The Shell Model of the Nucleus

4. Applications – nuclear spin

[Sec. 6.1 and 6.3 Dunlap]

Switching to the nucleus

2/3 1/302/3 2

3 500.16 10%

2(9 ) 938SOE V MeV

lA lAE Mc MeV

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 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

-

Predicting nuclear spinsUNDERSTANDING j-j COUPLING

nlj nljjm j

jm j

1 jm jnlj

jm j

jm j One nucleon in a j-state

Two nucleons in same j-state

A B

Both the states A and B (and any other mj substate) is ok by Pauli Principle. But state B is the lower energy state – because the strong interaction averaged over space is maximized.

HIGHER ENERGY STATE

LOW ENERGY STATE

Predicting nuclear spinsSPIN OF EVEN- EVEN NUCLEI

nljjm j

nljjm j

jm j

One nucleon in a j-state Two nucleons in same j-state

CORROLARY: If a nucleus has an even number of neutrons – then these will couple to give spin zero

If a nucleus has an even number of protons – then these will couple to give spin zero

If a nucleus has N=EVEN, and Z=EVEN then J=0

Predicting nuclear spinsSPIN OF ODD- EVEN NUCLEI

nljjm j

nljjm j

jm j

One nucleon in a j-state 2 nucleons in same j-state

nljjm j

jm j

3 nucleons in same j-state

1 jm j

CORROLARY: Since all EVEN-EVEN configurations are J=0 states – it follows that any EVEN-ODD nucleus must be getting its spin J from the single unpaired nucleon

unpairedjJ

Neutron and Proton levels

The potential as seen by a proton is different – especially in large nuclei – there is a Coulomb “tail” outside the nucleus and a “bump” in the center.

This does not effect the lower energy levels up to magic (special) number 50.

8

22

8

20

20

28

50

50

82

Neutron and Proton levelsFor large nuclei (Z>50) there can be some subtle changes in the energy level sequencing due to the Coulomb potential.

parity

PREDICTING NUCLEAR SPINS

Predicting nuclear spins

Nuclide J comment

0+ stable

105y E.C.

0+ stable

stable

0+ stable

165d -

0+ stable

4.5d -

0+ stable

Ca4020

Ca4120

Ca4220

Ca4320

Ca4420

Ca4520

Ca4620

Ca4720

Ca4820

2

7

2

7

2

7

2

7

PREDICTING NUCLEAR SPIN – Lead isotopes

L=4

PREDICTING NUCLEAR SPIN – Lead isotopes

Excited states of 41Ca

1?

Excited states of 17O and 17F

NOTE: As far as the strong force is concerned these two nuclei are THE SAME!

The strong force does not distinguish n and p. But the EM force does.