Nucleon Energy Levels in a Velocity-Dependent Potential

Phys. Rev. 104, 401 – Published 15 October 1956
A. A. Ross, R. D. Lawson, and Hans Mark


The neutron and proton level sequences in a diffuse, velocity-dependent potential have been investigated. A velocity-dependent interaction is used which manifests itself by attributing to a nucleon inside nuclear matter an "effective mass" which is a function of its position. Following Brueckner, Johnson and Teller, and Duerr, the effective mass in the center of the nucleus is chosen to be one-half the free-particle mass. The potential form was taken as V(r)=-V0{1+exp[α(r-a)]}, and for protons a Coulomb potential derived from a uniform charge distribution extending to r=a was added. The proper neutron shell structure and level sequence was obtained with the parameters α=1.16×1013 cm-1, a=1.3A13×10-13 cm, V0=69 Mev, and a spin-orbit coupling 33 times the Thomas term. For protons, using the same α, radius, and spin-orbit coupling, it was found that the potential depth had to be increased by roughly 13 Mev to bind the correct number of protons in Pb208. If Pauli principle correlations are included, then a deeper proton potential is obtained. This correction depends critically on the form of the nucleon densities. Since a self-consistent treatment has not been made, this effect has been estimated in two ways: (1) A Fermi-Thomas approximation was used to compute the densities. In this case, the correct neutron-proton ratio is obtained but the correct proton level sequence is destroyed. (2) The neutron well depth was increased by an amount (N+12Z)(Z+12N). In this approximation the correct proton level sequence is obtained but it is not possible to bind the proper number of protons.


  • Received 10 July 1956
  • Published in the issue dated October 1956

© 1956 The American Physical Society

Authors & Affiliations

A. A. Ross*, R. D. Lawson, and Hans Mark

  • Department of Physics, University of California, Berkeley, California

  • *Whiting Fellow in Physics. Submitted in partial fulfillment of the requirements for the Ph.D. degree, University of California, Berkeley, California.
  • Now at the Radiation Laboratory, University of California, Livermore, California.

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