THE INNER CRUST OF NEUTRON STARS IN RELATIVISTIC MEAN FIELD APPROACH

October 22, 2008 16:56 WSPC/INSTRUCTION FILE 01076

International Journal of Modern Physics EVol. 17, No. 9 (2008) 1765–1773c© World Scientific Publishing Company

THE INNER CRUST OF NEUTRON STARS IN

RELATIVISTIC MEAN FIELD APPROACH∗

JIGUANG CAO

China Institute of Atomic Energy, P.O. Box 275-18, Beijing 102413, P. R. China

[email protected]

ZHONGYU MA

China Institute of Atomic Energy, Beijing 102413, P. R. China†

NGUYEN VAN GIAI

Institut de Physique Nucleaire, IN2P3-CNRS, F-91406 Orsay Cedex, France

The microscopic properties and superfluidity of the inner crust in neutron stars areinvestigated in the framework of the relativistic mean field(RMF) model and BCS theory.The Wigner-Seitz(W-S) cell of inner crust is composed of neutron-rich nuclei immersedin a sea of dilute, homogeneous neutron gas. The pairing properties of nucleons in theW-S cells are treated in BCS theory with Gogny interaction. In this work, we emphasizeon the choice of the boundary conditions in the RMF approach and superfluidity of theinner crust. Three kinds of boundary conditions are suggested. The properties of the W-Scells with the three kinds of boundary conditions are investigated. The neutron densitydistributions in the RMF and Hartree-Fock-Bogoliubov(HFB) models are compared.

Keywords: Inner crust; Wigner-Seitz cell; RMF-BCS; Gogny force.

PACS Nos.: 26.60.+c,24.10.Jv,21.60.-n

1. Introduction

Neutron stars are important astrophysical objects whose study shed light on nu-

merous and diverse properties of physical systems, ranging from the evolution of

supernovae to the behavior of baryonic matter at various densities. The first exper-

imental fact pointing to the nuclear superfluidity in the neutron stars was the large

relaxation times which follow the sudden period variations (so called “glitches”)

of the neutron star rotation. Furthermore, the superfluidity of the inner crust has

∗Supported by the National Natural Science Foundation of China under Grant Nos 10475116,10535010, Major State Basic Research Development Program in China Under Contract Number2007CB81500 and the European Community Project Asia-Europe Link in Nuclear Physics andAstrophysics, CN/ASIA-LINK/008(94791).†Also Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator ofLanzhou, Lanzhou 730000, P. R. China.

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1766 J.-G. Cao, Z.-Y. Ma & N. Van Giai

important consequences on the cooling of neutron stars. The properties of the inner

crust of neutron stars have been investigated with an increasing interest in recent

years, especially its microscopic structure and the superfluid properties. According

to the standard model, the inner crust consists of a lattice of Wigner-Seitz(W-S)

cells, i.e., clusters of neutrons and protons localized in a region around the center

and surrounded by a neutron gas of approximately uniform density.

The first investigation of microscopic properties of the inner crust matter was

done by Negele and Vautherin in 1973.1 They applied the Hartree-Fock (HF) ap-

proximation to 11 representative densities of the inner crust matter, obtaining the

structure for each density by searching the lowest binding energy and β-stability of

the W-S cell. Recently Sandulescu et al.2,3 investigated the inner crust superfluid-

ity and specific heat in the Hartree-Fock-Bogoliubov(HFB) model with a density-

dependent, zero-range pairing interaction. Later, an energy functional method in-

cluding the pairing correlations of protons and neutrons was introduced by Baldo

et al.4 They calculated all the 11 cells and obtained an optimal structure at each

density. In fact, the results of these two investigations are more or less similar.

However, all of the previous investigations were done within the non-relativistic

framework. We know that relativistic models are as important as non-relativistic

models in the description of nuclear matter and finite nuclei. The relativistic mean

field theory (RMF) has been very successful in the description of ground state prop-

erties of finite nuclei. Therefore, it becomes very interesting to investigate the inner

crust properties in the relativistic models. In this work, we shall apply the RMF

model to investigate the structure of the W-S cells. Recently proposed parameter

set of DDME1 with density dependent coupling constants5 is adopted in the cal-

culations. The new parameter set not only gives good descriptions of the ground

state properties of finite nuclei, but also improves the behavior of the nuclear mat-

ter equation of state at higher densities and reproduces the empirical value of the

asymmetry energy at saturation density.5 The finite range Gogny force,7 which

has been proved to be very successful in the description of pairing correlation in

finite nuclei, is employed to investigate the pairing correlation of nucleons in the

Table 1. Nine regions of the inner crustconsidered in this work. ρ0=0.17 fm−3.

