Transverse isospin response function of asymmetric nuclear matter from a local isospin density functional

The time dependent local isospin density approximation (TDLIDA) has been extended to the study of the transverse isospin response function in nuclear matter with an arbitrary neutron-proton asymmetry parameter $\xi$. The energy density functional has been chosen in order to fit existing accurate quantum Monte Carlo calculations with a density dependent potential. The evolution of the response with $\xi$ in the $\mathrm{\Delta }{T}_z=\pm 1$ channels is quite different. While the strength of the $\mathrm{\Delta }{T}_z=+1$ channel disappears rather quickly by increasing the asymmetry, the $\mathrm{\Delta }{T}_z=-1$ channel develops a stronger and stronger collective mode that in the regime typical of neutron star matter at $\beta$ equilibrium almost completely exhausts the excitation spectrum of the system. The neutrino mean free paths obtained from the TDLIDA responses are strongly dependent on $\xi$ and on the presence of collective modes, leading to a sizable difference with respect to the prediction of the Fermi gas model.

Figure 1

Adimensional interaction parameter $G=2\nu \mathcal{W}(\rho ,m)$ as a function of the total nucleon density $\rho$.

Figure 2

Transverse response in units of $V\nu$ for $z=3q/\left(2k_F\xi \right)=6$. The gray area corresponds to the continuum of quasi-particle quasi-hole excitations. The poles at real values of $s$ correspond to the collective states in the system.

Figure 3

Same as Fig. 2 for $z=1.5$.

Figure 4

Same as Fig. 2 for $z=0.75$.

Figure 5

Same in Fig. 2 for $z=0.166$.

Figure 6

Excitation strengths in units of $V\nu$ for $z=3q/\left(2k_F\xi \right)=6$ as a function of the absolute value of the adimensional parameter $s$. The full and dashed lines indicate the quasi-particle quasi-hole and collective strengths in the $\mathrm{\Delta }{T}_z=-1$ and $\mathrm{\Delta }{T}_z=+1$ channels (which would correspond to negative values of $s$), respectively. Note that a fictitious width has been given to the collective state in order to correctly reproduce in the figure the collective contribution to the non-energy-weighted sum rule.

Figure 7

Same as in Fig. 6 for $z=1.5$. Notice that the collective state in the $\mathrm{\Delta }{T}_z=+1$ channel has disappeared, and all the residual strength is absorbed by quasi-particle quasi-hole excitations.

Figure 8

Same as in Fig. 6 for $z=0.75$. By now, Pauli blocking of the $\mathrm{\Delta }{T}_z=+1$ channel is total.

Figure 9

Same as in Fig. 6 for $z=0.166$. Notice once again that the Pauli blocking of the $\mathrm{\Delta }{T}_z=+1$ channel is total, and that the strength in the $\mathrm{\Delta }{T}_z=-1$ channel is almost completely exhausted by the collective state.

Figure 10

Neutrino mean free path for an interacting, asymmetric nuclear matter at saturation density computed considering only the contribution of the transverse isospin channel. The values of the asymmetry parameter are displayed in the figure.

Figure 11

NMFP computed including (solid lines) and excluding (dashed lines) the contributions from collective modes to the total strength.

Figure 12

Ratio of the NMFP computed for an interacting system and for a Fermi gas for different values of the density. The value of the asymmetry parameter is $\xi =0.1$. The densities considered are 0.5 ${\rho }_0$ (dotted line), ${\rho }_0$ (solid line), 1.5 ${\rho }_0$ (dashed line), and 2 ${\rho }_0$ (dotted-dashed line).

Figure 13

Same as Fig. 12, but for a value of the asymmetry parameter $\xi =0.9$.

Figure 14

NMFP at density $0.5{\rho }_0$ computed in a Fermi gas (dashed line) and in an interacting asymmetric nuclear matter (full line). The asymmetry parameter of the system is $\xi =0.1$.

Figure 15

Same as Fig. 14, but at saturation density ${\rho }_0$.