Remarks on the Theory of the Atomic Nucleus*

W. Heisenberg, Leipzig

The abundance of experimental material being collected on atomic nuclei reveals an ever clearer picture of the laws determining their structure. We therefore attempt below to give a brief comparison of contemporary theoretical views on the structure of the atomic nucleus, some of them already long known.

To this end we shall compare the empirical data on the atomic shell with the corresponding data on the nucleus, and shall seek a kind of analogy, in the spirit of the correspondence principle, between the already known laws of the atomic shell and the still unknown laws of the nucleus. The Rutherford–Bohr theory asserts that the atom consists of a nucleus and electrons. This proposition is confirmed by the experimental fact that, in collision experiments, the atom can be decomposed into its constituent parts—the nucleus and electrons. At the same time, light quanta are not regarded as constituent elements of the atom, despite the fact that they can be emitted by the atom and that, in the form of the Maxwellian field connecting the electrons with the nucleus, they constitute a considerable part of the total energy of the atom. To justify this manner of expression one may point out that, generally speaking, in collision experiments no light quanta leave the atom. More precisely, the light quanta that play some role in collision experiments are usually emitted by the atom only a long time after the impact process itself, after the normal lifetime of the excited states has elapsed. Their occurrence is not directly connected with the impact process. It is therefore more appropriate to say that light quanta are created by the atom in the transition from one stationary state to another; within the atom itself they do not exist as individuals—there exists only the field associated with them, the Maxwellian field of the nucleus and electrons. This fairly strict division of particles into those of which the atom consists and those which are created by it is facilitated by the smallness of the radiation forces in comparison with the Coulomb forces, the importance of which was emphasized by Bohr.

* Verhandelingen op 25 Mai 1935 aangeboden aan Prof. Dr. P. Seeman, M. Nijhoff, Hague. (From a collection of articles for the 70th birthday of P. Zeeman.) Translated by B. Levin.

The small magnitude of these forces is, in turn, a consequence of the smallness of the velocities of the electrons in the atom in comparison with the velocity of light, i.e., ultimately, of the smallness of the quantity \(\frac{e^2}{hc}\).

If one tries to describe in a similar way the structure of the atomic nucleus, then, on the basis of collision experiments in this case—experiments on the transformation of atoms—only heavy particles must be regarded as constituent parts of the nucleus: protons, neutrons, and possibly \(\alpha\)-particles. For only these particles appear after a collision as direct fragments of the nucleus. Light particles—electrons, positrons, and light quanta—according to experiments on artificial radioactivity and on the lifetimes of excited states of the nucleus, are usually emitted only a long time after the transformation. Thus their occurrence is not directly connected with the process of destruction of the nucleus. It is therefore expedient to speak of the appearance of electrons when the nucleus passes from one stationary state to another. That such a method of description is reasonable is convincingly proved by the experiments of Joliot and Curie, according to which positrons also can be emitted radioactively; and, of course, it is impossible simultaneously to regard both positive and negative electrons as constituent parts of the nucleus.

In exactly the same way as the occurrence of light quanta depends on the Maxwell field, which ensures the stability of the atomic shell, so too the possibility of the occurrence of light charged particles from the nucleus must be connected with the existence of a wave field which, in this case, must be essential for the stability of the atomic nucleus. This precisely agrees with the results of Fermi’s investigations[^1] on the theory of \(\beta\)-decay and of the investigations of Tamm[^2] and Ivanenko,[^3] which develop them and of which more will be said below. In nuclei, just as in shells, the distinction between particles that enter into the composition of the nucleus and particles that are created in transitions is facilitated by the smallness of the “forces of \(\beta\)-radiation” in comparison with the forces of interaction in the nucleus. The latter circumstance manifests itself in the magnitude of the natural lifetime of \(\beta\)-active nuclei. Such a state of affairs depends, as can be established, for example, on the basis of Fermi’s theory of \(\beta\)-decay, again on the fact that the transition energies of the nucleus, equal to the kinetic energies of the emitted particles, are small in comparison with the rest energy of the constituent parts of the nucleus, i.e., on the fact that the velocities of the protons and neutrons of the nucleus are small in comparison with the velocity of light.

