Nature of the Nuclear Field

An attempt to explain neutron-proton interaction made by Yukawa¹ in 1935 has been brought to general notice^2,3 in connexion with the new experimental evidence for the existence of a 'heavy electron'^4,5,6.

Although it is premature to draw definite theoretical conclusions from the present experimental knowledge, it is certainly suggestive that a Yukawa particle with a mass of the observed order of magnitude (100 m_el) does indeed give nuclear forces of the correct range. As nothing is otherwise known about the strength of the interaction of these particles with protons and neutrons, one can also assume it to be so large as to account for the known magnitude of the neutron-proton force.

Unfortunately, a theory based on the scalar relativistic Schrödinger equation, as suggested by Yukawa, cannot explain the experimentally known position of the ¹S level of the deuteron relative to its ³S ground state.

It has, however, been found that a more satisfactory result can be obtained if one admits a vector wave function for the new particle, such as was used by Proca⁷ in a different connexion. Proca's equations can be quantized on lines completely analogous to the Pauli-Weisskopf method⁸ for the scalar wave equation, and the resulting neutron-proton potential can easily be determined. Using the most general combination of possible interactions of the Yukawa and Proca type, the potential is found to be

where k is 2pc/h times the rest mass of the new particle, s _N and s _P the spin operators of neutron and proton respectively and r the distance between these particles. A, B and C are constants and are independent with regard to both sign and absolute magnitude. Therefore an interaction can obviously be chosen which gives the correct triplet-singlet separation.

The nuclear potential now considered most adequate for practical calculations⁹ has exactly the spin dependence one obtains from the above expression by putting C = 0, A : B = 3:5. It further includes the 'isotopic spin' factor (t _N t _P) which can be accounted for in the field theory here proposed exactly on the same lines as in the case of the Fermi field¹⁰, that is, by assuming that the new particle also has a charged and an uncharged state. It is a question yet to be considered whether a complete description of experimental data may also be possible without assuming this uncharged state to exist; also whether any specialized type of interaction of greater simplicity than the above general linear combination might perhaps suffice to fit the facts.

In any case, it is possible to give a field theory in which magnitude and range of the nuclear forces, as well as their dependence on spin and charge, are all accounted for equally well. It appears satisfactory to have such a model, even though its relation to experiment is admittedly quite uncertain.

The details of the calculation will be published elsewhere.

N. KEMMER
(Beit Scientific Research Fellow).

Imperial College of Science and Technology,
London, S.W.7.
Dec. 8.

1. Yukawa, Proc. Phys.-Math. Soc. Jap., 17, 48 (1935).
2. Oppenheimer and Serber, Phys. Rev., 51, 884 (1937).
3. Stueckelberg, Phys. Rev., 52, 41 (1937).
4. Neddermeyer and Anderson, Phys. Rev., 51, 885 (1937).
5. Street and Stevenson, Phys. Rev., 51, 1005 (1937).
6. Street and Stevenson, Phys. Rev., 52, 1003 (1937).
7. Proca. J. Phys., (vii), 7, 347 (1936).
8. Pauli and Weisskopf, Helv. phys. Acta, 7, 709 (1934).
9. Heisenberg, Naturwiss., 25, 749 (1937).
10. Kemmer, Phys. Rev., 52, 906 (1937).

Nuclear Forces, Heavy Electrons and the b-Decay

We have generalized a theory put forward by Yukawa¹ showing that nuclear forces can be explained by assuming the existence of new particles of mass about two hundred times that of the electron. Our theory is relativistically invariant, and in its present form gives results which we believe are of actual significance for cosmic ray and nuclear phenomena.

Our theory is based on the idea that the proton and neutron are two states of the same particle characterized by the eigenvalues –1 and +1 respectively of some operator, t say, and that transitions may occur from one state to the other with the emission or absorption of a particle of positive charge e and mass M_u. It follows from this that such a particle, which we shall call a U-particle, obeys Bose statistics. In order to explain the b-decay, we must then assume that in certain circumstances an electron may absorb a U-particle by becoming a neutrino or vice versa. Of course, the emission or absorption of a U-particle only takes place observably when it is consistent with the conservation of energy. In other cases the emission or absorption is merely virtual, as an intermediate state, as is the case in the quantum theory of radiation.

