Decomposition of scattering phase shifts and reaction cross sections using the complex scaling method

We apply the complex scaling method to the calculation of scattering phase shifts and extract the contributions of resonances in a phase shift and a cross section. The decomposition of the phase shift is shown to be useful in understanding the roles of resonant and nonresonant continuum states. As examples, we apply this method to several two-body systems: (i) a schematic model with the Gyarmati potential, which produces many resonances, (ii) the $\alpha \text{−}\alpha$ system, which has a Coulomb barrier potential in addition to an attractive nuclear interaction, and (iii) the $\alpha \text{−}n$ system, which has no barrier potential. Using different kinds of potentials, we discuss the reliability of this method to investigate the resonance structure in the phase shifts and cross sections.

Figure 1

The geometrical expressions for phase shifts: ${\delta }_r$, ${\delta }_c$, and ${\delta }_k$ as functions of the energy $E$. Both $E<E_r^{\mathrm{res}}$ and $E>E_r^{\mathrm{res}}$ cases are displayed in the upper panel (a) and lower panel (b), respectively. The details are explained in the text.

Figure 2

The resonance $\left({\delta }_R^{\theta }\right)$ and continuum $\left({\delta }_C^{\theta }\right)$ phase shifts and the sum ${\delta }_R^{\theta }+{\delta }_C^{\theta }$ for $\theta ={20}^{\circ }$.

Figure 3

The phase shifts of the Gyarmati potential for $J^{\pi }=0^{+}$ and the subtraction of the resonance terms one by one.

Figure 4

Upper panel: The decomposition of scattering phase shifts of the $\alpha \text{−}\alpha$ system for (a) $\ell =0$, (b) $2,$ and (c) 4, respectively. The dashed and dotted lines represent the contributions of the resonance and continuum terms, respectively. The solid lines display total scattering phase shifts. The experimental data [24] are shown with open circles. Lower panel: The distributions of eigenvalues are given in the complex energy plane for each partial wave.

Figure 5

Upper panel: The decomposition of the scattering phase shifts of $\alpha \text{−}n$ system for (a) $p_{3/2}$ and (b) $p_{1/2}$. The dashed and dotted lines represent the contributions of the resonance and continuum terms, respectively. The solid lines display total scattering phase shifts. The experimental data [27] are shown with open circles. Lower panel: The distributions of eigenvalues are displayed in the complex energy plane.

Figure 6

Total cross sections as functions of relative energy of the $\alpha \text{−}n$ system. The dotted line shows the present calculation and open circles are experimental data [28].

Figure 7

The partial cross sections of the (a) $p_{1/2}$ and (b) $p_{3/2}$ waves. The dashed, dotted, and solid lines denote the results of resonance, continuum contributions, and partial cross sections, respectively.

Figure 8

The partial cross sections in (a) $\ell =2$ and (b) 4 waves of the $\alpha \text{−}\alpha$ system. The dashed, dotted, and solid lines denote the results of resonance, continuum contributions, and partial cross sections, respectively.

Figure 9

The real part of complex phase shift on the $2\theta$ line for the $J^{\pi }=0^{+}$ wave at different scaling angles. Energy is given by the absolute value measured from the threshold.