The pasta phase and its consequences on neutrino opacities
aa r X i v : . [ nu c l - t h ] N ov The pasta phase and its consequences on neutrino opacities
M. D. Alloy ∗ and D. P. Menezes † Universidade Federal da Fronteira Sul, Chapec´o, SC, CEP 89.812-000, Brazil. Depto de F´ısica CFM, Universidade Federal de Santa Catarina,Florian´opolis, SC CP.476, CEP 88.040-900, Brazil (Dated: 31 de outubro de 2018)In this paper, we calculate the diffusion coefficients that are related to the neutrino opacities con-sidering the formation of nuclear pasta and homogeneous matter at low densities. Our results showthat the mean free paths are significantly altered by the presence of nuclear pasta in stellar matterwhen compared with the results obtained with homogeneous matter. These differences in neutrinoopacities certainly influence the Kelvin-Helmholtz phase of protoneutron stars and consequently theresults of supernova explosion simulations.
I. INTRODUCTION
When massive stars (8 M ⊙ < M < M ⊙ ) exhaustsits fuel supply, the forces that support the stars corequickly retreat, and the core is almost instantly crushedby gravity, which triggers a type II supernova explosion.The remnant of the gravitational collapse of the core of amassive star is a compact star or a black hole, dependingon the initial condition of the collapse. Newly-born pro-toneutron stars (PNS) are hot and rich in leptons, mostly e − and ν e and have masses of the order of 1 − M ⊙ [1, 2].During the very beginning of the evolution, most of thebinding energy, of the order of 10 ergs is radiated awayby the neutrinos.The composition of protoneutron and neutron starsremains a source of intense speculation in the litera-ture. Whether their internal structure is formed by nu-cleons and leptons, by other light baryons and leptons,by baryons, leptons and quarks (bearing or not a mixedphase), by baryons, leptons and kaons or by other pos-sible composition, is still unknown. The neutrino-signalsdetected by astronomers can be used as a constraint toinfer protoneutron star composition [2, 3]. For the samepurpose, theoretical studies involving different possibleequations of state obtained for all sorts of matter com-position have to be done because the temporal evolutionof the PNS in the so-called Kelvin-Helmholtz epoch, du-ring which the remnant compact object changes from ahot and lepton-rich PNS to a cold and deleptonized neu-tron star depends on two key ingredients: the equationof state (EoS) and its associated neutrino opacity at su-pranuclear densities [3, 4].Neutrinos already present or generated in the PNS hotmatter escape by diffusion (not free streaming) because ofthe very high densities and temperatures involved. Theneutrino opacity is calculated from the scattering andabsorption reactions that take place in the medium andhence, related to its mean free path, which is of the or-der of 10 cm and much smaller than the protoneutron ∗ E-mail me at:alloy@uffs.edu.br † E-mail me at:[email protected] star radius [5]. In the diffusion approximation used toobtain the temporal evolution of the PNS in the Kelvin-Helmholtz phase, the total neutrino mean free path de-pends on the calculation of diffusion coefficients, which,in turn, depend on the chosen EoS. At zero temperatureno trapped neutrinos are left in the star because theirmean free path would be larger than the compact starradius.A complete equation of state capable of describingmatter ranging from very low densities to few times sa-turation density and from zero temperature to around50 MeV is a necessary step towards the understanding ofPNS evolution. The constitution of the PNS crust playsa definite role in the emission of neutrinos. For this rea-son, the pasta phase, present in very low nuclear matteras the crust of PNS are included in the investigation ofthe neutrino opacity in the present work.A few words on the pasta phase follow. It is a frustra-ted system [6–10] present at densities of the order of 0.006- 0.1 fm − [11] in neutral nuclear matter and 0.029 - 0.065fm − [12, 13] in β -equilibrium stellar