Impact of strong magnetic fields on the inner crust of neutron stars
aa r X i v : . [ nu c l - t h ] J a n Impact of strong magnetic fields on the inner crust of neutron stars
S. S. Bao, ∗ J. N. Hu, † and H. Shen ‡ School of Physics and Information Engineering,Shanxi Normal University, Linfen 041004, China School of Physics, Nankai University, Tianjin 300071, China
We study the impact of strong magnetic fields on the pasta phases that are expected to exist in theinner crust of neutron stars. We employ the relativistic mean field model to describe the nucleoninteraction and use the self-consistent Thomas-Fermi approximation to calculate the nonuniformmatter in neutron star crust. The properties of pasta phases and crust-core transition are examined.It is found that as the magnetic field strength B is less than 10 G, the effects of magnetic fieldare not evident comparing with the results without magnetic field. As B is stronger than 10 G,the onset densities of pasta phases and crust-core transition density decrease significantly, and thedensity distributions of nucleons and electrons are also changed obviously.
I. INTRODUCTION
Neutron stars offer special natural laboratories for thestudy of nuclear physics and astrophysics due to theirextreme properties. Neutron stars consist of extremeneutron-rich matter and their densities can cover morethan 10 orders of magnitude from surface to center [1–3]. It is generally believed that a neutron star mainlyconsists of four parts, an outer crust of nuclei in a gas ofelectrons, an inner crust of neutron-rich nuclei with elec-tron and neutron gas, a liquid outer core of homogeneousnuclear matter, and an inner core of exotic matter withnon-nucleonic degrees of freedom [3–5]. From the neutrondrip to the crust-core transition, i.e., the density rangeof inner crust, the stable nuclear shape may change fromdroplet to rod, slab, tube, or bubble with increasing den-sity. As a result, the so-called nuclear pasta phases areexpected to appear in the inner crust of neutron stars [6–9], which play a significant role in interpreting a lot ofastrophysical observations, such as the giant flares andquasiperiodic oscillations from soft γ -ray repeaters, andglitches in the spin rate of pulsars [10–15]. The soft γ -ray repeaters and anomalous x-ray pulsars have alreadybeen confirmed as magnetars with very strong surfacemagnetic fields [16, 17], which can be as high as 10 -10 G [18, 19]. The magnetic field strength in the coreof a neutron star may even reach 10 G [20, 21]. So far,the mechanism and origin of strong magnetic fields inmagnetars remain unclear, and several hypotheses havebeen proposed (see Ref. [22] for a review and referencestherein). Duncan and Thompson [17] suggested that suchstrong fields could be generated by the dynamo mecha-nism in a rapidly rotating protoneutron star. It has alsobeen suggested that strong magnetic fields in neutronstars may result from magnetic flux conservation duringthe collapse of a massive progenitor [23]. It is still underdiscussion how strong the magnetic fields can be in the ∗ Electronic address: bao˙[email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] crust and interior of neutron stars.In past decades, great efforts have been devoted tostudy the effects of strong magnetic fields on the prop-erties of asymmetry nuclear matter and neutron starstructures, and the homogeneous stellar matter understrong magnetic fields has also been extensively stud-ied [18, 24–28]. The effects of Landau quantization canreduce the electron chemical potential and increase theproton fraction, which leads to the softening of equa-tion of state for neutron stars. The hyperonic matterappearing in the core of neutron stars under strong mag-netic fields were studied in Ref. [29], where it was foundthat the onset densities of hyperons could be observablychanged by strong magnetic fields. Furthermore, mag-netization and magnetic susceptibility properties of coldneutron star matter and even the warm stellar matterwere also examined within different methods [26, 30, 31].However, the studies on nonuniform crust matter understrong magnetic fields are rare due to the complex struc-tures of pasta phases. Recently, some researchers studiedthe density ranges and proton fractions in neutron starcrusts under strong magnetic fields by analyzing the dy-namical