# Neutron star crust (nuclear and plasma physics)

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Neutron star crust (nuclear and plasma physics)

Neutron star crust (nuclear and plasma physics) Lecture 1 Pawel Haensel Copernicus Astronomical Center (CAMK) Warszawa, Poland [email protected] La̧dek Zdrój, February, 2008 Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 1 / 23 Neutron star crust Problems and Methods White dwarfs and neutron star envelopes (below 108 g cm−3 : plasma and atomic physics (many-particle theory, non-ideal dense plasmas) Neutron star crusts: plasma and nuclear physics (108 g cm−3 -1014 g cm−3 (strongly coupled Coulomb plasma, unstable neutron-rich nuclei, nuclear many-body problem, ... ). Basic constituents - n p e like in “terrestrial physics”. Standard reference: S.L. Shapiro, S.A. Teukolsky Black Holes, White Dwarfs, and Neutron Stars: The Physics of Compact Objects (Wiley, Extreme physical conditions: density of the matter, high neutron excess due to dense electron gas. 1983). Alas, 25-years old. New up-to-date reference: P. Haensel, A.Y. Potekhin, and D.G. Yakovlev Neutron Stars 1. Equation of state and structure (Springer, 2007) Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 2 / 23 Structure of envelope and crust Schematic structure of an envelope of a neutron star with the internal temperature ∼ 108 K. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 3 / 23 Full pressure ionization mu = 1 12 12 M ( C) ≈ m0 = 1 56 56 M ( Fe) = 1.66 × 10−24 g mean distance between electrons - re −1 4π 3 Z ρ = r ne ≈ A m0 3 e radius of the innermost electron orbit in an (A, Z) atom a0 ~2 aZ0 = , a0 = = 0.5292 × 10−8 cm 2 Z me e expect complete ionization when re . aZ0 ρci ≈ 2.7 AZ 2 g cm−3 Examples ρci (56 Fe) ∼ 105 g cm−3 Pawel Haensel (CAMK) ρci (4 He) ∼ 102 g cm−3 Neutron star crust Lect.1 La̧dek Zdrój, Poland 4 / 23 Plasma parameters Simplifying approximation: one kind of nuclei (A, Z), A = N + Z. Model - one component plasma (OCP). Number density of nuclei (=ions) ni . Electron number density ne = Zni . Outer crust: all nucleons bound in nuclei =⇒ (macroscopic) nucleon (=baryon) number density nb = A ni Inner crust nb = A0 ni where A0 ≡ A + A00 . Here A00 is the number of free (unbound) nucleons per one atomic nucleus. We have A00 ni = nn (1 − w), where w is the fraction of volume occupied by atomic nuclei (be careful - neutron skin complicates things ...) . Outer crust A0 = A, inner crust A0 > A. Mass density ρ = E/c2 , but in the envelope and crust ρ ≈ mu nb , where mu = 1.6605 × 10−24 g is the atomic mass unit. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 5 / 23 Electrons Relativity parameter for electrons xr ≡ pF ≈ 1.00884 me c ρ6 Z A0 1/3 , (1) where pF = ~kF = ~ (3π 2 ne )1/3 , 6 is the electron Fermi momentum and ρ6 ≡ ρ/10 g cm q F = c2 (me c)2 + p2F (2) −3 . The Fermi energy (3) Electron rest energy me c2 is included. Electron Fermi temperature TF = Tr (γr − 1) , (4) where Tr = me c2 /kB ≈ 5.930 × 109 K , γr = p 1 + xr 2 , (5) is the relativistic temperature unit, kB - Boltzmann constant.Non-relativistic T Tr and xr 1. Ultra-relativistic - xr 1 or T Tr . Non-degenerate T TF . Strongly degenerate - T TF . Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 6 / 23 Electrons - continued p The electron Fermi velocity vF = (∂p /∂p)pF = cxr / 1 + x2r = cxr /γr . 4π 3 a 3 e The relative strength of the electron Coulomb interaction for ultra-relativistic degenerate electron plasma ne = e2 ≈ 2 × 10−3 ae F (6) The electron plasma temperature Tpe = ~ωpe /kB ≈ 3.300 × 108 xr p βr K . (7) where βr = vF /c and electron plasma frequency is given by ωpe = 4πe2 ne /m∗e 1/2 (8) Here, m∗e ≡ F /c2 = me γr is the effective dynamical mass of an electron at the Fermi surface. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 7 / 23 Ions - 1 The strength of the Coulomb interaction of ions is characterized by the Coulomb coupling parameter, Γ = (Ze)2 /(ai kB T ), (9) where ai = (3ni /4π)1/3 (10) is the ion-sphere radius. At sufficiently high temperatures, the ions form a classical Boltzmann gas. With decreasing T , the gas gradually, without a phase transition, becomes a strongly coupled Coulomb liquid, and then (with a phase transition) a Coulomb crystal. The gas and liquid constitute the neutron star ocean, the crystal is formed under the ocean. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 8 / 23 Ions - 2 The ion-sphere radius ai - ion-sphere model for strongly coupled Coulomb systems (T . Tl ) (liquid - solid). Nearly uniform electron background. In this model the matter is considered as an ensemble of ion spheres filled by the uniform electron background. Ion spheres are electrically neutral, ≈ non-interacting between themselves. 