(a) \[f\]=\[4,2\] ($\lambda,\mu$)=(2,2) (includes ground state).

Allowed states:

$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
0 0 1 $-96$ 0\
$2_1$ 0 1 $-78$ 2.32\
$2_2$ 0 1 $-78$ 2.32\
3 0 1 $-60$ 4.63\
4 0 1 $-36$ 7.72\

Note that the $2_1^+$ and $2_2^+$ states are degenerate in this scheme. This arises from the fact that, in a rotational picture, the $K$ values that are allowed are $\mu,~\mu-2,~...$. Thus we have a degeneracy of $J=2_1^+$ $K=0$ and $J=2_2^+$ $K=2$. This degeneracy does not correspond to the experimental situation -the $2_1^+$ and $2_2^+$ states are at $3.368~MeV$ and $5.960~MeV$ respectively -well separated.

As a practical matter this degeneracy gives problems for the shell model code $OXBASH$ [@oxbash]. Shell model routines often give wrong answers for transition rates when the states involved are degenerate. To overcome this difficulty, we have introduced weak additional terms in the Hamiltonian to remove the degeneracy e.g. we use a weak one-body spin-orbit interaction $-\xi \vec{l}\cdot\vec{s}$ with $\xi=0.1~MeV$.

\(b) Two degenerate bands \[f\]=\[4,1,1\] ($\lambda,\mu$)=(3,0), \[f\]=\[3,3\] ($\lambda,\mu$)=(0,3)

Allowed states:

$L$ = $S$ = $T$ = $\frac{E}{\bar{\chi}}$ = $E^*(MeV)$\
1 1 1 $-66$ 3.853\
3 1 1 $-36$ 7.716\

We get the low-lying $1^+$ states 