instability region of “ npe ” matter with variousmodels [32–35]. The neutron drip densities with strongmagnetic fields were calculated using Brussels-Montrealmicroscopic nuclear mass models in Ref. [36]. The mag-netic susceptibility and electron transport properties inthe neutron star crust with strong magnetic fields werereported in Refs. [20, 37, 38]. However, most studiesdo not take into account the nuclear pasta structures inthe inner crust of neutron stars. In Ref. [39], the nuclearpasta phases were studied using the relativistic mean field(RMF) models with NL3 [40] and TM1 [41] parametriza-tions under strong magnetic fields ≈ -10 G, wherethe proton fraction was fixed and the anomalous mag-netic moments of nucleons were neglected. The npe mat-ter satisfying β equilibrium condition was studied usingSkM nucleon-nucleon interaction in Ref. [42], where onlythe droplet phase was considered. Therefore, it is inter-esting and important to perform further investigationson the nonuniform matter in the inner crust of neutronstars under magnetic fields.In order to evaluate the influence of magnetic fields,the field strength at a given location in the star mustbe known. However, it is generally believed that themagnetic field configuration in a neutron star is verycomplex and difficult to determine [22, 43]. Only thesurface magnetic field can be obtained from related as-trophysical observations, whereas the internal magneticfield of the star cannot be directly accessible to observa-tions. Due to the complexity in dealing with Maxwell’sequations, a number of parameterized models have beenproposed to describe the magnetic field distribution inneutron stars [43–48]. In Ref. [43], the authors presenteda magnetic field profile from the surface to the interior ofthe star, where the magnetic field strength correspond-ing to the inner crust area could be as large as ≈ Gfor a central field strength of 5 × G. In the presentwork, we focus on the effects of strong magnetic fields inthe inner crust with a thickness of less than 1 km. Forsimplicity, we neglect the variation of the field strengthwithin this narrow range of radial distance and assume ahomogeneous magnetic field along the z direction.We employ the Wigner-Seitz (WS) approximationto describe the inner crust and use the self-consistentThomas-Fermi (TF) approximation to calculate thenonuniform matter with considering various pasta con-figurations. In the TF approximation, the surface en-ergy and the distributions of nucleons and electrons aretreated self-consistently. We adopt the RMF model to de-scribe nucleon-nucleon interaction. In the RMF model,nucleons interact with each other via the exchange ofscalar and vector mesons. We use two different RMFparametrizations, TM1 and IUFSU [49], which are suc-cessful in describing the ground-state properties of finitenuclei and compatible with maximum neutron-star mass ≈ M ⊙ . The TM1 model has been successfully used toconstruct the equation of state for neutrons stars and su-pernova simulations [50]. Compared with TM1 model, anadditional ω - ρ coupling term is added in IUFSU model,which plays an important role in modifying the densitydependence of symmetry energy and affects the neutronstar properties [49, 51]. The symmetry energy slope L inTM1 model is as large as 110 . L in IUFSUmodel is 40 . II. FORMALISM
We employ the TF approximation to study the in-ner crust of neutron stars with strong magnetic fields. The nucleon interaction is described by the RMF model,where the nucleons interact through the exchange of var-ious mesons, and the charged particles interact throughelectromagnetic field A µ . The isoscalar-scalar meson σ ,isoscalar-vector meson ω , and isovector-vector meson ρ are taken into account. For a system consisting of pro-tons, neutrons, and electrons, the Lagrangian density isgiven by L RMF = X i = p,n ¯ ψ i (cid:26) iγ µ ∂ µ − ( M + g σ σ ) − κ i σ µν F µν − γ µ h g ω ω µ + g ρ τ a ρ aµ + e τ ) A µ io ψ i + ¯ ψ e [ iγ µ ∂ µ − m e + eγ µ A µ ] ψ e + 12 ∂ µ σ∂ µ σ − m σ σ − g σ − g σ − W µν W µν + 12 m ω ω µ ω µ + 14 c ( ω µ ω µ ) − R aµν R aµν + 12 m ρ ρ aµ ρ aµ +Λ v (cid:0) g ω ω µ ω µ (cid:1) (cid:0) g ρ ρ aµ ρ aµ (cid:1) − F µν F µν , (1)where W µν , R aµν , and F µν are the antisymmetric fieldtensors corresponding to ω µ , ρ aµ , and A µ , respectively. κ i ( i = p, n ) denotes the anomalous