Role of magnetic fields on the outer crust in a magnetar

The origins of soft γ-ray repeaters and some anomalous x-ray pulsars are generally associated with a special type of neutron stars, the so-called magnetars, which possess very strong magnetic fields (MFs), of the order of 10$ ^{14} $$ - $10$ ^{15} $ G, on the surface [1−6]. The accurate description of the crust of a magnetar is essential to understand some astrophysical phenomena involving them, such as their oscillations [7−12] and cooling [13−15]. In the present work, we focus on the study of the outer crust in magnetars, where the fully ionized atoms are thought to be arranged in a Coulomb lattice embedded in a gas of degenerate electrons [16], and the high MF can have a non-negligible influence on the composition and equation of state due to the Landau-Rabi quantization of electron motion [17−21]. Moreover, an MF with $ B\gtrsim10^{17} $ G may modify appreciably the nuclear structure due to the alteration of the size and even the ordering of energy levels [22−25]. The calculations about the outer crust in neutron stars are based on the method proposed by Baym, Pethick, and Sutherland [26], where the masses of nuclei act as input parameters. Until now, very few studies have been conducted on the effects of the change of nuclear masses in the presence of an MF on the outer crust [27]. This is mainly due to the fact that the surface fields of magnetars inferred from the current observations are up to a few times 10$ ^{15} $ G [3, 6] and $ 3.2 \times 10^{16} $ G from the latest work [28], and fields with such order of magnitude play a small role in the nuclear masses. Nonetheless, considering that the upper limit of the interior MFs are of the order of 10$ ^{18} $ G according to the virial theorem [17] and numerical simulations [29−34], it is still worth investigating the nuclear masses under extremely high MFs and the corresponding effects on the crustal properties in magnetars prior to possible astronomical observations.

In the present work, to study the properties of nuclei in the presence of MFs, we use the publicly available Sky3D code based on the Skyrme density functional theory [35, 36] to solve the time-independent Hartree-Fock (HF) equations on a three-dimensional grid without further symmetry assumptions. This code has been used to study a wide range of problems, such as the nuclear structure, collective vibrational excitations, and heavy-ion collisions.

This article is organized as follows. In Sec. II, we describe the formulas for the present work. Sec. III provides the calculated results and some discussions. Finally, the summary is presented in Sec. IV.

The energy eigenvalue of an electron under MF presents the Landau-Rabi levels whose energies are

where c is the speed of light, $ m_e $ is the rest mass of the electron, $ p_z $ is the component of the momentum along the field, and $ \nu=n+1/2+s/2=0,1,2,... $, with n being the principal quantum number and s the spin along the MF axis with $ +1 $ for the spin-up and $ -1 $ for the spin-down cases. Note that $ \nu=0 $ corresponds to one single spin state, while all the $ \nu>0 $ states correspond to two spin states. Here, $ B_{\star}=B/B_c $ is the MF B measured in units of the critical field $ B_c $ defined as

at which the electronic cyclotron energy reaches the electron rest mass energy.

In the case of temperature $ T=0 $ K, the Fermi momenta of electrons $ k_{Fe} $ for different numbers of ν are related to the electronic chemical potential $ \mu_e $ in the magnetar's crust in equilibrium as $ \mu_e^2= c^2p_{Fe}(\nu)^2 + m_e^2c^4(1 + 2\nu B_{\star}) $. The requirement that $ p_{Fe}\geq 0 $ determines the maximum number of ν labeled as $\nu_{\rm max}$. Then, the expression of the electron density under MF can be given by

where $ \lambda_e=\hbar/{m_e}c $ is the Compton wavelength, the degeneracy $ g_{\nu} $ is 1 for $ \nu=0 $ and 2 for $ \nu\geq 1 $, and the dimensionless Fermi momentum is defined by $ x_e(\nu)=p_{Fe}(\nu)/m_ec $. The electronic energy density and pressure are respectively given by

where $ {\psi _ \pm }(x) = x\sqrt {1 + {x^2}} \pm \ln (x + \sqrt {1 + {x^2}}) $. In addition, the results in the absence of the MF ($ B = 0 $ G) corresponding to Eqs. (4) and (5) are given in this work by

where $ x=\lambda_e(3\pi^2n_e)^{1/3} $.

