Subatomic Physics

17.5. Generalization of the Single-Particle Model 533
The foreign states are not really foreign. While they cannot be described in terms
of the extreme single-particle shell model, they can be understood in terms of the
general shell model, through excitations from the core. In the case of 209 Pb, the
first such state appears at 2.15 MeV. The estimate based on Figs. 17.2 and 17.3
that core excitation will play a role at about 2 MeV is verified.
Figure 17.11: Excited states in 57 Ni and 209 Pb. The states
that allow an unambiguous shell-model assignment are labeled
with the corresponding quantum numbers.
We have discussed only two
properties of nuclei that are
well described by the singleparticle
model, spin and parity
of ground states and the
level sequence and quantum
numbers of the lowest excited
states. There are other
features that are explained
by the extreme single-particle
model, for instance the existence
of very-long-lived first
excited states in certain regions
of N and Z, the socalled
islands of isomerism.
However, the model applies only to a restricted class of nuclei—namely those with
only one nucleon outside a closed shell—and an extension to more general conditions
is necessary.
17.5 Generalization of the Single-Particle Model
The extreme single-particle shell model, discussed in the previous section, is based
on a number of rather unrealistic assumptions: The nucleons move in a spherical
fixed potential, no interactions among the particles are taken into account, and
only the last odd particle contributes to the level properties. These restrictions are
removed in various steps and to various degrees of sophistication; we briefly outline
some of the extensions.
1. All particles outside the closed major shells are considered. The angular momenta
of these particles can be combined in various ways to get the resulting
angular momentum. The two main schemes are the Russell–Saunders, or LS,
coupling, and the jj coupling. In the first, the orbital angular momenta are
assumed to be weakly coupled to the spins; spin and orbital angular momenta
of all nucleons in a shell are added separately to get the resulting L and