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
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
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