For the $ B=0 $ G case, the ions are believed to be arranged in a body-centered cubic lattice, and the lattice energy per baryon is approximately given by [27]

where the baryonic Fermi momentum $ p_{F_b} $ is in MeV. The associated lattice pressure is $ P_l= \varepsilon_l/3 $. The total pressure is therefore $ P=P_e+P_l $. Following the arguments in Ref. [27], this work still use Eq. (8) to estimate the contribution of the lattice energy in the presence of MFs.

The Gibbs free energy per nucleon is written as [27]

where $ M(A,Z,B) $ is the mass of the nuclei with proton number Z and atomic number A, and it is a function of B because we consider the effects of the MF on nuclear masses in this work. Then, the distribution of nuclei in the outer crust of neutron stars in equilibrium can be obtained by searching a series of nuclei for the minimum of these Gibbs free energies.

In this work, we follow the line of Ref. [25] where the Sky3D code is used to calculate the masses of nuclei under the MF. The effects of an external MF are included by introducing the B-field-related Hamilton operators $ \hat{H}_{p,n}^{(B)} $ into the original Hamiltonian in the Sky3D code, which are given by

where $ m_p $ and $ m_n $ are the proton and neutron masses, respectively, $ g_p=5.5856 $ and $ g_n=-3.8263 $ are the g-factors of a proton and neutron, and $ \vec L $ and $ \vec S $ are the orbital and spin angular momenta, respectively.

The following are some settings for the Sky3D code. We consider, in an isolated system, 24 grid points in each direction in three-dimensional Cartesian coordinates, with a distance of $ 1.0\ \text {fm} $ between each grid point. The force adopted in the type of Skyrme force is 'SLy6', which is mainly applied to neutron-rich nuclei and neutron matter in the field of astrophysics [37]. For the setting of the Newtonian gradient iteration method, the damping step is adjusted to $ x_0=0.4 $ and the damping adjustment parameter to $ E_0 = 100 $. The initial radii of the harmonic oscillator state are set to $ 3.1\ \text {fm} $. We begin the calculation in the absence of MF, which determines the input values to obtain the results at $ B=10^{16} $ G. Then, these results at $ B=10^{16} $ G are used for a higher MF and similar procedures are carried until $ B=10^{18} $ G. The maximum number of iterations are set to 1500 and the convergence criterion is that the energy fluctuation is less than $ 10^{-6}\ \text {MeV} $.

In Figs. 1−4, the calculations on 4110 nuclides from $ Z=5 $ to $ Z=82 $, ranging from neutron fraction 0.54 to 0.73, are made for the MF with $ B=0 $ G (left panels) and $ B=10^{18} $ G (right panels). Because the strong MF breaks the pairing in nuclei, these figures and the remainder of this work neglect the pairing energy in the $ B\neq 0 $ G cases for simplicity. Comparing Fig. 1 (a) and (b), we can observe that this super-strong MF increases the binding energies $BE$s of these nuclei by 8%$ \sim $15%. At $ B=10^{18} $ G, shown in Fig. 2, the root mean square radii R are found to be larger than those in the absence of MF.

Figure 1.  (color online) Binding energy per nucleon $BE/A$ (in MeV) for the nuclides with the neutron (N) and proton (Z) numbers at (a) $B = 0$ G (left panel) and (b) $B= 10^{18}$ G (right panel).

Figure 2.  (color online) Root mean square radii R (in fm) for the nuclides with the neutron (N) and proton (Z) numbers at (a) $B = 0$ G (left panel) and (b) $B= 10^{18}$ G (right panel).

Figure 3.  (color online) Total deformation β for the nuclides with the neutron (N) and proton (Z) numbers at (a) $B = 0$ G (left panel) and (b) $B= 10^{18}$ G (right panel).

The MF also plays an important role in the shape of the nucleus. In this work, we show the total deformation β and the triaxiality γ in Figs. 3−4, respectively. Here the parameters β and γ are known as Bohr-Mottelson parameters [38]. $ \beta = 0 $ refers to a spherical nucleus, which is non-deformed. When β is non-zero, the nucleus is deformed. The shape of a nucleus is prolate if $ \gamma=0 ^\circ $, oblate if $ \gamma=60 ^\circ $, and between the prolate and oblate deformations for $ 0 ^\circ<\gamma<60 ^\circ $. In Fig. 3 (a), the nuclides in the region near the magic numbers have β = 0, which means they are non-deformed. However, in Fig. 3 (b), this phenomenon does not appear and almost all the nuclides are deformed. The low β regions ($ \beta <0.05 $) mainly focus on the areas with N = 26$ - $36, 52$ - $68, and 96$ - $112. Note that some regions with large β in the absence of MF may become the small-β ones in the strong MF. For example, the blue area around $ Z=58-68 $ and $ N=92-102 $ in Fig. 3 (a), where β is greater than 0.2, cannot be observed in Fig. 3 (b). Thus, the strong MF may cause some nuclei with large deformation to become more 'spherical.' Compared with the results in Fig. 4 (a), the triaxialities vary considerably in Fig. 4 (b), and interestingly, we see $ \gamma=0^\circ $ around $ N=44, 74,114,172 $, between which $ \gamma=60^\circ $ appears. This means that in the super-strong MF, in general, the prolate and oblate deformations of a nucleus emerge alternately with increasing neutron numbers.

