Aziz Rabhi
email: rabhi@teor.fis.uc.pt

Constança Providência

Dense stellar matter with trapped neutrinos under strong magnetic fields

Keywords: numbers: 26.60.+c, 12.39.Ba, 21.65.+f, 97.60.Jd

We investigate the effects of strong magnetic fields on the equation of state of dense stellar neutrino-free and neutrino-trapped matter. Relativistic nuclear models both with constant couplings (NLW) and with density dependent parameters (DDRH) and including hyperons are considered. It is shown that at low densities neutrinos are suppressed in the presence of the magnetic field. The magnetic field reduces the strangeness fraction of neutrino-free matter and increases the strangeness fraction of neutrino-trapped matter. We have studied some properties of stars with trapped neutrinos and strong magnetic fields. A density dependent magnetic field with the magnitude 10 15 G at the surface was considered. The magnetic field reduces the strangeness content of the star, and, as a consequence, the possibility of formation of a black-hole after the outflow of neutrinos, since it is the appearance of exotic matter in neutrino free matter that may induce a black-hole formation, if the EOS becomes too soft.

I. INTRODUCTION

Neutron stars with very strong magnetic fields of the order of 10 14 -10 15 G are known as magnetars [1][2][3]. They are believed to be the sources of the intense γ and X rays (for a review refer to [4]). Although until presently only 16 magnetars have been identified as short γ-ray repeaters or anomalous X-ray pulsars [START_REF]SGR/APX online Catalogue[END_REF], according to Ref. [START_REF] Kouveliotou | [END_REF], a fraction as high as 10% of the neutron star population could be magnetars. Magnetars are warm, young stars, ∼ 1 kyear old.

The study of dense stellar matter under strong magnetic fields is, therefore of great interest. Our knowledge of neutron star composition and structure is still uncertain [START_REF] Glendenning | Compact Stars[END_REF]. For densities below twice normal nuclear matter density (ρ 0 ∼ 0.153 fm -3 ), the matter consists only of nucleons and leptons. For baryon densities above 2ρ 0 , the equation of state (EOS) and the composition of matter are much less certain and the strangeness degree of freedom should be taken into account either through the onset of hyperons, kaon condensation or a deconfinement phase transition into strange quark matter. The presence of hyperons in neutron stars, which has been studied by many authors [START_REF] Knorren | [END_REF][9][10][11][12][13][14][15][16][17][18], tends to soften the EOS at high density and lower the maximum mass of neutron stars [9,[14][15][16][17][18][19]. Cold dense stellar matter is neutrino free but the protoneutron star formed after a supernova explosion, with an entropy per baryon of the order of 1 to 2, contains trapped neutrinos. After 10 to 20 s, the star stabilizes at practically zero temperature and no trapped neutrinos are left [20].

In this article, we focus on the properties of hyperonic matter, which is composed of a chemically equilibrated and charge-neutral mixture of nucleons, hyperons, and leptons. The meson-hyperon couplings play an important role in the determination of the EOS and the composition of hyperonic matter. In the presence of strong magnetic fields, the pressure and composition of matter can be affected significantly [21]. In Ref. [21], the authors have investigated the effects of strong magnetic fields on the properties of neutron star matter including hyperons, but no neutrino trapping was considered. It was found that the EOS at high density could be significantly affected both by Landau quantization and by the magnetic moment interactions, but only for field strength B > 5 × 10 18 G. They have studied the neutron star structure using general relativistic magneto-hydrostatic calculations and have demonstrated that the maximum average fields within a stable neutron are limited to values B ∼ 1 -3 × 10 18 G. These values agree with the ones obtained from the scalar virial theorem according to which the maximum interior field strength could be as large as 10 18 G for a compact star [START_REF] Shapiro | Black holes, white dwarfs and neutron stars[END_REF].

In Ref. [21], it was also shown that the threshold densities of hyperons can be significantly altered by strong magnetic field. Similar conclusions were obtained in [START_REF] Yue | [END_REF] where the strangeness was included through an antikaon condensation or in [24] where not only hyperons but also the strange mesons σ * and φ were included in the EOS. We wish to investigate the influence of strong magnetic fields on hyperonic matter when neutrinos are trapped. Although, we should consider warm matter, it was shown [20,25] that the effect of the inclusion of trapped neutrinos is more important than the temperature effect. While the presence of neutrinos has a strong effect on the structure of the star, the temperature has almost no effect on the maximum star mass configuration and a calculation done at zero temperature predicts for hadronic stars just slightly larger gravitational masses, less than 1% larger, and slightly smaller radius, ∼ 1% smaller. We expect, therefore, that the main conclusions taken for T = 0 will still be valid at finite temperature. Similar studies at zero temperature have been done by other authors when investigating the effect of neutrinos on the compact star structure [26]. It should, however, be pointed out that the temperature will bring the onset of strangeness to lower densities although the overall fraction of strangeness will not change much. In a future work a finite temperature calculation will complete the present work.

For matter without a strong magnetic field, it has been shown in Ref. [20,25] that if stellar matter is described using an EOS with hyperons the inclusion of trapped neutrinos gives an harder EOS than the neutrino-free EOS, and, therefore, the maximum baryonic allowed mass of a stable star is higher when neutrinos are trapped. As a consequence, after the neutrino outflow has occurred the most massive stars may decay into a blackhole. This is not anymore true for a stellar matter EOS with no strangeness. In this case, neutrinos make the EOS softer [20]. However, we will not discuss this case in the present work.

The effect of the magnetic field on the structure and composition of a neutron star allowing quark-hadron phase transition with trapped neutrinos has been studied in Ref. [27]. They have concluded that the strong magnetic field makes the overall equation of state softer giving rise to much smaller maximum mass configurations which do not favor the formation of low mass black-holes. Their conclusion about the reduction of the maximum mass configuration is in contrast with the result from other works [21,28] where an increase is predicted due to the positive contribution of the magnetic field pressure to the total EOS. In Ref. [27] the magnetic field pressure enters the total EOS as a negative contribution, making the overall EOS softer.

This work is organized as follows: we make a brief review of the formalism used for the hadron matter with and without trapped neutrinos. Next we present and discuss the results obtained for the equation of state of stellar matter and the consequence on the properties of compact stars, for several values of the magnetic field. At the end we will draw some conclusions.

II. THE FORMALISM

For the description of the neutron star matter, we employ the standard mean-field theory (MFT) approach. A complete set of the equations and the description of the method can be found in Ref. [29,30]. The Lagrangian density of the relativistic TW model [31,32] is given by

L = b Ψb iγ µ ∂ µ -q b γ µ A µ -m b + Γ σb σ -Γ ωb γ µ ω µ -Γ ρb τ 3 b γ µ ρ µ - 1 2 µ N κ b σ µν F µν Ψ b + l ψl (iγ µ ∂ µ -q l γ µ A µ -m l ) ψ l + 1 2 ∂ µ σ∂ µ σ - 1 2 m 2 σ σ 2 + 1 2 m 2 ω ω µ ω µ - 1 4 Ω µν Ω µν - 1 4 F µν F µν + 1 2 m 2 ρ ρ µ ρ µ - 1 4 P µν P µν (1)
where Ψ b and ψ l are the baryon and lepton Dirac fields, respectively. The index b runs overs the eight lightest baryons n, p, Λ, Σ -, Σ 0 , Σ + , Ξ -and Ξ 0 (neglecting the Ω -and the ∆ quartet, which appear only at quite high densities, does not qualitatively affect our conclusions), and the sum on l is over electrons and muons (e -and µ -). σ, ω, and ρ represent the scalar, vector, and isovector-vector meson fields, which are exchanged for the description of nuclear interactions and A µ = (0, 0, Bx, 0) refers to a external magnetic field along the z-axis. The baryon mass and isospin projection are denoted by m b and τ 3 b , respectively. The mesonic and electromagnetic field strength tensors are given by their usual expressions:

Ω µν = ∂ µ ω ν -∂ ν ω µ , P µν = ∂ µ ρ ν -∂ ν ρ µ , and
F µν = ∂ µ A ν -∂ ν A µ .
The baryon anomalous magnetic moments (AMM) are introduced via the coupling of the baryons to the electromagnetic field tensor with

σ µν = i 2 [γ µ , γ ν ] and strength κ b = (µ b /µ N -q b m p /m b ).
We neglect the AMM of the leptons in this work, because their effect is very small as shown in Ref. [33]. For the electromagnetic field, only frozen-field configurations are considered and thus there is no associated field equation. The density dependent strong interaction couplings are denoted by Γ, the electromagnetic couplings by q and the baryons, mesons and leptons masses by m. The parameters of the model are the nucleon mass M = 939 MeV, the masses of mesons m σ , m ω , m ρ and the density dependent coupling parameters which are adjusted in order to reproduce some of the nuclear matter bulk properties and relations with the Dirac-Brueckner Hartree-Fock (DBHF) calculations [34], using the following parametrisation

Γ i (ρ) = Γ i (ρ 0 )f i (x), i = σ, ω (2)
with

f i (x) = a i 1 + b i (x + d i ) 2 1 + c i (x + d i ) 2 (3)
where x = ρ/ρ 0 and

Γ ρ (ρ) = Γ ρ (ρ 0 ) exp [-a ρ (x -1)] (4)
with the values of the parameters m i , Γ i , a i , b i , c i and d i , i = σ, ω, ρ given in [32]. The meson-hyperon couplings are assumed to be fixed fractions of the meson-nucleon couplings, Γ iH = x iH Γ iN , where for each meson i, the values of x iH are assumed equal for all hyperons H. Glendenning and Moszkwoski [35] have constrained the x iH values by neutron star maximum masses and the binding energy of the Λ at nuclear saturation. Two possible sets of x iH are given in Table II and will be identified by the x σ value, 0.6 and 0.8. Other hypernuclear potentials in nuclear matter, consistent with hypernuclear data [36][37][38], may be used to constrain further the sigma-hyperon couplings. This was done in [39] were, following Ref. [37] the values for the hypernuclear potentials, V Λ = -28 MeV, V Σ = 30 MeV, V Ξ = -18 MeV, have been used. While the main conclusions with respect to the strangeness content of the EOS do not differ when the last parametrization is used instead of the ones proposed in [35] with x σ = 0.6 or 0.8, the onset of each hyperon species will depend on the parametrization: with the parametrization of Glendenning and Moszkwoski [35] the Σ - and the Λ are the first hyperons to appear as the density increases while using the parametrization proposed in [39] the first hyperons are Λ and Ξ -. For GM1 model, we add to the Lagrangian density, Eq. ( 1), the scalar meson self-interaction terms

L nlσ = - 1 3 bm n (g σ σ) 3 - 1 4 c(g σ σ) 4 ,
where g i = Γ i , b and c are two dimensionless parameters.