Nzone ρ/ρ0 Z N RC(fm)

1 0.0016 40 140 53.62 0.0024 40 160 49.23 0.0035 40 210 46.34 0.0052 40 280 44.35 0.0094 40 460 42.26 0.022 50 900 39.37 0.034 50 1050 35.78 0.052 50 1300 33.19 0.12 50 1750 27.6

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The Inner Crust of Neutron Star 1767

W-S cells. The densities of the W-S cells in our calculations are chosen as those

suggested by Negele and Vautherin1 (see Table 1).

2. RMF-BCS Model

The Lagrangian of the RMF model with density-dependent couplings is

L = ψ (iγ · ∂ −m)ψ +1

2(∂σ)2 −

1

2m2

σσ2 −

1

4ΩµνΩµν +

1

2m2

ωω2 −

1

4~Rµν

~Rµν

+1

2m2

ρ~ρ2−

1

4FµνFµν

− gσ(ρv)ψσψ − gω(ρv)ψγµωµψ − gρ(ρv)ψγ

µ~ρµ · ~τψ

− eψγ ·A(1 − τ3)

2ψ , (1)

where the nucleon-meson couplings depend on the baryonic density ρv . The param-

eter set DDME1 of ref.5 will be used in the rest of this work.

In a spherical approximation of the W-S cells, the nucleon spinors can be de-

composed into their radial and spin-angular components,

ψi(r) =

iGiκ(r)

rY l

jm(r)

Fiκ(r)

r(σ · r)Y l

jm(r)

χi, (2)

where Y ljm(r) is a spin 1/2 spinor coupled with a spherical harmonic Yl to a total

angular momentum (jm). The radial parts of the nucleon spinors satisfy the Dirac

equation:

εiGiκ(r) = (−d

dr+κ

r)Fiκ(r) + [M + S(r) + V0(r) + ΣR

0 (r)]Giκ(r) ,

εiFiκ(r) = (d

dr+κ

r)Giκ(r) − [M + S(r) − V0(r) − ΣR

0 (r)]Fiκ(r) . (3)

S(r), V0(r) and ΣR0 (r) are the scalar, vector and rearrangement potentials, respec-

tively.5 The pairing properties are calculated in the BCS method6 with the finite

range Gogny force D1S as given in Ref. 7.

The nucleon wave functions in the central region of the W-S cells vanish at the

boundary. To describe the unbound and homogeneous neutron gas in the W-S cells,

we have to choose proper boundary conditions for the single-particle wave functions.

In the non-relativistic approach, a boundary condition in solving the Schroedinger

equation is usually chosen as follows:1 the even parity wave functions vanish at the

cell radius Rc while the first derivative of the odd-parity wave functions vanishes at

r = Rc. It is clear that with these mixed boundary conditions at the cell border all

single-particle wave functions solved in the non-relativistic Schroedinger equation

are orthogonal to one another. Therefore, it is natural to extend such boundary

conditions in the relativistic approach,Gnκ(Rc) = 0, l even

G′

nκ(Rc) = 0, l odd. (4)

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This is denoted as boundary condition C. In the relativistic approach the solutions

of the Dirac equation are two component Dirac spinors. The orthogonality of the

single-particle wave functions in the Dirac equation can be expressed as follows,

I ≡

∫ψnaκ(r)ψnbκ(r)dr = δ(na, nb)

= Gnaκ(Rc)Fnbκ(Rc) −Gnbκ(Rc)Fnaκ(Rc) . (5)

Unfortunately, it is seen that with the above boundary condition C the orthogo-

nality of the single-particle wave functions is not fulfilled. In order to satisfy the

orthogonality two kinds of boundary conditions in this work are suggested, bound-

ary condition A:Gnκ(Rc) = 0, l even

Fnκ(Rc) = 0, l odd, (6)

and boundary condition B:Gnκ(Rc) = 0, κ < 0

Fnκ(Rc) = 0, κ > 0. (7)

It is clear that the single-particle wave functions with the boundary conditions A

and B are orthogonal to one another. However, we have checked that the violation

of the orthogonality in case C is not large, generally less than 10−3. Therefore, in

our calculations we also adopt the boundary condition C for comparison.

