the shell model of the nucleus 3. the full model [sec. 5.5 and 5.6 dunlap]
TRANSCRIPT
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 proton
BL
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
m
esB 2
22μMagnetic moment
CV = the central potential
B = the Bohr magnetron
Atomic Spin-Orbit Coupling
ldr
dV
ermcB C
.
112
s
m
e.
l.sdr
dV
rmc
cE
sldr
dV
rmc
c
ldr
dV
ermc
s
m
eBE
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.sdr
dV
rmc
cE C1
.)(2
)(22
2
values of l \cdot s
J = L +s ,
J^2 = L^2+ s^2 +2 Ls
Ls =[J(J+1)-L(L+1)-s(s+1)]/2
For s=1/2, J = L+1/2 or L-1/2
If J=L+1/2, Ls= L/2
If J=L-1/2, Ls= -(L+1)/2
Spin-Orbit Coupling is general
l.sl.s )(1.
)(2
)(atom22
2
rWdr
dV
rmc
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
dr
dV
r
CrW 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
S
S Single Neutron and Proton Separation Energies
A