For neutrino-free stellar matter consisting of a β-equilibrium mixture of baryons and leptons, the following equilibrium conditions must be imposed:

µ b = q b µ n -q l µ e , ( 5
) µ µ = µ e , ( 6
)
where µ i is the chemical potential of species i. The electric charge neutrality condition is expressed by where ρ v i is the number density of particle i. If trapped neutrinos are included, we replace µ e → µ e -µ νe in the above equations,

b q b ρ v b + l q l ρ v l = 0, (7)
µ b = q b µ n -q l (µ e -µ νe ) , ( 8
)
and for leptons we have

µ µ -µ νµ = µ e -µ νe . ( 9
)
In the above equations µ µ , µ νe are the chemical potential, respectively, of the muon and electron neutrinos. The introduction of additional variables, the neutrino chemical potentials, requires additional constraints, which we supply by fixing the lepton fraction Y Le = Y e + Y νe = 0.4 [20,40]. Also, because no muons are present before and during the supernova explosion, the constraint Y Lµ = Y µ + Y νµ = 0 must be imposed.

For neutrino-trapped matter, the neutrino density is given by

ρ v νe = k νe F 3 6π 2 . ( 10
)
The chemical potentials of baryons and leptons are defined as

µ b = E b F + Γ ωb ω 0 + Γ ρb τ 3 b ρ 0 + Σ R 0 ( 11
)
µ l = E l F , (12)
where the Fermi energies for charged baryons, neutral baryons and leptons (electrons and muons) are given by

E i F = k 2 F + m2 i , i= b, l (13) (14)
with mi given by

mb,c = m * 2 b + 2ν|q b |B -sµ N κ b B, (charged baryons) (15) mb,0 = m * b -sµ N κ b B, (neutral baryons) (16) ml = m 2 l + 2ν|q l |B (charged leptons), (17)
where ν = n + 1 2 -sgn(q) s 2 = 0, 1, 2, . . . enumerates the Landau levels (LL) of the fermions with electric charge q, the quantum number s is +1 for spin up and -1 for spin down cases. The rearrangement term in (11) is given by

Σ R 0 = b ∂Γ ωb ∂ρ ρ v b ω 0 + ∂Γ ρb ∂ρ τ 3 b ρ v b ρ 0 - ∂Γ σb ∂ρ ρ s b σ = 1 Γ ωN ∂Γ ωN ∂ρ m 2 ω ω 2 0 + 1 Γ ρN ∂Γ ρN ∂ρ m 2 ρ ρ 2 0 - 1 Γ σN ∂Γ σN ∂ρ m 2 σ σ 2 . ( 18
)
The pressure of neutron star matter can be obtained by

P m = i µ i ρ v i -ε m = µ n b ρ v b -ε m ( 19
)
where the charge neutrality and β-equilibrium conditions are used to get the last equality. The expressions for the energy density ε m , scalar density ρ s b and the particle densities ρ v i may be found in [30]. If the stellar matter contains neutrinos trapped, their energy density and pressure contributions, respectively,

ε νe = k νe F 4 8π 2 , P νe = k νe F 4 24π 2 , ( 20
)
should be added to the stellar matter energy and pressure. The total energy density and pressure of the system includes the contribution of the magnetic field,

ε = ε m + B 2 2 , P = P m + B 2 2 . ( 21
)
With the obtained EOS, the mass-radius relation and other relevant quantities of neutron star can be derived by solving the Tolman-Oppenheimer-Volkoff (TOV) equations.

III. RESULTS AND DISCUSSION

A. Uniform magnetic field

In this section we consider that the external magnetic field is constant. The magnetic field will be defined in units of the critical field B c e = 4.414 × 10 13 G, so that B = B * B c e . In order to study the effect of strong magnetic fields on the structure of hyperonic matter we use two different relativistic mean-field approaches: the GM1 parametrisation of the NLW models [35], and the TW parametrization of the density-dependent relativistic hadronic (DDRH) models [31,32], given in Table II. We include the baryonic octet in the EOS and choose two sets of hyperon-meson coupling constants given in Table II. The static properties of the baryons considered are listed in Table I.

In Fig. 1, the EoS obtained with both relativistic mean-field models GM1 and TW are displayed for B * = 0, 10 5 , and 3 × 10 5 , for the relative hyperon coupling constant x σ = 0.6 (thick lines) and x σ = 0.8 (thin lines). In Fig. 1 (a) and Fig. 1 (c) the EOS does not include the contribution from the AMM and in Fig. 1 (b) and Fig. 1 (d) the AMM was included. The kink on each one of the curves identifies the onset of hyperons. Both for TW, which gives a softer EOS, and for GM1, the EOS at higher densities are harder for x σ = 0.8 than for x σ = 0.6 because a larger ω or ρ coupling constant makes the onset of hyperons only possible at larger densities. This is also valid in the presence of a strong magnetic field. The effects of the AMM are only noticeable for the stronger magnetic fields. It is seen that, in the absence of the AMM and in presence of a strong magnetic field, the EOS becomes softer for the smaller densities and harder at the larger densities due to the Landau quantization which affects charged particles.

The effect of magnetic field on the EOS, when trapped neutrinos are considered, is seen in Fig. 2 for GM1 and TW models, with x σ = 0.6 and without/with AMM. For B = 0 the presence of neutrinos makes the EOS harder when hyperons have been included [20,25]. A strong magnetic field without including the AMM, makes the EOS harder at high densities if neutrino trapping is enforced. Moreover, if the AMM are included the EOS become even harder. At low densities, however, the magnetic field softens the EOS even when the AMM are taken into account. We remark that for the largest field considered and taking into account AMM, it is not possible to get an EOS at low densities for the lepton fraction Y L = 0.4. In Fig. 3 we plot the baryon density threshold, at which the neutrino chemical potential vanishes, as function of the magnetic field, for GM1 and TW models using x σ = 0.6 and without/with including AMM. Below this threshold density, it is not possible to impose the lepton fraction Y L = 0.4. This happens because at low densities and for a very strong magnetic field the proton fraction of β equilibrium matter is very large. If the proton fraction is larger than 0.4, we must have an equal amount of electrons for charge neutrality and it will not be possible to get a lepton fraction 0.4 since the electrons already give a larger contribution than 0.4. The threshold density for neutrino onset increases very fast with the increase of the magnetic field, e.g. in Fig. 3(a) for B * = 5 × 10 5 corresponds a threshold density equal to 0.5ρ 0 for GM1 and 1.35ρ 0 for TW. The kink at ρ ∼ 2.5ρ 0 corresponds to the hyperon onset. However, at the surface of the magnetar the measured magnetic fields are at most B * ∼ 10 2 and it is probable that even if the magnetic field is stronger in the interior it will be weaker than the threshold values given in Fig. 3. In Fig. 4 we show, for several values of magnetic field without including the AMM, for GM1 and TW models and for the two values of the hyperon-meson coupling constants, the strangeness fraction defined as

r HP s = b q b s ρ b 3ρ ,
where q b s is the strange charge of baryon b, and is listed in Table I. The strangeness onset occurs around 2ρ 0 and it has almost reached 0.275 (0.24) for GM1 and 0.23 (0.18) for TW, neutrino-free matter and x σ = 0.6 (x σ = 0.8). For B = 0 and for the neutrino-free matter, increasing the hyperon-meson couplings decreases the strangeness contents, and this suppression of strangeness is stronger in TW than GM1. However, if the neutrino trapping is imposed the strangeness fraction is smaller and a stronger reduction occurs for GM1 and for x σ = 0.8.

For neutrino-free matter, the magnetic field reduces the strangeness fractions below a critical density that can be quite high, 6ρ 0 or larger for the fields shown. This effect starts to be detected already for B * = 10 5 , which represents the threshold for the magnetic field effects to become significant, independently of the EOS [21]. However, the magnetic field enhances the strangeness fraction of neutrino-trapped matter. If the AMM are included the strangeness fraction behaves in the same way.

In Figs. 5, 6, 7, and 8, we present the particle fractions Y i = ρ v i /ρ as function of the baryon density ρ/ρ 0 for several values of the magnetic fields, obtained with TW model using two values of the hyperon-meson coupling constants x σ = 0.6, 0.8 given in [35]. Figs. 5 and 7 are for neutrino-free matter and Figs. 6 and8 for neutrino-trapped matter. For GM1 we get similar results.

First, we study the results obtained with the hyperon-meson coupling constant equal to x σ = 0.6. For B = 0 neutrino free matter, shown in Fig. 5 (a), the proton fraction rises quickly with increasing density and reaches ∼ 0.1 around 2ρ 0 before the appearance of hyperons. Σ -is the first hyperon to appear. In Fig. 5 abundance due to the charge neutrality. In Fig. 9 we show for TW with x σ = 0.6 the neutron, proton, electron and neutrino chemical potentials as a function of density for different values of the magnetic field. It is seen that the proton and neutron chemical potentials decrease with the increase of B. This is also the general tendency of leptons except for the larger densities. In the absence of AMM the reduction of the neutron chemical potential with an increase of the magnetic field is due to a reduction of the isospin asymmetry. The AMM will further reduce the neutron chemical potential. Landau quantization is the main explanation for a reduction of the proton chemical potential. As a consequence the threshold for the appearance of hyperons like Σ -and Λ will occur at larger densities [21]. However, at densities above ∼ 5ρ 0 the threshold densities for Σ 0 , Σ + , Ξ -occur at smaller densities probably due to the smaller effective masses of these hyperons at large densities with strong magnetic fields. The inclusion of AMM, Fig. 5 (c) and (e), produces an even larger hyperon suppression at the lower densities.

TABLE III:

The threshold density of hyperons in fm -3 when AMM is not included. for several values of magnetic field, for both GM1 and TW models and for neutrino free and neutrino-trapped matter. We also identify, between brackets, the hyperon that appears at smaller densities. We consider now matter with trapped neutrinos. For B = 0, Fig. 6 (a), the proton and electron fractions are ∼ 0.3 from low densities until the appearance of hyperons at ∼ 3ρ 0 and, therefore, the neutrino fraction is ∼ 0.1 0.0 2.0×10 Below this density it is not possible to impose YLe = 0.4.

