Bohr model as an algebraic collective model

Developments and applications are presented of an algebraic version of Bohr's collective model. Illustrative examples show that fully converged calculations can be performed quickly and easily for a large range of Hamiltonians. As a result, the Bohr model becomes an effective tool in the analysis of experimental data. The examples are chosen both to confirm the reliability of the algebraic collective model and to show the diversity of results that can be obtained by its use. The focus of the paper is to facilitate identification of the limitations of the Bohr model with a view to developing more realistic, computationally tractable models.

Figure 1

Radial (β) wave functions, ${\mathcal{R}}_{\nu }^{\lambda }$, shown for $a=1$, with $\lambda ={\lambda }_0({\beta }_0)$, for ${\beta }_0=0$ and ${\beta }_0=6$. An appropriate choice of radial basis is with ${\beta }_0$ equal to the equilibrium $\beta$ deformation, in units of $1/a$, of the nucleus under consideration.

Figure 2

Potential energy $V_{\alpha }$ as a function of $\beta$ for different values of $\alpha$.

Figure 3

Energy levels of the rigid-beta Wilets-Jean model in units of $1/(2B{\beta }_0^2)$. (a) SO(5)-reduced $E2$ transition rates in units such that the $v=1$ to $v=0$ reduced transition rate is 100. (b) More detailed SO(3)-reduced $E2$ transition rates.

Figure 4

Energy levels and reduced $E2$ transition rates for the Hamiltonian $\hat{H}(B,\alpha ,\chi ,\kappa =0)$ of Eq. (89). (a) Near rigid-$\beta$ Wilets-Jean Hamiltonian with $B=40,\alpha =5.0,\chi =0$. (b) A relatively soft Wilets-Jean Hamiltonian with $B=40,\alpha =1.0,\chi =0$. (c) Hamiltonian of Eq. (89) with $B=40,\alpha =1.0,\chi =\pm 0.5$. Energy levels are given in each figure in units such that the lowest $L=2$ state has excitation energy of $L(L+1)=6$. The energy levels shown in (a) and (b) are degenerate SO(5) multiplets, and the $E2$ transition rates shown are SO(5)-reduced as in Fig. 3a; they are identical in value to the standard $B(E2;L_i=2v\to L_f=2v-1)$ transition rates between the states of maximum $L$ of each multiplet. The transition rates shown in (c) are standard SO(3)-reduced $B(E2;L_i\to L_f)$ transition rates. All transition rates are given in units such that $B(E2;2_1\to 0_1)=100$. Note that $0_1$ and $2_1$ refer to the lowest energy states of $L=0$ and 2, respectively.

Figure 5

(a) Low-energy spectrum of the Hamiltonian $\hat{H}(B,\alpha ,\chi ,\kappa =0)$ of Eq. (89) for $B=20$, $\alpha =1.5$, and $\chi =\pm 2.0$. Reduced $E2$ transition rates are shown in units for which $B(E2;2_1\to 0_1)=100$. Energy levels are given in units such that the lowest $L=2$ state has energy $E_{2_1}=6$. (b) Corresponding results for the adiabatic Bohr model in the axially symmetric limit showing a ground-state band, a one-phonon $\beta$ vibrational band, and one- and two-phonon $\gamma$ vibrational bands.

Figure 6

Low-energy spectrum of the Hamiltonian $\hat{H}(B,\alpha ,\chi =0,\kappa )$ for $B=22$, $\alpha =1.5$, and $\kappa =4.0$. The larger reduced $E2$ transition decay rates are shown in units for which $B(E2;2_1\to 0_1)=100$. Energy levels are given in units such that the lowest $L=2$ state has energy $E_{2_1}=6$.

Figure 7

Typical spectrum of the adiabatic Bohr model in the triaxial symmetric top limit in which ${\gamma }_0=\pi /6$. It shows the beginnings of a zero-vibrational-phonon $K=0,2,4,\dots$ rotational band sequence and parallel excited one-β-phonon and one-γ-phonon bands. The parameters of this model are chosen such that the excitation energy of the first-excited $L=2$ state, the lowest energy states of the one-phonon $\beta$ and $\gamma$ bands, and the $B(E2;0_{\beta }\to 2_1)$ and $B(E2;0_{\gamma }\to 2_2)$ reduced $E2$ transition rates are the same as those of Fig. 5.

Figure 8

$\gamma$ potentials of the form given by Eq. (100) with ${\beta }_0=1$ and for different values of $(c_1,c_2)$.

Figure 9

Low-energy spectrum of the Hamiltonian $\hat{H}(B,\alpha ,\chi ,\kappa )$ of Eq. (89) for $B=22,\alpha =1.5,\chi =\kappa =4.0$. The reduced $E2$ transition decay rates are in units for which $B(E2;2_1\to 0_1)=100$. Energy levels are given in units such that the lowest $L=2$ state has energy $E_{2_1}=6$.