Nuclear Structure 411
are designated by a prefix equal to the total quantum number n, a letter that indicatesl
for each particle in that level according to the usual pattern (s, p, d, f, g,...
corresponding, respectively, to l0, 1, 2, 3, 4,... ), and a subscript equal to j. The
spin-orbit interaction splits each state of given jinto 2j1 substates, since there are
2 j1 allowed orientations of Ji. Large energy gaps appear in the spacing of the levels
at intervals that are consistent with the notion of separate shells. The number of available
nuclear states in each nuclear shell is, in ascending order of energy, 2, 6, 12, 8, 22,
32, and 44. Hence shells are filled when there are 2, 8, 20, 28, 50, 82, and 126 neutrons
or protons in a nucleus.
The shell model accounts for several nuclear phenomena in addition to magic num-
bers. To begin with, the very existence of energy sublevels that can each be occupied
by two particles of opposite spin explains the tendency of nuclear abundances to favor
even Zand even Nas discussed in Sec. 11.3.
The shell model can also predict nuclear angular momenta. In even-even nuclei,
all the protons and neutrons should pair off to cancel out one another’s spin and
orbital angular momenta. Thus even-even nuclei ought to have zero nuclear angular
momenta, as observed. In even-odd and odd-even nuclei, the half-integral spin of the
single “extra” nucleon should be combined with the integral angular momentum of
the rest of the nucleus for a half-integral total angular momentum. Odd-odd nuclei
each have an extra neutron and an extra proton whose half-integral spins should yield
integral total angular momenta. Both these predictions are experimentally confirmed.Reconciling the ModelsIf the nucleons in a nucleus are so close together and interact so strongly that the
nucleus can be considered as analogous to a liquid drop, how can these same nucleons
be regarded as moving independently of each other in a common force field as required
by the shell model? It would seem that the points of view are mutually exclusive, since
a nucleon moving about in a liquid-drop nucleus must surely undergo frequent
collisions with other nucleons.
A closer look shows that there is no contradiction. In the ground state of a nucleus,
the neutrons and protons fill the energy levels available to them in order of increasing
energy in such a way as to obey the exclusion principle (see Fig. 11.8). In a collision,
energy is transferred from one nucleon to another, leaving the former in a state of
reduced energy and the latter in one of increased energy. But all the available levels of
lower energy are already filled, so such an energy transfer can take place only if the
exclusion principle is violated. Of course, it is possible for two indistinguishable
nucleons of the same kind to merely exchange their respective energies, but such a
collision is hardly significant since the system remains in exactly the same state it was
in initially. In essence, then, the exclusion principle prevents nucleon-nucleon collisions
even in a tightly packed nucleus and thereby justifies the independent-particle approach
to nuclear structure.
Both the liquid-drop and shell models of the nucleus are, in their very different
ways, able to account for much that is known of nuclear behavior. The collective
modelof Aage Bohr (Niels Bohr’s son) and Ben Mottelson combines features of both
models in a consistent scheme that has proved quite successful. The collective model
takes into account such factors as the nonspherical shape of all but even-even nuclei
and the centrifugal distortion experienced by a rotating nucleus. The detailed theory
is able to account for the spacing of excited nuclear levels inferred from the gamma-
ray spectra of nuclei and in other ways.bei48482_ch11.qxd 1/23/02 3:14 AM Page 411 RKAUL-9 RKAUL-9:Desktop Folder: