On a Phenomenological Neutron-Proton Interaction

Abstract

This paper describes the calculations and results obtained in fitting a phenomenological interaction to the properties of the deuteron and related low energy phenomena. This interaction is a linear combination of a central and tensor potential, where both potentials are of the Yukawa shape, but with different ranges. The calculations were based on the variation-iteration method which has the advantage of providing systematically improved trial wave functions, together with limits of error for the well depth eigenvalue at every stage of the process. The nonpositive definite character of the tensor potential complicates the nature of the convergence to the correct eigenvalue. Special methods are described to overcome this difficulty. The successive iterations were performed numerically on the Harvard Mark I calculator. The accuracy obtained for the well depth parameter was limited in practice by the finite intervals employed in the numerical integration, and is estimated to be one in ${10}^4$. The quadrupole moment, as a nonstationary quantity, does not exhibit the same degree of convergence; the accuracy obtained was here estimated as one in ${10}^3$. The results are presented in tabular form as values of the well depth parameter and of the quadrupole moment for four central potential ranges, five tensor ranges, and three values of the tensor strength. These tables also include values of the fractional amount of $D$ state and of the effective triplet range. The experimental magnitudes of the latter quantities serve to delimit the permissible values of the tensor range. The photoelectric cross section of the deuteron is shown to involve only the triplet effective range in addition to the familiar zero-range formula. The cross section for photomagnetic capture contains the singlet and triplet phenomenological parameters, and in addition, a mixed effective range which also includes the effect of an exchange magnetic moment. The value of the mixed range, as inferred from the experimental capture cross section, agrees with the average of the singlet and triplet effective ranges, within the rather large experimental uncertainties. For the energy domain in which it is appreciable, the photomagnetic cross section is almost uniquely fixed by the capture cross section. The comparison of these cross sections with experiment is satisfactory.