# the shell model of the nucleus 3. the full model [sec. 5.5 and 5.6 dunlap]

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The Shell Model of the Nucleus

3. The full model

[Sec. 5.5 and 5.6 Dunlap]

2820344058Note 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 levels26102146181022292

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 PhysicsIn 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 10-4 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 Physicse-ZeSituation as seen from the protonBLsSituation as seen from the electronHigh energy configuration shownMagnetic Interaction energyMagnetic 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/rMagnetic moment= the central potential= the Bohr magnetron

Atomic Spin-Orbit CouplingWhere l and s are dimensionless ang.momThe 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 sJ = L +s ,J^2 = L^2+ s^2 +2 LsLs =[J(J+1)-L(L+1)-s(s+1)]/2

For s=1/2, J = L+1/2 or L-1/2If J=L+1/2, Ls= L/2If J=L-1/2, Ls= -(L+1)/2

Spin-Orbit Coupling is generalAlthough 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 VC. Relativist motion always causes an effective B-like force. The above equation is expected to apply to a non-Coulomb force provided it is centralHighest energy stateLowest energy statelsTo find the energy of the atom one must of course take an average over the atom:nlIn the case of an atom this splitting is normally small ~ 10-4 eVTotal splitting =

Switching to the nucleusEven 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 bulkThe 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.

STRETCHJACKNIFELOW ENERGY STATEHIGH ENERGY STATEThe spin orbit energy of the nucleus is of opposite sign to the special relativity predictionTOTAL SPLITTING = nl

2820344058WOODS SAXON energy levels26102146181022292Which Energy levels are going to be depressed most?The ones with the largest angular momentumWHY?

1268250282082

The Full Energy Level diagram of the SHELL MODEL242642864210Remember the occupancy of each level nlj is (2j+1) only depends on jThe parity of each level nlj is only depends on l +-++-+++--++-+PARITY-

Fermi energySingle Neutron and Proton Separation Energies