The Shell Model of the Nucleus 3. The full model

1
The Shell Model of the Nucleus3. The full model

• Sec. 5.5 and 5.6 Dunlap

2
WOODS SAXON energy levels
92
22
2
10
58
Note 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.
18
40
6
34
14
20
2
10
8
6
2
2
3
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)
1963 Nobel Laureates
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Borrowing from Atomic Physics
In 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.
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Borrowing from Atomic Physics
Situation as seen from the proton
Situation as seen from the electron
B
L
s
Ze
e-
High energy configuration shown
?
Magnetic Interaction energy
Magnetic field B I/2r, with I the effective
current IZe/T, with T the period of the orbit.
The orbital angular momentum L mvr 2\pi
mr2/T. Note that B and L are pointing to the
same direction. So we have B ZeL/(mc2r3), in
CGS units, or B (L/mer c2) dV_c/dr, with V_c
-Ze2/r
the central potential
the Bohr magnetron
Magnetic moment
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Atomic Spin-Orbit Coupling
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
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values of l \cdot s

• J L s ,
• J2 L2 s2 2 Ls
• Ls J(J1)-L(L1)-s(s1)/2
• For s1/2, J L1/2 or L-1/2
• If JL1/2, Ls L/2
• If JL-1/2, Ls -(L1)/2

8
Spin-Orbit Coupling is general
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
To find the energy of the atom one must of course
take an average over the atom
Total splitting
Highest energy state
s
l
nl
Lowest energy state
In the case of an atom this splitting is normally
small 10-4 eV
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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.
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The spin-orbit effect comes from the nuclear
surface not the bulk
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.
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The spin orbit energy of the nucleus is of
opposite sign to the special relativity prediction
LOW ENERGY STATE
HIGH ENERGY STATE
STRETCH
JACKNIFE
nl
TOTAL SPLITTING
12
WOODS SAXON energy levels
92
22
2
10
58
18
Which Energy levels are going to be depressed
most?
40
6
34
14
20
2
10
The ones with the largest angular momentum WHY?
8
6
2
2
13
126
82
50
28
20
8
2
14

-

The Full Energy Level diagram of the SHELL
MODEL

-

10
2
-
4
6
Remember the occupancy of each level nlj is
(2j1) only depends on j

-
8

2
4

6
-
The parity of each level nlj is
only depends on l
2
-
4


2
PARITY
15
Single Neutron and Proton Separation Energies
Fermi energy