magnetic moment ofnucleons. In the RMF approximation, the meson fieldsare treated as classical fields, and the field operators arereplaced by their expectation values. For a static system,the nonvanishing expectation values are σ = h σ i , ω = (cid:10) ω (cid:11) , ρ = (cid:10) ρ (cid:11) , and A = (cid:10) A (cid:11) . From the Lagrangiandensity (1), we can obtain the equations of motion formeson fields and electromagnetic field, −∇ σ + m σ σ + g σ + g σ = − g σ (cid:0) n sp + n sn (cid:1) , (2) −∇ ω + m ω ω + c ω + 2Λ v g ω g ρ ρ ω = g ω ( n p + n n ) , (3) −∇ ρ + m ρ ρ + 2Λ v g ω g ρ ω ρ = g ρ n p − n n ) , (4) −∇ A = e ( n p − n e ) , (5)where n si and n i represent the scalar and vector densitiesof nucleons, respectively.For a nonuniform nuclear system at zero temperature,the local energy density including Coulomb energy isgiven by ε rmf ( r ) = X i = p,n,e ε i + g ω ω ( n p + n n ) + g ρ ρ ( n p − n n )+ 12 ( ∇ σ ) + 12 m σ σ + 13 g σ + 14 g σ −
12 ( ∇ ω ) − m ω ω − c ω −
12 ( ∇ ρ ) − m ρ ρ − Λ v g ω g ρ ω ρ −
12 ( ∇ A ) + eA ( n p − n e ) . (6)In order to study the effects of strong magnetic fields onneutron star crust, we assume that the nuclear system isin an external homogeneous magnetic field B along the z direction, A µ = (0 , , Bx, n sp and proton vector density n p are given by n sp = eBM ∗ π X ν X s √ M ∗ + 2 νeB − sκ p B √ M ∗ + 2 νeB × ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k pF,ν,s + E pF √ M ∗ + 2 νeB − sκ p B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! , (7) n p = eB π X ν X s k pF,ν,s , (8)and the proton energy density ε p in Eq. (6) is written as ε p = eB π X ν X s (cid:20) k pF,ν,s E pF + (cid:16)p M ∗ + 2 νeB − sκ p B (cid:17) × ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k pF,ν,s + E pF p M ∗ + 2 νeB − sκ p B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (9)where k pF,ν,s is the Fermi momentum of proton with spin s and Landau level ν , and M ∗ = M + g σ σ is the effectivenucleon mass. The Fermi energy of proton is given by E pF = r k p F,ν,s + (cid:16)p M ∗ + 2 νeB − sκ p B (cid:17) . (10)We notice that ν = 0 , , , . . . , ν max , ν max = " ( E pF + sκ p B ) − M ∗ eB , (11)where [ x ] means the largest integer which is not largerthan x . The neutron scalar density n sn and neutron vectordensity n n are given by n sn = M ∗ π X s h k nF,s E nF − ( M ∗ − sκ n B ) × ln (cid:12)(cid:12)(cid:12)(cid:12) k nF,s + E nF M ∗ − sκ n B (cid:12)(cid:12)(cid:12)(cid:12)(cid:21) , (12) n n = 12 π X s (cid:26) k n F,s − sκ n B (cid:2) ( M ∗ − sκ n B ) k nF,s + E n F (cid:18) arcsin M ∗ − sκ n BE nF − π (cid:19)(cid:21)(cid:27) , (13)and the neutron energy density ε n in Eq. (6) is writtenas ε n = 14 π X s (cid:26) k nF,s E nF − sκ n BE nF (cid:18) arcsin M ∗ − sκ n BE nF − π (cid:19) − (cid:18) sκ n B M ∗ − sκ n B (cid:19) × (cid:2) ( M ∗ − sκ n B ) k nF,s E nF + ( M ∗ − sκ n B ) ln (cid:12)(cid:12)(cid:12)(cid:12) k nF,s + E nF M ∗ − sκ n B (cid:12)(cid:12)(cid:12)(cid:12)(cid:21)(cid:27) , (14) where k nF,s is the Fermi momentum of neutron with spin s . The Fermi energy of neutron is given by E nF = q k n F,s + ( M ∗ − sκ n B ) . (15)The electron density is given by n e = eB π X ν X s k eF,ν,s , (16)and the electron energy density ε e in Eq. (6) is writtenas ε e = eB π X ν X s (cid:2) k eF,ν,s E eF + (cid:0) m e + 2 νeB (cid:1) × ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) k eF,ν,s + E eF p m e + 2 νeB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (17)where k eF,ν,s is the Fermi momentum of electron with spin s and Landau level ν , and the Fermi energy of electronis given by E eF = q k e F,ν,s + m e + 2 νeB. (18)For simplicity, the anomalous magnetic moment of elec-tron is neglected in our calculation. So, the largest Lan-dau level ν max of electron is given by ν max = (cid:20) E eF − m e eB (cid:21) , (19)where the meaning of [ x ] is the same as the case of pro-tons. We should point out that the energy density fromthe contribution of electromagnetic field, B / π , is ne-glected in our calculation, which does not affect the phasetransitions of different pasta phases and crust-core tran-sitions.We employ the WS approximation to describe the in-ner crust structure of neutron star, assuming that onlyone nucleus is included in a WS cell, where the nucleuscoexists with neutron and surrounded by electron gases.The β equilibrium and charge neutrality conditions aresatisfied in a WS cell, µ n = µ p + µ e , (20) N e = N p , (21)where the chemical potentials of nucleons and electronare written as µ n = E nF + g ω ω − g ρ ρ, (22) µ p = E pF + g ω ω + g ρ ρ + eA, (23) µ e = E eF − eA, (24)and the numbers of electrons and protons inside the WScell are given by N e = Z cell n e ( r ) d r, (25) N p = Z cell n p ( r ) d r. (26) TABLE I: Parameter sets used in this work. The masses are given in MeV.Model
