1.4 Shell Model
3
∗ level 0: 2 states (l = 0) = 2
∗ level 1: 6 states (l = 1) = 6
∗ level 2: 2 states (l = 0) + 10 states (l = 2) = 12
∗ level 3: 6 states (l = 1) + 14 states (l = 3) = 20
∗ level 4: 2 states (l = 0) + 10 states (l = 2) + 18 states (l = 4) = 30
∗ level 5: 6 states (l = 1) + 14 states (l = 3) + 22 states (l = 5) = 42
If nucleons are filled in this fashion for all possible shells, it is observed that some
shells contain some specific number of nucleons as shown here:
2
8 = 2 + 6
20 = 2 + 6 + 12
28 = 2 + 6 + 12 + 8
50 = 2 + 6 + 12 + 8 + 22
82 = 2 + 6 + 12 + 8 + 22 + 32
126 = 2 + 6 + 12 + 8 + 22 + 32 + 44
184 = 2 + 6 + 12 + 8 + 22 + 32 + 44 + 58
If protons and neutrons are filled in this fashion, it is observed that those nuclei
which have fully occupied shells will have the number of nucleons as 2, 8, 20, 28,
50, 82, 126, and 184. It is observed that the nucleus containing these specific number
of nucleons are highly stable compared to those which have one less or one more
nucleon. This number is recognized as Magic Number.
The shell model thus predicts the possibility of new isotopes which have not yet
been discovered like isotopes of magic numbers 184 and 126. It is observed that
nuclei containing even numbers of protons and neutrons are more stable than those
with odd numbers, which is due to the pairing effect. Moreover, nuclei which have
both neutron and proton numbers equal to one of the magic numbers are called
doubly magic, and are very stable. For example, Calcium (
40 Ca 20 ,
48 Ca 20 ) is a good
example to explain doubly magic number nuclei.
The shell model, thus, has given a better understanding of why some isotopes are
more stable than others and why a large number of isotopes are found with some
specific nucleon number. The shell model has been able to explain some properties
of a nucleus as enumerated here:
1. Large number of isotopes are found for nuclei of a magic number.
2. All naturally occurring radioactive series like Uranium 238 end with a stable
isotope Pb 206, and all of them belong to the magic number of neutrons or
protons.
3. Every isotope of nuclei has some specific values for the cross-section of neutron
absorption. Those isotopes belonging to magic numbers have a lower cross-section
than their surrounding isotopes.
4. Isotopes containing the last neutron of the neutron magic number nuclei have the
maximum binding energy. If one more neutron is added to it then the binding
energy of the isotope drops down sharply.
5. Nuclei of a magic number have nearly zero electric quadruple moments.
3
∗ level 0: 2 states (l = 0) = 2
∗ level 1: 6 states (l = 1) = 6
∗ level 2: 2 states (l = 0) + 10 states (l = 2) = 12
∗ level 3: 6 states (l = 1) + 14 states (l = 3) = 20
∗ level 4: 2 states (l = 0) + 10 states (l = 2) + 18 states (l = 4) = 30
∗ level 5: 6 states (l = 1) + 14 states (l = 3) + 22 states (l = 5) = 42
If nucleons are filled in this fashion for all possible shells, it is observed that some
shells contain some specific number of nucleons as shown here:
2
8 = 2 + 6
20 = 2 + 6 + 12
28 = 2 + 6 + 12 + 8
50 = 2 + 6 + 12 + 8 + 22
82 = 2 + 6 + 12 + 8 + 22 + 32
126 = 2 + 6 + 12 + 8 + 22 + 32 + 44
184 = 2 + 6 + 12 + 8 + 22 + 32 + 44 + 58
If protons and neutrons are filled in this fashion, it is observed that those nuclei
which have fully occupied shells will have the number of nucleons as 2, 8, 20, 28,
50, 82, 126, and 184. It is observed that the nucleus containing these specific number
of nucleons are highly stable compared to those which have one less or one more
nucleon. This number is recognized as Magic Number.
The shell model thus predicts the possibility of new isotopes which have not yet
been discovered like isotopes of magic numbers 184 and 126. It is observed that
nuclei containing even numbers of protons and neutrons are more stable than those
with odd numbers, which is due to the pairing effect. Moreover, nuclei which have
both neutron and proton numbers equal to one of the magic numbers are called
doubly magic, and are very stable. For example, Calcium (
40 Ca 20 ,
48 Ca 20 ) is a good
example to explain doubly magic number nuclei.
The shell model, thus, has given a better understanding of why some isotopes are
more stable than others and why a large number of isotopes are found with some
specific nucleon number. The shell model has been able to explain some properties
of a nucleus as enumerated here:
1. Large number of isotopes are found for nuclei of a magic number.
2. All naturally occurring radioactive series like Uranium 238 end with a stable
isotope Pb 206, and all of them belong to the magic number of neutrons or
protons.
3. Every isotope of nuclei has some specific values for the cross-section of neutron
absorption. Those isotopes belonging to magic numbers have a lower cross-section
than their surrounding isotopes.
4. Isotopes containing the last neutron of the neutron magic number nuclei have the
maximum binding energy. If one more neutron is added to it then the binding
energy of the isotope drops down sharply.
5. Nuclei of a magic number have nearly zero electric quadruple moments.