Subatomic Physics

532 The Shell Model
Table 17.2: Ground-State Spins and Parities
as Predicted by the Single-Particle Shell
Model and as Observed.
Shell-Model
Observed
Spin and
Nuclide Z N Assignment Parity
17O 8 9 d5/2
5 +
2
17F 9 8 d5/2
5 +
2
41Sc 21 20 f7/2
7 −
2
209Pb 82 127 g9/2
9 +
2
209Bi 83 126 h9/2
9 −
2
parities is complete. The quantum numbers of nuclei with a complete shell minus a
single particle can also be obtained from Fig. 17.9. Such a single-hole state can be
described in the language used in Section 5.10 for antiparticles; the hole appears as
an antiparticle, and Eq. (5.63) tells us that the angular momentum of the state must
be the same as that of the missing nucleon. Similarly, the parity of the hole state
must be the same as that of the missing nucleon state. (7) These properties of holes
also follow from the remark that a hole, together with the particle that can fill it,
couple together to give J =0 + for the closed shell. As a simple example, consider
4 4 3 He, shown in Fig. 17.7. Removing one neutron from He gives He. The removed
neutron was in an s1/2 state; the absence is indicated by the symbol (s1/2) −1 .The
corresponding spin-parity assignment of 3He is ( 1
2 )+ , in agreement with experiment.
Assignments for other single-hole nuclides can easily be given, and they also agree
with the experimental values.
Next we turn to excited states. In the spirit of the extreme single-particle model,
they are described as excitations of the valence nucleon alone; it moves into a higher
orbit. The core (closed shell) is assumed to remain undisturbed. Up to what energies
can such a picture be expected to hold? Figures 17.2 and 17.3 indicate that the
pairing energy is of the order of about 2 MeV. At an excitation energy of a few
MeV it is therefore possible that the valence nucleon remains in its ground state
but that a pair from the core is broken up and that one of the nucleons of the pair
is promoted to the next higher shell. It is also possible that a pair is excited to the
next higher shell. In either case, the resulting energy level is no longer describable
by the single-particle approach. It is consequently not surprising to find “foreign”
levels at a few MeV. Two examples are shown in Fig. 17.11, both doubly magic
nuclei plus one valence nucleon. In the case of 57Ni, the single-particle shell-model
assignments hold up to about 1 MeV, but above 2.5 MeV, foreign states appear.
7 A detailed discussion of hole states and of the particle–hole conjugation is given in A. Bohr
and B. R. Mottelson, Nuclear Structure, Benjamin, Reading, Mass., 1969. See Vol. I, p. 312 and
Appendix 3B.