Nuclear Physics
Instructor: Golam Dastegir Al-Quaderi
Nuclear Shell Model
• Both the infinite square-well potential and the
harmonic oscillator potentials reproduce a few of the
magic numbers.
• Recall that the nuclear magic numbers are:
𝑁𝑁 = 2, 8, 20, 28, 50, 82, 126
𝑍𝑍 = 2, 8, 20, 28, 50, 82
• Infinite square-well potential gives closed shells at 𝑁𝑁 or
𝑍𝑍 equal to 2,8,20,34,40,58, …
• Harmonic oscillator potential gives closed shells for
proton or neutron numbers of 2, 8, 20, 40, 70, etc.
Nuclear Shell Model
• Shell
structures:
• The capacity
of each level
is indicated to
its right.
• The circled
numbers
indicate the total
number of
nucleons.
Nuclear Shell Model
• How can we modify the potential to give the
proper magic numbers?
• We certainly cannot make a radical change in the
potential, because we do not want to destroy the
physical content of the model.
• It was fairly clear by the 1940s that a central
potential could not reproduce all the magic
numbers.
• Many unsuccessful attempts were made at
finding the needed correction.
Nuclear Shell Model
• The crucial breakthrough came in 1949 when Maria
Goeppert Mayer and Hans Jensen suggested - once
again following the lead from atomic physics - that
inside the nucleus, in addition to the central potential,
there is a strong spin-orbit interaction.
• The inclusion of a spin-orbit potential could give the
proper separation of the subshells.
• The spin-orbit interaction occurs because of the
electromagnetic interaction of the electron's magnetic
moment (spin part) with the magnetic field generated
by its motion about the nucleus (orbital part).
Nuclear Shell Model
• In atomic physics, a spin-orbit interaction splits
the two degenerate 𝑗𝑗 = 𝑙𝑙 ± 1/2 energy levels
and produces a fine structure.
• The effects are typically very small, one part in
105 in the spacing of atomic levels (hence the
name fine).
• No such electromagnetic interaction would be
strong enough to give the substantial changes in
the nuclear level spacing needed to generate the
observed magic numbers.
Nuclear Shell Model
• We adopt the concept of a nuclear spin-orbit
force of the same form as the atomic spin-orbit
force but certainly not electromagnetic in origin.
• The total potential sensed by a nucleon has the
form:
𝑉𝑉𝑡𝑡𝑡𝑡𝑡𝑡 𝑟𝑟 = 𝑉𝑉 𝑟𝑟 − 𝑓𝑓 𝑟𝑟 𝐿𝐿. 𝑆𝑆⃗
where 𝐿𝐿 and 𝑆𝑆⃗ are the orbital and the spin
angular momentum operators for a nucleon,
𝑓𝑓(𝑟𝑟) is an arbitrary function of the radial
coordinates
Nuclear Shell Model
• The sign of this interaction must be chosen to be
consistent with the data, so that the state with
𝑗𝑗 = 𝑙𝑙 + 1/2 can have a lower energy than the
state with 𝑗𝑗 = 𝑙𝑙 − 1/2 which is opposite to what
happens in atoms.
• Total angular momentum is: 𝐽𝐽⃗ = 𝐿𝐿 + 𝑆𝑆⃗ and
hence, 𝐽𝐽⃗. 𝐽𝐽⃗ = 𝐿𝐿2 + 𝑆𝑆⃗2 + 2𝐿𝐿. 𝑆𝑆⃗
1 2
𝐿𝐿. 𝑆𝑆⃗ = 𝐽𝐽⃗ − 𝐿𝐿2 − 𝑆𝑆⃗2
2
where we use: 𝐿𝐿, 𝑆𝑆⃗ = 0.
Nuclear Shell Model
• A quantum state can thus be labelled by :
𝑙𝑙, 𝑚𝑚𝑙𝑙 , 𝑠𝑠, 𝑚𝑚𝑠𝑠
𝑙𝑙, 𝑠𝑠, 𝑗𝑗, 𝑚𝑚𝑗𝑗
where we use the second choice of bases.