M m σ m ω m ρ g σ g ω g ρ g (fm − ) g c Λ v TM1 938.0 511.198 783.0 770.0 10.0289 12.6139 9.2644 − − E , K , E sym , and L are, respectively, the energy per nucleon, incompressibilitycoefficient, symmetry energy, and symmetry energy slope atsaturation density n .Model n (fm − ) E (MeV) K (MeV) E sym (MeV) L (MeV)TM1 0.145 − − B for TM1 and IUFSU models. The re-sults without the anomalous magnetic moments of nucleonsfor IUFSU model are also listed in the last two lines.Model B (G) Onset density (fm − )Rod Slab Tube Bubble Hom.TM1 0 — — — — 0.0618TM1 10 — — — — 0.0615TM1 10 — — — — 0.0610TM1 10 κ p , n = 0) 10 κ p , n = 0) 10 At a given average baryon density n b as well as radiusof WS cell r ws , we adopt the TF approximation to cal-culate the distributions of nucleons and electrons. Inpractice, we start with an initial guess for meson fields σ ( r ), ω ( r ), ρ ( r ), and electromagnetic field A ( r ), and thendetermine the chemical potentials, µ n , µ p , and µ e underthe constraints of Eqs. (20) and (21) and baryon numberconservation, n b V cell = Z cell [ n p ( r ) + n n ( r )] d r. (27)Once the chemical potentials are determined, it is easy tocalculate various densities and new mean fields by solving -8-4048120.00 0.02 0.04 0.06 0.08 0.10-8-4048 TM1IUFSU
B=0
B=10 G B=10 G B=10 G ( p, n =0) E / N ( M e V ) n b (fm -3 )drop. rod slab tub. bub. FIG. 1: (Color online) Binding energy per nucleon
E/N ofpasta phases as a function of baryon density n b for TM1 (up-per panel) and IUFSU (lower panel) models with differentmagnetic field strength, B = 0 (dashed line), B = 10 G(dotted line), and B = 10 G (solid line). The results with B = 10 G ignoring the anomalous magnetic moments of nu-cleons for IUFSU model are also plotted by dash-dotted linefor comparison. The onset densities of various nonsphericalpasta phases are indicated by the circle dots.
Eqs. (2)–(5). This procedure should be iterated untilconvergence is achieved. Furthermore, we calculate thetotal energy of WS cell E cell = Z cell ε rmf ( r ) d r, (28)and binding energy per nucleon E/N = E cell n b V cell − M. (29)We consider five nuclear pasta structures in this work.The volume of WS cell for different pasta shapes is given -4-3-2-100.00 0.02 0.04 0.06 0.08 0.10-4-3-2-1 TM1
B=0
B=10 G B=10 G B=10 G ( p, n =0) E ( M e V ) n b (fm -3 ) IUFSU FIG. 2: (Color online) Same as Fig. 1, but for binding energyper nucleon of pasta phases relative to that of homogeneousmatter ∆ E . by V cell = πr , for droplet and bubble ,lπr , for rod and tube , l r ws , for slab , (30)where l is the length for rod and tube and l is the widthfor slab. We notice that the value of l does not affectthe binding energy per nucleon E/N and is somewhatarbitrary.At a given average baryon density n b , we minimizethe binding energy per nucleon E/N with respect to thecell size r ws for all five pasta configurations and then wecompare E/N between different configurations in orderto determine the most stable shape that has the lowest
E/N . Besides, the binding energy per nucleon of ho-mogeneous matter at the same n b is also calculated andcompared to determine the crust-core transition where E/N of homogeneous matter becomes lower than that ofstable pasta phase. In the TF approximation, there is nodistinct boundary between the dense nuclear phase andthe dilute gas phase, so we prefer to adopt the definitionin Ref. [51], r in = r ws (cid:16) h n p i h n p i (cid:17) /D , for droplet, rod, and slab ,r ws (cid:16) − h n p i h n p i (cid:17) /D , for tube and bubble , (31) TM1IUFSU
B=0
B=10 G B=10 G ( p, n =0) n b (fm -3 ) Y p FIG. 3: (Color online) Proton fractions of pasta phases Y p asa function of baryon density n b for TM1 (upper panel) andIUFSU (lower panel) models with magnetic fields B = 10 G (solid line) and B = 0 (dashed line). The results with B = 10 G ignoring the anomalous magnetic moments ofnucleons are also plotted by dash-dotted line for comparison.Different colors correspond to various pasta structures. to measure the size of inner part in the WS cell, where theaverage values in brackets h· · · i are calculated over thecell volume V cell and the dimension of WS cell D = 1 , , III. RESULTS AND DISCUSSION
