Theory of Nuclear Matter for Neutron Stars and ... - Graduate Physics

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ContentsList of FiguresList of Mathematical and Physical SymbolsList of TablesAcknowledgementsixxixvxvi1 Introduction 11.1 Organization of Chapters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 The Finite-Range Force Model 62.1 Schematic Nuclear force model . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Realistic nuclear force model . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2.1 Non-Relativistic Potential Model . . . . . . . . . . . . . . . . . . . . 72.2.2 Relativistic Mean Field Model . . . . . . . . . . . . . . . . . . . . . . 92.3 Finite-Range Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.3.1 The Interaction Energy . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Thermodynamic Quantities . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Properties of Uniform Matter . . . . . . . . . . . . . . . . . . . . . . 142.3.4 The Zero-Temperature Limit . . . . . . . . . . . . . . . . . . . . . . . 142.3.5 Uniform Matter at Zero Temperature . . . . . . . . . . . . . . . . . . 162.4 Two-Phase Equilibrium of Bulk Matter . . . . . . . . . . . . . . . . . . . . . 192.4.1 Zero-Temperature Two-Phase Equilibrium . . . . . . . . . . . . . . . 202.4.2 Finite-Temperature Symmetric Matter . . . . . . . . . . . . . . . . . 212.4.3 Finite-Temperature Asymmetric Matter . . . . . . . . . . . . . . . . 222.5 The Nuclear Surface in the Semi-Infinite Approximation . . . . . . . . . . . 232.5.1 The Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Isolated Nuclei and Nuclei in Dense Matter . . . . . . . . . . . . . . . . . . . 322.6.1 Isolated Nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.6.2 Dense Matter and the Wigner-Seitz Approximation . . . . . . . . . . 352.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Modified model 393.1 Original Modification to Myers & Swiatecki Model . . . . . . . . . . . . . . . 393.2 Alternate Modification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42vi
3.3 Optimized Parameter Set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Nuclear Surface Tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494 Gaussian Type Finite Range model 524.1 Nuclear Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . 524.1.1 Energy density functional . . . . . . . . . . . . . . . . . . . . . . . . 524.1.2 Effective mass, Potential, and Thermodynamic properties . . . . . . . 544.2 Parameters for the Gaussian nuclear density functional . . . . . . . . . . . . 554.2.1 Determination of the 1+ǫ power . . . . . . . . . . . . . . . . . . . . 584.2.2 The effective range of the nuclear force : r 0 . . . . . . . . . . . . . . . 584.3 Nuclear matter and nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.1 Specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.3.2 Nuclei at T = 0 MeV . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.3.3 Heavy nuclei in the neutron star crust . . . . . . . . . . . . . . . . . 614.4 Phase transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.1 Uniform matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.4.2 Quark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.5 Astrophysical application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.5.1 Mass-radius relation of a cold neutron star . . . . . . . . . . . . . . . 694.5.2 Moment of inertia of the neutron star . . . . . . . . . . . . . . . . . . 694.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715 Nuclear Energy Density Functional and Neutron Stars 725.1 Nuclear energy density functional . . . . . . . . . . . . . . . . . . . . . . . . 725.2 Binding energy of single nucleus . . . . . . . . . . . . . . . . . . . . . . . . . 745.3 Neutron Star Crust . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.1 Neutron drip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.3.2 Phase transition to uniform nuclear matter . . . . . . . . . . . . . . . 765.3.3 Statistical formulae for heavy nuclei . . . . . . . . . . . . . . . . . . . 765.4 Mass-Radius of a cold neutron star . . . . . . . . . . . . . . . . . . . . . . . 785.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 806 Nuclear Equation of State for Hot Dense Matter 816.1 Construction of EOS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Liquid Droplet Model as a numerical technique . . . . . . . . . . . . . . . . 826.3 Choice of nuclear force model . . . . . . . . . . . . . . . . . . . . . . . . . . 866.3.1 Pure neutron matter . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3.2 Maximum mass of a cold neutron star . . . . . . . . . . . . . . . . . 876.3.3 Allowed region of mass and radii of neutron stars . . . . . . . . . . . 876.4 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886.4.1 Bulk Matter inside and outside nuclei . . . . . . . . . . . . . . . . . . 896.4.2 Coulomb energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 896.4.3 Nuclear surface energy . . . . . . . . . . . . . . . . . . . . . . . . . . 906.4.4 Nuclear translational energy . . . . . . . . . . . . . . . . . . . . . . . 946.4.5 Alpha particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94vii
Page 1 and 2: Theory of nuclear matter of neutron
Page 3 and 4: Abstract of the DissertationTheory
Page 5: To Jaenyeong, Jonghyun, and Jongbum
Page 9 and 10: List of Figures1.1 Mass and radius
Page 11 and 12: SymbolsNuclear Physics SymbolE ener
Page 13 and 14: ω δt 90−10QtT cTaylor expansion
Page 15 and 16: List of Tables1.1 Range of Tables .