Figure 4.  (color online) Triaxiality γ for the nuclides with the neutron (N) and proton (Z) numbers at (a) $B = 0$ G (left panel) and (b) $B= 10^{18}$ G (right panel).

In Fig. 5, we calculate the binding energy BE, radius R, and deformation parameters β and γ of six nuclides that may exist in the outer crust of a cold nonaccreting neutron star as a function of MF strength. In this figure, it is obvious that $BE$s are not simply positively correlated with the strength of the MF, but rise jaggedly. This occurs because the spin-up and spin-down states split under the MF, and the energy levels of a nucleus may undergo the rearrangement for $ B \gtrsim 10^{17} $ G [25]. The analogue in atomic physics is the transition from the Zeeman to Paschen-Back effect when the MF becomes very strong. In this figure, we can observe that R, β, and γ are nearly unchanged for $ B\lesssim 4.1\times 10^{17} $ G, but $BE$s still increase. At $ B\gtrsim 4.1\times 10^{17} $ G, Rs increase with increasing MF, whereas βs rise initially and then have small changes above some certain values of MF strength. In addition, at $ B\gtrsim 4.1\times 10^{17} $ G, the phenomenon that γs become $ \gamma=0^\circ $ rapidly when the MF increases, then increase to approximately $ \gamma=30^\circ $, and subsequently return to $ \gamma=0^\circ $ at a higher MF implies that the nuclei may not become more prolate with higher MF. This might relate to the alternate emergence of the prolate and oblate deformation of a nucleus with increasing neutron numbers, as can be observed in Fig. 4 (b).

Figure 5.  (color online) Binding energy (BE), radius R, and deformation parameters β and γ of six nuclides that may exist in the outer crust of a cold nonaccreting neutron star as a function of magnetic field strength.

To understand the jaggedness of the $BE$ in Fig. 5 more thoroughly, we display in Fig. 6 the neutron and proton energy levels of $ {}^{208} $Pb for $ B=0 $ G, $ B=10^{17} $ G, and $ B=10^{18} $ G. The interaction between the MF and magnetic moment for $ B\neq 0 $ splits the energy levels. The neutrons have no orbital magnetic moment, and thus the gaps between the neutron magic numbers are still obvious at $ B=10^{17} $ G. Higher MFs ($ B=10^{18} $ G here) are needed to modify or even eliminate the neutron shell structure. Meanwhile, the effects of the MF on the protons are sufficient owing to the orbital magnetic moment so that some outer and inner energy levels in the absence of MF cross over at $ B=10^{17} $ G and more energy levels cross over at $ B=10^{18} $ G. This rearrangement of energy levels leads to the appearance of the jaggedness for the $BE$.

Figure 6.  (color online) Energy levels of (a) neutron (upper panel) and (b) proton (lower panel) (in MeV) of ${}^{208}$Pb at $B=0$ G, $B=10^{17}$ G and $B=10^{18}$ G.

In Fig. 7, we illustrate the equilibrium composition of the outer crust in the absence of MF by minimizing Eq. (9). The nuclear masses are calculated with the Skyrme force SLy6 [37] by the Sky3D code. For comparison, the results based on the Hartree-Fock-Bogoliubov (HFB) method, labeled as HFB-21, and the relativistic mean-field model, labeled as DD-ME2, are also shown in this figure [18, 27]. We can observe some trends in common for these three models. The equilibrium element in the outermost part of crust is $ ^{56} $Fe. With the increasing pressure, two constant plateaus of neutron number appear, which correspond to two magic numbers, i.e., $ N=50 $ and $ 82 $, respectively. Accompanied by these two plateaus is the obvious increase in proton number with a subsequent decrease. All the maximum pressures for these models are similar, $ \sim 4.7\times 10^{-4} $ MeV fm$ ^{-3} $. At this value, the neutrons begin to drip out of the nuclei, indicating the boundary between the outer and inner crusts of the neutron stars. Nonetheless, some model-dependent features can be observed for these models. For example, DD-ME2 displays a decrease in the neutron and proton numbers in the interval between $ P\sim 10^{-7}-10^{-6} $ MeV fm$ ^{-3} $ and the $ N=50 $ plateau begins to appear in HFB-21 at higher pressure.

