# Comparison between the Thomas–Fermi and Hartree–Fock–Bogoliubov Methods in the Inner Crust of a Neutron Star: The Role of Pairing Correlations

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Wigner–Seitz Cell

#### 2.1. Hartree–Fock–Bogoliubov

#### 2.2. The ETFSI Method

#### 2.3. Choice of Functionals

#### Energy Minimization with ETFSI

#### 2.4. HFB vs. ETFSI+Pairing

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Neutron Condensation Energy

## References

- Chamel, N.; Haensel, P. Physics of neutron star crusts. Living Rev. Relativ.
**2008**, 11, 10. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Reinhard, P.G.; Bender, M. Mean Field: Relativistic versus Non-relativistic. In Lectures Notes in Physics “Extended Density Functionals in Nuclear Structure Physics”; Springer: Berlin/Heidelberg, Germany, 2004; Volume 268, pp. 249–268. [Google Scholar]
- Goriely, S.; Chamel, N.; Pearson, J.M. Skyrme-Hartree-Fock-Bogoliubov Nuclear Mass Formulas: Crossing the 0.6 MeV Accuracy Threshold with Microscopically Deduced Pairing. Phys. Rev. Lett.
**2009**, 102, 152503. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Pearson, J.M.; Chamel, N.; Potekhin, A.Y.; Fantina, A.F.; Ducoin, C.; Dutta, A.K.; Goriely, S. Unified Equations of State for Cold Non-Accreting Neutron Stars with Brussels–Montreal Functionals—I. Role of Symmetry Energy. Mon. Not. R. Astron. Soc.
**2018**, 481, 2994–3026, Erratum in**2019**, 486, 768. [Google Scholar] - Gendreau, K.C.; Arzoumanian, Z.; Okajima, T. The Neutron star Interior Composition ExploreR (NICER): An Explorer mission of opportunity for soft x-ray timing spectroscopy. Space Telescopes and Instrumentation 2012: Ultraviolet to Gamma Ray. Int. Soc. Opt. Photonics
**2012**, 8443, 844313. [Google Scholar] - Abbott, B.P.; Abbott, R.; Abbott, T.; Acernese, F.; Ackley, K.; Adams, C.; Adams, T.; Addesso, P.; Adhikari, R.; Adya, V.; et al. GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett.
**2017**, 119, 161101. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Most, E.R.; Weih, L.R.; Rezzolla, L.; Schaffner-Bielich, J. New constraints on radii and tidal deformabilities of neutron stars from GW170817. Phys. Rev. Lett.
**2018**, 120, 261103. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Blaschke, D.; Chamel, N. Phases of Dense Matter in Compact Stars. In The Physics and Astrophysics of Neutron Stars; Rezzolla, L., Pizzochero, P., Jones, D.I., Rea, N., Vidaña, I., Eds.; Springer International Publishing: Cham, Switzerland, 2018; Volume 457, pp. 337–400. [Google Scholar] [CrossRef] [Green Version]
- Stoecker, H.; Greiner, W. High energy heavy ion collisions—Probing the equation of state of highly excited hardronic matter. Phys. Rep.
**1986**, 137, 277–392. [Google Scholar] [CrossRef] - Danielewicz, P.; Lacey, R.; Lynch, W.G. Determination of the equation of state of dense matter. Science
**2002**, 298, 1592–1596. [Google Scholar] [CrossRef] [Green Version] - Steiner, A.W. Neutron star inner crust: Nuclear physics input. Phys. Rev. C
**2008**, 77, 035805. [Google Scholar] [CrossRef] [Green Version] - Lattimer, J.M.; Prakash, M. The physics of neutron stars. Science
**2004**, 304, 536–542. [Google Scholar] [CrossRef] [Green Version] - Alford, M.; Blaschke, D.; Drago, A.; Klähn, T.; Pagliara, G.; Schaffner-Bielich, J. Quark matter in compact stars? Nature
**2007**, 445, E7–E8. [Google Scholar] [CrossRef] [PubMed] - Chatterjee, D.; Vidaña, I. Do hyperons exist in the interior of neutron stars? Eur. Phys. J. A
**2016**, 52, 29. [Google Scholar] [CrossRef] [Green Version] - Vidana, I.; Bashkanov, M.; Watts, D.; Pastore, A. The d*(2380) in Neutron Stars—A New Degree of Freedom? Phys. Lett. B
**2018**, 781, 112–116. [Google Scholar] [CrossRef] - Li, J.J.; Sedrakian, A.; Weber, F. Competition between delta isobars and hyperons and properties of compact stars. Phys. Lett. B
**2018**, 783, 234–240. [Google Scholar] [CrossRef] - Baym, G.; Pethick, C.; Sutherland, P. The ground state of matter at high densities: Equation of state and stellar models. Astrophys. J.
