy is subatomic. There appears to be no escape from this conclusion; but since the hypothesis presents many difficulties when we study the details it is incumbent on us to examine carefully the alternatives.

  Formerly the contraction theory of Helmholtz and Kelvin held sway. This supposes that the supply is maintained by the conversion of gravitational energy into heat owing to the gradual contraction of the star. The energy obtainable from contraction is quite inadequate in view of the great age now attributed to the Sun.14

  We met Lord Kelvin in section 4.3, where his calculation of the age of the earth was discussed. He also worried about the likely age of the sun and the ways in which it could generate energy. In his 1862 ar
ticle “On the Age of the Sun's Heat,” Kelvin concludes:

  It seems, therefore, on the whole most probable that the Sun has not illuminated the Earth for 100,000,000 years, and almost certain that he has not done so for 500,000,000 years. As for the future, we may say, with equal certainty that inhabitants of the Earth cannot continue to enjoy the light and heat essential to their life for many million years longer unless sources now unknown to us are prepared in the general storehouse of creation.15

  Since the age of the sun is about 4.5 billion years, Eddington was quite right to dismiss Kelvin's theory. He responded to Kelvin's “sources now unknown to us” with the revolutionary idea that the source of the sun's energy is to be found in subatomic processes. We now know that this process is nuclear fusion. We next consider how this conclusion was arrived at and how fusion can operate in the sun since according to Eddington, “No source of energy is of avail unless it liberates energy in the deep interior of the star.”16

  11.6.1 Conditions for Nuclear Fusion

  Fusion requires two nuclei to come into contact so that they may join together, or fuse, to form a new nucleus. However, there is one immediate difficulty: nuclei are positively charged, and they repel one another through the long-range Coulomb force. Only when they overcome this repulsion can the strong but short-range nuclear forces take over and pull the constituent nucleons into one bigger, new nucleus. The electrical repulsion is said to produce a Coulomb barrier. The nuclei would need to be traveling at great speed to overcome the Coulomb barrier, and only a few particles in the sun can do that. Was this the end of the nuclear fusion in the sun idea?

  The situation was saved by the realization that at the subatomic level we must use quantum rather than classical mechanics. George Gamow developed a theory to show how α particles could overcome a Coulomb barrier; in quantum theory, a particle incident upon a barrier and not possessing enough energy to go over it, can, with some probability, tunnel through it to have the same energy on the inside. This concept was applied to fusion processes in the stars by Robert Atkinson and Fritz Houtermans, and calculations established that fusion was a viable process inside stars. Their 1929 paper paved the way for the final theory, which of course had to come later when more about nuclear physics was known—recall that it was not until 1932 that the neutron was discovered, and the positron followed in 1933.

  Houtermans had an interesting and adventurous life (he married four times). There are various versions of a charming story he told about events on the evening after he and Atkinson finished their paper:

  That evening, after we had finished our paper, I went for a walk with a pretty girl. As soon as it grew dark the stars came out, one after another, in all their splendor. “Don't they shine beautifully?” cried my companion. But I simply stuck out my chest and said proudly: “I've known since yesterday why it is that they shine.”17

  In some versions of the story, the girl was Charlotte Riefenstahl, who Houtermans married and divorced—twice!

  11.6.2 The Grand Solution

  In April 1938, George Gamow organized a conference bringing together astrophysicists and nuclear physicists, and the problem of energy generation inside stars was a central topic. Among the nuclear physicists was Hans Bethe (1906–2005). Soon after the conference, Bethe wrote his paper “Energy Production in Stars,” giving the solution to the mystery. (Of course there were others involved in this area of nuclear physics such as C. L. Critchfield and C. F. von Weizsäcker.) In 1967, Bethe was awarded the Nobel Prize “for his contribution to the theory of nuclear reactions, especially his discoveries concerning energy production in stars.”18 His 1939 paper is a masterpiece of nuclear physics, with a vast array of information about nuclear processes, including calculations of the details of these processes and the requirements for them to take place.

  The star of greatest interest to us is the sun, which is composed largely of hydrogen (over 90 percent) and helium. Therefore, any reactions we find to explain the sun's energy production must involve protons, the hydrogen nuclei. Bethe showed that the basic process involves the conversion of four protons into a He4 nucleus, which will involve energy production, again calculated using Einstein's E = mc2 equation. The helium nucleus He4 is stable. To understand the whole process and its energy creation, it is necessary to list the fundamental nuclear processes that must take place. These involve the creation of a deuteron d or H2, the bound state of a proton (p) and a neutron (n). (Recall that e+ is the positron, γ is a gamma ray, and ν is a neutrino, and here the electron neutrino is νe.) Here is Bethe's scheme:

  p + p → d + e+ + ν + 0.42MeV of energy,

  p + d → He3 + γ + 5.49MeV,

  after two of those steps: He3 + He3 → p + p + He4 + 12.86MeV,

  there are two e––e+ annihilations: e+ + e– → 2γ + 1.02MeV.