Figure 7.  (color online) Composition of the outer crust of a cold nonaccreting neutron star in the absence of a magnetic field $(B=0\ \text G)$ where the nuclear masses are obtained with the Skyrme force SLy6 [37] by the Sky3D code, the Hartree-Fock-Bogoliubov method labeled as HFB-21 [18], and the relativistic mean-field model labeled as DD-ME2 [27].

Figure 8 displays the neutron-drip transition pressure $ P_{\text{drip}} $ as a function of the MF, where $ P_{\text{drip}} $ is obtained by requiring that the Gibbs energy in Eq. (9) be equal to the neutron mass. It is shown that the effects of MF on $ P_{\text{drip}} $ are almost negligible, except for $ B>10^{16} $ G. Furthermore, it can be observed that when $ B\gtrsim10^{17} $ G, $ P_{\text{drip}} $ is nearly a linear function of the MF. These results are in agreement with those in Refs. [18, 27]. In addition, the variation in the nuclear masses due to the MF ($ B_n=B $) plays a negligible role for $ B\lesssim10^{17} $ G because the MF with such orders of magnitude cannot affect the nuclei obviously. For the extremely high MF, $ B=10^{18} $ G, $ P_{\text{drip}} $ in the $ B_n=B $ case increases approximately 20% in comparison with that when the change in nuclear masses is not taken into account ($ B_n=0 $).

Figure 8.  (color online) Neutron-drip transition pressure $P_{\text{drip}}$ versus magnetic field strength B, where the effect of the magnetic fields on the nuclear masses is included ($B_{n}=B$) and not included $(B_{n}=0)$.

In Fig. 9, we illustrate the equilibrium composition of the outer crust for (a) $ B= 10^{16} $ G, (b) $ B= 10^{17} $ G, (c) $ B=5\times 10^{17} $ G, and (d) $ B= 10^{18} $ G. The variation in the nuclear masses due to the MF is considered ($ B_{n} = B $) and not considered ($ B_{n} = 0 $). The nuclear masses are affected slightly by the MF for $ B\leq 10^{17} $ G, but we can still find obvious discrepancy between the $ B_{n} = B $ and $ B_{n} = B $ cases in Fig. 9 (a) and Fig. 9 (b). However, this result may be model dependent [27]. It can be observed that the plateaus with $ N=50 $ and $ 82 $ are present in Fig. 9 (a) and (b). There is a clear shift for the $ N=50 $ plateau to higher pressures in comparison with that in the absence of MF, presented in Fig. 1, but the pressure at which the $ N=82 $ plateau begins to emerge does not change significantly for $ B=10^{16} $ G. This is in agreement with the results in Ref. [27]. We also present the composition in detail for $ B= 10^{16} $ G and $ B= 10^{17} $ G in Tables 1 and 2, where only the $ B_{n} = B $ case is listed. In both tables, from $ P_{\text{ion}} $ to the maximum pressure in the outer crust $ P_{\text{drip}} $, $ ^{56} $Fe occurs first; then, $ ^{89} $Y on the $ N=50 $ plateau and $ ^{132} $Sn, $ ^{124} $Mo, $ ^{123} $Nb, $ ^{122} $Zr, $ ^{120} $Sr on the $ N=82 $ plateau are present. We notice that the same nuclides emerge at higher pressures in the stronger MF. This is mainly due to the reduced electronic Fermi energy under a high MF. In addition, some nuclides may appear or disappear under different MF strengths. $ ^{88} $Sr and $ ^{84} $Se, which exist in the $ B=10^{16} $ G case, disappear on the $ N=50 $ plateau for $ B=10^{17} $ G, whereas $ ^{90} $Zr appears on this plateau. The nuclides not observed on the $ N=82 $ plateau for $ B=10^{16} $ G, i.e., $ ^{131} $In, $ ^{130} $Cd, $ ^{129} $Ag, $ ^{128} $Pd, and $ ^{126} $Ru, can exist in the $ B=10^{17} $ G case. Moreover, we find that the nuclide at the neutron-drip transition in Fig. 9 (a) and (b) is the same, i.e., $ ^{120} $Sr, which is also that observed in the absence of MF (not shown in this paper). This is consistent with the observation in Ref. [18], but it is $ ^{124} $Sr in that study.

