Lesson 5
The Shell Model
Why models?
• Nuclear force not known!
• What do we know about the nuclear force?
(chapter 5)
• It is an exchange force, mediated by the virtual
exchange
h
off gluons
l
or mesons.
Electromagnetic force
• Virtual exchange of photons
ΔE • Δt ≥ h
ΔE
Δt ≈
h
hc hc
R = cΔt ≈
=
ΔE Eγ
Nuclear Force
• Virtual exchange of particles of mass m
h
Δt ≤
mc 2
h
R≤
mc
R = 1.4 fm ⇒ m ≥ 140 MeV / c 2
Assuming nuclear force is short range saturation property of nuclear forces
Nuclear Force
• Strongly attractive
• Repulsive core
• Not spherically
h i ll symmetric
i (deuteron
(d
quadrupole moment), has symmetric central
component and
d asymmetric
i tensor component.
• Spin dependent (deuteron ground state is
triplet, singlet state is unbound)
Nuclear potential
Other potentials of note
Other potentials of note
Table 5-1
Charge independence of nuclear forces
The nuclear force between a neutron and a proton is the same as the force between
two protons or two neutrons.
Nucleus
Total Binding
Energy (MeV)
Coulomb Energy
(MeV)
Net nuclear binding
energy (MeV)
3H
-8.486
0
-8.486
3He
-7.723
0.829
-8.552
13C
-97.10
7.631
-104.734
13N
-94.10
10.683
-104.770
23Na
-186.54
23.13
-209.67
23Ne
-181.67
27.75
-209.42
41Ca
-350.53
65.91
-416.44
41Sc
-343 79
-343.79
72 84
72.84
-416 63
-416.63
A
3
13
23
41
Isospin
Consider that the neutron and the proton are just two states of the nucleon.
Consider further that these two states are labeled by a quantum number, T,
called isospin.
For the nucleon, T
T=1/2.
1/2. There are two projections of T in isospace,
T3=+1/2 (proton) and T3=-1/2 (neutron)
For a nucleus containing Z protons and N neutrons, T3=(Z-N)/2.
Example
Consider the A=14 isobars, 14C, 14N and 14O. 14C and 14O have T3=±1.
Thus they must be part of an isospin multiplet, T=1. In 14N, T3=0, but there
must be a state with T
T=1.
1. This state is called the isobaric analog of the ground
states of 14C and 14O.
Getting back to the shell model
Evidence for nuclear shell structure:
•Peaks in binding energy /nucleon
•Changes
Changes in separation energies for certain numbers of neutron and protons
in separation energies for certain numbers of neutron and protons
•Magic numbers (2,8,20,28,50,82,….)
Zero quadrupole moments
Low neutron absorption cross sections
Abundance systematics
More Examples
p
Spin-orbit
Spin
orbit coupling
More on spin-orbit
spin orbit force
Consider the Woods-Saxon potential
Aufbau Prinzip
1. Fill in lowest energy first
2. Pair up particles as you add them (“Katz’s rule”)
3. Spin and parity of the ground state is the spin and parity of the last
nucleon for odd A nuclei. For e‐e nuclei, J,π =0+. For o‐o nuclei, use
Brennan‐Bernstein rules.
Reality Check
Brennan-Bernstein
Brennan
Bernstein Rules
1. When the odd nucleons are both particles or holes in their respective
subshells, Rule 1 states that when j1 = ℓ1  ½ and j2 = ℓ2 ∓ ½, then J = ∣j1 – j2∣.
2. Rule 2 states that when j1 = ℓ1  ½ and j2 = ℓ2  ½, then J = ∣j1 j2∣
3. Rule 3 states that for configurations in which the odd nucleons are a
combination of particles and holes, such as 36Cl, J=j1 + j2 +1
Example
Consider the o-o nuclei, 38Cl, 26Al and 56Co. Predict the ground state spin and parity
for these nuclei.
(a) 38Cl has 17 protons and 21 neutrons. The last proton is in a d3/2 level while
the last neutron is in an f7/2 level. (Figure 6-3).
jp = 2 – 1/2, jn = 3 + 1/2
J = |7/2 – 3/2| = 2
π=–
(b) 26Al has 13 protons and 13 neutrons. The last proton and the last neutron
are in d5/2 hole states, i.e., jp = jn = 2 + 1/2.
J = |5/2 + 5/2| = 5
π=+
(c) 56Co has 27 protons and 29 neutrons. The last proton is in a f7/2 hole state
and the last neutron is in a p3/2 state (1 + 1/2)
J = 7/2 + 3/2 – 1 = 4
π=+
Statistics on B
B-B
B rule.
rule
Successes of Shell Model
•Ground state spins and parities of nuclei
•Some information about excited states
Magnetic Moments; Schmidt Limits
Focus on a single particle
μi = gℓLi + gsSi
gℓ = ℓμ0, gs = 5.5845μ
5 5845μ0 for protons
and
gℓ = 0, gs = - 3.8263μ0 for neutrons
where μ0 is the nuclear magneton:
μ0 = eħ/2mpc
Application to Nuclei; Schmidt limits
For j = ℓ + s , μ = ℓgℓ + ½ gs
For j = ℓ - s, μ = (j/j+1)[(ℓ + 1) gℓ – ½ gs]
Islands of Isomerism
What is a nuclear isomer?
What causes an isomer?
Where do they occur?
Mirror nuclei
Definition: nuclear-pairs in which the numbers of protons and neutrons
are interchanged, for example, 3He and 3H
Failures of the Shell Model
•Positions and origin of low lying 2+ states in nuclei
•Rotational and vibration levels in deformed nuclei, like the rare earths
and the actinides.