In this section, we show the numerical results obtainedby using self-consistent TF approximation and discussthe effects of strong magnetic fields on the properties ofneutron star crust. The results obtained with differentintensity of magnetic fields in TM1 model are comparedwith that in IUFSU model. The parameter sets and sat-uration properties of these two RMF models are givenin Tables I and II, respectively. In Fig. 1, we plot thebinding energy per nucleon
E/N of pasta phases as afunction of average baryon density n b for TM1 (upperpanel) and IUFSU (lower panel) models with and with-out strong magnetic fields. We can see that the bindingenergy E/N with B = 10 G is slightly smaller than theone with B = 0, while both of them are obviously largerthan that with B = 10 G. This behavior is consistentwith the results in Ref. [39]. It is because the existence of
TM1IUFSU (a) n ( M e V ) n b ( fm -3 ) B=0
B=10 G TM1IUFSU (b) p ( M e V ) n b ( fm -3 ) TM1IUFSU (c) e ( M e V ) n b ( fm -3 ) FIG. 4: (Color online) Chemical potentials of neutrons, µ n (a), protons, µ p (b), and electrons, µ e (c), as functions of baryondensity n b for TM1 (upper panel) and IUFSU (lower panel) models with magnetic fields B = 10 G (solid line) and B = 0(dashed line). large degeneracy of the Landau levels in strong magneticfields can soften the equation of state. We also noticethat only the droplet configuration exists as B ≤ Gbefore the crust-core transition in the case of the TM1model. However, all pasta phases arise whether the mag-netic fields are considered in the case of IUFSU model.In order to check the effects of anomalous magnetic mo-ments of nucleons on pasta phase, we also calculate thepasta structures for different strength of magnetic fieldwith κ p , n = 0 in the IUFSU model. It is found that theanomalous magnetic moments of nucleons have very lit-tle impact on pasta structure as B ≤ G, while itshould not be neglected as B ∼ = 10 G, which is alsoplotted in Fig. 1 for comparison. One can see that
E/N with κ p , n = 0 is obviously larger than the result with theinclusion of anomalous magnetic moments. Besides, theonset densities of pasta phases are also changed.The transition densities of various pasta phases andcrust-core transition density with different intensity ofmagnetic fields are listed in detail in Table III. It is foundthat the results with B = 10 G for IUFSU model do notchange much whether the anomalous magnetic momentsof nucleons are considered. However, for B = 10 G,considerable differences are observed in the onset densi-ties of nonspherical pasta phases and the transition den-sity to homogeneous matter. For both TM1 and IUFSUmodels, one can see that the results with B = 10 G arequite similar to those with B = 0, so the effects of mag-netic fields on pasta structures can be neglected when the strength of magnetic fields B is not larger than ∼ = 10 G. So, we will not discuss the results with B = 10 Gin the following contents. Comparing the results of TM1and IUFSU models, the pasta structures are significantlydifferent for various values of B . In the TM1 model, thenonspherical structures such as rod, tube, and bubble ap-pear only in the case of B = 10 G; however, the slabstructure is absent. In the IUFSU model, all five kinds ofpasta structures occur with and without strong magneticfields. The differences between these two models shouldbe due to their different symmetry energy and its den-sity dependence. It has been found that a smaller sym-metry energy slope could result in more complex pastaphases [51]. On the other hand, as B increases, the onsetdensity of homogeneous matter, namely the crust-coretransition density, decreases both in TM1 and IUFSUmodels. The transition densities between different pastaphases also decrease with increasing B as observed in theIUFSU model. We also notice that the transition den-sity at the bubble-homogeneous matter is nonmonotonicwith increasing B in Ref. [39] using NL3 parametrizationto perform the calculation, where the proton fraction isfixed as Y p = 0 .