The motion of the U-particles is described by a 4-vector U _m. Denoting by Y the wave function for the proton-neutron, which in consequence has two components corresponding to the two states of this particle (each component itself being really composed of the four components belonging to a wave function for a particle obeying the Dirac equation), by y the similar wave function for the electron-neutrino, and by j _m the electromagnetic potentials, we can derive the complete equations of motion of the whole system in the usual way by the variation of a relativistically invariant Lagrange function with respect to Y, y, U and j. Using the methods of the second quantization, the equations can then be written in the Hamilton form necessary for quantum mechanics. Writing

the equations of motion for the U-particles are

and

and their conjugate complex equations. l, k = 1,2,3 and g, g′ represent the interaction of the U-particles with the proton-neutron and electron-neutrino respectively, and are the only two arbitrary constants in our theory. t_NP is a matrix representing the change of a proton into a neutron and t_en similarly represents the change of a neutrino into an electron. In addition, exactly as in electrodynamics, the equation

has to be imposed as a boundary condition. The equation of the proton-neutron is

where t_P and t_N are matrices which are unity only when the particle is a proton or neutron respectively, and zero otherwise. M_N and M_P are the masses of the neutron and proton. A similar equation holds for the electron-neutrino. The charge density of the U-particles is given by

and the commutation rules being

and the conjugate equations, the existence of U-particles of positive and negative charge follows automatically as in the case of the Klein-Gordon equation treated by Pauli and Weisskopf².

The purpose of this note is to point out the following consequences of the theory.

(1) A positive U-particle at rest may disintegrate spontaneously into a positive electron and a neutrino. This disintegration being spontaneous, the U-particle may be described as a 'clock', and hence it follows merely from considerations of relativity that the time of disintegration is longer when the particle is in motion. We believe that this may have to do with the fact observed by Blackett and others that below 2 × 10⁸ e.v. most cosmic ray particles are electrons, above this energy heavy electrons. In a previous paper³ we have shown that the experimental evidence requires that heavy electrons can apparently turn into ordinary electrons. Our U-particles are then to be identified with the heavy electrons, and it follows that most of the heavy electrons have been created either in the earth's atmosphere or not very far from it.

(2) The U-particles will suffer large energy losses in addition to the ionization and the Bremsstrahlung, due to collision with neutrons in which they are absorbed and re-emitted with less energy (analogue of the Compton effect).

(3) On colliding with a neutron they may be absorbed to form a fast proton, but only in the presence of other particles.

(4) It gives a relativistic interaction between a proton and a neutron which in the non-relativistic case reduces to the Heisenberg-Majorana exchange force of the type—.

(5) It may be objected that the use of the Dirac equation to describe the proton-neutron is incorrect inasmuch as it gives a wrong value for the magnetic moment. This objection, however, fails if the magnetic moments of the proton and neutron can arise through their interaction with the U-particles. It is quite easy to see without any calculation that our theory gives magnetic moments of opposite sign to the proton and neutron, as is required by experiment.

(6) When the maximum energy of the b-ray spectrum is not too large, it can be shown that our theory reduces exactly to Fermi's. In the general case, however, characteristic differences will appear which may improve the agreement with experiment.

Lastly, we would remark that the U-particles being charged, they cannot explain the close-range proton-proton interaction. To formulate this, we would have to introduce a neutral particle N of about the same mass M_u, which obeys similar equations to those for the U-particles and can be absorbed and emitted when a proton jumps from one energy state to another. The introduction of such a particle may not seem very arbitrary when we consider that it would give us a symmetrical state of affairs, with all the particles falling into three groups with masses of the order M_P ≈ 1840 m_e, M_u ≈ 200 m_e, and m_e, there being positive, negative and neutral particles of each group. The introduction of the N-particles would give, in addition to the processes mentioned above:

(7) The absorption of a positive U-particle by a neutron and the re-emission of an N-particle, and vice versa. This would give rise to a type of chain U–N–U, etc., such as we have already discussed⁴ in connexion with the proton-neutron interaction.⁵

The detailed calculations are being published elsewhere.

H. J. BHABHA.

Institute of Natural Philosophy, Edinburgh. Dec. 13.

1. Yukawa, H., Proc. Phys.-Math. Soc. Japan, 17, 48 (1935).
2. Pauli and Weisskopf, Helv. phys. Acta, 7, 709 (1935).
3. Bhabha, H. J., Proc. Roy. Soc., in the press.
4. Bhabha, H. J., Nature, 139, 1103 (1937).
5. Some of the points noted above were also mentioned by Dr. Heitler in a conversation with me on the original Yukawa theory. Cf. also Wenzel, G., Z. Phys., 104, 34 and 105, 738 (1937) for a different introduction of new Bose particles.

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