Figure 9.  (color online) Composition of the outer crust of a cold nonaccreting magnetar for the four cases, (a) $B = {10^{16}}$ G, (b) $B = {10^{17}}$ G, (c) $B=5\times 10^{17}$ G, and (d) $B = {10^{18}}$ G, where the effect of the magnetic fields on the nuclear masses is included ($B_{n}=B$) and not included $(B_{n}=0)$.

Nucleus	$P_{\text {min}}$	$P_{\text {max}}$	$n_{\text {min}}$	$n_{\text {max}}$
${}_{26}^{56}$Fe	$P_{\text{ion}}$	$4.27\cdot 10^{-8}$	$n_{\text{ion}}$	$5.54\cdot 10^{-7}$
${}_{26}^{58}$Fe	$4.28\cdot 10^{-8}$	$4.98\cdot 10^{-7}$	$5.74\cdot 10^{-7}$	$1.53\cdot 10^{-6}$
${}_{39}^{89}$Y	$4.99\cdot 10^{-7}$	$2.50\cdot 10^{-6}$	$1.59\cdot 10^{-6}$	$3.34\cdot 10^{-6}$
${}_{38}^{88}$Sr	$2.51\cdot 10^{-6}$	$4.40\cdot 10^{-6}$	$3.39\cdot 10^{-6}$	$4.44\cdot 10^{-6}$
${}_{34}^{84}$Se	$4.41\cdot 10^{-6}$	$7.51\cdot 10^{-6}$	$4.62\cdot 10^{-6}$	$5.99\cdot 10^{-6}$
${}_{30}^{76}$Zn	$7.52\cdot 10^{-6}$	$1.13\cdot 10^{-5}$	$6.28\cdot 10^{-6}$	$7.63\cdot 10^{-6}$
${}_{50}^{132}$Sn	$1.14\cdot 10^{-5}$	$6.07\cdot 10^{-5}$	$8.03\cdot 10^{-6}$	$3.64\cdot 10^{-5}$
${}_{42}^{124}$Mo	$6.08\cdot 10^{-5}$	$1.79\cdot 10^{-4}$	$3.80\cdot 10^{-5}$	$1.07\cdot 10^{-4}$
${}_{41}^{123}$Nb	$1.80\cdot 10^{-4}$	$2.02\cdot 10^{-4}$	$1.09\cdot 10^{-4}$	$1.18\cdot 10^{-4}$
${}_{40}^{122}$Zr	$2.03\cdot 10^{-4}$	$2.64\cdot 10^{-4}$	$1.20\cdot 10^{-4}$	$1.49\cdot 10^{-4}$
${}_{39}^{121}$Y	$2.65\cdot 10^{-4}$	$3.77\cdot 10^{-4}$	$1.51\cdot 10^{-4}$	$1.91\cdot 10^{-4}$
${}_{38}^{120}$Sr	$3.79\cdot 10^{-4}$	$4.49\cdot 10^{-4}$	$1.95\cdot 10^{-4}$	$2.31\cdot 10^{-4}$

Table 1.  Composition of the outer crust of a magnetar at $B=10^{16}$ G. $P_{\text {min}}$ ($P_{\text{max}}$) is the minimum (maximum) pressure of the appearing nuclide, in units of $ \text{MeV fm}^{-3}$, at which $n_{\text {min}}$ ($n_{\text {max}}$), in units of $ \text{fm}^{-3}$, is the corresponding average minimum (maximum) baryon number density. $P_{\text {ion}}$ and $n_{\text {ion}}$ are the complete ionization pressure and density, respectively. Here, only the $B_{n}=B$ case, which includes the effect of the magnetic fields on the nuclear masses, is shown.