**1971**, 170, 299. [Google Scholar] [CrossRef] - Pearson, J.M.; Goriely, S.; Chamel, N. Properties of the Outer Crust of Neutron Stars from Hartree-Fock-Bogoliubov Mass Models. Phys. Rev. C
**2011**, 83, 065810. [Google Scholar] [CrossRef] - Chamel, N.; Fantina, A.F. Binary and Ternary Ionic Compounds in the Outer Crust of a Cold Nonaccreting Neutron Star. Phys. Rev. C
**2016**, 94. [Google Scholar] [CrossRef] - Negele, J.W.; Vautherin, D. Neutron star matter at sub-nuclear densities. Nucl. Phys. A
**1973**, 207, 298–320. [Google Scholar] [CrossRef] - Chamel, N.; Fantina, A.; Zdunik, J.L.; Haensel, P. Neutron drip transition in accreting and nonaccreting neutron star crusts. Phys. Rev. C
**2015**, 91, 055803. [Google Scholar] [CrossRef] [Green Version] - Pastore, A.; Neill, D.; Powell, H.; Medler, K.; Barton, C. Impact of statistical uncertainties on the composition of the outer crust of a neutron star. Phys. Rev. C
**2020**, 101, 035804. [Google Scholar] [CrossRef] [Green Version] - Chamel, N.; Naimi, S.; Khan, E.; Margueron, J. Validity of the Wigner-Seitz approximation in neutron star crust. Phys. Rev. C
**2007**, 75, 055806. [Google Scholar] [CrossRef] [Green Version] - Chamel, N. Neutron conduction in the inner crust of a neutron star in the framework of the band theory of solids. Phys. Rev. C
**2012**, 85, 035801. [Google Scholar] [CrossRef] - Ring, P.; Schuck, P. The Nuclear Many-Body Problem; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2004. [Google Scholar]
- Baldo, M.; Saperstein, E.; Tolokonnikov, S. The role of the boundary conditions in the Wigner–Seitz approximation applied to the neutron star inner crust. Nucl. Phys. A
**2006**, 775, 235–244. [Google Scholar] [CrossRef] [Green Version] - Margueron, J.; Van Giai, N.; Sandulescu, N. Equation of state in the inner crust of neutron stars: Discusion of the unbound neutron states. In Exotic States of Nuclear Matter; Lombardo, U., Baldo, M., Burgio, F., Schulze, H.J., Eds.; World Scientific: Singapore, 2007; p. 362. [Google Scholar]
- Pastore, A.; Baroni, S.; Losa, C. Superfluid properties of the inner crust of neutron stars. Phys. Rev. C
**2011**, 84, 065807. [Google Scholar] [CrossRef] [Green Version] - Pastore, A.; Shelley, M.; Baroni, S.; Diget, C. A new statistical method for the structure of the inner crust of neutron stars. J. Phys. G Nucl. Part. Phys.
**2017**, 44, 094003. [Google Scholar] [CrossRef] [Green Version] - Fantina, A.F.; Zdunik, J.L.; Chamel, N.; Pearson, J.M.; Haensel, P.; Goriely, S. Crustal Heating in Accreting Neutron Stars from the Nuclear Energy-Density Functional Theory: I. Proton Shell Effects and Neutron-Matter Constraint. Astron. Astrophys.
