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Towards the Bottom of the Nuclear Binding Energy

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Part of the Astrophysics and Space Science Library book series (ASSL,volume 387)


Once it became known that hydrogen was the most abundant element in the Universe (in 1931), and once the binding energy per nucleon could be obtained as a function of the atomic weight (in 1935), it was quite natural to assume that the fusion of light elements into heavier ones was the source of stellar energy. But the mechanism remained mysterious. There is a basic nuclear difference between hydrogen and heavier elements in their neutron to proton ratio. While in hydrogen there are no neutrons, helium has two neutrons and two protons, and if four protons fuse to form a helium nucleus, then two protons must \(\upbeta \)-decay. We say that \(Y_\mathrm{ e}=n(p)/[n(p)+n(n)]\) changes from \(Y_\mathrm{ e}=1\) for hydrogen to \(Y_\mathrm{ e}=1/2\) for He and heavier nuclei. How does this change take place? Inside nuclei or during a collision? This is the basic difference between the CN cycle and the proton–proton chain.


  • Nuclear Reaction
  • Heavy Element
  • Coulomb Barrier
  • Stellar Evolution
  • Main Sequence Star

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  1. 1.

    In the CN cycle, the conversion of a proton into a neutron takes place inside a nucleus which absorbed an extra proton. In the pp chain, the conversion takes place ‘in flight’, while the two protons move one towards the other. A detailed description of the idea of nuclear reactions as energy source in stars can be found in Shaviv, G., The Life of Stars. The Controversial Inception and Emergence of the Theory of Stellar Structure, Springer, Heidelberg, and Magnes, Jerusalem (2009), so only a concise summary is presented in this chapter.

  2. 2.

    Gamow, G., Zeit. f. Phys. 51, 204 (1928). In his first attempt, Gamow assumed the \(\upalpha \) particle to be moving freely in the Coulomb field of the nucleus and got a continuous spectrum of emitted particles, in contrast with experiment. In his next attempt, Gamow combined the Coulomb and the nuclear forces to obtain an effective barrier.

  3. 3.

    Gurney, R.W. and Condon, E.U., Phys. Rev. 33, 127 (1929).

  4. 4.

    Gurney and Condon’s paper was received on 20 November and published in the February issue of Phys. Rev. However, the authors presented the results in several conference meetings before journal publication.

  5. 5.

    Condon, E., Proc. Natl Acad. Sci. U S A 11, 125 (1924).

  6. 6.

    Gamow, G. and Houtermans, F.G., Zeit. f. Phys. XX, 496 (1929).

  7. 7.

    The Coulomb barrier is due to the mutual repulsion by the positive charges of the nucleus. In stars the atoms are stripped of their electrons. The electrons move like free particles and form a sea of negative charges. The effects of the free streaming electrons were later considered by Salpeter, E.E., Aust. J. Phys. 7, 373 (1954), who found that they reduce the effective peak of the barrier and increase the tunneling.

  8. 8.

    Atkinson, who was an ex-student of Rutherford, went to study physics in Germany, where he married a German woman who knew Fritz Houtermans from high school. So when the Atkinsons and Houtermans moved from Göttingen to Berlin-Charlottenburg to accept new jobs, the two families met and the collaboration started.

    Charlotte Riefenstahl, a PhD student in physics at the University of Göttingen in 1927, was courted by Robert Oppenheimer and Houtermans. During a physics conference at the Black Sea resort of Batumi, Riefenstahl and Houtermans got married, with Wolfgang Pauli and Rudolf Peierls as witnesses to the ceremony.

