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The charge-dependence of nuclear forces

  • N. Kemmer (a1)

The theory of heavy electrons recently developed by several authors may be considered to give a satisfactory account of the empirically known neutron-proton interaction. However, it now seems well established that there exists a proton-proton interaction of comparable magnitude which is not accounted for equally well. Owing to the fact that the emission of a heavy electron involves the change of a neutron into a proton or vice versa, the first approximation of this theory gives only an exchange force between unlike particles, whereas a force between like particles must be due to double transitions and thus only appears in the second approximation. It is true that the expansion in terms of the number of particles emitted is actually so badly convergent that the second order proton-proton force at distances of about 10−13 cm. is found to be not essentially smaller than the first order neutron-proton force (see FHK), but nevertheless this does not explain experimental facts, since the calculated second order force is always repulsive.

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See for instance: Fröhlich, , Heitler, and Kemmer, , Proc. Roy. Soc. A 166 (1938), 154 referred to as FHK, and Kemmer, , Proc. Roy. Soc. A 166 (1938), 127, where a list of references is given; see also the recent paper by Yukawa, , Sakata, and Taketani, , Proc. Phys. Math. Soc. Japan, 20 (1938), 319.

Phys. Rev. 50 (1936), 806; 53 (1938), 239.

It has been previously shown (Kemmer, , Phys. Rev. 52 (1937), 906) that a similar extension of Fermi's theory of the β-field will also account for the symmetries of the CI-hypothesis. The writer would like to use this opportunity to point out an error of sign in the paper referred to. As direct calculation shows, the sign in equation (14) must be reversed. The alteration also affects a statement of Weizsäcker, v. (Zeits. f. Physik, 102 (1936), 572). In the case studied by him the force for large distances is actually attractive in the deuteron ground state, and thus not in contradiction with experience. The neutron-proton force for the general case given by Fierz, (Zeits. f. Physik, 104 (1937), 553) in his equation (3·4) has the correct sign only if the wave functions in that equation are considered to be anticommuting operators; there is a change of sign if we use their expectation values. The same is true for equation (4) of the writer's aforementioned paper. The results of the latter are only affected by this alteration in that the term γr −2(σr) (σ′r) cannot be omitted if agreement with experiment is to be established. There is, however, no necessity for this omission, and it is interesting to note that such a term must also be included in the new heavy electron theory.

Cassen, and Condon, , Phys. Rev. 50 (1936), 846.

Recently Majorana, (Nuovo Cimento, 14 (1937), 171) has shown that it is possible to eliminate the antiparticle in the same way in the case of any uncharged particle satisfying the Dirac equation. The simplest, though not the most general, way of proving this fact is by noting that a representation of Dirac's matrices can be chosen in which all three αi are real, while β is imaginary. Then Dirac's equation is seen to be a purely real differential equation, and we can confine ourselves to the consideration of its real solutions. Majorana shows that the quantized theory then gives a particle without its antiparticle, as in the case studied above (cf. also Kramers, , K. Akad. Amsterdam Proc. 40 (1937), 814).

The corresponding assumption for the Pauli-Weisskopf representation (38) is not compatible with the relativistic invariance of the interaction.

The probable form of the nuclear potential was derived from consideration of the binding energy of heavy nuclei by Volz, (Zeits. f. Physik, 105 (1937), 537) and still further determined and modified by Kemmer, (Nature, 140 (1937), 192). The modified form has since been taken as the basis of calculations by Heisenberg, (Naturwiss. 25 (1937), 749) and Flügge, (Zeits. f. Physik, 108 (1938), 545). It is interesting that the original constants of Volz do not correspond to the simpler form (32) so that their use would necessitate the acceptance of an antiparticle.

Helvetica Phya. Acta, 7 (1934), 709.

Proc. Phys.-Math. Soc. Japan, 19 (1937), 1084.

It is to be noted that the above value of the potential corresponds to the choice of the three-dimensional interaction as

where the suffix i refers to “isotopic spin space” and j refers to true space, and ψi is so normalized that the undisturbed energy is

This value was determined by Miss Littleton of Bristol who kindly solved the deuteron wave equation numerically for the particular potential here used, assuming m 0 = 137m el. Dr Bhabha has since informed me that the same integration has been performed mechanically at Cambridge for arbitrary m 0. The point determined by Miss Littleton lies exactly on the curve thereby found. Since these calculations still neglect the fact that the true theoretical force is not strictly central, no great weight can be attached to the numerical values.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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