Nine regions of the inner crust with densities and corresponding proton number

Z and neutron number N in the W-S cells have been determined in ref.,1 and they are

listed in Table 1. In finite nuclei there is always a maximum number of neutrons

which can be bound for a given number of protons. This neutron stability limit

defines the neutron drip line. Experimentally approaching the neutron drip line is

usually drastically limited since the neutron-rich nuclei are quickly β decaying. This

is not the case for the neutron-rich nuclei immersed in the inner crust since here

the β decay is blocked by the presence of the degenerate electron gas uniformly

distributed throughout the baryonic matter. Consequently, inside the inner crust

the nuclei can bind more neutrons than the nuclei in the vacuum. In addition,

their density and mean field can change significantly due to the presence of the

surrounding neutron gas.

3. Properties of Inner Crust Matter in RMF Approach

The calculated binding energies per nucleon and average neutron gas densities in

the RMF with DDME1 are given in Table 2. The results with the three kinds of

boundary conditions are compared. The differences of (E/A) in the three cases are

less than 2.5%, whereas the average neutron gas densities ρg are very similar. ρg

increases significantly with increasing inner crust density. The comparison of the

neutron density distributions of the 3 kinds of boundary conditions is shown in

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Table 2. Comparison of the binding energy per nucleon(E/A) and averageneutron gas density(ρg) obtained with boundary conditions A, B and C.

Nzone (E/A)A (E/A)B (E/A)C (ρg)A (ρg)B (ρg)C

(MeV) (MeV) (MeV) (fm)−3 (fm)−3 (fm)−3

1 −5.153 −5.143 −5.153 8.56E-05 8.56E-05 8.55E-052 −4.617 −4.569 −4.589 1.51E-04 1.51E-04 1.51E-043 −3.567 −3.513 −3.543 2.94E-04 2.94E-04 2.94E-044 −2.545 −2.512 −2.564 5.29E-04 5.29E-04 5.29E-045 −1.089 −1.073 −1.139 1.15E-03 1.15E-03 1.18E-036 0.300 0.265 0.145 3.04E-03 3.05E-03 3.05E-037 0.949 0.866 0.733 4.76E-03 4.77E-03 4.84E-038 1.707 1.619 1.450 7.50E-03 7.53E-03 7.55E-039 3.453 3.214 2.959 1.74E-02 1.74E-02 1.74E-02

Fig. 1. Comparison of neutron density distributions in 950Sn with boundary conditions A(solidcurve), B(dashed one) and C(dotted one).

Fig. 1. The density distributions of the neutron-rich nuclear cluster in the W-S

cell are very similar with the three boundary conditions. However, it is found that

the boundary condition C gives the flattest neutron gas density distribution and

the lowest binding energy per nucleon, although the orthogonality is violated. The

neutron gas density with boundary condition C is almost constant, it drops to about

half of its average value in case A, and to a very small value near the border in case

B. This can be understood because the wave functions with an odd parity reach a

maximum at the border of the W-S cell due to the vanishing of the first derivative

in case C.

In order to maintain the orthogonality of the wave functions in the rest of

our discussion we will stick to the boundary condition A. The nucleon density

distribution of each W-S cell are shown in Fig. 2. With increasing inner crust

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Fig. 2. The nucleon density distribution of Wigner-Seitz cells with the boundary condition A.Solid and dashed curves denote neutrons and protons, respectively.

Fig. 3. Comparison of the neutron density distributions of the cell 950Sn, with boundary conditionA in RMF (solid curve) and with HFB (dashed curve).

density, the nucleon density in the center of the cell decreases, the nuclear cluster

region becomes larger and the neutron gas density increases. It is illustrated that

the RMF approach could reasonably describe the structure of the W-S cell with a

lattice of neutron-rich nuclei immersed in a sea of unbound neutrons.

