Shell Model
Shortcomings of the SEMF „
• magic numbers for N and Z „
• spin & parity of nuclei
• unexplained „
magnetic moments of nuclei „
• value of nuclear density „
The nuclear shell model „
• General discussion
• Assumptions
• choosing a potential „
• L*S coupling „
• Nuclear “Spin” and Parity „
Shortfalls of the shell model
Magic Numbers

It is based on the observation that the semi empirical mass formula, which
describes the nuclear masses quite well on average, fails for certain “magic
numbers”. The nuclei whose proton number Z or neutron number N is one of
the magic numbers 2, 8, 20, 28, 50, 82, 126 are stable and they correspond
to the noble gases in the atomic world.

The doubly magic nuclei, those with both magic proton and magic neutron
numbers, are exceptionally stable. These are the following nuclides:

Further these magic number nuclei have higher abundance in nature. For example,
helium-4 is among the most abundant (and stable) nuclei in the universe.

Magic nuclei have many more stable isotopes and isotones than their neighbors
do.

Nuclei with N = magic number have much lower neutron absorption crosssections than surrounding isotopes.

These nuclei appear to be perfectly spherical in shape; they have zero quadruple
electric moments.

Magic number nuclei have higher first excitation energy.

Energy of alpha or beta particles emitted by magic numbered radioactive nuclei is
larger than that from other nuclei.
analogous to the atomic shell
ment of electrons in an atom, in
tability.
based on the Pauli exclusion
f the nucleus in terms of energy
tral potential around which the
ucleus.
h increase with energy that orbit
o describe many phenomena like
e spin and parity etc..
In atoms you know you have shells, major shell
chemistry is
based on that you have inert gases or noble gas
shells are
closed and then when one more electron you p
table and
new period is starts right. So, you have that hyd
that
right, below that you have sodium, potassium.
which a new
shell starts and there are many things common
So, there are many of the properties are comm
shell and
one more electron a new shell has just started.
•
The idea of the nuclear shell model had been proposed on the analogy of the atomic
shell structure. However, there was some doubt on this idea because there were
differences between the cases of atoms and nuclei. The differences are as follows:
•
In the case of an atom, there is a heavy nucleus at the central position of the
system, and it attracts light electrons around by the Coulomb interaction. In
contrast to this, there is nothing special at the nuclear center.
•
In the case of a nucleus, it is clear that the whole system can be described in the
image of a "raindrop".
Namely, the liquid drop model is valid for
nuclei. However, such an image is not valid for atoms.
•
Moreover, the biggest problem is the difference in the magic numbers of both the
systems, atoms and nuclei. Although the magic numbers in atoms are Z = 2, 10,
18, 36, 54, 86, the magic numbers in nuclei are Z or N = 2, 8, 20, 28, 50, 82,
126.
•
There is a big difference between these two kinds of systems, atoms and
nuclei. Even if the shell structure is held in both the systems, the average
potentials are presumed to be considerably different.
Basic assumption of Shell Model:
•
•
•
•
Each nucleon moves in potential: average of all other nucleons
Net effect of nuclear motions makes potential vary smoothly
Each nucleon is bound --» potential is a potential well type
Each nucleon moves in 'orbit' of that potential well
•
The basic assumption of the nuclear shell-model relies on the point that each
nucleon in a nucleus moves in an independent fashion under an average potential.
•
The first step will be to identify a suitable average potential for the nucleons.
•
Each nucleon moves independently in the nucleus uninfluenced by the motion of
the other nucleons.
•
Each nucleon moves in a potential well which is constant from the center of the
nucleus to its edge where it increases rapidly by several tens of MeV.
Choosing a Potential
•
•
•
According to this model, a given nucleon moves in an effective single particle
attractive potential produced by all other nucleons, so the first step is to identify a
suitable average potential for the nucleons.
Ideally, we would write down the Schrodinger equation for the nuclear force
potential and solve it to calculate the energy levels, as done for the hydrogen atom
in QM.
In the beginning the spin of the nucleon was neglected and therefore we will start in
spherical coordinates with a Hamiltonian corresponding to a spinless particle
moving in the central field U(r), i. e.
•
The eigenfunction of this Hamiltonian is a product of radial and angular coordinates
•
total eigenfunction
•
the eigenvalue problem acquires the simpler, one dimensional, form,
Harmonic Oscillator
The eigenvalues corresponding to this potential are
Energy depends only upon the number N = 2n+l where it is integer and N ≥ 0.
 Since ħω is a free parameter that defines the Harmonic oscillator potential, it can be
chosen large enough so that there are in the spectrum a series of bands, each band
labelled by the number N. The gap between bands was found to be related to the
mass number A by the relation,
 Each value of N implies several values of (n, l)
 The set of (n, l) values corresponding to a particular value of N would be clustered
together in bands separated by an energy ħω.
Harmonic oscillator… Continue
N=2n+l
n
l
Enl
Label
Label as
Degenercy
D=2(2l + 1)
∑D
n
0 1 2 3 4
1 2 3 4 5
l
0 1 2 3 4
s p d f g
0
0
0
3/2 ħω
1s
2
2
1
0
1
5/2 ħω
1p
6
8
2
0
1
2
0
7/2 ħω
1d
2s
10
2
20
3
0
1
3
1
9/2 ħω
1f
2p
14
6
40
4
0
1
2
4
2
0
11/2 ħω
1g
2d
3s
18
10
2
70
5
0
1
2
5
3
1
13/2 ħω
1h
2f
3p
22
14
6
112
Table: Energy levels corresponding to an Harmonic
oscillator potential of frequency ω
Harmonic oscillator… Continue
 Figure shows the Shell structure obtained
with Harmonic oscillator potential
 Capacity of each level is indicated of its
right
 Large Gaps occurs between the levels,
which we associate with shell closure.
 Circled numbers shows the total number
of nucleons at each shell closure.
 Neutrons and protons, being non-identical
particles are counted separately so 1s state
can holds 2 protons as well as 2 neutrons.
 The magic numbers of 2, 8, 20 emerged
but the higher levels do not corresponds at
all to the observed magic number.
Infinite Square Well Potential
•
•
•
Square well potential more closely approximates
the nuclear density, so it can be a better choice for
the mean potential.
Infinite square well in 1 dimension: a quick
reminder Particle trapped between x=0 and x=a
walls are impenetrable => ψ = 0 for x<0 and x>a
Potential energy assumed to be zero when
particle is inside the walls
 The harmonic oscillator potential as well
as the infinite well potential predict the
first few magic numbers.
Wood Saxon Potential… An Intermediate Form
The infinite square well and HO are not good
approximation to the nuclear potential for
several reasons:
 To separate a neutron or proton, we must
supply enough energy to take it out of these
potentials: infinite separation energy is
impossible.
 In addition, these potentials do not have a
sharp edge, but rather closely approximates
the nuclear charge and matter distribution,
falling smoothly to zero beyond the mean
radius R.
We choose a intermediate form:
V r  
Vo
1 expr  R/ a 
 The parameters R and a give, respectively,
the mean radius and skin thickness => R =
1.25A1/3and a = 0.524 fm
 The well depth V0 is adjusted to give the
proper separation energies and is of order
50MeV.
 This potential is called Wood Saxon
potential.
Sketch of the functional form of three popular phenomenological shell
model potentials: Woods-Saxon, harmonic oscillator, and the square well.
Continue…
Intermediate
Form
Intermediate potential: Filling the shells in order with 2(2l+1) nucleons, we again get the
magic numbers 2, 8, 20 but the higher numbers do not emerge from calculations.
Spin-Orbit potential
How we can modify the potential to give proper magic numbers?
 Inclusion of spin-orbit potential
Spin-orbit potential
 
