Subatomic Physics

17.4. The Single-Particle Shell Model 531
where r ≡|x| and
C = − 1
�
VLS(r)r
6
2 d 3 r. (17.18)
The nucleon–nucleon spin–orbit interaction leads to a spin–orbit interaction for
a nucleon moving in the average nuclear potential. As Eq. (17.17) shows, the
interaction vanishes where the density is constant, and it is strongest at the nuclear
surface. Numerical estimates with Eqs. (17.17) and (17.18) give the correct order
of magnitude of Vls.
17.4 The Single-Particle Shell Model
The simplest atomic system is hydrogen because it consists of only one electron
moving in the field of a heavy nucleus. Next in simplicity are the alkali atoms
which consist of a closed atomic shell plus one electron. In a first approximation
they are treated by assuming that the one valence electron moves in the field of the
nucleus shielded by the closed shells of electrons which form a spherically symmetric
system with zero angular momentum. The entire angular momentum of the atom is
provided by the valence electron (and the nucleus). In nuclear physics, the two-body
system (deuteron) has only one bound state and does not provide much insight. In
analogy to the atomic case, the next simplest cases then are nuclei with closed
shells plus one valence nucleon (or nuclides with closed shells minus one nucleon).
To discuss such nuclides we first return to closed shells.
What are the quantum numbers of nuclides with closed shells? In the shell
model, protons and neutrons are treated independently. Consider first a subshell
with a given value of the total angular momentum j, for instance, the proton subshell
1p1/2 (Fig. 17.9). There are 2j + 1 = 2 protons in this subshell. Since protons are
fermions, the total wave function must be antisymmetric. The spatial wave function
of two protons in the same shell is symmetric, and consequently the spin function
must be antisymmetric. Only one totally antisymmetric state can be formed from
two protons, but a state described by one wave function only must have spin J =0.
The same argument holds for any closed subshell or shell of protons or neutrons:
closed shells always have a total angular momentum of zero. The parity of a closed
shell is even because there are an even number of nucleons filling it.
Ground-state spin and parity of nuclides with closed shells plus or minus a single
particle are now straightforward to predict. Consider first a single proton outside a
closed shell. Because the closed shell has zero angular momentum and even parity,
angular momentum and parity of the nucleus are carried by the valence proton.
Angular momentum and parity of the proton can be read off from Fig. 17.9. The
corresponding level diagram for neutrons is very similar. A first example was already
given in Fig. 17.7 from which we deduced that the ground state assignment of 5He and negative parity. A few additional examples are shown
in Table 17.2. The agreement between predicted and observed values of spins and
should be p 3/2, orspin 3
2