3. This value is much larger than theresults of β equilibrium in this work.The behaviors in Table III can be understood fromFig. 2, where we plot the differences between the bindingenergy per nucleon of pasta phase and that of homo-geneous matter ∆ E as a function of baryon density n b with B = 0 , , G. We can see that a larger B TM1IUFSU r w s , r i n ( f m ) n b (fm -3 ) FIG. 5: (Color online) Radius of WS cell r ws (thick line)and nucleus r in (thin line) as a function of baryon density n b for TM1 (upper panel) and IUFSU (lower panel) modelswith magnetic fields B = 10 G (solid line) and B = 0(dashed line). The jumps in r ws and r in correspond to shapetransitions in pasta phases. results in a smaller ∆ E at lower baryon densities andthe results with B = 10 G are much lower than thosewith B = 0 , G. However, as n b increases, ∆ E with B = 10 G raises rapidly and then exceeds the resultswith B = 0 , G. As a result, ∆ E with larger B reaches “∆ E = 0” earlier, which leads to a smaller crust-core transition density.In Fig. 3, we plot the proton fraction of pasta phase Y p with B = 0 , G for TM1 and IUFSU models.The results with B = 10 , G will not be shown,considering no obvious differences from the results with B = 0. The results with B = 10 G neglecting theanomalous magnetic moments of nucleons for IUFSUmodel are also plotted. One can see that the protonfraction for κ p , n = 0 is slightly larger than that, includ-ing anomalous magnetic moments. It can be understoodfrom Eqs. (10) and (15). For κ p , n = 0, the proton Fermienergy E pF decreases while the neutron Fermi energy E nF increases, which leads to more proton energy levels oc-cupied. We can see in Fig. 3 that the proton fraction Y p with B = 10 G is much larger than the results with B = 0, especially at lower densities. It can be under-stood from Eq. (8). We notice that only the zeroth Lan-dau level is occupied, and eB is much larger than k p F,ν,s atlower densities, when the magnetic field B = 10 G is in- cluded. As a result, the proton fraction Y p with B = 10 G is larger than that with B = 0. As n b increases, k pF,ν,s increases rapidly, and higher Landau levels can be oc-cupied, so the difference of Y p with and without strongmagnetic fields becomes smaller at higher densities. Thisfeature plays an important role in affecting the chemicalpotentials. Compared with the IUFSU model, the TM1model has a larger symmetry energy slope L , which leadsto smaller proton fraction Y p with the same strength ofmagnetic fields. This behavior is consistent with thatobserved in the case without magnetic fields [51].In Fig. 4, we plot the chemical potentials of neutrons,protons, and electrons as a function of baryon density inpasta phases with B = 0 and B = 10 G. We can seethat the neutron chemical potential µ n with B = 10 Gis smaller than the results with B = 0 in all pasta phases,while the proton chemical potential µ p with B = 10 Gis larger than the one with B = 0 at lower densities,but µ p with B = 10 G is smaller than the results with B = 0 as baryon density n b increases. These behaviorscan be understood from the features of proton fraction.The proton fraction with B = 10 G is much largerthan the one with B = 0 at low densities (see Fig. 3),so the neutron fraction ( Y n ) with B = 10 G is muchlower than the one with B = 0 accordingly. As a result,proton (neutron) chemical potential with B = 10 G islarger (smaller) than the results with B = 0 obviouslyat low densities, which also results in the decrease of µ e according to the requirement of β equilibrium. Since thedifference of proton fraction with B = 10 G and B = 0becomes smaller at higher densities, the chemical poten-tials of neutrons and protons with B = 10 G are moreclose to the results with B = 0. The proton chemicalpotentials with B = 10 G are even lower than thosewith B = 0 for slab, tube, and bubble phases, while theneutron chemical potentials with B = 10 G and B = 0are close to each other.In Fig. 5, we show the radii of WS cell r ws and theradii of the inner part of WS cell r in with B = 10 G and B = 0 as a function of baryon density n b forboth TM1 and IUFSU models. It is seen that only fourkinds of pasta phases appear in strong magnetic fields B = 10 G for TM1 model, whereas all five pasta phasesarise for IUFSU model with or without strong magneticfields. One can see that for each solid pasta structure(droplet, rod, and slab), the radius of WS cell r ws de-creases with increasing baryon density n b , but the nucleusradius r in increases with n b . Such feature implies that asbaryon density n b increases, the size of nucleus becomeslarger and the distances between neighboring nuclei be-come shorter. In the hollow structure (tube and bubble),the size of inner gas phase r in decreases with n b . One cansee that as baryon density n b is close to the crust-coretransition, the radius r ws increases rapidly, however, wenotice that the binding energy per nucleon is not sensi-tive to the large r ws . The behaviors of r ws and r in withmagnetic fields B = 10 G are similar to the results with B = 0 as n b increases. r ws of solid structures (droplet, (a) Z d TM1IUFSU n b ( fm -3 ) B=0