    Houtermans was a Communist and a member of the German Communist Party. When Hitler came to power in 1933, Charlotte insisted that they leave Germany, and in 1935, after a short stay in England, they emigrated to the Soviet Union. In the Great Purge, Houtermans was arrested by the NKVD, tortured, and forced to confess to being a Trotskyist and German spy. After the Hitler–Stalin Pact of 1939, Houtermans was turned over to the Gestapo and imprisoned in Berlin (out of the frying pan into the fire). Through efforts of (the anti-Nazi) Max von Laue, Houtermans was released in 1940, and in 1944 accepted a position as a nuclear physicist at the Physikalisch-Technische Reichsanstalt. During this period he discovered that neptunium and plutonium are fissionable.

    While working in Forschunsinstitut Manfred von Ardenne, Houtermans showed that transuranic isotopes, such as neptunium and plutonium, could be used as fissionable fuels in substitution for uranium. During the war, Houtermans sent a telegram from Switzerland to Eugene Wigner, warning the USA of extensive German research on fission: Hurry up. We are on the track. After Khriplovich, I.B., The Eventful Life of Fritz Houtermans, Phys. Today 45, 29 (1992).

  9. 9.

    Atkinson, R.d’E., and Houtermans, F.G., Zeits. f. Physik 54, 656 (1929). The original paper had an error. Due to a lack of experimental data, the authors took the classical theoretical probability for the nuclear reaction, rather then the quantum value. This is a contradiction in terms, and it was Gamow who corrected them.

  10. 10.

    Here is Houtermans’s own account of the denouement: That evening, after we had finished our essay, 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. von Buttlar, in Leonium und andere Anekdoten um den Physik professor Dr. F.G. Houtermans, Bochum (1982). Strictly speaking, this self-glorification, rather typical of physicists, came slightly too soon!

  11. 11.

    Gamow, G. and Teller, E., Phys. Rev. 53, 608 (1938).

  12. 12.

    In fact, \(b=31.28Z_1Z_2A^{1/3}\), where \(Z_1\) and \(Z_2\) are the charges of the colliding particles and \(A\) is the reduced atomic weight \(A=A_1A_2/(A_1+A_2)=\mu /M_\mathrm{ u}\). The units are keV\(^{1/2}\) and \(M_\mathrm{ u}\) is the mass of 1 atomic mass unit.

  13. 13.

    In 1936, Atkinson called the tunneling expression the Gamow formula (Astrophys. J. 84, 73, 1936). As far as could be traced, Hoyle, F., Astrophys. J. Suppl. 1, 121 (1954) used the well-known Gamow peak, although it does not appear in the literature before. Subsequently, one finds the Gamow peak mentioned in A.G.W. Cameron’s series of lectures Stellar Evolution, Nuclear Astrophysics, and Nucleosynthesis CRL-41, Atomic energy of Canada Ltd, 1957. Cameron derived the peak without giving any reference whatsoever, and added: This peak in the integrand will be called the Gamow peak. No reason was given as to why the peak was not called the Houtermans, Atkinson, and Gamow peak. Independently, Satio Hayakawa, S., Hayashi, C., Imoto, M., and Kikuchi, K., Prog. Theor. Phys. 16, 507 (1956) treated the Gamow peak as the accepted term and discussed modifications due to resonances, ignoring the fact that Gamow and Teller had already done just that in 1938.

  14. 14.

    Poincaré, H., Lecons sur les hypothèses cosmogoniques, Librarie Scientifique, A. Hermann, 1811. This theorem is also the basis for the negative effective specific heat of stars. Only at this point do we need the Clausius connection between kinetic energy and temperature [Clausius, R.J.E., Phil. Mag. 2, 1 (1851); ibid. 102; ibid. 12, 81 (1856)]. The temperature is given by the kinetic energy divided by the Boltzmann constant. If the star loses energy \(L\), it must contract, i.e., reduce its radius, and consequently lower its negative gravitational energy \(E_\mathrm{ grav}\). The kinetic energy \(T\) is then more positive, so the temperature rises. In this way the star can lose energy and increases its temperature, in contrast to normal matter. This is one of Eddington’s famous paradoxes about stars: they lose energy and heat up.

  15. 15.

