The Theory of Nuclear Forces*

R. Peierls

Editor’s Note

Rudolph Peierls offers a summary of the emerging theory of nuclear forces. These forces could be most easily probed in two-particle interactions involving neutrons and protons. Scattering experiments had established the range of the force as about 1.2×10–13 cm, and supported the notion that the nuclear forces between neutrons and protons were identical, ignoring electrostatic differences. Nuclear forces could also depend strongly on direction in relation to the orientation of quantum-mechanical spins of the particles. The best theory proposed so far was that of Japanese physicist Hideki Yukawa, which viewed the nuclear force as originating from a nuclear field associated with a new quantum particle having mass of around 300 electron masses—later identified with the pion.ft　　中文

THE forces between the constituents of a nucleus are “short-range” forces, which have no appreciable effect over distances of more than a few times 10－13 cm. Hence it is impossible to find the laws of force by extrapolation from large-scale observations (as in the case of Coulomb’s law) and the only possible lines of attack are either, to deduce the law of force from direct observations on the properties of nuclei, or to derive them from some simpler, and more general, laws.ft　　中文

Using the direct experimental approach, it is best to start from phenomena which involve only two particles, since only these permit an unambiguous theoretical interpretation. This group consists of the properties of the deuteron, including its disintegration, and the scattering of protons and neutrons by hydrogen.ft　　中文

Practically all experiments are carried out in conditions in which the de Broglie wave-length of the particles is greater than the range of the forces, and this has the effect that one need consider only those states of motion in which the motion of one particle with respect to the other has no angular momentum. (In all other states of motion the centrifugal force prevents the particles from approaching sufficiently close for any interaction to take place.) Moreover, so long as the wave-length is greater than the range of the force, the effect of the field of force is largely independent of the exact variation of force with distance within the range, and depends approximately only on one constant, which may be called the “strength” of the field of force. (In the case of a potential of the type of a potential well, this strength is approximately proportional to the depth times the square of the width.)1ft　　中文

In the experiments on proton-neutron interaction two possible cases are discussed, according to whether the spins of the two colliding particles are parallel or opposite (in the case of like particles, Pauli’s principle ensures that, so long as there is no orbital angular momentum, the spins are necessarily opposite) and the strength of the interaction may be different for these two cases. For the case of a proton and a neutron with parallel spin; the strength constant can be obtained from the binding energy of the deuteron; for those with opposite spin, it follows from the scattering of neutrons by protons, since in this scattering the effect of the neutrons with parallel spin can be allowed for once their strength constant is known. The very high cross-section for slow neutrons is an indication that the strength constant of the interaction for neutrons with opposite spin is either just sufficient or just insufficient to give rise to a bound state of the deuteron with no resultant spin2. The choice between these alternatives can be made by means of the scattering of neutrons in para- and ortho-hydrogen where, because of interference, the results depend on the phase of the wave scattered by each proton. The result is that the force is just insufficient to give a bound state3.ft　　中文

For the proton-proton force, the strength can be derived from measurements of the scattering of protons in hydrogen.ft　　中文

Once these strength constants are known it is possible to calculate all other observable quantities, namely, the energy variation of the proton-proton and neutron-proton scattering, the photo-effect of the deuteron, etc., to the degree of approximation in which the ratio between the range of the forces and the de Broglie wave-length is negligible. The fact that these calculations give approximate agreement with observations serves as a check on the initial hypothesis of short-range forces.ft　　中文

Actually, all results of this simple theory have to be supplemented by correction terms involving the range. If the experimental data were accurate enough, it should thus be possible to estimate the range of the forces. The use of these correction terms requires greater accuracy than is at present available in the data on neutron-proton scattering, and the determination can only be carried out in the case of the proton-proton scattering, in which more accurate data have been obtained. The result of this is that the “range” for this interaction is about 1.2×10–13 cm. The experimental data are even accurate enough to yield a certain amount of information on the actual dependence of the force on distance4.ft　　中文

