Subatomic Physics

536 The Shell Model
symmetry term in the semiempirical mass formula. Equation (16.7) gives
or
∆sym = Esym(Z +1,N − 1) − Esym(Z, N) =−4asym
∆sym(in MeV) = −90
N − Z − 1
,
A
N − Z − 1
. (17.19)
A
The volume and surface terms are equal for the isobars, and thus the ground state
of the isobar with higher Z lies lower by the amount ∆sym. For the pair 209 Pb and
209 Bi, for instance, ∆sym ∼−19 MeV.
In the absence of the Coulomb interaction, isospin is a good quantum number.
As stated in Section 8.7, the isospin of a nuclear ground state assumes the smallest
allowed value. The isospin of the ground state of the isobar (Z, N) is thus given by
Eq. (8.34) as
N − Z
I> = , (17.20)
2
whereas for the isobar (Z +1,N − 1), the assignment is
I< =
N − Z
2 − 1=I> − 1. (17.21)
Because of charge independence, the levels of the parent nucleus (Z, N) alsoappear
with the same energy in the isobar (Z +1,N − 1). These analog states are shown
in Fig. 17.12. At this point, a crucial difference between light and heavy nuclei
appears. To appreciate it we return to Fig. 5.34, Table 5.11, and Eq. (17.3) and
note that nuclei have discrete levels (bound states) up to excitation energies of
about 8 MeV. Above about 8 MeV, emission of nucleons becomes possible, and the
spectrum is continuous.
Figure 17.12: Energy level diagram for the isobars (Z, N)
and (Z+1,N−1) in the absence of the Coulomb interaction.
In light nuclei, wherethesymmetryenergyissmall,theisobaric
analogs of the ground
state and of low-lying excited
states of the parent nucleus
lie in the discrete part
of the spectrum and consequently
are bound states. An
example is shown in Fig. 8.5
where the 0 + state in 14 Cis
the parent state, and the first
excited state in 14 Nistheisobaric
analog state.