J Margueron

H Sagawa

Extended Skyrme interaction (I): spin fluctuations in dense matter

Keywords: numbers: 21.30.Fe, 21.60.Jz, 21.65.-f, 26.50.+x

Most of the Skyrme interactions are known to predict spin or isospin instabilities beyond the saturation density of nuclear matter which contradict predictions based on realistic interactions. A modification of the standard Skyrme interaction is proposed so that the ferromagnetic instability is removed. The new terms are density dependent and modify only the spin p-h interaction in the case of spin-saturated system. Consequences for the nuclear response function and neutrino mean free path are shown. The overall effect of the RPA correlations makes dense matter more transparent for neutrino propagation by a factor of 2 to 10 depending of the density.

The understanding of dense nuclear matter and its application to nuclear physics and to astrophysics is strongly related to the properties of in-medium nuclear interaction. During the recent years, much efforts went into the fundamental understanding of the bare nuclear interaction, either through the developement of effective field theories derived from QCD symmetries [1] or through renormalisation group arguments applied to the Lippmann-Schwinger equation which generate a phase-equivalent low energy bare interaction named v low-k [2]. These theories aim at a separation of the long-range component of the nuclear forces dominated by the pion exchange, which is well under control, from the intermediate and short range components, which are dominated by correlated pion and heavy meson exchanges that are poorly known. A close agreement between the interaction v low-k and the effective Gogny interaction [3] have been found concerning the predictions of pairing properties in nuclear matter in a wide range of sub-nuclear densities [4].

Another modelisation of the nuclear interaction in dense matter based on the symmetries of QCD, the quark-meson coupling model, see Ref. [5] and references therein, has given a sound foundation to the Skyrme effective nuclear interaction [6,7]. The modern Skyrme forces are commonly adjusted to the empirical knowledge of nuclear matter at saturation density, and sometimes include also theoretical calculations of asymmetric or pure neutron matter, as well as a set of nuclei described within Hartree-Fock + BCS or Hartree-Fock Bogoliubov theories. So far the Skyrme interactions are successfully used to describe properties of nuclei such as binding energies, radii and excited states. Nevertheless, due to its very simple functional form, the standard Skyrme interaction is not suited for being extrapolated in astrophysical situation such as neutron stars. Indeed, the mean field solution obtained with such interactions become unstable for densities larger than the saturation density ρ 0 and also for neutron rich matter.

A recent extensive study of the symmetry energy deduced from Skyrme interactions [8] has shown that the symmetry energy becomes negative beyond the saturation density of nuclear matter for many of these interactions. This is related to the instability of the mean field with respect to the isospin density fluctuation δρ t where ρ t = ρ n -ρ p . Depending on the parameters of the interaction, instabilities can occurs in different channels, at various densities and at different isospins. It has been shown that there is a limited interval of densities between ρ 0 and 3ρ 0 for which the mean field is stable in symmetric matter and also in neutron matter [9]. The stability of the ground state toward small fluctuations can be studied within the HF+RPA framework. The fluctuations around the mean field are induced by the particle-hole interaction,

V ph = 1 N 0 ℓ (F ℓ + F ′ ℓ τ 1 • τ 2 + G ℓ σ 1 • σ 2 + G ′ ℓ (τ 1 • τ 2 )(σ 1 • σ 2 ))P l (cos θ) , (1)
in terms of the dimensionless Landau parameters

F ℓ , F ′ ℓ , G ℓ , G ′ ℓ and Legendre Polynomials P l (cos θ). N 0 = gm * k F /(2π 2 2
) is the density of state around the Fermi energy and g is the degeneracy. The matter is stable unless one of these parameters of multipolarity ℓ becomes lower than -2ℓ -1.

Recently, the ferromagnetic instability has regained interest [10,11] due to the observation of extremely high magnetic fields (∼ 10 15-16 G) in compact stars [12] and also of giant flares observed recently on 27 December 2004 [13]. We show in Fig. 1 the critical density ρ f at which the asymmetric matter becomes unstable with respect to the spin fluctuations. The proton fraction x p = ρ p /ρ is changed from symmetric matter (x p = 1/2) to neutron matter (x p = 0). We select various Skyrme interactions which are commonly used in the description of finite nuclei. Depending on the interaction, the density ρ f can be very close to ρ 0 (BSk16 [14], RATP [15]), around 2ρ 0 (SkM * [16], SLy5 [17]) or nearly reach 3ρ 0 (SGII [18], LNS [19]) in symmetric nuclear matter (x p =1/2). When the proton fraction is decreasing, the density ρ f either decrease (BSk16, RATP, SkM * , SGII) or increase (SLy5, LNS). From Fig. 1 it could be deduced that the prediction of ρ f is varying largely among the selected Skyrme interactions.

According to microscopic calculations with realistic interactions such as diffusion Monte-Carlo [20] or Brueckner Hartree-Fock (HF) calculations [21,22,23], there is no ferromagnetic instability up to substantially high densities. The onset of the ferromagnetic instability beyond a few times ρ 0 represents a serious limitation of the Skyrme interaction. This limitation should be cured in the calculation of nuclear matter properties such as equation of state, the response function and microscopic processes such as neutrino mean free path. It is then timely to enrich the effective Skyrme interaction in order to extend its domain of application and to take advantage of its simple form. To this end, extended Skyrme interactions shall mimic more accurately microscopic results of realistic interactions and preserve the accuracy of the original Skyrme interaction in the description of nuclei.

It is known that the spin and the spin-isospin excitations have strong impact on astronomical observables such as neutrino mean free path in neutron star, 0ν-and 2ν double beta decay processes. On the other hand, it has been recognized also that Skyrme interactions have a serious shortcoming in the spin channels. Our main purpose in this manuscript is to improve the spin dependent parts of Skyrme interactions keeping its simplicity and good properties for ground state properties. In this way, we will be able to extend the field of possible applications of Skyrme interactions for spin dependent excitations not only in finite nuclei but also infinite nuclei within the self-consistent theoretical model. It is important to make a bridge between finite and infinite systems by the self-consistent model without introducing any extra parameters. Top of realistic spin and spin-isospin interactions, we would like to introduce further tensor interactions and two-body spin-orbit interactions for constructing a global energy density functions of spin and spin-isospin channels. In this article, we first propose a simple way to get ride of the ferromagnetic instability by introducing a limited number of new terms to the standard Skyrme interaction in Sec. 1. In Sec. 2, we apply the new interaction to the calculation of the RPA response function in nuclear matter and then we calculate the neutrino mean free path. Conclusions are given in Sec. 3.

