The Nucleus
329
Shell Model
In the shell model, nucleons are assumed to move independently of each other
in an average, centrally-symmetric potential. They occupy discrete energy levels
in this potential, taking the Pauli exclusion principle into account. The grouping
together of some of the energy levels gives rise to a shell structure of the
nucleus. The independent motion or equivalently the long, mean free path is
partially justified by the Pauli principle which forbids transitions to states that
are already occupied.
Experimentally, it is found that nuclei with the number of neutrons or protons
equal to,
Z or (A – Z) = 2, 8, 20, 28, 50, 82, 126
(9.32)
are especially stable. These numbers are known as magic numbers. The stability
is particularly pronounced for nuclei with both the number of protons and
neutrons equal to magic numbers, e.g.
4
1 6
4 0
4 8
2 0 8
2
8
20
20
82
He, O, Ca, Ca, Pb . The stability
of
4
2 He leads to its being the only composite nucleus emitted in radioactive
decays. It is also found that the 3rd, 9th, 21st, 29th, 51st, 83rd and 127th neutron
or proton is loosely bound (in fact
5
2 He is unstable).
For describing the shell structure of nuclei, two of the potentials used are
the spherical-well potential and the simple harmonic oscillator potential. Since
the oscillator levels have already been deduced (Sec. 3.12, example 6), the
details for this case are given. The wave functions of the 3-dimensional oscillator
are products of the 1-dimensional wave functions and the energies are the sums
of the corresponding, 1-dimensional, equispaced energy levels:
E = ω (n + 3/2), n = n x + n y + n z
(9.33)
Now, corresponding to each n level, there are several degenerate states, e.g.
for n = 1, there are three states with n x or n y or n z equal to 1. These states
correspond to states with different angular momenta (n = 0, l = 0), (n = 1, l = 1),
(n = 2, l = 2, 0) etc. with the number of corresponding states (taking spin states
into account), being 2, 6, 12 etc. However, the actual potential cannot be
simulated by the oscillator potential at large distances. Since it goes to zero at
large distances, the larger angular momentum states are lowered with respect to
the smaller angular momentum states. The ordering of the energy levels with
the degeneracy removed is shown in Fig. (9.3). It should be noted that while
these levels explain the magic numbers 2, 8, 20 corresponding to closed shells,
they cannot explain the other magic numbers.
329
Shell Model
In the shell model, nucleons are assumed to move independently of each other
in an average, centrally-symmetric potential. They occupy discrete energy levels
in this potential, taking the Pauli exclusion principle into account. The grouping
together of some of the energy levels gives rise to a shell structure of the
nucleus. The independent motion or equivalently the long, mean free path is
partially justified by the Pauli principle which forbids transitions to states that
are already occupied.
Experimentally, it is found that nuclei with the number of neutrons or protons
equal to,
Z or (A – Z) = 2, 8, 20, 28, 50, 82, 126
(9.32)
are especially stable. These numbers are known as magic numbers. The stability
is particularly pronounced for nuclei with both the number of protons and
neutrons equal to magic numbers, e.g.
4
1 6
4 0
4 8
2 0 8
2
8
20
20
82
He, O, Ca, Ca, Pb . The stability
of
4
2 He leads to its being the only composite nucleus emitted in radioactive
decays. It is also found that the 3rd, 9th, 21st, 29th, 51st, 83rd and 127th neutron
or proton is loosely bound (in fact
5
2 He is unstable).
For describing the shell structure of nuclei, two of the potentials used are
the spherical-well potential and the simple harmonic oscillator potential. Since
the oscillator levels have already been deduced (Sec. 3.12, example 6), the
details for this case are given. The wave functions of the 3-dimensional oscillator
are products of the 1-dimensional wave functions and the energies are the sums
of the corresponding, 1-dimensional, equispaced energy levels:
E = ω (n + 3/2), n = n x + n y + n z
(9.33)
Now, corresponding to each n level, there are several degenerate states, e.g.
for n = 1, there are three states with n x or n y or n z equal to 1. These states
correspond to states with different angular momenta (n = 0, l = 0), (n = 1, l = 1),
(n = 2, l = 2, 0) etc. with the number of corresponding states (taking spin states
into account), being 2, 6, 12 etc. However, the actual potential cannot be
simulated by the oscillator potential at large distances. Since it goes to zero at
large distances, the larger angular momentum states are lowered with respect to
the smaller angular momentum states. The ordering of the energy levels with
the degeneracy removed is shown in Fig. (9.3). It should be noted that while
these levels explain the magic numbers 2, 8, 20 corresponding to closed shells,
they cannot explain the other magic numbers.