  The whole process may be summarized as

  4p → He4 + 2e+ + 2ν + energy.

  We can now calculate the total energy produced as 2 × (0.42 + 5.49 + 1.02) + 12.86 which gives 26.72MeV. About 0.52MeV is taken away by the neutrinos. This may seem like a small amount of energy, but there are an incredibly large number of protons in the sun to take part in the process thus explaining the energy that the sun radiates.

  The above process is called hydrogen burning, and Bethe's calculations show that it is a viable process and it does account for the energy produced by the sun. Andrew King calculates that if the sun burnt all its hydrogen, it would last for 1011 years, so only a relatively small amount of the sun's hydrogen has been used so far. The sun is essential for life on Earth, and through calculation 45, why the sun shines, we understand how the sun supports our planet.

  Bethe showed that other nuclear reactions could generate energy in the stars as they are classified in the Hertzsprung-Russell diagram (see King, chapter 1). In particular, Bethe showed that in stars much more massive and hotter than the sun, carbon could act as a sort of catalyst for the protons to helium process and the 26.72MeV energy generation. The chain of reactions involves nitrogen and oxygen and goes like this:

  p + C12 → N13 → C13 + e+ + ν

  p + C13 → N14

  p + N14 → O15 → N15 + e+ + ν

  p + N15 → C12 + He4

  Thus the carbon is preserved, which is good because it is not so common in stars. The existence of Bethe's “CNO cycle” is now well established. This theory was discovered independently by Carl von Weizsäcker. Bethe considered many other relevant nuclear reactions in his 1939 paper. (Details of Bethe's paper, along with other related papers mentioned in this section, can be found in the Source Book edited by Lang and Gingerich. The book Fusion: The Energy of the Universe by McCracken and Stott is a simple introduction to the subject.)

  11.6.3 The Story Continues

  The fusion process operating in the sun is a clean way to generate energy—there are no nasty radioactive by-products. Furthermore, the burning of one kilogram of hydrogen produces the staggering energy of 6 × 1014 joules whereas burning a kilogram of oil gives us about 4 × 107 joules. That is quite some difference! The obvious question then, is why not use energy generation by nuclear fusion here on Earth? There has been enormous effort to build a fusion reactor, but confining suitable matter to a region with the required temperatures and densities is an incredible challenge. It is not so easy to make a sun on the earth. The technological difficulties have yet to be overcome, although scientists always seem to be optimistic that it only needs a few more years’ work! (Interested readers could consult Dunlap's chapter 13 or the book by McCracken and Stott.)

  Although the energy-generation mechanism for the sun and other stars found acceptance and there was checking of the nuclear reactions in terrestrial laboratories, a problem arose when considering those pesky neutrinos introduced in section 11.4. Eventually, experimental methods were devised for detecting neutrinos, and then the “solar neutrino problem” emerged: the number of neutrinos coming from the sun is only half what was expected. Further experiments and the theory i
ntroduced in section 11.5 now suggest that the different neutrinos (νe, νμ, and ντ) associated with the electron, muon, and tau particles are linked in “neutrino oscillations,” and the number of electron neutrinos observed is just as it should be. This development also leads to the concept of a very small mass for neutrinos. Recent remarkable experiments have detected neutrinos, giving information about reactions deep in the solar interior. (Consult Dunlap's chapter 17 or the very readable article by Bahcall for more details.) Finally, before we get too diverted, it must be said that neutrinos are now part of astrophysics on an increasing scale (see the article by Halzen and Klein), so Pauli might feel vindicated at last.

  11.7 CALCULATIONS THAT FASCINATE AND FRIGHTEN

  By 1939, nuclear physicists had reported a discovery that would change the course of human history. They had discovered that heavy nuclei could split into different nuclei with the release of much energy. This process was named fission. The observed results were nothing like α decay; a large nucleus containing many protons splits into two nuclei that both have a large number of protons (in contrast to the two protons in the He4 nucleus or α particle). The most important nucleus involved in these studies was that of uranium, which contains 92 protons. Fission is observed when neutrons are incident on uranium.

  Conclusive experimental results were reported early in 1939 by Otto Hahn and Fritz Strassmann, although in their January Naturwissenschaften paper they write, “Now we still have to discuss some newer experiments, which we publish rather hesitantly due to their peculiar results.”19 (This paper and many others in the fission story are conveniently gathered together and discussed by Graetzer and Anderson.) In their February paper, Hahn and Strassmann are much more relaxed writing that “in a rather short time it has been possible to identify numerous new reaction products described above—with considerable certainty, we believe.”20 They talk about the fission by-products as barium, strontium, and yttrium.