• Now, 𝐿𝐿. 𝑆𝑆⃗ =
1
2
𝐽𝐽⃗2 − 𝐿𝐿2 − 𝑆𝑆⃗2
ℏ2
=
𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 − 𝑠𝑠(𝑠𝑠 + 1)
2
ℏ2
3
=
𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 −
4
2
ℏ2 /2
for 𝑗𝑗 = 𝑙𝑙 + 1/2
= � ℏ2
− (𝑙𝑙 + 1) for 𝑗𝑗 = 𝑙𝑙 − 1/2
2
Nuclear Shell Model
• The change of energies from their degenerate values
are:
𝑛𝑛𝑛𝑛𝑛𝑛𝑚𝑚𝑗𝑗 𝑉𝑉�𝑆𝑆𝑆𝑆 𝑛𝑛𝑛𝑛𝑛𝑛𝑚𝑚𝑗𝑗
2
3
ℏ
2
= 𝑓𝑓 𝑟𝑟 𝑛𝑛𝑛𝑛
𝑗𝑗 𝑗𝑗 + 1 − 𝑙𝑙 𝑙𝑙 + 1 −
4
2
ℏ2 /2
for 𝑗𝑗 = 𝑙𝑙 + 1/2
= � 𝑑𝑑3 𝑟𝑟 𝜓𝜓𝑛𝑛𝑛𝑛 𝑟𝑟⃗ 2 𝑓𝑓(𝑟𝑟) � ℏ2
− (𝑙𝑙 + 1) for 𝑗𝑗 = 𝑙𝑙 − 1/2
2
Nuclear Shell Model
• The total splitting between the two levels
becomes:
1
1
− Δ𝐸𝐸𝑛𝑛𝑛𝑛 𝑗𝑗 = 𝑙𝑙 +
Δ𝑆𝑆𝑆𝑆 = Δ𝐸𝐸𝑛𝑛𝑛𝑛 𝑗𝑗 = 𝑙𝑙 −
2
2
1
2
= ℏ 𝑙𝑙 +
� 𝑑𝑑 3 𝑟𝑟 𝜓𝜓𝑛𝑛𝑛𝑛 𝑟𝑟⃗ 2 𝑓𝑓(𝑟𝑟)
2
• Hence, the splitting due to the spin-orbit
interaction is larger for higher values of orbital
angular momentum.
Nuclear Shell Model
• Level Crossing: As the splitting between previously
degenerate energy levels due to spin-orbit coupling
increases as the orbital angular momentum increases,
we can have level crossing.
• Energy level with lower 𝑙𝑙 can be split to cross the
energy level with higher angular momentum level.
• For large 𝑙𝑙 , the splitting of any two neighboring
1
degenerate levels can shift the 𝑗𝑗 = 𝑙𝑙 − state of the
2
1
2
initially lower level to lie above the 𝑗𝑗 = 𝑙𝑙 + state of
the previously higher level.
Nuclear Shell Model
• For an appropriately chosen 𝑓𝑓(𝑟𝑟), the energy
levels for a central potential (like the infinite
square-well potential or the harmonic
oscillator potential) can split upon the
addition of a spin-orbit interaction.
• Including a spin-orbit interaction we can
reproduce the desired magic numbers, and
thereby accommodate a shell-like structure in
the nuclei.
Nuclear Shell Model
• The energy levels are labeled according to the
spectroscopic notation of atomic physics, as:
𝑛𝑛𝐿𝐿𝑗𝑗
1𝑆𝑆1 , 1𝑃𝑃1/2 , 1𝑃𝑃3/2 , 1𝐷𝐷5/2 , 2𝑆𝑆1 , …
2
2
• The multiplicity of any final level is given, as
usual, by (2𝑗𝑗 + 1).
• Adding spin-orbit interaction can thus have
energy level splitting of both the infinite squarewell potential as well as that of the harmonic
oscillator potential.
Nuclear Shell Model
• Infinite square-well +spin-orbit coupling:
Nuclear Shell Model
• HO+SO:
Nuclear Shell Model
• The predicted magic numbers correspond to
orbitals with a large gap separating them from
the next highest orbital.
