Elements of Modern Physics
334
where I 0 corresponds to the ground state angular momentum, and I takes on
values I 0 , I 0 + 1, I 0 +2, ... . The observed values of the positive parity Ω =
5
2
levels for
25
Al, are in good agreement [see Fig. 9.4(b) with these predictions
with I ≈ 1.36 × 10
–55
kg. m
2
. The analysis of Ω =
1
2
states is more complicated
and may be found in specialized books.
In addition to rotational states, collective motion leads to vibrational states
also. The vibrational levels may be analysed in terms of quanta of vibrational
energy, called phonons. These phonons carry two units of angular momentum
and an energy of about 1 MeV. In multi-phonon excitations, the angular momenta
of the phonons add vectorially and produce closely-spaced levels with the same
number of phonons but different angular momenta. Such levels have been
identified experimentally.
The deformation of the core, by the nucleons outside the closed shell, can
lead to large quadrupole moments. That this is the correct explanation of the
large observed quadrupole moments, is indicated by the fact that large
deformations may be expected odd A and odd Z nuclei as well as in odd A and
odd (A – Z) nuclei. This would lead to large quadrupole moments for odd Z as
well as odd (A – Z) nuclei as indeed it is observed experimentally.
All in all, collective motion is an important component of nuclear structure
and energy levels.
Degenerate Gas Model
The degenerate gas model is similar in spirit to the free-electron theory of metals
(Sec. 7.5). Here it is assumed that noninteracting nucleons are confined to the
volume of the nucleus. The predictions of the model are discussed here briefly
so as to emphasize the importance of the Pauli exclusion principle in determining
the properties of the nucleus.
As shown in Sec. 7.5, the number of fermions occupying the lowest energy
levels in a box of volume V is given by [Eq. (7.89)]
n =
3
3
8
3
V p
h
π
(9.43)
where p is the maximum momentum of the occupied states, p
2
= 2mE.
Substituting V =
3
0
4
,
3
r A
π
the maximum kinetic energy E m is
334
where I 0 corresponds to the ground state angular momentum, and I takes on
values I 0 , I 0 + 1, I 0 +2, ... . The observed values of the positive parity Ω =
5
2
levels for
25
Al, are in good agreement [see Fig. 9.4(b) with these predictions
with I ≈ 1.36 × 10
–55
kg. m
2
. The analysis of Ω =
1
2
states is more complicated
and may be found in specialized books.
In addition to rotational states, collective motion leads to vibrational states
also. The vibrational levels may be analysed in terms of quanta of vibrational
energy, called phonons. These phonons carry two units of angular momentum
and an energy of about 1 MeV. In multi-phonon excitations, the angular momenta
of the phonons add vectorially and produce closely-spaced levels with the same
number of phonons but different angular momenta. Such levels have been
identified experimentally.
The deformation of the core, by the nucleons outside the closed shell, can
lead to large quadrupole moments. That this is the correct explanation of the
large observed quadrupole moments, is indicated by the fact that large
deformations may be expected odd A and odd Z nuclei as well as in odd A and
odd (A – Z) nuclei. This would lead to large quadrupole moments for odd Z as
well as odd (A – Z) nuclei as indeed it is observed experimentally.
All in all, collective motion is an important component of nuclear structure
and energy levels.
Degenerate Gas Model
The degenerate gas model is similar in spirit to the free-electron theory of metals
(Sec. 7.5). Here it is assumed that noninteracting nucleons are confined to the
volume of the nucleus. The predictions of the model are discussed here briefly
so as to emphasize the importance of the Pauli exclusion principle in determining
the properties of the nucleus.
As shown in Sec. 7.5, the number of fermions occupying the lowest energy
levels in a box of volume V is given by [Eq. (7.89)]
n =
3
3
8
3
V p
h
π
(9.43)
where p is the maximum momentum of the occupied states, p
2
= 2mE.
Substituting V =
3
0
4
,
3
r A
π
the maximum kinetic energy E m is