Continuum random-phase approximation for $(n,\gamma )$ reactions on neutron-rich nuclei: Collective effects and resonances

We formulate a microscopic theory to calculate cross section of the radiative neutron capture reaction on neutron-rich nuclei using the continuum random-phase approximation (cRPA) to the time-dependent density functional theory (TDDFT). With an intention of applying to the $r$ process, for which the statistical reaction model may not be appropriate, we describe the transition between a initial state of incident neutron and a final state of the $\gamma$ decay by means of a single many-body framework of the cRPA-TDDFT. With the cRPA approach, it is possible to describe various excitation modes present in the $(n,\gamma )$ reaction, including soft dipole excitation, the giant resonances, as well as noncollective excitations and the single-particle resonances. Furthermore, it enables us to describe the $(n,\gamma )$ reaction where the the final states of the $\gamma$ transition are low-lying surface vibrational states. We demonstrate the theory by performing numerical calculation for the reaction ${}^{139}\mathrm{Sn}(n,\gamma ){}^{140}\mathrm{Sn}$. We discuss various new features which are beyond the single-particle model: the presence of narrow and wide resonances originating from noncollective and collective excitations and roles of low-lying quadrupole and octupole vibrational states.

Figure 1

The diagrams representing the matrix element $\langle ph|{\widehat{V}}_{\mathrm{scf}}(\widehat{F};\omega )|0\rangle$. (a) corresponds to the first term $\langle ph|\widehat{F}|0\rangle =\langle ph|\widehat{M}|i\rangle$ of the right-hand side (r.h.s.) of Eq. (16) while (b) corresponds to the second term. Note that the electromagnetic operator $\widehat{M}$ acts also on the hole line.

Figure 2

The calculated $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}+\gamma$ with $E1$ transitions populating different low-lying states in ${}^{140}\mathrm{Sn}$: the ground state (plotted with a green curve), the low-lying $2_1^{+}$ and $2_2^{+}$ states (red and blue curves respectively), and the octupole vibrational state $3_1^{-}$ (yellow curve). The horizontal axis is kinetic energy ${\epsilon }_{\mathrm{kin}}$ of the incident neutron. The smoothing parameter is $\eta ={10}^{-5}\mathrm{MeV}$.

Figure 3

The calculated partial $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}\left(1^{-}\right)\to {}^{140}\mathrm{Sn}(\mathrm{g}.\mathrm{s}.)+\gamma$ for $E1$ transitions, plotted separately for different partial waves of the incident neutron: $d_{5/2}$ (yellow curve), $g_{7/2}$ (green), and $g_{9/2}$ (dashed green). The horizontal axis is the neutron kinetic energy ${\epsilon }_{\mathrm{kin}}$. The smoothing parameter is $\eta ={10}^{-5}\mathrm{MeV}$.

Figure 4

The diagrams representing the $(n,\gamma )$ reaction of ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}\left(1^{-}\right)\to {}^{140}\mathrm{Sn}(\mathrm{g}.\mathrm{s}.)+\gamma$.

Figure 5

The same as Fig. 3, but a magnification in the range of ${\epsilon }_{\mathrm{kin}}=5.0–7.0\mathrm{MeV}$.

Figure 6

The calculated partial $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}(2^{-},3^{-})\to {}^{140}\mathrm{Sn}\left(2_1^{+}\right)+\gamma$ for $E1$ transitions, plotted separately for different partial waves of the incident neutron$;$ $s_{1/2}$ (blue curve), $d_{3/2}$ (yellow), $g_{7/2}$ (green), etc. Panel (a) is for the total spin $L^{\pi }=3^{-}$, and (b) for $L^{\pi }=2^{-}$. The horizontal axis is the neutron kinetic energy ${\epsilon }_{\mathrm{kin}}$. The smoothing parameter is $\eta ={10}^{-5}\mathrm{MeV}$.

Figure 7

The diagrams representing $(n,\gamma )$ reaction of ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}(2^{-},3^{-})\to {}^{140}\mathrm{Sn}\left(2_1^{+}\right)+\gamma$: (a) for the total spin $L^{\pi }=3^{-}$ and (b) for $L^{\pi }=2^{-}$.

Figure 8

The calculated partial $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}(2^{+},3^{+})\to {}^{140}\mathrm{Sn}\left(3_1^{-}\right)+\gamma$ for $E1$ transitions, plotted separately for different partial waves of the incident neutron: $p_{1/2}$ (blue curve), $f_{5/2}$ (yellow), $h_{9/2}$ (green), etc. Panel (a) is for the total quantum number $L^{\pi }=2^{+}$, and (b) for $L^{\pi }=3^{+}$. The horizontal axis is the neutron kinetic energy ${\epsilon }_{\mathrm{kin}}$. The smoothing parameter is $\eta ={10}^{-5}\mathrm{MeV}$.

Figure 9

The diagrams representing the $(n,\gamma )$ reaction ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}(2^{+},3^{+})\to {}^{140}\mathrm{Sn}\left(3_1^{-}\right)+\gamma$.

Figure 10

The calculated $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}+\gamma$ with $E1$ transitions populating different low-lying states in ${}^{140}\mathrm{Sn}$: the ground state (plotted with a green curve), the low-lying $2_1^{+}$ and $2_2^{+}$ states (red and blue curves respectively), and the octupole vibrational state $3_1^{-}$ (yellow curve). The horizontal axis is the neutron kinetic energy ${\epsilon }_{\mathrm{kin}}$. The smoothing parameter is $\eta =0.1\mathrm{MeV}$.

Figure 11

The calculated $(n,\gamma )$ cross sections for ${}^{139}\mathrm{Sn}\left({7/2}^{-}\right)+n\to {}^{140}\mathrm{Sn}+\gamma$ with $E2$ transitions populating different low-lying states in ${}^{140}\mathrm{Sn}$; the ground state (plotted with a green curve), the low-lying $2_1^{+}$ state (red curve), and the octupole vibrational state $3_1^{-}$ (yellow curve). The horizontal axis is the neutron kinetic energy ${\epsilon }_{\mathrm{kin}}$. The cross section for $2_2^{+}$ is not seen as it is smaller than the plotted range. The smoothing parameter is $\eta ={10}^{-5}\mathrm{MeV}$.

Figure 12

The same as Fig. 11, but for higher neutron kinetic energy and larger smoothing parameter $\eta =0.1\mathrm{MeV}$.