Subatomic Physics

17.2. The Closed Shells 525
solved in closed form. For many discussions the realistic potential is consequently
replaced by one that can be treated easily, either a square well or a harmonic oscillator
potential. We have encountered the latter in Section 15.7, and we can now
use the relevant information with very minor changes. The nuclear potential and
its approximation by the harmonic oscillator are shown in Fig. 17.4.
Figure 17.4: The more realistic potential
resembling the actual nuclear density distribution
is replaced by a harmonic oscillator
potential or a square well.
Consider first the harmonic oscillator whose
energy levels are shown in Fig. 15.7. The
group of degenerate levels corresponding to
one particular value of N is called an oscillator
shell. The degeneracy of each shell is
given by Eq. (15.30). In the application to
nuclei, each level can be occupied by two
nucleons, and consequently the degeneracy
is given by (N +1)(N + 2). In Table 17.1
the oscillator shells, their properties, and
the total number of levels up to the shell
N are listed. The orbitals are denoted by a
number and a letter; 2s, for instance, means
the second level with an orbital angular momentum
of zero.
Table 17.1: Oscillator Shells for the Three-
Dimensional Harmonic Oscillator.
N Orbitals Parity Degeneracy
Total Number
of Levels
0 1s + 2 2
1 1p − 6 8
2 2s, 1d + 12 20
3 2p, 1f − 20 40
4 3s, 2d, 1g + 30 70
5 3p, 2f, 1h − 42 112
6 4s, 3d, 2g, 1i + 56 168
Table 17.1 shows that the harmonic oscillator predicts shell closures at nucleons
numbers 2, 8, 20, 40, 70, 112, and 168. The first three agree with the magic
numbers, but after N = 2, the real shell closures differ from the predicted ones.
One of two conclusions is forced on us: either the agreement of the first three
numbers is fortuitous or an important feature is still missing. Of course by now it
is well understood that the second conclusion is correct. To introduce the missing
feature, we turn again to the level diagram.
The energy levels of the harmonic oscillator are degenerate for two different
reasons. Consider, for example, the level with N = 2, which contains the orbitals