B

TABLE IV:

The threshold density of hyperons in fm -3 when AMM is included. for several values of magnetic field, for both GM1 and TW models and for neutrino free and neutrino-trapped matter. We identify, between brackets, the hyperon that appears at smaller densities. since we are imposing Y L = 0.4. Λ is the first hyperon to appear for B=0. For B * = 10 5 (see Fig. 6 (b)), the proton fraction is larger (∼ 0.4) at the smaller densities, decreases to 0.3 for ρ ∼ ρ 0 and stays approximately constant until the appearance of hyperons at ∼ 3ρ 0 . The neutrino fraction rises quickly with increasing density and reaches ∼ 0.1 around ρ 0 and keeps, roughly, this value for larger densities. Λ is still the first hyperon to appear in this case. However, for B * = 3 × 10 5 (see Fig. 6 (d)), the proton fraction decreases slightly starting at ∼ 0.4 at lower densities until the appearance of hyperons at ∼ 3ρ 0 . Σ -is the first hyperon to appear almost at the same density as Λ. The neutrino fraction is almost zero at subsaturation densities, and rises, quickly, with increasing density, to ∼ 0.2 around 6ρ 0 . The net effect of magnetic field is a neutrino suppression due to the larger proton and therefore electron fractions and to move the threshold density of the negatively charged baryons (e.g. Σ -) to larger densities and the positively charged baryons to lower densities. This is due to a decrease (increase) of the chemical potentials of negatively charged baryons (positively charged baryons) when the neutrino chemical potential is taken into account.

In particular the onset of the Ξ -occurs for densities larger than 8ρ 0 . The neutral baryons are not affected by the presence of neutrinos. In the presence of a strong magnetic field the overall effect of the neutrinos is smaller than for B=0.

For x σ = 0.8, we obtain similar results and the main differences are: at high densities the onset of Ξ -occurs at ρ < 8ρ 0 , the onset of the Σ -and Λ occurs at larger densities and hyperon fractions are smaller.

The effect of the magnetic field on the onset of hyperons, in both cases for neutrino free and neutrino trapped matter, is clearly shown in Tables III andIV, where we give the threshold density of the first hyperon, for TW and GM1 models and for x σ = 0.6, and 0.8, without including the AMM (Table III)and including the AMM in (Table IV). The main conclusions for neutrino-free matter are: a) the hyperon onset occurs with the appearance of the Σ -meson, both with and without AMM; b) the onset density generally increases with the increase of the magnetic field but for GM1 it may decrease for B * = 10 5 ; c) the inclusion of AMM reduces the onset density of hyperons; d) the hyperon onset occurs at larger densities for x σ = 0.8. For neutrino-trapped matter we conclude that: a) the onset density of hyperons is generally smaller when the magnetic field magnitude increases; b) for small fields, Λ is the first hyperon to appear but for large fields it is either Σ -or Σ -and Λ together.

In order to better understand the effect of the magnetic field on the neutrino trapping, we show in Fig. 10 the fraction of neutrinos for several values of magnetic field. For B = 0, the neutrino fraction decreases at low densities and starts to increase after the onset of hyperons because with the hyperon onset the electron fraction gets smaller. For B * = 5 × 10 4 and B * = 10 5 , the neutrino fraction is affected by the Landau quantization of electrons, and oscillates around the B = 0 results, when AMM are not included. At low densities the main effect of magnetic field is the suppression of neutrinos. For B * > 10 5 , the neutrino fraction is lowered at low densities and enhanced at higher densities. However, when the AMM are included, the abundance of neutrinos is only slightly reduced for B * = 5× 10 4 and B * = 10 5 , while for B * > 10 5 the neutrino suppression is strong, e. g. for B * = 3 × 10 5 there are no neutrinos at a density below the saturation density. We conclude, therefore, that the abundance of neutrinos in the presence of strong magnetic fields has a strong suppression at low densities.

In order to study the effect of the magnetic field on the properties of the stars described by the EOS discussed above with neutrino trapping, we must consider a density dependent magnetic field, which is not larger than 10 15 G at the surface. We will do this in the next section.

B. Baryon density-dependent magnetic field

Since, to date, there is no information available on the interior magnetic field of the star, we will assume that the magnetic field is baryon density-dependent as suggested by Ref. [41]. The variation of the magnetic field B with the baryon density ρ from the center to the surface of a star is parametrized [30,41,42] by the following form

B ρ ρ 0 = B surf + B 0 1 -exp -β ρ ρ 0 γ , ( 22
)
where ρ 0 is the saturation density, B surf is the magnetic field at the surface taken equal to 10 15 G, in accordance with the values inferred from observations and B 0 represents the magnetic field at large densities. The parameters β and γ may be chosen in such way that the field decreases fast or slowly with the density from the center to the surface. In this work, we will use one set of value (β = 0.05 and γ = 2) allowing a slowly varying field. The magnetic field will be given in units of the critical field B c e = 4.414 × 10 13 G for the electron, so that B 0 = B * 0 B c e . We further take B 0 as a free parameter to check the effect of different fields. A detailed discussion about the variation of the magnetic field with the baryon density may be found in Ref. [30].

In this section, we will only present the results for the density-dependent relativistic model TW and we will take for the hyperon-meson coupling constants x σ = 0.6. Furthermore, in all the figures, we will only show the results obtained without including the baryonic AMM, because for the intensity of the magnetic fields considered, its contribution to the EOS is negligible. The effect of the AMM is non negligible for B * > 10 5 [43], and is larger for the smaller densities, but according to [21] these values will never be reached inside the star.

In Fig. 11, we show the total pressure P , see Eq. ( 21), as a function of the baryon density for the magnetic field strengths B * = 0, 10 5 , and 3 × 10 5 . The results with neutrino-trapped and neutrino-free matter are plotted with thick and thin lines, respectively. The EOS with neutrino trapping is stiffer than the neutrino free EOS, both with and without magnetic field. In Fig. 11(a), we only show the results without the AMM because for the fields considered almost no effect is introduced by the AMM. In Fig. 11(b) we show the difference between the EOS with and without AMM for neutrino free matter (thin line) and matter with trapped neutrinos (thick line). The differences are of the order of 20 MeV/fm 3 for a density 8ρ 0 . However, according to [21] the magnetic field inside a stable star cannot be larger than 1-3×10 18 G and therefore, for B * 0 = 3 × 10 5 the central density will not go beyond ρ ∼ 3ρ 0 , as seen from Fig. 11(c) where we have plotted the density dependent magnetic field according to the parametrization given in [START_REF] Shapiro | Black holes, white dwarfs and neutron stars[END_REF]. The differences introduced by AMM for ρ ∼ 3ρ 0 are smaller than 1%.

We have also studied the baryonic and leptonic composition of the stars. In Fig. 12 we plot the lepton and the baryon fractions for neutrino free matter (left column) and for neutrino trapped matter (right column) for TW model. We show the particle fractions for the smallest and the largest magnetic fields considered in the present work. The shaded regions in the figures represent the densities of the star for the maximum magnetic fields in stable star configurations obtained in [21] (B ∼ 1 -3 × 10 18 G) if it lies within the range of densities shown. We conclude that the main effect of magnetic fields with the intensity considered is the appearance at high densities of a larger leptonic fraction. Due to the contribution of the magnetic field to the total energy-momentum tensor, the contribution of matter becomes smaller when the magnetic field increases and, therefore, the fraction of strangeness in the star is strongly reduced because the larger contributions of hyperons for the star would come from the larger densities which are not reached in the presence of a strong magnetic field.

IV. CONCLUSIONS AND OUTLOOKS

In the present work we have studied the effect of a strong magnetic field on the EOS and the star properties when neutrinos are trapped in the star. We have used two different relativistic mean-field models, one with constant coupling parameters, GM1, and the other with density dependent coupling parameters, TW. We have also considered two sets of hyperon-coupling parameters. The main conclusions of the work do not depend either on the model or on the strength of the hyperon-couplings.

The phase of trapped neutrinos in the life of a proto-neutron star occurs while the stellar matter is still warm [20,40], and, therefore, a finite temperature calculation should have been done. However, we do not expect that temperature will change the main conclusions of the present work. In fact, in several works [20,44,45] it has been shown that the star properties such as mass and radius do not depend much on temperature.

We have shown that a strong magnetic field suppresses the presence of neutrinos at low densities. It was also shown that although strangeness is suppressed by the presence of neutrinos, if the star has a strong magnetic field this suppression is smaller. The magnetic field affects in a different way the charged and neutral baryons and it may affect the density order at which they appear. For neutrino free matter, Σ -is the first hyperon to appear at the smallest densities. However, for neutrino trapped matter, Λ is the first hyperon to appear except for the largest field considered when we may have Σ -or Σ -and Λ.

We have studied some properties of stars with trapped neutrinos and strong magnetic fields. We have considered a density dependent magnetic field with the magnitude 10 15 G at the surface, and determined the EOS of the star and the particle fractions as a function of the density. One of the main consequences of the presence of the magnetic field is the reduction of the strangeness fraction in the star. A similar conclusion with respect to other exotic components such as a quark core [30], or kaon condensation [START_REF] Yue | [END_REF] had already been pointed out. As a consequence the magnetic field reduces the possibility of formation of a black-hole after the outflow of neutrinos, since it is the onset of exotic matter in neutrino free matter that may induce a black-hole formation if the EOS becomes too soft.

FIG. 1 :

 1 FIG. 1: EOS obtained with TW and GM1 models for neutrino free matter, and for several values of the magnetic field without (a) and (c) and with (b) and (d) AMM. Thick lines correspond to xσ = 0.6 and thin lines to xσ = 0.8. The kink in each EOS represents the onset of hyperons.

FIG. 2 :

 2 FIG.2: EOS for TW and GM1 models and for xσ = 0.6, without (a) and (c) and with (b) and (d) AMM. Thick lines for neutrino-trapped matter and thin lines for neutrino-free matter.

6 FIG. 3 :

 63 FIG.3: Baryon density for which µν e = 0 versus B * obtained for TW and GM1 models, with/without AMM and for xσ = 0.6. Below this density it is not possible to impose YLe = 0.4.

FIG. 4 :

 4 FIG. 4: Strangeness fraction as a function of the baryonic density, for severals values of B, without AMM, and for GM1 and TW models. (a) and (b) for xσ = 0.6 and (c) and (d) for xσ = 0.8. Thick lines correspond to matter with trapped neutrinos and thin lines to neutrino-free matter.

FIG. 5 :

 5 FIG. 5: Baryon fraction versus baryon density obtained with TW model for xσ = 0.6 and neutrino-free matter.

FIG. 6 :

 6 FIG. 6: Baryon fraction versus baryon density obtained with TW model for xσ = 0.6, and neutrino-trapped matter.

FIG. 7 :FIG. 8 :

 78 FIG. 7: Baryon fraction versus baryon density obtained with TW model, for xσ = 0.8 and neutrino-free matter.

FIG. 9 :

 9 FIG. 9: Chemical potentials versus baryon density obtained with TW model, for several values of magnetic field, without AMM, and for xσ = 0.6 and for (a) neutrino-free matter; (b) neutrino-trapped matter.

FIG. 10 :

 10 FIG. 10: Neutrino fraction versus baryon density obtained with TW model, for several values of the magnetic field without/with including AMM and for xσ = 0.6.

