Exactly solvable configuration mixing scheme in the vibrational limit of the interacting boson model

An intruder configuration mixing scheme with $2n$-particle and $2n$-hole configurations from $n=0$ up to $n\to \infty$ in the U(5) (vibrational) limit of the interacting boson model is proposed. A simple Hamiltonian suitable to describe the intruder and normal configuration mixing is found to be exactly solvable, and its eigenstates can be expressed as the SU(1,1) coherent states built on the U(5) basis vectors of the interacting boson model. It is shown that the configuration mixing scheme keeps lower part of the vibrational spectrum unchanged and generates the intruder states due to the mixing. Some low-lying level energies and experimentally known $B\left(E2\right)$ ratios of ${}^{108,110}\mathrm{Cd}$ are fitted and compared with the experimental results.

Figure 1

Low-lying level pattern of the solvable configuration mixing U(5) model with the Hamiltonian given by (2) with $c=f=0$, where the lower-left ten levels belong to the $\zeta =1$ family and are the same as those generated from the U(5) limit Hamiltonian of the IBM ${\widehat{H}}_0$ only, but with the modified single $d$-boson energy ${\tilde{\epsilon }}_d$. The levels labeled with $\left(i\right)$ are intruder levels of the model with ${\mathrm{\Delta }}_s=2\sqrt{{\mathrm{\Delta }}^2-g^2}$ or ${\mathrm{\Delta }}_d=2\sqrt{{(\mathrm{\Delta }+{\epsilon }_d)}^2-g^2}$; $0_1^{+}\left(i\right)$, $2_1^{+}\left(i\right)$, $0_2^{+}\left(i\right)$, $2_2^{+}\left(i\right)$, $4_2^{+}\left(i\right)$ belong to the $\zeta =2$ family, and $0_3^{+}\left(i\right)$ belongs to the $\zeta =3$ family.

Figure 2

The ground-state expectation value of the total number of bosons ${\overline{N}}_{\zeta =1}$ (solid curve) and that of the first excited intruder $0_1^{+}\left(i\right)$ state ${\overline{N}}_{\zeta =2}$ (dashed curve) defined by (34) and (35), respectively, as functions of the mixing parameter $\overline{g}=g/{\epsilon }_d$, where $N=10$ and $\overline{\mathrm{\Delta }}=\mathrm{\Delta }/{\epsilon }_d=1$ is taken in the plot.