More information on these points would be obtained by measuring the neutron-proton scattering at medium energies with higher accuracy, or by observations at higher energies including, in particular, the angular distribution of the scattered particles.ft　　中文

An important contribution to the problem was the discovery by Rabi and others of a quadripole moment of the deuteron5. This proves that the neutron and proton have a tendency to have their spins in the direction of the line joining them, rather than at right angles to it. This implies obviously that the forces are not central, but that the potential energy depends on the angle between the spin direction and the line joining the two particles. Although all the calculations referred to above were carried out on the assumptions of central forces, the results remain practically unchanged. For opposite spins, which is the only case of interest for proton-proton interaction at low energies, the force must still be central, since the resultant spin is zero and hence does not set up a preferential direction; for parallel spin, the fact that the force is not central means that the angular momentum due to the motion of the particles is no longer constant but fluctuates. Hence the motion in the state of lowest energy is no longer one with zero orbital angular momentum; it can be shown that the state of motion is a mixture of a state with zero angular momentum and one with two units (S and D states in spectroscopic notation). From the electrical quadripole moment one can estimate that the contribution of the D state amounts to only a few percent6.ft　　中文

However, an appreciable non-central force is required to produce even this small effect, since in the D state a very strong centrifugal force has to be overcome. It seems probable, therefore, that the non-central forces must represent an appreciable fraction of the total force. Yet, again owing to the effect of the centrifugal force, no appreciable influence of this D state should be expected on the other observable phenomena, except possibly on some finer features which have not yet been thoroughly investigated. Thus the approximate agreement with the experimental evidence, and the estimate of the range remain practically unaffected by the correction terms.ft　　中文

The data on the strength and the estimate of the range give the same answer, within the experimental error, for the proton-proton and proton-neutron interactions for opposite spins, and the suggestion has therefore been made that these two forces are exactly the same, except for the electrostatic interaction between the protons (“charge-independence hypothesis”)7.ft　　中文

From the fact that the charge of stable light nuclei is about half their mass, and that only a small change is produced in the energy of a light nucleus if in its constituents the number of neutrons and the number of protons are interchanged, has further led to the belief that the nuclear forces are symmetric in proton and neutron, that is, the neutron-neutron interaction is exactly the same as the proton-proton interaction, except for the effect of the electric forces1.ft　　中文

Turning now to the evidence from nuclei containing more than two particles, this is usually discussed on the basis of the assumption that the forces are additive; that is, that the interaction force between two particles is not affected by the simultaneous presence of a third particle. Beyond its simplicity and the fact that it holds in the atom, this assumption is not founded on any evidence, and, in fact, there are certain theoretical arguments against it8. However, without this assumption the available evidence is insufficient to draw any conclusions at all, and one must therefore use the assumption of additivity as a working hypothesis, which may at a later stage be disproved. Moreover, most of the theoretical work on the nuclei of weight 3 and 4 was done before the discovery of the quadripole moment of the deuteron, and hence central forces were used. It is likely that the directional dependence of the force will not be negligible in these problems, but its precise effect has yet to be investigated.ft　　中文

So far as the calculations go, they show that the mass defects of the nuclei of mass 3 and 4 are very sensitive to the range of the force, and that the range required to give the right values is roughly of the same order of magnitude as that obtained from scattering experiments1,9. Quantitative agreement, however, was not obtained. The laws of force used in these calculations were usually restricted by adopting the charge-independence hypothesis, and also by assuming that the range of the force was the same for parallel and for opposite spins. Whether the remaining discrepancy is due to these restrictions, or to the assumption of central forces, or whether it represents a failure of the additivity, remains to be seen.ft　　中文