4 FIG. 11 :FIG. 12 :

 41112 FIG. 11: EOS for stellar matter for TW model without/with neutrino trapping, for xσ = 0.6, using a slowly varying parametrization of B (a) for several values of magnetic field and without AMM; (b) for B * 0 = 3 × 10 5 the difference between the EOS including AMM and the EOS without AMM. Thin lines correspond to neutrino-free matter and thick lines to neutrino-trapped matter. In (c) we plot the intensity of the magnetic field versus density for B * 0 = 5× 10 4 (2.20 × 10 18 G), B * 0 = 10 5 (4.41 × 10 18 G), and B * 0 = 3 × 10 5 (1.32 × 10 19 G).

TABLE I :

 I Static properties of the baryons considered in this study. The mass, electric charge and strange charge of the baryon b are denoted by m b , q b and q b s , respectively. The baryonic magnetic moment is denoted by µ b and the anomalous magnetic moment by κ b = (µ b /µN -q b mp/m b ), where µN is the nuclear magneton.

	baryon name	Mass	Charge	Strangeness	Magnetic moment	Anomalous magnetic moment
	b	(MeV)	q b (e)	q s b	µ b /µN	κ b
	p	938.27	1	0	2.97	1.79
	n	939.56	0	0	-1.91	-1.91
	Λ 0	1115.7	0	-1	-0.61	-0.61
	Σ +	1189.4	1	-1	2.46	1.67
	Σ 0	1192.6	0	-1	1.61	1.61
	Σ -	1197.4	-1	-1	-1.16	-038
	Ξ 0	1314.8	0	-2	-1.25	-1.25
	Ξ -	1321.3	-1	-2	-0.65	0.06

TABLE II :

 II Saturation properties of nuclear matter and the nucleon-meson coupling constants for the GM1 and TW models. The relative hyperon-meson coupling constants used in the calculation are also given.

		ρ0	-B/A		ΓσN /mσ	ΓωN /mω	ΓρN /mρ
	Model	(fm)	(MeV)	M * /M	(fm)	(fm)	(fm)	xσH	xωH	xρH	b	c
	GM1	0.153	16.30	0.70	3.434	2.674	2.100	0.600	0.653	0.600	0.002947	-0.001070
								0.800	0.913	0.800
	TW	0.153	16.30	0.56	3.84901	3.34919	1.89354	0.600	0.658	0.600	0.0	0.0
								0.800	0.905	0.800

Acknowledgments

We would like to thank João da Providência for many helpful and elucidating discussions. This work was partially supported by FEDER and Projects PTDC/FP/64707/2006 and CERN/FP/83505/2008, and by COMPSTAR, an ESF Research Networking Programme.


P Grygorov

P Gögelein

H Müther

Neutrino propagation in the nuclear "pasta phase" of neutron stars

The mean free path of neutrino in charged and neutral current reactions is calculated for inhomogeneous nuclear matter which is expected to appear in the inner crust of neutron stars. The relevant cross section depends on Fermi and Gamow-Teller strength distributions, which are derived from the self-consistent Skyrme-Hartree-Fock and a relativistic mean-field calculations. The inhomogeneous nuclear matter, which appears at the boundary between the inner crust and outer core is described in terms of cubic Wigner-Seitz cells. It allows for a microscopic description of the structures in the so-called pasta phase of nuclear configurations and provides a smooth transition to the limit of homogeneous matter. The influence of the pasta phase, its microscopical structure and geometrical shapes on neutrino propagation are considered.

I. INTRODUCTION

The transport properties of neutrino play an essential role in the physics of supernovae core collapse and in the evolution of the newly born neutron stars (NS). The most important ingredient of neutrino propagation calculations is the neutrino opacity in a wide range of densities. Both the charged current (CC) absorption and neutral current (NC) scattering reactions are important sources of the neutrino opacity. In earlier works on neutrino interactions with the homogeneous nuclear matter the noninteracting baryons were considered [1]. Later the strong interaction was taking into account both in the non-relativistic and relativistic calculations (see, e.g., [2][3][4][5][6][7] and ref. therein). It was shown that the neutrino opacities of interacting matter may significantly altered from those for the noninteracting case [4]. However, the use of homogeneous matter is a good approximation, while the exis-Confidential: not for distribution. Submitted to IOP Publishing for peer review 8 March 2010 tence of the stable quasi-nuclei in the crust of neutron stars is energetically favorable and must be taken into account.

At low densities, the nuclei in matter are expected to form the Coulomb lattice embedded in the neutron sea, that minimizes the Coulomb repulsion between the protons. With increase of density the nuclear pasta structures occur and the stable nuclear shape may change form from spherical droplet to rod, slab, tube and bubble shapes [8]. Roughly speaking, the favorable nuclear shape is determined by a balance between the surface and Coulomb energies. In the following under "pasta phase" we will assume quasi-nuclear structures with spherical as well as non-spherical shape, which are embedded in a neutron sea.

Various attempts have been made to describe the ground-state structure of pasta phase based on Thomas-Fermi approximation [9][10][11][12][13], coexisting phases method [13,14], Quantum Molecular Dynamics [15][16][17][18], Hartree-Fock and Relativistic Mean-Field calculations (RMF) within the Wigner-Seitz (WS) cell approximation [21][22][23][24][25]. Later the dynamical properties of pasta such as the response function and neutrino mean free path (NMFP) were investigated [15,[18][19][20]. It was found that the coherent scattering of neutrinos on inhomogeneous matter significantly reduces the mean free path.

The calculation of the NMFP in pasta phase presented in this work is based on Hartree-Fock calculations in cubic WS cell [22][23][24], which allows for the description of non-spherical quasi-nuclear structures such as rods or slabs and contains the limit of homogeneous matter in a natural way. The self-consistent calculations are performed for β-stable matter in a density range for which the quasi-nuclear structures discussed above are expected to appear. For the nuclear Hamiltonian we consider Skyrme forces (SLy4) [START_REF] Chabanat | [END_REF] but also perform calculations within the relativistic mean-field (Hartree) approximation. The stability of the pasta phase with increase of the temperature is also discussed.

The NMFP is extracted from the relevant cross sections of neutrino on different pasta structures. We pay special attention to the dependence of our results on the internal structure of the pasta phase and its geometrical shapes. The mean free paths obtained from these inhomogeneous structures are compared with those calculated for homogeneous nuclear matter at the same global density, thus one can estimate the influence of the inhomogeneous phase on the propagation of neutrinos.

After this introduction the details of Skyrme-Hartree-Fock approach with pairing will be outlined and a method to calculate the NMFP will be reviewed in section 2. In section 3 we discuss the density dependent relativistic mean-field (DDRMF) model. The numerical results are discussed in section 4 and the final section 5 contains the main conclusions.

II. SKYRME-HARTREE-FOCK CALCULATIONS

A. Energy functional

We perform self-consistent HF calculations with inclusion of pairing in a periodic lattice of WS cells of cubic shapes. The symmetry of such WS cell allows the formation of triaxial structures but also include rod-, and slab-like structures and provide a natural transition to homogeneous matter.

Since the Skyrme-Hartree-Fock approach has frequently been described in the literature [27][28][29][START_REF] Ring | The Nuclear Many Body Problem[END_REF] we will outline here only a few basic equations, which will define the nomenclature.

The Skyrme model is defined in terms of an energy density H(r), which can be split into various contributions [28,[START_REF] Chabanat | [END_REF]

H = H K + H 0 + H 3 + H eff + H fin + H so + H Coul , (1)
where H K is the kinetic energy term, H 0 a zero range term, H 3 a density dependent term, H eff an effective mass term, H fin a finite range term and H so a spin-orbit term. These terms are given by

H K = 2 2m τ, H 0 = 1 4 t 0 (2 + x 0 )ρ 2 -(2x 0 + 1)(ρ 2 p + ρ 2 n ) , H 3 = 1 24 t 3 ρ α (2 + x 3 )ρ 2 -(2x 3 + 1)(ρ 2 p + ρ 2 n ) , H eff = 1 8 t 1 (2 + x 1 ) + t 2 (2 + x 2 ) τ ρ + 1 8 t 2 (2x 2 + 1) -t 1 (2x 1 + 1) τ p ρ p + τ n ρ n , H fin = -1 32 3t 1 (2 + x 1 ) -t 2 (2 + x 2 ) ρ∆ρ + 1 32 3t 1 (2x 1 + 1) + t 2 (2x 2 + 1) ρ p ∆ρ p + ρ n ∆ρ n , H so = -1 2 W 0 ρ ∇J + ρ p ∇J p + ρ n ∇J n . (2)
The coefficients t i , x i , W 0 , and α are the parameters of a generalized Skyrme force [32]. The single-particle energies and wave functions are determined as solutions of the Hartree-Fock equations

-∇ 2 2m * q (r) ∇ + U q (r) -i W q (r) • (∇ × σ) ϕ q k (r, s) = ε q k ϕ q k (r, s) ( 3 )
where ϕ q k (r, s) is the single-particle wave function with orbital, spin and isospin quantum numbers k, s and q. The expression for the effective mass m * , the central potential U q (r) and spin orbin term W q (r) can be found in [22,[START_REF] Chabanat | [END_REF]. The HF equations have been solved by using the imaginary time step method [22,23,33] in discretized coordinate space within a cubic WS cell [32] of 20 fm length in each direction. The charge neutral matter contains protons, neutrons and electrons in β-equilibrium. To decrease the numerical effort in calculation of HF equations we assume two symmetries like in [32]:

• parity P ϕ k (r, s) = ϕ k (-r, s) = p k ϕ k (r, s), p k = ±1; (4)
• z-signature exp{iπ( Ĵz -1 2 )}ϕ k (x, y, z, s) = σϕ k (-x, -y, z, s) = η k ϕ k (x, y, z, s), η k = ±1. (5)
These symmetries still allow triaxial deformations and reduce the calculation to the positive coordinates in each direction. As additional symmetry time-reversal-invariance is assumed for the time-reversed pairs ϕ k , and ϕk:

ϕk(r, s) = ( T ϕ k )(r, s) = σϕ * k (r, -s). (6)
Summarising this symmetries it is sufficient to solve the HF equations for one wave function of the time-reversed pairs.

Pairing correlations are included in terms of the BCS approximation by assuming a density-dependent zero-range pairing force, which has been used in earlier calculations [22,24,34]. The code was tested for the set Skyrme III by comparing results for finite nuclei with those [32].

Figures 1,2 display density profiles of proton and neutron distributions at the global density of 0.0625 fm -3 and such quasi-nuclear structure can be characterized as rods along the z-axis. At a density of 0.0775 fm -3 the variational Hartree-Fock leads to a quasi-nuclear structure, which can be characterized as a set of parallel slabs. In figures 3 and 4, which show the proton and neutron density distributions at this density, we have chosen the orientation of the coordinates such that these slabs are orthogonal to the z-axis. At the density 0.0166 fm -3 we observe the spherical droplet phase. Its density distribution is symmetric in all directions therefore we do not illustrate it here. Thus we define three values of density, which profiles possess a clear spatial distribution and can be identified with the known pasta structures from other calculations [21]. Increasing density from 0.0166 fm -3 to 0.0775 fm -3 one obtains a variety of different intermediate density profiles which connect smoothly three representatives: droplet → rod → slab.