matter, where acompetition between the strong and the electromagneticinteractions takes place. The basic shapes of these com-plex structures were named [6] after well known types ofcheese and pasta: droplets = meat balls (bubbles = Swisscheese), rods = spaghetti (tubes = penne) and slabs =lasagna, for three, two and one dimensions respectively.The pasta phase is the ground state configuration if itsfree energy per particle is lower than the correspondingto the homogeneous phase at the same density.The evolution of PNS and simulation of supernova ex-plosion have already been considered for different mattercompositions, some with the inclusion of the pasta phase[3, 4, 7, 14, 15]. From [3, 4] one can see that the trans-port properties are significantly affected by the presenceor absence of hyperons and of the mixed phase in hy-brid stars. In [7] the linear response of the nuclear pastato neutrinos was calculated with a semi-classical simu-lation and the muon and taon neutrinos mean-free pathwere described by the static structure factor of the pastaevaluated with Metropolis Monte Carlo simulations. In[14] rod-like (two dimensions) and slab-like (one dimen-sion) pasta structures were included in the calculationof neutrino opacity within quantum molecular dynamics.A very interesting conclusion was that the pasta phaseoccupies 10-20% of the mass of the supernova core in thelater stage of the collapse.In the present work we investigate the influence of thepasta phase on the neutrino opacity by showing the ef-fects on the diffusion coefficients. The pasta phase iscalculated with the coexistence phases method (CP) in amean field approximation [11, 13, 16]. We consider onlynucleons and leptons in the EoS in β -equilibrium. In thepasta structure only electron neutrinos are considered.In section II we present the formalism used to obtainthe equation of state, in section III the recipe used for theconstruction of the pasta phase is outlined, in section IVthe expressions used to calculate the neutrino cross sec-tions and related mean free path are given and in sectionV our results are shown and the main conclusions arediscussed. II. FORMALISM
We consider a system of protons and neutrons withmass M interacting with and through an isoscalar-scalarfield φ with mass m s , an isoscalar-vector field V µ withmass m v and an isovector-vector field b µ with mass m ρ described by the well known non-linear Walecka model(NLWM) [17]. We impose β -equilibrium and charge neu-trality with neutrino trapping at finite temperature. Atzero temperature no neutrinos are left in the system.The Lagrangian density reads L = X j = p,n L j + L σ + L ω + L ρ + X l = e,ν L l , (1) where the nucleon Lagrangian reads L j = ψ j [ γ µ iD µ − M ∗ ] ψ j , (2)were M ∗ = M − g s φ is the effective baryon mass and iD µ = i∂ µ − g v V µ − g ρ τ · b µ . (3)The meson Lagrangian densities are given by L σ = 12 (cid:18) ∂ µ φ∂ µ φ − m s φ − κφ − λφ (cid:19) , (4) L ω = 12 (cid:18) −
12 Ω µν Ω µν + m v V µ V µ (cid:19) , (5) L ρ = 12 (cid:18) − B µν · B µν + m ρ b µ · b µ (cid:19) , (6)where Ω µν = ∂ µ V ν − ∂ ν V µ and B µν = ∂ µ b ν − ∂ ν b µ − g ρ ( b µ × b ν ). The lepton Lagrangian densities read L l = ψ l [ γ µ i∂ µ − m l ] ψ l , (7)where m e is the electron mass and the neutrino mass is m ν = 0.The parameters of the model are three coupling cons-tants g s , g v and g ρ of the mesons to the nucleons, thenucleon mass M , the electron mass m e , the masses ofthe mesons m s , m v and m ρ and self-interacting couplingconstants κ and λ . The numerical values of the parame-ters used in this work and usually referred to as NL3 [18]are shown in table I. They are fixed in such a way thatthe main nuclear matter bulk properties are the bindingenergy equal to 16.3 MeV at the saturation density 0.148fm − , the compressibility is 272 MeV and the effectivemass at the saturation density is 0.6 M. Model g s g v g ρ M m e m s m v m ρ κ/M λ NL3 10.217 12.868 8.948 939.0 0.511 508.194 782.501 763.0 4.377 -173.31Tabela I. Parameters set used in this work. All masses aregiven in MeV.
