# Inner crusts of neutron stars in strongly quantising magnetic fields

## Abstract

We study the properties and stability of nuclei in the inner crust of neutron stars in the presence of strong magnetic fields G. Nuclei coexist with a neutron gas and reside in a uniform gas of electrons in the inner crust. This problem is investigated within the Thomas-Fermi model. We extract the properties of nuclei based on the subtraction procedure of Bonche, Levit and Vautherin. The phase space modification of electrons due to Landau quantisation in the presence of strong magnetic fields leads to the enhancement of electron as well as proton fractions at lower densities fm. We find the equilibrium nucleus at each average baryon density by minimising the free energy and show that ,in the presence of strong magnetic fields, it is lower than that in the field free case. The size of the spherical cell that encloses a nucleus along with the neutron and electron gases becomes smaller in strong magnetic fields compared with the zero field case. Nuclei with larger mass and atomic numbers are obtained in the presence of strong magnetic fields as compared with cases of zero field.

## 1 Introduction

Strong surface magnetic fields G are found to exist in pulsars. Even stronger surface magnetic fields G were predicted by observations on soft gamma ray repeaters and anomalous x-ray pulsars (Kouveluotou et al., 1998, 1999). The latter class of neutron stars with very intense magnetic fields is known as magnetars (Thompson & Duncan, 1993, 1996). On the other hand, the interior magnetic field could be much higher than the surface field. The limiting interior field might be estimated from the scalar virial theorem (Lai & Shapiro, 1991). For a typical neutron star mass 1.5 and radius 15 km, the interior field could be as high as G.

Such strong magnetic fields quantise the motion of charged particles perpendicular to the field (Landau & Liftshitz, 1977). The effects of the phase space modification due to the Landau quantisation were studied on the composition and equation of state (EoS) in neutron stars extensively. Lai and Shapiro extended the Baym, Pethick and Sutherland (BPS) model (Baym et al., 1971b) to the magnetic field case and obtained equilibrium nuclei and the EoS in the outer crust in the presence of strong magnetic fields (Lai & Shapiro, 1991). The composition and EoS in the core of neutron stars in the presence of strongly quantising magnetic fields were investigated within a relativistic field theoretical model by Chakrabarty et al. (Chakrabarty et al., 1997; Bandyopadhyay et al., 1997). The transport properties such as thermal and electrical conductivities of neutron star crusts in magnetic fields were studied by several groups (Yakovlev & Kaminker, 1994; Hernquist, 1984). Recently the magnetised neutron star crust was studied using the Thomas-Fermi model and Baym-Bethe-Pethick (Baym et al., 1971a) and Harrison-Wheeler EoS for nuclear matter (Nag & Chakrabarty, 2010).

In the outer crust of a neutron star, neutrons and protons are bound inside nuclei and immersed in a uniform background of relativistic electron gas. As the density increases, nuclei become more and more neutron rich. Neutrons start to drip out of nuclei at a density g/cm. This is the beginning of the inner crust. The matter in the inner crust is made of nuclei embedded in a neutron gas along with the uniform electron gas. Further the matter is in -equilibrium and maintains charge neutrality. Nuclei are also in mechanical equilibrium with the neutron gas. The properties of nuclei in the inner crusts of neutron stars in zero magnetic field were studied by different groups. The early studies of the inner crust matter were based on the extrapolations of the semiempirical mass formula to the free neutron gas regime (Langer et al., 1969; Bethe et al., 1970). Baym, Bethe and Pethick considered the reduction of the nuclear surface energy due to the free neutron gas in their calculation (Baym et al., 1971a). The study of nuclei in the neutron star crust was carried out using the energy density of a many body system by Negele and Vautherin (Negele & Vautherin, 1973). With increasing density in the inner crust, unusual nuclear shapes might appear there (Ravenhall et al., 1983; Oyamatsu, 1993). The properties of nuclei in the inner crust were also investigated using a relativistic field theoretical model (Cheng et al., 1997).

There are two important aspects of the problem when nuclei are immersed in a neutron gas. On the one hand we have to deal with the coexistence of two phases of nuclear matter - denser phase inside a nucleus and low density phase outside it, in a thermodynamical consistent manner. On the other hand, the determination of the surface energy of the interface between two phases with good accuracy is needed. It was shown that this problem could be solved using the subtraction procedure of Bonche, Levit and Vautherin (Bonche et al., 1984, 1985; Suraud, 1987). The properties of a nucleus are isolated from nucleus plus neutron gas in a temperature dependent Hartree-Fock theory using the subtraction procedure. This same method was extended to isolated nuclei embedded in a neutron gas (De et al., 2001) as well as nuclei in the inner crust at zero temperature (Sil et al., 2002). This shows that it would be worth studying the properties of nuclei in the inner crust in the presence of strongly quantizing magnetic field relevant to magnetars using the subtraction procedure.

