Relativistic two-phonon model for the low-energy nuclear response

A two-phonon version of the relativistic quasiparticle time-blocking approximation introduces a new class of many-body models for nuclear structure calculations based on the covariant energy density functional. As a fully consistent extension of the relativistic quasiparticle random phase approximation, the relativistic two-phonon model implies fragmentation of nuclear states over two-quasiparticle and two-phonon configurations coupled to each other. In particular, we show how the lowest two-phonon $1^{-}$ state, identified as a member of the $[2^{+}\otimes 3^{-}]$ quintuplet, emerges from the coherent two-quasiparticle pygmy dipole mode in vibrational nuclei. The inclusion of the two-phonon configurations into the model space allows a quantitative description of the positions and the reduced transition probabilities of the lowest $1^{-}$ states in tin isotopes ${}^{112,116,120,124}$Sn as well as the low-energy fraction of the dipole strength below the giant dipole resonance without any adjustment procedures. The model is applied to the low-lying dipole strength in neutron-rich ${}^{68,70,72}$Ni isotopes. Recent experimental data for ${}^{68}$Ni are reproduced fairly well.

Figure 1

The Bethe-Salpeter equation for the response function $R$ of the many-body system in diagrammatic representation. The solid lines denote single-quasiparticle propagators. The integral part is divided into two terms, the small black circle represents the static effective interaction $\tilde{V}$ and the energy-dependent block $W\left(\omega \right)$ contains the dynamic contributions induced by the coupling to phonons.

Figure 2

The four-component Green's function in the diagrammatic representation.

Figure 3

The correspondence between the 2q$\otimes$phonon amplitude $\Phi$ of the conventional phonon coupling model and the two-phonon amplitude $\overline{\Phi }$ of the two-phonon model in a diagrammatic representation. Solid lines and Latin indices denote the single-quasiparticle nucleonic propagators, wavy curves with Greek indices show the phonon propagators, empty circles represent phonon vertices, and gray circles with the two attached nucleonic lines denote the RQRPA transition densities (see text).

Figure 4

Replacement of the uncorrelated two-nucleon propagator by the correlated one. Gray circles with the two attached nucleonic lines and wavy curve denote the RQRPA transition densities and the phonon propagator, respectively.

Figure 5

Low-lying dipole spectra of ${}^{116,120}$Sn calculated within RQRPA (dashed curves), RQTBA [blue solid curve, panels (a) and (c)] and RQTBA-2 [red solid curve, panels (b) and (d)]. A finite smearing parameter $\Delta =20$ keV has been used in the calculations. The inserts show the zoomed pictures of the spectra below 8 MeV with a smaller value $\Delta =2$ keV allowing us to see all the states in this energy region. The arrows indicate the neutron thresholds.

Figure 6

Energies and $B\left(E1\right)\uparrow$ values of the first $1^{-}$ states in the chain of tin isotopes ${}^{112,116,120,124}$Sn obtained within the relativistic two-phonon model (stars), compared with data from Refs. [32, 37, 60] (filled circles) and Refs. [38, 62] (open circles).

Figure 7

Low-lying dipole spectrum of ${}^{68,70,72}$Ni calculated within the RQRPA (black dotted curves), RQTBA (blue dashed curves), and RQTBA-2 (red solid curves) with a smearing parameter of 200 keV. The arrows show the neutron thresholds.

Figure 8

Low-lying dipole spectra of ${}^{68,70,72}$Ni calculated within RQTBA-2 with a smearing parameter of 20 keV (thin curves, light pink) and 200 keV (filled area). Panel (a) contains also the data from Ref. [14] (open circles, units on the right). The arrows show the neutron thresholds.