Subatomic Physics

534 The Shell Model
S : Σili = L, Σisi = S. The total orbital angular momentum L and the
total spin S of all nucleons in a shell are then added to form a given J. Inthe
jj coupling scheme, the spin–orbit force is assumed to be stronger than the
residual force between individual nucleons so that the spin and the angular
momentum of each nucleon are added first to give a total angular momentum
j; thesej’s are then combined to the total J. In most nuclei, the empirical
evidence indicates that the jj coupling is closer to the truth; in the lightest
nuclei (A � 16), the coupling scheme appears to be intermediate between the
LS and the jj coupling.
2. Residual forces between the particles outside the closed shells are introduced.
That such residual interactions are needed can be seen in many ways. Consider,
for instance, 69 Ga. It has three protons in state 2p 3/2 outside the
closed proton shell. These three protons can add their spins to get values
of J = 1 3 5 7
9
2 , 2 , 2 , 2 . (The state J = 2 is forbidden by the Pauli exclusion
principle.) In the absence of a residual interaction, these states are degenerate.
Experimentally, one state is observed to be lowest—quite often the
in this case). There must be an interaction that splits these
state J = j(= 3
2
degenerate states. In principle, one should derive the residual interaction as
what remains after the nucleon–nucleon interaction is replaced by an average
single-nucleon potential. In practice, such a program is too difficult, and the
residual interaction is determined empirically. However, many of the features
of the residual interaction can be understood on theoretical ground. Consider
as an example the pairing force described in Section 17.1. We have pointed out
there that two like nucleons prefer to be in an antisymmetric spin state, with
spins opposed and with a relative orbital angular momentum of zero ( 1S0). If
the residual force has a very short range and is attractive, this behavior can
be understood immediately. Consider for simplicity a zero-range force. The
two nucleons can take advantage of such a residual attraction only when they
are in a relative s state; the exclusion principle then forces their spins to be
opposed, as is observed in reality. Although the true nuclear forces are not of
such short range (indeed there is a repulsion at about 0.5 fm), the net effect is
unchanged. The energy gained by the action of the pairing force is called pairing
energy, and it is found empirically to be of the order of 12A−1/2 MeV. The
pairing energy leads to an understanding of the energies of the first excited
states of even–even nuclei: A pair must be broken, and the corresponding first
excited state lies roughly 1–2 MeV above the ground state.
3. Descriptions of nuclei with the inclusion of a dynamic treatment of closed
shells have become possible thanks to advanced computers. Such “extended”
shell model calculations allow the excitation of closed shell nucleons into open,
vacant ones, leaving behind holes. These extended shell models have been successful,
for instance in understanding level structure, electromagnetic transi-