Transcript Applications - Nuclear Spin

The Shell Model of the Nucleus
4. Applications – nuclear spin
[Sec. 6.1 and 6.3 Dunlap]
Switching to the nucleus
ESO
3
50MeV
2/3 V0
1/3

lA
 0.16  lA 
 10%
2/3
2
E
2(9 )
Mc
938MeV
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
Remember the
occupancy of each
level nlj is (2j+1) –
only depends on j
The parity of each
level nlj is (1) l
only depends on l
+
+
+
+
-
10
2
4
6
8
+
+
-
+
PARITY
2
4
6
-
2
4
+
2
Predicting nuclear spins
UNDERSTANDING j-j COUPLING
mj  j
HIGHER ENERGY
STATE
mj  j
 nlj
One nucleon in a j-state
m j  j 1
 nlj
LOW ENERGY STATE
mj  j
 nlj
mj   j
A
B
Two nucleons in same j-state
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.
Predicting nuclear spins
SPIN OF EVEN- EVEN NUCLEI
mj  j
mj  j
 nlj
 nlj
mj   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 spins
SPIN OF ODD- EVEN NUCLEI
mj  j
mj  j
mj  j
 nlj
 nlj
mj   j
One nucleon in a j-state
2 nucleons in same j-state
 nlj
m j  j 1
mj   j
3 nucleons in same j-state
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
J  junpaired
Neutron and Proton levels
82
50
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.
50
28
20
20
8
8
2
2
Neutron and Proton levels
For 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
40
20
Ca
41
20
Ca
42
20
Ca
43
20
Ca
44
20
Ca
45
20
Ca
46
20
47
20
J
0+
7
2

0+
7
2

0+
Ca
7
2
Ca
105y E.C.
stable
stable
Ca
48
20
stable

0+
7
2
comment

0+
stable
165d stable
4.5d -
stable
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.