The Shell Model of the Nucleus

3. The full model

[Sec. 5.5 and 5.6 Dunlap]

2

8

20

34

40

58Note 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

2

6

10

2

14

6

18

102

22 92

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)

slsl ˆ.ˆ)()(),,( rWrVrV WS

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 Physics

e-

Ze

Situation as seen from the protonBL

s

Βμ.E

Situation as seen from the electron

High energy configuration shownMagnetic 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

s

mes

B 222μMagnetic moment

CV = the central potential

B = the Bohr magnetron

Atomic Spin-Orbit Coupling

ldrdV

ermcB C

.11

2

sme .

l.sdrdVrmc

cE

sldrdVrmc

c

ldrdV

ermcs

meBE

C

C

C

1.)()(

/./1.)()(E

.11...

22

2

22

2

2

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:

l.sdrdVrmc

cE C1.)(2)(

22

2

values of l \cdot s

J = 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 general

l.sl.s )(1.)(2)(

atom22

2

rWdrdVrmc

cE CSO

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 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 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:

l.sl.s nlnlnlSO rdrRrWrRE 3atom )()()(

nl lnl .21

)1(21 lnl

In the case of an atom this splitting is normally small ~ 10-4 eV

Total splitting = )12(21 lnl

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

drdV

rCrW WS)(

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

21lj 2

1lj

)1(221 l

l221

TOTAL SPLITTING = )12.(21 l

21lj

21lj

nl

2

8

20

34

40

58

WOODS – SAXON energy levels

2

6

10

2

14

6

18

102

22 92

Which Energy levels are going to be depressed most?

The ones with the largest angular momentum

WHY?

126

82

50

28

20

8

2

The Full Energy Level diagram of the SHELL MODEL

2

42

6

42

8

642

10

Remember the occupancy of each level nlj is (2j+1) –only depends on j

The parity of each level nlj is only depends on l

l)1(

+

-

+

+

-

+

++

--

+

+

-+

PARITY

-

Fermi energy

p

n

SS Single Neutron and Proton Separation Energies

A