**2018**, 620, A105. [Google Scholar] [CrossRef] [Green Version] - Schuetrumpf, B.; Martínez-Pinedo, G.; Afibuzzaman, M.; Aktulga, H.M. Survey of Nuclear Pasta in the Intermediate-Density Regime: Shapes and Energies. Phys. Rev. C
**2019**, 100, 045806. [Google Scholar] [CrossRef] [Green Version] - Brack, M.; Jennings, B.; Chu, Y. On the extended Thomas-Fermi approximation to the kinetic energy density. Phys. Lett. B
**1976**, 65, 1–4. [Google Scholar] [CrossRef] [Green Version] - Pearson, J.M.; Chamel, N.; Goriely, S.; Ducoin, C. Inner Crust of Neutron Stars with Mass-Fitted Skyrme Functionals. Phys. Rev. C
**2012**, 85, 065803. [Google Scholar] [CrossRef] [Green Version] - Grill, F.; Margueron, J.; Sandulescu, N. Cluster structure of the inner crust of neutron stars in the Hartree-Fock-Bogoliubov approach. Phys. Rev. C
**2011**, 84, 065801. [Google Scholar] [CrossRef] [Green Version] - Pearson, J.; Chamel, N.; Pastore, A.; Goriely, S. Role of proton pairing in a semimicroscopic treatment of the inner crust of neutron stars. Phys. Rev. C
**2015**, 91, 018801. [Google Scholar] [CrossRef] - Mondal, C.; Viñas, X.; Centelles, M.; De, J.N. Structure and Composition of the Inner Crust of Neutron Stars from Gogny Interactions. Phys. Rev. C
**2020**, 102, 015802. [Google Scholar] [CrossRef] - Shelley, M.; Pastore, A. How accurately can the Extended Thomas-Fermi method describe the inner crust of a neutron star? arXiv
**2020**, arXiv:2002.01839. [Google Scholar] [CrossRef] - Pastore, A.; Margueron, J.; Schuck, P.; Viñas, X. Pairing in exotic neutron-rich nuclei near the drip line and in the crust of neutron stars. Phys. Rev. C
**2013**, 88, 034314. [Google Scholar] [CrossRef] [Green Version] - Shapiro, S.L.; Teukolsky, S.A. Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects; John Wiley & Sons: Hoboken, NJ, USA, 2008. [Google Scholar]
- Skyrme, T. CVII. The nuclear surface. Philos. Mag.
**1956**, 1, 1043–1054. [Google Scholar] [CrossRef] - Perlińska, E.; Rohoziński, S.; Dobaczewski, J.; Nazarewicz, W. Local density approximation for proton-neutron pairing correlations: Formalism. Phys. Rev. C
**2004**, 69, 014316. [Google Scholar] [CrossRef] [Green Version] - Pastore, A. Superfluid properties of the inner crust of neutron stars. II. Wigner-Seitz cells at finite temperature. Phys. Rev. C
**2012**, 86, 065802. [Google Scholar] [CrossRef] [Green Version] - Grammaticos, B.; Voros, A. Semiclassical approximations for nuclear hamiltonians. I. Spin-Independent Potentials. Ann. Phys.
**1979**, 123, 359. [Google Scholar] [CrossRef] - Bartel, J.; Bencheikh, K. Nuclear mean fields through self-consistent semiclassical calculations. Eur. Phys. J. A-Hadron. Nucl.
**2002**, 14, 179–190. [Google Scholar] [CrossRef] - Onsi, M.; Dutta, A.; Chatri, H.; Goriely, S.; Chamel, N.; Pearson, J. Semi-classical equation of state and specific-heat expressions with proton shell corrections for the inner crust of a neutron star. Phys. Rev. C
**2008**, 77, 065805. [Google Scholar] [CrossRef] [Green Version] - Chabanat, E.; Bonche, P.; Haensel, P.; Meyer, J.; Schaeffer, R. A Skyrme parametrization from subnuclear to neutron star densities Part II. Nuclei far from stabilities. Nucl. Phys. A
**1998**, 635, 231–256. [Google Scholar] [CrossRef] - Goriely, S.; Chamel, N.; Pearson, J. Further explorations of Skyrme-Hartree-Fock-Bogoliubov mass formulas. XIII. The 2012 atomic mass evaluation and the symmetry coefficient. Phys. Rev. C
**2013**, 88, 024308. [Google Scholar] [CrossRef] - Wiringa, R.B.; Fiks, V.; Fabrocini, A. Equation of State for Dense Nucleon Matter. Phys. Rev. C
**1988**, 38, 1010–1037. [Google Scholar] [CrossRef] [PubMed] - Wiringa, R.B. From Deuterons to Neutron Stars: Variations in Nuclear Many-Body Theory. Rev. Mod. Phys.
**1993**, 65, 231–242. [Google Scholar] [CrossRef] - Li, Z.H.; Schulze, H.J. Neutron Star Structure with Modern Nucleonic Three-Body Forces. Phys. Rev. C
**2008**, 78, 028801. [Google Scholar] [CrossRef] - Bertsch, G.; Esbensen, H. Pair correlations near the neutron drip line. Ann. Phys.