For nuclei beyond 4He, the observed binding energy ceases to rise rapidly with the number of particles. This fact, often briefly referred to as “saturation”, obviously means that the constituent particles of a large nucleus do not all attract each other with the same forces with which the constituents of a helium nucleus attract each other. This may be due to several reasons. For one thing, the additivity of the force might fail, and the attractive force between two particles may depend on the number of other particles in the immediate neighbourhood. This dependence might be such as to ensure that the total potential energy per particle always remained below a certain saturation value. A possible description on these lines has been put forward by Teller, Critchfield and Wigner10, but the idea has not yet been pursued very far.ft　　中文

Another possibility is that the attraction may turn into repulsion at very close approach, in analogy with the forces between the atoms of a liquefied inert gas. In this case the repulsion might ensure that any one particle can, within the range of its forces, be surrounded only by a small number of others, and hence the binding energy per particle is again limited. This possibility has not been fully explored, but it has the disadvantage that, owing to the wave mechanical penetrability of potential barriers, an extremely strong repulsion would have to be assumed to make this explanation possible.ft　　中文

The most attractive explanation is no doubt that the forces are “exchange forces” which depend on the symmetry of the wave function describing the motion of the particles, like the valency forces between the constituents of organic molecules. On this idea the neutron is trivalent, capable of forming a “bond” with one other neutron and two protons, and correspondingly for the proton. This would give a very natural explanation of the helium nucleus as a saturated structure1.ft　　中文

The directional properties of the forces, which must be inferred from the existence of the quadripole moment of the deuteron, may have a bearing on this question, since in a large nucleus different particles must necessarily be arranged in different directions relative to a given particle, and the forces may quite well be attractive for some pairs of particles and repulsive for others, if the directional dependence of the forces is strong6. This effect will give rise to some kind of saturation; whether this saturation is of the right kind, and in particular whether it leads naturally to the α-particle as a stable structure, remains to be seen.ft　　中文

Of the attempts to derive the nuclear forces from a more elementary phenomenon, the most interesting is the meson theory, which is based on an idea of Yukawa11. Yukawa supposes that the nuclear forces are due to a “nuclear field” in the same manner in which the electromagnetic forces are caused by the electromagnetic field. This field must, however, satisfy different field equations in order to produce short-range forces, and if this requirement is coupled with that of the principle of relativity, there is only one possible type of wave equation, and the law of interaction is limited to laws of a particular type, of which the simplest has the potential:

V ＝ g·e－kr/r,

(1)

where k and g are constants, and r is the distance between the particles. (Coulomb’s law is a special case of this with k = 0). In quantum theory, just as the electromagnetic field is associated with light quanta, the “nuclear field” will be associated with a new type of particle; the fact that the wave equation differs from that of light indicates that the rest mass of these particles is not zero, like that of light quanta, but has the finite value hk/2πc, where k is the same constant as in (1). In order to obtain a range of the right order of magnitude this mass has to be assumed to be a few hundred times that of the electron. The subsequent discovery of “mesons” of just such a mass in cosmic rays has very much increased our confidence in the “meson theory” of nuclear forces.ft　　中文

If we try to fit a law of the form of (1) to the observations on proton-proton scattering, we obtain very good agreement, but we have to choose a value of k which corresponds to a meson mass of about 300 electron masses. This is almost certainly higher than the mass of the mesons found in cosmic rays. The origin of this discrepancy is not clear.ft　　中文

On the meson theory of nuclear forces, mesons are capable of being absorbed and emitted in nuclear reactions provided sufficient energy is available for their creation, and they may be exchanged (that is, temporarily created by one particle and absorbed by another) even if the available energy is insufficient to liberate them. If this view is taken, conservation of angular momentum in the process requires the mesons, like light quanta, to have integral spins. (The electron, which has a half-integral spin, can only be absorbed or produced in pairs.) Zero spin would make the nuclear forces repulsive when they should be attractive, and the most likely assumption seems that of a spin of one unit, as for the photon12.ft　　中文

