Generalized coherent states satisfying the Pauli principle in a nuclear cluster model

We propose a new basis state, which satisfies the Pauli principle in the nuclear cluster model. The basis state is defined as the generalized coherent state of the harmonic oscillator wave function using a pair of the creation operators and is orthogonal to the Pauli-forbidden states having smaller quanta. In the coherent basis state, the range parameter is changeable and controls the radial dilation. This property is utilized for the precise description of the relative motion between nuclear clusters. We show the reliability of this framework for the $2\alpha$ system of ${}^8\mathrm{Be}$ in the semimicroscopic orthogonality condition model. We obtain the resonances and nonresonant continuum states of $2\alpha$ with complex scaling. The resonance solutions and the phase shifts of the $\alpha \text{−}\alpha$ scattering agree with those using the conventional projection operator method to remove the Pauli-forbidden states. We further discuss the extension of the present framework to the multi-$\alpha$ cluster systems using the SU(3) wave functions.

Figure 1

Diagonal energies of the relative motion in ${}^8\mathrm{Be}$ for $0^{+}$ (red) and $2^{+}$ (blue) states as functions of the HO length parameter $b$ (top) and dilation parameter $\beta$ (bottom) in the coherent basis method (CH) and the projection operator method (PO) without complex scaling.

Figure 2

Energy eigenvalues of ${}^8\mathrm{Be}$ (top: $0^{+}$ and $2^{+}$, bottom: $4^{+}$ and $6^{+}$) for the coherent basis method (CH, solid symbols) and the projection operator method (PO, open symbols) on the complex energy plane, measured from the $\alpha +\alpha$ threshold energy. Scaling angle $\theta$ is taken as ${16}^{\circ }$ ($0^{+}$), ${18}^{\circ }$ ($2^{+}$), ${20}^{\circ }$ ($4^{+}$), and ${25}^{\circ }$ ($6^{+}$). The eigenvalues deviated from the line of discretized continuum states are resonances.

Figure 3

Phase shifts of the $\alpha –\alpha$ scattering ($0^{+}, 2^{+}, 4^{+}$, and $6^{+}$) in the center-of-mass frame. The lines using dashed or dotted ones are the results in the coherent basis method and the gray solid lines for $0^{+}$, and $2^{+}$ are the ones in the projection operator method. The upper arrows from the bottom indicate the resonance energies of $0^{+}, 2^{+}, 4^{+}$, and $6^{+}$ in Table 2.