Nucleus	$P_{\text {min}}$	$P_{\text {max}}$	$n_{\text {min}}$	$n_{\text {max}}$
${}_{26}^{56}$Fe	$P_{\text{ion}}$	$8.07\cdot 10^{-7}$	$n_{\text{ion}}$	$7.54\cdot 10^{-6}$
${}_{40}^{90}$Zr	$8.08\cdot 10^{-7}$	$3.10\cdot 10^{-5}$	$8.26\cdot 10^{-6}$	$3.75\cdot 10^{-5}$
${}_{39}^{89}$Y	$3.11\cdot 10^{-5}$	$3.58\cdot 10^{-5}$	$3.81\cdot 10^{-5}$	$4.06\cdot 10^{-5}$
${}_{39}^{90}$Y	$3.59\cdot 10^{-5}$	$8.99\cdot 10^{-5}$	$4.12\cdot 10^{-5}$	$6.38\cdot 10^{-5}$
${}_{50}^{132}$Sn	$9.00\cdot 10^{-5}$	$4.05\cdot 10^{-4}$	$7.35\cdot 10^{-5}$	$1.52\cdot 10^{-4}$
${}_{49}^{131}$In	$4.06\cdot 10^{-4}$	$4.24\cdot 10^{-4}$	$1.54\cdot 10^{-4}$	$1.58\cdot 10^{-4}$
${}_{48}^{130}$Cd	$4.25\cdot 10^{-4}$	$5.01\cdot 10^{-4}$	$1.60\cdot 10^{-4}$	$1.74\cdot 10^{-4}$
${}_{47}^{129}$Ag	$5.02\cdot 10^{-4}$	$5.71\cdot 10^{-4}$	$1.76\cdot 10^{-4}$	$1.87\cdot 10^{-4}$
${}_{46}^{128}$Pd	$5.72\cdot 10^{-4}$	$6.38\cdot 10^{-4}$	$1.90\cdot 10^{-4}$	$2.00\cdot 10^{-4}$
${}_{44}^{126}$Ru	$6.39\cdot 10^{-4}$	$8.13\cdot 10^{-4}$	$2.06\cdot 10^{-4}$	$2.33\cdot 10^{-4}$
${}_{42}^{124}$Mo	$8.14\cdot 10^{-4}$	$8.86\cdot 10^{-4}$	$2.40\cdot 10^{-4}$	$2.50\cdot 10^{-4}$
${}_{41}^{123}$Nb	$8.87\cdot 10^{-4}$	$9.41\cdot 10^{-4}$	$2.54\cdot 10^{-4}$	$2.62\cdot 10^{-4}$
${}_{40}^{122}$Zr	$9.42\cdot 10^{-4}$	$9.79\cdot 10^{-4}$	$2.66\cdot 10^{-4}$	$2.71\cdot 10^{-4}$
${}_{38}^{120}$Sr	$9.80\cdot 10^{-4}$	$1.20\cdot 10^{-3}$	$2.81\cdot 10^{-4}$	$3.10\cdot 10^{-4}$

Table 2.  Same as Table 1, but at $B=10^{17}$ G.

For $ B= 5\times10^{17} $ G and $ B= 10^{18} $ G, the composition in the neutron star outer crust is significantly changed. First, we focus on the $ B_{n} = 0 $ case. Figure 9 (c) and (d) show that the outermost nuclide in the outer crust is not $ ^{56} $Fe, but $ ^{90} $Zr, which is just on the $ N=50 $ platform and spans in the broad range of pressures. The plateau with $ N=82 $ is still present and delays to higher pressures compared with that in the weak MF. It is surprising that the neutron plateau with a new magic number $ N=126 $ can be observed for $ B= 10^{18} $ G. This is in disagreement with the conclusion in Ref. [27], where the authors preset a medium-heavy nuclide as the upper limit of the composition in the outer crust. This is also pointed out by Ref. [39], where the HFB method was used and the effect of the MF on the nuclear masses was not included. Therefore, for the super-strong MF, such neutron-rich nuclide might be present in the outer crust. In the $ B_{n} = B $ case, it can be observed from Fig. 9 (c) and Table 3 that for $ B=5\times10^{17} $ G, the outermost nuclide is not on the $ N=50 $ platform, i.e., $ ^{92} $Nb, and the nuclide at $ P_{\text{drip}} $ is $ ^{182} $Ba, which is just on the $ N=126 $ plateau. For a stronger MF, i.e., $ B=10^{18} $ G ($B_{n} = B$), Fig. 4 (d) and Table 4 indicate that the platforms with $ N=82 $ and $ N=126 $ disappear. The outermost nuclide is $ ^{72} $Ge, and the nuclide at $ P_{\text{drip}} $ is $ ^{196} $Ce, which differs evidently from the $ N=126 $ plateau. This results from the fact that the binding energies increase jaggedly with increasing MF due to the rearrangement of the energy levels for $ B> 10^{17} $ G, as shown in Fig. 5. Thus, some concepts, e.g., magic numbers, should be reexamined when the effects of MF on nuclear masses are taken into account.