1 ni = 43 πa3i Ecell = WN (A, Z) + WL + We WL = Wee + Wei Wee = 3 Z 2 e2 5 ai WL = − Pawel Haensel (CAMK) 2 2 Wei = − 32 Zaie We = Ee (ne ) ni 9 Z 2 e2 10 ai Neutron star crust Lect.1 La̧dek Zdrój, Poland 9 / 23 Ions - 3 The crystallization of a Coulomb plasma occurs at a well defined temperature Tm = ρ 1/3 175 Z 2 e2 6 ≈ 1.3 × 105 Z 2 K, ai kB Γm A0 Γm (11) where Γm is the ”melting value” of Γ. For a classical OCP, Γm ≈ 175 . At low T the quantum effects on ion motion become important. They are especially pronounced at T Tpi , where Tpi ≡ ~ωpi ≈ 7.832 × 106 kB ρ6 Z 2 A0 A 1/2 K (12) is the plasma temperature of ions determined by the ion plasma frequency ωpi = 4πe2 ni Z 2 /mi 1/2 . (13) Notice: Tpi Tei . If Tpi Tm , huge zero-point ion vibrations suppress crystallization and reduce the melting temperature. Under these conditions, the actual melting temperature decreases with growing ρ and drops to zero at a certain ρ = ρm : not relevant for crust Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 10 / 23 ρ − T diagram for the outer envelope and crust Density-temperature diagram for the outer envelope composed of carbon (left) or iron (right). Electron Fermi temperature (TF ), electron and ion plasma temperatures (Tpe and Tpi ), temperature of the gradual gas-liquid transition (Tl ), and the temperature of the sharp liquid-solid phase transition (Tm ). Shaded: typical temperatures in the outer envelopes of middle-aged cooling neutron stars (age ∼ 104 − 106 ). Corner bounded by the dotted line: strong electron non-uniformity or bound-state formation (atoms!) in the iron plasma. Difficult region! Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 11 / 23 Pressure - WS cell, rigid e-background P = PL + P e , 1 PL = n i W L < 0 3 for ρ6 1 : Pawel Haensel (CAMK) Pe , PL ∝ ρ4/3 Neutron star crust Lect.1 La̧dek Zdrój, Poland 12 / 23 Crystal phase T < Tm - an infinite ion motion replaced by oscillations near equilibrium positions: a crystal is formed. Internal energy at T = 0 consists of two parts, U0 = UM + Uq , (14) UM = −Ni CM (Ze)2 /ai . (15) where UM - classical static-lattice energy, CM - Madelung constant. Uq accounts for the zero-point quantum vibrations. CM (bcc) = 0.895929 , CM (fcc) = 0.895874 , CM (hcp) = 0.895838 (Fuchs 1935) =⇒ ion-sphere model: UM = −0.9 Ni Z 2 e2 /ai is sufficiently accurate for the ion liquid and solid, as long as the ions are strongly coupled (Γ 1) Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 13 / 23 Melting - phase transition The solid-liquid phase transition occurs at Γ = Γm , where the free energies F (Γ) of liquid and solid phases intersect. Difficult to find Γm with high precision: intersection of F -s strongly affected by the thermal corrections which are small at Γ ∼ Γm (F - curves are nearly parallel). 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 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 ni kB Tm , 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 . Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 14 / 23 Latent heat at freezing The latent heat released Qlatent = Um,liq − Um,solid ≈ ∆Ui ≈ 0.76 Ni kB Tm , where and Um,liq and Um,solid are the corresponding values of the internal energy at T = Tm . Qlatent ≈ 5 × 10−3 Ui . Remember that Ui itself is a tiny fraction of the total internal energy U (mainly determined by degenerate electrons). Nevertheless, the absolute value of Qlatent (≈ 0.76kB T per ion) is a substantial fraction of thermal thermal energy of the ions. Note: dense matter may form a supercooled liquid and freeze at T < Tm . Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 15 / 23 Electrostatic corrections to electron presure (e) Approximate Pe = Pid (ideal fully degenerate electron gas), add PL ≈ −0.3 ni Z 2 e2 /ai . This gives Z 2 e2 Pr 2 2 x − 1 γ + ln(x + γ ) − 0.3 n . P = x r r r r i r 8π 2 3 ai where (16) Pr = me c2 /(~/me c)3 = 1.4218 × 1025 dyn cm−2 This EOS can be used in the outer envelope of NS not too close to the surface. It is also widely used in white dwarf cores. It is temperature independent and applies to any composition of the matter. In the ultrarelativistic electron gas (xr 1) both Pe and PL pressures have the same (polytropic) density dependence, Pe ∝ PL ∝ xr 4 . Therefore, their ratio is density independent being determined only by the ion charge number Z, PL 6 =− Pe 5 4 9π 1/3 Z 2/3 e2 ≈ −0.0046 Z 2/3 , ~c (17) (e) which gives PL /Pid ≈ −0.04 for the matter composed of iron. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 16 / 23 Ground state crust - T = 0 - 1 Notation: enthalpy per nucleon = h; energy per nucleon=E; energy density - E. Neglect envelope of NS with ρ < 108 g cm−3 (it has mass < 10−8 M ) - it could have been accreted, and the T -effects there might be significant. Consider crust at T = 0 and P = 0, i.e., h = E = E/nb . In this case the minimum energy per nucleon (ground state) is reached for a body-centered-cubic (bcc) lattice of 56 Fe, and is E(56 Fe) = 930.4 MeV. It corresponds to ρ = 7.86 g cm−3 and nb = 4.73 × 1024 cm−3 = 4.73 × 10−15 fm−3 . Remark56 Fe is not the most tightly bound free atomic nucleus. The maximum binding energy per nucleon b ≡ [(A − Z)mn c2 + Zmp c2 − M (A, Z)c2 ]/A in a nucleus with the ground-state mass M (A, Z) is reached for 62 Ni, b(62 Ni) = 8.7945 MeV, to be compared with b(56 Fe) = 8.7902 MeV. Let us notice that b(58 Fe) = 8.7921 MeV is also higher than b(56 Fe). Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 17 / 23 Ground state crust - T = 0 - 2 Ion-sphere model used - Γ 1. Terminology: ion sphere ≡ (spherical) unit cell, (spherical) Wigner-Seitz cell Charge neutrality of the unit cell, under pressure P , ne = Zni , P = Pe (ne , Z) + PL (ni , Z) , (18) where Pe is the electron pressure and PL is the “lattice” contribution (also called the electrostatic correction) resulting from the Coulomb interactions. Spherical ”unit cell” per one nucleus in a OCP. At T = 0 enthalpy of the cell=Gibbs free energy (G = H − T S) of the cell =Gcell . Gcell (A, Z) = WN (A, Z) + WL (Z, nN ) + [Ee (ne , Z) + P ]/ni , (19) where WN is the energy of the nucleus (including rest energy of nucleons), WL is the lattice (Coulomb) energy per cell, and Ee is the mean electron energy density. Neglecting quantum and thermal corrections and the nonuniformity of the electron gas, we have WL = −CM Z 2 e2 /rc , Pawel Haensel (CAMK) Neutron star crust Lect.1 CM ≈ 0.9. (20) La̧dek Zdrój, Poland 18 / 23 Ground state crust - T = 0 - 3 The lattice (Coulomb) contribution PL = 13 WL ni . The Gibbs free energy per nucleon h = g = Gcell /A is just the baryon (=nucleon) chemical potential µb (A, Z) for a given nuclide. To determine the ground state at a given P , one has to minimize µb (A, Z) with respect to A and Z. To a very good approximation, a density jump, at ground-state values (A, Z) change into (A0 , Z 0 ), is given by ∆ρ ∆nb Z A0 ≈ ≈ −1 . ρ nb A Z0 (21) This equation follows from the continuity of the pressure P ≈ Pe . A sharp discontinuity in ρ and nb is a consequence of the assumed one-component plasma model. Problem Derive formula for ∆ρ/ρ, Eq. (21) Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 19 / 23 Neutronization - simple argument g = Gcell /A T =0 Pawel Haensel (CAMK) g = h , µe = F (ne ) Neutron star crust Lect.1 La̧dek Zdrój, Poland 20 / 23 A model of ground state crust - 1 Use experimental masses if they are available. Because of pairing effect (nucleons in nuclei are superfluid!) only even-even nuclei are relevant for the ground-state problem. For the remaining isotopes, up to the last one stable with respect to the emission of a neutron pair, use theoretical masses. Example 1 (Haensel & Pichon 1994) The equilibrium nuclides present in the ”cold catalyzed matter” are listed in Table. In the fifth column one finds the maximum density ρmax at which a given nuclide is present. The sixth column gives the electron chemical potential µe at ρ = ρmax . The transition to the next nuclide has a character of a first-order phase transition. The corresponding fractional density jump ∆ρ/ρ is given in the last column. The last row above the horizontal line, which divides the table into two parts, corresponds to the maximum density, at which the ground state of dense matter contains a nucleus with mass measured in laboratory. The last row of Table corresponds to the neutron drip point which is determined theoretically. Pawel Haensel (CAMK) Neutron star crust Lect.1 La̧dek Zdrój, Poland 21 / 23 Ground state crust - T = 0 - nuclei - 1 Table: Nuclei in the ground state of cold dense matter (after Haensel & Pichon 1994), with slight modification. Upper part is obtained with experimentally measured nuclear masses. Lower part: from mass formula of Möller. The last line corresponds to the neutron drip point. element Z N Z/A ρmax (g cm−3 ) 56 Fe 62 Ni 64 Ni 66 Ni 86 Kr 84 Se 82 Ge 80 Zn 26 28 28 28 36 34 32 30 30 34 36 38 50 50 50 50 0.4643 0.4516 0.4375 0.4242 0.4186 0.4048 0.3902 0.3750 7.96 2.71 1.30 1.48 3.12 1.10 2.80 5.44 78 Ni 126 Ru 124 Mo 122 Zr 120 Sr 118 Kr 28 44 42 40 38 36 50 82 82 82 82 82 0.3590 0.3492 0.3387 0.3279 0.3167 0.3051 9.64 × 1010 1.29 × 1011 1.88 × 1011 2.67 × 1011 3.79 × 1011 (4.32 × 1011 ) Pawel Haensel (CAMK) × × × × × × × × 106 108 109 109 109 1010 1010 1010 Neutron star crust Lect.1 µe (MeV) ∆ρ/ρ (%) 0.95 2.61 4.31 4.45 5.66 8.49 11.4 14.1 2.9 3.1 3.1 2.0 3.3 3.6 3.9 4.3 16.8 18.3 20.6 22.9 25.4 (26.2 ) 4.0 3.0 3.2 3.4 3.6 La̧dek Zdrój, Poland 22 / 23 Ground state crust - T = 0 - nuclei - 2 Example 2 (Rüster, Hempel, & Schaffner-Bielich 2006) Table: Sequence of nuclei in the ground state of the outer crust of neutron star calculated by Rüster, Hempel, & Schaffner-Bielich (2006) using experimental nuclear data (upper part), and the theoretical mass table of the Skyrme model BSk8 (lower part). Table from Chamel & Haensel 2008 ρmax [g/cm3 ] 8.02 × 106 2.71 × 108 1.33 × 109 1.50 × 109 3.09 × 109 1.06 × 1010 2.79 × 1010 6.07 × 1010 8.46 × 1010 9.67 × 1010 1.47 × 1011 2.11 × 1011 2.89 × 1011 3.97 × 1011 4.27 × 1011 Pawel Haensel (CAMK) Element 56 Fe 62 Ni 64 Ni 66 Ni 86 Kr 84 Se 82 Ge 80 Zn 82 Zn 128 Pd 126 Ru 124 Mo 122 Zr 120 Sr 118 Kr Z 26 28 28 28 36 34 32 30 30 46 44 42 40 38 36 Neutron star crust