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The first excited $1/2^+$ state in $^9$Be has been observed as a sharp peak just above the $^8$Be+$n$ threshold energy in the photo-disintegration cross section of $^9$Be. Since the size of the peak has a strong influence on the reaction rate of the $^9$Be synthesis, experimental data have been investigated [1, 2].
We performed calculations using an $\alpha+\alpha+n$ three-cluster model [3, 4] together with the complex scaling method (CSM) [5, 6], which well reproduces the measured photo-disintegration cross section. The results indicate that the virtual-state character of the $1/2^+$ state plays an important role in formation of the peak structure appearing in the cross section observed above the $^8$Be+$n$ threshold. However, the virtual state cannot be directly obtained as an isolated pole solution in the CSM, because the scaling angle in the CSM cannot be increased over the position of the virtual state pole on the negative imaginary axis of the complex momentum plane.
The pole position of the virtual state is discussed calculating the continuum level density, the scattering phase shift, and scattering length calculated in the CSM [7, 8]. In the next step, we need to distinguish a virtual state from a resonant state and to calculate their contributions in the photo-disintegration cross sections. For this purpose, we apply the CSM to a simple schematic two-body model which simulates the 8Be+n system [8]. In this schematic two-body model, both clusters ($^8$Be and $n$) are assumed to be structureless and spinless. To understand the origin of a low-energy peak in the photodisintegration cross section just above the breakup threshold energy in 9Be, we calculate the contributions of the components of the strength function. The decomposed $E1$ transition strength is shown in Fig. 1. From the results, we can see that the real part of the matrix element $M^2$ and the imaginary part of $\rho(E)$ are dominant. This property is consistent with the $R$-matrix description [8], in which the peak is described as results from a resonant state. The asymmetric peak, displayed as solid curve in Fig. 1, implies the existence of a virtual state or a resonant state near zero energy.