Figure 3 shows a comparison of the neutron density distributions in the 950Sn

cell using the RMF model and the non-relativistic HFB model of Ref. 2. Although

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the main features of the W-S cell are very similar, an obvious difference in the two

models can be observed from this figure. The density value inside the neutron-rich

cluster in the HFB is higher than that in the RMF, whereas the neutron gas density

in RMF is larger. This may partly be attributed to the difference of the asymmetry

energy and its density dependence in the two approaches.

The superfluid properties of the W-S cell is studied in the RMF + BCS model

with Gogny pairing force. In Fig. 4, the density distributions with and without

Fig. 4. Comparison of neutron density distributions of the cell 1800Sn in RMF and RMF-BCSmodels.

Fig. 5. The neutron pairing gaps ∆n calculated using Gogny interaction as a function of neutronsingle-particle energies in the cell 1800Sn.

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pairing correlations for ρ = 0.12ρ0 and Z = 50,N = 1750(Nzone = 9) are compared.

One can notice that the core and the tail of the neutron density distribution with

pairing are flatter than those without pairing.

The state-dependent neutron pairing gaps are shown in Fig. 5. The overall

dependence of the gaps of the neutron gas on the energy is simply linear, which is

quite close to the behavior of the pairing gap in neutron matter.8 One can observe

that the maximum of ∆n is about 4 MeV and the corresponding single-particle

energy is about −10 MeV, which corresponds to the depth of the neutron gas

potential. The presence of the core nucleus in the W-S cell leads to a reduction of

the gap by about 0.5 MeV. This behavior of the neutron pairing gap is quite similar

to that obtained in a non-relativistic mean field approach with the Gogny pairing

force by Pizzochero et al.8 We also found that the pairing gaps in this W-S cell are

larger than those obtained in finite nuclei.9

4. Summary

The properties of the inner crust of neutron stars are studied in the relativistic

mean field framework for the first time. We must mention a recent work10 where

the RMF is used for calculating the properties of cubic W-S cells in neutron star

crust. We use the RMF-BCS model with the finite-range Gogny pairing interaction

in the present work. The calculation is performed with the recently proposed pa-

rameter set DDME1, which could reproduce the empirical value of the asymmetry

energy at saturation density. To describe the unbound and homogeneous neutron

gas in the W-S cells three kinds of boundary conditions are proposed in our cal-

culations. It is found that the boundary condition C among the three boundary

conditions produces the flattest neutron gas density distribution and the lowest

binding energy per nucleon, although with a small violation of the orthogonality

of the wave functions. The boundary conditions A and B with full orthogonality

give more or less similar neutron gas density distributions. In comparison with the

non-relativistic HFB model, we found that the neutron gas densities of the W-S

cells obtained in the RMF model are somewhat higher. The neutron density dis-

tribution is flatter when pairing correlations are included. The overall dependence

of the gaps of the neutron gas on the energy is quite close to the behavior of the

pairing gap in neutron matter.

Acknowledgments

We gratefully acknowledge N. Sandulescu and J. Margueron for many stimulating

discussions.

References

1. J. W. Negele and D. Vautherin, Nucl. Phys. A 207 (1973) 298.2. N. Sandulescu, N. Van Giai and R. J. Liotta, Phys. Rev. C 69 (2004) 045802.

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3. N. Sandulescu, Phys. Rev. C 70 (2004) 025801.4. M. Baldo, E. E. Saperstein and S. T. Tolokonnikov, Nucl. Phys. A 749 (2005) 42.5. T. Niksic, D. Vretenar, P. Finelli and P. Ring, Phys. Rev. C 66 (2002) 024306.6. P. Ring and P. Schuck, The Nuclear Many-Body Problem, Springer, New York (1980).7. J. Decharge and D. Gogny, Phys. Rev. C 21 (1980) 1568.8. P. M. Pizzochero, F. Barranco and R. A. Broglia, The Astrophysical Journal 569

(2002) 381–394.9. M. Serra and P. Ring, Phys. Rev. C 65 (2002) 064324.

10. P. Gogelein and H. Muther, arXiv:nucl-th/07041984.

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THE INNER CRUST OF NEUTRON STARS IN RELATIVISTIC MEAN FIELD APPROACH