The spin-orbit term derived from the surface gradient of the mean field has
PHYSICS · Multi-part · Fermi University Physics · Nuclear Physics · The nuclear force and nuclear models · The nuclear force and nuclear models
The Nucleus Resolved, Ch The shell model: a mean field, a spin-orbit term and the magic numbers
Problem
The spin-orbit term derived from the surface gradient of the mean field has λso>0, which makes the j=ℓ+21 member of a split orbital lie lower in energy, with splitting ΔEso=Cℏ2(2ℓ+1)/2. Suppose instead, for the sake of the argument, that λso had the opposite sign, so that the j=ℓ−21 member lay lower instead, by the same magnitude of splitting. The oscillator shell Nosc=3 contains the orbitals 1f7/2 (8 states), 1f5/2 (6 states), 2p3/2 (4 states) and 2p1/2 (2 states), a total of 20, sitting above the closed shell of 20 states from Nosc≤2.
((a)) With the correct sign, which orbital of the Nosc=3 shell drops low enough to join the Nosc≤2 group, and what running total (magic number) results?
((b)) With the sign reversed, which orbital would drop instead, and what running total would result?
((c)) State what this shows about the sign required to reproduce the measured magic number 28.