Subatomic Physics

18.4. One-Particle Motion in Deformed Nuclei (Nilsson Model) 557
A state is consequently denoted
by K π . Actually, three
partially conserved quantum
numbers are used to describe
a given level further. We
shall not need these asymptotic
quantum numbers here.
As an application of the
Nilsson model, we consider
the ground states of some nuclides
with a neutron or proton
number around 11. Figure
18.1 shows that these nuclides
are expected to have
a deformation of the order
of 0.1, and consequently the
Nilsson model should be applicable.
Figure 18.10: In a nonspherical nucleus, the total angular
momentum, j, of a nucleon is no longer a conserved quantity.
Only its component, K, along the nuclear symmetry
axis is conserved. A nucleon with spin j (in the spherical
case) gives rise to K values j, j − 1,..., 1
. States K and
2
−K have the same energy.
The relevant properties of a number of nuclides are summarized in Table 18.2.
If it is assumed that the nuclides are described by the single-particle spherical shell
model, their ground-state spin-parity assignment can be read from Fig. 17.9: only
the last odd nucleon is assumed to determine the moments. The listed nuclides
have one or three nucleons outside the closed shell 8: According to Fig. 17.9, they
should all have an assignment ( 5
2 )+ . In reality, the spins are different, even for
19F, which has only one proton outside the magic number 8. The quadrupole
moment has been measured for two of the listed nuclides, and 〈r2 〉 can be taken
from Eq. (6.26); Eq. (18.31) then provides the value of the deformation parameter
δ(≈ ɛ). In agreement with the estimate from Fig. 18.1, δ is of the order of 0.1. The
value δ =0.1 is indicated in Fig. 18.9. Following this line the predicted assignments
can be read: for one nucleon outside the closed shell 8, ( 1
2 )+ is predicted. Three
nucleons outside the shell lead to an assignment ( 3
2 )+ . As Table 18.2 shows, these
values agree with experiment and demonstrate that the Nilsson model can explain
at least some of the properties of deformed nuclei. (In all these assignments it is
assumed that the even number of nucleons, for instance, the 10 neutrons in 19F, remain coupled to zero.)
The prediction of ground-state moments is only one of the successes of the
Nilsson model. It is also able to correlate a great many other observed properties
of deformed nuclei. (16,17)
16B.R. Mottelson and S.G. Nilsson, Kgl. Danske Videnskab. Selskab, Mat-fys. Medd. 1, No.8
(1959).
17M.E. Bunker and C.W. Reich, Rev. Mod. Phys. 43, 348 (1971).