From the Euler-Lagrange formalism we obtain theequations of motion for the nucleons and for the mesonfields: ∇ φ = m s φ + 12 κφ + 13! λφ − g s ρ s , (8) ∇ V = m v V − g v ρ B , (9) ∇ b = m ρ b − g ρ ρ , (10)where ρ s , ρ B and ρ are defined next. By replacing the meson fields by their mean values φ → h φ i = φ , (11) V µ → h V µ i = V , (12) b µ → h b µ i = b , (13)the equations of motion read φ = − κ m s φ − λ m s φ + g s m s ρ s , (14) V = g v m v ρ B , (15) b = g ρ m ρ ρ , (16)where ρ B = ρ p + ρ n is the baryonic density and ρ = ρ p − ρ n , ρ p and ρ n are the proton and neutron densitiesgiven by ρ j = 2 Z d p (2 π ) ( f j + − f j − ) , j = p, n (17)where f j ± = 1 / (1 + exp [( ǫ j ∓ ν j ) /T ]), ǫ j = p p + M ∗ and ν j = µ j − g v V − g ρ τ b , where τ is the appropriateisospin projector for the baryon charge states and µ j arethe nucleon chemical potentials. The scalar density ρ s isgiven by ρ s = 2 X j = p,n Z d p (2 π ) M ∗ ǫ j ( f j + + f j − ) . (18)The thermodynamic quantities of interest are given interms of the meson fields. They are the total energydensity E T = E + X l = e,ν E l , (19)with E = 1 π X j = p,n Z dp p p p + M ∗ ( f j + + f j − ) (20)+ m v V + m ρ b + m s φ (21)+ κ φ + λ φ , (22)the total pressure is P T = P + X l = e,ν P l , (23)with P = 13 π X j = p,n Z dp p p p p + M ∗ ( f j + + f j − ) (24)+ m v V + m ρ b − m s φ (25) − κ φ − λ φ , (26)and the total entropy density S = 1 T ( E T + P T − X j = p,n µ j ρ j − X l = e,ν µ l ρ l ) , (27) where the electron and electron neutrino energy densitiesare E l = g l π Z dp p q p + m l ( f l + + f l − ) , (28)and electron and electron neutrino pressure are P l = g l π Z dp p p p + m l ( f l + + f l − ) . (29)The electron density ρ e and electron neutrino density ρ ν are given by ρ l = g l Z d p (2 π ) ( f l + − f l − ) , (30)where g e = 2, g ν = 1, f l ± = 1 / (1 + exp [( ǫ l ∓ µ l ) /T ])with ǫ l = p p + m l and µ e is the electron chemical po-tential, ǫ ν is the electron neutrino energy and µ ν is theelectron neutrino chemical potential. The condition of β equilibrium in a system of protons, neutrons, electronsand trapped electron neutrinos is µ p = µ n − µ e + µ ν . (31)We impose neutrality of charge as ρ p = ρ e and fix thelepton fraction Y L = ρ e + ρ ν ρ B . (32)Notice that muons are not considered in the presentcalculation. III. COEXISTING PHASES: NEUTRALNUCLEAR MATTER WITH NEUTRINOTRAPPING
The formation of pasta phase has been studied latelywith great interest [7, 19]. Next we show the main stepsfor the calculation of the pasta phase with the coexistencephases method based on [20, 21]. For further details,please refer to [11, 13].For a given total density ρ B and lepton fraction Y L webuild pasta structures with different geometrical forms ina background nucleon gas with β stability and neutrinotrapping. This is achieved by calculating from the Gibbsconditions the density and the particle fractions of thepasta and of the background gas so that in the whole wehad to solve simultaneously the following seven equations P I ( ν Ip , ν In , M ∗ I ) = P II ( ν IIp , ν
IIn , M ∗ II ) , (33) µ In = µ IIn , (34) µ Ie = µ IIe , (35) µ Iν = µ IIν , (36) m s φ I + κ φ I ) + λ φ I ) = g s ρ Is , (37) m s φ II + κ φ II ) + λ φ II ) = g s ρ IIs , (38) f ( ρ Ip − ρ Ie ) + (1 − f )( ρ IIp − ρ IIe ) = 0 , (39)where I and II label each of the phases, f is the volumefraction of phase I f = ρ B − ρ II ρ I − ρ II . (40)The total pressure is given by P T = P I + P e + P ν . Thetotal energy density of the system is given by E = f E I + (1 − f ) E II + E e + E ν + E surf + E Coul , (41)with E surf = 2 E Coul [22, 23], and E Coul = 2 α / ( e π Φ) / [ σD ( ρ Ip − ρ IIp )] / , (42)where α = f for droplets, rods