The analogy under consideration between the behavior of the atomic shell and the behavior of the atomic nucleus may, in general, be represented by the following scheme (see p. 3).

Together with electrons and positrons, Pauli’s neutrino is also mentioned here. As is known, its existence is highly probable on the basis of continuous \(\beta\)-spectra and empirical rules concerning the statistics and spin of nuclei. In exactly the same relations—

REMARKS ON THE THEORY OF THE ATOMIC NUCLEUS

The discrepancy between the lifetime of β-active nuclei and the lifetime of excited states in the emission of γ-rays indicates that in β-radiation two particles leave the nucleus simultaneously.

If it is desired to form an idea of the character of the interactions leading to the radiation of electrons and neutrinos, then, in the spirit of the analogy set forth above, one should imagine the interaction energy as depending on the product of the wave functions of the heavy particles and the corresponding light particles at one and the same point of space. For, just as light quanta and electrons can enter into interaction only at one and the same point of space—since Maxwell’s theory is a theory of “action at contact”—so too light charged elementary particles and the heavy constituent parts of the nucleus can influence one another, apart from the action of their Maxwellian fields, only at one and the same place. And indeed, Fermi made this assumption the basis of his theory of β-decay. It turns out here that the assumption of the simultaneous emission of one electron and one neutrino is a necessary premise for the mathematical formulation of the theory of β-decay. The interaction energy is represented by a volume integral of the product of four wave functions, which always belong to the proton, neutron, electron, and neutrino. The wave function of the neutrino is necessary here in order to obtain an expression that behaves relativistically, like an energy density.

Fermi assumed, moreover, that only the wave functions, and not their derivatives, enter into the expression for the interaction energy. However, as was pointed out by Bethe and Peierls,* this assumption is apparently too special.

* The results of Bethe and Peierls were reported by Bethe in September 1934 at the conference in Copenhagen. See also the reports of the congress in London, autumn 1934.

If the interaction energy is known, then by this very fact not only the laws of β-decay are determined. Just as Maxwell’s equations determine all electrical force interactions between material particles, so the interaction energy between the heavy constituent parts of the nucleus and the light elementary particles determines all the interaction forces between the heavy particles that are associated with the field of electrons and neutrinos. Approximately, if one may neglect the reaction of radiation, i.e. when the electrons move slowly in comparison with the velocity of light, the action of the Maxwell field may be replaced by an action-at-a-distance force, namely the Coulomb force. In a similar way, in the case where the motion of the heavy particles takes place slowly in comparison with the velocity of light, the Fermi interaction leads to an action-at-a-distance force between neutrons and protons, as was independently shown by Fermi,* Tamm, and Ivanenko (loc. cit.). The forces arising in this way are, in their character, exchange forces. Thus, Fermi’s theory admits in principle a mathematical expression of the idea that from the possibility of β-decay there follows the existence of exchange forces5. The dependence of these forces on the distance between the heavy particles and on their spin coordinates depends, of course, on the exact form of the expression for the interaction energy according to Fermi and therefore for the time being cannot be determined definitively. Tamm and Ivanenko showed that if one adopts the form of the interaction energy specially chosen by Fermi, then the exchange forces turn out to be too small to explain the structure of the nucleus. This difficulty, however, can be removed if derivatives of the wave functions of the electron and neutrino are introduced into the expression for the interaction energy, as was proposed by Bethe and Peierls. Owing to this uncertainty in the expression for the interaction energy it is not yet possible to decide whether the exchange forces have the form proposed by Majorana,6 or else the form proposed earlier by the author. Empirical data on mass defects speak rather in favor of the Majorana form of interaction.

