In this paper, we study the role of the symmetry energy on the composition of the outer crust of a neutron star. Although some correlations can be observed at the neutron-drip transition, the composition of the outer crust is mainly sensitive to the details of the nuclear structure far from the valley of stability rather than to the symmetry energy only

Chamel, N.

Fantina, A. F.

Goriely, S.

Pearson, J. M.

International audienceIn this paper, we study the role of the symmetry energy on the composition of the outer crust of a neutron star. Although some correlations can be observed at the neutron-drip transition, the composition of the outer crust is mainly sensitive to the details of the nuclear structure far from the valley of stability rather than to the symmetry energy only

F. Fantina, A.

M. Pearson, J.

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Symmetry energy and composition of the outer crust of neutron stars

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DOI 10.1393/ncc/i2016-16400-1Colloquia: IWM-EC 2016IL NUOVO CIMENTO 39 C (2016) 400Symmetry energy and composition of the outer crustof neutron starsA. F. Fantina(1)(2), N. Chamel(2), J. M. Pearson(3) and S. Goriely(2)(1) Grand Accélérateur National d’Ions Lourds (GANIL), CEA/DRF, CNRS/IN2P3Bvd Henri Becquerel, 14076 Caen, France(2) Institut d’Astronomie et d’Astrophysique, CP-226, Université Libre de Bruxelles (ULB)1050 Brussels, Belgium(3) Dépt. de Physique, Université de Montréal - Montréal (Québec), H3C 3J7, Canadareceived 10 January 2017Summary. — In this paper, we study the role of the symmetry energy on thecomposition of the outer crust of a neutron star. Although some correlations can beobserved at the neutron-drip transition, the composition of the outer crust is mainlysensitive to the details of the nuclear structure far from the valley of stability ratherthan to the symmetry energy only.1. – IntroductionNeutron stars (NSs) are among the most compact objects in the Universe, with centraldensities that can reach up to several times the nuclear matter density, n0 ≈ 0.16 fm−3.They result from the gravitational core collapse of stars whose mass is greater than about8–10 M (M being the mass of our Sun), at the end point of their evolution (see, e.g.,ref. [1]). Apart from a thin atmospheric plasma layer of light elements (mainly hydrogenand helium) possibly surrounding a Coulomb liquid of electrons and ions, the interiorof a NS can be qualitatively divided into three regions: i) the outer crust, at densitiesabove ρ ≈ 104 g cm−3, made of a crystal lattice of fully ionised atoms arranged in a body-centered cubic (bcc) lattice neutralised by a uniform degenerate electron gas (see, e.g.,ref. [2]); ii) the inner crust, above the neutron-drip density ρdrip ≈ 4× 1011 g cm−3 (see,e.g., refs. [1-3]), composed of neutron-proton clusters immersed in a neutron liquid andneutralised by the degenerate electron gas; iii) a core, at densities above ρ ≈ 1014 g cm−3(about half saturation density), composed by a homogenous liquid mixture of neutrons,protons, electrons, and, at higher densities, muons. In the innermost part, the compo-sition is still a matter of debate, and additional particles like hyperons or deconfinedquarks may exist (see, e.g., ref. [1]).The NS crust is also a unique nuclear physics “laboratory” to probe the propertiesof asymmetric nuclear matter, like the symmetry energy, at subsaturation density (see,Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0) 12 A. F. FANTINA et al.e.g., ref. [4]). The symmetry energy in infinite homogeneous nuclear matter is usuallydefined as(1) S1(n) =12∂2(E/n)∂η2∣∣∣∣η=0,where E(n, η) is the energy density of homogeneous nuclear matter with proton (neutron)density np (nn), baryon density n = nn + np, and charge asymmetry η = (nn − np)/n.The symmetry energy can be also defined as the difference between the energy of pureneutron matter and that of symmetric matter. However, the two definitions do notexactly coincide, because of higher order terms in E(n, η) (see, e.g., ref. [5]). Here, wewill adopt the first definition, eq. (1), that can be expanded around n0 as(2) S1(n) ≈ J +13L(n − n0n0)+118Ksym(n − n0n0)2.While the value of the symmetry energy at saturation, J ≈ 30MeV, is rather wellconstrained by nuclear physics experiments, the value of its slope, L, or that of higher-order coefficients like Ksym are still poorly known (see, e.g., refs. [6, 7]). Nevertheless,various studies have shown that the symmetry energy plays a role on the properties ofthe NS crust, like its composition, the crust-core transition, or the transition betweenthe outer and inner crust (see, e.g., ref. [8] and references therein).In this paper, we study the role of the symmetry energy on the composition of theouter crust, for non-accreting and unmagnetised NSs, using a recent set of Brussels-Montreal microscopic nuclear mass models [9].2. – Model of neutron-star crustIn the region of the crust considered here, the pressure is high enough that atomsare fully ionised [2]. Moreover, we assume that the temperature T is lower than thecrystallisation temperature, so that nuclei are arranged in a regular bcc crystal lattice,that we consider made of only one type of ion AZX, with mass number A and atomicnumber Z. The crystallisation temperature is usually much lower than the electronFermi temperature. Therefore, electrons are highly degenerate, and can be treated asa uniform ideal relativistic Fermi gas. We also set T = 0. Expressions for the electronenergy density and pressure, Ee and Pe, can be found in ref. [1]. The main correctionsarise from electron-ion interactions (see, e.g., ref. [10] for a discussion of other corrections;see also ref. [11] for a recent discussion on electron exchange and polarisation correctionsto Ee and Pe that have not been included in the present work). Neglecting the finitesize of ions and the quantum zero-point motion of ions off their equilibrium position,the lattice contribution to the energy density is given by EL = Ce2n4/3e Z2/3, where C isa crystal structure constant (C = −1.444231 for a bcc lattice [12]), e is the elementarycharge, and ne is the electron number density (see, e.g., ref. [1]). The total pressure readsP = Pe + PL, where the lattice pressure is PL = EL/3. The only microscopic inputs forthe description of the outer crust are nuclear masses, which can be calculated from thecorresponding atomic masses after subtracting out the binding energy of the atomicelectrons (see eq. (A4) of ref. [13]). For the masses that have not yet been measured, wehave used the microscopic mass tables computed by the Brussels-Montreal group (see,e.g., ref. [14] for a recent review of these models).SYMMETRY ENERGY AND COMPOSITION OF THE OUTER CRUST OF NEUTRON STARS 3The family of Brussels-Montreal nuclear mass models used here [9] are based on thenuclear energy density functional (EDF) theory using generalised Skyrme zero-range ef-fective interactions [15], supplemented with a microscopic contact pairing interaction [16].For these models, the masses of nuclei were obtained by adding to the Hartree-Fock-Bogoliubov (HFB) energy a phenomenological Wigner term and a correction term forthe rotational and vibrational spurious collective energy [5]. The EDFs BSk22, BSk23,BSk24, BSk25, and BSk26 underlying the nuclear mass models HFB-22, HFB-23, HFB-24, HFB-25, and HFB-26, respectively, were fitted to the 2353 measured masses of nucleiwith N , Z ≥ 8 from the 2012 Atomic Mass Evaluation [17], with a root-mean-square(rms) deviation of about 0.5–0.6 MeV. Moreover, the incompressibility Kv of infinite ho-mogeneous symmetric nuclear matter at saturation was required to fall in the range240 ± 10MeV [18], and the isoscalar effective mass was fixed to the realistic valueM∗s = 0.8M . These EDFs were also constrained to reproduce the equation of state(EoS) of homogeneous neutron matter, as obtained by many-body calculations usingrealistic interactions. Moreover, the EoSs of symmetric nuclear matter obtained fromthese EDFs are also compatible with the constraints inferred from the analysis of heavy-ion collision experiments [19, 20]. In constructing these five EDFs, BSk22 to BSk26,different values of the symmetry energy coefficient J were imposed thus making themsuitable for a systematic study of the role of the symmetry energy on the