and slabs and α = 1 − f for bubbles and tubes, σ is the surface energy coefficient, D is the dimension of the system. For droplets, rods andslabs, Φ is given byΦ = (cid:20)(cid:18) − Df − D D − + f (cid:19) D +2 (cid:21) , D = 1 , , f − − ln( f ) D +2 , D = 2 , (43)and for bubbles the above expressions are valid with f replaced by 1 − f . The surface tension plays a signifi-cant role on the appearance of the pasta phase. In ourtreatment of the surface tension we essentially follow theprescription given in [11, 13], but some comments onthe importance of the surface energy on the calculationof the pasta phase are mandatory. It has been shown thatthe existence of the pasta phase as the lowest free energymatter and of its internal structures essentially dependson the value of the surface tension [9, 11, 13, 16, 24]. Inthe present paper the surface energy coefficient is para-metrized in terms of the proton fraction according to thefunctional proposed in [25], obtained by fitting Thomas-Fermi and Hartree-Fock numerical values with a Skyrmeforce. The same prescription was used in [11, 13]. Howe-ver, a better recipe is to consider the surface energy coef-ficient in a consistent way, in terms of relativistic models.In [16] the surface energy was parametrized according tothe Thomas-Fermi calculations for three parametrizati-ons of the relativistic NLWM. The Gibbs prescriptionwas used to obtain the σ coefficient which is the appro-priate surface tension coefficient to be used [26, 27]. Thisimprovement will be added to our calculations in a forth-coming work. IV. NEUTRINO CROSS SECTIONS
To calculate neutrino opacities and mean free paths weconsider [5] neutral current scattering reactions ν e + n → ν e + n, (44) ν e + p → ν e + p, (45) and charged current absorption reactions ν e + n → e − + p. (46) ν e + p → e + + n. (47)The cross sections for reactions (44), (45), (46) and(47) employed in this study follow [5]:Reaction (44): σ n = σ ref = (cid:0) σ (cid:1) (cid:16) ǫ ν m e c (cid:17) , nN D , νD or νN D,σ ref (cid:16) ǫ ν p F c (cid:17) (cid:16) (1+4 g A )5 (cid:17) , nD , νN D , [28] ,σ ref (cid:0) (cid:1) (cid:16) π (1+2 g A )8 (cid:17) × (cid:16) Tǫ ν (cid:17) (cid:16) Tp F c (cid:17) (cid:16) M ∗ c ǫ F (cid:17) , nD , νD , [29, 30] . (48)Reaction (45): σ p = σ n , pN D , νD or νN D,σ n (cid:16) Y n Y p (cid:17) / , pD , νN D,σ n (cid:16) Y n Y p (cid:17) , pD , νD , [29] . (49)Reactions (46) and (47): σ a = σ ref (1 + 3 g A ) , nN D , νN D,σ ref (1 + 3 g A ) (cid:16) Y p Y n + Y p (cid:17) , nN D , νD or νN D [31] ,σ ref (1 + 3 g A ) (cid:0) (cid:1) (cid:16) π (cid:17) × (cid:16) Tǫ ν (cid:17) (cid:16) M ∗ c ǫ F (cid:17) (cid:16) Y e Y n (cid:17) / , nD , νD , [28] , , nD , Y L < . . (50)In this expressions p F and ǫ F mean the Fermi momen-tum and Fermi energy of the degenerate neutron. Y e , Y n , Y p , Y L , are the electron, neutron, proton and leptonfractions. ND denotes the non degenerate regime, whileD denotes the validity in case of degenerate particles. σ = 1 . × − cm and g A = 1 . λ ν = 1 ρ n σ n + ρ p σ p + ρ B σ a . (51)Rosseland neutrino mean free paths are related with dif-fusion coefficients D j [4] by λ kν = D k R ∞ dǫ ν ǫ kν f ν (1 − f ν ) , (52)where D k = Z ∞ dǫ ν ǫ kν λ ν f ν (1 − f ν ) . k = 2 , , Figura 1. free energy per particle with the NL3 parametriza-tion obtained for T = 5 MeV and Y L = 0 . F / A - M ( M e V ) (fm -3 ) Homogeneous matter Slabs Rods Droplets
T=5 MeVY L =0.4NL3 All contributions from neutrino opacities are related withthe diffusion coefficients and can to be used as inputto the solution of the transport equations in the equi-librium diffusion approximation to simulate the Kelvin-Helmholtz phase of the protoneutron stars [32].
