Subatomic Physics

18.3. Rotational Families 551
A spinless nucleus can have a permanent deformation (intrinsic quadrupole moment),
and its effect can be seen in the existence of rotational levels and also in the
rates of transitions leading to and from the J = 0 level. However, the quadrupole
moment cannot be observed directly because the absence of a finite spin does not
permit singling out a particular axis. In any measurement, an average over all directions
is involved, and the permanent deformation appears only as a particularly
large skin thickness.
18.3 Rotational Families
Deformed nuclei with spin zero in their ground state give rise to a rotational band,
with spin-parity assignments 0 + , 2 + ,... Since many deformed nuclei with spins different
from zero exist, the treatment of rotations must be extended to this more
general case. The situation then becomes considerably more complicated, and we
shall only treat the simplest case, namely a nucleus consisting of a deformed, axially
symmetric, spinless core and one valence nucleon, and we shall neglect the
interaction between the intrinsic and the collective (rotational) motion.
Figure 18.6: (a) The deformed core gives rise to a collective
angular momentum R; the valence nucleon produces
an angular momentum j. (b) R and j add up
to the total nuclear angular momentum J. The eigenvalue
of the component of J along the symmetry axis
3 is denoted by �K.
We assume that the valence nucleon
does not affect the core so
that it behaves like the deformed
spinless nucleus treated in the
previous section. The core then
gives rise to a rotational angular
momentum R perpendicular
to the symmetry axis, 3, so that
R3 = 0. The valence nucleon produces
an angular momentum j; R
and j are shown in Fig. 18.6(a);
they add up to the total nuclear
angular momentum J:
J = R + j. (18.19)
The total angular momentum J and its component, J3, along the nuclear symmetry
axis are conserved, and they satisfy the eigenvalue equations
J 2 opψ = J(J +1)� 2 ψ, J3,opψ = K�ψ. (18.20)
Because R3 = 0, the eigenvalue of j3,op is also given by �K.
If, as assumed, the state of the valence nucleon is not affected by the collective
rotation, then it is to be expected that each state of the valence nucleon can form
the base (head) of a separate rotational band. In the following we shall compute the