Page 17 and 18: Chapter 1IntroductionStars give us
Page 19 and 20: Since the mass of neutron star is d
Page 21 and 22: shown in Fig. 1.2, LS220 is the onl
Page 23 and 24: density:p = ρ2ρ o[ K9µ n = −B
Page 25 and 26: nuclear energy density functional i
Page 27 and 28: 2.3.2 Thermodynamic QuantitiesThe t
Page 29 and 30: point r and with momentum p isU n (
Page 31 and 32: Thus, in the case of zero temperatu
Page 33 and 34: For standard nuclear matter, u = 1
Page 35 and 36: a α L α U β L β U σ0.5882 1.11
Page 37 and 38: Figure 2.2: Bulk equilibrium of T =
Page 39 and 40: Figure 2.4: Coexistence curves for
Page 41 and 42: proton fraction x are irrelevant fo
Page 43 and 44: potential model analogue for the Ha
Page 45 and 46: with the convention that z II is th
Page 47 and 48: Figure 2.10: The surface tension of
Page 49 and 50: Figure 2.12: Contour plots of surfa
Page 51 and 52: 2.6.2 Dense Matter and the Wigner-S
Page 53 and 54: Figure 2.15: Proton number per unit
Page 55 and 56: Chapter 3Modified model† The trun
Page 57 and 58:
We choose the value ǫ = 1/3. One m
Page 59 and 60:
where∆V t = T oρ o[( ρρ o) ǫ[
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Q 1 to a small number also results
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where p = (S v ,L) and M ij = 1 ∂
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3.4 Nuclear Surface TensionCompared
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1.41.2FRTF II ω 0 , T=0 MeVFRTF II
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For the finite-range term, we use a
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Landau’s quasi-particle formula g
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The symmetry energy in nuclear matt
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Figure 4.2: The left figure shows t
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Figure 4.4: This figure shows the b
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Figure 4.5: In a neutron star, heav
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The energy exchange from the gradie
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Figure 4.8: The left panel shows th
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4.5 Astrophysical application4.5.1
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0.550.5GaussianFRTF IIFRTF I0.450.4
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neutron matter respectively. Those
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Table 5.3: Nuclear properties of si
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which aref Rc = a 0 +a 1 x+a 2 x 2
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parametrized[ht]504540EDF, Z nucFit
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Chapter 6Nuclear Equation of State
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Coulomb, and, symmetry part [24], i
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We can eliminate N s , leading to t
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pressure from pure neutron matter,
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2.52GSRsSGISIVSkaSkI1SkI2SkI3SkI5St
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HNsn, p, α,eFigure 6.7: In the Wig
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Table 6.1: Surface tension analytic
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free energy density for the alpha p
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Here s ′ = ∂s(u)/∂u.The minim
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6.5.2 Determination of Coulomb surf
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gives analytic derivatives of therm
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200180160Atomic number, Y p =0.45,
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for Fermi integral. This fitting fu
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A.1 Taylor expansion integrationThe
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thenwhere0u − i+1 = fu− i +J
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We separate the two integrals as in
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Each w i can be obtained[ae −∆/
Page 131 and 132:
By expanding e r 0/a −e −r 0/a
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M 3 2a 0.433 1p 0 (e 2 /a) √ π/3
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of find exact polynomial expression
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Appendix CPhase coexistenceBulk equ
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To summarize, we need to solve 5 eq
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D.1 Non-uniform electron density ap
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Appendix ENuclear Quantities in Non
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Table E.1: Non-relativistic Skyrme
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[20] C.J. Pethick and D.G. Ravenhal
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