As mentioned above the Hartree-Fock calculations were performed within the WS cell approximation, assuming that the inner crust is divided into independent cubic cells, and the wave functions of dripped, unbound neutrons satisfy the periodic boundary conditions.

However from the standard model of neutron stars [35] it is known, that the inner crust is a perfect crystal and therefore the band theory of solids should be applied for the unbound neutrons. In this case the energy spectrum of unbound neutrons will be formed of "bands" in momentum space, thus contain more information than in WS cell approximation. However as it was shown in recent investigations by Chamel et al. [36], the differences between the full band theory and WS approximation are expected to be essential when the processes under considerations involve energy transfer, which is comparable with the level spacing induced by discretization. In our calculation such level spacing in the energy spectrum of unbound neutrons does not exceed 200 keV, therefore we suppose that the WS approximation is a good starting point for calculation of NMFP with energies of the incoming neutrinos 10≤ E ν ≤100 MeV.

The single-particle energies and wave functions for protons and neutrons resulted from the Hartree-Fock calculations were used to evaluate the NMFP by using the method outlined in the following subsection.

B. Neutrino mean free path in Skyrme-Hartree-Fock model

The matrix element for the neutrino-nucleon reactions ν

+ n → ν + n ( ν + n → p + e )
is given by

M = G F C √ 2 J µ j µ , (7)
density ρp [fm where

J µ = iū n(p) (V γ µ + Aγ µ γ 5 )u n , (8)
j µ = -iū ν γ µ (1 -γ 5 )u ν(e) (9)
are hadronic and leptonic currents, respectively. The parameters A, V must be replaced by the respective values of coupling constants and C stands for the Cabbibo factor in the charged-current reaction [37]. The total cross section can be written as where within this non-relativistic approach we neglect the lower components in Dirac spinors u ( ϕ 0 ). In the charged current reaction the influence of the Coulomb field on the outgoing electron can be taken into account by multiplying the cross section by the Fermi function [START_REF] Eisenberg | Excitation Mechanisms of the Nucleus[END_REF]. For this reaction M 1 and M 2 stand for the Fermi and Gamow-Teller matrix elements, respectively. The integration is performed over the spatial angle ϑ between the momenta of incoming and the outgoing leptons. The single-particle wave functions ϕ( r) and single-particle energies ε f (ε i ) are obtained from the solution of the HF equations. Note that these single-particle energies enter (10) as the energy for the outgoing lepton is defined as

σ = f p ν(e) E ν(e) 1 2 1 -1 d(cos ϑ)|M| 2 , ( 10
)
|M| 2 = G 2 F C 2 π V 2 (1 + cos ϑ)|M 1 | 2 + A 2 (1 - 1 3 cos ϑ)|M 2 | 2 , (11)
M 1 = ϕ 4 |e i q r |ϕ 2 , M 2 = ϕ 4 | σe i q r |ϕ 2 , (12)
F (Z f , E e )
E ν(e) = E in ν + ε i -ε f ,
where E in ν is the energy of the incoming neutrino. The formalism described so far is appropriate for the neutrino-nucleus interaction. With some extensions it may also be used to evaluate the interaction of neutrinos with the quasinuclear structures in the crust of neutron stars. Unlike spherical nuclei and the case of the droplet phase, the cross section of neutrino on rods and slabs, generally speaking, depends on the spatial orientation of momentum transfer q in (12), since the density distributions of rod and slab phases are non spherical, as it is shown on figures 1-4. The precise averaging over all possible mutual orientations of vectors q and r requires additional numerical efforts. Thus, in order to reduce this effort we considered three particular cases, with the vector q along the direction of the x, y and z-axis. Doing so, we determine the averaged cross section as

σ = 1 3 (σ x + σ y + σ z ), (13)
where σ x(y,z) represents the cross section calculated for the momentum transfer along x(y, z)axis.

In contrast to a finite nucleus, the WS cell of the inhomogeneous nuclear matter contains a large number of unbound neutrons, which give nonzero contribution to the total cross section.

Thus, the cross section consists of two parts: the cross section due to the interaction with the nucleons bound in the quasi-nuclear structure and the cross section due to the interaction with unbound neutrons. Therefore, one can consider (10) as a cross section of neutrinos with all nucleons in a given volume V cell of a WS cell. The reverse NMFP can then be written as

1 λ = σ V cell . ( 14
)
Another important distinction of pasta structures in the crust of neutron stars from the finite, isolated nuclei consists in the existence of the electron sea in the volume of the WS cell. Therefore the nonzero chemical potential of electrons must be taken into account in the evaluation of charged current reactions by a blocking of final states for electrons with energies below the respective Fermi energy µ e .

III. RELATIVISTIC MEAN-FIELD CALCULATIONS

In order to test the sensitivity of the results on the underlying nuclear model and the choice of the NN interaction we also investigated the NMFP in the inhomogeneous nuclear matter evaluated within a relativistic mean-field (Hartree) approximation by using a model of density-dependent meson-nucleon coupling constants. The parameterization of these constants has been fitted to reproduce the properties of the nucleon self-energy evaluated in Dirac-Brueckner-Hartree-Fock (DBHF) calculations of asymmetric nuclear matter but has also been adjusted to provide a good description for bulk properties of finite nuclei [40][41][42][43].

The density-dependent relativistic mean-field (DDRMF) approach has also been used to describe the properties of inhomogeneous nuclear matter in the crust of neutron stars [24].

A. Density dependent relativistic mean-field approach

The Density Dependent Relativistic Mean Field approach is an effective field theory of interacting mesons and nucleons. Following the usual notation we consider scalar (σ, δ)

and vector mesons (ω, ρ), which with respect to the isospin correspond to isoscalar (σ, ω)

and isovector (δ, ρ), respectively. The Lagrangian density consists of three parts: the free baryon Lagrangian density L B , the free meson Lagrangian density L M and the interaction Lagrangian density L int :

L = L B + L M + L int , (15)
which take the explicit form

L B = Ψ( iγ µ ∂ µ -M)Ψ, L M = 1 2 ι=σ,δ ∂ µ Φ ι ∂ µ Φ ι -m 2 ι Φ 2 ι -1 2 κ=ω,ρ,γ 1 2 F (κ)µν F µν (κ) -m 2 κ A (κ)µ A µ (κ) , L int = -g σ ΨΦ σ Ψ -g δ Ψτ Φ δ Ψ -g ω Ψγ µ A µ (ω) Ψ -g ρ Ψτ γ µ A µ (ρ) Ψ -e Ψγ µ 1 2 (1 + τ 3 )A µ (γ) Ψ, (16)
with the field strength tensor

F (κ)µν = ∂ µ A (κ)ν -∂ ν A (κ)µ
for the vector mesons. In the above Lagrangian density the nucleon field consisting of Dirac-spinors in isospin space is denoted by Ψ and the nucleon rest mass by M = 938.9 MeV. The scalar meson fields are Φ σ and Φ δ , the vector meson fields A (ω) and A (ρ) . Bold symbols denote vectors in the isospin space acting between the two species of nucleons. The mesons have rest masses m κ for each meson κ and couple to the nucleons with the strength of the coupling constants g κ , which depend on a density of the nucleon field Ψ. This density dependence of the coupling constants was parametrized by

g κ (ρ B ) = a κ + b κ + d κ x 3 exp(-c κ x), (17)
where x = ρ B /ρ 0 , and ρ 0 = 0.16 fm -3 is the saturation density of symmetric nuclear matter.

The values obtained for the fit of the coupling functions are summarized in table I.

Applying the variational principle to the Lagrangian we obtain a Dirac equation for the nucleons and Klein-Gordon and Proca equations for the meson fields. Due to densitydependent vertices the variation principle changes to

δL δ Ψ = ∂L ∂ Ψ + ∂L ∂ρ δρ δ Ψ , (18)
where the second expression creates the so-called rearrangement contribution Σ R to the selfenergies of the nucleon field. These rearrangement contributions contribute only to the zero component of the vector self-energy. Including these additional contributions we denote the Dirac equation for the nucleonic single-particle wave function ψ α in Hartree approximation

αp + (Σ 0 + Σ R ) + β(M + Σ S ) ψ α = α ψ α , (19)
where the self-energy contributions read

Σ S = g σ Φ σ + g δ Φ δ τ 3 , Σ 0 = g ω A (ω) 0 + g ρ A (ρ) 0 τ 3 + e 1 2 (1 -τ 3 )A (γ) 0 , (20)
and the rearrangement self-energy contribution Σ R is obtained by

Σ R = ∂g σ ∂ρ Φ σ ρ s + ∂g δ ∂ρ Φ δ ρ s 3 + ∂g ω ∂ρ γ µ A (ω) 0 ρ + ∂g ρ ∂ρ A (ρ) 0 ρ 3 . (21)
The various densities are obtained from the nucleon single-particle wave functions in the "no-sea" approximation as

ρ s (x) = α η α ψα (x)ψ α (x) ρ s 3 (x) = α η α ψα (x)τ 3 ψ α (x) ρ(x) = α η α ψα (x)γ 0 ψ α (x) ρ 3 (x) = α η α ψα (x)γ 0 τ 3 ψ α (x) ρ (em) (x) = α η α ψα (x) 1 2 (1 -τ 3 )ψ α (x) [-ρ e (x)]. ( 22
)
where ρ s is the scalar density, ρ the baryon density, ρ s 3 the scalar isovector density, ρ 3 the vector isovector density, and ρ (em) the charge density. The occupation factors η α have to be determined from the desired scheme of occupation.

Neglecting retardation effects the Klein-Gordon equations reduce to inhomogeneous Helmholtz equations with source terms [START_REF] Fritz | [END_REF]

(-∆ + m 2 σ ) Φ σ = -g σ ρ s (-∆ + m 2 δ ) Φ δ = -g δ ρ s 3 (-∆ + m 2 ω ) A (ω) 0 = g ω ρ (-∆ + m 2 ρ ) A (ρ) 0 = g ρ ρ 3 -∆ A (γ) 0 = e ρ (em) , (23)
from which the self-energy contributions (20) are obtained. The Dirac equation for the nucleons (19), the evaluation of the resulting densities (22), these meson field equations (23) and the calculation of the resulting self-energy contributions (20) form a set of equations, which have to be solved in a self-consistent way.

For the description of nuclear matter in a Wigner-Seitz cell the Dirac equation (19) and the meson field equations ( 23) are solved in spatial representation. The numerical procedure to solve the Dirac equation in the cubic box is the same as for non-relativistic Skyrme-Hartree-Fock approach described above. Pairing correlations are included in terms of the BCS approximation assuming a density dependent zero-range pairing force [34].

