Virtual-state character of the ${}^9\mathrm{Be}1/2^{+}$ state in the ${}^9\mathrm{Be}\left(\gamma ,n\right){}^8\mathrm{Be}$ reaction

We study the character of the first excited $1/2^{+}$ state of ${}^9\mathrm{Be}$, which is observed as a low-lying sharp peak in the cross section of ${}^9\mathrm{Be}(\gamma ,n)2{}^4\mathrm{He}$ just above the ${}^8\mathrm{Be}+n$ threshold. Using the $\alpha +\alpha +n$ three-body model, we describe the ground and excited unbound states of ${}^9\mathrm{Be}$ above the $\alpha +\alpha +n$ threshold. Applying the complex scaling method to the three-body model, we find no $1/2^{+}$ resonant solutions with the scaling angle of $\theta \le {15}^{\circ }$, while the low-lying peak in the photodisintegration cross section is reproduced in the present calculation. It is found that the ${}^8\mathrm{Be}+n$ continuum states dominate the low-lying peak in the cross section. Furthermore, using the analytical continuation of the coupling constant of the three-body interaction for the $\alpha +\alpha +n$ system, we discuss the virtual-state character of the $1/2^{+}$ state.

Figure 1

Calculated photodisintegration cross sections in comparison with experimental data. The red (dashed) and blue (dotted) lines are results with and without the three-body potential, where $v_3=1.10$ MeV and $\mu =0.02$ fm. The black (solid) line represents the cross section calculated by using an attractive three-body potential with $v_3=-1.02$ MeV. The experimental data below $E_{\gamma }=2.2$ MeV are taken from Refs. [6] and [8]. The arrow indicates the threshold energy of the ${}^8\mathrm{Be}\left(0^{+}\right)+n$ channel.

Figure 2

Distribution of energy eigenvalues of $1/2^{+}$ states measured from the $\alpha +\alpha +n$ threshold with scaling angle $\theta ={15}^{\circ }$. We here employ the three-body potential with $v_3=-1.02$ MeV and $\mu =0.02{\mathrm{fm}}^{-2}$. The red (solid), blue (dashed), and green (dotted) lines represent the branch cuts for $\alpha +\alpha +n,{}^8\mathrm{Be}(0^{+})$ + $n$, and ${}^5\mathrm{He}(3/2^{-})+\alpha$ continua, respectively.

Figure 3

Decomposed photodisintegration cross sections. The red (solid) and blue (dashed) lines are contributions of the $\alpha +\alpha +n$ and ${}^8\mathrm{Be}+n$ continuum states. The black thin line is the same as that in Fig. 1.

Figure 4

Pole trajectory of the ${}^9\mathrm{Be}1/2^{+}$ state in a complex energy plane by changing the three-body potential. The closed circles represent the poles obtained as isolated three-body resonances in CSM. The open circles and crosses are speculated pole positions for the virtual states and broad resonances, respectively.

Figure 5

Decomposed photodisintegration cross sections by using the three-body potential with $v_3=-1.3$. The red (solid) and blue (dashed) lines are contributions of the $\alpha +\alpha +n$ and ${}^8\mathrm{Be}+n$ continuum states. The green (dotted) line is that of the resonance pole. The black thin line represents the sum of them. The arrow indicates the threshold energy of the ${}^8\mathrm{Be}\left(0^{+}\right)+n$ channel.