Subatomic Physics

552 Collective Model
energy levels of these bands. The Hamiltonian is the sum of the rotational energy
and the energy of the valence nucleon,
or, with Eqs. (18.11) and (18.19),
H = Hrot + Hnuc,
H = R2 op
2I + Hnuc = 1
2I (J op − j op ) 2 + Hnuc.
The physical meaning becomes clearer if the Hamiltonian is written as the sum of
three terms,
H = HR + Hp + Hc, HR = 1
2I (J 2 op − 2J3,opj3,op),
Hp = Hnuc + 1
2I j2 op , Hc = − 1
I (J1,opj1,op + J2,opj2,op).
(18.21)
The third term, Hc, resembles the classical Coriolis force,anditiscalledtheCoriolis,
or rotation–particle coupling, term. It can be neglected except for the special
case K = 1
2 .(13) The second term, Hp, is independent of the rotational state of the
nucleus, and its contribution to the energy can be found by solving
Hpψ = Epψ.
The first term describes the energy of the rotational motion. With Eq. (18.20), the
energy eigenvalues of this term are given by
The total energy is then (13)
ER = �2
2I [J(J +1)− 2K2 ], J ≥ K. (18.22)
EJ,K = �2
2I [J(J +1)− 2K2 ]+Ep. (18.23)
This relation describes a sequence of levels, similar to the one given in Eq. (18.14)
for spinless nuclei. Following the terminology in molecular physics, the sequence
belonging to a particular value of K is called a rotational band, and the state with
lowest spin is called the band head. Characteristic differences exist between the case
K =0andK �= 0:
1. The spins for the case K = 0 are the even integers, while the spins for K �= 0
are given by
J = K, K +1,K+2,..., K �= 0. (18.24)
13 For the treatment of the case K = 1
2 ,seeref.(6)or(9).