The Shell Model of the Nucleus 3. The full model [Sec. 5.5 and 5.6 Dunlap]

Note that magic numbers 8, 20, 28, 50, 82 for neutron number in nucleus from exp. Apart from 2,8 and 20 all the other numbers predicted by the primitive shell model are WRONG. WOODS – SAXON energy levels

What makes the magic numbers? The answer is SPIN-ORBIT coupling energy. This term was much bigger than had been expected. It was discovered in 1949 independently by two research groups. Maria Goeppert Mayer (1906 – 1972) J. Hans D. Jensen (1907 – 1973) (with Haxel and Suess) 1963 Nobel Laureates

Borrowing from Atomic Physics In atomic physics the splitting of the p and d levels is known as FINE STRUCTURE. This FINE STRUCTURE is produced by SPIN-ORBIT coupling. In the H – spectrum the size of the SPIN ORBIT SPLITTING is eV which is very small compared to the gross structure on the 10eV scale. Let us look at what is causing this splitting in more detail.

Borrowing from Atomic Physics e-e- Ze Situation as seen from the proton B L s  Situation as seen from the electron High energy configuration shown Magnetic Interaction energy Magnetic field B = I/2r, with I the effective current I=Ze/T, with T the period of the orbit. The orbital angular momentum L = mvr = 2\pi mr^2/T. Note that B and L are pointing to the same direction. So we have B =ZeL/(mc^2r^3), in CGS units, or B = (L/mer c^2) dV_c/dr, with V_c = -Ze^2/r Magnetic moment = the central potential = the Bohr magnetron

Atomic Spin-Orbit Coupling Where l and s are dimensionless ang.mom The above derivation is not quite right though. We have failed to take into consideration the effect of “Thomas Precession” which is a relativist effect that arises from switching back to the frame in which the nucleus is at rest: Correctly we have:

values of l \cdot s J = L +s, J^2 = L^2+ s^2 +2 Ls Ls =[J(J+1)-L(L+1)-s(s+1)]/2 For s=1/2, J = L+1/2 or L-1/2 If J=L+1/2, Ls= L/2 If J=L-1/2, Ls= -(L+1)/2

Spin-Orbit Coupling is general Although derived for Atomic Physics, this equation for the spin orbit coupling energy is in fact quite GENERAL. It applies to the SPIN-ORBIT energy for any centralized potential V C. Relativist motion always causes an effective “B-like” force. The above equation is expected to apply to a non-Coulomb force – provided it is central Highest energy state l s Lowest energy state l s To find the energy of the atom one must of course take an average over the atom: nl In the case of an atom this splitting is normally small ~ eV Total splitting =

Switching to the nucleus Even for large A and large l one is not going to get a big splitting. This was not able to change the magic numbers. Mayer, Jensen, Haxel and Suess knew this – they had done this calculation – it had no effect on the magic numbers. What they discovered was that this relativistic Spin Orbit energy was being swamped by another Spin-Orbit energy that was coming from a non-relativistic source.

The spin-orbit effect comes from the nuclear surface not the bulk The reason is that the strong spin direction forces are matched by equal and opposite forces in the center of the nucleus. On the surface nucleons can only pass the shown nucleon in one direction only.

STRETCH JACKNIFE LOW ENERGY STATE HIGH ENERGY STATE The spin orbit energy of the nucleus is of opposite sign to the special relativity prediction TOTAL SPLITTING = nl

WOODS – SAXON energy levels Which Energy levels are going to be depressed most? The ones with the largest angular momentum WHY?

The Full Energy Level diagram of the SHELL MODEL Remember the occupancy of each level nl j is (2j+1) – only depends on j The parity of each level nl j is only depends on l PARITY -

Fermi energy Single Neutron and Proton Separation Energies