V. RESULTS AND CONCLUSIONS
Before we tackle the problem of the consequences ofthe pasta phase on the diffusion coefficients, we displaya characteristic figure of the free energy for the homoge-neous and pastalike matter obtained for T = 5 MeV and Y L = 0 . β -equilibrium presentsa pasta phase smaller than matter with trapped neutri-nos [13, 16] as a consequence of the fact that the latterpresents a larger fraction of protons. According to stu-dies on binodals and spinodals underlying the conditionsfor phase coexistence and phase transitions [11, 33, 34],non-homogeneous matter with trapped neutrinos is ex-pected to be found until temperatures around T = 12MeV, depending on the model considered.We next show the diffusion coefficients D , D and D as function of the baryon density for different temperatu-res obtained for both homogeneous matter and the pastaphase. According to [11, 13] the densities where mat-ter becomes homogeneous depend on the proton fraction and on the temperatures involved, but it is always smal-ler than 0.1 fm − for the NL3 parametrization and forthe σ values we consider in the present work.In obtaining the diffusion coefficients, the EoS wascalculated as a grid where temperature ranges are inbetween 0 and 50 MeV and densities vary from 0.005 to0.5 fm − . In our codes we have implemented a prescrip-tion given in [35] to evaluate the Fermi integrals so thatthe same codes run from zero (10 − ) to high temperatu-res. We have calculated the diffusion coefficientes onlyfor baryonic densities above 0 . f m − because the in-tegrals of type (53) are very difficult to converge at lowerdensities. We show results for lepton fractions equal to0.2 and 0.4 because those are typical values necessary inthe numerical simulation of protoneutron star evolution.In all figures the diffusion coefficients obtained withhomogeneous matter join the curves obtained with thepasta phase at densities higher than the ones shown. For D calculated at T=5 MeV and Y L = 0 .
4, for instance,they cross each other at ρ = 0 .
12 fm − . Our codes inter-rupt the calculation once homogeneous matter becomesthe ground state configuration, as depicted in Fig. 1.This means that there will always be a gap in the dif-fusion coefficients when the transport equations are cal-culated with the inclusion of the pasta phase. The samebehavior is found at the pressure values for homogeneousand pasta phases at the transition density.From figures 2, 3 and 4 we can see that only threestructures are found inside the pasta phase for the presentmodel: droplets, rods and slabs as far as Y L = 0 .
