Dipole response in ${}^{208}\mathrm{Pb}$ within a self-consistent multiphonon approach

Background: The electric dipole strength detected around the particle threshold and commonly associated with the pygmy dipole resonance offers unique information on neutron skin and symmetry energy, and is of astrophysical interest. The nature of such a resonance is still under debate.

Purpose: We intend to describe the giant and pygmy resonances in ${}^{208}\mathrm{Pb}$ by enhancing their fragmentation with respect to the random-phase approximation.

Method: We adopt the equation of motion phonon method to perform a fully self-consistent calculation in a space spanned by one-phonon and two-phonon basis states using an optimized chiral two-body potential. A phenomenological density-dependent term, derived from a contact three-body force, is added to get single-particle spectra more realistic than the ones obtained by using the chiral potential only. The calculation takes into full account the Pauli principle and is free of spurious center-of-mass admixtures.

Results: We obtain a fair description of the giant resonance and obtain a dense low-lying spectrum in qualitative agreement with the experimental data. The transition densities as well as the phonon and particle-hole composition of the most strongly excited states support the pygmy nature of the low-lying resonance. Finally, we obtain realistic values for the dipole polarizability and the neutron skin radius.

Conclusions: The results emphasize the role of the two-phonon states in enhancing the fragmentation of the strength in the giant resonance region and at low energy, consistently with experiments. For a more detailed agreement with the data, the calculation suggests the inclusion of the three-phonon states as well as a fine tuning of the single-particle spectrum to be obtained by a refinement of the nuclear potential.

Figure 1

TDA $E1$ cross section in ${}^{208}\mathrm{Pb}$ computed using $V_{\chi }$ (a) only and $V_{\chi }+V_{\rho }$ (b). The experimental data (black squares) are taken from [19, 20, 58, 59, 60].

Figure 2

Comparison with experiments [70] of the HF charge densities obtained by using for the strength of $V_{\rho }$ the values (a) $C_{\rho }=0$, (b) $C_{\rho }=2000$ MeV ${\mathrm{fm}}^6$, (c) $C_{\rho }=3000$ MeV ${\mathrm{fm}}^6$.

Figure 3

Experimental [19, 20, 58, 59, 60] versus EMPM $E1$ cross sections in ${}^{208}\mathrm{Pb}$. A Lorentzian of width $\mathrm{\Delta }=0.5$ MeV was adopted. The EMPM cross section is computed using a HF (continuous red line) and a correlated (dashed blue line) ground state.

Figure 4

TDA (a) versus EMPM (b) $E1$ strength distributions in ${}^{208}\mathrm{Pb}$. The different scales used for the two plots are to be noticed.

Figure 5

TDA (a) versus EMPM (b) $E1$ low-lying spectra plotted on an amplified (logarithmic) scale.

Figure 6

EMPM versus experimental $E1$ low-lying spectra. The EMPM spectra are compute with (a) and without (b) ground-state correlations. The experimental data (c) are taken from [20].

Figure 7

Polarizability as function of energy. The EMPM calculation is carried out using a HF (red line) and a correlated ground state (black line).

Figure 8

Neutron skin thickness versus the strength of $V_{\rho }$.

Figure 9

EMPM isovector (a) versus isoscalar (b) dipole low-lying spectra plotted on a logarithm scale so as to make clearly visible the many small peaks not discernible otherwise. The isovector strength is nothing but the low-lying sector of the spectrum shown in Fig. 4. The isoscalar one is obtained by using the isoscalar operator (56).

Figure 10

$E1$ transition density for a low-lying $1^{-}$ state (a) and one in the GDR region (b).