Nuclear response for the Skyrme effective interaction with zero-range tensor terms. II. Sum rules and instabilities

The formalism of linear response theory for Skyrme forces including tensor terms presented by Davesne et al. [Phys. Rev. C 80, 024314 (2009)] is generalized for the case of a Skyrme energy density functional in infinite matter. We also present analytical results for the odd-power sum rules, with particular attention to the inverse energy weighted sum rule, $M_{-1}$, as a tool to detect instabilities in Skyrme functionals.

Figure 1

Nuclear response function for two given values of the transferred momentum $q$ (in fm${}^{-1}$) for the T44 tensor parametrization. On the left (a) we show the response function in the channel $(1,1,1)$ with the discrete p-h transition corresponding to the zero sound. On the right (b) we show the response function in the channel $(1,1,0)$ in the proximity of a pole. ${\rho }_{\mathrm{sat}}$ is the saturation density of the system.

Figure 2

EWSR (in MeV) as a function of the transferred momentum $q$ (in fm${}^{-1}$) for the T44 tensor parametrization. The red dashed and black solid lines correspond to the integral [Eq. (8)] and analytical expressions [Eqs. (11, 12, 13)], respectively. Results are shown for the saturation density and for each $(S,M,I)$ channel.

Figure 3

CEWSR (in MeV${}^3$) as a function of the transferred momentum $q$ (in fm${}^{-1}$) for the T44 tensor parametrization. See Fig. 2 for other details.

Figure 4

IEWSR (in MeV${}^{-1}$) as a function of the transferred momentum $q$ (in fm${}^{-1}$) for the T44 tensor parametrization. Contrary to Fig. 2 we modified the order of the panel for reasons of clarity.

Figure 5

Critical densities (in fm${}^{-3}$) as functions of the transferred momentum $q$ (in fm${}^{-1}$) for the T44 tensor parametrization. Open circles show the critical densities extracted from the pole of the $M_{-1}$ moment while the dashed lines correspond to the pole of the corresponding strength function.

Figure 6

Critical densities, ${\rho }_c$ (in fm${}^{-3}$), as functions of the transferred momentum $q$ (in fm${}^{-1}$) for some usual Skyrme EDFs: SkP [38], SkM* [39], SGII [14], SLy4 [40, 41, 42], BSk8 [43], and SkO [44]. The horizontal dashed-dotted line represents the saturation density of the system.

Figure 7

Critical densities, ${\rho }_c$ (in fm${}^{-3}$), in the $S=1$ channels as functions of the transferred momentum $q$ (in fm${}^{-1}$) for the T11 to T16 [top four panels (a)–(d)] and for the T11 to T61 [bottom four panels (e)–(h)] tensor parametrizations. The horizontal dashed-dotted line represents the saturation density of the system.

Figure 8

$\ell =0$ Landau parameters as a function of the ratio $\rho /{\rho }_{\mathrm{sat}}$, where $\rho$ is the density of the system and ${\rho }_{\mathrm{sat}}$ is the saturation density, for some usual Skyrme parametrizations. The T44 tensor parametrization does not appear on the $G_0,G_0^{\prime }$ [panels (c) and (d)] plots, since in that case the inequalities that should be satisfied in the presence of a tensor interaction are Eqs. (23, 24, 25, 26, 27).

Figure 9

$\ell =1$ Landau parameters as a function of the ratio $\rho /{\rho }_{\mathrm{sat}}$, where $\rho$ is the density of the system and ${\rho }_{\mathrm{sat}}$ is the saturation density, for some usual Skyrme parametrizations. See Fig. 8 for other details.

Figure 10

On the left panel (a) we show the left-hand side of Eqs. (23, 24, 25, 26, 27), while on the right panel (b), adopting the same color code, the same but replacing in Eqs. (23, 24, 25, 26, 27) the terms $G_0,H_0$ with $G_0^{\prime },H_0^{\prime }$. The T44 tensor parametrization is considered here. See text for details.