**1991**, 209, 327–363. [Google Scholar] [CrossRef] - Gandolfi, S.; Illarionov, A.Y.; Fantoni, S.; Pederiva, F.; Schmidt, K. Equation of State of Superfluid Neutron Matter and the Calculation of the S 0 1 Pairing Gap. Phys. Rev. Lett.
**2008**, 101, 132501. [Google Scholar] [CrossRef] [Green Version] - Bulgac, A.; Yu, Y. Renormalization of the Hartree-Fock-Bogoliubov equations in the case of a zero range pairing interaction. Phys. Rev. Lett.
**2002**, 88, 042504. [Google Scholar] [CrossRef] [Green Version] - Chamel, N. Effective Contact Pairing Forces from Realistic Calculations in Infinite Homogeneous Nuclear Matter. Phys. Rev. C
**2010**, 82, 014313. [Google Scholar] [CrossRef] [Green Version] - Cao, L.; Lombardo, U.; Schuck, P. Screening effects in superfluid nuclear and neutron matter within Brueckner theory. Phys. Rev. C
**2006**, 74, 064301. [Google Scholar] [CrossRef] - Barranco, F.; Broglia, R.; Esbensen, H.; Vigezzi, E. Role of finite nuclei on the pairing gap of the inner crust of neutron stars. Phys. Lett. B
**1997**, 390, 13–17. [Google Scholar] [CrossRef] - Dean, D.; Hjorth-Jensen, M. Pairing in nuclear systems: From neutron stars to finite nuclei. Rev. Mod. Phys.
**2003**, 75, 607. [Google Scholar] [CrossRef] [Green Version] - Sandulescu, N.; Van Giai, N.; Liotta, R. Superfluid properties of the inner crust of neutron stars. Phys. Rev. C
**2004**, 69, 045802. [Google Scholar] [CrossRef] [Green Version] - Maurizio, S.; Holt, J.W.; Finelli, P. Nuclear pairing from microscopic forces: Singlet channels and higher-partial waves. Phys. Rev. C
**2014**, 90, 044003. [Google Scholar] [CrossRef] [Green Version] - Watanabe, G.; Pethick, C.J. Superfluid density of neutrons in the inner crust of neutron stars: New life for pulsar glitch models. Phys. Rev. Lett.
**2017**, 119, 062701. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bennemann, K.H.; Ketterson, J.B. Superconductivity: Volume 1: Conventional and Unconventional Superconductors Volume 2: Novel Superconductors; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Burrello, S.; Gulminelli, F.; Aymard, F.; Colonna, M.; Raduta, A.R. Heat Capacity of the Neutron Star Inner Crust within an Extended Nuclear Statistical Equilibrium Model. Phys. Rev. C.
**2015**, 92, 055804. [Google Scholar] [CrossRef] - Pizzochero, P.M.; Viverit, L.; Broglia, R.A. Vortex-Nucleus Interaction and Pinning Forces in Neutron Stars. Phys. Rev. Lett.
**1997**, 79, 3347–3350. [Google Scholar] [CrossRef] [Green Version] - Pastore, A.; Barranco, F.; Broglia, R.; Vigezzi, E. Microscopic calculation and local approximation of the spatial dependence of the pairing field with bare and induced interactions. Phys. Rev. C
**2008**, 78, 024315. [Google Scholar] [CrossRef] [Green Version] - Virtanen, P.; Gommers, R.; Oliphant, T.E.; Haberland, M.; Reddy, T.; Cournapeau, D.; Burovski, E.; Peterson, P.; Weckesser, W.; Bright, J.; et al. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nat. Methods
**2020**, 17, 261–272. [Google Scholar] [CrossRef] [Green Version] - Pearson, J.M.; Chamel, N.; Potekhin, A.Y. Unified Equations of State for Cold Nonaccreting Neutron Stars with Brussels-Montreal Functionals. II. Pasta Phases in Semiclassical Approximation. Phys. Rev. C
**2020**, 101, 015802. [Google Scholar] [CrossRef] [Green Version] - Carreau, T.; Gulminelli, F.; Margueron, J. Bayesian analysis of the crust-core transition with a compressible liquid-drop model. Eur. Phys. J. A
**2019**, 55, 188. [Google Scholar] [CrossRef]

