Nuclear response functions with finite-range Gogny force: Tensor terms and instabilities

A fully antisymmetrized random phase approximation calculation employing the continued fraction technique is performed to study nuclear matter response functions with the finite-range Gogny force. The most commonly used parameter sets of this force, as well as some recent generalizations that include the tensor terms, are considered and the corresponding response functions are shown. The calculations are performed at first and second order in the continued fraction expansion and the explicit expressions for the second-order tensor contributions are given. Comparisons between first- and second-order continued fraction expansion results are provided. The differences between the responses obtained at the two orders turn out to be more pronounced for the forces including tensor terms than for the standard Gogny ones. In the vector channels the responses calculated with Gogny forces including tensor terms are characterized by a large heterogeneity, reflecting the different choices for the tensor part of the interaction. For the sake of comparison the response functions obtained considering a $G$-matrix-based nuclear interaction are also shown. As a first application of the present calculation, the possible existence of spurious finite-size instabilities of the Gogny forces with or without tensor terms has been investigated. The positive conclusion is that all the Gogny forces but the GT2 one are free of spurious finite-size instabilities. In perspective, the tool developed in the present paper can be inserted in the fitting procedure to construct new Gogny-type forces.

Figure 1

Feynman diagrams that enter the second-order CF expansion of the RPA propagator: (a) first-order direct, (b) first-order exchange, and (c) second-order exchange.

Figure 2

RPA response function in symmetric nuclear matter at $k_F=270\mathrm{MeV}/c$ and $q=27\mathrm{MeV}/c$ calculated with the continued fraction technique at first order (dashed line) and at second order (solid line) in the CF expansion by considering different parametrizations of the Gogny interaction. The different spin and isospin channels $(S,T)$ are plotted in different colors. The black dotted lines represent the HF response. The collective modes above the continuum are represented by vertical lines.

Figure 3

As in Fig. 2, but for $q=270\text{MeV}/c$.

Figure 4

RPA response function in symmetric nuclear matter at $k_F=270\text{MeV}/c$ and $q=27\text{MeV}/c$ calculated with the continued fraction technique at first order (dashed line) and at second order (solid line) in the CF expansion by considering different parametrizations of the Gogny interaction with tensor terms. The different spin and isospin channels $(S,M,T)$ are plotted in different colors. The black dotted lines represent the HF response. The collective modes above the continuum are represented by vertical lines. For comparison, we also display the response functions calculated with a $G$-matrix nuclear interaction.

Figure 5

As in Fig. 4, but for $q=270\text{MeV}/c$.

Figure 6

Critical densities ${\rho }_c$ divided by a constant value of the saturation density ${\rho }_0=0.16{\mathrm{fm}}^{-3}$ as a function of the transferred momentum $q$ (in ${\mathrm{fm}}^{-1}$) for the most commonly used parametrizations of the Gogny force. The calculations of $R_{(S,T)}(q,\omega =0)$, through which the critical densities are deduced, are performed at first and second order in the continued fraction expansion.

Figure 7

As in Fig. 6, but for the Gogny forces with tensor terms.