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Lectures 4 & 5
The end of the SEMF and the
nuclear shell model
Oct 2006, Lectures 4&5
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4.1 Overview
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4.2 Shortcomings of the SEMF
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4.3 The nuclear shell model
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Oct 2006, Lectures 4&5
magic numbers for N and Z
spin & parity of nuclei unexplained
magnetic moments of nuclei
value of nuclear density
values of the SEMF coefficients
choosing a potential
L*S coupling
Nuclear “Spin” and Parity
Shortfalls of the shell model
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4.2 Shortcomings of the SEMF
Oct 2006, Lectures 4&5
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4.2 Shortcomings of the SEMF
(magic numbers in Ebind/A)
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SEMF does not apply for A<20
There are systematic deviations from SEMF for A>20
(N,Z)
(10,10)
(6,6)
(2,2)
(8,8)
2*(2,2)
= Be(4,4)
Ea-a=94keV
Oct 2006, Lectures 4&5
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Z
4.2 Shortcomings
of the SEMF
Neutron Magic
Numbers
(magic numbers in numbers of
stable isotopes and isotones)
Proton
Magic
Numbers
• Magic Proton Numbers
(stable isotopes)
• Magic Neutron Numbers
(stable isotones)
N
Oct 2006, Lectures 4&5
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4.2 Shortcomings of the SEMF
(magic numbers in separation energies)
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Neutron separation
energies
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saw tooth from
pairing term
step down when N
goes across magic
number at 82
Oct 2006, Lectures 4&5
Ba Neutron separation energy in MeV
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4.2 Shortcomings of the SEMF
(abundances of elements in the solar system)
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Complex plot due to dynamics of creation, see
lecture on nucleosynthesis
Z=82
N=126
N=82
Z=50
N=50
iron mountain
no A=5 or 8
Oct 2006, Lectures 4&5
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4.2 Shortcomings of the SEMF
(other evidence for magic numbers, Isomers)
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Nuclei with N=magic have abnormally small n-capture cross
sections (they don’t like n’s) First excitation energy
Close to magic numbers
nuclei can have “long lived”
excited states (tg>O(10-6 s)
called “isomers”. One speaks
of “islands of isomerism”
[Don’t make holydays there!]
They show up as nuclei with
very large energies for their
first excited state (a nucleon
has to jump across a shell
closure)
Oct 2006, Lectures 4&5
208Pb
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4.2 Shortcomings of the SEMF
(others)
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spin & parity of nuclei do not fit into a drop
model
magnetic moments of nuclei are incompatible
with drops
actual value of nuclear density is unpredicted
values of the SEMF coefficients except
Coulomb and Asymmetry are completely
empirical
Oct 2006, Lectures 4&5
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4.2 Towards a nuclear shell model
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How to get to a quantum mechanical model of the
nucleus?
Can’t just solve the n-body problem because:
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we don’t know if a two body model makes sense (it does
not make much sense for a normal liquid drop)
if it did make sense we don’t know the two body potentials
(yet!)
and if we did, we could not even solve a three body
problem
But we can solve a two body problem!
Need simplifying assumptions
Oct 2006, Lectures 4&5
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4.3 The nuclear shell model
This section follows Williams, Chapters 8.1 to 8.4
Oct 2006, Lectures 4&5
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4.3 Making a shell model
(Assumptions)
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Assumptions:
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Each nucleon moves in an averaged potential
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Each nucleon moves in single particle orbit corresponding to its state in
the potential
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neutrons see average of all nucleon-nucleon nuclear interactions
protons see same as neutrons plus proton-proton electric repulsion
the two potentials for n and p are wells of some form (nucleons are
bound)
 We are making a single particle shell model
Q: why does this make sense if nucleus full of nucleons and typical mean
free paths of nuclear scattering projectiles = O(2fm)
A: Because nucleons are fermions and stack up. They can not loose energy
in collisions since there is no state to drop into after collision
Use Schroedinger Equation to compute Energies (i.e. non-relativistic),
justified by simple infinite square well energy estimates
Aim to get the correct magic numbers (shell closures) and be content
Oct 2006, Lectures 4&5
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4.3 Making a shell model
(without thinking, just compute)
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Try some potentials; motto: “Eat what you know”
desired
magic
numbers
126
82
50
28
20
8
2
Oct 2006,Coulomb
Lectures 4&5
infin. square
harmonic
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4.3 Making a shell model
(with thinking)
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We know how potential should look like!
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It must be of finite depth and …
If we have short range nucleon-nucleon potential then …
… the average potential must look like the density
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flat in the middle (you don’t know where the middle is if you are
surrounded by nucleons)
steep at the edge (due to short range nucleon-nucleon potential)
R ≈ Nuclear Radius
d ≈ width of the edge
Oct 2006, Lectures 4&5
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4.3 Making a shell model
(what to expect when rounding off a potential well)
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Higher L solutions get larger “angular momentum barrier” 
Higher L wave functions are “localised” at larger r and thus closer to “edge”
Rounding the edge
Radial Wavefunction U(r)=R(r)*r for the finite square well
affects high L states
most because they are
closer to the edge then
low L ones.
High L states drop in
energy because
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can now spill out
across the “edge”
this reduces their
curvature
which reduces their
energy
So high L states drop
rounding the well!!
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4.3 Making a
shell model
The “well improvement program”
(with thinking)
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Harmonic is bad
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Even realistic well
does not match
magic numbers
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Need more splitting
of high L states
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Include spin-orbit
coupling a’la atomic
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magnetic coupling
much too weak and
wrong sign
Two-nucleon
potential has nuclear
spin orbit term
deep in nucleus it
averages away
at the edge it has
biggest effect
the higher L the
bigger the split
Oct 2006, Lectures 4&5
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4.3 Making a shell model
(spin orbit terms)
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and using spherical symmetry gives:
r r 1 dV (r )
E = L gS
mr
Q: how does the spin orbit term look like?
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LS analogy to atomic physics:
r r r
r r
r r
E =  gB and B = v  E and E = V (r )
r
r r r
r
r
r
and  = gS and p = mv and L = r  p
dr
Spin S and orbital angular momentum L in our model are that of single
nucleon in the assumed average potential
In the middle the two-nucleon interactions average to a flat potential and the
two-nucleon spin-orbit terms average to zero
Reasonable to assume that the average spin-orbit term is strongest in the non
symmetric environment near the edge  W (r ) : 1 dV (r )
r dr
V (r )  V (r )  W (r )L gS
2
 h  1 dV (r )
Dimension: Length2
where: W (r ) = - VLS 