The resulting single-particle energies and spinors were used in the calculation of NMFP as described in the next Subsection.

B. Neutrino mean free path in relativistic mean-field model

First, let us consider the charged current reaction. Here we will exploit the most general form for the nucleonic current, which is allowed due to the Lorentz, parity and isospin invariances [37]

J CC µ = i ψp [F v 1 (q 2 )γ µ + F v 2 (q 2 )σ µν q ν + F A (q 2 )γ 5 γ µ -iF p (q 2 )γ 5 q µ ]ψ n , (24)
where F v 1 and F v 2 are isovector electromagnetic formfactors, F A is the axial-vector formfactor, F p is the induced pseudoscalar formfactor. Following the common practice we ignore the contribution of the second-class currents. The leptonic current has the same structure as in (9). Analogously to (7), (11) the averaged squared matrix element for the charged current reaction can be written in the form

|M| 2 = G 2 F C 2 2 [ |M 1 | 2 (1 - p l 3E l cos ϑ) + |M 2 | 2 (1 + p l E l cos ϑ) ( 2 5
)

+|M 3 | 2 (p 2 l + E 2 ν -2p l E ν cos ϑ - p 3 l 3E l cos ϑ - p l 3E l E 2 ν cos ϑ + 2p 2 l 3E l E ν ) +|M 4 | 2 p l E l ((p 2 l + E 2 ν ) cos ϑ -2E l E ν cos ϑ + p l E l + E l p l E 2 ν -2p l E ν )],
where

M 1 = F 1 ψp γψ n + F A ψp γ 5 γψ n , M 2 = F 1 ψp γ 0 ψ n + F A ψp γ 5 γ 0 ψ n -iF p ψp γ 5 q 0 ψ n , M 3 = F 2 ψp Σψ n , M 4 = F p ψp γ 5 ψ n , and
Σ =   σ 0 0 σ   .
Dirac spinors ψ and the respective single-particle energies are obtained from the solution of the Dirac equation (19).

The hadronic part of the neutral current involves additionally isoscalar electromagnetic form factors F s 1 and F s 2 , so that

J NC µ = i 2 ψn [F A (q 2 )γ 5 γ µ -iF p (q 2 )γ 5 q µ (26) +(1 -2 sin 2 θ W )(F v 1 (q 2 )γ µ + F v 2 (q 2 )σ µν q ν ) -2sin 2 θ W (F s 1 (q 2 )γ µ + F s 2 (q 2 )σ µν q ν )]ψ n ,
where θ W is the Weinberg angle. The respective matrix element for this reaction looks like

|M| 2 = G 2 F 8 [ |N 1 | 2 (1 -1 3 cos ϑ) + |N 2 | 2 (1 + cos ϑ) ( 2 7 ) +|N 3 | 2 (E 2 ν + E 2 ν -2E ν E ν cos ϑ -1 3 E 2 ν cos ϑ -1 3 E 2 ν cos ϑ + 2 3 E ν E 2 ν ) +|N 4 | 2 (E ν -E ν ) 2 (1 + cos ϑ)],
where

N 1 = ((1 -2 sin 2 θ W )F v 1 -2 sin 2 θ W F s 1 ) ψn γψ n + F A ψn γ 5 γ, N 2 = -F A ψn γ 5 γ 0 ψ n -((1 -2 sin 2 θ W )F v 1 -2 sin 2 θ W F s 1 ) ψn ψ n + iF P ψn γ 5 ψ n , N 3 = ((1 -2 sin 2 θ W )F v 2 -2 sin 2 θ W F s 2 ) ψn Σψ n , N 4 = F p ψn γ 5 ψ n .
Substituting ( 25) and ( 27) in ( 10) one obtains the mean free path of neutrinos in relativistic mean-field model for charged and neutral current reactions, respectively.

IV. RESULTS AND DISCUSSIONS

Before we start the discussion of the transport properties of neutrinos in the crust of neutron stars let us review some details of self-consistent Hartree-Fock and relativistic meanfield calculations. For all the results presented in this manuscript a temperature T= 1 MeV was considered. This temperature is high enough to take into account some effects of finite temperatures and low enough to maintain the pairing correlations and stable quasi-nuclear structures. It was found that all results discussed below are not very sensitive to temperature changes in the range 0≤T≤1 MeV. The increase of temperature from 0 to 1 MeV leads to a smearing of levels in vicinity of the Fermi surface, however the number of such levels is small in comparison to the total number.

With increase of temperature the pasta phase structures become smoother and at some critical temperature they disappear. For the Skyrme-Hartree-Fock approach this critical temperature is around 5 MeV and 10 MeV for slab and rod structures, respectively, while the droplet structure disappears at a temperature higher than 15 MeV. Thus, the spherically symmetric droplet phase will play the main role in different simulations containing the temperature evolution. Employing the relativistic approach the pasta phase structures turn One can notice that the chosen temperature T=1 MeV is relatively small in order to connect our calculations of the mean free path with a certain astrophysical scenario, e.g., the neutrino cooling of NS [44]. However, as we emphasized in the Introduction, the main goal of our work is to investigate the difference of NMFPs in the pasta and homogeneous matter due to the different spatial density distributions. This difference will be largest at low temperature as at higher temperatures the inhomogeneous mass distributions tend to disappear. Therefore the low temperature considered here determines the maximal effect of the spatial density distribution on the mean free path. For higher temperatures this influence becomes weaker and vanishes at the melting temperature, which is around T=15

MeV.

For temperatures below 15 MeV and global densities below 0.08 fm -3 the variational calculations yield structures with inhomogeneous density distributions of protons and neu-trons (see figures 1-4). Comparing the spectra of single-particle energies obtained for the homogeneous and inhomogeneous solutions one observes that the single-particle energies for the localized states in inhomogeneous matter show more negative energies than the lowest single-particle states for the homogeneous approach. In the β-equilibrium all proton states are localized and therefore tend to have more attractive single-particle energies in the inhomogeneous as compared to the homogeneous density calculation. The variational calculations allowing for pasta structure yield larger proton fractions than obtained for the β-equilibrium of homogeneous matter at the same global density.

This can be seen from inspecting figure 5. The upper panel of this figure contains results of the proton abundances for baryonic matter in β-equilibrium resulted from non-relativistic Skyrme-Hartree-Fock calculations. The proton abundance of homogeneous matter is a monotonically increasing function of total density and it reaches the value of 4% at the density 0.1 fm -3 . Allowing for inhomogeneous matter distribution one obtains a significant increase of the proton fraction at densities below 0.03 fm -3 , while in the density region from 0.03 to 0.08 fm -3 its value is almost constant around 3.2%. The lower panel of figure 5 displays the corresponding results derived from the relativistic mean field approach. This relativistic approach seems to provide a smaller symmetry energy at these low densities, which leads to smaller proton abundances in the inhomogeneous as well as the homogeneous solution.

In figure 6 we want to demonstrate the dependence of the neutrino cross section for the charge current reaction on the spatial orientation of the momentum transfer q. This is displayed in terms of the corresponding neutrino mean free path, which has been calculated according to (14) from σ x (solid line) and σ z (dashed line), respectively. Note that due to our choice of the coordinate system the results for σ y are identical to those for σ x for the rod as well as the slab structures.

For the density ρ of 0.0625 fm -3 , which leads to a rod structure, we obtain results for the NMFP ranging 20 km for neutrinos with an energy of 10 MeV down to 30 cm for neutrinos with an energy of 100 MeV. For low-energy neutrinos the NMFP for reactions with a momentum transfer parallel to the x-axis is larger by a factor of 2 as compared to a momentum transfer parallel to the z-axis, difference which disappears for neutrinos with larger energies. This factor of 2 is non-negligible but small on the scale of variations for the NMFP as a function of the neutrino energy. Therefore the simple averaging procedure of (13) seems to be adequate. Similar results are obtained for the slab configuration as can be seen from the lower panel of figure 6. Note that the results for the NMFP are considerably smaller at low neutrino energies (by a factor of 10) and even at neutrino energies as large as 100 MeV smaller by a factor 2, although the ratio of the inverse densities is only about 1.2.

The NMFP calculated in CC (left panel) and NC (middle panel) reactions for homogeneous and inhomogeneous matter distributions are shown in figure 7. First, let us compare the NMFP of homogeneous matter for both types of reactions. The main influence on NMFP's results from the available phase space for each reaction. In fact, the proton fraction of homogeneous matter does not exceed 1% for the densities considered here. Thus we have to consider a much larger blocking effect for the neutrons in the final states NC reactions than for the protons in the CC reactions. Therefore the cross section of CC absorption is larger than in NC scattering, and consequently, the mean free path is shorter, as it is shown by the red dashed lines in figure 7. Due to the small proton abundances in homogeneous matter the Pauli blocking factor of final electron states affects the result for the CC reaction only at very small neutrino energies E ν < 10 MeV.

Figure 7 also presents results for the NMFP of inhomogeneous matter for both types The cross section for neutrino scattering in homogeneous matter increases with the baryonic density in a non-linear way (see discussion above). Therefore one may expect that the mean free path in the inhomogeneous matter is shorter than the corresponding one for homogeneous of the same global density, since the scattering on the quasi-nuclear structures shall enhance the respective cross section. Nevertheless, the NMFP obtained for the charged current reaction, which is shown in left column of figure 7 demonstrates the opposite behavior, specially at low densities, where the droplet phase occurs. The NMFP obtained from absorption in inhomogeneous matter is longer than the respective result derived from homogeneous matter calculations.

In order to explain this effect one should consider difference in proton fractions of homogeneous and inhomogeneous matter discussed in the beginning of this Section. At a typical density 0.0165 fm -3 , where the droplet phase occurs the proton abundance in inhomogeneous matter is significantly larger than the respective value obtained in the homogeneous matter.

This difference in the proton fractions has two effects: first, the homogeneous matter contains less protons in comparison with the inhomogeneous one. Consequently, the number of unoccupied final proton states is larger and more transitions, which contribute to the total cross section, are possible. Secondly, the chemical potential of electrons compensating the charge of the protons in matter is lower in case of homogeneous matter and the respective Pauli blocking factor for the produced electrons is lower than those obtained for the inhomogeneous matter. This effect again modifies the cross section considerably at low E ν . With increase of the energy of incoming neutrinos the Pauli blocking of electrons rapidly drops and more transitions become possible, so that the differences between homogeneous and inhomogeneous matter distributions are getting less significant and the respective NMFP's become closer one to another. At higher densities of matter, where the rod and slab phases occur, the difference in proton abundances are less important, therefore the resulted mean free paths are very similar and the effect of inhomogeneous structure becomes negligible.

At the end we should notice that at neutrino energies less than 10 MeV the NMFP's of homogeneous and inhomogeneous matter distributions calculated in CC reaction significantly exceed the typical neutron star radius. Therefore one can conclude that the charged current reaction is kinematically suppressed [45].