Extended Skyrme interaction

Some of the most recent Skyrme interactions are fitted so as to reproduce the theoretical binding energy in symmetric and neutron matter up to a density as much as few times ρ 0 . As a result, the instability in the isospin channel, which occurs only if the binding energy of neutron matter becomes more attractive than that of symmetric matter, is usually either totally removed or pushed to very large densities. It is then possible to get ride of the isospin instability without modifying the standard Skyrme interaction. In the following, we focus on the spin instability and test the new terms added to the existing Skyrme interactions which do not show the instability in the isospin channel. We adopt three different Skyrme interactions, SLy5, LNS and BSk16 for this study. SLy5 is an interaction suited by construction to make predictions in neutron rich nuclei, LNS is an interaction which reproduces global features of G-matrix in symmetric and asymmetric nuclear matter and BSk16 reproduces known masses of nuclei with the best accuracy of rms deviation 632 keV. Some nuclear matter properties of these interactions are listed in Tab. 1. The monopole Landau parameters in symmetric (SM) and neutron (NM) matter are drawn in Fig. 2 as a function of the density. As expected, these interactions has no instability in the isospin channel. Notice however that in the isospin channel, the Landau parameters F ′ 0,SM obtained from the three interaction are more attractive than the one deduced from the microscopic G-matrix calculation [24] (filled squares in Fig. 2), especially in the high density region. However, it is not necessary to correct this channel since there are no instabilities in the density range considered in Fig. 2. In the spin channels, the Landau parameters G 0,SM , G ′ 0,SM and G 0,NM become more attractive as the density increases. The three Skyrme interactions have pathological behaviors in the spin channels G 0 and G ′ 0 for which we now propose a prescription to cure. From the density dependence of the Landau parameters represented in Fig. 2, it is clear that the standard Skyrme interactions are not repulsive enough in the spin channel, and most probably also in the isospin channel at higher density than the saturation density. The repulsion shall come from the effect of the three body force [24]. Through the density matrix expansion, it has been shown that in-medium many-body correlations give rize to density dependent terms in the nuclear functional [25]. These terms could be directly included in an effective density dependent nuclear interaction such as Skyrme or Gogny. The most important contribution comes from the scalar-density and it is the only density which has been considered. However, spin, isospin and spin-isospin density dependent terms shall also be considered from the density matrix expansion.

Here, we will study the impact of the new density dependent terms such as the spin and spin-isospin densities:

V add. (r 1 , r 2 ) = 1 6 t s 3 (1 + x s 3 P σ )[ρ s (R)] γs δ(r) + 1 6 t st 3 (1 + x st 3 P σ )[ρ st (R)] γst δ(r) (2)
where P σ = (1+σ 1 •σ 2 )/2 is the spin-exchanged operator, r = r 1 -r 2 and R = (r 1 +r 2 )/2. In Eq. ( 2), we have introduced the spin density ρ s = ρ ↑ -ρ ↓ and the spin-isospin density ρ st = ρ n↑ -ρ n↓ -ρ p↑ + ρ p↓ . Spin symmetry is satisfied if the power of the density dependent terms γ s and γ st are even.

In the literature, it has already been proposed to add new terms to the standard Skyrme effective interaction. One of the motivation is that the HF single particle levels are usually too widely spaced at the Fermi surface. This shall be corrected by including the coupling of single particles to vibrations which increase the effective mass of the single particle states from m * /m < 1 to m * /m ≥ 1 at the Fermi surface. This dynamical effect could be mimic by introducing a gradient term in the density dependent interaction (see Refs. [26,27] and references therein). Such a term has also been included in global fits of mass formula to make nucleosynthesis calculations as accurate as possible [28]. However, the removal of the spin instability makes symmetric matter unstable at high density [29]. Up to now, there is no satisfactory additional term which prevent the matter to fall into spin instabilities.

In the following, we adopt the notations of Ref. [17] where the density functional H is expressed as a sum of the kinetic term K, a zero range term H 0 , a density dependent term H 3 , an effective-mass term H eff and some additional terms coming from spin-orbit coupling, spin and gradient coupling and coulomb interaction. The additional terms (2) modify the density dependent part, H 3 to be H 3 +H s 3 +H st 3 , where the additional density dependent terms reads

H s 3 = t s 3 12 ρ γs s (1 + x s 3 2 )ρ 2 + x s 3 2 ρ 2 s -(x s 3 + 1 2 )(ρ 2 n + ρ 2 p ) - 1 2 (ρ 2 sn + ρ 2 sp ) , (3)
H st 3 = t st 3 12 ρ γst st (1 + x st 3 2 )ρ 2 + x st 3 2 ρ 2 s -(x st 3 + 1 2 )(ρ 2 n + ρ 2 p ) - 1 2 (ρ 2 sn + ρ 2 sp ) , (4)
where ρ sn = ρ n↑ -ρ n↓ and ρ sp = ρ p↑ -ρ p↓ . The mean field U q , where q = n, p, is then corrected to be

U add. q = t s 3 12 ρ γs s {(2 + x s 3 )ρ -(1 + 2x s 3 )ρ q } + t st 3 12 ρ γst st {(2 + x st 3 )ρ -(1 + 2x st 3 )ρ q }. (5)
In symmetric nuclear matter the Landau parameters G 0 and G ′ 0 [30] are also modified by the following additional terms

G add. 0 N 0 = t s 3 48 γ s (γ s -1)[3ρ 2 ρ γs-2 s -(2x s 3 + 1)ρ 2 t -ρ 2 st ] + t st 3 12 (x st 3 - 1 2 )ρ γst st + t s 3 24 (x s 3 - 1 2 ) (γ s + 1) (γ s + 2) ρ γs s , (6)
G ′add. 0 N 0 = t st 3 48 γ st (γ st -1)[3ρ 2 + (2x s 3 -1) ρ 2 s -(2x st 3 + 1)ρ 2 t ]ρ γst-2 st - t s 3 24 ρ γs s - t st 3 48 (γ st + 2)(γ st + 1)ρ γst st . (7)
In pure neutron matter, the additional terms for G 0 are

G add. 0 N 0 = t s 3 24 (1 -x s 3 )γ s (γ s -1)ρ 2 ρ γs-2 s + t st 3 12 x st 3 -1 ρ γst st + t s 3 24 (x s 3 -1) (γ s + 2)(γ s + 1)ρ γs s . (8)
Non-trivial result for the Landau parameters G add. 0 and G ′add. 0 imposes a condition on the powers of the density dependence in Eq. ( 2) to be uniquely fixed as γ s = 2 and γ st = 2. Notice that spin symmetry is also satisfied by these selections. It turns out that the additional contributions (5) to the mean field are simply null in spin-saturated systems, as the densities ρ s = 0 and ρ st = 0. It is thus possible to add the new terms (2) to the existing Skyrme forces without destroying good properties of the original Skyrme interactions. We obtain the contributions to the Landau parameters

G add. 0 = N 0 8 t s 3 ρ 2 , (9)
G ′add. 0 = N 0 8 t st 3 ρ 2 , (10)
in spin-saturated symmetric matter and