    See Jeans, Astronomy and Cosmogony, Cambridge University Press, 1929, p. 67.

  16. 16.

    For comparison, the potential energy of a proton on the surface of the Earth is \(9.7\times 10^{-6}\) keV.

  17. 17.

    To the skeptic, Eddington retorted: If you don’t think the center of the Sun is hot enough, go and look for a hotter place. That being said, once tunneling had been discovered, it is a puzzle as to why Eddington did not attempt to see how his original hypothesis from 1919 could be realized. Furthermore, there was apparently no reaction from Eddington to Bethe’s discovery of the CN cycle, which vindicated his 20 year old hypothesis.

  18. 18.

    Atkinson, R.d’E., Astrophys. J. 73, 250, 308 (1931); Astrophys. J. 84, 73 (1936).

  19. 19.

    In 1960, Atkinson was awarded the Eddington Medal by the Royal Astronomical Society in recognition for the idea of a regenerative process, having the essential property of the CN cycle later proposed by Bethe and von Weizsäcker. Redman, R.O., President RAS, QJRAS 1, 26, 1960. It is interesting to note that the seminal papers which won the Eddington medal were cited less than 10 times until 2010. Explanations of the regenerative process appear in the next section.

  20. 20.

    Eddington, A.S., The Internal Constitution of the Stars, Cambridge, p. 301.

  21. 21.

    Gamow, G., Proc. R. Soc. 126, 632 (1930).

  22. 22.

    Chadwick, J., Nature, 27 February, 312 (1932). The full paper is: Proc. R. Soc. A 136, 692 (1932).

  23. 23.

    Birge, R.T., Menzel, D.H., Phys. Rev. 37, 1669 (1931).

  24. 24.

    Urey was awarded the Nobel Prize for Chemistry in 1934 for the discovery of deuterium. The high-minded Urey publicly acknowledged the crucial role played by Brickwedde and Murphy in the discovery, and gave each of them one-quarter of the Nobel prize money.

  25. 25.

    Urey, H.C., Brickwedde, F.G., and Murphy, G.M., Phys. Rev. 39, 164 (1932). Brickwedde produced the first sample of hydrogen in which the spectrum of deuterium was observed. The idea of a heavy hydrogen isotope was rejected by Aston who, in 1927 [Aston, F.W., Proc. R. Soc. Lond. A 115, 487 (1927)], used mass spectrometric evidence to set an upper limit for the deuterium abundance ratio of \(^2\mathrm{ D}/^1\mathrm{ H}<1/5{,}000\). The spectroscopic evidence was not sufficient to convince the scientific community of the existence of a heavy hydrogen nucleus, although it was extremely clear. The history with the discovery of helium on the Sun was not sufficiently convincing for the existence of a new isotope in water, and Urey and Brickwedde were obliged to distill the water and provide a ‘real proof’. This concise description does not do justice to the story, which is a chronicle of trial and error, false ideas, and misleading opinions by well-known scientists. See Brickwedde, F.G. Harold Urey and the Discovery of Deuterium, Phys. Today 35, 34 (1982).

  26. 26.

    The discovery was not credited because of misidentifications by this list of distinguished scientists.

  27. 27.

    Carl Anderson won the 1936 Nobel Prize for Physics for the discovery of the positron.

  28. 28.

    Livingood, J.J. and Snell, A.H., Phys. Rev. 48, 851 (1935).

  29. 29.

    Willy Fowler used to give the following example in his lectures on nuclear astrophysics. With a 10 MW beam of protons in an accelerator which produces this power and accelerates 1 MeV protons before causing them to impinge on other protons, the result will be on average one reaction per year of operation. Now note that the Gamow peak for the pp reaction in the Sun is at 4–5 keV, so the rate of this reaction in the Sun will be significantly smaller.

  30. 30.

    Bethe, H., and Peierls, R., Proc. R. Soc. A 148, 146 (1935).

  31. 31.

