Spin instabilities of infinite nuclear matter and effective tensor interactions

We study the effects of the tensor force, present in modern effective nucleon-nucleon interactions, in the spin instability of nuclear and neutron matter. Stability conditions of the system against certain very low energy excitation modes are expressed in terms of Landau parameters. It is shown that in the spin case, the stability conditions are equivalent to the condition derived from the spin susceptibility, which is obtained as the zero-frequency and long-wavelength limit of the spin response function calculated in the random phase approximation. Zero-range forces of the Skyrme type and finite-range forces of M3Y and Gogny type are analyzed. It is shown that for the Skyrme forces considered, the tensor effects are sizable and tend to increase the spin instability, which appears at smaller densities than in the case that the tensor is not taken into account. On the contrary, the tensor contribution of finite-range forces to the spin susceptibility is small or negligible for both isospin channels of symmetric nuclear matter as well as for neutron matter. A comparison with the spin susceptibility provided by realistic interactions is also presented.

Figure 1

Inverse of the spin susceptibility respect to the inverse Hartree-Fock spin susceptibility as a function of density for the Skyrme interactions SIII (the solid line includes the tensor effects and the full circles are without) and SLy5 (the dot-dashed line include the tensor effects and the triangles are without). Panels (a) and (b) correspond to symmetric nuclear matter in the singlet and triplet isospin channels respectively. Panel (c) shows the results for neutron matter.

Figure 2

Inverse of the spin susceptibility respect to the inverse Hartree-Fock spin susceptibility as a function of density for the Skyrme interactions T44 (the solid line includes the tensor effects and full circles are without) and T61 (the dot-dashed line include the tensor effects and the triangles are without). Panels (a) and (b) correspond to symmetric nuclear matter in the singlet and triplet isospin channels respectively. Panel (c) shows the results for neutron matter.

Figure 3

Instability inequalities of Eqs. (3, 4, 5, 6, 7, 8) for the singlet isospin channel as a function of density for symmetric nuclear matter obtained for the finite range interactions: (a) M3Y-P2, (b) Gogny-D1S and (c) Gogny-D1M. Solid, dashed and dotted lines all with empty circles correspond to $L=1,J=0^{-},1^{-}$ and $2^{-}$ respectively. Solid, dashed and dotted lines all with full diamonds correspond to $L=2,J=1^{+},2^{+}$ and $3^{+}$. The effects of the tensor forces are included in all cases.

Figure 4

Inverse of the spin susceptibility respect to the inverse Hartree-Fock spin susceptibility in the singlet isospin channel for symmetric nuclear matter as a function of density for the finite range effective forces including the tensor component: (a) M3Y-P2, (b) Gogny D1S and (c) Gogny D1M. Empty circles stand for the results taking into account only $G_0$ in the Eq. (27), with all $H$'s equal to zero. Then the different curves incorporate step by step the effects of the $H$'s. Curve with up triangles ($G_0,H_0$), triangle oriented to the left ($G_0,H_0,H_1$), triangles oriented to the right ($G_0,H_0,H_1,H_2$) and the solid line stands for the full $\mathit{RPA}$ calculation.

Figure 5

Inverse of the spin susceptibility respect to the inverse Hartree-Fock spin susceptibility in the triplet isospin channel for symmetric nuclear matter as a function of density for the finite range effective forces including the tensor component: (a) M3Y-P2, (b) Gogny D1S and (c) Gogny D1M. Empty circles stand for the results taking into account only $G_0^{\prime }$ in the Eq. (27), with all $H^{\prime }$'s equal to zero. Then the different curves incorporate step by step the effects of the $H^{\prime }$'s. Curve with up triangles ($G_0^{\prime },H_0^{\prime }$), triangle oriented to the left ($G_0^{\prime },H_0^{\prime },H_1^{\prime }$), triangles oriented to the right ($G_0^{\prime },H_0^{\prime },H_1^{\prime },H_2^{\prime }$) and the solid line stand for the full $\mathit{RPA}$ calculation.

Figure 6

Inverse of the spin susceptibility respect to the inverse Hartree-Fock spin susceptibility for neutron matter as a function of density for the finite range effective forces including the tensor component: (a) M3Y-P2, (b) Gogny D1S and (c) Gogny D1M. Empty circles stand for the results taking into account only $G_0^{\left(n\right)}$ in the Eq. (27), with all $H^{\left(n\right)}$'s equal to zero. Then the different curves incorporate step by step the effects of the $H^{\left(n\right)}$'s. Curve with up triangles ($G_0^{\left(n\right)},H_0^{\left(n\right)}$), triangle oriented to the left ($G_0^{\left(n\right)},H_0^{\left(n\right)},H_1^{\left(n\right)}$), triangles oriented to the right ($G_0^{\left(n\right)},H_0^{\left(n\right)},H_1^{\left(n\right)},H_2^{\left(n\right)}$) and the solid line stand for the full $\mathit{RPA}$ calculation.

Figure 7

Spin susceptibility of neutron matter measured respect to the free spin susceptibility as a function of density for several interactions containing tensor forces. SLy5 (squares), M3Y-P2 (solid line), Gogny D1ST (dashed), Gogny D1MT (Dashed-dashed-dot), Av6'-AFDMC (full circles joint by dots), and Reid6-CBF (empty squares joint by dots).