G add. 0 = N 0 12 t s 3 (1 -x s 3 )ρ 2 . ( 11
)
in spin-saturated neutron matter. The contributions to the Landau parameters G 0 and G ′ 0 in symmetric matter in Eq. ( 9) and ( 10) depend only on the parameters t s 3 and t st 3 , while that for the Landau parameter G 0 in neutron matter in Eq. ( 11) depends on t s and x s 3 . We shall then first fix t s 3 and t st 3 so as to reproduce the Landau parameters in symmetric matter, and then we fix x s 3 in neutron matter. Since the Landau parameters G 0 and G ′ 0 are independent of x st 3 , this parameter can be set as x st 3 =0. We show in Fig. 3 the contributions of the new terms for the Landau parameters G 0 and G ′ 0 in symmetric matter and for G 0 in neutron matter added to the three effective interactions SLy5, LNS and BSk16. We compare the original Landau parameters in Fig. 2 (solid lines) with the ones which include the contributions of the new terms ( 6), (7) and (8). The dashed lines correspond to different values of the new parameters t s 3 , t st 3 and x s 3 with the step indicated in each graph. The values for the parameters t s 3 , t st 3 and x s 3 resulting from the best adjustment for asymptotic behavior at very high density to the Brueckner HF calculations are given in Tab. 2 and are labelled SLy5st, LNSst and BSk16st respectivelly. With the parameters given in Tab. 2, it is confirmed that the correction terms to the Landau parameters ( 9)-( 11) are repulsive enough to remove the spin instabilities. There are however still differences between the corrected Landau parameters and the ones given by the G-matrix which could be originated from the tensor interaction. We can see that the LNS interaction, fitted originally to the equation of state deduced from the G-matrix, gives the best fitted Landau parameters among the three interactions. Finally, it could be noticed that the corrected Landau parameters G ′new 0 are increased by about +0.3 which makes it closer to the empirical value. Notice that the empirical value is estimated to be 1.3±0.2 from Wood-Saxon single-particle states plus one-pion and rho meson models (see for instance Tab.I of Ref. [31] and references therein and also Ref. [32]) and 0.7±0.1 from self-consistent Skyrme mean-field models [33].

RPA response functions and neutrino propagation

As a consequence of the new terms the spin channels are more repulsive than the original Skyrme functional. The effect of the new terms could then modify the response function of collective spin modes already at ρ 0 . For instance, we show in Tab. 2 the Landau parameters G 0 and G ′ 0 deduced from the original Skyrme interactions and the one including the new terms G new 0 and G ′new 0 . The increased of the Landau parameters may have significant influence on spin and spin-isospin excitations such as Gamow-Teller states.

In the following, we calculate the RPA response function in nuclear matter at finite temperature,

S (S,T ) (q, ω, T ) = - 1 π 1 1 -e -ω/T Imχ (S,T ) (q, ω, T ) ( 12
)
where χ (S,T ) (q, ω, T ) is the susceptibility [START_REF] Fetter | quantum Theory of Many-Particle Systems[END_REF] obtained as the solution of the Bethe-Salpeter equation [START_REF] Garcia-Recio | [END_REF]. In Fig. ( 4), we show the response functions in the spin channels (S = 1) at T = 0 MeV and at densities ρ = ρ 0 and ρ = 2ρ 0 calculated by the LNS and LNSst interactions. The HF solution (dotted line) is compared with the RPA using the original LNS Skyrme interaction (dashed line) and also with the RPA including the new terms in LNSst (solid line). According to the semi-classical Steinwedel-Jensen model [START_REF] Ring | The nuclear many body problem[END_REF] the optimal transfered momentum to compare with nuclei shall be q = π/2R where R is the radius of the nuclei. For 208 Pb, one thus obtain q=0.22 fm -1 . For ρ = ρ 0 , the effect of the new terms is to move the collective mode to slightly higher energies by 0.5-1 MeV. For ρ = 2ρ 0 , the (S = 1, T = 1) Gamow-Teller channel is not far from being unstable from the original Skyrme interaction and the low energy collective mode is being formed. By the new term t st 3 , the low energy mode is suppressed and the strength is reduced substantially at ω > 0. A reduction of the strength at low energy is also observed for the (S = 1, T = 0) channel. This effect shall be seen in the calculation of microscopic processes such as the neutrino cross section.

Neutrinos play a crucial role in physics of supernova explosions [START_REF] Bethe | [END_REF] and in the early evolution of their compact stellar remnants [38,39,40]. To clarify the effect coming from the additional interaction (2), we consider the case of pure neutron matter. The scattering of neutrinos on neutrons is then mediated by the neutral current of the electroweak interaction. In the non-relativistic limit, the mean free path of neutrino with initial energy E ν is given by [41]

λ -1 (E ν , T ) = G 2 F 16π 2 dk 3 c 2 V (1 + cos θ) S (S=0) (q, ω, T ) + c 2 A (3 -cos θ) S (S=1) (q, ω, T ) , (13)
where G F is the Fermi constant, c V (c A ) is the vector (axial) coupling constant, k 1 = (E ν , k 1 ) and k 3 are the initial and final neutrino four-momenta, while q = k 1 -k 3 is the transferred four-momentum, and cos θ = k1 • k3 . In the following, we impose the average energy E ν = 3T in MeV [40]. In Eq. ( 13), the contribution of the response function S (S) (q, ω, T ) is clearly identified. It describes the response of neutron matter to excitations induced by neutrinos, and contain the relevant information on the medium. The vector (axial) part of the neutral current gives rise to density (spindensity) fluctuations, corresponding to the spin S = 0 (S = 1) channel. The neutrino mean free path is shown in Fig. 5 as a function of the baryonic density and for different temperatures (T =1, 10, 20 MeV). The neutrino mean free path calculated with the original LNS Skyrme interaction (dashed line) is strongly reduced at high density as a consequence of the onset of the spin instabilities [43,[START_REF] Margueron | The Nuclear Many-Body Problem[END_REF]. The inclusion of the new density dependent interaction (2) in LNSst removes the spin instability and reduces the response function as shown in Fig. 4. As a consequence, the mean free path is increased and the matter is more transparent to neutrino as the density increases. This result is similar to a previous calculation of neutrino mean free path deduced directly from a microscopic G-matrix [44]. It illustrates the important contribution of the spin channel to the neutrino mean free path at high density. It is also clear from Fig. 5 that the effects of the correlations are not washed out by the temperature in the range going up to T = 20 MeV.

Conclusions

We have proposed the extension of the Skyrme interaction (2) which conserves its simplicity and extends its domain of stability to large densities and large isospin asymmetries. The parameters of the new terms have been adjusted based on microscopic G-matrix calculations. For the three interactions SLy5st, LNSst and BSk16st, the spin channels become more repulsive compared to the original interactions SLy5, LNS and BSk16; and the dimensionless Landau parameter G ′ 0 is increased by about +0.3. The simplicity of the extended Skyrme interaction makes possible extensive calculations not only in asymmetric and dense matter but also in asymmetric finite nuclei. We have calculated RPA response functions of spin channels and also neutrino mean free path in dense matter. The collapse of the neutrino mean free path at high density is suppressed by the newly added terms in the Skyrme interaction and the overall effect of the RPA correlations makes dense matter more transparent for neutrino propagation by a factor of 2 to 10 depending on the density.

We would like to extend further the applications of our new spin-density dependent interactions for the systematic study of odd mass nuclei, the spin-dependent excitations in finite nuclei and also infinite nuclear matter. Eventually our ultimate goal is to construct the global energy density functional for the spin and spin-isospin dependent interactions including tensor and two-body spin-orbit forces. Concomitantly, an extensive experimental campaign dedicated to the measurement of Gamow-Teller and also other spin-isospin modes is proposed at RIKEN [45]. The new data will be very usefull to compare with our global energy density functional.

