Effect of Assumed Range of Tensor Force on the Neutron-Proton Interaction

Abstract

The work of Rarita and Schwinger, while successful in accounting for the experimental value of the quadrupole moment of the deuteron, reduces the central field part of the interaction potential by roughly 30 percent and implies a thorough revision of the nuclear binding energy calculations of Wigner, Feenberg and others. In view of a suggestion by Breit that the experimental deuteron properties might be realized without extreme modification of the central interaction by increasing the range of the tensor interaction, the problem was reinvestigated. The calculations were simplified when it was noticed that closed solutions of the radial wave equations are obtainable if the tensor potential is confined to an infinitely thin spherical shell.For attractive central and tensor interactions it was found that the quadrupole moment can be accounted for by placing the shell beyond ${r}_{1}=0.7(\frac{{e}^{2}}{m{c}^{2}})$, and for a central field square well interaction of range either ${r}_{0}=\frac{{e}^{2}}{m{c}^{2}}$ or ${r}_{0}=\frac{{e}^{2}}{2m{c}^{2}}$ if the shell is placed at ${r}_{1}=1.5(\frac{{e}^{2}}{m{c}^{2}})$ the central field need be reduced by only about two percent. For a repulsive tensor interaction the quadrupole moment can be accounted for with ${r}_{1}<0.7(\frac{{e}^{2}}{m{c}^{2}})$ but the central interaction had to be increased by a factor of 4 for ${r}_{0}=\frac{{e}^{2}}{m{c}^{2}}$. Investigation of neutron-proton scattering with interaction parameters determined by the deuteron quadrupole moment and binding energy shows that for the case of attractive interaction the zero-energy cross section changes from 4.40 to 4.46 barns as ${r}_{1}$ changes from $1\mathrm{to}1.5\frac{{e}^{2}}{m{c}^{2}}$ when ${r}_{0}=\frac{{e}^{2}}{m{c}^{2}}$ and from 3.94 to 3.62 barns when ${r}_{0}=\frac{{e}^{2}}{2m{c}^{2}}$. For the case of repulsive tensor force with ${r}_{0}=\frac{{e}^{2}}{m{c}^{2}}$, the zero-energy cross section changes from 4.48 to 2.97 barns as ${r}_{1}$ changes from $0.56\mathrm{to}0.61(\frac{{e}^{2}}{m{c}^{2}})$. It may be noted that this latter range includes the experimental value of the cross section. Extension of the calculations to high energies shows that experimental results for the total cross section are reproduced up to an incident particle energy of 15 Mev and the angular anisotropy is not in disagreement with experiment for the attractive interaction. The theoretical cross section is smaller than experimental values for higher energies except for repulsive tensor force, and the anisotropy is only in qualitative agreement with the recent work at Berkeley.