4. For Y L = 0 . − . The interpolation procedure we use depends onthe quantities η i = ( µ i − M ∗ ) /T, i = p, n . Whenevereither η p or η n inverts its sign, these kinks appear, i.e.,they are the result of the effective nucleon mass beinggreater than the corresponding chemical potential. Mo-reover, the pasta phase diffusion coefficients are alwayslower than the corresponding coefficients obtained withhomogeneous matter.Our results for the diffusion coefficients D and D areone order of magnitude larger than the results obtainedin [32]. This difference can be explained because in thepresent paper all diffusion coefficients are calculated atvery low baryonic densities. For larger densities the re-sults coincide.In summary we point out that in the present paper wehave investigated the influence of the pasta phase on theneutrino opacity by calculating the diffusion coefficients.The homogeneous EoS was obtained with the NL3 para-metrization of the NLWM in a mean field approximation.The pasta phase was obtained with the coexistence pha-ses method (CP).Recent calculations for the pasta phase within theThomas-Fermi approximation at finite temperatures [36]show that the internal pasta structure is much richer ascompared with the CP method we have employed in thepresent work. Hence, the dependence on the structureof the pasta phase is also of interest and this calculationis planned for different parametrizations of the NLWM.More sophisticated matter, which includes the α -particlesshould also be considered [16].We have checked that the neutrino interactions inwarm and low baryonic densities with pasta formationshow significant differences when compared with homo-geneous matter. Next the temporal evolution of the PNSwill be calculated and, in face of the present results, weexpect that the cooling and deleptonization eras will be influenced by the presence of the pasta phase at low den-sities.An obvious improvement is the inclusion of hyperons inthe EoS. However, the pasta phase can still be computedjust with protons, neutrons and light clusters becausehyperons are expected to appear only at densities wherethe pasta phase is no longer present. ACKNOWLEDGMENTS
This work was partially supported by the Braziliansponsoring bodies CNPq and CAPES (M.D. Alloy scho-larship). [1] J. M. Lattimer and M. Prakash, Science , 536 (2004).[2] W. Keil and H. Janka, Astron. Astrophys. , 145(1995).[3] J.A. Pons, A.W. Steiner, M. Prakash and J.M. Lattimer,Phys.Rev.Lett. (2001) 5223-5226.[4] J.A. Pons, S. Reddy, M. Prakash, J.M. Lattimer and J.A.Miralles, Astrophys. J. 513, 780 (1999).[5] A. Burrows and J. M. Lattimer, Astrophs. J 307, 178(1986).[6] D. Ravenhall, C.J. Pethick and J.R. Wilson, Phys. Rev.Lett. , 2066 (1983).[7] C. J. Horowitz, M. A. Perez-Garcia, and J. Piekarewicz,Phys. Rev. C , 045804 (2004).[8] C.J. Horowitz, M.A. P´erez-Garcia, D.K. Berry and J.Piekarewicz, Phys. Rev. C 72 , 035801 (2006).[9] T. Maruyama, T. Tatsumi, D.N. Voskresensky, T. Tani-gawa and S. Chiba, Phys. Rev.
C 72 , 015802 (2005).[10] G. Watanabe and H. Sonoda, cond-mat / 0502515[11] S.S. Avancini, D.P. Menezes, M.D. Alloy, J.R. Marinelli,M.M.W. de Moraes and C. Providˆencia, Phys. Rev.
C78 , 015802 (2008).[12] J. Xu, L.W. Chen, B.A. Li and H.R. Ma,arXiv:0807.4477v1 [nucl-th].[13] S.S. Avancini, L. Brito, J.R. Marinelli, D.P. Menezes,M.M.W. de Moraes, C. Providˆencia and A.M. Santos -Phys. Rev.
C 79 , 035804 (2009).[14] H. Sonoda, G. Watanabe, K. Sato, T. Takiwaki, K. Ya-suoka, T. Ebisuzaki, Phys. Rev. C 75, 042801 (2007).[15] G. Watanabe, H. Sonoda, T. Maruyama, K. Sato, K. Ya-suoka, T. Ebisuzaki, Phys. Rev. Lett. 103, 121101 (2009).[16] S.S. Avancini, C.C. Barros, D.P. Menezes and C. Pro-vidˆencia, Phys. Rev. C 82, 025808 (2010).[17] B. Serot and J.D. Walecka,
Advances in Nuclear Physics
16, Plenum-Press, (1986) 1.[18] G. A. Lalazissis, J. K¨onig and P. Ring, Phys. Rev.