Table 1. Nuclear matter properties of the Skyrme interactions SLy5, LNS, BSk16 at saturation density ρ 0 ; the binding energy a v , the incompressibility K ∞ , the symmetry energy a s , the density ρ f of the ferromagnetic instability. 2) are set to be γ s = γ st = 2, and x st 3 = 0. Are also shown the dimensionless Landau parameters G 0 , G new 0 , G ′ 0 G ′ new 0 deduced from the original Skyrme interactions and the new interactions including the additional terms, respectively.

ρ 0 a v K ∞ a s ρ f /ρ 0 (fm -3 ) (MeV) (MeV) (MeV ) SLy5
t s 3 t st 3 x s 3 G 0 G new 0 G ′ 0 G ′ new 0 SLy5st 0.6×10 4 2×10 4 -3 1.

Figure 1 .Figure 2 . 1 Figure 3 ., t st 3 and x s 3 . 3 3 )Figure 4 .Figure 5 .

 121333345 Figure1. Ferromagnetic phase diagram for various effective Skyrme interactions. The horizontal axis shows the critical density ρ f at which the asymmetric matter becomes unstable, while the vertical axis shows the proton fraction x p = ρ p /ρ. The matter is spin symmetric for smaller density than ρ f , while it becomes ferromagnetic for larger density than ρ f . See the text for details.

Table 2 .

 2 Parameters of the additional terms t s 3 (in MeV.fm 3γs-2 ), t st 3 (in MeV.fm 3γst-2 ) and x s 3 for the interactions SLy5st, LNSst and BSk16st in order to reproduce the realistic Landau parameters by Brueckner HF calculations. The powers of the density dependence in Eq. (

		0.16	-15.97	229.9	32	2.09
	LNS	0.1746	-15.32	210.9	33.4	2.48
	BSk16	0.1586	-16.05	241.6	30	1.24

Acknowledgments

This work was supported by the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the Program number C(2) 20540277.


J Margueron

S Goriely

M Grasso

G Colò

H Sagawa

Extended Skyrme interaction (II): ground state of nuclei and of nuclear matter

Keywords: numbers: 21.30.Fe, 21.10.Dr, 21.65.-f, 26.60.-c

We study the effect of time-odd components of the Skyrme energy density functionals on the ground state of finite nuclei and in nuclear matter. The spin-density dependent terms, which have been recently proposed as an extension of the standard Skyrme interaction, are shown to change the total binding energy of odd-nuclei by only few tenths of keV, while the time-odd components of standard Skyrme interactions give an effect that is larger by one order of magnitude. The HFB-17 mass formula based on a Skyrme parametrization is adjusted including the new spin-density dependent terms. A comprehensive study of binding energies in the whole mass table of 2149 nuclei gives a root mean square (rms) deviation of 0.575 MeV between experimental data and the calculated results, which is slightly better than the original HFB-17 mass formula. From the analysis of the spin instabilities of nuclear matter, restrictions on the parameters governing the spin-density dependent terms are evaluated. We conclude that with the extended Skyrme interaction, the Landau parameters G 0 and G ′ 0 could be tuned with a large flexibility without changing the ground-state properties in nuclei and in nuclear matter.

Introduction

Despite many theoretical and experimental investigations, the spin and the spin-isospin channels in either the ground and the excited states of nuclei are still widely open for future study [1,2,3,4,5,6]. It is indeed difficult to probe the spin and the spin-isospin channels of nuclear interaction since the ground states of nuclei are non-spin polarized in the case of even-even nuclei and at most polarized by the last unpaired nucleons in odd nuclei.

The analysis of spin and spin-isospin collective modes such as magnetic dipole (M1) and Gamow-Teller (GT) states gives access to the nuclear interaction in these channels. The Landau parameter G ′ 0 has been deduced from the analysis of the GT mode: a model based on Woods-Saxon single-particle states plus one-pion and rho meson exchange interactions gives G ′ 0 = 1.3 ± 0.2 (see Ref. [2,3] and references therein). A slightly different value G ′ 0 = 1.0 ± 0.1 was derived from observed GT and M1 strength distributions using the phenomenological energy density functionals DF3 [1,7]. Anyway, in both cases empirical single-particle energies are used (that is, for states close to the Fermi energy the effective mass m * /m is ≈ 1).

Self-consistent Hartree-Fock (HF) plus Random Phase Approximation (RPA) calculations constitute a somewhat different framework, in which the density of states around the Fermi energy is lower (or, equivalently, the effective mass is about 30% smaller). As compared with the empirical case, the unperturbed particle-hole transitions have larger energies and as a consequence one needs a smaller residual repulsive effect to fit the observed GT peak. It is not surprising, therefore, that self-consistent calculations of the GT resonance, performed using different Skyrme interactions in Ref. [6], point to G ′ 0 ∼ 0.6. Skyrme interactions are characterized by a spin-isospin G ′ 1 parameter as well, and specific terms of the effective mean field, like the spin-orbit potential, may also break the simple correlation between the GT properties and the parameter G ′ 0 . However, as widely used interaction like SLy5 have unrealistic (negative) values of G ′ 0 , it is undeniable that adding more flexibility to the spin-isospin part of Skyrme forces is useful.

The spin and spin-isospin component of the nuclear interaction is also reflected into the time-odd component of the mean field. The properties of even nuclei give constraints to the time-even component of the mean field while very little is known about properties of the time-odd mean fields. Time-odd components of the mean field compete with pairing correlations in determining the odd-even mass staggering [8]. Fast rotation induces time-odd components in the mean field [9] which could be probed from the measurement of the dynamical moments of superdeformed bands.

In Ref. [10], new spin-density dependent terms have been introduced on top of standard Skyrme forces in order to remove the ferromagnetic instability associated with all Skyrme parameterizations [5]. The new terms retain the simplicity and the good properties of Skyrme interactions for nuclear matter and the ground states of eveneven nuclei. However, these new terms slightly change the properties of odd systems. This work aims at studying quantitatively the effects of these new terms on the ground state properties of odd nuclei, in particular the total binding energy and the density distribution. Since these terms contribute only in odd nuclei, both odd-even and oddodd nuclei are considered in the present study. To provide an approximate maximal estimate of the effects while keeping our model simple, we perform HF calculations with the following approximation to treat odd nuclei. We use the equal filling approximation, so that the time-reversal symmetry is not broken, but with an additional ansatz. As within this scheme the spin-densities would be by definition equal to zero, when constructing the spin-densities with the wave function of the odd nucleon, we assume that the spin-up state is completely filled while the spin-down state is empty between the two possible spin orientations (or, equivalently, the opposite). We call this procedure for the construction of spin-densities the one-spin polarized approximation (OSPA). The OSPA gives an upper value of the contribution of the new terms.

The article is organized as follows: in Sec. II we remind the newly proposed spindensity dependent terms as in Ref. [10]. In Sec. III, we estimate how much these new spin-density dependent terms affect the total binding energy of odd nuclei. In Sec. IV, it will be shown that, introducing these terms in the most predictive HF-Bogoliubov (HFB) mass formula [11], the quality of the mass fit can be recovered with an optimal renormalization of the Skyrme parameters. In Sec. V we will analyze the ground state properties of infinite nuclear matter with respect to spin-polarization and give a range for the parameter of newly introduced spin-density x s 3 . Finally, conclusions and outlook are given in Sec. VI.