The most important task of the theory of nuclear structure for the immediate future will be to determine the exact expression for Fermi’s interaction energy between heavy and light particles by comparing all the empirical data—on the form of the continuous spectrum of β-rays, on mass defects, and so forth.

True, even when this interaction is known, there remains in our knowledge of nuclear structure one more gap, which has no analogies in the theory of the atomic shell. Owing to the Coulomb form of the interaction, the structure of the atomic shell is quite negligibly affected by the small deviations from Coulomb’s law that arise at very small distances between particles. By contrast, the exchange forces in the theory of the nucleus depend on the distance between particles to a much higher degree than

* I consider it my duty to thank Fermi for kindly communicating his considerations. See also Wick4.

forces are Coulomb forces (if one takes the expression for the interaction according to Fermi, then, according to Tamm and Ivanenko, it is proportional to \(r^{-5}\); if derivatives of the wave functions enter into the interaction energy, then possibly it is proportional to \(r^{-7}\) or \(r^{-9}\)); thus precisely the deviations from this simple force law at small distances between particles are of primary importance for the structure of the nucleus. The determination of these deviations is closely connected with the problem of the infinite self-energy which is obtained from the Fermi field for a heavy particle in exactly the same way as it is obtained from the Maxwell field for a charged particle. However, no method has as yet been proposed for solving this question.

One can avoid this difficulty by introducing, in a purely formal way, suitably chosen radii of protons or neutrons; however, the correctness of such an operation remains doubtful. Then the interaction leading to the formation of electrons and neutrinos, according to Wick’s investigations \(^{7}\), has yet another important consequence: it leads to a certain addition to the magnetic moment of the proton and to the magnetic moment of the neutron.

The question arises how far the analogy between the magnetic moments of these heavy particles and the magnetic moments of the electrons in the shell of an atom can be carried. For a complete description of the analogy between the atomic shell and the nucleus it is especially important to establish to what extent the magnetic moments of the heavy particles of the nucleus can, with the aid of the vector model, be combined into the magnetic moment of the entire nucleus. Wick’s results are based on considerations analogous to those of Tamm and Ivanenko. We shall briefly repeat them here.

If one carries out a perturbation calculation, with the Fermi interaction energy between light and heavy particles as the perturbation, and calculates the magnetic moment of the proton or neutron, then additional terms arise, corresponding to the second approximation. In form they are exactly equal to the self-energy of these particles and correspond to the possible creation and annihilation of a positron—neutrino pair at the proton, or an electron—neutrino pair at the neutron. These additional moments would be infinitely large if a finite radius of the heavy particles were not introduced, as is done also in the case of self-energies. If this radius is chosen so that the self-energy is of the order of the relativistic rest energy, then one obtains the largest addition to the magnetic moments of the electron—neutrino and positron—neutrino pairs, whose energy is a hundred to a thousand times greater than the rest energy of the electron. The exact value again depends on the exact form of the Fermi interaction. If for this interaction one chooses a form giving the correct order of magnitude for the exchange forces, then the additional magnetic moments are obtained of the order of a nuclear magneton.

Nevertheless, for the neutron this magnetic moment could be compensated to zero if the neutron could transform not

not only by the emission of an electron and a neutrino into a proton, but also by the emission of a positron and a neutrino into a hypothetical negative proton. The empirical results8 for the magnetic moments of \({}^{1}\mathrm{H}\) and \({}^{2}\mathrm{D}\) seem to speak against such a possibility. Apparently, the neutron possesses a magnetic moment.

If one assumes that only the first of the emission processes named exists, then, because of the sign of the charges of the particles that can arise, the additional moments of the proton and neutron must be equal and opposite. A small deviation from this equality is obtained only if the difference between the masses of the proton and neutron is taken into account. The relative magnitude of this deviation is given by the ratio of the mass difference (in energy units) to an energy, increased by a factor of a hundred or a thousand, of the rest energy of the electron. The deviation should amount to only a few percent, and it turns out that the magnetic moment of the neutron must be somewhat greater in absolute value if the mass of the neutron is greater than the mass of the proton, as must be assumed according to Chadwick. This additional moment must be added to the ordinary magnetic moment of the proton, which according to Dirac’s theory corresponds to a spin of \(\frac{1}{2}\hbar\).

