Equation of state for compact stars Lecture 1 - LUTh

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$J. Phys. A: Math. Gen. 32 (1999) - LUTH - Observatoire de Paris$

Crystallization of ions At T < Tm, an infinite (unlimited, unbound) ion motion is replaced by oscillations near equilibrium positions: a crystal is formed. Internal energy at T = 0 consists of two parts, where U0 = UM + Uq, , (26) UM = −N N CM(Ze) 2 /ai. (27) Here, UM is the classical static-lattice energy, CM is the Madelung constant. Uq accounts for the zero-point quantum vibrations. CM(bcc) = 0.895929 (Fuchs 1935) bcc - body-centered-cubic Klaus Fuchs - famous for his role in the Soviet A-bomb program in 1940s CM(fcc) = 0.895874, fcc - face-centered-cubic CM(hcp) = 0.895838 fcc - hexagonal closest packing, see, e.g., Kittel 1986. The ion-sphere expression UM = −0.9 N N Z 2 e 2 /ai is sufficiently accurate for the ion liquid and solid, as long as the ions are strongly coupled (Γ ≫ 1) Pawe̷l Haensel (CAMK) EOS for compact stars Lecture 1, IHP Paris, France 22 / 54
Melting - phase transition The solid-liquid phase transition occurs at Γ = Γm, where the free energies F (Γ) of liquid and solid phases intersect. It is difficult to find Γm with high precision, because the intersection of the free energy curves is strongly affected by the thermal corrections which are small at Γ ∼ Γm (the free energy curves are nearly parallel). For instance, a 0.1% error in Fii shifts the intersection by ∆Γ ≈ 15 (Pollock & Hansen 1973)! The phase transition is 1st order but very weak. Fix the number density of ions and decrease the temperature =⇒ discontinuous changes of the ion energy Ui and the ion pressure Pi at the melting point. One can easily show that both quantities undergo discontinuous drops at Γ = Γm. The drop of the ion energy density Ui/V , associated with the crystallization to the bcc lattice at Γm = 175, is ∆Ui/V ≈ 0.76 n N kBTm, with ∆Ui/Ui ≈ 0.5%, and the drop of the ion pressure ∆Pi/Pi is approximately three times smaller. Because the ion pressure is only a small part of the total pressure P (the leading part Pe is provided by the electrons), the fractional drop of the total pressure ∆P/P is much smaller than the fractional drop of the ion pressure ∆Pi/Pi. Pawe̷l Haensel (CAMK) EOS for compact stars Lecture 1, IHP Paris, France 23 / 54
Page 1 and 2: Equation of state for compact stars
Page 3 and 4: Anticipation and prediction of neut
Page 5 and 6: EOS - an overview EOS of neutron st
Page 7 and 8: Plasma parameters Outer envelope: o
Page 9 and 10: Electrons - continued The density o
Page 11 and 12: ni = nN Ions The strength of the Co
Page 13 and 14: Ions - continued The scale-length t
Page 15 and 16: ρ − T diagram for the outer enve
Page 17 and 18: Pressure - electrons - main term (i
Page 19 and 20: Coulomb gas of ions (id)i Ideal cla
Page 21: N = N N = Ni = n N V Strongly coupl
Page 25 and 26: Quantum zero-point vibrations and m
Page 27 and 28: Other corrections relevant to elect
Page 29 and 30: Ground state crust - T = 0 Notation
Page 31 and 32: Ground state crust - T = 0 The latt
Page 33 and 34: Ground state crust - T = 0 - nuclei
Page 35 and 36: Matter should be in equilibrium wit
Page 37 and 38: Effective nucleon-nucleon interacti
Page 39 and 40: Nucleon density distribution in the
Page 41 and 42: Semi-classical - Thomas-Fermi metho
Page 43 and 44: Thomas-Fermi method - continued Neu
Page 45 and 46: Spherical unit cell radius rc, the
Page 47 and 48: Funny nuclei - ”nuclear pasta”
Page 49 and 50: Overall inner crust EOSs - SLy and
Page 51 and 52: Melting temperature Melting tempera
Page 53 and 54: Shear modulus of the crust Effectiv

Equation of state for compact stars Lecture 1 - LUTh

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