Extended Skyrme interactions

As described in Ref. [10], the new spin-density dependent terms added to the conventional Skyrme force are of the following form:

V s,st (r 1 , r 2 ) = 1 6 t s 3 (1 + x s 3 P σ )[ρ s (R)] γs δ(r) + 1 6 t st 3 (1 + x st 3 P σ )[ρ st (R)] γst δ(r) (1)
where P σ = (1+σ 1 •σ 2 )/2 is the spin-exchange operator, r = r 1 -r 2 and R = (r 1 +r 2 )/2. In Eq. ( 1), we have introduced the spin-density ρ s ≡ ρ ↑ -ρ ↓ and the spin-isospin-density ρ st ≡ ρ n↑ -ρ n↓ -ρ p↑ + ρ p↓ . Spin symmetry is satisfied if the power of the densitydependent terms γ s and γ st is even. The total energy E TOT in finite nuclei is related to the (local) energy density H(r) through

E TOT ≡ d 3 r H(r) . (2)
In the following, we adopt the notation of Ref. [12] where H is expressed as the sum of a kinetic term K, a zero-range term H 0 , a density-dependent term H 3 , an effectivemass term H eff , a finite-range term H fin , spin-orbit and spin-gradient terms (H so and H sg ), and eventually the Coulomb term H Coul . However, the expression for the energy density provided in [12] holds in the case of time-reversal symmetry; in odd nuclei, the energy density H acquires also a dependence on the spin-densities ρ s and ρ st , hereafter named H odd , even without additional terms in the force depending on these densities.

Expressions for H odd have been derived for instance in Refs. [9,13] or in Appendix I of Ref. [14]. For the reader's convenience we repeat here, in Appendix A, the expression for H odd .

The additional terms (1) modify the standard H 3 contribution to be H 3 + H s 3 + H st 3 , where the last two terms read

H s 3 = t s 3 12 ρ γs s (1 + x s 3 2 )ρ 2 + x s 3 2 ρ 2 s -(x s 3 + 1 2 )(ρ 2 n + ρ 2 p ) - 1 2 (ρ 2 sn + ρ 2 sp ) , (3)
H st 3 = t st 3 12 ρ γst st (1 + x st 3 2 )ρ 2 + x st 3 2 ρ 2 s -(x st 3 + 1 2 )(ρ 2 n + ρ 2 p ) - 1 2 (ρ 2 sn + ρ 2 sp ) , (4)
with ρ sn = ρ n↑ -ρ n↓ and ρ sp = ρ p↑ -ρ p↓ .

The additional terms to the mean field coming from H s 3 and H st 3 are

U s,st q = t s 3 12 ρ γs s (2 + x s 3 )ρ -(1 + 2x s 3 )ρ q + t st 3 12 ρ γst st (2 + x st 3 )ρ -(1 + 2x st 3 )ρ q . ( 5
)
where q = n, p.

As already mentioned, the additional contributions to the mean field are zero in even-even nuclei. Since most of the Skyrme interactions are adjusted on (few) eveneven nuclei, it is thus possible to add for these interactions the new terms (1) in a perturbative manner. The new four parameters t s 3 , x s 3 , t st 3 and x st 3 in Eq. ( 1) have been adjusted in Ref. [10] in order to reproduce the Landau parameters extracted from a Gmatrix calculation in uniform matter, while γ s = γ st = 2 is imposed by spin symmetry. Some other interactions, in particular those produced by the Brussels-Montréal group, are globally adjusted by fitting the properties (essentially the nuclear masses) of both even and odd nuclei. In this case, when including the new spin-density terms, a global re-adjustment of the interaction might be necessary.

In the following, we first analyze the effect of the new spin-density dependent terms for a few selected nuclei using the SLy5 Skyrme interaction for which the new spin-density dependent terms are added perturbatively. Later, on the basis of the latest BSk17 Skyrme parameter set, we study the global impact of our new terms on nuclear masses and show that the Skyrme parameters can be refitted to provide almost equivalent properties.

Ground states of Ca nuclei

According to the OSPA, we define the densities ρ s (r) and ρ st (r) in odd nuclei as

ρ s (r) = 1 4πr 2 i ϕ 2 i (r)m s (i) , (6)
ρ st (r) = 1 4πr 2 i ϕ 2 i (r)m s (i)m t (i), (7)
where m s (i) and m t (i) are the spin and isospin z-component for each single nucleon having the wave function ϕ i (r). The last occupied state fully contributes to the spindensity. It is then clear that the OSPA corresponds to maximizing the spin-density and its effects. Note that if the time-reversal symmetry is not broken (in the filling approximation, for instance) both spin-up and spin-down states must be degenerate and the densities ρ s and ρ st are zero. Since in this work we aim at an approximate and maximal estimate and not at a precise prediction of the effects, we use the HF method with the OSPA to treat odd nuclei.

All calculations in this Section are performed with the SLy5 parameterization for the conventional Skyrme force and the additional parameters for the spin-density terms given in Table 2 of [10]. We choose two systems: 41 Ca and 49 Ca. Being built on top of double magic nuclei, pairing is neglected for these nuclei in this section. Both nuclei are odd-even, then the proton spin-density is zero and the spin-isospin-density is equal to the spin-density for neutrons. For the nucleus 41 Ca the spin-density is built with the neutron 1f 7/2 wave function while in 49 Ca it is constructed with the neutron 2p 3/2 wave function. The choice of these nuclei has been driven by the fact that one of the nuclei, 41 Ca, has a spin-density which probes the external region of the nucleus, while for the other nucleus, 49 Ca, the spin-density may probe more the central part. We could not use the OSPA if the single-particle wave functions changed when adding one or two nucleons. Therefore, as a necessary step to proceed with the evaluation of the impact of spin-density dependent terms within the OSPA, we have first checked that the wave function of the neutron state 1f 7/2 does not change appreciably when passing from the even nucleus 40 Ca to the next even isotope with N + 2 neutrons, 42 Ca. An analogous check has been made for the wave function of the proton state 1f 7/2 in 40 Ca and in the Z + 2 isotone 42 Ti. Also, the spin-density calculated with the OSPA for 41 Ca has been compared with half the difference of the neutron densities of 42 Ca and 40 Ca, where effects coming from the rearrangement of the deeper state are also included. Very small differences are found, mainly in the central region. Analogous results are obtained for 48 Ca, 49 Ca and 50 Ca.

The spin-densities in 41 Ca and 49 Ca are plotted in the two panels of Fig. 1 together with the neutron densities of both nuclei. For comparison, the neutron densities of the nearest even-even nuclei 42 Ca and 50 Ca are also displayed. The spin-densities and the neutron densities have been calculated in the two odd-even systems. Two independent calculations have been actually performed with and without the new spindensity dependent terms (1) and only negligible differences have been found so that they are not appreciable in the figures.