Recently the stability of nuclei embedded in an electron gas was investigated within a relativistic mean field model in zero magnetic field (Bürvenich et al., 2007). It was observed in their calculation that nuclei became more stable against decay and spontaneous fission with increasing electron number density. It is worth mentioning here that the electron number density is enhanced in the presence of strong magnetic fields due to Landau quantisation compared with the zero field case. The question is whether the nuclear system in the inner crust of magnetars would be more stable than those of the field free case. This is the focus of our calculation in this article.

The paper is organised in the following way. In section 2, the formalism for the calculation of nuclei of the inner crust immersed in a neutron as well as an electron gas in the presence of strongly quantising magnetic fields is described. Results of our calculation are discussed in section 3. Section 4 contains the summary and conclusions.

## 2 Formalism

We investigate the properties of nuclei and their stability in the inner crust in the presence of strong magnetic fields using the Thomas-Fermi (TF) model. In this case nuclei are immersed in a nucleonic gas as well as a uniform background of electrons and may be arranged in a lattice. Each lattice volume is replaced by a spherical cell with a nucleus at its center in the Wigner-Seitz (WS) approximation. Each cell is taken to be charge neutral such that the number of electrons is equal to the number of protons in it. The Coulomb interaction between cells is neglected. Electrons are assumed to be uniformly distributed within a cell. The system maintains the -equilibrium. We assume that the system is placed in a uniform magnetic field. Electrons are affected by strongly quantizing magnetic fields. Protons in the cell are affected by magnetic fields only through the charge neutrality condition. The interaction of nuclear magnetic moment with the field is negligible in a magnetic field G (Lattimer et al., 2000).

The calculation below is performed in a zero temperature TF model. In the WS cell, a nucleus is located at the centre and immersed in a low density neutron gas whereas protons are trapped in the nucleus. However, the spherical cell does not define a nucleus. The nucleus is realised after subtraction of the gas part from the cell as shown by Bonche, Levit and Vautherin (Bonche et al., 1984, 1985). In an earlier calculation, it was demonstrated that the TF formalism at finite temperature gave two solutions (Suraud, 1987). One solution corresponds to the nucleus plus neutron gas and the second one represents only the neutron gas. The density profiles of the nucleus plus neutron gas as well as that of the neutron gas are obtained self-consistently in the TF formalism. Finally the nucleus is obtained as the difference of two solutions. This formalism is adopted in our calculation at zero temperature as described below.

The nucleus plus gas solution coincides with the gas solution at large distance leading to the definition of the thermodynamic potential () of the nucleus as (Bonche et al., 1984, 1985)

(1) |

where is the thermodynamic potential of the nucleus plus gas phase and is that of the gas only. The thermodynamic potential is defined as

(2) |

where and are the chemical potential and number of q-th species, respectively. The free energy is given by

(3) |

where is nuclear energy density functional, is the Coulomb energy density and is the energy density of electrons. The free energy is a function of average baryon density () and proton fraction (). The nuclear energy density is calculated using the SKM nucleon-nucleon interaction and given by (Sil et al., 2002; Brack et al., 1985; Stone et al., 2003)

(4) | |||||

The first two terms of the nuclear energy density are kinetic energy densities of neutrons and protons, respectively. The third term originates from the zero range part of the Skyrme interaction whereas the last term is the contribution of the density dependent part of the nucleon-nucleon interaction. The effective mass of nucleons is given by

(5) | |||||

where total baryon density is .

The Coulomb energy densities for the NG and G phases are:

(6) | |||||

where and are proton densities in the nucleus plus gas and only gas. Here the coulomb energy densities () represent the direct part. We do not consider the exchange part because its contribution is small.

So far the formalism described above is applicable for the zero magnetic field. However, we study the effects of magnetic fields on electrons which , in turn, influence the properties of nuclei in the inner crust. The Coulomb energy density and the energy density of electrons in Eq. (3) would be modified in strongly quantising magnetic fields. In the presence of a magnetic field, the motion of electrons is quantized in the plane perpendicular to the field. Protons in nuclei would be influenced by a magnetic field through the charge neutrality condition. We take the magnetic field () along Z-direction and assume that it is uniform throughout the inner crust. If the field strength exceeds a critical value G, then electrons become relativistic (Lai, 2001). The energy eigenvalue of relativistic electrons in quantizing magnetic field is given by

(7) |

where is the Z-component of momentum, , is the Landau quantum number. The Fermi momentum of electrons, , is obtained from

(8) |

where is the direct part of the single particle Coulomb potential.