B=10 G A d TM1 (b)IUFSU n b ( fm -3 ) FIG. 6: (Color online) Properties of spherical nuclei in droplet phase, such as the charge number Z d (a) and the nucleon number A d (b), as a function of baryon density n b for TM1 (upper panel) and IUFSU (lower panel) models with magnetic fields B = 0G (dashed line) and B = 10 G (solid line). rod, and slab) with B = 10 G is smaller than that with B = 0, while r in of solid structures with B = 10 Gis larger. For tube and bubble phases, r ws ( r in ) with B = 10 G is smaller (larger) than the results with B = 0. As a result, the nuclear radius becomes largerwhile the separation distance is smaller with B = 10 G compared to the results with B = 0, which leads tothe volume fraction of dense liquid phase in WS cell in-creasing more quickly with strong magnetic fields. Ac-cordingly, the crust-core transition happens at a smallerbaryon density n b . The behavior of r in can be under-stood from the liquid-droplet model. We know that thecompetition of Coulomb energy and surface energy playsan important role in determining the sizes of WS cell andnucleus inside it. In Ref. [39], the authors found that thesurface tension increased with the strength of magneticfields. A larger surface tension leads to a larger size andmore protons of the nucleus inside a WS cell. As a result, r in of droplet, rod, and slab with B = 10 G are largerthan results of B = 0. Furthermore, the radius of WScell r ws also depends on the volume fraction of the innerpart, so its behavior is more complex.We present in Fig. 6 the charge number Z d and nu-cleon number A d of the spherical nucleus as a functionof baryon density n b in the droplet configuration, wherethe background neutron gas is subtracted for defining A d within the subtraction procedure. Note that the resultsof nonspherical configurations are not presented due toarbitrariness in the definition of the nucleus. It is shownthat the charge number Z d with B = 10 G is largerthan the one with B = 0 at fixed baryon density n b . Thereason is that the strong magnetic fields lead to largerproton fraction of WS cell and larger surface tension,both resulting in more protons in the nucleus. As baryondensity n b increases, both charge number Z d and nucleonnumber A d increase first and then decrease in the TM1model with or without strong magnetic fields. However,the behaviors in the IUFSU model are different, whereboth Z d and A d increase with increasing baryon density n b .In order to study further the properties of sphericalnucleus of the droplet phase, we show the distributionsof proton ρ p , neutron ρ n , and baryon ρ b in the WS cell atdifferent average baryon densities n b in Figs. 7 and 8 forTM1 and IUFSU model, respectively. In Fig. 7 (a), onecan see that the proton density ρ p with B = 0 decreaseswith increasing average baryon density n b , which directlylead to the reduction of Z d with n b in Fig. 6 (a), consider-ing the radius of nucleus r in hardly changed at n b ≤ . − for TM1 model (see Fig. 5). The behavior of ρ p with B = 10 G is similar to the result with B = 0, butthe proton with B = 10 G has larger range of distri- proton(a)n b =0.01 fm -3 n b =0.02 fm -3 p ( f m - ) n b =0.03 fm -3 r (fm) neutron(b) TM1
B=0
B=10 G n ( f m - ) r (fm) baryon(c) b ( f m - ) r (fm) FIG. 7: (Color online) Density distributions of protons, ρ p (a), neutrons, ρ n (b), and baryons, ρ b (c), in the WS cell at n b = 0 . , . , .
03 fm − (top to bottom) obtained in the TF approximation for TM1 model with magnetic fields B = 0(dashed line) and B = 10 G (solid line). The cell boundary is indicated by the hatching. bution with increasing n b , which implies larger nucleusradius. As a result, the charge number of nucleus Z d with B = 10 G in Fig. 6 is nonmonotonic for the TM1model. At lower density, n b = 0 .
01 fm − , proton density ρ p with B = 10 G is larger than the one with B = 0at fixed radius r , while as n b increases, ρ p with B = 10 G in the center part of nucleus decreases rapidly and islower than the result with B = 0, but ρ p with B = 10 G in the outer part of nucleus is always larger than theresult with B = 0. In general, the nucleus with B = 10 G includes more protons compared to the results with B = 0. For the same reason, it is easy to understand thebehavior of nucleon number A d in Fig. 6 (b) from thebaryon density distribution in Fig. 7 (c). Besides, we cansee in Fig. 7 that as n b increases, the reduction of ρ b atthe center of WS cell comes mainly from the decrease of ρ p , while the increment of ρ b at the boundary of WS cellis due to the augment of ρ n in the gas phase.By comparing Fig. 8 with Fig. 7, we can see that theeffects of strong magnetic fields on nucleon distributionsare quite similar in IUFSU and TM1 models. The pres-ence of strong magnetic fields can enhance the chargenumber in the nucleus and reduce the neutron density ρ n and baryon density ρ b both at the center and bound-ary of WS cell. It is shown that the nucleon distributionsin WS cell for IUFSU model are different from the resultsfor TM1 model. From Fig. 8, we can see that ρ p in the center of nucleus decreases more quickly with increasing n b than the results for TM1 model, especially the resultswith strong magnetic fields B = 10 G, which decreasesabout 50% from n b = 0 .
01 to 0 .
03 fm − . The incrementof charge number in nucleus Z d with increasing n b is dueto the increase of nuclear radius r in and larger ρ p in theboundary area of nucleus.In order to investigate the effects of strong mag-netic fields on the distributions of nucleons and leptonsof various pasta phases, we show in Fig. 9 the den-sity distributions of neutrons, protons, and electrons inWS cell at five different average baryon densities n b =0 . , . , . , . , .
