Subatomic Physics

18.5. Vibrational States in Spherical Nuclei 561
Figure 18.12: Vibrational states. The vibrational
phonon carries an angular momentum 2 and positive
parity. The states are characterized by the number, N,
of phonons. The energy of the ground state has been
set equal to zero.
Nuclear physicists have borrowed
the expression phonons from their solidstate
colleagues, (26) and the situation is
described by saying that the phonon angular
momentum is 2�, and that one
phonon is present in the first excited
state, two phonons in the second excited
state, and so forth.
Since the ground states of even–even
nuclei always have spin 0, the first excited
vibrational states should have assignments
2 + . Two phonons have an energy
2�ω and they can couple to form
states 0 + , 2 + ,and4 + . The states with
spin 1 and 3 are forbidden by the requirement
that the wave function of two
identical bosons must be symmetric under
exchange. The expected spectrum
is sketched in Fig. 18.12. Even–even nuclei
near closed shells indeed show spectra
with the characteristics predicted by
the vibrational model. (27)
The angular dependence of the
shape oscillations is described by
the spherical harmonics Y m
2 ,and
we know from Eq. (15.24) that
these are eigenfunctions of the
total angular momentum with
quantum number l = 2. The
vibration carries an angular momentum
of 2 and positive parity.
Figure 18.13: Plot of the excitation energy of
the first 4 + state versus the excitation energy
of the first 2 + state for a large range of nuclei
[From ref. (28).]
26 C. Kittel, Introduction to Solid State Physics, 6th ed., Wiley, New York, 1986, Chapters4 and
5; J.M. Ziman, Electrons and Phonons, Clarendon Press, Oxford University, 1960. J.A. Reisland,
The Physics of Phonons, Wiley, New York, 1973.