propertiesof the NS crust. In particular, BSk22, BSk23, BSk24, and BSk25 have J = 32, 31,30 and 29 MeV, respectively, and L = 68.5, 57.8, 46.4, and 36.9 MeV, respectively, andthey were fitted to the realistic neutron-matter EoS labelled “V18” in ref. [21]. BSk26was fitted to the EoS labelled “A18 + δ v + UIX∗” in ref. [22] under the constraintJ = 30MeV, leading to a different value of L = 37.5MeV at saturation with respect toBSk24. The values of J and L at saturation for all these EDFs are also consistent withthose obtained from different theoretical and experimental constraints [6, 7]. Additionalinformation on the value of J can be obtained from the recent measurement of ref. [23].Indeed, Tarbert et al. inferred the diffuseness an and the half-height radius Cn of theneutron two-parameter Fermi distribution in 208Pb, and thus the 208Pb neutron-skinthickness Δrnp = 0.15± 0.03(stat) fm, from coherent pion photoproduction cross sectionmeasurements. In fig. 1, we show this “experimental” constraint, together with the val-ues of an and Cn predicted by the Brussels-Montreal EDFs BSk22 to BSk26 (dots). Theoutcome of these measurements tends to disfavour higher values of J . Incidentally, com-paring various constraints from both nuclear physics and astrophysics [24, 25], BSk24(BSk22) was found to be the best (worst) in the series of EDFs BSk22-BSk26. We willthus take BSk24 as “reference” model, while we will not consider the BSk26 EDF inthe following analysis since it has been constrained to a different neutron-matter EoS.Furthermore, the values of the neutron-skin thickness in 208Pb predicted by the EDFsBSk22 to BSk25 vary from 0.18 fm (BSk22) to 0.12 fm (BSk25), in agreement with thevalue extracted in ref. [23]. For all these reasons, we believe that these EDFs can bereliably applied to describe the NS crust.3. – Numerical resultsWe have computed the properties of the outer crust of non-accreting unmagnetisedNSs minimising the Gibbs free energy per nucleon at fixed pressure, as described inref. [10]. We have increased the pressure with a step ΔP = 0.003 P . We have made use ofthe experimental masses from the 2012 Atomic Mass Evaluation [17] whenever available,complemented with the nuclear mass tables HFB-22 to HFB-25 from the BRUSLIB4 A. F. FANTINA et al.Fig. 1. – Diffuseness an versus half-height radius Cn for208Pb extracted from ref. [23]. Dotscorrespond to the values of an and Cn obtained with the Brussels-Montreal energy-densityfunctionals BSk22 to BSk26 underlying the nuclear mass models HFB-22 to HFB-26.database [26]. The composition of the outer crust predicted by the Brussels-Montrealmodels HFB-22 to HFB-25 is shown in table I. In bold are listed elements whose atomicmass is experimentally known [17]. Comparing the sequence of nuclei in table I, it canTable I. – Composition of the neutron-star outer crust as predicted by the Brussels-Montrealnuclear mass models HFB-22 to HFB-25 [9]. In bold are elements whose atomic mass is exper-imentally known [17].HFB-22 HFB-23 HFB-24 HFB-2556Fe 56Fe 56Fe 56Fe62Ni 62Ni 62Ni 62Ni58Fe 58Fe 58Fe 58Fe64Ni 64Ni 64Ni 64Ni66Ni 66Ni 66Ni 66Ni86Kr 86Kr 86Kr 86Kr84Se 84Se 84Se 84Se82Ge 82Ge 82Ge 82Ge80Zn 80Zn 80Zn 80Zn79Cu – – –78Ni 78Ni 78Ni 78Ni80Ni 80Ni 80Ni –124Mo 124Mo 124Mo 124Mo122Zr 122Zr 122Zr 122Zr121Y – 121Y 121Y120Sr 120Sr 120Sr 120Sr122Sr 122Sr 122Sr 122Sr124Sr – 124Sr –126Sr 126Sr – –122Kr – – –SYMMETRY ENERGY AND COMPOSITION OF THE OUTER CRUST OF NEUTRON STARS 5be noted that it is similar for the different models, apart from some missing nuclides.The discrepancies in the predicted nuclei arise from the uncertainties in the masses ofneutron-rich nuclei. For example, HFB-22 predicts the existence of odd nucleus 79Cu,contrarily to the other models. On the other hand, HFB-23 (HFB-25) does not supportthe presence of 121Y (80Ni), unlike the other models. The masses of 79Cu calculated withHFB-22 and HFB-24 differ by only 700 keV, while for 80Ni the difference in the theoreticalmass calculated with HFB-25 and HFB-24 amounts to 750 keV. Finally, for 121Y, thedifference in the predicted mass between HFB-23 and HFB-24 amounts to 1.8 MeV. Inref. [9], it has been discussed how the masses