Pygmy and giant dipole resonances in the nitrogen isotopes

The configuration-interaction shell model with the WBP10 effective interaction has been used to investigate the pygmy and giant dipole resonances in the nitrogen isotopes. Large enhancement of low-lying dipole strength, i.e., pygmy dipole resonances (PDRs), is predicted in the neutron-rich ${}^{17,18,19,20}\mathrm{N}$. The nature of the PDRs is analyzed via the transition densities and transition matrix elements. It turns out these PDRs involve a larger amount of excitations between the $2s1d$ and loosely bound $1f2p$ shells. Combining with the transition densities, it is concluded that the PDRs in ${}^{17,18,19,20}\mathrm{N}$ are collective and due to the oscillation between the excess neutrons and the isospin saturated core. The isospin dependence of energy splitting and sum rule of isospin doublets is discussed. The theoretical energy splitting of isospin doublets can significantly deviate from the systematic values when nucleus is far away from the $\beta$-stability line. The ratios of $T_{<}$ and $T_{>}$ energy-weighted sum rule (EWSR) are consistently larger than the systematic values, and it is noticed that the calculated EWSR ratio over the systematic ratio increases with increasing isospin almost linearly. We also calculated the photoabsorption cross sections for the nitrogen isotopes. We proposed the normalization factors for $0–1\hslash \omega$ and $2–3\hslash \omega$ calculations. After the normalization, the shell model has well reproduced the experimental photoabsorption cross sections in ${}^{14,15}\mathrm{N}$, especially the detailed structure of resonances.

Figure 1

Neutron and proton separation energies of the nitrogen isotopes. The solid circles are experimental values. The open squares and stars are the shell-model results in $0–1\hslash \omega$ and $2–3\hslash \omega$ spaces, respectively. In panel (b), the normalized proton separation energies which takes into account of the Coulomb barrier are also shown by open squares. The lines are only guides for the eyes.

Figure 2

Mass radii of the nitrogen isotopes. The solid cycles indicate the experimental data compiled by Ozawa et al.; see Ref. [40] and references in. The open squares indicate the experimental results given in Ref. [41]. The solid line gives the shell-model results in $0–1\hslash \omega$ space using the SHF single-particle wave functions.

Figure 3

The calculated electric dipole strengths from the excited to the ground states for ${}^{14,15,16}\mathrm{N}$ in the shell-model space up to $0–1\hslash \omega$ and $2–3\hslash \omega$ excitations. The thin drop lines are the discrete $B\left(E1\right)$ values in the shell-model calculations. The dashed (red), dotted (blue), and solid (black) lines represent the transitions from $T=T_z,T=T_z+1$ states (labeled as $T_{<}$ and $T_{>})$ and their sums, respectively. The arrows separate the continuum from the discrete states.

Figure 4

Same as Fig. 3 but for ${}^{17-20}\mathrm{N}$ in the $spsdpf$ space up to $0–1\hslash \omega$ excitations.

Figure 5

Shell-model transition densities of discrete dipole transitions in ${}^{17}\mathrm{N}$. The solid red and dashed blue lines are for proton and neutron, respectively. The dotted brown line gives the location of the mass radius in the shell-model calculations.

Figure 6

Shell-model transition densities of discrete dipole transitions in ${}^{18}\mathrm{N}$.

Figure 7

Shell-model transition densities of discrete dipole transitions in ${}^{19}\mathrm{N}$.

Figure 8

Shell-model transition densities of discrete dipole transitions in ${}^{20}\mathrm{N}$.

Figure 9

Contributions to the total matrix elements of valence orbitals for the de-excitations from the selected PDR states in ${}^{17-20}\mathrm{N}$. The horizontal axis is the single-particle excitation energies of valence orbitals. The magenta bars with slanted lines, the solid red bars, and the open blue bars indicate the transitions between the valence orbitals of $1s\leftrightarrow 1p,1p\leftrightarrow 2s1d$, and $2s1d\leftrightarrow 1f2p$, respectively.

Figure 10

(a) GDR energies and (b) splitting of isospin doublets in the nitrogen isotopes. The stars indicate the systematic values obtained from Eq. (11). The open circles and solid squares represent the shell-model results in the $0–1\hslash \omega$ and $2–3\hslash \omega$ spaces.

Figure 11

Ratios of $T_{<}$ and $T_{>}$ EWSR values in the nitrogen isotopes. The subfigure shows the shell-model ratios over the systematic values.

Figure 12

The EWSR values of the PDRs in nitrogen, carbon, and oxygen isotopes. The EWSR values are summed up to (a) 10 and (b) 15 MeV. The experimental values of oxygen isotopes are taken from Ref. [22].

Figure 13

Theoretical and experimental photo absorption cross sections in ${}^{14,15,16}\mathrm{N}$. The solid lines with error bars are experimental data. The solid without error bars are total photoabsorption cross sections in the shell-model calculations, and dashed and dotted lines are contributions from $T_{<}$ and $T_{>}$ states. The experimental data of ${}^{14}\mathrm{N}$ are taken from the evaluated version [54], and we add 10% of the data as the error bars. The experimental data of ${}^{15}\mathrm{N}$ are taken from Ref. [55]. For the reduction factors in the legends, see the text for details.

Figure 14

Same as Fig. 13, but for ${}^{17,18,19,20}\mathrm{N}$ in the $0–1\hslash \omega$ space calculations. The same set of reduction factors is used in all subfigures; see the text for details.