• For the lowest levels, the spin-orbit splitting is
sufficiently small that the original magic
numbers, 2, 8, and 20, are retained.
• For the higher levels, the splitting becomes
important and the gaps now appear at the
numbers 28, 50, 82 and 126.
Predictions of the Shell Model
• 1. Magic Numbers: The shell model correctly
predicts the magic numbers of the nuclei.
• 2. Spin-Parity: The shell model also correctly
predicts the spins and parities of many nuclear
states.
• According to the model, the proton and the
neutron levels fill up independently and, due to
the Pauli exclusion principle, only two neutrons
or two protons can occupy any given level (with
their intrinsic spins anti-parallel).
Predictions of the Shell Model
• Nucleons pair off in every filled level, yielding
zero total angular momentum.
• Thus the last unpaired nucleon must
determine the spin-parity of the ground state.
• As a result, the ground states of even–even
nuclei are expected to be 0+ because all
nucleons are paired with a partner of opposite
angular momentum (i.e. the ground states of
all even-even nuclei have zero spin).
Predictions of the Shell Model
• The ground states of odd–even nuclei should
then take the quantum numbers of the one
unpaired nucleon.
• Example-1: 17𝐹𝐹8 ( 17𝐹𝐹 9 ) and 17𝑂𝑂9 ( 17𝑂𝑂8 ) have one
unpaired nucleon outside a doubly magic 16𝑂𝑂8
core.
• The nine neutrons for 17𝑂𝑂8 and nine protons for
17 9
𝐹𝐹 will fill the following levels
1𝑆𝑆1 2 , 1𝑃𝑃3 4 , 1𝑃𝑃1 2 , 1𝐷𝐷5
2
2
2
2
1
Predictions of the Shell Model
• The total angular momentum of the last unpaired
nucleon, is in 𝑙𝑙 = 2, 𝑗𝑗 = 5/2.
5+
2
• The spin parity of the nucleus is predicted to be
since the parity is −1 𝑙𝑙 .
• This agrees with observation.
• Example-2:The first excited states of 17𝐹𝐹8 and 17𝑂𝑂9 ,
corresponding to raising the unpaired nucleon to the
next higher orbital, are predicted to be
1+
.
2
• Once again in agreement with observation.
Predictions of the Shell Model
• Example-3: Consider the isobars 13𝐶𝐶 6 and 13𝑁𝑁7 . (Note
that these are, in fact, mirror nuclei.)
• The six protons in 12𝐶𝐶 and the six neutrons in 13𝑁𝑁
should be completely paired off.
• The remaining seven nucleons in both cases should fill
the following shells:
1𝑆𝑆1 2 , 1𝑃𝑃3 4 , 1𝑃𝑃1
2
2
2
1
• The last unpaired nucleon - a neutron for 13𝐶𝐶 6 and a
1
13
7
proton for 𝑁𝑁 - has total angular momentum 𝑗𝑗 =
2
and orbital angular momentum 𝑙𝑙 = 1.
Predictions of the Shell Model
• Hence, according to the shell model, the spin-parity of
1−
the ground state for these nuclei is expected to be .
• Example-4: For
2
33𝑆𝑆 ,
16
the measured value of the
, 1𝑃𝑃
2
ground state spin-parity is
3 +
.
2
• According to the shell model, the seventeen neutrons
will fill up the levels as follows:
1𝑆𝑆
1
2
2
, 1𝑃𝑃
3
2
4
1
2
6
, 1𝐷𝐷5 , 2𝑆𝑆
2
1
2
2
, 1𝐷𝐷
3
2
1
once again leading to a prediction consistent with
experiment
Predictions of the Shell Model
• Example-5: 15𝑁𝑁7 and 15𝑂𝑂7 have one “hole” in
their 16𝑂𝑂8 core.
• The ground state quantum numbers should
then be the quantum numbers of the hole
1
which are 𝑙𝑙 = 1, 𝑗𝑗 = .
