Probing resonances in the Dirac equation with quadrupole-deformed potentials with the complex momentum representation method

Resonances play critical roles in the formation of many physical phenomena, and many techniques have been developed for the exploration of resonances. In a recent letter [Phys. Rev. Lett. 117, 062502 (2016)], we proposed a new method for probing single-particle resonances by solving the Dirac equation in complex momentum representation for spherical nuclei. Here, we develop the theoretical formalism of this method for deformed nuclei. We elaborate numerical details and calculate the bound and resonant states in ${}^{37}\mathrm{Mg}$. The results are compared with those from the coordinate representation calculations with a satisfactory agreement. In particular, the present method can expose clearly the resonant states in a complex momentum plane and determine precisely the resonance parameters for not only narrow resonances but also broad resonances that were difficult to obtain before.

Figure 1

Single-particle spectra in ${}^{37}\mathrm{Mg}$ for the states ${\mathrm{\Omega }}^{\pi }=1/2^{-}$ with ${\beta }_2=0.4$ in the complex $k$ plane with four different contours. The open red circles, blue squares, olive diamonds, and magenta dots represent the resonances obtained in four different contours, respectively. The smaller labels with a dot inside represent the continuum. The gray lines represent the corresponding contours of momentum integration.

Figure 2

Single-particle spectra in ${}^{37}\mathrm{Mg}$ for the states ${\mathrm{\Omega }}^{\pi }={\frac{1}{2}}^{\pm },{\frac{3}{2}}^{\pm },...,{\frac{9}{2}}^{\pm }$ with ${\beta }_2=-0.2$ in the complex momentum plane. The bound states, resonant states with different quantum numbers, and continuum are marked with different labels, while the purple solid line represents the contour of momentum integration in complex momentum plane.

Figure 3

Single-particle resonances for the states ${\mathrm{\Omega }}^{\pi }={\frac{1}{2}}^{\pm },{\frac{3}{2}}^{\pm },...,{\frac{9}{2}}^{\pm }$ in the complex momentum plane with several different deformations. The outstanding labels denote the resonant states with the corresponding quantum numbers, while the black open circles and purple solid line represent the continuum and the integration contour in the complex momentum plane, respectively.

Figure 4

The evolution of single-particle energies to deformation for all the bound and resonant states concerned, where the bound states are marked by the solid lines and the resonant states by the dashed lined with the Nilsson labels on the lines and the corresponding spherical labels in the position ${\beta }_2=0$.

Figure 5

The evolution of widths to deformation for all the concerned resonant states. They are marked by the color dashed lines with the Nilsson labels on the lines and the corresponding spherical labels in the position ${\beta }_2=0$.

Figure 6

Radial-momentum probability distributions for the states ${\mathrm{\Omega }}^{\pi }=5/2^{+}$ with ${\beta }_2=0.3$, where the blue dashed line and red solid line correspond, respectively, to the bound state and resonant state, and the others correspond to the background of the continuum.

Figure 7

Radial density distributions in the coordinate space for the bound states 1/2[110] and 1/2[310], and the resonant state 1/2[301] with ${\beta }_2=0.4$. Here, the left four lines belong to 1/2[110], the top four lines on the right belong to 1/2[310], and the other four lines belong to 1/2[301]. The four contours are the same as those in Fig. 1.