**Figure 1.**Densities (top row) and fields (bottom row), for neutrons (

**left column**) and protons (

**right column**), for a Wigner–Seitz cell with ${\rho}_{b}=0.02$ fm${}^{-3}$, $Z=40$ and ${R}_{WS}=24.0$ fm. Black lines show the matter densities and central potentials; orange lines show the kinetic densities and effective masses; and purple lines show the spin current densities and spin-orbit fields. Solid lines show the results of using the HFB method, and dotted lines show those from using the ETFSI method; both used the SLy4 functional. Note the different scale used in panels a and b for the neutron and proton densities, to make clearer the bump observed in the proton densities; see the text for details. Note also that all spin current densities and spin-orbit fields are multiplied by 5 to make them more visible.

**Figure 3.**The pairing gap in pure neutron matter (PNM), as a function of density, for the two pairing strengths used with the SLy4 functional (green dash-dotted line shows weak, red dashed line shows strong), and for the BSk24 functional (orange dotted line). See text for details.

**Figure 4.**Individual contributions to energy per particle, e, for a Wigner Seitz cell with ${\rho}_{b}=0.02$ fm${}^{-3}$. Results are shown for the two pairing strengths used with the SLy4 functional (green dash-dotted line shows weak, red dashed line shows strong). The top left panel shows only ${e}_{Sky}$, while the bottom left shows ${e}_{Sky}+{e}_{e}$; see Equation (2) and text for details. In the top-right panel for neutron pairing, showing ${e}_{cond}$ (Equation (8)), the weak result has been multiplied by 10 to make clear the variations for weak and strong on the same scale.

**Figure 5.**Selected slices of the energy surface at fixed baryonic densities ${\rho}_{b}$, showing the variation of e with Z. Black solid lines show the HFB results, blue dotted lines the ETFSI results and the red dashed lines the results for ETFSI with neutron and proton pairing included. Results with the SLy4 functional (with strong pairing) are shown in the left column, and those with BSk24 are shown in the right column. The HFB calculations use the optimum values of ${R}_{WS}$ found with the ETFSI+pairing method.

**Figure 6.**Energy per particle, e, for the inner crust, using the SLy4 functional. The result for ETFSI is shown by the blue dotted line, and the red dashed line shows the result with neutron and proton pairing included (using the strong pairing).

**Figure 7.**The variation through the inner crust of: the total particle number A (panel a), proton fraction ${Y}_{p}$ (panel b), Wigner–Seitz cell radius ${R}_{WS}$ (panel c) and pressure P (panel d). ETFSI results are shown with blue dotted lines, and those for ETFSI+pairing are shown with red dashed lines.

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Shelley, M.; Pastore, A.
Comparison between the Thomas–Fermi and Hartree–Fock–Bogoliubov Methods in the Inner Crust of a Neutron Star: The Role of Pairing Correlations. *Universe* **2020**, *6*, 206.
https://doi.org/10.3390/universe6110206

**AMA Style**

Shelley M, Pastore A.
Comparison between the Thomas–Fermi and Hartree–Fock–Bogoliubov Methods in the Inner Crust of a Neutron Star: The Role of Pairing Correlations. *Universe*. 2020; 6(11):206.
https://doi.org/10.3390/universe6110206

**Chicago/Turabian Style**

Shelley, Matthew, and Alessandro Pastore.
2020. "Comparison between the Thomas–Fermi and Hartree–Fock–Bogoliubov Methods in the Inner Crust of a Neutron Star: The Role of Pairing Correlations" *Universe* 6, no. 11: 206.
https://doi.org/10.3390/universe6110206