The magnetic moments of the neutron and proton must, with sufficient accuracy, be added vectorially into the magnetic moment of the whole nucleus, just as this occurs for the shell of an atom. For, as was indicated above, the principal addition to the magnetic moments comes from those orbits of the virtual electron–neutrino pair whose energy is a hundred to a thousand times greater than the rest energy of the electron. The presence nearby of other heavy particles only slightly perturbs these orbits. The relative magnitude of the deviations from strict additivity of magnetic moments is given approximately by the ratio of the Coulomb interactions and mutual energies to a hundred- to thousand-fold value of the electron rest energy. Thus it proves possible to calculate the magnetic moments of nuclei with an error of a few percent by vector summation of the orbital moments and spin moments. Such attempts are contained in the works of Landé9, Schüler and Kallmann10, Tamm and Altschuler11, and Inglis12. However, it should be kept in mind that the vector model for atomic nuclei, owing to the absence of a central force, may be considerably more complicated than for the atomic shell. Therefore the conclusions drawn in the above-mentioned works cannot be required to have the same degree of accuracy as in determining electron configurations on the basis of the Zeeman effect.

Returning again to the analogy described above between the atomic nucleus and the atomic shell, it should be said that considerations concerning magnetic moments nicely complement the picture of the structure of the nucleus from protons and neutrons, which can be successfully described with the aid of the laws of quantum mechanics. This description is correct only to the same degree of accuracy with which the velocity of the heavy particles can be regarded as very small in comparison with the

speed of light. According to empirical data on the dimensions of the atomic nucleus and on mass defects, this accuracy is considerably lower than the accuracy of analogous calculations for the atomic shell. However, if the considerations given above concerning magnetic moments are correct, then, as in the atomic shell, one can take another step forward in accuracy and approximately take into account relativistic effects, in particular spin. Thanks to the rich experimental material which the Zeeman effect has added to our knowledge of the energy levels and quantum numbers of the atomic shell, it has proved possible to give an exact theoretical explanation of its structure. Apparently, one may hope that, in a similar way, the magnetic moments of nuclei will in the future give us very precise information about the structure of the atomic nucleus.

References Cited

1. E. Fermi, Z. Physik, 88, 161, 1934.
2. I. Tamm, Nature, 133, 981, 1934.
3. D. Iwanenko, Nature, 133, 981, 1934.
4. C. Wick, Rendic. R. Nat. Acad Lincei, 19, 319, 1934.
5. W. Heisenberg, Z. Physik, 77, 1; 78, 156; 80, 587, 1932.
6. E. Majorana, Z. Physik, 82, 137, 1933.
7. C. Wick, Rend. R. Nat. Acad. Lincei (in press).
8. J. Estermann, R. Frisch and O. Stern, Nature, 132, 169, 1933; J. Estermann and O. Stern, Nature, 133, 911, 1934; J. Rabi, J. Kellog, J. Zacharias, Phys. Rev., 46, 157 and 163, 1934; F. Kalckar and E. Teller. Nature, 134, 183, 1934.
9. A. Landé, Phys. Rev., 44, 1028, 1933; D. Inglis and A. Landé, Phys. Rev., 45, 842; 46, 76, 1934.
10. H. Kallmann and H. Schüler, Z. Physik, 88, 210, 1934; H. Schüler, Z. Physik, 88, 323, 1934.
11. I. Tamm and Altschuler, Dokl. Acad. Sci. USSR, 1, 455, 1934.
12. D. Inglis, Phys. Rev., 47, 84, 1935; G. Breit, Phys. Rev. 46, 230, 1934.