The results of neutral current reaction are shown on the middle panel of figure 7. It is obvious that the appearance of pasta phase in this case has no important influence on neutrino propagation, since this type of reaction does not depend on Pauli blocking of neutrino in final state (no trapped neutrinos). The only small difference in NMFP's of homogeneous matter and droplet phase may be explained by different values of matrix elements in (12), since the s.p. wave functions of bound neutrons in droplet significantly differ from wave functions of homogeneous matter. However even this small effect becomes negligible if the global density increases and the density profiles become smoother and transition to the homogeneous phase approaches.

The total NMFP, which combines both reactions, is shown on the right panel. One can see, that the total mean free path is slightly shorter than the mean free path in charged current reaction. Therefore we conclude that this reaction is dominant in the total neutrino response of the system.

A comparison of NMFP's of charged and neutral currents in case of pasta phase based on a relativistic mean-field model in a WS cell is displayed in figure 8. It is worth mentioning that within the relativistic model we could not find any formation of slab structures. Therefore only results for droplet and rod structures are shown. Also, the global density, at which the droplet phase occurs in the relativistic mean-field model is two times larger than the respective density in the nonrelativistic model. The difference between proton fractions of homogeneous matter and pasta phase is not so significant. In fact, the values of proton abundance around ρ = 0.02 fm -3 , displayed in the lower panel of figure 5, are about 40% smaller than the corresponding values obtained in the Skyrme model (the upper panel).

Therefore we omit the comparison between NMFP's of homogeneous and inhomogeneous matter however we compare the mean free paths of pasta phase for both types of reactions.

One can see that at E ν < 20MeV the behavior of CC curves is determined by the Pauli blocking, while at higher energies the result becomes sensitive to the structure of phase space available for the reactions. Both charged and neutral current mean free paths decrease if the global density of matter rises.

Up to now we considered the nuclear matter in β-equilibrium and found that the main role in NMFP of charged current reaction plays the electron blocking factor and the difference in phase spaces, caused by different proton numbers in pasta phase and homogeneous matter at the same global density. Therefore all effects due to different spatial distribution were hidden.

In order to investigate the influence of density distributions on NMFP one can perform calculations in homogeneous and inhomogeneous matter with the same proton fraction. For comparison we choose the non-relativistic droplet phase and homogeneous matter at the same density with fixed proton fraction 3.5%, ignoring the electron blocking factor. The results are shown in figure 9. The mean free path is shorter for inhomogeneous density distributions, however the difference between two curves is only about 20% at the neutrino energy E ν =10 MeV. Since the core of droplet phase, which is localed at the center of WS cell originated from the deeply bound states, its contribution becomes less significant with increase of energy E ν . In fact, when energy and momentum transfers rise, the contribution of deeply bound states becomes less important and on the contrary the contribution of transitions between weakly bound and free states of neutrons and protons increases. In this study we considered two interaction models: a nonrelativistic Skyrme force and a relativistic density-dependent mean field model. The dependence of the NMFP's on these interaction models turns out to be rather weak. At small neutrino energies (E ν < 25

MeV) the RMF yields a NMFP which is up to 20 percent larger than the corresponding result derived from the Skyrme model, whereas at larger values of E ν the Skyrme model predicts NMFP's which are up to 10 percent larger than those resulting from the RMF. So we conclude that the model-dependence of the NMFP is generally larger than the difference resulting from inhomogeneous versus homogeneous matter. Only at small densities and small neutrino energies one observes significant effects from inhomogeneous matter (see Fig. 7).

Summarizing we conclude that the NMFP is determined by three different factors. The first of them -the Pauli blocking effect of final electrons in CC reaction play the most important role at low neutrino energy and drops exponentially if the energy increases. The second factor is the difference in baryonic phase spaces of different reactions. The phase space of CC absorption is larger than in NC scattering, because the Fermi energy of final (proton) states is considerably lower than the neutron Fermi energy. The last factor, related to a significant difference of the spatial density distributions, leads to the reduction of the NMFP in inhomogeneous matter, however it plays a smaller role in comparison with other effects.

V. SUMMARY AND CONCLUSION

The aim of this study was to examine the role of the inhomogeneous baryonic density structures in the crust of neutron stars on the propagation of neutrinos. Our calculations of neutrino mean free paths (NMFP) are based on microscopic descriptions of the so-called "pasta structures" derived from 3D Hartree-Fock calculations with the SLy4 parameterization of the Skyrme potential as well as density-dependent relativistic mean-field calculations, which reproduce the empirical properties of normal nuclei with good accuracy. We find that the evaluated NMFP due to charged current reactions significantly depend on the structure of the pasta phase. This is mainly due to fact that the proton abundances derived from the β-equilibrium in inhomogeneous matter are larger than the corresponding values determined for symmetric matter. The effects of inhomogeneous baryonic density distributions is less pronounced for the neutral current contribution.

Recent studies show that the weakly bound neutrons may play an important role in formation of collective modes in the crust of neutron stars [26]. In our calculations the role of such collective features of neutrons has not yet been considered and an accurate calculation of nuclear response functions should be done in the future.

FIG. 2 :

 2 FIG. 2: Neutron density distribution for Skyrme HF + BCS calculation at an average density of ρ = 0.0625 fm -3 (rod).

density ρp [fm - 3 ]FIG. 3 : 3 ] 12 FIG. 4 :

 333124 FIG. 3: Proton density distribution for Skyrme HF + BCS calculation at an average density of ρ = 0.0775 fm -3 (slab).

FIG. 5 :

 5 FIG. 5: Proton abundance in the case of uniform matter (dashed line) and pasta phase (solid line). The symbols refer to specific calculations, whereas the lines have been added to guide the eye. The results for Skyrme-Hartree-Fock calculations are shown in the upper panel and the relativistic mean-field results in the lower one. The dashed arrows indicate typical densities leading to pasta structures of droplet, rod and slab shape.

3 FIG. 6 :

 36 FIG.6:The neutrino mean free path (NMFP) calculated for the charged current reaction in case of rod and slab configurations demonstrate the dependence of the result on spatial orientation of the momentum transfer q.

3 FIG. 7 :FIG. 8 :

 378 FIG. 7: Skyrme-Hartree-Fock calculations of NMFP for pasta phases (solid curves) and the respective results for homogeneous matter at the same global density (dashed curves). The results for the charged current reaction are shown in the left column, the neutral current NMFP in the middle, and the total NMFP is shown in the right panel.

FIG. 9 :

 9 FIG. 9: Skyrme-Hartree-Fock calculations of NMFP for droplet phase (solid curves) and the respective results for homogeneous matter (dashed curves) at the same global density ρ =0.0166fm -3 . The effects of electron blocking factor and β-equilibrium were ignored.

TABLE I :

 I Parameter set from DBHF by van Dalen et al.[42] for the density dependent relativistic mean field approach.

VI. ACKNOWLEDGMENTS

One of us, P. Grygorov, would like to thank Dr. V. Rodin for useful discussions on weak interaction with nuclei. This work has been supported by the European Graduate School "Hadrons in Vacuum, in Nuclei and Stars" (Basel, Graz, Tübingen), which obtains financial support by the DFG.


D N Poenaru
email: d.poenaru@fias.uni-frankfurt.de

R A Gherghescu

Walter Greiner

Individual and collective properties of fermions in nuclear and atomic cluster systems

Keywords: numbers: 25.60.Pj, 23.70.+j, 34.80.Ht, 36.40.Wa

Introduction

The electronic shell structure in alkali metal clusters [1] has shown a strong analogy with the single-particle states of atomic nuclei since delocalized electrons of a metallic cluster are fermions in a confined space like the nucleons (protons and neutrons). This analogy was practically exploited to adapt the nuclear macroscopic-microscopic method (MMM) [2] to atomic cluster physics [3][4][5][6][7][8], (see also the review papers [9][10][11] and the references therein) as an alternative to the density functional theory or quantum molecular dynamics used with great success to study the ground state properties and the fission process of atomic clusters. The detailed calculations for triaxial shapes [4] explained very well the experimental ground state properties of metallic (potassium, copper, and sodium) clusters, as well as the fission energetics of doubly ionized clusters. The electronic-entropy effects in the size-evolutionary patterns of small clusters have been properly investigated [6].

Within MMM the collective properties of fermions are well accounted for by the liquid drop model (LDM). The shell correction adds the contribution of the individual properties based on a quantum single-particle shell model. We contributed to this field by investigating within MMM the nanophysics of hemispheroidal clusters deposited on planar surfaces [12][13][14][15][16].

We would like to emphasize the importance of microscopical corrections and of the phenomenological LDM deformation energies for the synthesis of superheavy nuclei, cluster radioactivity, as well as for the singly charged trimer emission from a multiply ionized metallic cluster. By comparing these processes one can see the uniqueness of the latter one, for which both the LDM deformation energy and the shell correction are reaching the minima simultaneously at the same mass asymmetry. We hope that this remarkable property could be exploited in applications using the released energy in a similar way to the rich variety of applications of nuclear α-decay.

Superheavy nuclei

This year on February 19, the Nicolaus Copernicus birth day, the International Union of Pure and Applied Chemistry (IUPAC) officially recognized the sixth superheavy element, Copernicium (symbol Cn, Z = 112) discovered at GSI by an international team of scientists during the past 30 years. As shown in table 1 the "cold fusion" method of producing superheavy elements, leading to low excitation of the compound nucleus followed by one evaporated neutron at GSI Darmstadt [17,[START_REF] Hofmann | On Beyond Uranium[END_REF] and RIKEN [START_REF] Morita | [END_REF], is based on closed shell nuclear targets 208 Pb and 209 Bi (the cold valley idea [20,[START_REF] Greiner | Cluster Structure of Atomic Nuclei[END_REF]). The excited compund nucleus, A Z, is produced by bombarding the target, At Z t , with the projectile, Ap Z p :

At Z t + Ap Z p → A Z (1)
where the hadronic numbers are conserved: A = A p + A t and Z = Z p + Z t . The neutron number N = A -Z. The hot fusion method based on 48 Ca induced reactions, promoted at JINR Dubna [START_REF] Oganessian | [END_REF]23], allows to produce less neutron-defficient superheavies closer to the valley of beta-stability. The magic numbers of nucleons are marked with bold face fonts in table 1. For all seven superheavy elements the target neutron number is magic, N t = 126, for 3 of them (even atomic number Z) the target atomic number is magic, Z t = 82, and for Ds and Rg also the projectile atomic number, Z p = 28, is magic.