Since the contributions to the total energy E TOT coming from the spin-density dependent terms (3) and ( 4) are proportional to the square of the spin densities represented in Fig. 1, these contributions are expected to be negligible as compared to the usual density dependent term H 3 . To be more quantitative, we have made several calculations for the ground-state energies that are summarized in Tables 1 and2. The partial contributions to the ground-state energy (2) are written

E MF ≡ d 3 r (H 0 + H 3 + H eff + H fin + H sg ), E so ≡ d 3 r H so , E Coul ≡ d 3 r H
Coul and E kin ≡ d 3 r H kin , using the notations of Ref. [12]. In Table 1 the total energy, the mean field, the spin-orbit, the Coulomb and the kinetic contributions to the total energy and the single-particle energy (for the neutron states 1f 7/2 or 2p 3/2 ) are provided for the odd nuclei 41 Ca and 49 Ca and for the nearest even-even nuclei 42 Ca and 50 Ca. The calculation for the odd nuclei are performed either with or without the spin-density dependent interaction (1). The difference between the total energy, and its contributions, without and with the corrections ∆E is shown to be less than 50 keV in both odd nuclei 41 Ca and 49 Ca. It is interesting to notice that the kinetic energy contributes to reduce the impact of the spin-dependent interaction (1). The spin-orbit and the Coulomb energies are very weakly affected by the spin-density terms.

In Table 2 the total and separate contributions coming from the time-odd terms of the Skyrme interaction (see Eqs. (A.2)-(A.5) in Appendix A) are provided, in the case of the odd nuclei 41 Ca and 49 Ca: these contributions are calculated perturbatively within the OSPA. These terms are classified according to the standard notations and labeled as E odd 0 , E odd eff+fin+sg and E odd 3 which are the contributions from the central, the momentum dependent and the density dependent terms, respectively. The corrections E odd 0 and E odd 3 give a dominant and repulsive contribution which increases the total energy while the correction E odd eff+fin+sg is smaller and attractive. The total correction remains quite small, that is, of the order of 0.15-0.3 MeV for both nuclei. Note that the sign of these corrections could change from one Skyrme interaction to another, but such corrections in Table 2 remain larger than ∆E TOT in Table 1. From the quantitative comparison shown in Tables 1 and2, we can infer, as expected, that the new spin-density dependent terms (1) modify the ground state energies of odd nuclei much smaller than those coming from the time-odd terms of the standard Skyrme interaction.

Global adjustment on the nuclear chart

Some Skyrme interactions have been determined by fitting the parameters to essentially all of the available mass data and therefore are constrained to even as well as odd systems. In this case, the new spin-density dependent terms (1) added to the standard Skyrme interaction may modify the quality of the fit. To study the impact of the new terms on the prediction of nuclear masses, we consider now the latest and most accurate HFB-17 mass formula (with a rms deviation of 0.581 MeV on the 2149 measured masses of [17]) obtained with the BSk17 Skyrme force [11].

If we consider the parameters of the additional spin-density dependent interaction (1) determined in Ref. [10], namely t s 3 = 2 × 10 4 MeV fm 4 , t st 3 = 1.5 × 10 4 MeV fm 4 , x s 3 = -2, x st 3 = 0 and γ s = γ st = 2, we find that the impact of the new terms on nuclear masses are relatively small, as already discussed in Sect. III. Fig. 2 shows the mass difference obtained by a spherical HFB calculation when the spin terms are added or not. The nuclear masses of odd-A and odd-odd nuclei are globally increased by a value of the order of 100 keV. For light nuclei this correction is the largest and can reach at most 350 keV. As already shown in Ref. [10], the spin-density terms are repulsive and leads to an increase of the rms deviation with respect to all the 2149 measured masses from 0.581 MeV to 0.591 MeV, keeping the good quality of the mass fit. Deterioration can potentially be avoided if the force parameters are re-adjusted to the nuclear mass data.

We have determined a new BSk17st Skyrme force which essentially corresponds to the BSk17 force, but for which the Skyrme as well as the pairing parameters have been slightly renormalized by a new fit on the whole set of mass data. The new parameter set has been built with the idea of evaluating a maximum effect of the new spin-density dependent term (in the same spirit of the OSPA discussed above). The values adopted for the parameters are t s 3 = 4 × 10 4 MeV fm 4 , t st 3 = 3 × 10 4 MeV fm 4 , x s 3 = -0.5, x st 3 = 0 and γ s = γ st = 2. t s 3 and t st 3 are twice larger than in Ref. [10] and this leads to Landau parameters G 0 = -0.03 and G ′ 0 = 0.99 in symmetric matter at saturation density, whereas the parameter set of Ref. [10] is associated with lower values, namely G 0 = -0.36 and G ′ 0 = 0.75. As explained in the Introduction, the value of G ′ 0 may be considered large in keeping with the effective mass 0.8 of BSk17, yet still quite acceptable for the purpose of the present study. With a value of t s 3 twice larger than BSk16st, the value of x s 3 = -2 in Ref. [10] has been modified consistently to be -0.5 to keep the contribution to the Landau parameter G 0 in spin-saturated infinite neutron matter identical to the one determined in Ref. [10] [see Eq. ( 11) of Ref. [10]]. As far as the parameter x st 3 is concerned, there is so far no constraint that could guide us in fixing its value. For this reason the zero value was assumed in Ref. [10]. However, as shown in Sec V, the value of x st 3 influences the stability of the partially polarized neutron matter, i.e., the lower its value, the higher the barrier between the S = 0 and S = 1 configuration. For this reason, as discussed in Sec. V, the value of x st 3 is set to -3. The strategy of the mass fit is the same as the one described in Ref. [11,16]. In particular, the mass model is given from deformed HFB and the pairing force is constructed from the microscopic pairing gaps of symmetric nuclear matter and neutron matter calculated from realistic two-and three-body forces, with mediumpolarization effects included. To accelerate the fit, a first estimation of the energy gained by deformation is performed and subtracted to the experimental masses. A first series of parameters are then obtained from the comparison of spherical HFB mass model with the corrected experimental masses. The corrections due to deformation are then reevaluated and a new fit is performed. This fast procedure is repeated until convergence. The isoscalar effective mass m * s /m is constrained to 0.80 and the symmetry energy at saturation J to be 30 MeV in order to reproduce at best the energy-density curve of neutron matter [15] from realistic two-and three-nucleon forces. Note that the parameters of the additional spin-density interaction are not fitted in this procedure.

The final force parameters labeled BSk17st and resulting from a fit to essentially all mass data are given in Table 3. It can be seen that there is little difference between the parameters of the BSk17 and BSk17st forces; note that the parameters of the rotational and vibrational corrections [16], are identical for both forces. The same holds for the parameters of infinite matter with the incompressibility coefficient K v = 241.7 MeV, the volume energy coefficient a v = -16.053 MeV and the isovector effective mass of m * v /m = 0.784, as in Ref. [11].

The rms residuals for the BSk17 and BSk17st sets are compared in Table 4. The inclusion of the new spin-density dependent terms (1) which slightly deteriorated the accuracy of the BSk17 force leads now even to a small improvement of the predictions by about 6 keV, with the BSk17st force parameters. Both forces have been used to estimate the mass of the 8508 nuclei with 8 ≤ Z ≤ 110 and lying between the proton and neutron drip lines. Differences of no more than roughly ±0.5 MeV are found on the entire set.

