Subatomic Physics

524 The Shell Model
has an excitation energy that is nearly 2 MeV larger than that of the other isotopes.
Figure 17.3: Ground and first-excited states of the even-A
isotopes of Pb.
17.2 The Closed Shells
Furthermore, unlike all the
other isotopes for which the
spins and parities of the first
excited states are 2 + ,thatof
208 Pb is 3 − . The closed shell
affects not only the energy of
the first excited state but also
its spin and parity.
The first task in the construction of the shell model is the explanation of the magic
numbers. In the independent-particle model it is assumed that the nucleons move
independently in the nuclear potential. Because of the short range of the nuclear
forces, this potential resembles the nuclear density distribution. To see the resemblance
explicitly, we consider a two-body force of the type
V12 = V0f(x1 − x2), (17.4)
where V0 is the central depth of the potential and f describes its shape. The
function f is assumed to be smooth and of very short range. A crude estimate
of the strength of the central potential acting on nucleon 1 in the nucleus can be
obtained by averaging over nucleon 2. Such an averaging represents the action of all
nucleons (except 1) on 1. Averaging is performed by multiplying V12 by the density
distribution of nucleon 2 in the nucleus, ρ(x2),
�
V (1) = V0 d 3 x2f(x1 − x2)ρ(x2).
If f is of sufficiently short range, ρ(x2) can be approximated by ρ(x1), and V (1)
becomes
�
V (1) = CV0ρ(x1), C = d 3 xf(x). (17.5)
The potential seen by a particle is indeed proportional to the nuclear density distribution.
The density distribution, in turn, is approximately the same as the charge
distribution. The charge distribution of spherical nuclei was studied in Section 6.4,
and it was found that it can be represented in a first approximation by the Fermi
distribution, Fig. 6.4. It would therefore be appropriate to start the investigation
of the single-particle levels by using a potential that has the form of a Fermi distribution
but is attractive. The Schrödinger equation for such a potential cannot be