We illustrate the idea of cold valley for production of superheavy nuclei on potential energy surfaces (PES) versus the relative distance between centers ξ = (R-R i )/(R t -R i ) and the mass asymmetry η = (A t -A p )/(A) shown in figure 1 for 300 120. Here R is the separation distance of the fragments with an initial value R i (equal to R 0 -R p for spherical shapes) and the touching point value R t (equal to R p + R t for spherical fragments). R 0 , R t , R p are the radii of compound nucleus, of the target, and of the projectile. The phenomenological (LDM-like) Yukawa-plus-exponential (Y+E) model deformation energy, E Y +E , is shown at the bottom of this figure, where one can see an almost zero fission barrier, and the Businaro-Gallone mountains. The calculations using an advanced two-center shell model [24,25] are outlined elsewhere [26]. At the center is plotted the shell plus pairing corrections, δE sh+p , producing a finite fission barrier which makes possible the existence of this superheavy nucleus. Different valleys at various mass asymmetry are obtained whenever the nucleons of one or the two fragments are reaching a magic value during the deformation process. By adding E Y +E + δE sh+p we get the MMM deformation energy E def plotted at the top.

Figure 2 is a plot of cuttings through the three PES at the touching point configuration, R = R t , ξ = 1. The most important cold valley for synthesis of a superheavy nucleus with Z = 120 and A = 300 corresponds to the reaction 208 82 Pb 126 + 92 38 Sr 54 → 300 120, for which η 0.4. This is indeed the deepest valley of the shell and pairing corrections (the dotted curve in figure 2) because the doubly magic 208 Pb has a very strong shell effect. Nevertheless, after adding the Y+E model deformation energy, having a steep variation toward the Businaro-Gallone peak, this valley becomes shallower in the final result E def (the solid curve). The deepest valley of the total deformation energy remains that produced by the doubly magic light fragment 132 50 Sn 82 at η 0.12, which is also responsible, as a heavy fragment, for the asymmetric mass distribution in the region of heavy nuclei lighter than Fm.

Cold fission and cluster radioactivity

Cluster radioactivity was predicted in 1980 [27] and the first experimental confirmation was reported in 1984 [28]. A typical PES for heavy nuclei decaying by fission or cluster radioactivity

A Z → A d Z d + Ae Z e (2)
within LDM or Y+E model will always have a minimum at mass symmetry η = (A d -A e )/A = 0, so that collective and single-particle properties in nuclei are driving the system toward different mass asymmetry. One exception is 264 Fm which fissions symmetrically in two identical doubly magic fragments 132 Sn [29]. In this case both phenomenological deformation energy and the shell corrections are exhibiting a minimum value at η = 0. Other exceptions of symmetrical mass distribution of fission fragments were experimentally observed in fission of few Fm, Md, No, and Rf isotopes [30,[START_REF] Hoffman | chap 10 in Nuclear Decay Modes[END_REF].

In the region of nuclei with Z = 87 -96, where cluster radioactivity was confirmed [START_REF] Bonetti | [END_REF] (experiments performed in Oxford, Moscow, Orsay, Argonne, Berkeley, Dubna, Livermore, Geneva, Milano, Vienna, Beijing), the 208 Pb valley proved again to be of practical importance, as can be seen from the table 2.

Even for alpha decay it is possible to see such a valley if the emitter is 212 Po or 106 Te [33]. In the latter case the heavy fragment 102 Sn with proton magic number Z = 50 plays the important role.

As can be seen from the table 2, in all successful measurements of cluster decay modes performed until now the daughter has at least one magic number of nucleons, either Z d = 82, N d = 126, or both (daughter 208 Pb). There is one exception: the 28 Mg radioactivity of 236 U where the daughter has an atomic number 80 and a neutron number 128, not very far from the magicity. Also two of the emitted clusters 14 C, and 34 Si possess a magic neutron number (N e = 8 and N e = 20, respectively), and there is one emitted cluster, 20 O, with magic number of protons Z e = 8.

Fission of multiply charged metallic clusters

Now let us turn to the metallic cluster fission process [5,11,[34][35][36][37][38][START_REF]Latest advances in atomic cluster collisions: fission, fusion, electron, ion and photon impact[END_REF][START_REF] Lyalin | [END_REF][41] :

M z+ N → M z 1 + N 1 + M z 2 + N 2 (3)
where the excess charge z and the number of atoms N are conserved: z = z 1 + z 2 and N = N 1 + N 2 . Also the corresponding number of delocalized electrons is conserved:

n e = n e1 + n e2 , where n e = N -z, n e1 = N 1 -z 1 and n e2 = N 2 -z 2 .
The experiments on charged metallic cluster fission are showing a clear high yield for a singly charged trimer light fragment, z 2 = 1, N 2 = 3, n e2 = 2 in eq. 3, the analog of the α-decay, having a magic number of 2 delocalized electrons. Since the minima of the two valleys coincide we may say that in atomic cluster physics the "ideal" conditions of a superasymmetric fission (alpha-like emission) are frequently fulfilled. We present in figure 3 the large asymmetry part of our calculations for fission M 6+ 146 → M 5+ 143 + M 1+ 3 of alkali metal clusters Cs, K, Na, Li (left-hand side) and transition metal clusters Al, Ag, Au, Cu (right-hand side) with a number of atoms 146, and a positive charge excess z = 6. The particular split leading to a singly charged trimer emission M 1+ 3 illustrates, particularly for the alkali metals, how the minmum of the LDM energy takes place at the same mass asymmetry, corresponding to p = n e1 = 2 or to n e2 = n -p = 138, where n = n e . Both numbers of fission fragments delocalized electrons 143 -5 = 138 and 3 -1 = 2 are magic numbers. A simple explanation of the existence of a minimum of LDM deformation energy at the large mass asymmetry corresponding to p = 2 can be obtained by plotting in figure 4 the derivative dQ LD /dp of the released (dissociation) energy calculated within LDM for spherical shapes [36] as a sum of three contributions coming from the Coulomb, surface, and ionization energies:

Q LD = Q C + Q s + Q IP , dQ LD dp = e 2 6r s z 2 1 p 4/3 - (z -z 1 ) 2 (n -p) 4/3 + 2a s 3 1 (n -p) 1/3 - 1 p 1/3 + e 2 24r s z -z 1 (n -p) 4/3 - z 1 p 4/3 ( 4
)
where a s = 4πr 2 s σ is the surface energy constant proportional to the surface tension σ, e 2 /2 = 7.1998259 eV• Å, e is the electron charge and r s is the Wigner-Seitz radius. Due to the smooth variation with p of the interaction energy between the separated fragments, the fission barrier is minimum when the Q-value is maximum, dQ LD /dp = 0, which happens very close to p = 2 for alkali clusters. Transition metals (right-hand side of figure 4) may not have a minimum fission barrier at p = 2, but anyhow the slope of deformation energy variation with p is very small in this region.

We can see in figure 5 the influence of deformation and shell effects (points) and the spherical LDM values (lines), increasing the Q 2 -values particularly for any parent number of delocalized electrons leading to a magic daughter. Spherical magic numbers: 2, 8, 40, 58, 92, 138, 198, 264, 344, 442, 554, 680, ... The points were obtained by making the calculations after minimization within MMM [16] of the parent and daughter deformation energies E def = E LDM + δE sh+p vs. spheroidal deformation.

Possible applications in nanotechnology may be envisaged in which the kinetic energy of the singly charged trimer can be used in analogy with the wide spread applications of nuclear α-decay. The Q-value for metallic cluster fission increases with the charge z (see figure 5). It is large when the surface tension, σ, is large and the Wigner-Seitz radius, r s , is small (larger for transition metallic clusters and smaller for alkali metal clusters). At a given value of z the larger Q-values are obtained for smaller n e when the fissility parameter X = E 0 C /(2E 0 s ) = (ez) 2 /(16πr 3 s σn e ) approaches unity. In conclusion the valleys produced by shell effects on PES of superheavy nuclei and cluster radioactive heavy nuclei are usually shallower because the minima of shell corrections occur at a mass asymetry η = 0, while the minimum of LDM energy is placed at η = 0. Charged clusters are ideally "alpha" emitters because both LDM and shell corrections are reaching a minimum for the corresponding mass asymmetry. When the number of delocalized electrons of both fragments are magic, the Q-value exhibits local maxima. In order to have large absolut values one has to choose metallic clusters with high σ and low r s (transition metals). UEFISCSU, and partially within PN09370102 of the Nucleu programme of Ministry of Education and Research, Bucharest.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .

 12345 Figure 1. Y+E model deformation energy (bottom), shell plus pairing corrections (center), and the total deformation energy (top) PES of 300 120 vs ξ = (R-R i )/(R t -R i ) and η = (A t -A p )/(A).

Table 1 .

 1 Magic numbers of nucleons in the target and projectile producing cold valleys for synthesis of superheavy nuclei at GSI Darmstadt and RIKEN

		Element	Projectile	Target
	Z	Symbol	Name	N p Z p	N t	Z t
	107	Bh	Bohrium	54 Cr 30 24 209 Bi 126 83
	108	Hs	Hassium	58 Fe 32 26 208 Pb 126 82
	109	Mt	Meitnerium	58 Fe 32 26 209 Bi 126 83
	110	Ds	Darmstadtium 62 Ni 34 28 208 Pb 126 82
	111	Rg	Roentgenium	64 Ni 36 28 209 Bi 126 83
	112	Cn	Copernicium	70 Zn 40 30 208 Pb 126 82
	113			70 Zn 40 30 209 Bi 126 83

Table 2 .

 2 Magic numbers of nucleons in the daughter and emitted cluster of the experimentally confirmed cluster radioactivities.

		Cluster		Parent, Daughter	Cluster		Parent, Daughter
		Z e N e		Z d N d		Z e N e		Z d N d
	14 C	6	8	221 Fr	81 126	14 C	6	8	221 Ra 82 125
				222 Ra 82 126				223 Ra 82 127
				224 Ra 82 128				226 Ra 82 130
				223 Ac 83 126				225 Ac 83 128
	20 O	8	12 228 Th 82 126	23 F	9	14 231 Pa 82 126
	22 Ne 10 12	230 U	82 126 24 Ne 10 14 231 Pa 81 126
	24 Ne 10 14	232 U	82 126				233 U 82 127
				234 U	82 128				235 U 82 129
	25 Ne 10 15	233 U	82 128 25 Ne 10 15	235 U 82 128
	26 Ne 10 16	234 U	82 126 28 Mg 12 16	234 U	80 126
	28 Mg 12 16	236 U	80 128				236 Pu 82 126
				238 Pu 82 128 30 Mg 12 18	236 U	80 126
	30 Mg 12 18 238 Pu 82 126 32 Si	14 18 238 Pu 80 126
	34 Si	14 20 242 Cm 82 126

of alkali clusters (left-hand side) and transition metal clusters (right-hand side).

Acknowledgments

This work is partially supported by Deutsche Forschungsgemeinschaft Bonn, partially within IDEI Programme under contracts 123/01.10.2007 and 124/01. 10.2007 with