Finally note that the nuclear binding energies remain extremely insensitive to the value adopted for the parameters x s 3 and x st 3 . In particular, setting x s 3 = -2 instead of -0.5 decreases the binding energy by no more than 10 keV. Similarly, a change of x st 3 from zero to -3 impact the masses by maximum 8 keV. Therefore, these two parameters should be constrained rather by stability conditions of polarized or non-polarized infinite nuclear matter, as shown in the next Section.

Ground state of infinite nuclear matter

In Ref. [10], it has been shown that the new spin-density dependent interaction (1) stabilize non-polarized matter with respect to spin-fluctuations. As shown in Fig. 3, with the new terms, the Landau parameters G 0 and G ′ 0 remains larger than -1 at all densities for SLy5st, LNSst [19] (which include the new terms as parametrized in Ref. [10]) and the BSk17st forces However, it has not been checked if the true ground state is really that of nonpolarized matter. To do so, the energy for different spin-polarizations should be compared with that of non-polarized matter. This is done in the next two subsections for both symmetric nuclear matter and neutron matter

Symmetric nuclear matter

The difference of the binding energy of spin-polarized matter to that of spin-symmetric matter, E/A(δ S , ρ) -E/A(δ S = 0, ρ), is represented in Fig. 4 as a function of the polarization δ S = (ρ ↑ -ρ ↓ )/ρ. In the left panel, we have represented the binding energy of the BSk17 Skyrme interaction without the new spin-density dependent terms (1). The instability occurs between ρ=0.18 and 0.2 fm -3 . At ρ=0.2 fm -3 the minimum energy is obtained for a polarization δ S =0.76.

The binding energy of BSk17st which includes the spin-density dependent terms is represented in the right panel of Fig. 4. As expected, the energy of non-polarized matter is convex around δ S = 0, but there is a change of convexity for large values of δ S ∼ 0.8. We have indeed observed a large influence of the parameter x s 3 on the binding energy of fully polarized matter. In Fig. 5 are represented the binding energies E/A(δ S , ρ) -E/A(δ S = 0, ρ) for the three modified Skyrme interactions BSk17st, LNSst and SLy5st for which we changed the values of the parameter x s 3 . Its values are indicated in the legend of Fig. 5. We fixed the density ρ=0.6 fm -3 to be the highest value where the nuclear Skyrme interaction is applied. For values of the parameter x s 3 =-3 the ground state of nuclear matter is fully polarized (δ S = 1) for the interactions BSk17st and LNSst. Increasing the value of the parameter x s 3 from -3 to 0, the binding energy of fully polarized matter is going up in Fig. 5. There is then a critical value above which non-polarized matter is the ground-state of nuclear matter.

One could obtain an estimate of this critical value by analyzing the contribution of the new spin-density terms (1) in spin-polarized symmetric matter. It reads

H s 3 (sym.) = t s 3 16 ρ 2 ρ γs s 1 + 2x s 3 -1 3 δ 2 S , (8)
and H st 3 (sym.) = 0 since ρ st = 0. The term ( 8) is zero for the spin-symmetric matter with ρ s = 0 and is always positive for δ S = 1 if one chooses x s 3 > -1. It is thus clear that one necessary condition for the spin-symmetric matter to be the absolute ground state at all densities is x s 3 > -1. This is the case for the adopted value of x s 3 = -0.5 for BSk17st. Nevertheless, as shown in Fig. 5 for instance for SLy5st, the stability of spin-symmetric matter could be obtained even if x s 3 < -1 at the density ρ=0.6 fm -3 . At lower density, SLy5st is also stable, but not at higher density.

We remind that from the analysis of the Landau parameters the stability around spin-symmetric matter requires that x s 3 < 1 (see Eq. ( 11) of Ref. [10]). As a conclusion, one could adjust the parameter x s 3 inside the range -1 x s 3 < 1.

Neutron matter

The case of pure neutron matter is somehow very peculiar. The correction due to the spin-density dependent terms reads

H s 3 (neut.) = t s 3 24 ρ 2 ρ γs s (1 -x s 3 ) 1 -δ 2 S . (9)
It is then clear that the correction is zero for δ S = 0 and also for δ S = 1. This property is related to the anti-symmetrization of the interacting nucleons. Indeed, in fully polarized neutron matter, the quantum numbers for spin and isospin are S = 1 and T = 1 while the new spin-density dependent interaction (1) act in the L = 0 channel. The new spin-density dependent interaction (1) have thus no effect at all in the purely spinpolarized neutron matter. Only odd L terms could play a role in the fully polarized neutron matter. This property has been used to provide a necessary condition to remove the spin instabilities and lead to the condition -5/4 < x 2 < -1 [20]. This condition has been used in the fitting procedure of SLy5 [12] and it explains the robustness of the spin-symmetric ground state for this interaction. However if more flexibility in the Skyrme parameters is necessary, it might be interesting to introduce an interaction of the following form

t s 5 (1 + x s 5 P σ ) k ′ ρ s (R) • δ(r)k . (10)
This L = 1 term will not contribute to spin-symmetric matter and could be adjusted to fit the energy of fully polarized matter. In Fig. 6, it is shown that the new spin-density terms (1) contribute to the binding energy for partially polarized matter and tend to stabilize the state δ S = 0. After the ferromagnetic transition, the state δ S = 0 is not any more the absolute ground state in pure neutron matter. However, the new spin-density terms generate a potential barrier between the non-polarized and fully polarized states. The height of the barrier depends on the chosen parameters of the new spin terms (1) as well as on the neutron matter density. In the right panel of Fig. 6, the top of the barrier at the ferromagnetic transition is located around the polarization δ S = 1/ √ 2 with a height (in MeV per nucleon) of 1 96

t s 3 (1 -x s 3 ) + t st 3 (1 -x st 3 ) ρ 3 . (11)
So, the higher the density, the higher the barrier. This barrier height is always positive since the curvature of the binding energy around δ S = 0 is related to the Landau parameter G 0,N M in neutron matter which is larger than -1 at all densities (see Fig. 3).

For the BSk17st force, at ρ ≃ ρ f = 0.19 fm -3 this barrier amounts to about 10 MeV per nucleon. From Eq. ( 11), it can also be seen that the parameter x st 3 influences the height of the barrier. The lower x st 3 , the larger the barrier. For BSk17st, we set x st 3 = -3 to get a barrier above the non-polarized ground state of the order of 10 MeV per nucleon. This condition is chosen with respect to the theoretical predictions [21,22,23,24] that nuclear matter is spin-symmetric up to reasonable high densities (see for instance discussion in the introduction of Ref. [10]). With a barrier height of the order of 10 MeV per nucleon, newly born neutron stars with typical temperature going from 1 to 5 MeV might not be the site of a ferromagnetic phase transition. The transition towards a non-polarized cold neutron star shall then be stable and the remaining neutron star spin-symmetric. Notice however that despite the theoretical predictions that dense matter is not spin-polarized [21,22,23,24], there are no strong evidences against the occurrence of ferromagnetic phase transition from observation of neutron stars. Indeed, a spin-polarized phase in the core of neutron stars might induce the very huge magnetic fields 10 15-16 G yet unexplained that have been proposed as the driving force for the braking of magnetars [25].