Nucleus	$P_{\text {min}}$	$P_{\text {max}}$	$n_{\text {min}}$	$n_{\text {max}}$
${}_{41}^{92}$Nb	$P_{\text{ion}}$	$3.53\cdot 10^{-4}$	$n_{\text{ion}}$	$2.85\cdot 10^{-4}$
${}_{36}^{88}$Kr	$3.54\cdot 10^{-4}$	$6.44\cdot 10^{-4}$	$3.09\cdot 10^{-4}$	$4.10\cdot 10^{-4}$
${}_{50}^{130}$Sn	$6.45\cdot 10^{-4}$	$8.27\cdot 10^{-4}$	$4.42\cdot 10^{-4}$	$4.97\cdot 10^{-4}$
${}_{48}^{130}$Cd	$8.28\cdot 10^{-4}$	$3.25\cdot 10^{-3}$	$5.17\cdot 10^{-4}$	$1.00\cdot 10^{-3}$
${}_{44}^{126}$Ru	$3.26\cdot 10^{-3}$	$6.16\cdot 10^{-3}$	$1.05\cdot 10^{-3}$	$1.44\cdot 10^{-3}$
${}_{56}^{182}$Ba	$6.17\cdot 10^{-3}$	$6.27\cdot 10^{-3}$	$1.65\cdot 10^{-3}$	$1.66\cdot 10^{-3}$

Table 3.  Same as Table 1, but at $B=5\times 10^{17}$ G.

Nucleus	$P_{\text {min}}$	$P_{\text {max}}$	$n_{\text {min}}$	$n_{\text {max}}$
${}_{32}^{72}$Ge	$P_{\text{ion}}$	$8.41\cdot 10^{-4}$	$n_{\text{ion}}$	$6.23\cdot 10^{-4}$
${}_{37}^{85}$Rb	$8.42\cdot 10^{-4}$	$1.18\cdot 10^{-3}$	$6.41\cdot 10^{-4}$	$7.48\cdot 10^{-4}$
${}_{37}^{87}$Rb	$1.19\cdot 10^{-3}$	$4.54\cdot 10^{-3}$	$7.69\cdot 10^{-4}$	$1.46\cdot 10^{-3}$
${}_{50}^{136}$Sn	$4.55\cdot 10^{-3}$	$5.49\cdot 10^{-3}$	$1.71\cdot 10^{-3}$	$1.87\cdot 10^{-3}$
${}_{50}^{140}$Sn	$5.50\cdot 10^{-3}$	$8.09\cdot 10^{-3}$	$1.93\cdot 10^{-3}$	$2.33\cdot 10^{-3}$
${}_{46}^{136}$Pd	$8.10\cdot 10^{-3}$	$1.36\cdot 10^{-2}$	$2.45\cdot 10^{-3}$	$3.14\cdot 10^{-3}$
${}_{58}^{196}$Ce	$1.37\cdot 10^{-2}$	$1.43\cdot 10^{-2}$	$3.63\cdot 10^{-3}$	$3.71\cdot 10^{-3}$

Table 4.  Same as Table 1, but at $B=10^{18}$ G.

In Fig. 10, we compare the behavior of the pressure P as a function of the baryonic density n for the three magnetic field cases. The result for $ B=0 $ G is also shown for comparison. It can be observed that the effects of the variation on the nuclear masses due to the MF are negligible. In the low density region, the phenomenon that the density is almost unchanged over a wide range of pressures is more obvious under a higher MF. This indicates that the outermost material of the magnetar is almost incompressible. This occurs because the electrons occupy only the lowest Landau level at low densities. The electrons may occupy a growing number of levels rapidly with increasing densities, and the effects of the MF will become unimportant. This may result in a slope of these lines at higher densities similar to that for $ B=0 $ G, which is approximately linear, as can be observed in this figure.

Figure 10.  (color online) Pressure P versus average nucleon number density n (equation of state) in the outer crust of a cold nonaccreting neutron star at four magnetic fields $B = 0\ \text G$, $B = {10^{16}}\ \text G$, $B = {10^{17}}\ \text G$, and $B = {10^{18}}\ \text G$, where the effect of the magnetic fields on the nuclear masses is included ($B_{n}=B$) and not included $(B_{n}=0)$.