2
• The quantum numbers of the ground state are
1−
then predicted to be , in agreement with
2
observation.
Predictions of the Shell Model
• 3. Nuclear Magnetic Moment: The shell model
also makes predictions for nuclear magnetic
moments.
• As for the total angular momentum, the magnetic
moments results from a combination of the spin
and orbital angular momentum.
• The weighting is different because the
gyromagnetic ratio of the spin differs from that of
the orbital angular momentum.
Predictions of the Shell Model
• As measurements show, the proton and the
neutron have intrinsic dipole moments of
2.79 𝜇𝜇𝑁𝑁 and −1.91𝜇𝜇𝑁𝑁 , respectively.
• Thus, we expect the intrinsic magnetic moment
of any unpaired nucleon to contribute to the total
magnetic moment of the nucleus.
• In addition, since protons are charged, the orbital
motion of any unpaired proton can also
contribute to the magnetic moment of the
nucleus.
Predictions of the Shell Model
• Example-1: For the deuteron, for example, if we
assume that the proton and the neutron are in
1𝑆𝑆1/2 states.
• Then without orbital angular momentum for the
proton 𝑙𝑙 = 0 , we expect the magnetic
moment of the deuteron to be the sum of the
intrinsic dipole moments of the proton and the
neutron:
𝜇𝜇𝑑𝑑 = 2.79 𝜇𝜇𝑁𝑁 − 1.91 𝜇𝜇𝑁𝑁 = 0.88 𝜇𝜇𝑁𝑁
• The observed magnetic moment of the deuteron
is 0.88 𝜇𝜇𝑁𝑁 - in good agreement with expectation.
Predictions of the Shell Model
• Example-2: The nucleus of tritium 3𝐻𝐻1 has two
neutrons and one proton, all in the 1𝑆𝑆1/2 state.
• Since the neutrons are paired, they should not
contribute to the magnetic moment.
• The unpaired proton, having 𝑙𝑙 = 0, will have no
contribution from its orbital motion.
• Consequently, the total magnetic moment of 3𝐻𝐻1
should be the same as that of the unpaired
proton i.e. 2.79 𝜇𝜇𝑁𝑁 .
• The measured value is 2.98 𝜇𝜇𝑁𝑁 .
Predictions of the Shell Model
• Example-3: For 3𝐻𝐻𝐻𝐻 2 , the unpaired nucleon is a
neutron in a 1𝑆𝑆1/2 state.
• Consequently, the total magnetic moment should
be the same as that of the neutron, −1.91 𝜇𝜇𝑁𝑁
(the measured value is −2.13 𝜇𝜇𝑁𝑁 ).
• Example-4: 4𝐻𝐻𝐻𝐻 2 (𝛼𝛼-particle) has a closed shell
structure (in fact, it is doubly magic).
• The shell model would therefore predict no spin
and no magnetic moment, which is indeed
experimentally correct.
Predictions of the Shell Model
• Example-5: In 10𝐵𝐵 5 , the five protons and the five
neutrons have the same level structure, namely,
2
1𝑆𝑆1 , 1𝑃𝑃3
2
2
3
• Thus, there is one unpaired proton and one
unpaired neutron.
• The unpaired proton will be in an 𝑙𝑙 = 1 state, and
therefore the orbital motion will contribute:
2ℏ
𝜇𝜇 =
𝑙𝑙 = 𝜇𝜇𝑁𝑁
2𝑚𝑚𝑁𝑁 𝑐𝑐
Predictions of the Shell Model
• Thus the total magnetic moment will be:
2.79 𝜇𝜇𝑁𝑁 − 1.91 𝜇𝜇𝑁𝑁 + 𝜇𝜇𝑁𝑁 = 1.88 𝜇𝜇𝑁𝑁
comparing quite well with the measured value of
1.80 𝜇𝜇𝑁𝑁 .
• We see therefore that the shell model, in addition
to providing the known magic numbers, also
describes other important properties of light
nuclei.
• For heavy nuclei, however, there is marked
difference between the predictions of the shell
model and the measured quantities.