Conclusions

The occurrence of the spin instability beyond the saturation density is a common feature shared by different effective mean-field approaches such as Skyrme HF, Gogny HF [26] or relativistic HF [27]. The analysis of the spin component of the Skyrme interaction as well as its extensions might thus guide us to a wider understanding of the spin channel in general for nuclear interaction. There are many reason for looking at this channel. For instance for its competition with pairing correlations in the odd-even-mass staggering [8], for rotating superdeformed nuclei [9], for a better description of GT response, and for all applications in astrophysics such as for instance predictions of β-decay half-lives of very neutron rich nuclei produced during the r-process nucleosynthesis [7], reliable calculation of neutron star crust properties such as ground-states and collective motion [28], for 0νand 2ν double beta decay processes, for URCA fast cooling, and also for neutrino mean free path in proto-neutron stars.

In this paper we have carefully analyzed the ground state properties of finite nuclei and infinite matter obtained by the extended Skyrme interactions with the spin-density terms proposed in Ref. [10]. In finite even nuclei, the new spin-density interaction (1) are simply zero and from the OSPA we have shown that these terms has only negligible contributions to the ground-state of odd nuclei. These results has been obtained either by introducing the new spin-density dependent terms in a perturbative way to existing Skyrme interactions such as SLy5st or performing a global adjustment of the parameter set on the nuclear chart. A new mass formula HFB-17st adjusted in the whole isotope chart (2149 nuclei) is obtained with the rms deviation of about 575 keV. From the analysis of the ground state of nuclear matter, a range for the parameter x s 3 is restricted to -1 x s 3 < 1 in order to stabilize the spin-symmetric matter. The case of neutron matter is also discussed and it is shown that the new terms (1) with the relative angular momentum L = 0 have no contribution to fully polarized neutron matter. Thus, it has been shown that by using the extended Skyrme interactions [10], the Landau parameters G 0 and G ′ 0 could be tuned to realistic values without altering the ground-state properties in odd nuclei as well as of nuclear matter.

Table 1. Total energy, mean field, spin-orbit, Coulomb and kinetic contributions to the total energy (third column) and single-particle energy of the neutron state 1f 7/2 for 41-42 Ca and 2p 3/2 for 49-50 Ca calculated, for the nearest even nuclei and for the odd nuclei, without/with the spin-dependent terms (1) in the mean field. ∆E is the difference of energy with and without the spin-dependent terms (1).

Nucleus

E

Figure 1 .

 1 Figure 1. Top panel: neutron density of 42 Ca (solid line) and 41 Ca (dashed line) and spin-density in 41 Ca (squares). The densities are given in units of fm -3 . Bottom panel: the same as top panel, but for 50 Ca and 49 Ca.

Figure 2 . 17 0Figure 3 .

 2173 Figure 2. Difference between the nuclear mass of odd-A and odd-odd nuclei obtained with the BSk17 force with and without the additional spin-density dependent terms. The calculation is made here assuming spherical symmetry.

Figure 4 .Figure 5 .

 45 Figure 4. Difference between the binding energies E/A(δ S , ρ) -E/A(δ S = 0, ρ) in symmetric matter for BSk17 and BSk17st Skyrme interaction as a function of the spin polarization δ S = (ρ ↑ -ρ ↓ )/ρ for the different densities, as indicated in the legend (in units of fm -3 ).

Figure 6 .

 6 Figure 6. Same as Fig.4, but for neutron matter.

Table 2 .

 2 Total and separate contributions to the energy from the time-odd (spin symmetry breaking) terms of the SLy5 Skyrme interaction[14,13].

		TOT	E MF	E so	E Coul	E kin s.p. energy
		(MeV)	(MeV) (MeV) (MeV)	(MeV)	(MeV)
	42 Ca	-362.591 -1111.434	-9.173 72.023 685.993	-9.66
	41 Ca	-352.942 -1081.395	-5.259 72.116 661.596	-9.64
	41 Ca with (1) -352.918 -1081.359	-5.259 72.115 661.584	-9.64
	∆E	0.024	0.036	0.000 -0.001	-0.012
	50 Ca	-429.654 -1326.381 -33.958 70.905 859.779	-5.84
	49 Ca	-423.876 -1305.865 -33.639 71.105 844.523	-5.70
	49 Ca with (1) -423.825 -1305.754 -33.634 71.102 844.461	-5.70
	∆E	0.051	0.111	0.005 -0.003	-0.062

Table 3 .

 3 Parameter sets for BSk17 and BSk17st: the first 15 lines give the Skyrme and additional spin parameters, the 16th the spin-orbit term, the 17th to 20th lines the pairing parameters and the last 4 lines the Wigner correction[16]. See the text and Ref.[11] for more details

		BSk17	BSk17st
	t 0	-1837.33	-1837.19
	t 1	389.102	388.916
	t 2	-3.1742	-5.3076
	t 3	11523.8	11522.7
	t s 3	0	40000
	t st 3	0	30000
	x 0	0.411377 0.410279
	x 1 -0.832102 -0.834832
	x 2	49.4875	29.0669
	x 3	0.654962 0.655322
	x s 3	0	-0.5
	x st 3	0	-3
	γ	0.3	0.3
	γ s	-	2
	γ st	-	2
	W 0	145.885	146.048
	f + n	1.000	1.000
	f -n	1.044	1.045
	f + p	1.055	1.059
	f -p	1.050	1.059
	V W	-2.00	-2.06
	λ	320	410
	V ′ W	0.86	084
	A 0	28	28

Table 4 .

 4 Rms (σ) deviations between experimental data[17] and HFB-17 or HFB17st predictions. The first line refers to all the 2149 measured masses M , the second to the masses M nr of the subset of 185 neutron-rich nuclei with S n ≤ 5.0 MeV, the third to the 1988 measured neutron separation energies S n and the fourth to 1868 measured beta-decay energies Q β . The fifth line shows the comparison with the 782 measured charge radii[18]. Note that units for energy and radius are MeV and fm, respectively.

		HFB-17 HFB-17st
	σ(2149 M )	0.581	0.575
	σ(M nr )	0.729	0.738
	σ(S n )	0.506	0.495
	σ(Q β )	0.583	0.585
	σ(R c )	0.0300	0.0302

Acknowledgments S.G. acknowledges financial support from FNRS. The work is partially supported by COMPSTAR, an ESF Research Networking Programme and the Japanese Ministry of Education, Culture, Sports, Science and Technology by Grant-in-Aid for Scientific Research under the program numbers (C) 20540277.

Appendix A. Time-odd components in the mean field of the Skyrme interaction

In odd nuclei, the energy density H acquires a dependence on the spin-densities ρ s and ρ st [9,13]. Respecting the decomposition of the Skyrme energy functional proposed in Ref. [12], the components of H odd are

2)

3)

where τ s (τ st ) is the spin (spin-isospin) kinetic density energy defined as τ s = τ ↑ -τ ↓ (τ st = τ n↑ -τ n↓ -τ p↑ + τ p↓ ) and j 2 s (j 2 st ) is the spin (spin-isospin) current.