C 55 ,540 (1997).[19] G. Watanabe et al., Physical Review Letters , 031101(2005).[20] , M. Barranco and J. R. Buchler, Phys. Rev. C 22, 1729(1980).[21] D. P. Menezes and C. Providˆencia, Phys. Rev. C 60,024313 (1999).[22] , D. G. Ravenhall, C. J. Pethick and J. R. Wilson,Phys.Rev. Lett 50, 2066 (1983). [23] T. Maruyama, T. Tatsumi, D. N. Voskresensky, T. Tani-gawa and S. Chiba, Phys. Rev. C 72, 015802 (2005).[24] G. Watanabe, K. Iida and K. Sato, Nucl. Phys. A676,455 (2000); G. Watanabe, K. Iida and K. Sato, Nucl.Phys. A687, 512 (2001); G. Watanabe, K. Iida and K.Sato, Nucl. Phys. A726, 357 (2003).[25] J.M. Lattimer, C.J. Pethick, D.G. Ravenhall, D.Q.Lamb, Nucl. Phys. A432 , 646 (1985).[26] M. Centelles, M. Del Estal and X. Vi˜nas, Nucl. Phys.
A635 , 193 (1998).[27] W. D. Mayers and W. J. Swiatecki, Phys. Rev. C ,034318 (2001).[28] N. Iwamoto, Ph.D. thesis, AA(Illinois Univ., Urbana-Champaign.), 1981.[29] B. T. Goodwin and C. J. Pethick, Astrophys. J. , 816(1982).[30] R. F. Sawyer and A. Soni, Astrophys. J. , 859 (1979).[31] S. A. Bludman and K. A. van Riper, Astrophys. J. ,631 (1978).[32] S. Reddy, M. Prakash and J.M. Lattimer, Phys. Rev. D58 , 013009 (1998).[33] C. Ducoin, C. Providˆencia, A. M. Santos, L. Brito andPh. Chomaz, Phys. Rev.
C 78 , 055801 (2008).[34] H. Pais, A. Santos and C. Providˆencia, Phys. Rev. C ,045808 (2009).[35] J.M. Aparicio, Astrophys. J. Suppl. 117 (1998) 627.[36] S.S. Avancini, S. Chiacchiera, D.P. Menezes and C. Pro-vidˆencia, arXiv:1010.3644v1 [nucl-th]. Figura 2. Diffusion coefficient D as function of baryon den-sity for different temperature and proton fraction values forhomogeneous matter and pasta phase. ( fm -3 ) D ( M e V m ) Rods Droplets Homogeneous matter
T=3 MeVYL=0.2NL3 D ( M e V m ) Slabs Rods Droplets Homogeneous matter
T=3 MeVYL=0.4NL3 ( fm -3 )0.01 0.02 0.03 0.04 0.052468101214 D ( M e V m ) ( fm -3 ) Rods Droplets Homogeneous matter
T=5 MeVYL=0.2NL3 ( fm -3 ) D ( M e V m ) T=5 MeVYL=0.4NL3
Rods Droplets Slabs Homogeneous matter
Figura 3. Diffusion coefficient D as function of baryon den-sity for different temperature and proton fraction values forhomogeneous matter and pasta phase. ( fm -3 ) D ( M e V m ) Rods Droplets Homogeneous matter
T=3 MeVYL=0.2NL3
Slabs Rods Droplets Homogeneous matter
T=3 MeVYL=0.4NL3 D ( M e V m ) ( fm -3 )0.01 0.02 0.03 0.04 0.0502468101214 D ( M e V m ) ( fm -3 ) Rods Droplets Homogeneous matter
T=5 MeVYL=0.2NL3 ( fm -3 ) D ( M e V m ) T=5 MeVYL=0.4NL3
Rods Droplets Slabs Homogeneous matter
Figura 4. Diffusion coefficient D as function of baryon den-sity for different temperature and proton fraction values forhomogeneous matter and pasta phase. ( fm -3 ) D ( M e V m ) Rods Droplets Homogeneous matter
T=3 MeVYL=0.2NL3
T=3 MeVYL=0.4NL3
Slabs Rods Droplets Homogeneous matter ( fm -3 ) D ( M e V m ) D ( M e V m ) ( fm -3 ) Rods Droplets Homogeneous matter