    Fermi, E., PRL 48, 570 (1935).

  32. 32.

    Weizsäcker, C.F., Physik. Zeits. 38, 176 (1937) Paper I; ibid. 633 (1937) Paper II.

  33. 33.

    Bethe, H.A. and Bacher, R.F., Nuclear Physics A. Stationary States of Nuclei, Rev. Mod. Phys. 8, 82 (1936); Bethe, H.A., Nuclear Physics B. Nuclear Dynamics, Theoretical, Rev. Mod. Phys. 9, 69 (1937); Livingston, M.S., and Bethe, H.A., Nuclear Physics C. Nuclear Dynamics, Experimental, Rev. Mod. Phys. 9, 245 (1937).

  34. 34.

    Bethe, H.A. and Critchfield, C.L., Phys. Rev. 54, 248 (1938).

  35. 35.

    von Weizsäcker, C.F., Physik. Zeits. 38, 176 (1937).

  36. 36.

    Gamow, G., and Teller, E., Phys. Rev. 49, 895 (1936).

  37. 37.

    Fierz, M., Zeit. für Phys. 194, 553 (1937).

  38. 38.

    Similarly, in the cases of \(^5 \mathrm{ Li}\), \(^5\mathrm{ He}\), and \(^8\mathrm{ Be}\), the effective potential wells of the protons and the neutrons is not sufficiently deep to allow for a bound state, and these nuclei do not have a stable state and consequently decay quickly. On the other hand, \(\mathrm{ ^4He}\) has the largest binding energy per nucleon and only one bound state, the ground state.

  39. 39.

    The reference given is Goldhaber, Physical Review, to be published. However, no such paper by Goldhaber could be found in the Phys. Rev. A year later, Margenau [Phys. Rev. 55, 1173 (1939)] investigated the structure of \(\mathrm{ ^6He}\), and Grönblom at Cornell University, where Bethe was based, investigated the \(\upbeta \)-decay of \(\mathrm{ ^6He}\) [Phys. Rev. 56, 508 (1939)]. Bethe was aware of Grönblom’s results since he suggested the problem to him. As a matter of fact, Grönblom found that the strength of the interaction between the protons, the factor which determines the speed of the reaction, was significantly stronger than what Bethe and Critchfield had assumed, and hence the rate of energy production was greater. Bethe and Critchfield estimated their data from the reaction \(\mathrm{ ^{13}N}\rightarrow \mathrm{ ^{12}C}+\mathrm{ e}^+ + \upnu \).

  40. 40.

    On the basis of a footnote in the Physical Review paper, one could guess that the commentator was Robert Oppenheimer.

  41. 41.

    Bethe, H.A. and Critchfield, C.L., PRL 54, 862 (1938).

  42. 42.

    Alvarez, L.W. and Cornog, R., Phys. Rev. 56, 379 (1939).

  43. 43.

    Gamow, G., PRL 53, 907 (1938).

  44. 44.

    Legend has it that Bethe attended a conference in Washington organized by Teller and Gamow on stellar energy sources, and solved the CN cycle in the train on his way back to Cornell. Bethe made a point of diffusing this story, but admitted that figuring out the workings of the CN cycle required two weeks. Hans Bethe and His Physics, ed. by Brown, G.E. and Lee, C-H., World Scientific, 2006.

  45. 45.

    Burcham, W.E., and Smith, C.L., Nature 143, 795 (1939) investigated short-range \(\upalpha \) particles from oxygen, nitrogen, and fluorine bombarded with protons, and did not bother to derive the probability for the reaction.

  46. 46.

    This is true only if the cycle operates at equilibrium, which is a good assumption when the star is on the main sequence. But, in an explosion, for example, this is not the case.

  47. 47.

    Holloway, M.G. and Bethe, H.A., Phys. Rev. 57, 747 (1940). Even today the investigation of the reaction continues. For a recent paper, see Mukhamedzhanov, A.M., and 15 coauthors, Journal of Physics Conference 15, 202, 012017 (2010).