of drip-line nuclei are correlated with J .Since mass is the only nuclear quantity on which the composition of the crust depends,it can be surprising that no correlation between J and composition is apparent. Thiscorrelation may be masked by the noise arising from the different errors with which thedifferent EDFs fit the data and the numerical errors with which masses are calculatedwith a given EDF. Nevertheless, it is possible to establish some correlations betweenthe properties of the crust at the neutron-drip transition and J (or L), as discussedin ref. [27]. Indeed, the Z/A ratio of the neutron-drip nucleus decreases with J , from0.311 for HFB-25 to 0.295 for HFB-22. On the contrary, the neutron-drip density andpressure increase almost linearly with J , ranging from ndrip = 2.51×10−4 fm−3 for HFB-25 to 2.71× 10−4 fm−3 for HFB-22 (the corresponding neutron-drip pressure varies from4.83 × 10−4 MeV fm−3 for HFB-25 to 4.99 × 10−4 MeV fm−3 for HFB-22) (see ref. [27]for details).4. – ConclusionsWe have studied the role of the symmetry energy on the composition of the outer crustof a NS, using experimental atomic masses, complemented with accurately calibratedBrussels-Montreal nuclear mass models, from HFB-22 to HFB-25. The composition ofthe outer crust of a NS is very sensitive to the details of the nuclear structure far fromthe valley of stability. Around the neutron-drip transition, the composition is mainlydetermined by the masses of neutron-rich Sr and Kr isotopes. The small deviations inthe compositions predicted by the different EDFs do not appear to be correlated with J .Nevertheless, correlations between the slope of the symmetry energy (or equivalently J)and the properties of the NS crust at the neutron drip can be observed.∗ ∗ ∗This work has been supported by Fonds de la Recherche Scientifique - FNRS(Belgium), NSERC (Canada), and the COST Action MP1304 “NewCompStar”.REFERENCES[1] Haensel P., Potekhin A. Y. and Yakovlev D. G., Neutron Stars 1: Equation of Stateand Structure (Springer, Berlin) 2007.[2] Chamel N. and Haensel P., Living Rev. Relativ., 11 (2008) 10.[3] Chamel N., Fantina A. F., Zdunik J. L. and Haensel P., Phys. Rev. C, 91 (2015)055803.[4] Steiner A. W., Prakash M., Lattimer J. and Ellis P. J., Phys. Rep., 411 (2005)325.[5] Goriely S., Chamel N. and Pearson J. M., Phys. Rev. C, 82 (2010) 035804.[6] Tsang M. B., Stone J. R., Camera F. et al., Phys. Rev. C, 86 (2012) 015803.6 A. F. FANTINA et al.[7] Lattimer J. M. and Lim Y., Astrophys. J., 771 (2013) 51.[8] Baldo M. and Burgio G. F., Prog. Part. Nucl. Phys., 91 (2016) 203.[9] Goriely S., Chamel N. and Pearson J. M., Phys. Rev. C, 88 (2013) 024308.[10] Pearson J. M., Goriely S. and Chamel N., Phys. Rev. C, 83 (2011) 065810.[11] Chamel N. and Fantina A. F., Phys. Rev. D, 93 (2016) 063001.[12] Baiko D. A., Potekhin A. Y. and Yakovlev D. G., Phys. Rev. E, 64 (2001) 057402.[13] Lunney D., Pearson J. M. and Thibault C., Rev. Mod. Phys., 75 (2003) 1021.[14] Chamel N., Pearson J. M., Fantina A. F. et al., Acta Phys. Pol. B, 46 (2015) 349.[15] Chamel N., Goriely S. and Pearson J. M., Phys. Rev. C, 80 (2009) 065804.[16] Chamel N., Phys. Rev. C, 82 (2010) 061307(R).[17] Audi G., Wang M., Wapstra A. H. et al., Chin. Phys. C, 36 (2012) 1287.[18] Colò G., Giai N. V., Meyer J. et al., Phys. Rev. C, 70 (2004) 024307.[19] Danielewicz P., Lacey R. and Lynch W. G., Science, 298 (2002) 1592.[20] Lynch W. G. et al., Prog. Part. Nucl. Phys., 62 (2009) 427.[21] Li Z. H. and Schulze H.-J., Phys. Rev. C, 78 (2008) 028801.[22] Akmal A., Pandharipande V. R. and Ravenhall D. G., Phys. Rev. C, 58 (1998) 1804.[23] Tarbert C. M., Watts D. P., Glazier D. I. et al., Phys. Rev. Lett., 112 (2014) 242502.[24] Pearson J. M., Chamel N., Fantina A. F. and Goriely S., Eur. Phys. J. A, 50 (2014)43.[25] Fantina A. F., Chamel N., Pearson J. M. and Goriely S., AIP Conf. Proc., 1645(2015) 92.[26] http://www.astro.ulb.ac.be/bruslib.[27] Fantina A. F., Chamel N., Mutafchieva Y. D. et al., Phys. Rev. C, 93 (2016) 015801.

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Symmetry energy and composition of the outer crust of neutron stars

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