Effects of the continuum on the spin-orbit splitting

A major step in the development of the shell model was the introduction

by Mayer and Jensen [1,2] of a rather strong spin-orbit potential component to the

nuclear mean field.   This coupling splits the levels with j = l +/- 1/2,

which are degenerate in a spin-independent potential, lowering those

with parallel coupling j = l + 1/2.   The sequence of single-particle levels obtained

when the spin-orbit force is added to the central potential provides a natural

explanation of the major magic numbers and has been confirmed by

a large body of  experimental evidence.   The average one-body potential

results from a strong two-body spin-orbit force  V_LS(1,2) [3].  

In this work we study the possible effects of the continuum on the

spin-orbit splitting when the initial levels are near threshold.  We do this 

by considering a simple model of two nucleons a in single-l shell

interacting by the two-body interaction above, which we solve in the LS basis.

The results show an apparent reduction of the spin-orbit splitting as a function of the 

width of the j=l-1/2  level,  which is located into the continuum.  

[1] Mayer, M. G., Phys. Rev. 75, 1969, (1949).

[2] Haxel, O., Jensen, J. H. D., and Suess, H. E. , Phys. Rev. 75, 1766, (1949).

[3] A. Bohr and B. R. Mottelson,  Nuclear Structure Vol. I, World Scientific Publishing Co (1998).