In Fig. 11, we compare the neutron fraction $y_{n}$ in the outer crust for the cases $ B=0 $ G, $ 5\times10^{17} $ G, and $ 10^{18} $ G. It is shown that $ y_{n} $ is positively correlated with the pressure P in all cases. This is consistent with the physical intuition as the inverse $ \beta\text{-} $decay reaction becomes more intense with increasing depth of the magnetar, and the nuclei will become increasingly neutron-rich. For $ B \neq 0 $ G, it is shown that no essential difference can be observed between the $B_{n}= 0$ and $B_{n}= B$ cases, and $ y_{n} $ is smaller than that in the $ B=0 $ G case at the same pressure (these ultra-strong MFs, i.e., $ B>10^{17} $ G, can make the outermost nuclide become richer in neutrons than $ ^{56} $Fe, and thus may lead to a neutron-richer region in the outer crust). This indicates that, except in the outermost region for the ultra-strong MF case, the nuclei in the outer crust of a magnetar are more symmetric than those in an ordinary neutron star with weak MF at the same pressure.

Figure 11.  (color online) Neutron fraction $y_{n}$ ($y_{n}$=$N/A$) versus pressure P in the outer crust of a neutron star for the magnetic field strength of $B=0\ \text G$, $B = {5\times 10^{17}}\ \text G$, and $B = {10^{18}}\ \text G$, where the effect of the magnetic fields on the nuclear masses is included ($B_{n}=B$) and not included $(B_{n}=0)$.

In summary, we explore the properties of 4110 nuclides from $ Z=5 $ to $ Z=82 $ with the Sky3D code and the composition of the outer crust of a cold non-accreting magnetar. The effects of the variation in the nuclear masses due to the MF have been investigated. The MF play an important role in the binding energies, size, and shape of the nucleus. A strong MF increases the binding energies and nuclear radii. For the MF $ B \lesssim 10^{17} $ G, the nuclear shapes are nearly unchanged. At higher MF ($ B=10^{18} $ G in this work), almost all the nuclides are deformed, whereas some nuclei with large deformation can become more 'spherical.' Moreover, in general, the prolate and oblate deformations of a nucleus emerge alternately with increasing neutron numbers. For the MF $ B \lesssim 10^{17} $ G, there are some trends in common with the case in the absence of MF, i.e., $ ^{56} $Fe occurs first, the plateaus with $ N=50 $ and $ 82 $ are present, and the nuclide at the neutron-drip transition might not change, but these trends may be altered for $ B> 10^{17} $ G. If the effects of the MF on the nuclear masses are not taken into account ($ B_{n} = 0 $), the MF only influences the electrons, but the outermost nuclide in the outer crust is $ ^{90} $Zr, not $ ^{56} $Fe. The $ N=50 $ and $ N=82 $ platforms are still present. Even the neutron plateau with a new magic number $ N=126 $ can be observed for $ B= 10^{18} $ G. When the variation in the nuclear masses due to the MF is considered, the spin-up and spin-down states split. Thus, the energy levels of a nucleus may undergo a rearrangement, and the situation may become more complex. In the case of $ B_n=B $, the outermost nuclide is $ ^{92} $Nb for $ B= 5\times10^{17} $ G and $ ^{72} $Ge for $ B= 10^{18} $ G. Moreover, the $ N=82 $ and $ N=126 $ plateaus disappear for $ B= 10^{18} $ G. Thus, some concepts, e.g., magic numbers, should be reexamined in the $ B_n=B $ case. In addition, we also study the neutron-drip transition pressure, equation of state, and neutron fraction of the outer crust in the presence of the MF. The effects of the MF on $ P_{\text{drip}} $ are almost negligible, except for $ B>10^{16} $ G. At the extremely high MF, $ B=10^{18} $ G, $ P_{\text{drip}} $ in the $ B_{n}=B $ case increases approximately 20% in comparison with that for the case of $ B_{n}=0 $. The equation of state is affected by the MF mainly in the low density region, where the density is almost unchanged over a wide range of pressures, whereas the variation in the nuclear masses due to the MF plays a negligible role. The neutron fractions do not show a significant difference between the $ B_{n}=0 $ and $ B_{n}=B $ cases.

The present work shows that nuclei with a neutron number $ N\geq126 $ emerge in both the $ B_{n}=0 $ and $ B_{n}=B $ cases for $ B= 10^{18} $ G, whereas the $ N=126 $ nuclei emerge when $ B_{n}=B $ and not when $ B_{n}=0 $ for $ B= 5\times10^{17} $ G. Thus, considering the effects of the MF on the nuclear masses or not may modify the conclusion about the composition of the outer crust under a strong MF. This needs to be studied in more detail later.