compensate 1/r * d/dr
m
c
   r dr
1
and VLS = VLS (E nucleon ) and V (r ) =
(Woods-Saxon)
r
a


1  exp 

 d 
with a = R 0 A
1
`3
Oct 2006, Lectures 4&5
and R 0  1.2 fm and d  0.75 fm = "thickness of edge"
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4.3 Making a shell model
(spin orbit terms)
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Good quantum numbers without LS term :
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With LS term need operators commuting with new H
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J=L+S & Jz=Lz+Sz with quantum numbers j, jz, l, s
Since s=½ one gets j=l+½ or j=l-½ (l≠0)
Giving eigenvalues of LS [ LS=(L+S)2-L2-S2 ]
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l, lz & s=½ , sz from operators L2, Lz, S2, Sz with
Eigenvalues of l(l+1)ħ2, s(s+1)ħ2, lzħ, szħ
½[j(j+1)-l(l+1)-s(s+1)]ħ2
So potential becomes:
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V(r) + ½l ħ2 W(r)
for j=l+½
V(r) - ½(l+1) ħ2 W(r)
for j=l -½
we can see this asymmetric splitting on slide 16
Oct 2006, Lectures 4&5
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4.3 Making a shell model
(fine print)
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There are of course two wells with different
potentials for n and p
We currently assume one well for all nuclei but …
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The shape of the well depends on the size of the nucleus
and this will shift energy levels as one adds more nucleons
Using a different well for each nucleus is too long
winded for us though perfectly doable
So lets not use this model to precisely predict exact
energy levels but to make magic numbers and …
Oct 2006, Lectures 4&5
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4.3 Predictions from the shell model
(total nuclear “spin” in ground states)
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Total nuclear angular momentum is called nuclear
spin = Jtot
Just a few empirical rules on how to add up all
nucleon J’s to give Jtot of the whole nucleus
Two identical nucleons occupying same level (same
n,j,l) couple their J’s to give J(pair)=0
Jtot(even-even ground states) = 0
Jtot(odd-A; i.e. one unpaired nucleon) = J(unpaired
nucleon) Carefull: Need to know which level nucleon
occupies. I.e. more or less accurate shell model wanted!
|Junpaired-n-Junpaired-p|<Jtot(odd-odd)< Junpaired-n+Junpaired-p
there is no rule on how to combine the two unpaired J’s
Oct 2006, Lectures 4&5
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4.3 Predictions from the shell model
(nuclear parity in groundstates)
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Parity of a compound system (nucleus):
P ( A -nucleon system) =
Pintrinsic (nucleon i )  P  i (nucleon i )

i
A
i
A
=1,
=1,
where P  i = (-1)l i and Pintrinsic (nucleon i ) = 1 
P ( A -nucleon system) =
( -1)l

i
A
i
=1,
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P(even-even groundstates) = +1 because all
levels occupied by two nucleons
P(odd-A groundstates) = P(unpaired nucleon)
No prediction for parity of odd-odd nuclei
Oct 2006, Lectures 4&5
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4.3 Shortcomings of the shell model
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The fact that we can not predict spin or parity for odd-odd
nuclei tells us that we do not have a very good model for the
LS interactions
A consequence of the above is that the shell model
predictions for nuclear magnetic moments are very imprecise
We can not predict accurate energy levels because:
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we only use one “well” to suit all nuclei
we ignore the fact that n and p should have separate wells of different
shape
As a consequence of the above we can not reliably predict
much (configuration, excitation energy) about excited states
other then an educated guess of the configuration of the
lowest excitation
Oct 2006, Lectures 4&5
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