J2  J J
  
J  Ls
 
   
2
J  L  L  2L  s  s  s
 
2
2
2
J  L  s  2L  s
  J 2  L2  s 2
Ls 
2
Ls 
𝑉𝑠𝑜 𝑟 𝐿. 𝑠
J 2  j  j  1
2
  1
2
s 2  s s  1
2
L2 
Q.M.
1
j  j  1 

2
  1  s s  1
2
Equation (1)
𝟏
𝟏
s= ½ and Possible values of
j are 𝒋 = 𝒍 + 𝟐 𝒐𝒓 𝒋 = 𝒍 − 𝟐
𝟏

Put s= ½ and 𝒋 = 𝒍 + 𝟐 in equation (1) will give < 𝐿. 𝑠 > =
𝟏
ħ2
𝑙
2
Put s= ½ and 𝒋 = 𝒍 − 𝟐 in equation (1) will give < 𝐿. 𝑠 > = −
ħ2
(𝑙
2
+ 1)
• Consider a level such as 1f (l=3) which has
degeneracy of 2(2l+1)=14, possible j values
1
5
7
are 𝑗 = 𝑙 ± 2 = 2 𝑜𝑟 2
• So levels are 1𝑓5/2 and 1𝑓7/2
 The degeneracy of each level is 2j+1
• The capacity of 1𝑓5/2 state is 6 and 1𝑓7/2
state is 8. Again total 14.
• 1𝑓5/2 and 1𝑓7/2 are known as spin-orbit
doublets which has some energy separation.
 For pair of states we can find energy
separation as
𝟏
𝑳. 𝒔
−
𝑳.
𝒔
=
𝟐𝒍 + 𝟏 ħ𝟐
𝟏
𝟏
𝒋=𝒍+𝟐
𝒋=𝒍−𝟐
𝟐
 The energy splitting increases with l
 The effect of Vso(r) is negative sign so in the
spin-orbit doublet larger j is pushed
downward.
𝑉𝑠𝑜 𝑟 𝐿. 𝑠
To Understand….
1g9/2 appears close to
the lower major level
so its capacity (10
nucleons) will be
added with the lower
levels. It will give
magic number 50
1f7/2 appears in between
2nd and 3rd shell…
(same gap) and it gives
the magic number 28
A similar effect occurs at the top of each major shells so we will get all magic numbers.
Next Lecture
In the next lecture we will use shell model to
predict the properties (e.g., parity, magnetic
dipole moments) of nuclei…….
Thank You