09 fm − for IUFSU model with B = 0 and 10 G, respectively. From Fig. 9 (a), wecan see that the neutron density ρ n at the center of theWS cell is larger than that at the boundary for droplet,rod, and slab phases, while it is opposite for the tubeand bubble phases. We also notice that the difference of ρ n between the center and the boundary of the WS celldecreases with increasing n b , which implies that nucleardistribution in the WS cell becomes more diffuse as closeto the crust-core transition. A similar tendency is alsoobserved in Fig. 9 (c), where the electron distributionsin the WS cell are plotted. We can see that the electrondensity ρ e is close to uniform distribution with increas-ing n b . With the strong magnetic field B = 10 G, theelectron density in the whole WS cell obviously increases0 (a)protonn b =0.01 fm -3 n b =0.02 fm -3 n b =0.03 fm -3 p ( f m - ) r (fm) (b)neutronIUFSU B=0
B=10 G n ( f m - ) r (fm) (c)baryon b ( f m - ) r (fm) FIG. 8: (Color online) Same as Fig. 7, but for IUFSU model. comparing to the results with B = 0. This is differentfrom the effects of strong magnetic fields on proton dis-tribution ρ p . From Fig. 9 (b), we can see that ρ p at theboundary of nucleus with B = 10 G is larger than theresults with B = 0. However, ρ p in the center of nucleuswith B = 10 G is lower than the one with B = 0.We can clearly see that the proton disappears in the gasphase due to its chemical potential smaller than its mass.Note that some kinks in ρ p with B = 10 G cor-respond to the changes of Landau level. For clarity, theneutron density in the center of nuclear liquid phase n n,L ,the one in the gas phase n n,G , and the proton densityin the center of nuclear liquid phase n p,L are plotted inFig. 10 as a function of baryon density n b with B = 0and B = 10 G for IUFSU model. One can see thatas n b increases, the neutron density of liquid phase n n, L does not change much; however, the neutron density ofgas phase n n, G increases with n b obviously. Comparingto the results with B = 0, both n n, L and n n, G with B = 10 G decrease. On the other hand, the protondensity of liquid phase n p,L decreases with increasing n b .At lower baryon densities, such as in the droplet phase, n p, L with B = 10 G is higher than that with B = 0,while at higher baryon densities, the behavior is opposite. IV. CONCLUSION
In this work, we have studied the influence of strongmagnetic fields on the properties of nuclear pasta phasesand crust-core transition in the inner crust of neutronstar by using the RMF model and the self-consistentTF approximation. The distributions of nucleons andelectrons in the WS cell are determined self-consistently,in which the charge neutrality and β equilibrium con-ditions are satisfied. It has been found that the pastaphase structures and the crust-core transition densitywere changed obviously when the magnetic field strengthis as large as B = 10 G, where the binding energy pernucleon
E/N is lower than the results with B = 0, andthe onset densities of various pasta phases and crust-coretransition density become smaller. However, the protonfraction Y p with B = 10 G is larger than that with B = 0, since the protons occupy the lowest Landau level.The impacts of anomalous magnetic moments of nucle-ons are almost invisible in the case of B = 10 G, butthey have to be taken into account for a stronger mag-netic field as B = 10 G. In general, the radius of WScell decreases with increasing B , while the size of nucleusincreases with B , which results in the charge number andnucleon number of the nucleus varying with B . The den-sity distributions of nucleons and electrons with B = 10 G are clearly different from the results with B = 0.In order to check the model dependence of the re-sults obtained, we adopt two successful RMF models,1 neutron(a) droplet n b =0.03 fm -3 rod n b =0.05 fm -3 n ( f m - ) slab n b =0.07 fm -3 tube n b =0.08 fm -3 r (fm) bubble n b =0.09 fm -3 proton(b) B=0
B=10 G p ( f m - ) r (fm) electron(c) e ( - f m - ) r (fm) FIG. 9: (Color online) Density distributions of neutrons, ρ n (a), protons, ρ p (b), and electrons, ρ e (c), in the WS cell at n b = 0 . , . , . , . , .
09 fm − (top to bottom) obtained in the TF approximation for IUFSU model with magneticfields B = 0 (dashed line) and B = 10 G (solid line). The cell boundary is indicated by the hatching. i.e., TM1 and IUFSU, with different symmetry energiesand their slopes, which play an important role in deter-mining the properties of inner crust of neutron star withstrong magnetic fields. The features with strong mag-netic fields due to the symmetry energy and its densityslope are similar to the results with B = 0, which are con-sistent with our earlier study [51]. A smaller slope L leadsto more complex pasta structures. For the TM1 modelwith a larger slope L , only droplet appears in the innercrust of neutron star for B = 0. However, some non-spherical pasta phases arise before crust-core transitionfor B = 10 G, even though the crust-core transitiondensity becomes smaller. It would be interesting to fur-ther study the nuclear pasta phase with strong magneticfields and their impacts on the observations of neutronstar.
Acknowledgment
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