This assumption fixes the equations of the meson field completely, but not its interaction with the proton and neutron. This interaction is governed by two terms which, by analogy with the electric charge and magnetic moment, one may term the “meson charge” and the “meson moment” of the heavy particles. The law (1), in particular, is obtained if the heavy particles have only a meson charge g, but no meson moment. This law does not agree with experiment in detail, since it gives neither a spin dependence of the force (as required to explain the properties of the neutron-proton scattering) nor a directional dependence that would account for the quadripole moment of the deuteron. The introduction of a “meson moment” would help to give the right dependence12, but the force would then increase so rapidly at short distances that the proton and the neutron would fall into another, producing an infinite binding energy. Probably this result should not be taken too seriously, since the methods of quantum theory are likely to fail for too close approach, but it would mean in any event that the quantitative study of nuclear forces would have to be abandoned until an exact description of this failure of quantum theory is available.ft　　中文

It has also been suggested that two kinds of mesons might exist, of which one has the spin one, the other zero spin, and with such properties that the singular terms in the interaction energy just cancel. In the crude approximation to which such calculations are usually carried out, the directional dependence of the forces would then also just cancel, together with the infinities, but it is possible that finer effects would give a sufficiently large directional variation13.ft　　中文

Lastly, there arises the question as to the electric charge of the meson. The mesons observed in cosmic rays are charged, and if one of them is emitted by one nuclear particle and absorbed by another, this will involve an exchange of charge, thus ensuring that the forces are of the exchange type, as required by the most widely accepted explanation of the saturation of nuclear forces. Such an exchange will, however, be possible only if one of the heavy particles is a proton and the other a neutron, but not for two like particles. In order to account for the equally strong forces between like particles, one has to assume the existence of neutral mesons in addition to the charged ones14. There is certain independent evidence for the existence of neutral mesons in cosmic rays15. If we take the view that the saturation of the nuclear forces is due to their directional dependence and not to their exchange character, it is, in fact, possible to attribute all nuclear forces to neutral mesons6. This would have the advantage of removing the discrepancy between the mass of the charged mesons from cosmic rays, and the range of the forces as determined from proton-proton scattering. It would mean, on the other hand, that the particles, the discovery of which was hailed as a confirmation of Yukawa’s theory, had actually no connexion with the particles postulated by Yukawa.ft　　中文

This summary of the present state of the theory16 closes with a number of questions to which the answer is not known. But the fact that it is possible to ask these questions at all is a sign of the rapid progress that has been made in this field in the last few years.ft　　中文

(145, 687-690; 1940)

Rudolph Peierls: University of Birmingham.

References:

  1. Bethe and Bacher, Rev. Mod. Phys., 8, 82 (1936).

  2. Simons, Phys. Rev., 55, 793 (1939).

  3. Brickwedde, Dunning, Hoge and Manley, Phys. Rev., 54. 266 (1938).

  4. Hoisington, Share and Breit, Phys. Rev., 56, 884 (1939).

  5. Kellogg, Rabi, Ramsay and Zacharias, Phys. Rev., 55, 318 (1939).

  6. Bethe, Phys. Rev., 55, 1261 (1939).

  7. Breit, Condon and Present, Phys. Rev., 50, 825 (1936).

  8. Primakoff and Holstein, Phys. Rev., 55, 1218 (1939).

  9. Rarita and Present, Phys. Rev., 51, 788 (1937); Rarita and Slawsky, Phys. Rev., 54, 1053 (1938).

  10. Wigner, Critchfield and Teller, Phys. Rev., 56, 530 (1939).

  11. Yukawa, Proc. Phys. Math. Soc. Japan, 17, 48 (1935).

  12. Fröhlich, Heitler and Kemmer, Proc. Roy. Soc., A, 166, 154 (1938).

  13. Möller and Rosenfeld, Nature, 144, 241 (1939); 144, 476 (1939).

  14. Kemmer, Proc. Cam. Phil. Soc., 34, 354 (1938).

  15. Arley and Heitler, Nature, 142, 158 (1938).

  16. Cf. also Peierls, R., Ann. Rep. Chem. Soc., in the press.

* Based on lectures given at the Royal Institution on January 17, 24 and 31.