The number density of electrons in a magnetic field is calculated as

(9) |

Here the spin degeneracy is for the lowest Landau level () and for all other levels.

The maximum Landau quantum number () is given by

(10) |

The energy density of electrons is obtained from,

(11) |

We minimise the thermodynamic potential in the TF approximation with the condition of number conservation of each species. The density profiles of neutrons and protons with or without magnetic fields are obtained from

(12) |

This results in the following coupled equations (Sil et al., 2002; De et al., 2001)

(13) |

where is the effective mass of q-th species , and are the single particle potentials of nucleons in the nucleus plus gas as well as gas phases (Brack et al., 1985). On the other hand, and are direct parts of the single particle Coulomb potential corresponding to the nucleus plus gas and only gas solutions and both are given by

(14) |

The average chemical potential for q-th nucleon is

(15) |

where refers to or of the cell which is defined by the average baryon density and proton fraction . The -equilibrium condition is written as

(16) |

The average electron chemical potential in magnetic fields is given by

(17) |

where denotes the average of the single particle Coulomb potential.

Density profiles of neutrons and protons in the cell are constrained as

(18) |

where and are neutron and proton numbers in the cell, respectively.

Finally, number of neutrons () and protons () in a nucleus with mass number are obtained using the subtraction procedure as

(19) |

## 3 Results and Discussion

We find out the equilibrium nucleus at each density point minimising the free energy of the system within a WS cell maintaining charge neutrality and -equilibrium. The variables of this problem are the average baryon density (), the proton fraction () and the radius of a cell (). For a fixed value of , and , the total number of nucleons () is given by where the volume of a cell is . The proton number in the cell is and the neutron number is . Now we obtain density profiles of neutrons and protons in the cell using Eqs. (13) and (18) at a given average baryon density and . Consequently, we calculate chemical potentials of neutrons and protons and free energy per nucleon. Next we vary the proton fraction, calculate chemical potentials and density profiles and obtain the -equilibrium in the cell. Finally we adust the cell size () and repeat the above mentioned steps to get the minimum of the free energy. These values of and are then used to calculate neutron and proton numbers in a nucleus at an average baryon density corresponding to the free energy minimum with the help of Eq. (19). This procedure is repeated for each average baryon density.

The minimum of the free energy originates from the interplay of different contributions. The free energy per nucleon is given by

(20) |

The nuclear energy including the Coulomb interaction among protons is denoted by , is the lattice energy which involves the Coulomb interaction between electrons and protons and the electron kinetic energy is . The free energy per nucleon in the presence of magnetic field G is shown as a function of the cell size for an average baryon density fm in Fig. 1. We note that the nuclear energy increases with . On the other hand, the lattice energy and electron kinetic energy both decrease with increasing cell size. The competition of with the sum of and determines the free energy minimum. The cell radius corresponding to the free energy minimum is 32.1 fm for the zero field case (not shown in the figure) and 31.9 fm for G. The corresponding proton fraction for G is 0.03.

In Figure 2, the cell size corresponding to the free energy minimum is plotted as a function of average baryon density for magnetic fields , , and G. For magnetic fields , several Landau levels are populated by electrons. Consequently we do not find any change in the cell size in the magnetic fields compared with the zero field case. However we find some change in the cell size for G when only the zeroth Landau level is populated by electrons for , whereas first two levels are populated in the density range 0.005 to 0.015 fm. However the cell size is increased due to the population of only zeroth Landau level in the presence of the magnetic field G compared with the zero field case. The size of the cell always decreases with increasing average baryon density.

The proton fraction in the presence of magnetic fields is shown as a function of average baryon density in Fig. 3. Protons in nuclei are affected by the Landau quantisation of electrons through the charge neutrality condition in a cell. For magnetic fields G, the proton fraction is the same as that of the zero field case over the whole density range considered here. We find some changes in the proton fraction below fm when the field is G. Though electrons populate the zeroth Landau level for fm, the proton fraction decreases below the corresponding proton fraction of the zero field case. However, for magnetic field G, the proton fraction is strongly enhanced due to the population of the zeroth Landau level for fm compared with the zero field case.

The density profile of neutrons in the nucleus plus gas (NG) as well as gas (G) phases corresponding to fm with and without magnetic fields are exhibited as a function of distance () within the cell in Fig. 4. The solid line denotes the zero field case where as the dashed line represents the density profile with the field G. The horizontal lines imply the uniform gas phases in both cases. The proton fraction is 0.040 for the magnetic field case whereas it is 0.022 for the zero field case. Though neutrons are not directly affected by the magnetic field, the neutron chemical potential is modified through the -equilibrium due to Landau quantisation of electrons. Consequently, the number density of neutrons is altered. We find that the neutron density is higher in the gas phase for the zero field case than that of the situation with the magnetic field. We show the subtracted density profiles of neutrons with magnetic field G in Fig. 5. The neutron density profiles in the nucleus phase with and without magnetic field are different. Further we note that less number of neutrons drip out of a nucleus in the presence of the magnetic field than the situation without the field. This may be attributed to the shift in the -equilibrium in strong magnetic fields. We find a similar situation in the calculation of the outer crust in magnetic fields that the neutron drip point is shifted to higher densities (Nandi & Bandyopadhyay, 2010).