  48. 48.

    Bethe, H.A., Phys. Rev. 47, 747 (1935).

  49. 49.

    Caughlan, G.R. and Fowler, W.A., Astrophys. J. 136, 453 (1962).

  50. 50.

    Here Bethe cited a private communication with Gamow.

  51. 51.

    Goldhaber, M., Proc. Camb. Phil. Soc. 30, 560 (1934).

  52. 52.

    Schatzman, E., Ann. Ap. 12, 281 (1949).

  53. 53.

    Schatzman, E., Compt. Rend. 232, 1740 (1951); Ann. Ap. 16, 162 (1953).

  54. 54.

    Fowler, W.A., Phys. Rev. 81, 655 (1951). This was a brief report at the annual meeting of the American Physical Society.

  55. 55.

    Bethe, H.A. and Critchfield, C.L., Phys. Rev. 54, 248 (1938).

  56. 56.

    Good, W.M., Kunz, W.E., and Moak, C.D., PRL 83, 845 (1951).

  57. 57.

    Agnew, H.M. and 6 coauthors, Phys. Rev. 84, 862 (1951).

  58. 58.

    Salpeter, E.E., Phys. Rev. 88, 547 (1952); Astrophys. J. 116, 649 (1952).

  59. 59.

    Salpeter’s great idea was as follows: consider helium 3 as composed of \(2\mathrm{ p}+\mathrm{ n}\) and tritium as \(2\mathrm{ n}+\mathrm{ p}\). In the mutual collision, the extra particle of the second nucleus is captured to form the \(\mathrm{ ^4He}\) nucleus in both reactions. The experiment was carried out by Allen, K.W., and 4 coauthors, PRL 82, 262 (1951). Among others, this experiment showed that a di-neutron does not form in this experiment. Two neutrons, like two protons, do not have a bound state.

  60. 60.

    Junker, M., Phys. Rev. C 57, 2700 (1998); Arpesella, C., et al., Phys. Lett. B 389, 452 (1996).

  61. 61.

    Greenstein, J.L., Astrophys. J. 113, 531 (1951).

  62. 62.

    Alpher, R.A. and Herman, R.C., Rev. Mod. Phys. 22, 153 (1950).

  63. 63.

    Good, W.M., Kunz, W.E., and Moak, C.D., Phys. Rev. 94, 87 (1954). The paper was published on 1 April 1954.

  64. 64.

    Friedman, E.A. and Motz, L., Phys. Rev. 89, 648 (1953).

  65. 65.

    Epstein, I., Astrophys. J. 112, 207 (1950).

  66. 66.

    The neutrinos created in the core escape from the Sun without any interaction with the solar matter. The Sun is practically transparent to neutrinos. In contrast, photons released in the core of the Sun interact with the matter and diffuse out gradually, losing all memory of what happened in the core.

  67. 67.

    Holmgren, H.D., and Johnston, R.L., Bull. Am. Phys. Soc. Ser. II 3, 26 (1958).

  68. 68.

    Fowler, W.A., Astrophys. J. 127, 551 (1958), sent for publication 24 February 1958.

  69. 69.

    Cameron, A.G.W., Bull. Am. Phys. Soc. II 3, 227 (1958); Ann. Rev. Nucl. Sci. 8, 299 (1958); Chalk River Report CRL-41, 2nd edn., 1958, unpublished.

  70. 70.

    The story of the solar neutrinos is unfolded in Shaviv, G., The Life of Stars. The Controversial Inception and Emergence of the Theory of Stellar Structure, Springer, Heidelberg, and Magnes, Jerusalem (2009).

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Shaviv, G. (2012). Towards the Bottom of the Nuclear Binding Energy. In: The Synthesis of the Elements. Astrophysics and Space Science Library, vol 387. Springer, Berlin, Heidelberg.

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