Lect.1 N 30 34 36 38 50 50 50 50 52 82 82 82 82 82 82 ai [fm] 1404.05 449.48 266.97 259.26 222.66 146.56 105.23 80.58 72.77 80.77 69.81 61.71 55.22 49.37 47.92 La̧dek Zdrój, Poland 23 / 23 Neutron star crust (nuclear and plasma physics) Lecture 2 Pawel Haensel Copernicus Astronomical Center (CAMK) Warszawa, Poland [email protected] La̧dek Zdrój, February, 2008 Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 1 / 10 Approximate calculation of neutron drip point - 1 ”Simplified nuclear mass formula”. Neglecting Coulomb, surface and all other finite-size terms and keeping only quadratic term in δ = (N − Z)/A. Energy per nucleon in an atomic nucleus (A, Z) (subtracting the rest energy and neglecting neutron-proton mass difference) EN (A, Z)/A ' E0 + S0 δ 2 , E0 = energy per nucleon in symmetric nuclear matter and S0 = symmetry energy, both calculated at saturation density. The neutron and proton chemical potentials (without rest energy contribution) µ0n = (∂EN /∂N )Z = E0 + (2δ + δ 2 ) S0 , µ0p = (∂EN /∂Z)N = E0 + (−2δ + δ 2 ) S0 . Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland (1) 2 / 10 Approximate calculation of neutron drip point - 2 δ corresponding to ρND - calculated from µ0n = 0, p δND = 1 − (E0 /S0 ) − 1 . (2) Experimental values E0 = −16 MeV, S0 = 32 MeV =⇒ δND = 0.225. Beta equilibrium condition: µn = µp + µe , therefore µe = µn − µp ' 4S0 δ . (3) Using µe = 0.516 (ρ6 Z/A)1/3 MeV (valid for ρ6 1), we get bulk approximation : ρND ≈ 2.2 × 1011 g cm−3 , (4) quite close to the refined theoretical value. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 3 / 10 Inner crust - bulk approximation - 1 Coulomb and surface effects, etc. - neglected. At a given (macroscopic, mean) nucleon density nb , nucleons in “i” and “o” coexisting fluids. Densities: nn,i , np,i , nn,o . np,o 6= 0 only in the proton drip regime (nb > nPD ). Equilibrium “i” - “o” ensured by (strong) NN interaction. Implies equality of the chemical potentials of nucleons and the equality of the nucleon pressures, µn,i = µn,o , µp,i = µp,o , PN,i = PN,o , (5) N - nucleon contribution. Condition on µp applies only at nb > nPD . At a given nb , Eqs. (5) =⇒ nn,i , np,i , nn,o , and np,o . Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 4 / 10 Inner crust - bulk approximation - 2 w - volume fraction occupied by the “i” phase. At a given mean nucleon number density nb = w ni + (1 − w) no , the total energy density is E = w EN,i + (1 − w) EN,o + Ee , (6) where Ee is the electron energy density. Beta equilibrium implies: µn = µp + µe . (7) ne = w np,i + (1 − w) np,o . (8) Overall charge neutrality Eqs. (5)–(8) determine the equilibrium state at a given nb . Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 5 / 10 Inner crust - bulk approximation - 3 The two-fluid state: stable at nb at which the energy per nucleon is lower than in the uniform npe matter. Stability lost at n2↔1 . While nb −→ n2↔1 from lower density side, w → 1. Therefore 2 ↔ 1 transition is continuous (no density jump). Surface and Coulomb contributions increase E =⇒ the crust-core occurs at a baryon number density ncc < n2↔1 upper bound. Problem: Propose a procedure for numerical calculation of bulk equilibrium approximation of dense matter from a given function E(nn , np ) at densities 108 − 3 × 1014 g cm−3 Example: For the SLy4 effective NN interaction: nPD = 0.085 fm−3 , n2↔1 = 0.091 fm−3 , ncc = 0.078 fm−3 . w(ncc ) ≈ 0.3 (agreement with Compressible Liquid Drop Model - CLDM) Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 6 / 10 Hartree-Fock calculation of ground state crust Put A = N + Z nucleons and Z electrons in a box of volume Vcell (=unit cell) and calculate ground state of the system at fixed nb = A/Vcell . The A − body problem - solved using nuclear many-body theory. A ∼ 50 − 1000. ”Effective nuclear” Hamiltonian eff ĤN = A X j=1 t̂j + X eff v̂jk , (9) k<j≤A eff where t̂j is the kinetic energy operator of j-th nucleon, while v̂jk is an operator of an effective two-body interaction between a jk nucleon pair. eff ĤN has to reproduce – as accurate as possible within the Hartree-Fock approximation – relevant properties of the ground state of a many-nucleon system, particularly, the ground state energy E0 . This last condition can be eff written as hΦ0 |ĤN |Φ0 i ' hΨ0 |ĤN |Ψ0 i , where Φ0 and Ψ0 are, respectively, the Hartree-Fock and exact wave functions, and ĤN is the exact nuclear Hamiltonian. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 7 / 10 Effective nucleon-nucleon interaction Most popular: Skyrme interaction The basic assumption which justifies a Skyrme-type effective NN interaction (Skyrme 1956) is that its range is small as compared with the internucleon distances. This means, that in momentum ˆ eff (k, k0 ) can be representation the effective NN interaction ve approximated by a momentum independent term plus terms quadratic in the initial and final relative momenta of an interacting nucleon pair, k and k0 , with the appropriate spin dependence. More in Ring & Schuck 1980 eff Parametrizing ĤN Numerical values of the parameters of effective interaction are to be determined from fitting masses of laboratory nuclei in ground states and low-lying excited states. The complete Hamiltonian of the unit eff cell is Ĥcell = ĤNeff + VCoul + Ĥe , where VCoul describes Coulomb interaction between protons and electrons, and Ĥe corresponds to a uniform electron gas. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 8 / 10 Nucleon wave functions - 1 Box=unit cell=ion sphere. The Hartree-Fock approximation for the many-body nucleon wave function is h i (n) ΦN Z = CN Z det ϕ(p) (ξ ) det ϕ (ζ ) , k l αi βj (n) (10) (p) where ϕβj (ζl ) and ϕαi (ξk ) are single-particle wave functions (orbitals) for neutrons (j, l = 1, . . . , N ) and protons (i, k = 1, . . . , Z), respectively, and CN Z is a normalization constant. The space and spin coordinates of a k-th proton are represented by ξk ; ζl is the same for an l-th neutron; {αi } and {βj } are sets of quantum numbers of occupied single-particle states for protons and neutrons, respectively. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 9 / 10 Nucleon wave functions - 2 The Hartree-Fock equations for ϕ(p) and ϕ(n) were derived by minimizing the Hartree-Fock energy functional at a fixed volume Vc of the unit cell, h i (n) (p) eff Ecell ϕα , ϕβ = hΦN Z Φe |Ĥcell | ΦN Z Φe i = minimum , (11) where Φe is the plane-wave Slater determinant for an ultra-relativistic electron gas of constant density ne = Z/Vc . The minimization performed at fixed average neutron and proton densities, nn = N/Vc , np = Z/Vc = ni Z. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 10 / 10 Determining the ground state of inner crust unit cell = ion sphere Ground state at (N, Z), β-equilibrium (n) (p) Once the H-F orbitals ϕβ and ϕα are determined, one finds the minimum (ground state) value of Ecell (N, Z), filling the lowest N neutron and Z proton states. Then, the absolute ground state configuration is found by minimizing Ecell (N, Z) at a fixed A = N + Z. Neutron drip in H-F αZ and βN correspond to “Fermi levels” for protons and neutrons, respectively. In terms of the single-nucleon orbitals, the neutron drip point corresponds to the threshold density at which the neutron Fermi (n) level becomes unbound, i.e., ϕβN extends over the entire unit cell. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 11 / 10 Nucleon density distribution in the inner crust - HF Density profiles of neutron and protons, at several average densities ρ, along a line joining the centers of two adjacent unit cells. Based on Fig. 3 of Negele & Vautherin (1973). Remains to be added: nucleon pairing =⇒ HF+Bogoliubov ≡ HFB Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 12 / 10 Semi-classical - Thomas-Fermi method - 1 ∼ 103 nucleons in unit cell - full quantum description is ”too detailed” for getting ground state energy - semiclassical approximation is quite precise In HF method energy was a functional of nucleon wave functions. In TF (more refided: Extended TF, ...) the energy is a functional of nucleon densities Z EN = EN [nn (r), np (r), ∇nn (r), ∇np (r)] + mn c2 nn (r) cell +mp c2 np (r) d3 r . (12) Z 1 ECoul = e [np (r) − ne ] φ(r) d3 r , (13) 2 cell where φ(r) is the electrostatic potential. Used in condensed matter, and recently adapted to nuclear structure - EDFT - Energy Density Functional Theory based on Kohn-Sham theorem. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 13 / 10 Thomas-Fermi method - 2 φ(r) should be calculated from the Poisson equation, ∇2 φ(r) = 4πe [np (r) − ne ] . (14) To determine the ground state at a given nb , one has to find nn (r) and np (r), which minimize Ecell /Vc under the constraints Z Z 3 [nn (r) + np (r)] d r , [np (r) − ne ] d3 r = 0 . Vc nb = cell cell (15) The problem is simplified assuming spherical symmetry; in this case the unit cell is approximated by a sphere of the radius rc = (3Vc /4π)1/3 . The boundary conditions are such that far from the nucleus surface the nucleon densities are uniform. This requires the nuclear radius to be significantly smaller than rc . Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 14 / 10 Thomas-Fermi method - continued Neutron and proton density profiles at two average mass densities along a line joining the centers of adjacent unit cells. Based on Fig. 5 of Cheng et al. (1997), with the permission of the authors. No wiggles in profiles - because no quantum interference between nucleon wave functions. Pawel Haensel (CAMK) Neutron star crust Lect.2 La̧dek Zdrój, Poland 15 / 10 Neutron star crust (nuclear and plasma physics) Lecture 3 Pawel Haensel Copernicus Astronomical Center (CAMK) Warszawa, Poland [email protected] La̧dek Zdrój, February, 2008 Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 1 / 16 H-F uniform nuclear matter - Hamiltonian - 1 Put N neutrons, Z protons (A = N + Z nucleons) in a large box of volume V and calculate ground state of the system at fixed nn = N/V , np = Z/V . Coulomb interactions switched-off. Assume that nuclear matter is uniform =⇒ translational invariance. Approximation: mn ≈ mp = m Effective nuclear Hamiltonian ĤNeff = A X j=1 t̂j + X eff v̂jk , (1) k<j≤A eff where t̂j is the kinetic energy operator of j-th nucleon, while v̂jk is an operator of an effective two-body interaction between a jk nucleon pair. Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 2 / 16 H-F - uniform nuclear matter - Hamiltonian - 2 The effective nuclear Hamiltonian ĤNeff has to reproduce – within the Hartree-Fock approximation – relevant properties of the ground state of a many-nucleon system: the ground state energy E0 , etc. This last condition can be written as hΦ0 |ĤNeff |Φ0 i ≈ hΨ0 |ĤN |Ψ0 i (2) where Φ0 and Ψ0 are, respectively, the Hartree-Fock and exact wave functions, and ĤN is the exact nuclear Hamiltonian. Single-particle wave-functions - 0 Nuclear matter is uniform and spin non-polarised. Single-particle state labels α, β = p , sz . Coordinate-space representation: r ), ϕ(p) r) , ϕ(n) α (r β (r (3) Isospin formalism is not used - we distinguish n and p. Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 3 / 16 Single-particle wave-functions Eigenfunctions of p̂p and t̂ Coordinate representation. Spin sz - omitted. ∇, p̂p = −i~∇ r ) = p ϕ(n) r ) =⇒ p̂pϕ(n) α (r α (r r ) = Ceipp·rr /~ . (4) ϕ(n) α (r It is also eigenfuction of t̂ = −~2∇ 2 /2m: r) = t̂ϕ(n) α (r p2 (n) ϕ (rr ) . 2m α (5) (n) Generalization: ϕα (rr ) will be also the eigenfunctions of ĥ = t̂ + Ûn (p̂p2 ), with eigenvalue p2 /2m + Un (pp2 ). Normalization in periodicity box V = L3 Allowed values of px = pjx are j × 2π~/L, j = 1, 2, 3, . . ., same for py √ (n) and pz . ϕα (rr ) are orthogonal and normalized to 1 with C = 1/ V . Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 4 / 16 Miscellaneous Replacing P p by R d3 p Summation of allowed p can be replaced by integration over p , because for L −→ ∞ Z X dpx F (px ) = F (pjx ) . 2π~/L j (6) It is sufficient that V is macroscopic (but much weaker conditions can be fine see Thomas-Fermi). In 3-D we get Z X d3 p L −→ ∞ =⇒ F (pp) = V F (pp) . (7) (2π~)3 p (n) fα (p) , fβ Single-particle wavefunctions, occupation numbers Z ? 1 ipp·rr /~ √ r r )ϕ(n) r ) = δm,n . ϕ(n) (r ) = e , d3 rϕ(n) α αn (r αm (r V P (n) P (p) = 0, 1 α fα = N , β fβ = Z . Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland (8) 5 / 16 Single-particle energies Do not depend on sz - two-fold degenerate. tα = hα|t̂|αi = p2 . 2m (9) HF single-particle energies: X (n) fα0 [hαα0 |v nn |αα0 i − hαα0 |v nn |α0 αi] enα = tα + 0 + epβ = tβ + α X β X (p) fβ hαβ|v np |αβi , (p) fβ 0 [hββ 0 |v pp |ββ 0 i − hββ 0 |v pp |β 0 βi] β0 + X fα(p) hβα|v np |βαi . (10) α Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 6 / 16 Skyrme interaction - matrix elements ∇1 v̂12 - in spatially uniform nuclear matter. k 1 = p 1 /~ , k̂k 1 = −i∇ hα1 α2 |v̂|α1 α2 i = V −2 Z 3 d r1 Z 3 d r2 e k 1r 1 +k k 2r 2 ) −i(k v̂(rr 1 − r 2 )ei(kk 1r 1 +kk 2r 2 ) Zero-range A0 δ(rr 1 − r 2 ). Matrix elements of δ(rr 1 − r 2 ) Z −2 V d3 r1 = V −1 . (11) (12) Momentum-dependent gradient-terms: A1 δ(rr 1 − r 2 )(k̂k 1 − k̂k 2 )2 , . . . matrix element ∝ (kk 1 − k 2 )2 /V , . . . Helpful: r 1 , r 2 −→ R = (rr 1 + r 2 )/2 , r = r 1 − r 2 , k 1 , k 2 −→ k = (kk 1 − k 2 )/2 , P = k 1 + k 2 , Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 7 / 16 eα , E for Skyrme e(n) α1 = X p21 + hα1 α2 |v̂|α1 α2 i + · · · = 2m α 2 p2 2 X (n) = 1 + f (C1 + C2 p22 + C3 p21 . . .) + . . . 2m V p p 2 (13) 2 (n) Ground state occupation numbers: fp 2 = Θ(p2 − pFn ) P(n)occ 3 /(3π 2 ), same for protons. 2 p = N =⇒ nn = kFn (n) 2/3 eα contains terms with factors: nn , nn , np , nn p2 , ... Ground state energy density - kinetic part: (n)occ ( p)occ 2 2 X 1 X p = ~ (τn + τp ) Ekin = 2 + V 2m 2m p p τn = Pawel Haensel (CAMK) (14) 3 3 2 2 nn kFn , τp = np kFp , 5 5 Neutron star crust Lect.3 (15) La̧dek Zdrój, Poland 8 / 16 HF - EN,int Spin factors omitted for simplicity (only one spin state included): hΦ0 | (n)occ (n)occ A 1 X X 1 X nn |pp1p 2 ia hpp1p 2 |v̂12 v̂ij |Φ0 i = 2 i6=j=1 2 p p 1 2 (p)occ (p)occ + 1 X X pp hpp1p 2 |v̂12 |pp1p 2 ia 2 p p 1 2 (p)occ (n)occ + X X p1 pn hpp1p2 |v̂12 |pp1p2 i (16) p2 |pp1p 2 ia ≡ |pp1p 2 i − |pp2p 1 i Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 9 / 16 E(nn , nn ) for Skyrme - 1 Final form E(nn , np ) = ~2 (τn + τp ) 2m 1 + t0 (2 + x0 )(nn + np )2 − (2x0 + 1)(n2n + n2p ) 4 1 + t3 (nn + np )γ (2 + x3 )(nn + np )2 − (2x3 + 1)(n2n + n2p ) 24 1 + [t1 (2 + x1 ) + t2 (2 + x2 )] (τn + τp )(nn + np ) 8 1 + [t2 (2x2 + 1) − t1 (2x1 + 1)] (τn nn + τp np ) . 8 Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland (17) 10 / 16 E(nn , np ) for Skyrme - 2 Parameter values of the Skyrme forces. SLy4 used in these lectures. force SLy4 SLy7 SkM∗ Sk10 t0 (MeV fm3 ) t1 (MeV fm5 ) t2 (MeV fm5 ) t3 (MeV fm3+3γ ) γ x0 x1 x2 x3 W0 (MeV fm5 ) -2488.91 486.82 -546.39 13777.0 -2482.41 457.97 -419.85 13677.0 -2645.0 410.0 -135.0 15595.0 -1057.3 235.9 -100.00 14463.5 1 0.2885 0 0 0.2257 120.0 1 6 1 6 1 6 0.834 -0.344 -1.000 1.354 123.0 0.846 -0.511 -1.000 1.391 126.0 0.09 0 -1.000 0 130.0 Applications to bulk equilibrium and surface energy in neutron star crust: in Douchin, Haensel, & Meyer (2000). Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 11 / 16 Equation of state for bulk matter - general Thermodynamics at T = 0. Chemical potential without rest energies. ∂E ∂E µn = , µp = . (18) ∂nn np ∂np nn Pressure of nucleons P = −E + nn µn + np µp . (19) Mass-energy density and baryon (number) density: ρ = nn mn + np mp + E/c2 , Pawel Haensel (CAMK) Neutron star crust Lect.3 nb = nn + np . La̧dek Zdrój, Poland (20) 12 / 16 Equation of state of symmetric nuclear matter m = (mn + mp )/2 ~2 (~c)2 = = 20 .276 MeV fm2 2m 2mc2 energy per nucleon (without rest energy) = E/nb SLy4 Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 13 / 16 Saturation parameters of nuclear matter Self-bound symmetric nuclear matter. P = 0 at nb = n0 . nn = (1 + δ)nb /2, np = (1 − δ)nb /2. Nuclear matter parameters at saturation point SkI0 SkM∗ SLy4 SLy7 n0 [fm−3 ] 0.1553 0.1603 0.1595 0.1581 E0 [MeV] -15.99 -15.77 -15.97 -15.89 K0 [MeV] 370.4 216.6 229.9 229.7 S0 [MeV] 32.64 30.03 32.00 31.99 Pawel Haensel (CAMK) Charge symmetry of nuclear forces=⇒ E(nb , δ) = E(nb , −δ) Neglecting higher order terms: 2 nb − n0 , n0 (21) K0 - incompressibility, S0 - symmetry energy. K0 E(nb , δ) ' E0 +S0 δ + 9 Neutron star crust Lect.3 2 La̧dek Zdrój, Poland 14 / 16 Energy-density for different δ E/nb vs. nb δ = (nn − np )/nb . Pawel Haensel (CAMK) SLy4 δ = 1 − 2xp , xp = (1 − δ)/2 . Neutron star crust Lect.3 La̧dek Zdrój, Poland 15 / 16 Further study - Problems Notation r = r 1 − r 2 , ∇ = ∂/∂rr nb = 3 2kF . 3π 2 Problem Consider a model of nuclear matter with effective interaction v̂(rr ) = [a + b nb ] δ(rr ) . (22) Derive formula for energy per nucleon as a function of kF . Show that nuclear matter can have a stable bound state (saturation). Find values of a and b that yield saturation at E0 = −16 MeV and at n0 = 0.16 fm−3 . Problem Consider a model of nuclear matter with effective interaction ← − → − ∇ ∇ v̂(rr ) = a0 δ(rr ) − a1 δ(rr ) . i i (23) Derive formula for energy per nucleon as a function of kF . Show that nuclear matter can have a stable bound state (saturation). Find values of a0 and a1 that yield saturation at E0 = −16 MeV and at n0 = 0.16 fm−3 . Problem Using E(nn , np ) for SLy4 force, calculate P (nb ) for symmetric nuclear matter and check it with Table. Calculate n0 , E0 , S0 and K0 for SLy4. Pawel Haensel (CAMK) Neutron star crust Lect.3 La̧dek Zdrój, Poland 16 / 16 Neutron star crust (nuclear and plasma physics) Lecture 4 Pawel Haensel Copernicus Astronomical Center (CAMK) Warszawa, Poland [email protected] La̧dek Zdrój, February, 2008 Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 1 / 15 Compressible Liquid Drop Model - basics The model is classical (non-quantum) par excellence. Nucleon subsystems: bulk nuclear matter inside nuclei - (i), neutron gas outside nuclei (o), nucleons in the transition layer between (i) and (o) - the surface (s) or (surf). Ecell = EN,bulk + EN,surf + ECoul + Ee , (1) EN,bulk = Vc [w EN,i + (1 − w) EN,o ] . (2) EN,surf and ECoul depend on sizes and shapes of nuclear structures. Input quantities calculated using many-body microscopic theories. Unknown parameters: nuclear radii - rp , rn , density of neutron and protons inside nuclei nn,i , np,i , density of neutrons outside nuclei - nn,o , surface tension at the nuclear surface - σs , number of neutrons ”adsorbed at the nuclear surface” - Ns , thickness of ”neutron skin” - sn , radius of the unit (ion) cell- rc , . . . . Equilibrium conditions: mechanical and ”chemical” equilibrium, charge neutrality of the unit cell =⇒ structure of the ground state of the inner crust at a given nb . Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 2 / 15 Surface tension - plane ”i-o” interface - 1 V = L3 . Fixed Nn ,Np , L. Variables: µn , µp (adjusted to get correct Nn ,Np ). Then position of the interface (z0 ), densities nn (z), np (z) and pressure P will result from minimizing Ω(µn , µp ) = E − Nn µn − Np µp . dV = L2 dz Z Ω= dV ω(z) = Ωbulk + Ωsurf ω(z) = E − µn nn (z) − µp np (z) Ωbulk = −P L3 −P = Eo − nn,o = Ei − nn,i µn,i − np,i µp,i Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 3 / 15 Surface tension - plane ”i-o” interface - 2 Ω = min. ⇔ ωtot = ωbulk + ωsurf = Ω/L2 = min. ωbulk = −P L does not depend on surface properties ωtot = min. ⇔ ωsurf = min Z ωtot 0 [ω(z) − Ei + nn,i µn,i + np,i µp,i ] dz = −∞ +∞ Z [ω(z) − Eo + nn,o µn,o ] dz (3) 0 ωref (z) = Θ(−z) [Ei − nn,i µn,i − np,i µp,i ] + Θ(z) [Eo − nn,o µn,o ] Z (4) +∞ (ω − ωref ) dz . ωsurf = (5) −∞ Plane interface: ωref = −P is z0 -independent =⇒ ωsurf does not depend on z0 . Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 4 / 15 Plane interface - surface layer - 1 Example of parametrization of densities: nn (z) = nn,o + [nn,i − nn,o ] f (z; an , bn ) , np (z) = np,i f (z; ap , 0) , h i z−b f (z; a, b) = 1/ 1 + e a . We have to find ωsurf (ap , bp , an , bn ) = min. The value at minimum is equal σs . It is a function of only one variable, e.g. xi or nn,o . Definition z0 just within “surface layer”. Only zn − zp has physical meaning. Z +∞ [f (z; ap ) − Θ(zp − z)] dz = 0 , −∞ Z +∞ [nn (z; an , bn ) − nn,i Θ(zn − z) − nn,o Θ(z − zn )] dz = 0 . (6) −∞ Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 5 / 15 Plane interface - surface layer - 2 Neutron skin, surface density (=column density) of adsorbed neutrons Neutron skin thickness sn = zn − zp . Surface density of adsorbed neutrons νn = sn (nn,i − nn,o ) Number of adsorbed neutrons - plane surface of area A: Ns = Aνs . Bookkeeping of neutrons and protons - L3 box + plane surface Surface at z = zp defines Vi ≡ L2 Lp : Z +L 2 np (z)dz = L2 Lp np,i , Np = L −L Z +L Nn = L 2 nn (z)dz = Nn,i + Ns + Nn,o , Nn,i = Vi nn,i , −L 3 Nn,o = (L − Vi )nn,o , Ns = L2 sn (nn,i − nn,o ) . Pawel Haensel (CAMK) Neutron star crust Lect.4 (7) La̧dek Zdrój, Poland 6 / 15 Relation between σs and es Separation of surface contribution EN = Ebulk + L2 es = Ωbulk + L2 σs +(Nn,o + Nn,i )µn + Ns µn + Np µp , (8) es,zp = σs + sn (nn,i − nn,o ) µn . (9) Third surface of reference: Z +∞ [nn (z) + np (z) − Θ(zN − z)(nn,i + np,i ) − Θ(z − zN )nn,o ] dz = 0 . −∞ Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 7 / 15 σs and es - results for SLy [Douchin,Haensel,Meyer 2000] Crossing of σs and es at neutron drip where µn = 0. Before neutron drip, nn,o = 0 and µn < 0 =⇒ σs < es . For small δi < 0.3 both σs , es ∝ δi2 . Strong dependence of es on zref . Monotonic decrease of σs . For small δi < 0.3 sn ∝ δi . Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 8 / 15 Coulomb energy - simplest approx. - spherical nuclei - 1 Constant charge densities, sharp proton surface. Charge units: e = qp = −qe . For r < rc ρel (r) = Θ(rp −r)(np,i −ne )−Θ(r−rp )ne . (10) Coulomb energy of a W-S cell ECoul = Z I= Z d r r<rc Pawel Haensel (CAMK) 3 Neutron star crust Lect.4 r 0 <rc 1 2 e I , 2 d3 r0 (11) ρel (r)ρel (r0 ) . |rr − r 0 | La̧dek Zdrój, Poland (12) 9 / 15 Coulomb energy - simplest approx. - spherical nuclei - 2 Problem: Calculate ECoul (rp , npi , w) Aswer: rp5 n2p f3 (w) , w = (rp /rc )3 6 1 f3 (w) = (2 − 3w1/2 + w) 5 ECoul = 2πw(enp,i rp )2 f3 (w) ECoul = (4π)2 e2 Hint Use multipole expansion 1 r − r0| |r Pawel Haensel (CAMK) = 4π ∞ X m=l X 1 l r< l+1 l=0 m=−l 2l + 1 r> Neutron star crust Lect.4 ? n0 n) , Ylm (n̂ )Ylm (n̂ (13) La̧dek Zdrój, Poland 10 / 15 Variable, constraints, equilibrium For the sake of simplicity - no neutron skin Five independent variables. Possible set ne , nn,i , np,i , nn,o , rc rp Constraints - charge neutrality and nb = const.: ne = wnp,i , w(nn,i + np,i ) + (1 − w)nn,o = nb = const. Thermod. potential density to be minimized: E = wEi + (1 − w)Eo + 3σs w + ECoul + Ee rp w = (rp /rc )3 ECoul = 2πw(ene rp )2 f3 (w) four independent variables Pawel Haensel (CAMK) Neutron star crust Lect.4 Ee = a(ne )4/3 La̧dek Zdrój, Poland 11 / 15 Minimize in rc Choice of variables: [nn,i np,i nn,o ] rc E = wEi + (1 − w)Eo + w= nb −nn,o nn,i +np,i −nn,o [.....] fixed. 3σs w + ECoul + Ee rp rp = w1/3 rc ∂ECoul 2ECoul ∂Esurf Esurf = , =− rc rc rc rc ”Virial Theorem” Esurf = 2ECoul Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 12 / 15 Minimize in nn,i Choice of variables: nn,i [np,i rp rc ] E = wEi + (1 − w)Eo + [.....] fixed. 3σs w + ECoul + Ee rp w = (rp /rc )3 is also fixed. From nb -constraint: nn,o = nb − w(nn,i + np,i ) −w =⇒ δnn,o = δnn,i 1−w 1−w ∂E ∂E w =− ∂nn,i 1 − w ∂nn,o ∂E = w(µn,i − µn,o ) = 0 =⇒ ∂nn,i µn,i = µn,o Pawel Haensel (CAMK) Neutron star crust Lect.4 La̧dek Zdrój, Poland 13 / 15 Minimize in np,i Choice of variables: np,i [nn,o rp rc ] E = wEi + (1 − w)Eo + [.....] fixed. 3σs w + ECoul + Ee rp w = (rp /rc )3 is also fixed. From nb -constraint: nn,i = nb − (1 − w)(nn,0 ) − np,i w ∂E ∂E =− ∂nn,i ∂np,o ∂E = 0 =⇒ ∂np,i Pawel Haensel (CAMK) µn,i − µp,i − µe = Neutron star crust Lect.4 8π 2 e np,i rp2 f3 (w) . 5 La̧dek Zdrój, Poland 14 / 15 Minimize in rp Choice of variables: rp [Np Nn rc ] [.....] fixed. E = wEi + (1 − w)Eo + 3σs w + ECoul + Ee rp rp varied =⇒ w varied. nb fixed by construction. nn = (1 − w)nn,o + wnn,i , np = wpn,i ∂E ∂E =− ∂nn,i ∂np,o ∂E = 0 =⇒ ∂np,i Pawel Haensel (CAMK) Pi − Po = 2σs 4π 2 2 2 − e np,i rp (1 − w) . rp 15 Neutron star crust Lect.4 La̧dek Zdrój, Poland 15 / 15 SLy4 - curvature included - inner crust Spherical unit cell radius rc , the proton radius of nuclei rp , and the fraction of volume w filled by protons (in percent) vs. average matter density ρ. Douchin & Haensel (2000) Pawel Haensel (CAMK) Nuclei: A and Z vs. ρ. The dotted line gives number of the nucleons in nuclei after subtracting the neutrons belonging to neutron skin. Douchin & Haensel (2000) Neutron star crust Lect.4 La̧dek Zdrój, Poland 16 / 15