Now we know the density profiles of neutron and protons in the nucleus plus gas phase as well as in the nucleus at each average baryon density. We immediately calculate the total number of neutrons and protons in the nucleus plus gas phase and in a nucleus using Eqs. (18) and (19). We show total number () in a cell for magnetic fields , and G with average baryon density in Fig. 6. The dotted line denotes the zero field case. In all cases, growing with the density reaches a maximum and then decreases. Such a trend was observed in the calculation of Negele and Vautherin in the absence of a magnetic field (Negele & Vautherin, 1973). We note that our predictions for G do not change from the field free results because a large number of Landau levels is populated in that magnetic field. For magnetic field G, the total number of nucleons decreases in the density regime 0.005 - 0.02 fm compared with the corresponding results of the field free case. This can be understood from the behaviour of the cell size around that density regime in Fig. 2. For G, the zeroth Landau level is populated by electrons for densities fm. This modifies the -equilibrium and the charge neutrality conditions which, in turn, impact the size of the cell and the total number of nucleons in a cell. This effect is pronounced in the case of G. In this case, is significantly reduced compared with the zero field case for densities fm.

We obtain neutron (), proton () and total nucleon numbers () in the nucleus at each average baryon density following the subtraction procedure. Total nucleon and proton numbers are shown in Fig. 7 for the above mentioned magnetic fields. When the magnetic field is G or more, It is noted that our results in certain cases start oscillating from the field free results. This may be attributed to the fact that the population of Landau levels jumps from a few levels to zero in the above mentioned fields as baryon density decreases from higher to lower values. In contrast to Fig. 6, we find total nucleon and proton numbers inside the nucleus at each density point beyond 0.002 upto 0.04 fm are significantly enhanced in case of G compared with the field free case as well as other magnetic fields considered here. This clearly demonstrates that more neutrons are inside the nucleus in the presence of strong magnetic fields G than in the gas phase in that density regime. This is opposite to the situation in the zero magnetic field. This can be easily understood from the density profiles with and without magnetic fields in Fig. 4 and Fig. 5.

We plot the free energy per nucleon of the system with average baryon density in Fig. 8. Our results for G do not change much form the field free results. However, for G, the free energy per nucleon is reduced at lower densities ( 0.004 fm) as compared with the field free case. We find more pronounced reduction in the free energy per nucleon in the field G almost over the whole density regime considered here.

## 4 Summary and Conclusions

We have investigated properties of nuclei in the inner crust and their stability in the presence of strong magnetic fields or more. Nuclei are immersed in a neutron gas and uniform background of electrons. We have adopted the SKM interaction for the nuclear energy density functional and studied this problem in the Thomas-Fermi model. Electrons are affected through Landau quantisation in strong magnetic fields because much less Landau levels can be occupied in these cases. Consequently, electron number density and energy density are modified in strongly quantising magnetic field and the -equilibrium condition is altered compared with the field free case. The enhancement of electron number density in magnetic fields G due to the population of the zeroth Landau level leads to enhancement in proton fraction through the charge neutrality condition. We minimise the free energy of the system within a Wigner-Seitz cell to obtain the nucleus at each average baryon density. In this connection we used the subtraction procedure to obtain the density profiles of a nucleus from the nucleus plus gas and only gas solutions at each average baryon density point. We note that less number of neutrons drip out of a nucleus in the presence of strong fields than the situation without magnetic field. This results in larger mass and proton numbers in a nucleus in the presence of magnetic field G compared with the corresponding nucleus in the field free case. Further the free energy per nucleon of the system is reduced in magnetic fields G.

Magnetars might eject crustal matter due to tremendous magnetic stress on the crust (Gelfand et al., 2005). The ejected matter of the inner crust might expand to much lower densities. The decompressed crustal matter has long been considered as an important site for -process nuclei (Lattimer & Schramm, 1977; Arnould et al., 2007). It would be worth studying the -process in the decompressed crustal matter of magnetars using the results of our calculation as an input.

### Footnotes

- affiliation: Centre for Astroparticle Physics, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India
- affiliation: Centre for Astroparticle Physics, Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Kolkata-700064, India

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