Velocity-Dependent Forces and Nuclear Structure. II. Spin-Dependent Forces

Abstract

Recent investigations have shown the presence of velocity-dependent forces in the two-nucleon interaction, increasing the number of parameters in the two-nucleon potential, and making more difficult the determination of these parameters in a unique way. In view of the success of the shell model and of the assumptions that the same interactions hold between nucleons inside nuclear matter as between free nucleons, it is of interest to explore the possible restrictions on the two-nucleon potential that follow from level arrangements and separations in nuclear shell theory. In the present paper we carry out this exploration for velocity-dependent forces, starting with the two-body spin-orbit force, two forms of which have recently been proposed, and considering also the simplest velocity-dependent forces that depend on the second power of the momentum, which include the velocity-dependent tensor force recently introduced by Breit. The two-body spin-orbit force of Gammel and Thaler has a very short range, and, taking advantage of this fact, we show in Sec. 2 that the interaction energy for two nucleons in the same shell is proportional to the interaction energy for the zero-range velocity-dependent central force discussed previously by the author. The simple expression for the interaction energy allows us to compare in Sec. 3 the level separation due to the spin-orbit force of Gammel and Thaler and that due to the spin-orbit force of Signell and Marshak. We also compare the effects of both types of spin-orbit forces with the interaction energy due to the central even singlet force of Gammel, Christian, and Thaler. In Sec. 4 we analyze a velocity-dependent central force that acts only in the triplet state. In Sec. 5 we discuss the velocity-dependent tensor potential in the long-range approximation, and show the restrictions that follow on the strength of this potential from the assumption that the separation between levels should be small compared with the separation between shells. In Sec. 6 we discuss the velocity-dependent tensor potential in the short-range approximation, and obtain restrictions on the product of strength and range of this potential. The interaction energies for all short-range velocity-dependent potentials show similarities, suggesting the possibility of finding simple closed expressions for the interaction energies for all short-range forces.